2013 中国数学奥林匹克（CMO）

若 S 只有一个元素 a，则对任意 m,n∈S，必有 m=n=a，于是 3m−2n=3a−2a=a∈S。条件自动满足。因此所有单元素集 {a}（a∈Z）都是解。

现设 |S|≥2。定义 S 中不同元素之间的最小距离 d = min{|m−n| : m,n∈S, m≠n}。由定义，d 是正整数，且存在 a∈Z 使得 a+d 与 a+2d 都属于 S（取 S 中某对距离为 d 的元素，令较小者为 a+d，较大者为 a+2d）。

由 a+d, a+2d∈S，反复应用封闭性 3m−2n∈S： 取 m=a+2d, n=a+d，得 3(a+2d)−2(a+d)=a+4d∈S； 取 m=a+d, n=a+2d，得 3(a+d)−2(a+2d)=a−d∈S； 取 m=a+d, n=a−d，得 3(a+d)−2(a−d)=a+5d∈S； 取 m=a+2d, n=a+4d，得 3(a+2d)−2(a+4d)=a−2d∈S。 一般地，若 a+kd, a+(k+1)d∈S，则 3(a+(k+1)d)−2(a+kd)=a+(k+3)d∈S，以及 3(a+kd)−2(a+(k+1)d)=a+(k−2)d∈S。由此可归纳证明：对所有满足 3∤k 的整数 k，都有 a+kd∈S。

令 S₀ = {a+kd | k∈Z, 3∤k}。对任意 m=a+k₁d, n=a+k₂d∈S₀（即 3∤k₁, 3∤k₂），有 3m−2n = a+(3k₁−2k₂)d。需证 3∤(3k₁−2k₂)。由于 3k₁−2k₂ ≡ −2k₂ ≡ k₂ (mod 3)，而 3∤k₂，故 3∤(3k₁−2k₂)。因此 3m−2n∈S₀，S₀ 满足条件。

现已知 S₀⊆S。若 S=S₀，则得到第二类解。若 S≠S₀，取 b∈S\S₀。由 S₀ 的结构，存在整数 l 使得 a+ld ≤ b < a+(l+1)d。由于 l 和 l+1 中至少有一个不被 3 整除，故 a+ld 与 a+(l+1)d 中至少有一个属于 S₀⊆S。 情形一：a+ld∈S₀。则 0 ≤ b−(a+ld) < d。由 d 的最小性，必有 b−(a+ld)=0，即 b=a+ld。但 b∉S₀，故 3|l。 情形二：a+(l+1)d∈S₀。则 0 < |a+(l+1)d−b| ≤ d。由 d 的最小性，必有 b=a+(l+1)d 或 b=a+ld。若 b=a+(l+1)d，则 b∈S₀，矛盾；故 b=a+ld，同样 3|l。 两种情形都得到 b=a+ld 且 3|l。此时 a+ld, a+(l+1)d, a+(l+2)d∈S（后两者因 3∤(l+1), 3∤(l+2) 而属于 S₀）。应用封闭性：3(a+(l+1)d)−2(a+ld)=a+(l+3)d∈S，3(a+(l−1)d)−2(a+ld)=a+(l−3)d∈S（注意 a+(l−1)d, a+(l−2)d∈S₀）。归纳可得对所有 j∈Z，a+(l+3j)d∈S。结合 S₀，得 {a+kd | k∈Z}⊆S。

令 S₁ = {a+kd | k∈Z}。对任意 m=a+k₁d, n=a+k₂d∈S₁，有 3m−2n = a+(3k₁−2k₂)d∈S₁（因为 3k₁−2k₂ 是整数）。故 S₁ 满足条件。 现证 S=S₁。若存在 x∈S\S₁，则 x 不在等差数列 {a+kd} 中。取 y∈S₁ 使得 |x−y| 最小，则 0<|x−y|<d（因为 S₁ 中相邻元素距离为 d）。但 x,y∈S 且 x≠y，这与 d 的最小性矛盾。故 S=S₁。

综合以上讨论，所有满足条件的非空整数集 S 恰好有三类： (1) 单元素集 S={a}，a∈Z； (2) S={a+kd | k∈Z, 3∤k}，a∈Z, d∈Z, d>0； (3) S={a+kd | k∈Z}，a∈Z, d∈Z, d>0。 这三类集合互不相同（(1) 是有限集，(2)(3) 是无限集；(2) 中元素下标不被 3 整除，(3) 中所有下标都出现），且都已验证满足条件。

取 λ ∈ (0, (1-2t)/(2(1+t)))，注意 0 < t < 1/2 时 (1-2t)/(2(1+t)) > 0，故这样的 λ 存在。令 x_i = λ^i（i = 1, 2, 3, …），X = {x_1, x_2, …}。由于 0 < λ < 1，X 是严格递减的正数无穷集。

假设存在 a ∈ ℝ、d > 0 以及 x_i, x_j, x_k ∈ X 使得 max{|x_i-(a-d)|, |x_j-a|, |x_k-(a+d)|} ≤ td。展开绝对值不等式得到三个区间约束： -td ≤ x_i - (a-d) ≤ td，即 x_i + (1-t)d ≤ a ≤ x_i + (1+t)d； -td ≤ x_j - a ≤ td，即 x_j - td ≤ a ≤ x_j + td； -td ≤ x_k - (a+d) ≤ td，即 x_k - (1+t)d ≤ a ≤ x_k - (1-t)d。 这三个区间必须有公共交集（都包含 a）。

三个区间有公共点，故任意两个区间的左端点 ≤ 另一区间的右端点。取三组关键配对： (1) 第三区间左端点 ≤ 第一区间右端点：x_k - (1+t)d ≤ x_i + (1+t)d，得 d ≥ (x_k - x_i)/(2(1+t))； (2) 第一区间左端点 ≤ 第二区间右端点：x_i + (1-t)d ≤ x_j + td，即 x_i + (1-2t)d ≤ x_j，因 1-2t > 0，得 d ≤ (x_j - x_i)/(1-2t)； (3) 第二区间左端点 ≤ 第三区间右端点：x_j - td ≤ x_k - (1-t)d，即 x_j + (1-2t)d ≤ x_k，得 d ≤ (x_k - x_j)/(1-2t)。 由 (2)(3) 及 d > 0 知 x_j > x_i 且 x_k > x_j，故 x_i < x_j < x_k。由于 x_n = λ^n 严格递减，这意味着 i > j > k。

由 i > j > k 及 x_n = λ^n，计算比值： (x_j - x_i)/(x_k - x_i) = (λ^j - λ^i)/(λ^k - λ^i) = λ^j(1 - λ^{i-j}) / [λ^k(1 - λ^{i-k})]。 因 i > j > k，有 i-j < i-k，故 1 - λ^{i-j} < 1 - λ^{i-k}（因 0 < λ < 1），所以 (x_j - x_i)/(x_k - x_i) < λ^j/λ^k = λ^{j-k} ≤ λ（因 j > k，j-k ≥ 1）。 另一方面，由 d 的下界和上界 (1)(2)： (x_k - x_i)/(2(1+t)) ≤ d ≤ (x_j - x_i)/(1-2t)， 故 (x_j - x_i)/(x_k - x_i) ≥ (1-2t)/(2(1+t))。 但 λ 的选取满足 λ < (1-2t)/(2(1+t))，与上式矛盾。故假设不成立，原命题得证。

设 X 为任意无穷集。在 X 中任取三个不同元素，不妨设 x < y < z。令 d = (z-x)/2 > 0。此时 a-d 和 a+d 的中点为 a，而 x 和 z 的中点为 (x+z)/2。关键观察：当 a = (x+z)/2 时，x = a-d，z = a+d，此时 |x-(a-d)| = |z-(a+d)| = 0。

令 a = max{x + (1-t)d, y - td}。需要验证三个区间约束同时成立： (1) x + (1-t)d ≤ a ≤ x + (1+t)d：由 a 的定义，a ≥ x + (1-t)d 显然。需证 a ≤ x + (1+t)d，即证 y - td ≤ x + (1+t)d。因 y < z = x + 2d，故 y - td < x + 2d - td = x + (2-t)d。当 t ≥ 1/2 时，2-t ≤ 1+t（即 t ≥ 1/2），故 y - td < x + (1+t)d。✓ (2) y - td ≤ a ≤ y + td：由 a 的定义，a ≥ y - td 显然。需证 a ≤ y + td，即证 x + (1-t)d ≤ y + td。因 x < y，故 x + (1-t)d < y + (1-t)d。当 t ≥ 1/2 时，1-t ≤ t（即 t ≥ 1/2），故 x + (1-t)d < y + td。✓ (3) z - (1+t)d ≤ a ≤ z - (1-t)d：由 (1) 中已证 x + (1-t)d = z - (1+t)d，故 a ≥ x + (1-t)d = z - (1+t)d。需证 a ≤ z - (1-t)d = x + (1+t)d，这在 (1) 中已证。✓ 三个区间约束全部成立，故 max{|x-(a-d)|, |y-a|, |z-(a+d)|} ≤ td。

设 $\htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{2}\htmlData{tutor-start=5,tutor-end=6}{k}\htmlData{tutor-start=6,tutor-end=7}{+}\htmlData{tutor-start=7,tutor-end=8}{1}$（$\htmlData{tutor-start=0,tutor-end=1}{k} \htmlData{tutor-start=2,tutor-end=6}{\ge }\htmlData{tutor-start=6,tutor-end=7}{1}$）。定义集合族如下：

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逐一验证对称差条件。分五种情况讨论 $\htmlData{tutor-start=0,tutor-end=1}{1} \htmlData{tutor-start=2,tutor-end=6}{\le }\htmlData{tutor-start=6,tutor-end=7}{i} \htmlData{tutor-start=8,tutor-end=9}{<} \htmlData{tutor-start=10,tutor-end=11}{j} \htmlData{tutor-start=12,tutor-end=16}{\le }\htmlData{tutor-start=16,tutor-end=17}{2}\htmlData{tutor-start=17,tutor-end=18}{k}\htmlData{tutor-start=18,tutor-end=19}{+}\htmlData{tutor-start=19,tutor-end=20}{1}$：

（1）$\htmlData{tutor-start=0,tutor-end=1}{1} \htmlData{tutor-start=2,tutor-end=6}{\le }\htmlData{tutor-start=6,tutor-end=7}{i} \htmlData{tutor-start=8,tutor-end=9}{<} \htmlData{tutor-start=10,tutor-end=11}{j} \htmlData{tutor-start=12,tutor-end=16}{\le }\htmlData{tutor-start=16,tutor-end=17}{k}$：$\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{i}} \htmlData{tutor-start=6,tutor-end=16}{\setminus }\htmlData{tutor-start=16,tutor-end=17}{A}_{\htmlData{tutor-start=19,tutor-end=20}{j}} \htmlData{tutor-start=22,tutor-end=23}{=} \htmlData{tutor-start=24,tutor-end=26}{\{}\htmlData{tutor-start=26,tutor-end=27}{i}\htmlData{tutor-start=27,tutor-end=28}{,} \htmlData{tutor-start=29,tutor-end=30}{i}\htmlData{tutor-start=30,tutor-end=31}{+}\htmlData{tutor-start=31,tutor-end=32}{1}\htmlData{tutor-start=32,tutor-end=33}{,} \dots\htmlData{tutor-start=39,tutor-end=40}{,} \htmlData{tutor-start=41,tutor-end=42}{j}\htmlData{tutor-start=42,tutor-end=43}{-}\htmlData{tutor-start=43,tutor-end=44}{1}\htmlData{tutor-start=44,tutor-end=46}{\}}$，$\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{j}} \htmlData{tutor-start=6,tutor-end=16}{\setminus }\htmlData{tutor-start=16,tutor-end=17}{A}_{\htmlData{tutor-start=19,tutor-end=20}{i}} \htmlData{tutor-start=22,tutor-end=23}{=} \htmlData{tutor-start=24,tutor-end=33}{\emptyset}$，故 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{i}} \htmlData{tutor-start=7,tutor-end=14}{\Delta }\htmlData{tutor-start=14,tutor-end=15}{A}_{\htmlData{tutor-start=17,tutor-end=18}{j}}\htmlData{tutor-start=19,tutor-end=20}{|} \htmlData{tutor-start=21,tutor-end=22}{=} \htmlData{tutor-start=23,tutor-end=24}{j} \htmlData{tutor-start=25,tutor-end=26}{-} \htmlData{tutor-start=27,tutor-end=28}{i}$。

（2）$\htmlData{tutor-start=0,tutor-end=1}{1} \htmlData{tutor-start=2,tutor-end=6}{\le }\htmlData{tutor-start=6,tutor-end=7}{i} \htmlData{tutor-start=8,tutor-end=12}{\le }\htmlData{tutor-start=12,tutor-end=13}{k}$，$\htmlData{tutor-start=0,tutor-end=1}{j} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{k}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{1}$：$\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{i}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=10}{\{}\htmlData{tutor-start=10,tutor-end=11}{i}\htmlData{tutor-start=11,tutor-end=12}{,} \dots\htmlData{tutor-start=18,tutor-end=19}{,} \htmlData{tutor-start=20,tutor-end=21}{k}\htmlData{tutor-start=21,tutor-end=23}{\}}$，$\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{k}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{1}} \htmlData{tutor-start=8,tutor-end=9}{=} \htmlData{tutor-start=10,tutor-end=12}{\{}\htmlData{tutor-start=12,tutor-end=13}{k}\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=16}{k}\htmlData{tutor-start=16,tutor-end=17}{+}\htmlData{tutor-start=17,tutor-end=18}{1}\htmlData{tutor-start=18,tutor-end=20}{\}}$。$\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{i}} \htmlData{tutor-start=6,tutor-end=16}{\setminus }\htmlData{tutor-start=16,tutor-end=17}{A}_{\htmlData{tutor-start=19,tutor-end=20}{k}\htmlData{tutor-start=20,tutor-end=21}{+}\htmlData{tutor-start=21,tutor-end=22}{1}} \htmlData{tutor-start=24,tutor-end=25}{=} \htmlData{tutor-start=26,tutor-end=28}{\{}\htmlData{tutor-start=28,tutor-end=29}{i}\htmlData{tutor-start=29,tutor-end=30}{,} \dots\htmlData{tutor-start=36,tutor-end=37}{,} \htmlData{tutor-start=38,tutor-end=39}{k}\htmlData{tutor-start=39,tutor-end=40}{-}\htmlData{tutor-start=40,tutor-end=41}{1}\htmlData{tutor-start=41,tutor-end=43}{\}}$（共 $\htmlData{tutor-start=0,tutor-end=1}{k}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{i}$ 个），$\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{k}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{1}} \htmlData{tutor-start=8,tutor-end=18}{\setminus }\htmlData{tutor-start=18,tutor-end=19}{A}_{\htmlData{tutor-start=21,tutor-end=22}{i}} \htmlData{tutor-start=24,tutor-end=25}{=} \htmlData{tutor-start=26,tutor-end=28}{\{}\htmlData{tutor-start=28,tutor-end=29}{k}\htmlData{tutor-start=29,tutor-end=30}{+}\htmlData{tutor-start=30,tutor-end=31}{1}\htmlData{tutor-start=31,tutor-end=33}{\}}$（共 1 个），故 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{i}} \htmlData{tutor-start=7,tutor-end=14}{\Delta }\htmlData{tutor-start=14,tutor-end=15}{A}_{\htmlData{tutor-start=17,tutor-end=18}{k}\htmlData{tutor-start=18,tutor-end=19}{+}\htmlData{tutor-start=19,tutor-end=20}{1}}\htmlData{tutor-start=21,tutor-end=22}{|} \htmlData{tutor-start=23,tutor-end=24}{=} \htmlData{tutor-start=25,tutor-end=26}{k} \htmlData{tutor-start=27,tutor-end=28}{-} \htmlData{tutor-start=29,tutor-end=30}{i} \htmlData{tutor-start=31,tutor-end=32}{+} \htmlData{tutor-start=33,tutor-end=34}{1} \htmlData{tutor-start=35,tutor-end=36}{=} \htmlData{tutor-start=37,tutor-end=38}{(}\htmlData{tutor-start=38,tutor-end=39}{k}\htmlData{tutor-start=39,tutor-end=40}{+}\htmlData{tutor-start=40,tutor-end=41}{1}\htmlData{tutor-start=41,tutor-end=42}{)} \htmlData{tutor-start=43,tutor-end=44}{-} \htmlData{tutor-start=45,tutor-end=46}{i}$。

（3）$\htmlData{tutor-start=0,tutor-end=1}{1} \htmlData{tutor-start=2,tutor-end=6}{\le }\htmlData{tutor-start=6,tutor-end=7}{i} \htmlData{tutor-start=8,tutor-end=12}{\le }\htmlData{tutor-start=12,tutor-end=13}{k}$，$\htmlData{tutor-start=0,tutor-end=1}{j} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{k}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{m}$（$\htmlData{tutor-start=0,tutor-end=1}{m} \htmlData{tutor-start=2,tutor-end=6}{\ge }\htmlData{tutor-start=6,tutor-end=7}{2}$）：$\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{i}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=10}{\{}\htmlData{tutor-start=10,tutor-end=11}{i}\htmlData{tutor-start=11,tutor-end=12}{,} \dots\htmlData{tutor-start=18,tutor-end=19}{,} \htmlData{tutor-start=20,tutor-end=21}{k}\htmlData{tutor-start=21,tutor-end=23}{\}}$，$\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{k}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{m}} \htmlData{tutor-start=8,tutor-end=9}{=} \htmlData{tutor-start=10,tutor-end=12}{\{}\htmlData{tutor-start=12,tutor-end=13}{k}\htmlData{tutor-start=13,tutor-end=14}{+}\htmlData{tutor-start=14,tutor-end=15}{1}\htmlData{tutor-start=15,tutor-end=16}{,} \dots\htmlData{tutor-start=22,tutor-end=23}{,} \htmlData{tutor-start=24,tutor-end=25}{k}\htmlData{tutor-start=25,tutor-end=26}{+}\htmlData{tutor-start=26,tutor-end=27}{m}\htmlData{tutor-start=27,tutor-end=28}{-}\htmlData{tutor-start=28,tutor-end=29}{1}\htmlData{tutor-start=29,tutor-end=31}{\}}$。两者不相交，故 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{i}} \htmlData{tutor-start=7,tutor-end=14}{\Delta }\htmlData{tutor-start=14,tutor-end=15}{A}_{\htmlData{tutor-start=17,tutor-end=18}{k}\htmlData{tutor-start=18,tutor-end=19}{+}\htmlData{tutor-start=19,tutor-end=20}{m}}\htmlData{tutor-start=21,tutor-end=22}{|} \htmlData{tutor-start=23,tutor-end=24}{=} \htmlData{tutor-start=25,tutor-end=26}{(}\htmlData{tutor-start=26,tutor-end=27}{k} \htmlData{tutor-start=28,tutor-end=29}{-} \htmlData{tutor-start=30,tutor-end=31}{i} \htmlData{tutor-start=32,tutor-end=33}{+} \htmlData{tutor-start=34,tutor-end=35}{1}\htmlData{tutor-start=35,tutor-end=36}{)} \htmlData{tutor-start=37,tutor-end=38}{+} \htmlData{tutor-start=39,tutor-end=40}{(}\htmlData{tutor-start=40,tutor-end=41}{m} \htmlData{tutor-start=42,tutor-end=43}{-} \htmlData{tutor-start=44,tutor-end=45}{1}\htmlData{tutor-start=45,tutor-end=46}{)} \htmlData{tutor-start=47,tutor-end=48}{=} \htmlData{tutor-start=49,tutor-end=50}{k} \htmlData{tutor-start=51,tutor-end=52}{+} \htmlData{tutor-start=53,tutor-end=54}{m} \htmlData{tutor-start=55,tutor-end=56}{-} \htmlData{tutor-start=57,tutor-end=58}{i} \htmlData{tutor-start=59,tutor-end=60}{=} \htmlData{tutor-start=61,tutor-end=62}{j} \htmlData{tutor-start=63,tutor-end=64}{-} \htmlData{tutor-start=65,tutor-end=66}{i}$。

（4）$\htmlData{tutor-start=0,tutor-end=1}{i} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{k}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{1}$，$\htmlData{tutor-start=0,tutor-end=1}{j} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{k}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{m}$（$\htmlData{tutor-start=0,tutor-end=1}{m} \htmlData{tutor-start=2,tutor-end=6}{\ge }\htmlData{tutor-start=6,tutor-end=7}{2}$）：$\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{k}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{1}} \htmlData{tutor-start=8,tutor-end=9}{=} \htmlData{tutor-start=10,tutor-end=12}{\{}\htmlData{tutor-start=12,tutor-end=13}{k}\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=16}{k}\htmlData{tutor-start=16,tutor-end=17}{+}\htmlData{tutor-start=17,tutor-end=18}{1}\htmlData{tutor-start=18,tutor-end=20}{\}}$，$\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{k}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{m}} \htmlData{tutor-start=8,tutor-end=9}{=} \htmlData{tutor-start=10,tutor-end=12}{\{}\htmlData{tutor-start=12,tutor-end=13}{k}\htmlData{tutor-start=13,tutor-end=14}{+}\htmlData{tutor-start=14,tutor-end=15}{1}\htmlData{tutor-start=15,tutor-end=16}{,} \dots\htmlData{tutor-start=22,tutor-end=23}{,} \htmlData{tutor-start=24,tutor-end=25}{k}\htmlData{tutor-start=25,tutor-end=26}{+}\htmlData{tutor-start=26,tutor-end=27}{m}\htmlData{tutor-start=27,tutor-end=28}{-}\htmlData{tutor-start=28,tutor-end=29}{1}\htmlData{tutor-start=29,tutor-end=31}{\}}$。$\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{k}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{1}} \htmlData{tutor-start=8,tutor-end=18}{\setminus }\htmlData{tutor-start=18,tutor-end=19}{A}_{\htmlData{tutor-start=21,tutor-end=22}{k}\htmlData{tutor-start=22,tutor-end=23}{+}\htmlData{tutor-start=23,tutor-end=24}{m}} \htmlData{tutor-start=26,tutor-end=27}{=} \htmlData{tutor-start=28,tutor-end=30}{\{}\htmlData{tutor-start=30,tutor-end=31}{k}\htmlData{tutor-start=31,tutor-end=33}{\}}$，$\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{k}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{m}} \htmlData{tutor-start=8,tutor-end=18}{\setminus }\htmlData{tutor-start=18,tutor-end=19}{A}_{\htmlData{tutor-start=21,tutor-end=22}{k}\htmlData{tutor-start=22,tutor-end=23}{+}\htmlData{tutor-start=23,tutor-end=24}{1}} \htmlData{tutor-start=26,tutor-end=27}{=} \htmlData{tutor-start=28,tutor-end=30}{\{}\htmlData{tutor-start=30,tutor-end=31}{k}\htmlData{tutor-start=31,tutor-end=32}{+}\htmlData{tutor-start=32,tutor-end=33}{2}\htmlData{tutor-start=33,tutor-end=34}{,} \dots\htmlData{tutor-start=40,tutor-end=41}{,} \htmlData{tutor-start=42,tutor-end=43}{k}\htmlData{tutor-start=43,tutor-end=44}{+}\htmlData{tutor-start=44,tutor-end=45}{m}\htmlData{tutor-start=45,tutor-end=46}{-}\htmlData{tutor-start=46,tutor-end=47}{1}\htmlData{tutor-start=47,tutor-end=49}{\}}$（共 $\htmlData{tutor-start=0,tutor-end=1}{m}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{2}$ 个），故 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{k}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{1}} \htmlData{tutor-start=9,tutor-end=16}{\Delta }\htmlData{tutor-start=16,tutor-end=17}{A}_{\htmlData{tutor-start=19,tutor-end=20}{k}\htmlData{tutor-start=20,tutor-end=21}{+}\htmlData{tutor-start=21,tutor-end=22}{m}}\htmlData{tutor-start=23,tutor-end=24}{|} \htmlData{tutor-start=25,tutor-end=26}{=} \htmlData{tutor-start=27,tutor-end=28}{1} \htmlData{tutor-start=29,tutor-end=30}{+} \htmlData{tutor-start=31,tutor-end=32}{(}\htmlData{tutor-start=32,tutor-end=33}{m}\htmlData{tutor-start=33,tutor-end=34}{-}\htmlData{tutor-start=34,tutor-end=35}{2}\htmlData{tutor-start=35,tutor-end=36}{)} \htmlData{tutor-start=37,tutor-end=38}{=} \htmlData{tutor-start=39,tutor-end=40}{m} \htmlData{tutor-start=41,tutor-end=42}{-} \htmlData{tutor-start=43,tutor-end=44}{1} \htmlData{tutor-start=45,tutor-end=46}{=} \htmlData{tutor-start=47,tutor-end=48}{j} \htmlData{tutor-start=49,tutor-end=50}{-} \htmlData{tutor-start=51,tutor-end=52}{i}$。

（5）$\htmlData{tutor-start=0,tutor-end=1}{k}\htmlData{tutor-start=1,tutor-end=2}{+}\htmlData{tutor-start=2,tutor-end=3}{2} \htmlData{tutor-start=4,tutor-end=8}{\le }\htmlData{tutor-start=8,tutor-end=9}{i} \htmlData{tutor-start=10,tutor-end=11}{<} \htmlData{tutor-start=12,tutor-end=13}{j} \htmlData{tutor-start=14,tutor-end=18}{\le }\htmlData{tutor-start=18,tutor-end=19}{2}\htmlData{tutor-start=19,tutor-end=20}{k}\htmlData{tutor-start=20,tutor-end=21}{+}\htmlData{tutor-start=21,tutor-end=22}{1}$：设 $\htmlData{tutor-start=0,tutor-end=1}{i} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{k}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{a}$，$\htmlData{tutor-start=0,tutor-end=1}{j} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{k}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{b}$（$\htmlData{tutor-start=0,tutor-end=1}{2} \htmlData{tutor-start=2,tutor-end=6}{\le }\htmlData{tutor-start=6,tutor-end=7}{a} \htmlData{tutor-start=8,tutor-end=9}{<} \htmlData{tutor-start=10,tutor-end=11}{b} \htmlData{tutor-start=12,tutor-end=16}{\le }\htmlData{tutor-start=16,tutor-end=17}{k}\htmlData{tutor-start=17,tutor-end=18}{+}\htmlData{tutor-start=18,tutor-end=19}{1}$）。$\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{i}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=10}{\{}\htmlData{tutor-start=10,tutor-end=11}{k}\htmlData{tutor-start=11,tutor-end=12}{+}\htmlData{tutor-start=12,tutor-end=13}{1}\htmlData{tutor-start=13,tutor-end=14}{,} \dots\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{k}\htmlData{tutor-start=23,tutor-end=24}{+}\htmlData{tutor-start=24,tutor-end=25}{a}\htmlData{tutor-start=25,tutor-end=26}{-}\htmlData{tutor-start=26,tutor-end=27}{1}\htmlData{tutor-start=27,tutor-end=29}{\}}$，$\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{j}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=10}{\{}\htmlData{tutor-start=10,tutor-end=11}{k}\htmlData{tutor-start=11,tutor-end=12}{+}\htmlData{tutor-start=12,tutor-end=13}{1}\htmlData{tutor-start=13,tutor-end=14}{,} \dots\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{k}\htmlData{tutor-start=23,tutor-end=24}{+}\htmlData{tutor-start=24,tutor-end=25}{b}\htmlData{tutor-start=25,tutor-end=26}{-}\htmlData{tutor-start=26,tutor-end=27}{1}\htmlData{tutor-start=27,tutor-end=29}{\}}$。$\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{i}} \htmlData{tutor-start=6,tutor-end=14}{\subset }\htmlData{tutor-start=14,tutor-end=15}{A}_{\htmlData{tutor-start=17,tutor-end=18}{j}}$，$\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{j}} \htmlData{tutor-start=6,tutor-end=16}{\setminus }\htmlData{tutor-start=16,tutor-end=17}{A}_{\htmlData{tutor-start=19,tutor-end=20}{i}} \htmlData{tutor-start=22,tutor-end=23}{=} \htmlData{tutor-start=24,tutor-end=26}{\{}\htmlData{tutor-start=26,tutor-end=27}{k}\htmlData{tutor-start=27,tutor-end=28}{+}\htmlData{tutor-start=28,tutor-end=29}{a}\htmlData{tutor-start=29,tutor-end=30}{,} \dots\htmlData{tutor-start=36,tutor-end=37}{,} \htmlData{tutor-start=38,tutor-end=39}{k}\htmlData{tutor-start=39,tutor-end=40}{+}\htmlData{tutor-start=40,tutor-end=41}{b}\htmlData{tutor-start=41,tutor-end=42}{-}\htmlData{tutor-start=42,tutor-end=43}{1}\htmlData{tutor-start=43,tutor-end=45}{\}}$（共 $\htmlData{tutor-start=0,tutor-end=1}{b}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{a}$ 个），故 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{i}} \htmlData{tutor-start=7,tutor-end=14}{\Delta }\htmlData{tutor-start=14,tutor-end=15}{A}_{\htmlData{tutor-start=17,tutor-end=18}{j}}\htmlData{tutor-start=19,tutor-end=20}{|} \htmlData{tutor-start=21,tutor-end=22}{=} \htmlData{tutor-start=23,tutor-end=24}{b} \htmlData{tutor-start=25,tutor-end=26}{-} \htmlData{tutor-start=27,tutor-end=28}{a} \htmlData{tutor-start=29,tutor-end=30}{=} \htmlData{tutor-start=31,tutor-end=32}{j} \htmlData{tutor-start=33,tutor-end=34}{-} \htmlData{tutor-start=35,tutor-end=36}{i}$。

$\htmlData{tutor-start=0,tutor-end=1}{i} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{j}$ 时显然成立；$\htmlData{tutor-start=0,tutor-end=1}{i} \htmlData{tutor-start=2,tutor-end=3}{>} \htmlData{tutor-start=4,tutor-end=5}{j}$ 时由对称差对称性归为 $\htmlData{tutor-start=0,tutor-end=1}{i} \htmlData{tutor-start=2,tutor-end=3}{<} \htmlData{tutor-start=4,tutor-end=5}{j}$ 情形。因此对所有 $\htmlData{tutor-start=0,tutor-end=1}{i}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{j}$ 均有 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{i}} \htmlData{tutor-start=7,tutor-end=14}{\Delta }\htmlData{tutor-start=14,tutor-end=15}{A}_{\htmlData{tutor-start=17,tutor-end=18}{j}}\htmlData{tutor-start=19,tutor-end=20}{|} \htmlData{tutor-start=21,tutor-end=22}{=} \htmlData{tutor-start=23,tutor-end=24}{|}\htmlData{tutor-start=24,tutor-end=25}{i} \htmlData{tutor-start=26,tutor-end=27}{-} \htmlData{tutor-start=28,tutor-end=29}{j}\htmlData{tutor-start=29,tutor-end=30}{|}$。

计算总和：$\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{i}}\htmlData{tutor-start=6,tutor-end=7}{|} \htmlData{tutor-start=8,tutor-end=9}{=} \htmlData{tutor-start=10,tutor-end=11}{k} \htmlData{tutor-start=12,tutor-end=13}{-} \htmlData{tutor-start=14,tutor-end=15}{i} \htmlData{tutor-start=16,tutor-end=17}{+} \htmlData{tutor-start=18,tutor-end=19}{1}$（$\htmlData{tutor-start=0,tutor-end=1}{i} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{1}\htmlData{tutor-start=5,tutor-end=6}{,} \dots\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{k}$），$\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{k}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{1}}\htmlData{tutor-start=8,tutor-end=9}{|} \htmlData{tutor-start=10,tutor-end=11}{=} \htmlData{tutor-start=12,tutor-end=13}{2}$，$\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{k}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{j}}\htmlData{tutor-start=8,tutor-end=9}{|} \htmlData{tutor-start=10,tutor-end=11}{=} \htmlData{tutor-start=12,tutor-end=13}{j} \htmlData{tutor-start=14,tutor-end=15}{-} \htmlData{tutor-start=16,tutor-end=17}{1}$（$\htmlData{tutor-start=0,tutor-end=1}{j} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{2}\htmlData{tutor-start=5,tutor-end=6}{,} \dots\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{k}\htmlData{tutor-start=15,tutor-end=16}{+}\htmlData{tutor-start=16,tutor-end=17}{1}$）。

$\htmlData{tutor-start=0,tutor-end=1}{S}_{\htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{k}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{1}} \htmlData{tutor-start=9,tutor-end=10}{=} \sum_{\htmlData{tutor-start=17,tutor-end=18}{i}\htmlData{tutor-start=18,tutor-end=19}{=}\htmlData{tutor-start=19,tutor-end=20}{1}}^{\htmlData{tutor-start=23,tutor-end=24}{k}}\htmlData{tutor-start=25,tutor-end=26}{(}\htmlData{tutor-start=26,tutor-end=27}{k}\htmlData{tutor-start=27,tutor-end=28}{-}\htmlData{tutor-start=28,tutor-end=29}{i}\htmlData{tutor-start=29,tutor-end=30}{+}\htmlData{tutor-start=30,tutor-end=31}{1}\htmlData{tutor-start=31,tutor-end=32}{)} \htmlData{tutor-start=33,tutor-end=34}{+} \htmlData{tutor-start=35,tutor-end=36}{2} \htmlData{tutor-start=37,tutor-end=38}{+} \sum_{\htmlData{tutor-start=45,tutor-end=46}{j}\htmlData{tutor-start=46,tutor-end=47}{=}\htmlData{tutor-start=47,tutor-end=48}{2}}^{\htmlData{tutor-start=51,tutor-end=52}{k}\htmlData{tutor-start=52,tutor-end=53}{+}\htmlData{tutor-start=53,tutor-end=54}{1}}\htmlData{tutor-start=55,tutor-end=56}{(}\htmlData{tutor-start=56,tutor-end=57}{j}\htmlData{tutor-start=57,tutor-end=58}{-}\htmlData{tutor-start=58,tutor-end=59}{1}\htmlData{tutor-start=59,tutor-end=60}{)} \htmlData{tutor-start=61,tutor-end=62}{=} \frac{\htmlData{tutor-start=69,tutor-end=70}{k}\htmlData{tutor-start=70,tutor-end=71}{(}\htmlData{tutor-start=71,tutor-end=72}{k}\htmlData{tutor-start=72,tutor-end=73}{+}\htmlData{tutor-start=73,tutor-end=74}{1}\htmlData{tutor-start=74,tutor-end=75}{)}}{\htmlData{tutor-start=77,tutor-end=78}{2}} \htmlData{tutor-start=80,tutor-end=81}{+} \htmlData{tutor-start=82,tutor-end=83}{2} \htmlData{tutor-start=84,tutor-end=85}{+} \frac{\htmlData{tutor-start=92,tutor-end=93}{k}\htmlData{tutor-start=93,tutor-end=94}{(}\htmlData{tutor-start=94,tutor-end=95}{k}\htmlData{tutor-start=95,tutor-end=96}{+}\htmlData{tutor-start=96,tutor-end=97}{1}\htmlData{tutor-start=97,tutor-end=98}{)}}{\htmlData{tutor-start=100,tutor-end=101}{2}} \htmlData{tutor-start=103,tutor-end=104}{=} \htmlData{tutor-start=105,tutor-end=106}{k}\htmlData{tutor-start=106,tutor-end=107}{(}\htmlData{tutor-start=107,tutor-end=108}{k}\htmlData{tutor-start=108,tutor-end=109}{+}\htmlData{tutor-start=109,tutor-end=110}{1}\htmlData{tutor-start=110,tutor-end=111}{)} \htmlData{tutor-start=112,tutor-end=113}{+} \htmlData{tutor-start=114,tutor-end=115}{2}\htmlData{tutor-start=115,tutor-end=116}{.}$

取上述 $\htmlData{tutor-start=0,tutor-end=1}{2}\htmlData{tutor-start=1,tutor-end=2}{k}\htmlData{tutor-start=2,tutor-end=3}{+}\htmlData{tutor-start=3,tutor-end=4}{1}$ 个集合中的前 $\htmlData{tutor-start=0,tutor-end=1}{2}\htmlData{tutor-start=1,tutor-end=2}{k}$ 个，即 $\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{A}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \dots\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=22}{A}_{\htmlData{tutor-start=24,tutor-end=25}{2}\htmlData{tutor-start=25,tutor-end=26}{k}}$。由于原族中任意两个集合的对称差条件已满足，去掉 $\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{k}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{1}}$ 后剩余集合仍满足条件。

总和为： $\htmlData{tutor-start=0,tutor-end=1}{S}_{\htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{k}} \htmlData{tutor-start=7,tutor-end=8}{=} \htmlData{tutor-start=9,tutor-end=10}{S}_{\htmlData{tutor-start=12,tutor-end=13}{2}\htmlData{tutor-start=13,tutor-end=14}{k}\htmlData{tutor-start=14,tutor-end=15}{+}\htmlData{tutor-start=15,tutor-end=16}{1}} \htmlData{tutor-start=18,tutor-end=19}{-} \htmlData{tutor-start=20,tutor-end=21}{|}\htmlData{tutor-start=21,tutor-end=22}{A}_{\htmlData{tutor-start=24,tutor-end=25}{2}\htmlData{tutor-start=25,tutor-end=26}{k}\htmlData{tutor-start=26,tutor-end=27}{+}\htmlData{tutor-start=27,tutor-end=28}{1}}\htmlData{tutor-start=29,tutor-end=30}{|} \htmlData{tutor-start=31,tutor-end=32}{=} \htmlData{tutor-start=33,tutor-end=34}{k}\htmlData{tutor-start=34,tutor-end=35}{(}\htmlData{tutor-start=35,tutor-end=36}{k}\htmlData{tutor-start=36,tutor-end=37}{+}\htmlData{tutor-start=37,tutor-end=38}{1}\htmlData{tutor-start=38,tutor-end=39}{)} \htmlData{tutor-start=40,tutor-end=41}{+} \htmlData{tutor-start=42,tutor-end=43}{2} \htmlData{tutor-start=44,tutor-end=45}{-} \htmlData{tutor-start=46,tutor-end=47}{k} \htmlData{tutor-start=48,tutor-end=49}{=} \htmlData{tutor-start=50,tutor-end=51}{k}^{\htmlData{tutor-start=53,tutor-end=54}{2}} \htmlData{tutor-start=56,tutor-end=57}{+} \htmlData{tutor-start=58,tutor-end=59}{2}\htmlData{tutor-start=59,tutor-end=60}{.}$

其中 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{2}\htmlData{tutor-start=5,tutor-end=6}{k}\htmlData{tutor-start=6,tutor-end=7}{+}\htmlData{tutor-start=7,tutor-end=8}{1}}\htmlData{tutor-start=9,tutor-end=10}{|} \htmlData{tutor-start=11,tutor-end=12}{=} \htmlData{tutor-start=13,tutor-end=14}{|}\htmlData{tutor-start=14,tutor-end=16}{\{}\htmlData{tutor-start=16,tutor-end=17}{k}\htmlData{tutor-start=17,tutor-end=18}{+}\htmlData{tutor-start=18,tutor-end=19}{1}\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=22}{k}\htmlData{tutor-start=22,tutor-end=23}{+}\htmlData{tutor-start=23,tutor-end=24}{2}\htmlData{tutor-start=24,tutor-end=25}{,} \dots\htmlData{tutor-start=31,tutor-end=32}{,} \htmlData{tutor-start=33,tutor-end=34}{2}\htmlData{tutor-start=34,tutor-end=35}{k}\htmlData{tutor-start=35,tutor-end=37}{\}}\htmlData{tutor-start=37,tutor-end=38}{|} \htmlData{tutor-start=39,tutor-end=40}{=} \htmlData{tutor-start=41,tutor-end=42}{k}$。

**事实 1**：对任意有限集 $\htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Y}$，有 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{X}\htmlData{tutor-start=2,tutor-end=3}{|} \htmlData{tutor-start=4,tutor-end=5}{+} \htmlData{tutor-start=6,tutor-end=7}{|}\htmlData{tutor-start=7,tutor-end=8}{Y}\htmlData{tutor-start=8,tutor-end=9}{|} \htmlData{tutor-start=10,tutor-end=14}{\ge }\htmlData{tutor-start=14,tutor-end=15}{|}\htmlData{tutor-start=15,tutor-end=16}{X} \htmlData{tutor-start=17,tutor-end=24}{\Delta }\htmlData{tutor-start=24,tutor-end=25}{Y}\htmlData{tutor-start=25,tutor-end=26}{|}$。

证明：$\htmlData{tutor-start=0,tutor-end=1}{X} \htmlData{tutor-start=2,tutor-end=9}{\Delta }\htmlData{tutor-start=9,tutor-end=10}{Y} \htmlData{tutor-start=11,tutor-end=12}{=} \htmlData{tutor-start=13,tutor-end=14}{(}\htmlData{tutor-start=14,tutor-end=15}{X} \htmlData{tutor-start=16,tutor-end=26}{\setminus }\htmlData{tutor-start=26,tutor-end=27}{Y}\htmlData{tutor-start=27,tutor-end=28}{)} \htmlData{tutor-start=29,tutor-end=34}{\cup }\htmlData{tutor-start=34,tutor-end=35}{(}\htmlData{tutor-start=35,tutor-end=36}{Y} \htmlData{tutor-start=37,tutor-end=47}{\setminus }\htmlData{tutor-start=47,tutor-end=48}{X}\htmlData{tutor-start=48,tutor-end=49}{)}$，而 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{X}\htmlData{tutor-start=2,tutor-end=3}{|} \htmlData{tutor-start=4,tutor-end=5}{=} \htmlData{tutor-start=6,tutor-end=7}{|}\htmlData{tutor-start=7,tutor-end=8}{X} \htmlData{tutor-start=9,tutor-end=14}{\cap }\htmlData{tutor-start=14,tutor-end=15}{Y}\htmlData{tutor-start=15,tutor-end=16}{|} \htmlData{tutor-start=17,tutor-end=18}{+} \htmlData{tutor-start=19,tutor-end=20}{|}\htmlData{tutor-start=20,tutor-end=21}{X} \htmlData{tutor-start=22,tutor-end=32}{\setminus }\htmlData{tutor-start=32,tutor-end=33}{Y}\htmlData{tutor-start=33,tutor-end=34}{|} \htmlData{tutor-start=35,tutor-end=39}{\ge }\htmlData{tutor-start=39,tutor-end=40}{|}\htmlData{tutor-start=40,tutor-end=41}{X} \htmlData{tutor-start=42,tutor-end=52}{\setminus }\htmlData{tutor-start=52,tutor-end=53}{Y}\htmlData{tutor-start=53,tutor-end=54}{|}$，同理 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{Y}\htmlData{tutor-start=2,tutor-end=3}{|} \htmlData{tutor-start=4,tutor-end=8}{\ge }\htmlData{tutor-start=8,tutor-end=9}{|}\htmlData{tutor-start=9,tutor-end=10}{Y} \htmlData{tutor-start=11,tutor-end=21}{\setminus }\htmlData{tutor-start=21,tutor-end=22}{X}\htmlData{tutor-start=22,tutor-end=23}{|}$，相加即得。

**事实 2**：对任意非空有限集 $\htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Y}$，若 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{X} \htmlData{tutor-start=3,tutor-end=10}{\Delta }\htmlData{tutor-start=10,tutor-end=11}{Y}\htmlData{tutor-start=11,tutor-end=12}{|} \htmlData{tutor-start=13,tutor-end=14}{=} \htmlData{tutor-start=15,tutor-end=16}{1}$，则 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{X}\htmlData{tutor-start=2,tutor-end=3}{|} \htmlData{tutor-start=4,tutor-end=5}{+} \htmlData{tutor-start=6,tutor-end=7}{|}\htmlData{tutor-start=7,tutor-end=8}{Y}\htmlData{tutor-start=8,tutor-end=9}{|} \htmlData{tutor-start=10,tutor-end=14}{\ge }\htmlData{tutor-start=14,tutor-end=15}{3}$。

证明：$\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{X} \htmlData{tutor-start=3,tutor-end=10}{\Delta }\htmlData{tutor-start=10,tutor-end=11}{Y}\htmlData{tutor-start=11,tutor-end=12}{|} \htmlData{tutor-start=13,tutor-end=14}{=} \htmlData{tutor-start=15,tutor-end=16}{1}$ 意味着 $\htmlData{tutor-start=0,tutor-end=1}{X}$ 与 $\htmlData{tutor-start=0,tutor-end=1}{Y}$ 恰好差一个元素。不妨设 $\htmlData{tutor-start=0,tutor-end=1}{X} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{Y} \htmlData{tutor-start=6,tutor-end=11}{\cup }\htmlData{tutor-start=11,tutor-end=13}{\{}\htmlData{tutor-start=13,tutor-end=14}{a}\htmlData{tutor-start=14,tutor-end=16}{\}}$ 或 $\htmlData{tutor-start=0,tutor-end=1}{Y} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{X} \htmlData{tutor-start=6,tutor-end=11}{\cup }\htmlData{tutor-start=11,tutor-end=13}{\{}\htmlData{tutor-start=13,tutor-end=14}{a}\htmlData{tutor-start=14,tutor-end=16}{\}}$。由于 $\htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Y}$ 均非空，较小的那个至少含 1 个元素，较大的至少含 2 个元素，故 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{X}\htmlData{tutor-start=2,tutor-end=3}{|} \htmlData{tutor-start=4,tutor-end=5}{+} \htmlData{tutor-start=6,tutor-end=7}{|}\htmlData{tutor-start=7,tutor-end=8}{Y}\htmlData{tutor-start=8,tutor-end=9}{|} \htmlData{tutor-start=10,tutor-end=14}{\ge }\htmlData{tutor-start=14,tutor-end=15}{1} \htmlData{tutor-start=16,tutor-end=17}{+} \htmlData{tutor-start=18,tutor-end=19}{2} \htmlData{tutor-start=20,tutor-end=21}{=} \htmlData{tutor-start=22,tutor-end=23}{3}$。

等号成立当且仅当较小集合恰含 1 个元素，较大集合恰含 2 个元素。

设 $\htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{2}\htmlData{tutor-start=5,tutor-end=6}{k}$（$\htmlData{tutor-start=0,tutor-end=1}{k} \htmlData{tutor-start=2,tutor-end=6}{\ge }\htmlData{tutor-start=6,tutor-end=7}{1}$）。将 $\htmlData{tutor-start=0,tutor-end=1}{2}\htmlData{tutor-start=1,tutor-end=2}{k}$ 个集合按下标对称配对：$\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{1}}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{A}_{\htmlData{tutor-start=11,tutor-end=12}{2}\htmlData{tutor-start=12,tutor-end=13}{k}}\htmlData{tutor-start=14,tutor-end=15}{)}$，$\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{2}}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{A}_{\htmlData{tutor-start=11,tutor-end=12}{2}\htmlData{tutor-start=12,tutor-end=13}{k}\htmlData{tutor-start=13,tutor-end=14}{-}\htmlData{tutor-start=14,tutor-end=15}{1}}\htmlData{tutor-start=16,tutor-end=17}{)}$，…，$\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{k}\htmlData{tutor-start=5,tutor-end=6}{-}\htmlData{tutor-start=6,tutor-end=7}{1}}\htmlData{tutor-start=8,tutor-end=9}{,} \htmlData{tutor-start=10,tutor-end=11}{A}_{\htmlData{tutor-start=13,tutor-end=14}{k}\htmlData{tutor-start=14,tutor-end=15}{+}\htmlData{tutor-start=15,tutor-end=16}{2}}\htmlData{tutor-start=17,tutor-end=18}{)}$，以及中间一对 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{k}}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{A}_{\htmlData{tutor-start=11,tutor-end=12}{k}\htmlData{tutor-start=12,tutor-end=13}{+}\htmlData{tutor-start=13,tutor-end=14}{1}}\htmlData{tutor-start=15,tutor-end=16}{)}$。

对 $\htmlData{tutor-start=0,tutor-end=1}{i} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{1}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{2}\htmlData{tutor-start=8,tutor-end=9}{,} \dots\htmlData{tutor-start=15,tutor-end=16}{,} \htmlData{tutor-start=17,tutor-end=18}{k}\htmlData{tutor-start=18,tutor-end=19}{-}\htmlData{tutor-start=19,tutor-end=20}{1}$，由事实 1： $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{i}}\htmlData{tutor-start=6,tutor-end=7}{|} \htmlData{tutor-start=8,tutor-end=9}{+} \htmlData{tutor-start=10,tutor-end=11}{|}\htmlData{tutor-start=11,tutor-end=12}{A}_{\htmlData{tutor-start=14,tutor-end=15}{2}\htmlData{tutor-start=15,tutor-end=16}{k}\htmlData{tutor-start=16,tutor-end=17}{+}\htmlData{tutor-start=17,tutor-end=18}{1}\htmlData{tutor-start=18,tutor-end=19}{-}\htmlData{tutor-start=19,tutor-end=20}{i}}\htmlData{tutor-start=21,tutor-end=22}{|} \htmlData{tutor-start=23,tutor-end=27}{\ge }\htmlData{tutor-start=27,tutor-end=28}{|}\htmlData{tutor-start=28,tutor-end=29}{A}_{\htmlData{tutor-start=31,tutor-end=32}{i}} \htmlData{tutor-start=34,tutor-end=41}{\Delta }\htmlData{tutor-start=41,tutor-end=42}{A}_{\htmlData{tutor-start=44,tutor-end=45}{2}\htmlData{tutor-start=45,tutor-end=46}{k}\htmlData{tutor-start=46,tutor-end=47}{+}\htmlData{tutor-start=47,tutor-end=48}{1}\htmlData{tutor-start=48,tutor-end=49}{-}\htmlData{tutor-start=49,tutor-end=50}{i}}\htmlData{tutor-start=51,tutor-end=52}{|} \htmlData{tutor-start=53,tutor-end=54}{=} \htmlData{tutor-start=55,tutor-end=56}{(}\htmlData{tutor-start=56,tutor-end=57}{2}\htmlData{tutor-start=57,tutor-end=58}{k}\htmlData{tutor-start=58,tutor-end=59}{+}\htmlData{tutor-start=59,tutor-end=60}{1}\htmlData{tutor-start=60,tutor-end=61}{-}\htmlData{tutor-start=61,tutor-end=62}{i}\htmlData{tutor-start=62,tutor-end=63}{)} \htmlData{tutor-start=64,tutor-end=65}{-} \htmlData{tutor-start=66,tutor-end=67}{i} \htmlData{tutor-start=68,tutor-end=69}{=} \htmlData{tutor-start=70,tutor-end=71}{2}\htmlData{tutor-start=71,tutor-end=72}{k} \htmlData{tutor-start=73,tutor-end=74}{+} \htmlData{tutor-start=75,tutor-end=76}{1} \htmlData{tutor-start=77,tutor-end=78}{-} \htmlData{tutor-start=79,tutor-end=80}{2}\htmlData{tutor-start=80,tutor-end=81}{i}\htmlData{tutor-start=81,tutor-end=82}{.}$

对中间一对 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{k}}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{A}_{\htmlData{tutor-start=11,tutor-end=12}{k}\htmlData{tutor-start=12,tutor-end=13}{+}\htmlData{tutor-start=13,tutor-end=14}{1}}\htmlData{tutor-start=15,tutor-end=16}{)}$，由 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{k}} \htmlData{tutor-start=7,tutor-end=14}{\Delta }\htmlData{tutor-start=14,tutor-end=15}{A}_{\htmlData{tutor-start=17,tutor-end=18}{k}\htmlData{tutor-start=18,tutor-end=19}{+}\htmlData{tutor-start=19,tutor-end=20}{1}}\htmlData{tutor-start=21,tutor-end=22}{|} \htmlData{tutor-start=23,tutor-end=24}{=} \htmlData{tutor-start=25,tutor-end=26}{1}$ 及事实 2： $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{k}}\htmlData{tutor-start=6,tutor-end=7}{|} \htmlData{tutor-start=8,tutor-end=9}{+} \htmlData{tutor-start=10,tutor-end=11}{|}\htmlData{tutor-start=11,tutor-end=12}{A}_{\htmlData{tutor-start=14,tutor-end=15}{k}\htmlData{tutor-start=15,tutor-end=16}{+}\htmlData{tutor-start=16,tutor-end=17}{1}}\htmlData{tutor-start=18,tutor-end=19}{|} \htmlData{tutor-start=20,tutor-end=24}{\ge }\htmlData{tutor-start=24,tutor-end=25}{3}\htmlData{tutor-start=25,tutor-end=26}{.}$

将以上不等式相加： $\htmlData{tutor-start=0,tutor-end=1}{S}_{\htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{k}} \htmlData{tutor-start=7,tutor-end=8}{=} \htmlData{tutor-start=9,tutor-end=10}{(}\htmlData{tutor-start=10,tutor-end=11}{|}\htmlData{tutor-start=11,tutor-end=12}{A}_{\htmlData{tutor-start=14,tutor-end=15}{k}}\htmlData{tutor-start=16,tutor-end=17}{|} \htmlData{tutor-start=18,tutor-end=19}{+} \htmlData{tutor-start=20,tutor-end=21}{|}\htmlData{tutor-start=21,tutor-end=22}{A}_{\htmlData{tutor-start=24,tutor-end=25}{k}\htmlData{tutor-start=25,tutor-end=26}{+}\htmlData{tutor-start=26,tutor-end=27}{1}}\htmlData{tutor-start=28,tutor-end=29}{|}\htmlData{tutor-start=29,tutor-end=30}{)} \htmlData{tutor-start=31,tutor-end=32}{+} \sum_{\htmlData{tutor-start=39,tutor-end=40}{i}\htmlData{tutor-start=40,tutor-end=41}{=}\htmlData{tutor-start=41,tutor-end=42}{1}}^{\htmlData{tutor-start=45,tutor-end=46}{k}\htmlData{tutor-start=46,tutor-end=47}{-}\htmlData{tutor-start=47,tutor-end=48}{1}}\htmlData{tutor-start=49,tutor-end=50}{(}\htmlData{tutor-start=50,tutor-end=51}{|}\htmlData{tutor-start=51,tutor-end=52}{A}_{\htmlData{tutor-start=54,tutor-end=55}{i}}\htmlData{tutor-start=56,tutor-end=57}{|} \htmlData{tutor-start=58,tutor-end=59}{+} \htmlData{tutor-start=60,tutor-end=61}{|}\htmlData{tutor-start=61,tutor-end=62}{A}_{\htmlData{tutor-start=64,tutor-end=65}{2}\htmlData{tutor-start=65,tutor-end=66}{k}\htmlData{tutor-start=66,tutor-end=67}{+}\htmlData{tutor-start=67,tutor-end=68}{1}\htmlData{tutor-start=68,tutor-end=69}{-}\htmlData{tutor-start=69,tutor-end=70}{i}}\htmlData{tutor-start=71,tutor-end=72}{|}\htmlData{tutor-start=72,tutor-end=73}{)} \htmlData{tutor-start=74,tutor-end=78}{\ge }\htmlData{tutor-start=78,tutor-end=79}{3} \htmlData{tutor-start=80,tutor-end=81}{+} \sum_{\htmlData{tutor-start=88,tutor-end=89}{i}\htmlData{tutor-start=89,tutor-end=90}{=}\htmlData{tutor-start=90,tutor-end=91}{1}}^{\htmlData{tutor-start=94,tutor-end=95}{k}\htmlData{tutor-start=95,tutor-end=96}{-}\htmlData{tutor-start=96,tutor-end=97}{1}}\htmlData{tutor-start=98,tutor-end=99}{(}\htmlData{tutor-start=99,tutor-end=100}{2}\htmlData{tutor-start=100,tutor-end=101}{k}\htmlData{tutor-start=101,tutor-end=102}{+}\htmlData{tutor-start=102,tutor-end=103}{1}\htmlData{tutor-start=103,tutor-end=104}{-}\htmlData{tutor-start=104,tutor-end=105}{2}\htmlData{tutor-start=105,tutor-end=106}{i}\htmlData{tutor-start=106,tutor-end=107}{)}\htmlData{tutor-start=107,tutor-end=108}{.}$

计算求和： $\htmlData{tutor-start=0,tutor-end=7}{\sum_{i}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{1}}^{\htmlData{tutor-start=12,tutor-end=13}{k}\htmlData{tutor-start=13,tutor-end=14}{-}\htmlData{tutor-start=14,tutor-end=15}{1}}\htmlData{tutor-start=16,tutor-end=17}{(}\htmlData{tutor-start=17,tutor-end=18}{2}\htmlData{tutor-start=18,tutor-end=19}{k}\htmlData{tutor-start=19,tutor-end=20}{+}\htmlData{tutor-start=20,tutor-end=21}{1}\htmlData{tutor-start=21,tutor-end=22}{-}\htmlData{tutor-start=22,tutor-end=23}{2}\htmlData{tutor-start=23,tutor-end=24}{i}\htmlData{tutor-start=24,tutor-end=25}{)} \htmlData{tutor-start=26,tutor-end=27}{=} \htmlData{tutor-start=28,tutor-end=29}{(}\htmlData{tutor-start=29,tutor-end=30}{2}\htmlData{tutor-start=30,tutor-end=31}{k}\htmlData{tutor-start=31,tutor-end=32}{-}\htmlData{tutor-start=32,tutor-end=33}{1}\htmlData{tutor-start=33,tutor-end=34}{)} \htmlData{tutor-start=35,tutor-end=36}{+} \htmlData{tutor-start=37,tutor-end=38}{(}\htmlData{tutor-start=38,tutor-end=39}{2}\htmlData{tutor-start=39,tutor-end=40}{k}\htmlData{tutor-start=40,tutor-end=41}{-}\htmlData{tutor-start=41,tutor-end=42}{3}\htmlData{tutor-start=42,tutor-end=43}{)} \htmlData{tutor-start=44,tutor-end=45}{+} \dots \htmlData{tutor-start=52,tutor-end=53}{+} \htmlData{tutor-start=54,tutor-end=55}{3} \htmlData{tutor-start=56,tutor-end=57}{=} \sum_{\htmlData{tutor-start=64,tutor-end=65}{m}\htmlData{tutor-start=65,tutor-end=66}{=}\htmlData{tutor-start=66,tutor-end=67}{1}}^{\htmlData{tutor-start=70,tutor-end=71}{k}\htmlData{tutor-start=71,tutor-end=72}{-}\htmlData{tutor-start=72,tutor-end=73}{1}}\htmlData{tutor-start=74,tutor-end=75}{(}\htmlData{tutor-start=75,tutor-end=76}{2}\htmlData{tutor-start=76,tutor-end=77}{m}\htmlData{tutor-start=77,tutor-end=78}{+}\htmlData{tutor-start=78,tutor-end=79}{1}\htmlData{tutor-start=79,tutor-end=80}{)} \htmlData{tutor-start=81,tutor-end=82}{=} \htmlData{tutor-start=83,tutor-end=84}{(}\htmlData{tutor-start=84,tutor-end=85}{k}\htmlData{tutor-start=85,tutor-end=86}{-}\htmlData{tutor-start=86,tutor-end=87}{1}\htmlData{tutor-start=87,tutor-end=88}{)}\htmlData{tutor-start=88,tutor-end=89}{(}\htmlData{tutor-start=89,tutor-end=90}{k}\htmlData{tutor-start=90,tutor-end=91}{+}\htmlData{tutor-start=91,tutor-end=92}{1}\htmlData{tutor-start=92,tutor-end=93}{)} \htmlData{tutor-start=94,tutor-end=95}{=} \htmlData{tutor-start=96,tutor-end=97}{k}^{\htmlData{tutor-start=99,tutor-end=100}{2}} \htmlData{tutor-start=102,tutor-end=103}{-} \htmlData{tutor-start=104,tutor-end=105}{1}\htmlData{tutor-start=105,tutor-end=106}{.}$

（这里用了等差数列求和：首项 3，末项 $\htmlData{tutor-start=0,tutor-end=1}{2}\htmlData{tutor-start=1,tutor-end=2}{k}\htmlData{tutor-start=2,tutor-end=3}{-}\htmlData{tutor-start=3,tutor-end=4}{1}$，共 $\htmlData{tutor-start=0,tutor-end=1}{k}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{1}$ 项，和为 $\frac{\htmlData{tutor-start=6,tutor-end=7}{(}\htmlData{tutor-start=7,tutor-end=8}{k}\htmlData{tutor-start=8,tutor-end=9}{-}\htmlData{tutor-start=9,tutor-end=10}{1}\htmlData{tutor-start=10,tutor-end=11}{)}\htmlData{tutor-start=11,tutor-end=12}{(}\htmlData{tutor-start=12,tutor-end=13}{3} \htmlData{tutor-start=14,tutor-end=15}{+} \htmlData{tutor-start=16,tutor-end=17}{2}\htmlData{tutor-start=17,tutor-end=18}{k} \htmlData{tutor-start=19,tutor-end=20}{-} \htmlData{tutor-start=21,tutor-end=22}{1}\htmlData{tutor-start=22,tutor-end=23}{)}}{\htmlData{tutor-start=25,tutor-end=26}{2}} \htmlData{tutor-start=28,tutor-end=29}{=} \htmlData{tutor-start=30,tutor-end=31}{(}\htmlData{tutor-start=31,tutor-end=32}{k}\htmlData{tutor-start=32,tutor-end=33}{-}\htmlData{tutor-start=33,tutor-end=34}{1}\htmlData{tutor-start=34,tutor-end=35}{)}\htmlData{tutor-start=35,tutor-end=36}{(}\htmlData{tutor-start=36,tutor-end=37}{k}\htmlData{tutor-start=37,tutor-end=38}{+}\htmlData{tutor-start=38,tutor-end=39}{1}\htmlData{tutor-start=39,tutor-end=40}{)} \htmlData{tutor-start=41,tutor-end=42}{=} \htmlData{tutor-start=43,tutor-end=44}{k}^{\htmlData{tutor-start=46,tutor-end=47}{2}} \htmlData{tutor-start=49,tutor-end=50}{-} \htmlData{tutor-start=51,tutor-end=52}{1}$。）

因此： $\htmlData{tutor-start=0,tutor-end=1}{S}_{\htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{k}} \htmlData{tutor-start=7,tutor-end=11}{\ge }\htmlData{tutor-start=11,tutor-end=12}{3} \htmlData{tutor-start=13,tutor-end=14}{+} \htmlData{tutor-start=15,tutor-end=16}{(}\htmlData{tutor-start=16,tutor-end=17}{k}^{\htmlData{tutor-start=19,tutor-end=20}{2}} \htmlData{tutor-start=22,tutor-end=23}{-} \htmlData{tutor-start=24,tutor-end=25}{1}\htmlData{tutor-start=25,tutor-end=26}{)} \htmlData{tutor-start=27,tutor-end=28}{=} \htmlData{tutor-start=29,tutor-end=30}{k}^{\htmlData{tutor-start=32,tutor-end=33}{2}} \htmlData{tutor-start=35,tutor-end=36}{+} \htmlData{tutor-start=37,tutor-end=38}{2}\htmlData{tutor-start=38,tutor-end=39}{.}$

设 $\htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{2}\htmlData{tutor-start=5,tutor-end=6}{k}\htmlData{tutor-start=6,tutor-end=7}{+}\htmlData{tutor-start=7,tutor-end=8}{1}$（$\htmlData{tutor-start=0,tutor-end=1}{k} \htmlData{tutor-start=2,tutor-end=6}{\ge }\htmlData{tutor-start=6,tutor-end=7}{1}$）。将 $\htmlData{tutor-start=0,tutor-end=1}{2}\htmlData{tutor-start=1,tutor-end=2}{k}\htmlData{tutor-start=2,tutor-end=3}{+}\htmlData{tutor-start=3,tutor-end=4}{1}$ 个集合按下标对称配对：$\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{1}}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{A}_{\htmlData{tutor-start=11,tutor-end=12}{2}\htmlData{tutor-start=12,tutor-end=13}{k}\htmlData{tutor-start=13,tutor-end=14}{+}\htmlData{tutor-start=14,tutor-end=15}{1}}\htmlData{tutor-start=16,tutor-end=17}{)}$，$\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{2}}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{A}_{\htmlData{tutor-start=11,tutor-end=12}{2}\htmlData{tutor-start=12,tutor-end=13}{k}}\htmlData{tutor-start=14,tutor-end=15}{)}$，…，$\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{k}\htmlData{tutor-start=5,tutor-end=6}{-}\htmlData{tutor-start=6,tutor-end=7}{1}}\htmlData{tutor-start=8,tutor-end=9}{,} \htmlData{tutor-start=10,tutor-end=11}{A}_{\htmlData{tutor-start=13,tutor-end=14}{k}\htmlData{tutor-start=14,tutor-end=15}{+}\htmlData{tutor-start=15,tutor-end=16}{3}}\htmlData{tutor-start=17,tutor-end=18}{)}$，以及中间三个 $\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{k}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{A}_{\htmlData{tutor-start=10,tutor-end=11}{k}\htmlData{tutor-start=11,tutor-end=12}{+}\htmlData{tutor-start=12,tutor-end=13}{1}}\htmlData{tutor-start=14,tutor-end=15}{,} \htmlData{tutor-start=16,tutor-end=17}{A}_{\htmlData{tutor-start=19,tutor-end=20}{k}\htmlData{tutor-start=20,tutor-end=21}{+}\htmlData{tutor-start=21,tutor-end=22}{2}}$。

对 $\htmlData{tutor-start=0,tutor-end=1}{i} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{1}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{2}\htmlData{tutor-start=8,tutor-end=9}{,} \dots\htmlData{tutor-start=15,tutor-end=16}{,} \htmlData{tutor-start=17,tutor-end=18}{k}\htmlData{tutor-start=18,tutor-end=19}{-}\htmlData{tutor-start=19,tutor-end=20}{1}$，由事实 1： $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{i}}\htmlData{tutor-start=6,tutor-end=7}{|} \htmlData{tutor-start=8,tutor-end=9}{+} \htmlData{tutor-start=10,tutor-end=11}{|}\htmlData{tutor-start=11,tutor-end=12}{A}_{\htmlData{tutor-start=14,tutor-end=15}{2}\htmlData{tutor-start=15,tutor-end=16}{k}\htmlData{tutor-start=16,tutor-end=17}{+}\htmlData{tutor-start=17,tutor-end=18}{2}\htmlData{tutor-start=18,tutor-end=19}{-}\htmlData{tutor-start=19,tutor-end=20}{i}}\htmlData{tutor-start=21,tutor-end=22}{|} \htmlData{tutor-start=23,tutor-end=27}{\ge }\htmlData{tutor-start=27,tutor-end=28}{|}\htmlData{tutor-start=28,tutor-end=29}{A}_{\htmlData{tutor-start=31,tutor-end=32}{i}} \htmlData{tutor-start=34,tutor-end=41}{\Delta }\htmlData{tutor-start=41,tutor-end=42}{A}_{\htmlData{tutor-start=44,tutor-end=45}{2}\htmlData{tutor-start=45,tutor-end=46}{k}\htmlData{tutor-start=46,tutor-end=47}{+}\htmlData{tutor-start=47,tutor-end=48}{2}\htmlData{tutor-start=48,tutor-end=49}{-}\htmlData{tutor-start=49,tutor-end=50}{i}}\htmlData{tutor-start=51,tutor-end=52}{|} \htmlData{tutor-start=53,tutor-end=54}{=} \htmlData{tutor-start=55,tutor-end=56}{(}\htmlData{tutor-start=56,tutor-end=57}{2}\htmlData{tutor-start=57,tutor-end=58}{k}\htmlData{tutor-start=58,tutor-end=59}{+}\htmlData{tutor-start=59,tutor-end=60}{2}\htmlData{tutor-start=60,tutor-end=61}{-}\htmlData{tutor-start=61,tutor-end=62}{i}\htmlData{tutor-start=62,tutor-end=63}{)} \htmlData{tutor-start=64,tutor-end=65}{-} \htmlData{tutor-start=66,tutor-end=67}{i} \htmlData{tutor-start=68,tutor-end=69}{=} \htmlData{tutor-start=70,tutor-end=71}{2}\htmlData{tutor-start=71,tutor-end=72}{k} \htmlData{tutor-start=73,tutor-end=74}{+} \htmlData{tutor-start=75,tutor-end=76}{2} \htmlData{tutor-start=77,tutor-end=78}{-} \htmlData{tutor-start=79,tutor-end=80}{2}\htmlData{tutor-start=80,tutor-end=81}{i}\htmlData{tutor-start=81,tutor-end=82}{.}$

对中间三个集合，注意 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{k}} \htmlData{tutor-start=7,tutor-end=14}{\Delta }\htmlData{tutor-start=14,tutor-end=15}{A}_{\htmlData{tutor-start=17,tutor-end=18}{k}\htmlData{tutor-start=18,tutor-end=19}{+}\htmlData{tutor-start=19,tutor-end=20}{1}}\htmlData{tutor-start=21,tutor-end=22}{|} \htmlData{tutor-start=23,tutor-end=24}{=} \htmlData{tutor-start=25,tutor-end=26}{1}$，由事实 2： $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{k}}\htmlData{tutor-start=6,tutor-end=7}{|} \htmlData{tutor-start=8,tutor-end=9}{+} \htmlData{tutor-start=10,tutor-end=11}{|}\htmlData{tutor-start=11,tutor-end=12}{A}_{\htmlData{tutor-start=14,tutor-end=15}{k}\htmlData{tutor-start=15,tutor-end=16}{+}\htmlData{tutor-start=16,tutor-end=17}{1}}\htmlData{tutor-start=18,tutor-end=19}{|} \htmlData{tutor-start=20,tutor-end=24}{\ge }\htmlData{tutor-start=24,tutor-end=25}{3}\htmlData{tutor-start=25,tutor-end=26}{.}$

又 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{k}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{1}} \htmlData{tutor-start=9,tutor-end=16}{\Delta }\htmlData{tutor-start=16,tutor-end=17}{A}_{\htmlData{tutor-start=19,tutor-end=20}{k}\htmlData{tutor-start=20,tutor-end=21}{+}\htmlData{tutor-start=21,tutor-end=22}{2}}\htmlData{tutor-start=23,tutor-end=24}{|} \htmlData{tutor-start=25,tutor-end=26}{=} \htmlData{tutor-start=27,tutor-end=28}{1}$，由事实 1（$\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{k}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{2}}$ 非空，$\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{k}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{2}}\htmlData{tutor-start=8,tutor-end=9}{|} \htmlData{tutor-start=10,tutor-end=14}{\ge }\htmlData{tutor-start=14,tutor-end=15}{1}$）： $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{k}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{2}}\htmlData{tutor-start=8,tutor-end=9}{|} \htmlData{tutor-start=10,tutor-end=14}{\ge }\htmlData{tutor-start=14,tutor-end=15}{1}\htmlData{tutor-start=15,tutor-end=16}{.}$

因此： $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{|}\htmlData{tutor-start=2,tutor-end=3}{A}_{\htmlData{tutor-start=5,tutor-end=6}{k}}\htmlData{tutor-start=7,tutor-end=8}{|} \htmlData{tutor-start=9,tutor-end=10}{+} \htmlData{tutor-start=11,tutor-end=12}{|}\htmlData{tutor-start=12,tutor-end=13}{A}_{\htmlData{tutor-start=15,tutor-end=16}{k}\htmlData{tutor-start=16,tutor-end=17}{+}\htmlData{tutor-start=17,tutor-end=18}{1}}\htmlData{tutor-start=19,tutor-end=20}{|}\htmlData{tutor-start=20,tutor-end=21}{)} \htmlData{tutor-start=22,tutor-end=23}{+} \htmlData{tutor-start=24,tutor-end=25}{|}\htmlData{tutor-start=25,tutor-end=26}{A}_{\htmlData{tutor-start=28,tutor-end=29}{k}\htmlData{tutor-start=29,tutor-end=30}{+}\htmlData{tutor-start=30,tutor-end=31}{2}}\htmlData{tutor-start=32,tutor-end=33}{|} \htmlData{tutor-start=34,tutor-end=38}{\ge }\htmlData{tutor-start=38,tutor-end=39}{3} \htmlData{tutor-start=40,tutor-end=41}{+} \htmlData{tutor-start=42,tutor-end=43}{1} \htmlData{tutor-start=44,tutor-end=45}{=} \htmlData{tutor-start=46,tutor-end=47}{4}\htmlData{tutor-start=47,tutor-end=48}{.}$

将以上不等式相加： $\htmlData{tutor-start=0,tutor-end=1}{S}_{\htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{k}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{1}} \htmlData{tutor-start=9,tutor-end=10}{=} \htmlData{tutor-start=11,tutor-end=12}{(}\htmlData{tutor-start=12,tutor-end=13}{|}\htmlData{tutor-start=13,tutor-end=14}{A}_{\htmlData{tutor-start=16,tutor-end=17}{k}}\htmlData{tutor-start=18,tutor-end=19}{|} \htmlData{tutor-start=20,tutor-end=21}{+} \htmlData{tutor-start=22,tutor-end=23}{|}\htmlData{tutor-start=23,tutor-end=24}{A}_{\htmlData{tutor-start=26,tutor-end=27}{k}\htmlData{tutor-start=27,tutor-end=28}{+}\htmlData{tutor-start=28,tutor-end=29}{1}}\htmlData{tutor-start=30,tutor-end=31}{|} \htmlData{tutor-start=32,tutor-end=33}{+} \htmlData{tutor-start=34,tutor-end=35}{|}\htmlData{tutor-start=35,tutor-end=36}{A}_{\htmlData{tutor-start=38,tutor-end=39}{k}\htmlData{tutor-start=39,tutor-end=40}{+}\htmlData{tutor-start=40,tutor-end=41}{2}}\htmlData{tutor-start=42,tutor-end=43}{|}\htmlData{tutor-start=43,tutor-end=44}{)} \htmlData{tutor-start=45,tutor-end=46}{+} \sum_{\htmlData{tutor-start=53,tutor-end=54}{i}\htmlData{tutor-start=54,tutor-end=55}{=}\htmlData{tutor-start=55,tutor-end=56}{1}}^{\htmlData{tutor-start=59,tutor-end=60}{k}\htmlData{tutor-start=60,tutor-end=61}{-}\htmlData{tutor-start=61,tutor-end=62}{1}}\htmlData{tutor-start=63,tutor-end=64}{(}\htmlData{tutor-start=64,tutor-end=65}{|}\htmlData{tutor-start=65,tutor-end=66}{A}_{\htmlData{tutor-start=68,tutor-end=69}{i}}\htmlData{tutor-start=70,tutor-end=71}{|} \htmlData{tutor-start=72,tutor-end=73}{+} \htmlData{tutor-start=74,tutor-end=75}{|}\htmlData{tutor-start=75,tutor-end=76}{A}_{\htmlData{tutor-start=78,tutor-end=79}{2}\htmlData{tutor-start=79,tutor-end=80}{k}\htmlData{tutor-start=80,tutor-end=81}{+}\htmlData{tutor-start=81,tutor-end=82}{2}\htmlData{tutor-start=82,tutor-end=83}{-}\htmlData{tutor-start=83,tutor-end=84}{i}}\htmlData{tutor-start=85,tutor-end=86}{|}\htmlData{tutor-start=86,tutor-end=87}{)} \htmlData{tutor-start=88,tutor-end=92}{\ge }\htmlData{tutor-start=92,tutor-end=93}{4} \htmlData{tutor-start=94,tutor-end=95}{+} \sum_{\htmlData{tutor-start=102,tutor-end=103}{i}\htmlData{tutor-start=103,tutor-end=104}{=}\htmlData{tutor-start=104,tutor-end=105}{1}}^{\htmlData{tutor-start=108,tutor-end=109}{k}\htmlData{tutor-start=109,tutor-end=110}{-}\htmlData{tutor-start=110,tutor-end=111}{1}}\htmlData{tutor-start=112,tutor-end=113}{(}\htmlData{tutor-start=113,tutor-end=114}{2}\htmlData{tutor-start=114,tutor-end=115}{k}\htmlData{tutor-start=115,tutor-end=116}{+}\htmlData{tutor-start=116,tutor-end=117}{2}\htmlData{tutor-start=117,tutor-end=118}{-}\htmlData{tutor-start=118,tutor-end=119}{2}\htmlData{tutor-start=119,tutor-end=120}{i}\htmlData{tutor-start=120,tutor-end=121}{)}\htmlData{tutor-start=121,tutor-end=122}{.}$

计算求和： $\htmlData{tutor-start=0,tutor-end=7}{\sum_{i}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{1}}^{\htmlData{tutor-start=12,tutor-end=13}{k}\htmlData{tutor-start=13,tutor-end=14}{-}\htmlData{tutor-start=14,tutor-end=15}{1}}\htmlData{tutor-start=16,tutor-end=17}{(}\htmlData{tutor-start=17,tutor-end=18}{2}\htmlData{tutor-start=18,tutor-end=19}{k}\htmlData{tutor-start=19,tutor-end=20}{+}\htmlData{tutor-start=20,tutor-end=21}{2}\htmlData{tutor-start=21,tutor-end=22}{-}\htmlData{tutor-start=22,tutor-end=23}{2}\htmlData{tutor-start=23,tutor-end=24}{i}\htmlData{tutor-start=24,tutor-end=25}{)} \htmlData{tutor-start=26,tutor-end=27}{=} \htmlData{tutor-start=28,tutor-end=29}{2}\htmlData{tutor-start=29,tutor-end=30}{k} \htmlData{tutor-start=31,tutor-end=32}{+} \htmlData{tutor-start=33,tutor-end=34}{(}\htmlData{tutor-start=34,tutor-end=35}{2}\htmlData{tutor-start=35,tutor-end=36}{k}\htmlData{tutor-start=36,tutor-end=37}{-}\htmlData{tutor-start=37,tutor-end=38}{2}\htmlData{tutor-start=38,tutor-end=39}{)} \htmlData{tutor-start=40,tutor-end=41}{+} \dots \htmlData{tutor-start=48,tutor-end=49}{+} \htmlData{tutor-start=50,tutor-end=51}{4} \htmlData{tutor-start=52,tutor-end=53}{=} \htmlData{tutor-start=54,tutor-end=55}{2}\sum_{\htmlData{tutor-start=61,tutor-end=62}{m}\htmlData{tutor-start=62,tutor-end=63}{=}\htmlData{tutor-start=63,tutor-end=64}{1}}^{\htmlData{tutor-start=67,tutor-end=68}{k}\htmlData{tutor-start=68,tutor-end=69}{-}\htmlData{tutor-start=69,tutor-end=70}{1}}\htmlData{tutor-start=71,tutor-end=72}{(}\htmlData{tutor-start=72,tutor-end=73}{m}\htmlData{tutor-start=73,tutor-end=74}{+}\htmlData{tutor-start=74,tutor-end=75}{1}\htmlData{tutor-start=75,tutor-end=76}{)} \htmlData{tutor-start=77,tutor-end=78}{=} \htmlData{tutor-start=79,tutor-end=80}{2} \htmlData{tutor-start=81,tutor-end=87}{\cdot }\frac{\htmlData{tutor-start=93,tutor-end=94}{(}\htmlData{tutor-start=94,tutor-end=95}{k}\htmlData{tutor-start=95,tutor-end=96}{-}\htmlData{tutor-start=96,tutor-end=97}{1}\htmlData{tutor-start=97,tutor-end=98}{)}\htmlData{tutor-start=98,tutor-end=99}{(}\htmlData{tutor-start=99,tutor-end=100}{k}\htmlData{tutor-start=100,tutor-end=101}{+}\htmlData{tutor-start=101,tutor-end=102}{2}\htmlData{tutor-start=102,tutor-end=103}{)}}{\htmlData{tutor-start=105,tutor-end=106}{2}} \htmlData{tutor-start=108,tutor-end=109}{=} \htmlData{tutor-start=110,tutor-end=111}{(}\htmlData{tutor-start=111,tutor-end=112}{k}\htmlData{tutor-start=112,tutor-end=113}{-}\htmlData{tutor-start=113,tutor-end=114}{1}\htmlData{tutor-start=114,tutor-end=115}{)}\htmlData{tutor-start=115,tutor-end=116}{(}\htmlData{tutor-start=116,tutor-end=117}{k}\htmlData{tutor-start=117,tutor-end=118}{+}\htmlData{tutor-start=118,tutor-end=119}{2}\htmlData{tutor-start=119,tutor-end=120}{)} \htmlData{tutor-start=121,tutor-end=122}{=} \htmlData{tutor-start=123,tutor-end=124}{k}^{\htmlData{tutor-start=126,tutor-end=127}{2}} \htmlData{tutor-start=129,tutor-end=130}{+} \htmlData{tutor-start=131,tutor-end=132}{k} \htmlData{tutor-start=133,tutor-end=134}{-} \htmlData{tutor-start=135,tutor-end=136}{2}\htmlData{tutor-start=136,tutor-end=137}{.}$

（等差数列：首项 4，末项 $\htmlData{tutor-start=0,tutor-end=1}{2}\htmlData{tutor-start=1,tutor-end=2}{k}$，共 $\htmlData{tutor-start=0,tutor-end=1}{k}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{1}$ 项，和为 $\frac{\htmlData{tutor-start=6,tutor-end=7}{(}\htmlData{tutor-start=7,tutor-end=8}{k}\htmlData{tutor-start=8,tutor-end=9}{-}\htmlData{tutor-start=9,tutor-end=10}{1}\htmlData{tutor-start=10,tutor-end=11}{)}\htmlData{tutor-start=11,tutor-end=12}{(}\htmlData{tutor-start=12,tutor-end=13}{4}\htmlData{tutor-start=13,tutor-end=14}{+}\htmlData{tutor-start=14,tutor-end=15}{2}\htmlData{tutor-start=15,tutor-end=16}{k}\htmlData{tutor-start=16,tutor-end=17}{)}}{\htmlData{tutor-start=19,tutor-end=20}{2}} \htmlData{tutor-start=22,tutor-end=23}{=} \htmlData{tutor-start=24,tutor-end=25}{(}\htmlData{tutor-start=25,tutor-end=26}{k}\htmlData{tutor-start=26,tutor-end=27}{-}\htmlData{tutor-start=27,tutor-end=28}{1}\htmlData{tutor-start=28,tutor-end=29}{)}\htmlData{tutor-start=29,tutor-end=30}{(}\htmlData{tutor-start=30,tutor-end=31}{k}\htmlData{tutor-start=31,tutor-end=32}{+}\htmlData{tutor-start=32,tutor-end=33}{2}\htmlData{tutor-start=33,tutor-end=34}{)} \htmlData{tutor-start=35,tutor-end=36}{=} \htmlData{tutor-start=37,tutor-end=38}{k}^{\htmlData{tutor-start=40,tutor-end=41}{2}} \htmlData{tutor-start=43,tutor-end=44}{+} \htmlData{tutor-start=45,tutor-end=46}{k} \htmlData{tutor-start=47,tutor-end=48}{-} \htmlData{tutor-start=49,tutor-end=50}{2}$。）

因此： $\htmlData{tutor-start=0,tutor-end=1}{S}_{\htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{k}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{1}} \htmlData{tutor-start=9,tutor-end=13}{\ge }\htmlData{tutor-start=13,tutor-end=14}{4} \htmlData{tutor-start=15,tutor-end=16}{+} \htmlData{tutor-start=17,tutor-end=18}{(}\htmlData{tutor-start=18,tutor-end=19}{k}^{\htmlData{tutor-start=21,tutor-end=22}{2}} \htmlData{tutor-start=24,tutor-end=25}{+} \htmlData{tutor-start=26,tutor-end=27}{k} \htmlData{tutor-start=28,tutor-end=29}{-} \htmlData{tutor-start=30,tutor-end=31}{2}\htmlData{tutor-start=31,tutor-end=32}{)} \htmlData{tutor-start=33,tutor-end=34}{=} \htmlData{tutor-start=35,tutor-end=36}{k}^{\htmlData{tutor-start=38,tutor-end=39}{2}} \htmlData{tutor-start=41,tutor-end=42}{+} \htmlData{tutor-start=43,tutor-end=44}{k} \htmlData{tutor-start=45,tutor-end=46}{+} \htmlData{tutor-start=47,tutor-end=48}{2} \htmlData{tutor-start=49,tutor-end=50}{=} \htmlData{tutor-start=51,tutor-end=52}{k}\htmlData{tutor-start=52,tutor-end=53}{(}\htmlData{tutor-start=53,tutor-end=54}{k}\htmlData{tutor-start=54,tutor-end=55}{+}\htmlData{tutor-start=55,tutor-end=56}{1}\htmlData{tutor-start=56,tutor-end=57}{)} \htmlData{tutor-start=58,tutor-end=59}{+} \htmlData{tutor-start=60,tutor-end=61}{2}\htmlData{tutor-start=61,tutor-end=62}{.}$

对任意正整数 $\htmlData{tutor-start=0,tutor-end=1}{n}$，将其写成二进制表示 $\htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{2}^{\htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{1}}} \htmlData{tutor-start=14,tutor-end=15}{+} \htmlData{tutor-start=16,tutor-end=17}{2}^{\htmlData{tutor-start=19,tutor-end=20}{a}_{\htmlData{tutor-start=22,tutor-end=23}{2}}} \htmlData{tutor-start=26,tutor-end=27}{+} \cdots \htmlData{tutor-start=35,tutor-end=36}{+} \htmlData{tutor-start=37,tutor-end=38}{2}^{\htmlData{tutor-start=40,tutor-end=41}{a}_{\htmlData{tutor-start=43,tutor-end=44}{k}}}$，其中 $\htmlData{tutor-start=0,tutor-end=1}{0} \htmlData{tutor-start=2,tutor-end=12}{\leqslant }\htmlData{tutor-start=12,tutor-end=13}{a}_{\htmlData{tutor-start=15,tutor-end=16}{1}} \htmlData{tutor-start=18,tutor-end=19}{<} \htmlData{tutor-start=20,tutor-end=21}{a}_{\htmlData{tutor-start=23,tutor-end=24}{2}} \htmlData{tutor-start=26,tutor-end=27}{<} \cdots \htmlData{tutor-start=35,tutor-end=36}{<} \htmlData{tutor-start=37,tutor-end=38}{a}_{\htmlData{tutor-start=40,tutor-end=41}{k}}$。定义集合 $\htmlData{tutor-start=0,tutor-end=1}{T}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{n}\htmlData{tutor-start=3,tutor-end=4}{)} \htmlData{tutor-start=5,tutor-end=6}{=} \htmlData{tutor-start=7,tutor-end=9}{\{}\htmlData{tutor-start=9,tutor-end=10}{2}^{\htmlData{tutor-start=12,tutor-end=13}{a}_{\htmlData{tutor-start=15,tutor-end=16}{1}}}\htmlData{tutor-start=18,tutor-end=19}{,} \htmlData{tutor-start=20,tutor-end=21}{2}^{\htmlData{tutor-start=23,tutor-end=24}{a}_{\htmlData{tutor-start=26,tutor-end=27}{2}}}\htmlData{tutor-start=29,tutor-end=30}{,} \htmlData{tutor-start=31,tutor-end=37}{\ldots}\htmlData{tutor-start=37,tutor-end=38}{,} \htmlData{tutor-start=39,tutor-end=40}{2}^{\htmlData{tutor-start=42,tutor-end=43}{a}_{\htmlData{tutor-start=45,tutor-end=46}{k}}}\htmlData{tutor-start=48,tutor-end=50}{\}}$，并约定 $\htmlData{tutor-start=0,tutor-end=1}{T}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{0}\htmlData{tutor-start=3,tutor-end=4}{)} \htmlData{tutor-start=5,tutor-end=6}{=} \htmlData{tutor-start=7,tutor-end=18}{\varnothing}$。

由 Lucas 定理，$\htmlData{tutor-start=0,tutor-end=1}{C}_{\htmlData{tutor-start=3,tutor-end=4}{n}}^{\htmlData{tutor-start=7,tutor-end=8}{i}}$ 为奇数当且仅当在二进制下 $\htmlData{tutor-start=0,tutor-end=1}{i}$ 的每一位都不超过 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 的对应位，这等价于 $\htmlData{tutor-start=0,tutor-end=1}{T}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{i}\htmlData{tutor-start=3,tutor-end=4}{)} \htmlData{tutor-start=5,tutor-end=15}{\subseteq }\htmlData{tutor-start=15,tutor-end=16}{T}\htmlData{tutor-start=16,tutor-end=17}{(}\htmlData{tutor-start=17,tutor-end=18}{n}\htmlData{tutor-start=18,tutor-end=19}{)}$。因此 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{n}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{q}\htmlData{tutor-start=5,tutor-end=6}{)} \htmlData{tutor-start=7,tutor-end=8}{=} \sum_{\htmlData{tutor-start=15,tutor-end=16}{A} \htmlData{tutor-start=17,tutor-end=27}{\subseteq }\htmlData{tutor-start=27,tutor-end=28}{T}\htmlData{tutor-start=28,tutor-end=29}{(}\htmlData{tutor-start=29,tutor-end=30}{n}\htmlData{tutor-start=30,tutor-end=31}{)}} \htmlData{tutor-start=33,tutor-end=34}{q}^{\htmlData{tutor-start=36,tutor-end=42}{\sigma}\htmlData{tutor-start=42,tutor-end=43}{(}\htmlData{tutor-start=43,tutor-end=44}{A}\htmlData{tutor-start=44,tutor-end=45}{)}} \htmlData{tutor-start=47,tutor-end=48}{=} \prod_{\htmlData{tutor-start=56,tutor-end=57}{a} \htmlData{tutor-start=58,tutor-end=62}{\in }\htmlData{tutor-start=62,tutor-end=63}{T}\htmlData{tutor-start=63,tutor-end=64}{(}\htmlData{tutor-start=64,tutor-end=65}{n}\htmlData{tutor-start=65,tutor-end=66}{)}} \htmlData{tutor-start=68,tutor-end=69}{(}\htmlData{tutor-start=69,tutor-end=70}{1} \htmlData{tutor-start=71,tutor-end=72}{+} \htmlData{tutor-start=73,tutor-end=74}{q}^{\htmlData{tutor-start=76,tutor-end=77}{a}}\htmlData{tutor-start=78,tutor-end=79}{)}\htmlData{tutor-start=79,tutor-end=80}{,}$ 其中 $\htmlData{tutor-start=0,tutor-end=6}{\sigma}\htmlData{tutor-start=6,tutor-end=7}{(}\htmlData{tutor-start=7,tutor-end=8}{A}\htmlData{tutor-start=8,tutor-end=9}{)}$ 表示集合 $\htmlData{tutor-start=0,tutor-end=1}{A}$ 中所有元素之和。