2010 中国国家集训队选拔考试（China TST）

books/competition_archive/china_tst/2010_china_tst.pdf · HS-MATH-1024-v2.1-solution-aware

24 题0 个小问/题组

Quiz 1 / Day 1 · 代数

Assume real numbers ai, bi (i = 0, 1, · · · , 2n) satisfy the following conditions: (1) for i = 0, 1, · · · , 2n −1, we have ai + ai+1 ≥0; (2) for j = 0, 1, · · · , n −1, we have a2j+1 ≤0; (2) for any integer p, q, 0 ≤p ≤q ≤n, we have P2q i=0(−1)iaibi ≥0, and determine when the equality holds. k=2p bk > 0. Prove that P2n

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题目标签：QUIZ1-DAY1-P1

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Quiz 1 / Day 1 · 平面几何

Let ABCD be a convex quadrilateral. Assume line AB and CD intersect at E, and B lies between A and E. Assume line AD and BC intersect at F, and D lies between A and F. Assume the circumcircles of △BEC and △CFD intersect at C and P. Prove that ∠BAP = ∠CAD if and only if BD ∥EF.

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题目标签：QUIZ1-DAY1-P2

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Quiz 1 / Day 1 · 数论

Fine all positive integers m, n ≥2, such that (1) m + 1 is a prime number of type 4k −1; (2) there is a (positive) prime number p and nonnegative integer a, such that m2n−1 −1 m −1 = mn + pa.

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题目标签：QUIZ1-DAY1-P3

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Quiz 1 / Day 2 · 平面几何

Let △ABC be an acute triangle with AB > AC, let I be the center of the incircle. Let M, N be the midpoint of AC and AB respectively. D, E are on AC and AB respectively such that BD ∥IM and CE ∥IN. A line through I parallel to DE intersects BC in P. Let Q be the projection of P on line AI. Prove that Q is on the circumcircle of △ABC.

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Quiz 1 / Day 2 · 组合数学

Let M = {1, 2, · · · , n}, each element of M is colored in either red, blue or yellow. Set A = {(x, y, z) ∈M × M × M|x + y + z ≡0 mod n, x, y, z are of same color}, B = {(x, y, z) ∈ M × M × M|x + y + z ≡0 mod n, x, y, z are of pairwise distinct color}. Prove that 2|A| ≥|B|.

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题目标签：QUIZ1-DAY2-P2

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Quiz 1 / Day 2 · 组合数学

Let A be a finite set, and A1, A2, · · · , An are subsets of A with the following conditions: (1) |A1| = |A2| = · · · = |An| = k, and k > |A| 2 ; (2) for any a, b ∈A, there exist Ar, As, At (1 ≤r < s < t ≤n) such that a, b ∈Ar ∩As ∩At; (3) for any integer i, j (1 ≤i < j ≤n), |Ai ∩Aj| ≤3. Find all possible value(s) of n when k attains maximum among all possible systems (A1, A2, · · · , An, A).

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题目标签：QUIZ1-DAY2-P3

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Quiz 2 / Day 1 · 平面几何

Let ABCD be a convex quadrilateral with A, B, C, D concyclic. Assume ∠ADC is acute and AB BC = DA CD. Let Γ be a circle through A and D, tangent to AB, and let E be a point on Γ and inside ABCD. Prove that AE ⊥EC if and only if AE AB −ED AD = 1.

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题目标签：QUIZ2-DAY1-P1

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Quiz 2 / Day 1 · 数论

Given positive integer n, find the largest real number λ = λ(n), such that for any degree n polynomial with complex coefficients f(x) = anxn + an−1xn−1 + · · · + a0, and any permutation x0, x1, · · · , xn of 0, 1, · · · , n, the following inequality holds Pn k=0 |f(xk) − f(xk+1)| ≥λ|an|, where xn+1 = x0. 1≤i<j≤n(ai + aj) has at least k + 1 different prime divisors.

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题目标签：QUIZ2-DAY1-P2

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Quiz 2 / Day 1 · 数论

Let k > 1 be an integer, set n = 2k+1. Prove that for any positive integers a1 < a2 < · · · < an, the number Q

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Quiz 2 / Day 2 · 平面几何

Let △ABC be an acute triangle, and let D be the projection of A on BC. Let M, N be the midpoints of AB and AC respectively. Let Γ1 and Γ2 be the circumcircles of △BDM and △CDN respectively, and let K be the other intersection point of Γ1 and Γ2. Let P be an arbitrary point on BC and E, F are on AC and AB respectively such that PEAF is a parallelogram. Prove that if MN is a common tangent line of Γ1 and Γ2, then K, E, A, F are concyclic. i=1 i=1

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题目标签：QUIZ2-DAY2-P1

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Quiz 2 / Day 2 · 代数

Find all positive real numbers λ such that for all integers n ≥2 and all positive real numbers a1, a2, · · · , an with a1 + a2 + · · · + an = n, the following inequality holds: Pn 1 ai −λ Qn 1 ai ≤ n −λ.

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题目标签：QUIZ2-DAY2-P2

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Quiz 2 / Day 2 · 数论

For integers n > 1, define f(n) to be the sum of all postive divisors of n that are less than n. Prove that for any positive integer k, there exists a positive integer n > 1 such that n < f(n) < f2(n) < · · · < fk(n), where fi(n) = f(fi−1(n)) for i > 1 and f1(n) = f(n).

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Quiz 3 / Day 1 · 代数

Given integer n ≥2 and positive real number a, find the smallest real number M = M(n, a), such that for any positive real numbers x1, x2, · · · , xn with x1x2 · · · xn = 1, the following inequality holds: n X ≤M 1 a + S −xi i=1 i=1 xi. where S = Pn

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Quiz 3 / Day 1 · 组合数学

In a football league, there are n ≥6 teams. Each team has a homecourt jersey and a road jersey with different color. When two teams play, the home team always wear homecourt jersey and the road team wear their homecourt jersey if the color is different from the home team’s homecourt jersey, or otherwise the road team shall wear their road jersey. It is required that in any two games with 4 different teams, the 4 teams’ jerseys have at least 3 different color. Find the least number of color that the n teams’ 2n jerseys may use. n

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题目标签：QUIZ3-DAY1-P2

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Quiz 3 / Day 1 · 数论

Given positive integer k, prove that there exists a positive integer N depending only on k such that for any integer n ≥N, has at least k different prime divisors. k

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Quiz 3 / Day 2 · 平面几何

Let ω be a semicircle and AB its diameter. ω1 and ω2 are two different circles, both tangent to ω and to AB, and ω1 is also tangent to ω2. Let P, Q be the tangent points of ω1 and ω2 to AB respectively, and P is between A and Q. Let C be the tangent point of ω1 and ω. Find tan ∠ACQ.

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题目标签：QUIZ3-DAY2-P1

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Quiz 3 / Day 2 · 数论

Prove that there exists a sequence of unbounded positive integers a1 ≤a2 ≤a3 ≤· · · , such that there exists a positive integer M with the following property: for any integer n ≥M, if n + 1 is not prime, then any prime divisor of n! + 1 is greater than n + an.

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题目标签：QUIZ3-DAY2-P2

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Quiz 3 / Day 2 · 数论

An (unordered) partition P of a positive integer n is an n-tuple of nonnegative integers P = (x1, x2, · · · , xn) such that Pn k=1 kxk = n. For positive integer m ≤n, and a partition Q = (y1, y2, · · · , ym) of m, Q is called compatible to P if yi ≤xi for i = 1, 2, · · · , m. Let S(n) be the number of partitions P of n such that for each odd m < n, m has exactly one partition compatible to P and for each even m < n, m has exactly two partitions compatible to P. Find S(2010).

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题目标签：QUIZ3-DAY2-P3

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TST 1 / 未标注日 · 平面几何

Given acute triangle ABC with AB > AC, let M be the midpoint of BC. P is a point in triangle AMC such that ∠MAB = ∠PAC. Let O, O1, O2 be the circumcenters of △ABC, △ABP, △ACP respectively. Prove that line AO passes through the midpoint of O1O2.

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题目标签：TST1-P1

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TST 1 / 未标注日 · 代数

Let A = {a1, a2, · · · , a2010} and B = {b1, b2, · · · , b2010} be two sets of complex numbers. Suppose X X 1≤i<j≤2010 (ai + aj)k = 1≤i<j≤2010 (bi + bj)k holds for every k = 1, 2, · · · , 2010. Prove that A = B.

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题目标签：TST1-P2

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TST 1 / 未标注日 · 数论

Let n1, n2, · · · , n26 be pairwise distinct positive integers satisfying (1) for each ni, its digits belong to the set {1, 2}; (2) for each i, j, ni can’t be obtained from nj by adding some digits on the right. Find the smallest possible value of P26 i=1 S(ni), where S(m) denotes the sum of all digits of a positive integer m.

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题目标签：TST1-P3

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TST 2 / 未标注日 · 组合数学

Let G = G(V, E) be a simple graph with vertex set V and edge set E. Suppose |V | = n. A map f : V →Z is called good, if f satisfies the followings: (1) P v∈V f(v) = |E|; (2) color arbitarily some vertices into red, one can always find a red vertex v such that f(v) is no more than the number of uncolored vertices adjacent to v. Let m(G) be the number of good maps. Prove that if every vertex in G is adjacent to at least one another vertex, then n ≤m(G) ≤n!.

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题目标签：TST2-P1

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TST 2 / 未标注日 · 数论

Given integer a1 ≥2. For integer n ≥2, define an to be the smallest positive integer which is not coprime to an−1 and not equal to a1, a2, · · · , an−1. Prove that every positive integer except 1 appears in this sequence {an}.

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题目标签：TST2-P2

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TST 2 / 未标注日 · 代数

Given integer n ≥2 and real numbers x1, x2, · · · , xn in the interval [0, 1]. Prove that there exist real numbers a0, a1, · · · , an satisfying the following conditions: (1) a0 + an = 0; (2) |ai| ≤1, for i = 0, 1, · · · , n; (3) |ai −ai−1| = xi, for i = 1, 2, · · · , n.

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题目标签：TST2-P3

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