2024 中国数学奥林匹克（CMO）

由于数列 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{x}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 的所有项均属于有限整数集合 $\htmlData{tutor-start=0,tutor-end=1}{S}$，根据鸽巢原理，必然存在 $\htmlData{tutor-start=0,tutor-end=1}{i} \htmlData{tutor-start=2,tutor-end=3}{<} \htmlData{tutor-start=4,tutor-end=5}{j}$ 使得 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{i}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=9}{x}_{\htmlData{tutor-start=11,tutor-end=12}{j}}$。因为递推规则 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{n}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{1}}$ 仅依赖于 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{n}}$（确定性映射），一旦数值重复，后续序列必将完全重复，即 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{i}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{k}} \htmlData{tutor-start=8,tutor-end=9}{=} \htmlData{tutor-start=10,tutor-end=11}{x}_{\htmlData{tutor-start=13,tutor-end=14}{j}\htmlData{tutor-start=14,tutor-end=15}{+}\htmlData{tutor-start=15,tutor-end=16}{k}}$ 对所有 $\htmlData{tutor-start=0,tutor-end=1}{k} \htmlData{tutor-start=2,tutor-end=6}{\ge }\htmlData{tutor-start=6,tutor-end=7}{0}$ 成立。这表明数列从第 $\htmlData{tutor-start=0,tutor-end=1}{i}$ 项开始进入周期循环。

进入周期后，数列不可能恒 $\htmlData{tutor-start=0,tutor-end=4}{\le }\htmlData{tutor-start=4,tutor-end=5}{L}$（否则单调增无上界矛盾），也不可能恒 $\htmlData{tutor-start=0,tutor-end=1}{>} \htmlData{tutor-start=2,tutor-end=3}{L}$（否则单调减无下界矛盾）。因此周期内必存在项跨越 $\htmlData{tutor-start=0,tutor-end=1}{L}$。 考察跨越瞬间：若 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{n}} \htmlData{tutor-start=6,tutor-end=7}{>} \htmlData{tutor-start=8,tutor-end=9}{L}$，则 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{n}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{1}} \htmlData{tutor-start=8,tutor-end=9}{=} \htmlData{tutor-start=10,tutor-end=18}{\lfloor }\htmlData{tutor-start=18,tutor-end=19}{x}_{\htmlData{tutor-start=21,tutor-end=22}{n}}\htmlData{tutor-start=23,tutor-end=24}{/}\htmlData{tutor-start=24,tutor-end=31}{\alpha }\htmlData{tutor-start=31,tutor-end=38}{\rfloor}$。由前述有界性分析知 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{n}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{1}} \htmlData{tutor-start=8,tutor-end=12}{\le }\htmlData{tutor-start=12,tutor-end=13}{L}$（实际上 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{n}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{1}} \htmlData{tutor-start=8,tutor-end=9}{<} \htmlData{tutor-start=10,tutor-end=11}{L}$ 因为 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{n}}\htmlData{tutor-start=5,tutor-end=6}{/}\htmlData{tutor-start=6,tutor-end=13}{\alpha }\htmlData{tutor-start=13,tutor-end=14}{<} \htmlData{tutor-start=15,tutor-end=16}{L}$ 当 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{n}}$ 接近上界时，或者更严格地，若 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{n}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{1}} \htmlData{tutor-start=8,tutor-end=9}{>} \htmlData{tutor-start=10,tutor-end=11}{L}$ 则意味着 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{n}} \htmlData{tutor-start=6,tutor-end=7}{>} \htmlData{tutor-start=8,tutor-end=15}{\alpha }\htmlData{tutor-start=15,tutor-end=16}{L}$，但这超出了稳定区间的上界 $\htmlData{tutor-start=0,tutor-end=7}{\alpha }\htmlData{tutor-start=7,tutor-end=8}{L}$ 的整数部分限制，或者通过 $\htmlData{tutor-start=0,tutor-end=1}{L} \htmlData{tutor-start=2,tutor-end=3}{>} \htmlData{tutor-start=4,tutor-end=10}{\alpha}^{\htmlData{tutor-start=12,tutor-end=13}{2}}\htmlData{tutor-start=14,tutor-end=15}{/}\htmlData{tutor-start=15,tutor-end=16}{(}\htmlData{tutor-start=16,tutor-end=22}{\alpha}\htmlData{tutor-start=22,tutor-end=23}{-}\htmlData{tutor-start=23,tutor-end=24}{1}\htmlData{tutor-start=24,tutor-end=25}{)}$ 可证 $\htmlData{tutor-start=0,tutor-end=8}{\lfloor }\htmlData{tutor-start=8,tutor-end=9}{(}\htmlData{tutor-start=9,tutor-end=10}{L}\htmlData{tutor-start=10,tutor-end=11}{+}\htmlData{tutor-start=11,tutor-end=12}{1}\htmlData{tutor-start=12,tutor-end=13}{)}\htmlData{tutor-start=13,tutor-end=14}{/}\htmlData{tutor-start=14,tutor-end=21}{\alpha }\htmlData{tutor-start=21,tutor-end=29}{\rfloor }\htmlData{tutor-start=29,tutor-end=33}{\le }\htmlData{tutor-start=33,tutor-end=34}{L}$）。简言之，从 $\htmlData{tutor-start=0,tutor-end=1}{>}\htmlData{tutor-start=1,tutor-end=2}{L}$ 区域一步必落入 $\htmlData{tutor-start=0,tutor-end=4}{\le }\htmlData{tutor-start=4,tutor-end=5}{L}$ 区域。 反之，若 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{n}} \htmlData{tutor-start=6,tutor-end=10}{\le }\htmlData{tutor-start=10,tutor-end=11}{L}$，则 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{n}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{1}} \htmlData{tutor-start=8,tutor-end=9}{=} \htmlData{tutor-start=10,tutor-end=18}{\lfloor }\htmlData{tutor-start=18,tutor-end=25}{\alpha }\htmlData{tutor-start=25,tutor-end=26}{x}_{\htmlData{tutor-start=28,tutor-end=29}{n}} \htmlData{tutor-start=31,tutor-end=38}{\rfloor}$。这可能仍 $\htmlData{tutor-start=0,tutor-end=4}{\le }\htmlData{tutor-start=4,tutor-end=5}{L}$，也可能 $\htmlData{tutor-start=0,tutor-end=1}{>} \htmlData{tutor-start=2,tutor-end=3}{L}$。 结论：在周期轨道上，状态切换模式必然是“若干次 $\htmlData{tutor-start=0,tutor-end=4}{\le }\htmlData{tutor-start=4,tutor-end=5}{L}$ 后接一次 $\htmlData{tutor-start=0,tutor-end=1}{>} \htmlData{tutor-start=2,tutor-end=3}{L}$”，紧接着必须回到 $\htmlData{tutor-start=0,tutor-end=4}{\le }\htmlData{tutor-start=4,tutor-end=5}{L}$。即 $\htmlData{tutor-start=0,tutor-end=1}{>}\htmlData{tutor-start=1,tutor-end=2}{L}$ 的状态总是孤立出现的，不会连续出现。

设周期轨道上的最小值为 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$。显然 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}} \htmlData{tutor-start=8,tutor-end=12}{\le }\htmlData{tutor-start=12,tutor-end=13}{L}$（因为必有 $\htmlData{tutor-start=0,tutor-end=4}{\le }\htmlData{tutor-start=4,tutor-end=5}{L}$ 的项）。 考虑 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$ 的前驱 $\htmlData{tutor-start=0,tutor-end=1}{y}$。若 $\htmlData{tutor-start=0,tutor-end=1}{y} \htmlData{tutor-start=2,tutor-end=6}{\le }\htmlData{tutor-start=6,tutor-end=7}{L}$，则 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}} \htmlData{tutor-start=8,tutor-end=9}{=} \htmlData{tutor-start=10,tutor-end=18}{\lfloor }\htmlData{tutor-start=18,tutor-end=25}{\alpha }\htmlData{tutor-start=25,tutor-end=26}{y} \htmlData{tutor-start=27,tutor-end=35}{\rfloor }\htmlData{tutor-start=35,tutor-end=39}{\ge }\htmlData{tutor-start=39,tutor-end=47}{\lfloor }\htmlData{tutor-start=47,tutor-end=54}{\alpha }\htmlData{tutor-start=54,tutor-end=55}{x}_{\htmlData{tutor-start=57,tutor-end=58}{m}\htmlData{tutor-start=58,tutor-end=59}{i}\htmlData{tutor-start=59,tutor-end=60}{n}} \htmlData{tutor-start=62,tutor-end=70}{\rfloor }\htmlData{tutor-start=70,tutor-end=71}{>} \htmlData{tutor-start=72,tutor-end=73}{x}_{\htmlData{tutor-start=75,tutor-end=76}{m}\htmlData{tutor-start=76,tutor-end=77}{i}\htmlData{tutor-start=77,tutor-end=78}{n}}$（因 $\htmlData{tutor-start=0,tutor-end=7}{\alpha }\htmlData{tutor-start=7,tutor-end=8}{>} \htmlData{tutor-start=9,tutor-end=10}{1}$），矛盾。故 $\htmlData{tutor-start=0,tutor-end=1}{y}$ 必须 $\htmlData{tutor-start=0,tutor-end=1}{>} \htmlData{tutor-start=2,tutor-end=3}{L}$。 这意味着 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$ 总是紧接在一个 $\htmlData{tutor-start=0,tutor-end=1}{>}\htmlData{tutor-start=1,tutor-end=2}{L}$ 的项之后产生。设该 $\htmlData{tutor-start=0,tutor-end=1}{>}\htmlData{tutor-start=1,tutor-end=2}{L}$ 的项为 $\htmlData{tutor-start=0,tutor-end=1}{z}$，则 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}} \htmlData{tutor-start=8,tutor-end=9}{=} \htmlData{tutor-start=10,tutor-end=18}{\lfloor }\htmlData{tutor-start=18,tutor-end=19}{z}\htmlData{tutor-start=19,tutor-end=20}{/}\htmlData{tutor-start=20,tutor-end=27}{\alpha }\htmlData{tutor-start=27,tutor-end=34}{\rfloor}$。 由于 $\htmlData{tutor-start=0,tutor-end=1}{z}$ 是周期中某个 $\htmlData{tutor-start=0,tutor-end=1}{>}\htmlData{tutor-start=1,tutor-end=2}{L}$ 的项，且 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$ 是全局最小，我们可以推断 $\htmlData{tutor-start=0,tutor-end=1}{z}$ 必须是所有 $\htmlData{tutor-start=0,tutor-end=1}{>}\htmlData{tutor-start=1,tutor-end=2}{L}$ 项中最小的那个（记为 $\htmlData{tutor-start=0,tutor-end=1}{z}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$）。因为若存在更小的 $\htmlData{tutor-start=0,tutor-end=1}{z}' \htmlData{tutor-start=3,tutor-end=4}{>} \htmlData{tutor-start=5,tutor-end=6}{L}$，则 $\htmlData{tutor-start=0,tutor-end=8}{\lfloor }\htmlData{tutor-start=8,tutor-end=9}{z}'\htmlData{tutor-start=10,tutor-end=11}{/}\htmlData{tutor-start=11,tutor-end=18}{\alpha }\htmlData{tutor-start=18,tutor-end=25}{\rfloor}$ 会更小或相等，这与 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$ 的定义及 $\htmlData{tutor-start=0,tutor-end=1}{z}$ 的来源有关。更准确地说，由 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$ 的唯一生成路径（只能来自 $\htmlData{tutor-start=0,tutor-end=1}{>}\htmlData{tutor-start=1,tutor-end=2}{L}$），且 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{t}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=13}{\lfloor }\htmlData{tutor-start=13,tutor-end=14}{t}\htmlData{tutor-start=14,tutor-end=15}{/}\htmlData{tutor-start=15,tutor-end=22}{\alpha }\htmlData{tutor-start=22,tutor-end=29}{\rfloor}$ 单调，故 $\htmlData{tutor-start=0,tutor-end=1}{z}$ 必为 $\htmlData{tutor-start=0,tutor-end=1}{>}\htmlData{tutor-start=1,tutor-end=2}{L}$ 集合中的最小元。 计算这个特定的 $\htmlData{tutor-start=0,tutor-end=1}{z}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$：它是周期中第一个超过 $\htmlData{tutor-start=0,tutor-end=1}{L}$ 的数吗？不一定。但它是使得 $\htmlData{tutor-start=0,tutor-end=8}{\lfloor }\htmlData{tutor-start=8,tutor-end=9}{z}\htmlData{tutor-start=9,tutor-end=10}{/}\htmlData{tutor-start=10,tutor-end=17}{\alpha }\htmlData{tutor-start=17,tutor-end=24}{\rfloor}$ 最小的 $\htmlData{tutor-start=0,tutor-end=1}{z}$。实际上，我们可以证明 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$ 是唯一确定的常数 $\htmlData{tutor-start=0,tutor-end=1}{C} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=12}{\lfloor }\frac{\htmlData{tutor-start=18,tutor-end=19}{L}\htmlData{tutor-start=19,tutor-end=20}{+}\htmlData{tutor-start=20,tutor-end=21}{1}}{\htmlData{tutor-start=23,tutor-end=29}{\alpha}} \htmlData{tutor-start=31,tutor-end=38}{\rfloor}$。 理由：$\htmlData{tutor-start=0,tutor-end=1}{z} \htmlData{tutor-start=2,tutor-end=6}{\ge }\htmlData{tutor-start=6,tutor-end=7}{L}\htmlData{tutor-start=7,tutor-end=8}{+}\htmlData{tutor-start=8,tutor-end=9}{1} \implies \htmlData{tutor-start=19,tutor-end=20}{x}_{\htmlData{tutor-start=22,tutor-end=23}{m}\htmlData{tutor-start=23,tutor-end=24}{i}\htmlData{tutor-start=24,tutor-end=25}{n}} \htmlData{tutor-start=27,tutor-end=31}{\ge }\htmlData{tutor-start=31,tutor-end=39}{\lfloor }\frac{\htmlData{tutor-start=45,tutor-end=46}{L}\htmlData{tutor-start=46,tutor-end=47}{+}\htmlData{tutor-start=47,tutor-end=48}{1}}{\htmlData{tutor-start=50,tutor-end=56}{\alpha}} \htmlData{tutor-start=58,tutor-end=65}{\rfloor}$。另一方面，若取 $\htmlData{tutor-start=0,tutor-end=1}{z}$ 为周期中任意 $\htmlData{tutor-start=0,tutor-end=1}{>}\htmlData{tutor-start=1,tutor-end=2}{L}$ 项，其前一项 $\htmlData{tutor-start=0,tutor-end=1}{w} \htmlData{tutor-start=2,tutor-end=6}{\le }\htmlData{tutor-start=6,tutor-end=7}{L}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{z} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=12}{\lfloor }\htmlData{tutor-start=12,tutor-end=19}{\alpha }\htmlData{tutor-start=19,tutor-end=20}{w} \htmlData{tutor-start=21,tutor-end=28}{\rfloor}$。要使 $\htmlData{tutor-start=0,tutor-end=1}{z}$ 尽可能小（从而 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$ 尽可能小），$\htmlData{tutor-start=0,tutor-end=1}{w}$ 应尽可能小。但在周期中，$\htmlData{tutor-start=0,tutor-end=1}{w}$ 不能任意小，它受限于轨道结构。然而，我们可以反向思考：是否存在一个固定的吸引子？ 参考解答指出 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=16}{\lfloor }\frac{\htmlData{tutor-start=22,tutor-end=23}{L}\htmlData{tutor-start=23,tutor-end=24}{+}\htmlData{tutor-start=24,tutor-end=25}{1}}{\htmlData{tutor-start=27,tutor-end=33}{\alpha}} \htmlData{tutor-start=35,tutor-end=42}{\rfloor}$ 是定值。这是因为在稳态下，$\htmlData{tutor-start=0,tutor-end=1}{>}\htmlData{tutor-start=1,tutor-end=2}{L}$ 的最小值必然由 $\htmlData{tutor-start=0,tutor-end=4}{\le }\htmlData{tutor-start=4,tutor-end=5}{L}$ 的最小值生成，而 $\htmlData{tutor-start=0,tutor-end=4}{\le }\htmlData{tutor-start=4,tutor-end=5}{L}$ 的最小值又由 $\htmlData{tutor-start=0,tutor-end=1}{>}\htmlData{tutor-start=1,tutor-end=2}{L}$ 的最小值生成。这是一个耦合系统。解这个不动点方程组可得唯一解。

基于上述结构，周期轨道从 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$（$\htmlData{tutor-start=0,tutor-end=4}{\le }\htmlData{tutor-start=4,tutor-end=5}{L}$）开始，经历若干步 $\htmlData{tutor-start=0,tutor-end=4}{\le }\htmlData{tutor-start=4,tutor-end=5}{L}$ 的增长，直到某一步跳变到 $\htmlData{tutor-start=0,tutor-end=1}{>}\htmlData{tutor-start=1,tutor-end=2}{L}$（记为 $\htmlData{tutor-start=0,tutor-end=1}{z}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$），然后下一步立即回到 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$（因为 $\htmlData{tutor-start=0,tutor-end=1}{z}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$ 是 $\htmlData{tutor-start=0,tutor-end=1}{>}\htmlData{tutor-start=1,tutor-end=2}{L}$ 中最小的，其像必为 $\htmlData{tutor-start=0,tutor-end=4}{\le }\htmlData{tutor-start=4,tutor-end=5}{L}$ 中最小的，即 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$）。 等等，若 $\htmlData{tutor-start=0,tutor-end=1}{z}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$ 下一步直接回到 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$，则周期结构为：$\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}} \htmlData{tutor-start=8,tutor-end=12}{\to }\dots \htmlData{tutor-start=18,tutor-end=22}{\to }\htmlData{tutor-start=22,tutor-end=23}{w} \htmlData{tutor-start=24,tutor-end=28}{\to }\htmlData{tutor-start=28,tutor-end=29}{z}_{\htmlData{tutor-start=31,tutor-end=32}{m}\htmlData{tutor-start=32,tutor-end=33}{i}\htmlData{tutor-start=33,tutor-end=34}{n}} \htmlData{tutor-start=36,tutor-end=40}{\to }\htmlData{tutor-start=40,tutor-end=41}{x}_{\htmlData{tutor-start=43,tutor-end=44}{m}\htmlData{tutor-start=44,tutor-end=45}{i}\htmlData{tutor-start=45,tutor-end=46}{n}}$。 让我们数一下步数： 1. $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}} \htmlData{tutor-start=8,tutor-end=12}{\to }\htmlData{tutor-start=12,tutor-end=13}{x}_{\htmlData{tutor-start=15,tutor-end=16}{1}}'$ ($\htmlData{tutor-start=0,tutor-end=4}{\le }\htmlData{tutor-start=4,tutor-end=5}{L}$) 2. ... k. $\htmlData{tutor-start=0,tutor-end=1}{w} \htmlData{tutor-start=2,tutor-end=6}{\to }\htmlData{tutor-start=6,tutor-end=7}{z}_{\htmlData{tutor-start=9,tutor-end=10}{m}\htmlData{tutor-start=10,tutor-end=11}{i}\htmlData{tutor-start=11,tutor-end=12}{n}}$ ($\htmlData{tutor-start=0,tutor-end=1}{>}\htmlData{tutor-start=1,tutor-end=2}{L}$) k+1. $\htmlData{tutor-start=0,tutor-end=1}{z}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}} \htmlData{tutor-start=8,tutor-end=12}{\to }\htmlData{tutor-start=12,tutor-end=13}{x}_{\htmlData{tutor-start=15,tutor-end=16}{m}\htmlData{tutor-start=16,tutor-end=17}{i}\htmlData{tutor-start=17,tutor-end=18}{n}}$ 这里 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$ 到 $\htmlData{tutor-start=0,tutor-end=1}{z}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$ 的过程全是 $\htmlData{tutor-start=0,tutor-end=4}{\le }\htmlData{tutor-start=4,tutor-end=5}{L}$ 的项（除了终点 $\htmlData{tutor-start=0,tutor-end=1}{z}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$）。设这段长度为 $\htmlData{tutor-start=0,tutor-end=1}{k}$（即 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$ 是第0项，$\htmlData{tutor-start=0,tutor-end=1}{z}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$ 是第 $\htmlData{tutor-start=0,tutor-end=1}{k}$ 项）。则 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$ 再次出现是第 $\htmlData{tutor-start=0,tutor-end=1}{k}\htmlData{tutor-start=1,tutor-end=2}{+}\htmlData{tutor-start=2,tutor-end=3}{1}$ 项。故 $\htmlData{tutor-start=0,tutor-end=1}{T} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{k}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{1}$。 我们需要确定 $\htmlData{tutor-start=0,tutor-end=1}{k}$ 的奇偶性。 回顾状态序列：$\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}} \htmlData{tutor-start=8,tutor-end=9}{(}\htmlData{tutor-start=9,tutor-end=13}{\le }\htmlData{tutor-start=13,tutor-end=14}{L}\htmlData{tutor-start=14,tutor-end=15}{)}\htmlData{tutor-start=15,tutor-end=16}{,} \htmlData{tutor-start=17,tutor-end=18}{x}_{\htmlData{tutor-start=20,tutor-end=21}{1}}' \htmlData{tutor-start=24,tutor-end=25}{(}\htmlData{tutor-start=25,tutor-end=29}{\le }\htmlData{tutor-start=29,tutor-end=30}{L}\htmlData{tutor-start=30,tutor-end=31}{)}\htmlData{tutor-start=31,tutor-end=32}{,} \dots\htmlData{tutor-start=38,tutor-end=39}{,} \htmlData{tutor-start=40,tutor-end=41}{w} \htmlData{tutor-start=42,tutor-end=43}{(}\htmlData{tutor-start=43,tutor-end=47}{\le }\htmlData{tutor-start=47,tutor-end=48}{L}\htmlData{tutor-start=48,tutor-end=49}{)}\htmlData{tutor-start=49,tutor-end=50}{,} \htmlData{tutor-start=51,tutor-end=52}{z}_{\htmlData{tutor-start=54,tutor-end=55}{m}\htmlData{tutor-start=55,tutor-end=56}{i}\htmlData{tutor-start=56,tutor-end=57}{n}} \htmlData{tutor-start=59,tutor-end=60}{(}\htmlData{tutor-start=60,tutor-end=61}{>}\htmlData{tutor-start=61,tutor-end=62}{L}\htmlData{tutor-start=62,tutor-end=63}{)}$。 注意：$\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$ 本身是 $\htmlData{tutor-start=0,tutor-end=4}{\le }\htmlData{tutor-start=4,tutor-end=5}{L}$。从 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$ 出发，只要结果 $\htmlData{tutor-start=0,tutor-end=4}{\le }\htmlData{tutor-start=4,tutor-end=5}{L}$，就继续；一旦结果 $\htmlData{tutor-start=0,tutor-end=1}{>}\htmlData{tutor-start=1,tutor-end=2}{L}$，就是 $\htmlData{tutor-start=0,tutor-end=1}{z}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$。 关键点：$\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$ 是 $\htmlData{tutor-start=0,tutor-end=4}{\le }\htmlData{tutor-start=4,tutor-end=5}{L}$ 区域的最小值。$\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{n}\htmlData{tutor-start=4,tutor-end=5}{e}\htmlData{tutor-start=5,tutor-end=6}{x}\htmlData{tutor-start=6,tutor-end=7}{t}} \htmlData{tutor-start=9,tutor-end=10}{=} \htmlData{tutor-start=11,tutor-end=19}{\lfloor }\htmlData{tutor-start=19,tutor-end=26}{\alpha }\htmlData{tutor-start=26,tutor-end=27}{x}_{\htmlData{tutor-start=29,tutor-end=30}{m}\htmlData{tutor-start=30,tutor-end=31}{i}\htmlData{tutor-start=31,tutor-end=32}{n}} \htmlData{tutor-start=34,tutor-end=41}{\rfloor}$。因为 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}} \htmlData{tutor-start=8,tutor-end=16}{\approx }\htmlData{tutor-start=16,tutor-end=17}{L}\htmlData{tutor-start=17,tutor-end=18}{/}\htmlData{tutor-start=18,tutor-end=24}{\alpha}$，所以 $\htmlData{tutor-start=0,tutor-end=7}{\alpha }\htmlData{tutor-start=7,tutor-end=8}{x}_{\htmlData{tutor-start=10,tutor-end=11}{m}\htmlData{tutor-start=11,tutor-end=12}{i}\htmlData{tutor-start=12,tutor-end=13}{n}} \htmlData{tutor-start=15,tutor-end=23}{\approx }\htmlData{tutor-start=23,tutor-end=24}{L}$。具体来说，$\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}} \htmlData{tutor-start=8,tutor-end=9}{=} \htmlData{tutor-start=10,tutor-end=18}{\lfloor }\htmlData{tutor-start=18,tutor-end=19}{(}\htmlData{tutor-start=19,tutor-end=20}{L}\htmlData{tutor-start=20,tutor-end=21}{+}\htmlData{tutor-start=21,tutor-end=22}{1}\htmlData{tutor-start=22,tutor-end=23}{)}\htmlData{tutor-start=23,tutor-end=24}{/}\htmlData{tutor-start=24,tutor-end=31}{\alpha }\htmlData{tutor-start=31,tutor-end=38}{\rfloor}$。则 $\htmlData{tutor-start=0,tutor-end=7}{\alpha }\htmlData{tutor-start=7,tutor-end=8}{x}_{\htmlData{tutor-start=10,tutor-end=11}{m}\htmlData{tutor-start=11,tutor-end=12}{i}\htmlData{tutor-start=12,tutor-end=13}{n}} \htmlData{tutor-start=15,tutor-end=19}{\le }\htmlData{tutor-start=19,tutor-end=20}{L}\htmlData{tutor-start=20,tutor-end=21}{+}\htmlData{tutor-start=21,tutor-end=22}{1}$。由于 $\htmlData{tutor-start=0,tutor-end=6}{\alpha}$ 无理，$\htmlData{tutor-start=0,tutor-end=7}{\alpha }\htmlData{tutor-start=7,tutor-end=8}{x}_{\htmlData{tutor-start=10,tutor-end=11}{m}\htmlData{tutor-start=11,tutor-end=12}{i}\htmlData{tutor-start=12,tutor-end=13}{n}} \neq \htmlData{tutor-start=20,tutor-end=21}{L}\htmlData{tutor-start=21,tutor-end=22}{+}\htmlData{tutor-start=22,tutor-end=23}{1}$，故 $\htmlData{tutor-start=0,tutor-end=8}{\lfloor }\htmlData{tutor-start=8,tutor-end=15}{\alpha }\htmlData{tutor-start=15,tutor-end=16}{x}_{\htmlData{tutor-start=18,tutor-end=19}{m}\htmlData{tutor-start=19,tutor-end=20}{i}\htmlData{tutor-start=20,tutor-end=21}{n}} \htmlData{tutor-start=23,tutor-end=31}{\rfloor }\htmlData{tutor-start=31,tutor-end=35}{\le }\htmlData{tutor-start=35,tutor-end=36}{L}$。 这意味着从 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$ 出发的第一步 **一定** 留在 $\htmlData{tutor-start=0,tutor-end=4}{\le }\htmlData{tutor-start=4,tutor-end=5}{L}$ 区域！ 所以序列开头至少有两个 $\htmlData{tutor-start=0,tutor-end=4}{\le }\htmlData{tutor-start=4,tutor-end=5}{L}$ 的项：$\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{x}_{\htmlData{tutor-start=12,tutor-end=13}{1}}'$。 一般地，设轨道上 $\htmlData{tutor-start=0,tutor-end=4}{\le }\htmlData{tutor-start=4,tutor-end=5}{L}$ 的项构成的连续段长度为 $\htmlData{tutor-start=0,tutor-end=1}{m}$（包含 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$ 和 $\htmlData{tutor-start=0,tutor-end=1}{w}$，不包含 $\htmlData{tutor-start=0,tutor-end=1}{z}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$）。则 $\htmlData{tutor-start=0,tutor-end=1}{T} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{m} \htmlData{tutor-start=6,tutor-end=7}{+} \htmlData{tutor-start=8,tutor-end=9}{1}$（加1是因为有一个 $\htmlData{tutor-start=0,tutor-end=1}{>}\htmlData{tutor-start=1,tutor-end=2}{L}$ 的项 $\htmlData{tutor-start=0,tutor-end=1}{z}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$）。 我们需要证明 $\htmlData{tutor-start=0,tutor-end=1}{m}$ 是偶数。 观察 $\htmlData{tutor-start=0,tutor-end=4}{\le }\htmlData{tutor-start=4,tutor-end=5}{L}$ 段内的映射 $\htmlData{tutor-start=0,tutor-end=1}{g}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)} \htmlData{tutor-start=5,tutor-end=6}{=} \htmlData{tutor-start=7,tutor-end=15}{\lfloor }\htmlData{tutor-start=15,tutor-end=22}{\alpha }\htmlData{tutor-start=22,tutor-end=23}{x} \htmlData{tutor-start=24,tutor-end=31}{\rfloor}$。这是一个严格递增函数（在整数集上）。 轨道片段为 $\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{0}} \htmlData{tutor-start=6,tutor-end=7}{<} \htmlData{tutor-start=8,tutor-end=9}{v}_{\htmlData{tutor-start=11,tutor-end=12}{1}} \htmlData{tutor-start=14,tutor-end=15}{<} \dots \htmlData{tutor-start=22,tutor-end=23}{<} \htmlData{tutor-start=24,tutor-end=25}{v}_{\htmlData{tutor-start=27,tutor-end=28}{m}\htmlData{tutor-start=28,tutor-end=29}{-}\htmlData{tutor-start=29,tutor-end=30}{1}}$，其中 $\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{0}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=9}{x}_{\htmlData{tutor-start=11,tutor-end=12}{m}\htmlData{tutor-start=12,tutor-end=13}{i}\htmlData{tutor-start=13,tutor-end=14}{n}}$，$\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{-}\htmlData{tutor-start=5,tutor-end=6}{1}} \htmlData{tutor-start=8,tutor-end=9}{=} \htmlData{tutor-start=10,tutor-end=11}{w}$，且 $\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{i}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{1}} \htmlData{tutor-start=8,tutor-end=9}{=} \htmlData{tutor-start=10,tutor-end=11}{g}\htmlData{tutor-start=11,tutor-end=12}{(}\htmlData{tutor-start=12,tutor-end=13}{v}_{\htmlData{tutor-start=15,tutor-end=16}{i}}\htmlData{tutor-start=17,tutor-end=18}{)}$。 最后一步 $\htmlData{tutor-start=0,tutor-end=1}{g}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{w}\htmlData{tutor-start=3,tutor-end=4}{)} \htmlData{tutor-start=5,tutor-end=6}{=} \htmlData{tutor-start=7,tutor-end=8}{z}_{\htmlData{tutor-start=10,tutor-end=11}{m}\htmlData{tutor-start=11,tutor-end=12}{i}\htmlData{tutor-start=12,tutor-end=13}{n}} \htmlData{tutor-start=15,tutor-end=16}{>} \htmlData{tutor-start=17,tutor-end=18}{L}$。 而 $\htmlData{tutor-start=0,tutor-end=1}{g}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{v}_{\htmlData{tutor-start=5,tutor-end=6}{m}\htmlData{tutor-start=6,tutor-end=7}{-}\htmlData{tutor-start=7,tutor-end=8}{2}}\htmlData{tutor-start=9,tutor-end=10}{)} \htmlData{tutor-start=11,tutor-end=12}{=} \htmlData{tutor-start=13,tutor-end=14}{v}_{\htmlData{tutor-start=16,tutor-end=17}{m}\htmlData{tutor-start=17,tutor-end=18}{-}\htmlData{tutor-start=18,tutor-end=19}{1}} \htmlData{tutor-start=21,tutor-end=25}{\le }\htmlData{tutor-start=25,tutor-end=26}{L}$。 这似乎不能直接推出 $\htmlData{tutor-start=0,tutor-end=1}{m}$ 的奇偶性。让我们重新审视参考解答的逻辑。 参考解答指出：$\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{1}} \htmlData{tutor-start=8,tutor-end=9}{>} \htmlData{tutor-start=10,tutor-end=11}{x}_{\htmlData{tutor-start=13,tutor-end=14}{m}\htmlData{tutor-start=14,tutor-end=15}{+}\htmlData{tutor-start=15,tutor-end=16}{3}} \htmlData{tutor-start=18,tutor-end=19}{>} \dots$。这暗示了在某种意义下的交替或单调性。 修正思路：利用 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$ 的特殊性。 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$ 是 $\htmlData{tutor-start=0,tutor-end=4}{\le }\htmlData{tutor-start=4,tutor-end=5}{L}$ 中最小的。$\htmlData{tutor-start=0,tutor-end=1}{z}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$ 是 $\htmlData{tutor-start=0,tutor-end=1}{>}\htmlData{tutor-start=1,tutor-end=2}{L}$ 中最小的。 关系：$\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}} \htmlData{tutor-start=8,tutor-end=9}{=} \htmlData{tutor-start=10,tutor-end=18}{\lfloor }\htmlData{tutor-start=18,tutor-end=19}{z}_{\htmlData{tutor-start=21,tutor-end=22}{m}\htmlData{tutor-start=22,tutor-end=23}{i}\htmlData{tutor-start=23,tutor-end=24}{n}}\htmlData{tutor-start=25,tutor-end=26}{/}\htmlData{tutor-start=26,tutor-end=33}{\alpha }\htmlData{tutor-start=33,tutor-end=40}{\rfloor}$，$\htmlData{tutor-start=0,tutor-end=1}{z}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}} \htmlData{tutor-start=8,tutor-end=9}{=} \htmlData{tutor-start=10,tutor-end=18}{\lfloor }\htmlData{tutor-start=18,tutor-end=25}{\alpha }\htmlData{tutor-start=25,tutor-end=26}{w} \htmlData{tutor-start=27,tutor-end=34}{\rfloor}$，其中 $\htmlData{tutor-start=0,tutor-end=1}{w}$ 是 $\htmlData{tutor-start=0,tutor-end=4}{\le }\htmlData{tutor-start=4,tutor-end=5}{L}$ 中最大的（在到达 $\htmlData{tutor-start=0,tutor-end=1}{z}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$ 之前）。 其实有一个更简单的奇偶性论证： 考虑序列在 $\htmlData{tutor-start=0,tutor-end=4}{\le }\htmlData{tutor-start=4,tutor-end=5}{L}$ 和 $\htmlData{tutor-start=0,tutor-end=1}{>}\htmlData{tutor-start=1,tutor-end=2}{L}$ 之间的跳转。每次从 $\htmlData{tutor-start=0,tutor-end=4}{\le }\htmlData{tutor-start=4,tutor-end=5}{L}$ 到 $\htmlData{tutor-start=0,tutor-end=1}{>}\htmlData{tutor-start=1,tutor-end=2}{L}$ 消耗1步，从 $\htmlData{tutor-start=0,tutor-end=1}{>}\htmlData{tutor-start=1,tutor-end=2}{L}$ 到 $\htmlData{tutor-start=0,tutor-end=4}{\le }\htmlData{tutor-start=4,tutor-end=5}{L}$ 消耗1步。 在一个完整周期内，进出 $\htmlData{tutor-start=0,tutor-end=1}{>}\htmlData{tutor-start=1,tutor-end=2}{L}$ 区域的次数必须相等。设次数为 $\htmlData{tutor-start=0,tutor-end=1}{k}$。 总步数 $\htmlData{tutor-start=0,tutor-end=1}{T} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{(}\text{\htmlData{tutor-start=11,tutor-end=12}{在}} \htmlData{tutor-start=14,tutor-end=18}{\le }\htmlData{tutor-start=18,tutor-end=19}{L} \text{\htmlData{tutor-start=26,tutor-end=27}{的}\htmlData{tutor-start=27,tutor-end=28}{步}\htmlData{tutor-start=28,tutor-end=29}{数}}\htmlData{tutor-start=30,tutor-end=31}{)} \htmlData{tutor-start=32,tutor-end=33}{+} \htmlData{tutor-start=34,tutor-end=35}{(}\text{\htmlData{tutor-start=41,tutor-end=42}{在}} \htmlData{tutor-start=44,tutor-end=45}{>}\htmlData{tutor-start=45,tutor-end=46}{L} \text{\htmlData{tutor-start=53,tutor-end=54}{的}\htmlData{tutor-start=54,tutor-end=55}{步}\htmlData{tutor-start=55,tutor-end=56}{数}}\htmlData{tutor-start=57,tutor-end=58}{)}$。 已知每次进入 $\htmlData{tutor-start=0,tutor-end=1}{>}\htmlData{tutor-start=1,tutor-end=2}{L}$ 后必立即离开，所以在 $\htmlData{tutor-start=0,tutor-end=1}{>}\htmlData{tutor-start=1,tutor-end=2}{L}$ 的步数恰好等于跳转次数 $\htmlData{tutor-start=0,tutor-end=1}{k}$。 而在 $\htmlData{tutor-start=0,tutor-end=4}{\le }\htmlData{tutor-start=4,tutor-end=5}{L}$ 的步数呢？ 参考解答中提到 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{1}} \htmlData{tutor-start=8,tutor-end=9}{>} \htmlData{tutor-start=10,tutor-end=11}{x}_{\htmlData{tutor-start=13,tutor-end=14}{m}\htmlData{tutor-start=14,tutor-end=15}{+}\htmlData{tutor-start=15,tutor-end=16}{3}} \htmlData{tutor-start=18,tutor-end=19}{>} \dots$ 这种下降子列，这通常出现在 $\htmlData{tutor-start=0,tutor-end=1}{T}$ 为奇数的构造中。 让我们采用反证法或构造法。 事实上，对于此类 $\htmlData{tutor-start=0,tutor-end=6}{\alpha}$-shift 系统，若 $\htmlData{tutor-start=0,tutor-end=1}{L}$ 足够大，周期通常为奇数。 严格推导： 设周期内 $\htmlData{tutor-start=0,tutor-end=1}{>}\htmlData{tutor-start=1,tutor-end=2}{L}$ 的项集为 $\htmlData{tutor-start=0,tutor-end=1}{H}$，$\htmlData{tutor-start=0,tutor-end=4}{\le }\htmlData{tutor-start=4,tutor-end=5}{L}$ 的项集为 $\htmlData{tutor-start=0,tutor-end=1}{S}$。 映射 $\htmlData{tutor-start=0,tutor-end=4}{\phi}\htmlData{tutor-start=4,tutor-end=5}{:} \htmlData{tutor-start=6,tutor-end=7}{H} \htmlData{tutor-start=8,tutor-end=12}{\to }\htmlData{tutor-start=12,tutor-end=13}{S}$ 为 $\htmlData{tutor-start=0,tutor-end=1}{x} \htmlData{tutor-start=2,tutor-end=10}{\mapsto }\htmlData{tutor-start=10,tutor-end=18}{\lfloor }\htmlData{tutor-start=18,tutor-end=19}{x}\htmlData{tutor-start=19,tutor-end=20}{/}\htmlData{tutor-start=20,tutor-end=27}{\alpha }\htmlData{tutor-start=27,tutor-end=34}{\rfloor}$。这是单射（因 $\htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{/}\htmlData{tutor-start=2,tutor-end=8}{\alpha}$ 严格增且间距 $\htmlData{tutor-start=0,tutor-end=1}{>}\htmlData{tutor-start=1,tutor-end=2}{1}$? 不一定，但在极小元处成立）。 映射 $\htmlData{tutor-start=0,tutor-end=4}{\psi}\htmlData{tutor-start=4,tutor-end=5}{:} \htmlData{tutor-start=6,tutor-end=7}{S} \htmlData{tutor-start=8,tutor-end=12}{\to }\htmlData{tutor-start=12,tutor-end=13}{S} \htmlData{tutor-start=14,tutor-end=19}{\cup }\htmlData{tutor-start=19,tutor-end=20}{H}$ 为 $\htmlData{tutor-start=0,tutor-end=1}{x} \htmlData{tutor-start=2,tutor-end=10}{\mapsto }\htmlData{tutor-start=10,tutor-end=18}{\lfloor }\htmlData{tutor-start=18,tutor-end=25}{\alpha }\htmlData{tutor-start=25,tutor-end=26}{x} \htmlData{tutor-start=27,tutor-end=34}{\rfloor}$。 在周期轨道上，$\htmlData{tutor-start=0,tutor-end=4}{\phi}$ 是 $\htmlData{tutor-start=0,tutor-end=1}{H}$ 到 $\htmlData{tutor-start=0,tutor-end=1}{S}$ 的双射吗？是的，因为每个 $\htmlData{tutor-start=0,tutor-end=1}{s} \htmlData{tutor-start=2,tutor-end=6}{\in }\htmlData{tutor-start=6,tutor-end=7}{S}$ 都有唯一前驱（若在周期内）。若 $\htmlData{tutor-start=0,tutor-end=1}{s}$ 的前驱在 $\htmlData{tutor-start=0,tutor-end=1}{S}$，则 $\htmlData{tutor-start=0,tutor-end=1}{s} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=12}{\lfloor }\htmlData{tutor-start=12,tutor-end=19}{\alpha }\htmlData{tutor-start=19,tutor-end=20}{s}' \htmlData{tutor-start=22,tutor-end=29}{\rfloor}$；若前驱在 $\htmlData{tutor-start=0,tutor-end=1}{H}$，则 $\htmlData{tutor-start=0,tutor-end=1}{s} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=12}{\lfloor }\htmlData{tutor-start=12,tutor-end=13}{h}\htmlData{tutor-start=13,tutor-end=14}{/}\htmlData{tutor-start=14,tutor-end=21}{\alpha }\htmlData{tutor-start=21,tutor-end=28}{\rfloor}$。 关键引理：在最小周期轨道上，$\htmlData{tutor-start=0,tutor-end=1}{S}$ 中恰有一个元素的前驱在 $\htmlData{tutor-start=0,tutor-end=1}{H}$ 中，那就是 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$。 证明：若有两个元素 $\htmlData{tutor-start=0,tutor-end=1}{s}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{s}_{\htmlData{tutor-start=10,tutor-end=11}{2}}$ 前驱在 $\htmlData{tutor-start=0,tutor-end=1}{H}$，设为 $\htmlData{tutor-start=0,tutor-end=1}{h}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{h}_{\htmlData{tutor-start=10,tutor-end=11}{2}}$。则 $\htmlData{tutor-start=0,tutor-end=1}{s}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=16}{\lfloor }\htmlData{tutor-start=16,tutor-end=17}{h}_{\htmlData{tutor-start=19,tutor-end=20}{1}}\htmlData{tutor-start=21,tutor-end=22}{/}\htmlData{tutor-start=22,tutor-end=29}{\alpha }\htmlData{tutor-start=29,tutor-end=36}{\rfloor}\htmlData{tutor-start=36,tutor-end=37}{,} \htmlData{tutor-start=38,tutor-end=39}{s}_{\htmlData{tutor-start=41,tutor-end=42}{2}} \htmlData{tutor-start=44,tutor-end=45}{=} \htmlData{tutor-start=46,tutor-end=54}{\lfloor }\htmlData{tutor-start=54,tutor-end=55}{h}_{\htmlData{tutor-start=57,tutor-end=58}{2}}\htmlData{tutor-start=59,tutor-end=60}{/}\htmlData{tutor-start=60,tutor-end=67}{\alpha }\htmlData{tutor-start=67,tutor-end=74}{\rfloor}$。由于 $\htmlData{tutor-start=0,tutor-end=1}{h}_{\htmlData{tutor-start=3,tutor-end=4}{i}}$ 来自 $\htmlData{tutor-start=0,tutor-end=1}{S}$ 中某元素 $\htmlData{tutor-start=0,tutor-end=1}{w}_{\htmlData{tutor-start=3,tutor-end=4}{i}}$ 的像 $\htmlData{tutor-start=0,tutor-end=8}{\lfloor }\htmlData{tutor-start=8,tutor-end=15}{\alpha }\htmlData{tutor-start=15,tutor-end=16}{w}_{\htmlData{tutor-start=18,tutor-end=19}{i}} \htmlData{tutor-start=21,tutor-end=28}{\rfloor}$，这会导致轨道分裂或重合。在最小周期假设下，只能有一个“入口”进入 $\htmlData{tutor-start=0,tutor-end=1}{S}$ 来自 $\htmlData{tutor-start=0,tutor-end=1}{H}$。 既然只有一个入口，那么 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{H}\htmlData{tutor-start=2,tutor-end=3}{|} \htmlData{tutor-start=4,tutor-end=5}{=} \htmlData{tutor-start=6,tutor-end=7}{1}$？不对。$\htmlData{tutor-start=0,tutor-end=1}{H}$ 中每个元素都必须有前驱。若 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{H}\htmlData{tutor-start=2,tutor-end=3}{|} \htmlData{tutor-start=4,tutor-end=5}{>} \htmlData{tutor-start=6,tutor-end=7}{1}$，则必须有多个入口？ 不。$\htmlData{tutor-start=0,tutor-end=1}{H}$ 中元素 $\htmlData{tutor-start=0,tutor-end=1}{h}$ 的前驱必须在 $\htmlData{tutor-start=0,tutor-end=1}{S}$ 中（因为 $\htmlData{tutor-start=0,tutor-end=1}{>}\htmlData{tutor-start=1,tutor-end=2}{L}$ 不能来自 $\htmlData{tutor-start=0,tutor-end=1}{>}\htmlData{tutor-start=1,tutor-end=2}{L}$）。所以 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{H}\htmlData{tutor-start=2,tutor-end=3}{|}$ 个元素都需要 $\htmlData{tutor-start=0,tutor-end=1}{S}$ 提供前驱。 而 $\htmlData{tutor-start=0,tutor-end=1}{S}$ 提供前驱的方式有两种：1. $\htmlData{tutor-start=0,tutor-end=1}{S}$ 内部流转；2. $\htmlData{tutor-start=0,tutor-end=1}{S}$ 跳转到 $\htmlData{tutor-start=0,tutor-end=1}{H}$。 $\htmlData{tutor-start=0,tutor-end=1}{S}$ 跳转到 $\htmlData{tutor-start=0,tutor-end=1}{H}$ 的次数必须等于 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{H}\htmlData{tutor-start=2,tutor-end=3}{|}$（因为每个 $\htmlData{tutor-start=0,tutor-end=1}{h}$ 都要被生成一次）。 $\htmlData{tutor-start=0,tutor-end=1}{H}$ 跳转回 $\htmlData{tutor-start=0,tutor-end=1}{S}$ 的次数也必须等于 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{H}\htmlData{tutor-start=2,tutor-end=3}{|}$（因为每个 $\htmlData{tutor-start=0,tutor-end=1}{h}$ 都要离开一次）。 现在看 $\htmlData{tutor-start=0,tutor-end=1}{S}$ 内部。$\htmlData{tutor-start=0,tutor-end=1}{S}$ 的元素排成若干条链，每条链的终点跳转到 $\htmlData{tutor-start=0,tutor-end=1}{H}$，起点来自于 $\htmlData{tutor-start=0,tutor-end=1}{H}$。 因为是从 $\htmlData{tutor-start=0,tutor-end=1}{H}$ 回到 $\htmlData{tutor-start=0,tutor-end=1}{S}$ 再跳回 $\htmlData{tutor-start=0,tutor-end=1}{H}$，所以链的条数等于 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{H}\htmlData{tutor-start=2,tutor-end=3}{|}$。 设这 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{H}\htmlData{tutor-start=2,tutor-end=3}{|}$ 条链的长度分别为 $\htmlData{tutor-start=0,tutor-end=1}{l}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{l}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \dots\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=22}{l}_{\htmlData{tutor-start=24,tutor-end=25}{|}\htmlData{tutor-start=25,tutor-end=26}{H}\htmlData{tutor-start=26,tutor-end=27}{|}}$（长度指 $\htmlData{tutor-start=0,tutor-end=1}{S}$ 中元素个数）。 则 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{S}\htmlData{tutor-start=2,tutor-end=3}{|} \htmlData{tutor-start=4,tutor-end=5}{=} \sum \htmlData{tutor-start=11,tutor-end=12}{l}_{\htmlData{tutor-start=14,tutor-end=15}{j}}$。 总周期 $\htmlData{tutor-start=0,tutor-end=1}{T} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{|}\htmlData{tutor-start=5,tutor-end=6}{S}\htmlData{tutor-start=6,tutor-end=7}{|} \htmlData{tutor-start=8,tutor-end=9}{+} \htmlData{tutor-start=10,tutor-end=11}{|}\htmlData{tutor-start=11,tutor-end=12}{H}\htmlData{tutor-start=12,tutor-end=13}{|} \htmlData{tutor-start=14,tutor-end=15}{=} \sum \htmlData{tutor-start=21,tutor-end=22}{l}_{\htmlData{tutor-start=24,tutor-end=25}{j}} \htmlData{tutor-start=27,tutor-end=28}{+} \htmlData{tutor-start=29,tutor-end=30}{|}\htmlData{tutor-start=30,tutor-end=31}{H}\htmlData{tutor-start=31,tutor-end=32}{|} \htmlData{tutor-start=33,tutor-end=34}{=} \sum \htmlData{tutor-start=40,tutor-end=41}{(}\htmlData{tutor-start=41,tutor-end=42}{l}_{\htmlData{tutor-start=44,tutor-end=45}{j}} \htmlData{tutor-start=47,tutor-end=48}{+} \htmlData{tutor-start=49,tutor-end=50}{1}\htmlData{tutor-start=50,tutor-end=51}{)}$。 我们要证 $\htmlData{tutor-start=0,tutor-end=1}{T}$ 是奇数。 考察每条链 $\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{0}} \htmlData{tutor-start=6,tutor-end=10}{\to }\htmlData{tutor-start=10,tutor-end=11}{v}_{\htmlData{tutor-start=13,tutor-end=14}{1}} \htmlData{tutor-start=16,tutor-end=20}{\to }\dots \htmlData{tutor-start=26,tutor-end=30}{\to }\htmlData{tutor-start=30,tutor-end=31}{v}_{\htmlData{tutor-start=33,tutor-end=34}{l}\htmlData{tutor-start=34,tutor-end=35}{-}\htmlData{tutor-start=35,tutor-end=36}{1}} \htmlData{tutor-start=38,tutor-end=42}{\to }\htmlData{tutor-start=42,tutor-end=43}{h}$。 其中 $\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{0}}$ 来自 $\htmlData{tutor-start=0,tutor-end=1}{H}$，即 $\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{0}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=16}{\lfloor }\htmlData{tutor-start=16,tutor-end=17}{h}_{\htmlData{tutor-start=19,tutor-end=20}{p}\htmlData{tutor-start=20,tutor-end=21}{r}\htmlData{tutor-start=21,tutor-end=22}{e}\htmlData{tutor-start=22,tutor-end=23}{v}}\htmlData{tutor-start=24,tutor-end=25}{/}\htmlData{tutor-start=25,tutor-end=32}{\alpha }\htmlData{tutor-start=32,tutor-end=39}{\rfloor}$。 $\htmlData{tutor-start=0,tutor-end=1}{h} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=12}{\lfloor }\htmlData{tutor-start=12,tutor-end=19}{\alpha }\htmlData{tutor-start=19,tutor-end=20}{v}_{\htmlData{tutor-start=22,tutor-end=23}{l}\htmlData{tutor-start=23,tutor-end=24}{-}\htmlData{tutor-start=24,tutor-end=25}{1}} \htmlData{tutor-start=27,tutor-end=34}{\rfloor}$。 对于最小周期轨道，可以证明所有链长 $\htmlData{tutor-start=0,tutor-end=1}{l}_{\htmlData{tutor-start=3,tutor-end=4}{j}}$ 都是偶数？或者 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{H}\htmlData{tutor-start=2,tutor-end=3}{|}$ 是奇数且 $\htmlData{tutor-start=0,tutor-end=1}{l}_{\htmlData{tutor-start=3,tutor-end=4}{j}}$ 有特定性质？ 实际上，对于本题特定的 $\htmlData{tutor-start=0,tutor-end=1}{L}$ 范围，可以证明 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{H}\htmlData{tutor-start=2,tutor-end=3}{|}\htmlData{tutor-start=3,tutor-end=4}{=}\htmlData{tutor-start=4,tutor-end=5}{1}$ 且 $\htmlData{tutor-start=0,tutor-end=1}{l}_{\htmlData{tutor-start=3,tutor-end=4}{1}}$ 是偶数。 若 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{H}\htmlData{tutor-start=2,tutor-end=3}{|}\htmlData{tutor-start=3,tutor-end=4}{=}\htmlData{tutor-start=4,tutor-end=5}{1}$，设 $\htmlData{tutor-start=0,tutor-end=1}{H}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{z}\htmlData{tutor-start=5,tutor-end=7}{\}}$。则 $\htmlData{tutor-start=0,tutor-end=1}{S}$ 是一条链 $\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{0}} \htmlData{tutor-start=6,tutor-end=10}{\to }\dots \htmlData{tutor-start=16,tutor-end=20}{\to }\htmlData{tutor-start=20,tutor-end=21}{v}_{\htmlData{tutor-start=23,tutor-end=24}{l}\htmlData{tutor-start=24,tutor-end=25}{-}\htmlData{tutor-start=25,tutor-end=26}{1}} \htmlData{tutor-start=28,tutor-end=32}{\to }\htmlData{tutor-start=32,tutor-end=33}{z} \htmlData{tutor-start=34,tutor-end=38}{\to }\htmlData{tutor-start=38,tutor-end=39}{v}_{\htmlData{tutor-start=41,tutor-end=42}{0}}$。 $\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{0}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=16}{\lfloor }\htmlData{tutor-start=16,tutor-end=17}{z}\htmlData{tutor-start=17,tutor-end=18}{/}\htmlData{tutor-start=18,tutor-end=25}{\alpha }\htmlData{tutor-start=25,tutor-end=33}{\rfloor }\htmlData{tutor-start=33,tutor-end=34}{=} \htmlData{tutor-start=35,tutor-end=36}{x}_{\htmlData{tutor-start=38,tutor-end=39}{m}\htmlData{tutor-start=39,tutor-end=40}{i}\htmlData{tutor-start=40,tutor-end=41}{n}}$。 $\htmlData{tutor-start=0,tutor-end=1}{z} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=12}{\lfloor }\htmlData{tutor-start=12,tutor-end=19}{\alpha }\htmlData{tutor-start=19,tutor-end=20}{v}_{\htmlData{tutor-start=22,tutor-end=23}{l}\htmlData{tutor-start=23,tutor-end=24}{-}\htmlData{tutor-start=24,tutor-end=25}{1}} \htmlData{tutor-start=27,tutor-end=34}{\rfloor}$。 且 $\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{i}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{1}} \htmlData{tutor-start=8,tutor-end=9}{=} \htmlData{tutor-start=10,tutor-end=18}{\lfloor }\htmlData{tutor-start=18,tutor-end=25}{\alpha }\htmlData{tutor-start=25,tutor-end=26}{v}_{\htmlData{tutor-start=28,tutor-end=29}{i}} \htmlData{tutor-start=31,tutor-end=38}{\rfloor}$。 我们需要 $\htmlData{tutor-start=0,tutor-end=1}{l}$ 是偶数。 注意到 $\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{0}}$ 是 $\htmlData{tutor-start=0,tutor-end=1}{S}$ 中最小，$\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{l}\htmlData{tutor-start=4,tutor-end=5}{-}\htmlData{tutor-start=5,tutor-end=6}{1}}$ 是 $\htmlData{tutor-start=0,tutor-end=1}{S}$ 中最大（在链意义上，因 $\htmlData{tutor-start=0,tutor-end=8}{\lfloor }\htmlData{tutor-start=8,tutor-end=15}{\alpha }\htmlData{tutor-start=15,tutor-end=16}{x} \htmlData{tutor-start=17,tutor-end=24}{\rfloor}$ 增）。 $\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{0}} \htmlData{tutor-start=6,tutor-end=14}{\approx }\htmlData{tutor-start=14,tutor-end=15}{L}\htmlData{tutor-start=15,tutor-end=16}{/}\htmlData{tutor-start=16,tutor-end=22}{\alpha}$。$\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{l}\htmlData{tutor-start=4,tutor-end=5}{-}\htmlData{tutor-start=5,tutor-end=6}{1}} \htmlData{tutor-start=8,tutor-end=16}{\approx }\htmlData{tutor-start=16,tutor-end=17}{L}$。 增长倍数约为 $\htmlData{tutor-start=0,tutor-end=6}{\alpha}^{\htmlData{tutor-start=8,tutor-end=9}{l}} \htmlData{tutor-start=11,tutor-end=19}{\approx }\htmlData{tutor-start=19,tutor-end=20}{L} \htmlData{tutor-start=21,tutor-end=22}{/} \htmlData{tutor-start=23,tutor-end=24}{(}\htmlData{tutor-start=24,tutor-end=25}{L}\htmlData{tutor-start=25,tutor-end=26}{/}\htmlData{tutor-start=26,tutor-end=32}{\alpha}\htmlData{tutor-start=32,tutor-end=33}{)} \htmlData{tutor-start=34,tutor-end=35}{=} \htmlData{tutor-start=36,tutor-end=42}{\alpha}$。故 $\htmlData{tutor-start=0,tutor-end=1}{l} \htmlData{tutor-start=2,tutor-end=10}{\approx }\htmlData{tutor-start=10,tutor-end=11}{1}$？ 不对。$\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{0}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=16}{\lfloor }\htmlData{tutor-start=16,tutor-end=17}{(}\htmlData{tutor-start=17,tutor-end=18}{L}\htmlData{tutor-start=18,tutor-end=19}{+}\htmlData{tutor-start=19,tutor-end=20}{1}\htmlData{tutor-start=20,tutor-end=21}{)}\htmlData{tutor-start=21,tutor-end=22}{/}\htmlData{tutor-start=22,tutor-end=29}{\alpha }\htmlData{tutor-start=29,tutor-end=36}{\rfloor}$。$\htmlData{tutor-start=0,tutor-end=7}{\alpha }\htmlData{tutor-start=7,tutor-end=8}{v}_{\htmlData{tutor-start=10,tutor-end=11}{0}} \htmlData{tutor-start=13,tutor-end=21}{\approx }\htmlData{tutor-start=21,tutor-end=22}{L}\htmlData{tutor-start=22,tutor-end=23}{+}\htmlData{tutor-start=23,tutor-end=24}{1}$。所以 $\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=16}{\lfloor }\htmlData{tutor-start=16,tutor-end=23}{\alpha }\htmlData{tutor-start=23,tutor-end=24}{v}_{\htmlData{tutor-start=26,tutor-end=27}{0}} \htmlData{tutor-start=29,tutor-end=36}{\rfloor}$ 很可能就是 $\htmlData{tutor-start=0,tutor-end=1}{L}$ 或者接近 $\htmlData{tutor-start=0,tutor-end=1}{L}$ 的数。 如果 $\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=7}{>} \htmlData{tutor-start=8,tutor-end=9}{L}$，则 $\htmlData{tutor-start=0,tutor-end=1}{l}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{1}$（链长为1：$\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{0}} \htmlData{tutor-start=6,tutor-end=10}{\to }\htmlData{tutor-start=10,tutor-end=11}{z}$）。此时 $\htmlData{tutor-start=0,tutor-end=1}{T} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{1}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{1}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{2}$，偶数！ 但这与结论矛盾。说明 $\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{1}}$ 必须 $\htmlData{tutor-start=0,tutor-end=4}{\le }\htmlData{tutor-start=4,tutor-end=5}{L}$。 即 $\htmlData{tutor-start=0,tutor-end=8}{\lfloor }\htmlData{tutor-start=8,tutor-end=15}{\alpha }\htmlData{tutor-start=15,tutor-end=23}{\lfloor }\htmlData{tutor-start=23,tutor-end=24}{(}\htmlData{tutor-start=24,tutor-end=25}{L}\htmlData{tutor-start=25,tutor-end=26}{+}\htmlData{tutor-start=26,tutor-end=27}{1}\htmlData{tutor-start=27,tutor-end=28}{)}\htmlData{tutor-start=28,tutor-end=29}{/}\htmlData{tutor-start=29,tutor-end=36}{\alpha }\htmlData{tutor-start=36,tutor-end=44}{\rfloor }\htmlData{tutor-start=44,tutor-end=52}{\rfloor }\htmlData{tutor-start=52,tutor-end=56}{\le }\htmlData{tutor-start=56,tutor-end=57}{L}$。 这等价于 $\htmlData{tutor-start=0,tutor-end=7}{\alpha }\htmlData{tutor-start=7,tutor-end=15}{\lfloor }\htmlData{tutor-start=15,tutor-end=16}{(}\htmlData{tutor-start=16,tutor-end=17}{L}\htmlData{tutor-start=17,tutor-end=18}{+}\htmlData{tutor-start=18,tutor-end=19}{1}\htmlData{tutor-start=19,tutor-end=20}{)}\htmlData{tutor-start=20,tutor-end=21}{/}\htmlData{tutor-start=21,tutor-end=28}{\alpha }\htmlData{tutor-start=28,tutor-end=36}{\rfloor }\htmlData{tutor-start=36,tutor-end=37}{<} \htmlData{tutor-start=38,tutor-end=39}{L}\htmlData{tutor-start=39,tutor-end=40}{+}\htmlData{tutor-start=40,tutor-end=41}{1}$。这恒成立。 关键是 $\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{1}}$ 是否可能 $\htmlData{tutor-start=0,tutor-end=1}{>} \htmlData{tutor-start=2,tutor-end=3}{L}$？ $\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=16}{\lfloor }\htmlData{tutor-start=16,tutor-end=23}{\alpha }\htmlData{tutor-start=23,tutor-end=24}{v}_{\htmlData{tutor-start=26,tutor-end=27}{0}} \htmlData{tutor-start=29,tutor-end=36}{\rfloor}$。若 $\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=7}{>} \htmlData{tutor-start=8,tutor-end=9}{L}$，则 $\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=10}{\ge }\htmlData{tutor-start=10,tutor-end=11}{L}\htmlData{tutor-start=11,tutor-end=12}{+}\htmlData{tutor-start=12,tutor-end=13}{1}$。 这意味着 $\htmlData{tutor-start=0,tutor-end=7}{\alpha }\htmlData{tutor-start=7,tutor-end=8}{v}_{\htmlData{tutor-start=10,tutor-end=11}{0}} \htmlData{tutor-start=13,tutor-end=17}{\ge }\htmlData{tutor-start=17,tutor-end=18}{L}\htmlData{tutor-start=18,tutor-end=19}{+}\htmlData{tutor-start=19,tutor-end=20}{1}$。 但 $\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{0}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=16}{\lfloor }\htmlData{tutor-start=16,tutor-end=17}{(}\htmlData{tutor-start=17,tutor-end=18}{L}\htmlData{tutor-start=18,tutor-end=19}{+}\htmlData{tutor-start=19,tutor-end=20}{1}\htmlData{tutor-start=20,tutor-end=21}{)}\htmlData{tutor-start=21,tutor-end=22}{/}\htmlData{tutor-start=22,tutor-end=29}{\alpha }\htmlData{tutor-start=29,tutor-end=37}{\rfloor }\htmlData{tutor-start=37,tutor-end=41}{\le }\htmlData{tutor-start=41,tutor-end=42}{(}\htmlData{tutor-start=42,tutor-end=43}{L}\htmlData{tutor-start=43,tutor-end=44}{+}\htmlData{tutor-start=44,tutor-end=45}{1}\htmlData{tutor-start=45,tutor-end=46}{)}\htmlData{tutor-start=46,tutor-end=47}{/}\htmlData{tutor-start=47,tutor-end=53}{\alpha}$。 所以 $\htmlData{tutor-start=0,tutor-end=7}{\alpha }\htmlData{tutor-start=7,tutor-end=8}{v}_{\htmlData{tutor-start=10,tutor-end=11}{0}} \htmlData{tutor-start=13,tutor-end=17}{\le }\htmlData{tutor-start=17,tutor-end=18}{L}\htmlData{tutor-start=18,tutor-end=19}{+}\htmlData{tutor-start=19,tutor-end=20}{1}$。 由于 $\htmlData{tutor-start=0,tutor-end=6}{\alpha}$ 无理，$\htmlData{tutor-start=0,tutor-end=7}{\alpha }\htmlData{tutor-start=7,tutor-end=8}{v}_{\htmlData{tutor-start=10,tutor-end=11}{0}} \neq \htmlData{tutor-start=18,tutor-end=19}{L}\htmlData{tutor-start=19,tutor-end=20}{+}\htmlData{tutor-start=20,tutor-end=21}{1}$。故 $\htmlData{tutor-start=0,tutor-end=7}{\alpha }\htmlData{tutor-start=7,tutor-end=8}{v}_{\htmlData{tutor-start=10,tutor-end=11}{0}} \htmlData{tutor-start=13,tutor-end=14}{<} \htmlData{tutor-start=15,tutor-end=16}{L}\htmlData{tutor-start=16,tutor-end=17}{+}\htmlData{tutor-start=17,tutor-end=18}{1}$。 所以 $\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=16}{\lfloor }\htmlData{tutor-start=16,tutor-end=23}{\alpha }\htmlData{tutor-start=23,tutor-end=24}{v}_{\htmlData{tutor-start=26,tutor-end=27}{0}} \htmlData{tutor-start=29,tutor-end=37}{\rfloor }\htmlData{tutor-start=37,tutor-end=41}{\le }\htmlData{tutor-start=41,tutor-end=42}{L}$。 这说明链长 $\htmlData{tutor-start=0,tutor-end=1}{l}$ 至少为 2（$\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{v}_{\htmlData{tutor-start=10,tutor-end=11}{1}}$ 都在 $\htmlData{tutor-start=0,tutor-end=1}{S}$ 中）。 接下来 $\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{2}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=16}{\lfloor }\htmlData{tutor-start=16,tutor-end=23}{\alpha }\htmlData{tutor-start=23,tutor-end=24}{v}_{\htmlData{tutor-start=26,tutor-end=27}{1}} \htmlData{tutor-start=29,tutor-end=36}{\rfloor}$。$\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{1}}$ 接近 $\htmlData{tutor-start=0,tutor-end=1}{L}$。$\htmlData{tutor-start=0,tutor-end=7}{\alpha }\htmlData{tutor-start=7,tutor-end=8}{v}_{\htmlData{tutor-start=10,tutor-end=11}{1}}$ 远大于 $\htmlData{tutor-start=0,tutor-end=1}{L}$。 所以 $\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{2}}$ 几乎肯定 $\htmlData{tutor-start=0,tutor-end=1}{>} \htmlData{tutor-start=2,tutor-end=3}{L}$。 若 $\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{2}} \htmlData{tutor-start=6,tutor-end=7}{>} \htmlData{tutor-start=8,tutor-end=9}{L}$，则链为 $\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{0}} \htmlData{tutor-start=6,tutor-end=10}{\to }\htmlData{tutor-start=10,tutor-end=11}{v}_{\htmlData{tutor-start=13,tutor-end=14}{1}} \htmlData{tutor-start=16,tutor-end=20}{\to }\htmlData{tutor-start=20,tutor-end=21}{z}$。长度 $\htmlData{tutor-start=0,tutor-end=1}{l}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{2}$。 此时 $\htmlData{tutor-start=0,tutor-end=1}{T} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{l} \htmlData{tutor-start=6,tutor-end=7}{+} \htmlData{tutor-start=8,tutor-end=9}{1} \htmlData{tutor-start=10,tutor-end=11}{=} \htmlData{tutor-start=12,tutor-end=13}{3}$，奇数！ 能否 $\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{2}} \htmlData{tutor-start=6,tutor-end=10}{\le }\htmlData{tutor-start=10,tutor-end=11}{L}$？ $\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{2}} \htmlData{tutor-start=6,tutor-end=10}{\le }\htmlData{tutor-start=10,tutor-end=11}{L} \iff \htmlData{tutor-start=17,tutor-end=25}{\lfloor }\htmlData{tutor-start=25,tutor-end=32}{\alpha }\htmlData{tutor-start=32,tutor-end=33}{v}_{\htmlData{tutor-start=35,tutor-end=36}{1}} \htmlData{tutor-start=38,tutor-end=46}{\rfloor }\htmlData{tutor-start=46,tutor-end=50}{\le }\htmlData{tutor-start=50,tutor-end=51}{L} \iff \htmlData{tutor-start=57,tutor-end=64}{\alpha }\htmlData{tutor-start=64,tutor-end=65}{v}_{\htmlData{tutor-start=67,tutor-end=68}{1}} \htmlData{tutor-start=70,tutor-end=71}{<} \htmlData{tutor-start=72,tutor-end=73}{L}\htmlData{tutor-start=73,tutor-end=74}{+}\htmlData{tutor-start=74,tutor-end=75}{1}$。 但 $\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=16}{\lfloor }\htmlData{tutor-start=16,tutor-end=23}{\alpha }\htmlData{tutor-start=23,tutor-end=24}{v}_{\htmlData{tutor-start=26,tutor-end=27}{0}} \htmlData{tutor-start=29,tutor-end=37}{\rfloor }\htmlData{tutor-start=37,tutor-end=38}{>} \htmlData{tutor-start=39,tutor-end=46}{\alpha }\htmlData{tutor-start=46,tutor-end=47}{v}_{\htmlData{tutor-start=49,tutor-end=50}{0}} \htmlData{tutor-start=52,tutor-end=53}{-} \htmlData{tutor-start=54,tutor-end=55}{1}$。 $\htmlData{tutor-start=0,tutor-end=7}{\alpha }\htmlData{tutor-start=7,tutor-end=8}{v}_{\htmlData{tutor-start=10,tutor-end=11}{1}} \htmlData{tutor-start=13,tutor-end=14}{>} \htmlData{tutor-start=15,tutor-end=21}{\alpha}\htmlData{tutor-start=21,tutor-end=22}{(}\htmlData{tutor-start=22,tutor-end=29}{\alpha }\htmlData{tutor-start=29,tutor-end=30}{v}_{\htmlData{tutor-start=32,tutor-end=33}{0}} \htmlData{tutor-start=35,tutor-end=36}{-} \htmlData{tutor-start=37,tutor-end=38}{1}\htmlData{tutor-start=38,tutor-end=39}{)} \htmlData{tutor-start=40,tutor-end=41}{=} \htmlData{tutor-start=42,tutor-end=48}{\alpha}^{\htmlData{tutor-start=50,tutor-end=51}{2}} \htmlData{tutor-start=53,tutor-end=54}{v}_{\htmlData{tutor-start=56,tutor-end=57}{0}} \htmlData{tutor-start=59,tutor-end=60}{-} \htmlData{tutor-start=61,tutor-end=67}{\alpha}$。 代入 $\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{0}} \htmlData{tutor-start=6,tutor-end=14}{\approx }\htmlData{tutor-start=14,tutor-end=15}{L}\htmlData{tutor-start=15,tutor-end=16}{/}\htmlData{tutor-start=16,tutor-end=22}{\alpha}$，得 $\htmlData{tutor-start=0,tutor-end=6}{\alpha}^{\htmlData{tutor-start=8,tutor-end=9}{2}} \htmlData{tutor-start=11,tutor-end=12}{(}\htmlData{tutor-start=12,tutor-end=13}{L}\htmlData{tutor-start=13,tutor-end=14}{/}\htmlData{tutor-start=14,tutor-end=20}{\alpha}\htmlData{tutor-start=20,tutor-end=21}{)} \htmlData{tutor-start=22,tutor-end=23}{-} \htmlData{tutor-start=24,tutor-end=31}{\alpha }\htmlData{tutor-start=31,tutor-end=32}{=} \htmlData{tutor-start=33,tutor-end=40}{\alpha }\htmlData{tutor-start=40,tutor-end=41}{L} \htmlData{tutor-start=42,tutor-end=43}{-} \htmlData{tutor-start=44,tutor-end=50}{\alpha}$。 我们需要 $\alpha L - \alpha < L+1 \iff L(\alpha-1) < \alpha+1 \iff L < \frac{\alpha+1}{\alpha-1}$。 但题设 $\htmlData{tutor-start=0,tutor-end=1}{L} \htmlData{tutor-start=2,tutor-end=3}{>} \frac{\htmlData{tutor-start=10,tutor-end=16}{\alpha}^{\htmlData{tutor-start=18,tutor-end=19}{2}}}{\htmlData{tutor-start=22,tutor-end=28}{\alpha}\htmlData{tutor-start=28,tutor-end=29}{-}\htmlData{tutor-start=29,tutor-end=30}{1}}$。显然 $\frac{\htmlData{tutor-start=6,tutor-end=12}{\alpha}^{\htmlData{tutor-start=14,tutor-end=15}{2}}}{\htmlData{tutor-start=18,tutor-end=24}{\alpha}\htmlData{tutor-start=24,tutor-end=25}{-}\htmlData{tutor-start=25,tutor-end=26}{1}} \htmlData{tutor-start=28,tutor-end=29}{>} \frac{\htmlData{tutor-start=36,tutor-end=42}{\alpha}\htmlData{tutor-start=42,tutor-end=43}{+}\htmlData{tutor-start=43,tutor-end=44}{1}}{\htmlData{tutor-start=46,tutor-end=52}{\alpha}\htmlData{tutor-start=52,tutor-end=53}{-}\htmlData{tutor-start=53,tutor-end=54}{1}}$ （因 $\htmlData{tutor-start=0,tutor-end=6}{\alpha}^{\htmlData{tutor-start=8,tutor-end=9}{2}} \htmlData{tutor-start=11,tutor-end=12}{>} \htmlData{tutor-start=13,tutor-end=19}{\alpha}\htmlData{tutor-start=19,tutor-end=20}{+}\htmlData{tutor-start=20,tutor-end=21}{1}$ 对 $\htmlData{tutor-start=0,tutor-end=6}{\alpha}\htmlData{tutor-start=6,tutor-end=7}{>}\htmlData{tutor-start=7,tutor-end=8}{1}\htmlData{tutor-start=8,tutor-end=9}{.}\htmlData{tutor-start=9,tutor-end=10}{6}\htmlData{tutor-start=10,tutor-end=11}{1}\htmlData{tutor-start=11,tutor-end=12}{8}$ 成立，若 $\htmlData{tutor-start=0,tutor-end=6}{\alpha}$ 接近1呢？$\htmlData{tutor-start=0,tutor-end=6}{\alpha}^{\htmlData{tutor-start=8,tutor-end=9}{2}} \htmlData{tutor-start=11,tutor-end=12}{-} \htmlData{tutor-start=13,tutor-end=20}{\alpha }\htmlData{tutor-start=20,tutor-end=21}{-} \htmlData{tutor-start=22,tutor-end=23}{1}$ 的根是黄金比例。若 $\htmlData{tutor-start=0,tutor-end=1}{1} \htmlData{tutor-start=2,tutor-end=3}{<} \htmlData{tutor-start=4,tutor-end=11}{\alpha }\htmlData{tutor-start=11,tutor-end=12}{<} \htmlData{tutor-start=13,tutor-end=14}{1}\htmlData{tutor-start=14,tutor-end=15}{.}\htmlData{tutor-start=15,tutor-end=16}{6}\htmlData{tutor-start=16,tutor-end=17}{1}\htmlData{tutor-start=17,tutor-end=18}{8}$，则 $\htmlData{tutor-start=0,tutor-end=6}{\alpha}^{\htmlData{tutor-start=8,tutor-end=9}{2}} \htmlData{tutor-start=11,tutor-end=12}{<} \htmlData{tutor-start=13,tutor-end=19}{\alpha}\htmlData{tutor-start=19,tutor-end=20}{+}\htmlData{tutor-start=20,tutor-end=21}{1}$。此时 $\htmlData{tutor-start=0,tutor-end=1}{L}$ 的下界 $\frac{\htmlData{tutor-start=6,tutor-end=12}{\alpha}^{\htmlData{tutor-start=14,tutor-end=15}{2}}}{\htmlData{tutor-start=18,tutor-end=24}{\alpha}\htmlData{tutor-start=24,tutor-end=25}{-}\htmlData{tutor-start=25,tutor-end=26}{1}}$ 可能小于 $\frac{\htmlData{tutor-start=6,tutor-end=12}{\alpha}\htmlData{tutor-start=12,tutor-end=13}{+}\htmlData{tutor-start=13,tutor-end=14}{1}}{\htmlData{tutor-start=16,tutor-end=22}{\alpha}\htmlData{tutor-start=22,tutor-end=23}{-}\htmlData{tutor-start=23,tutor-end=24}{1}}$ 吗？ 比较 $\htmlData{tutor-start=0,tutor-end=6}{\alpha}^{\htmlData{tutor-start=8,tutor-end=9}{2}}$ 与 $\htmlData{tutor-start=0,tutor-end=6}{\alpha}\htmlData{tutor-start=6,tutor-end=7}{+}\htmlData{tutor-start=7,tutor-end=8}{1}$。当 $\htmlData{tutor-start=0,tutor-end=7}{\alpha }\htmlData{tutor-start=7,tutor-end=11}{\in }\htmlData{tutor-start=11,tutor-end=12}{(}\htmlData{tutor-start=12,tutor-end=13}{1}\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=19}{\phi}\htmlData{tutor-start=19,tutor-end=20}{)}$ 时，$\htmlData{tutor-start=0,tutor-end=6}{\alpha}^{\htmlData{tutor-start=8,tutor-end=9}{2}} \htmlData{tutor-start=11,tutor-end=12}{<} \htmlData{tutor-start=13,tutor-end=19}{\alpha}\htmlData{tutor-start=19,tutor-end=20}{+}\htmlData{tutor-start=20,tutor-end=21}{1}$。 此时题设 $\htmlData{tutor-start=0,tutor-end=1}{L} \htmlData{tutor-start=2,tutor-end=3}{>} \frac{\htmlData{tutor-start=10,tutor-end=16}{\alpha}^{\htmlData{tutor-start=18,tutor-end=19}{2}}}{\htmlData{tutor-start=22,tutor-end=28}{\alpha}\htmlData{tutor-start=28,tutor-end=29}{-}\htmlData{tutor-start=29,tutor-end=30}{1}}$ 并不排斥 $\htmlData{tutor-start=0,tutor-end=1}{L} \htmlData{tutor-start=2,tutor-end=3}{<} \frac{\htmlData{tutor-start=10,tutor-end=16}{\alpha}\htmlData{tutor-start=16,tutor-end=17}{+}\htmlData{tutor-start=17,tutor-end=18}{1}}{\htmlData{tutor-start=20,tutor-end=26}{\alpha}\htmlData{tutor-start=26,tutor-end=27}{-}\htmlData{tutor-start=27,tutor-end=28}{1}}$。 如果 $\htmlData{tutor-start=0,tutor-end=1}{L}$ 落在这个区间，可能出现 $\htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{2}} \htmlData{tutor-start=6,tutor-end=10}{\le }\htmlData{tutor-start=10,tutor-end=11}{L}$，导致链更长。 但题目要求证明 $\htmlData{tutor-start=0,tutor-end=1}{T}$ **总是** 奇数。这意味着无论链多长，$\htmlData{tutor-start=0,tutor-end=1}{l}$ 总是偶数。 这提示我们 $\htmlData{tutor-start=0,tutor-end=1}{l}$ 的奇偶性是由 $\htmlData{tutor-start=0,tutor-end=6}{\alpha}$ 的无理性及取整函数的性质决定的拓扑不变量。 参考解答中的 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{1}} \htmlData{tutor-start=8,tutor-end=9}{>} \htmlData{tutor-start=10,tutor-end=11}{x}_{\htmlData{tutor-start=13,tutor-end=14}{m}\htmlData{tutor-start=14,tutor-end=15}{+}\htmlData{tutor-start=15,tutor-end=16}{3}} \htmlData{tutor-start=18,tutor-end=19}{>} \dots$ 实际上是在描述 $\htmlData{tutor-start=0,tutor-end=1}{S}$ 中元素的某种配对或交错性质，或者是在模2意义下的行为。 不过，针对竞赛解答的复现，我们只需忠实呈现其逻辑： 1. 确定 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$ 为定值 $\htmlData{tutor-start=0,tutor-end=8}{\lfloor }\htmlData{tutor-start=8,tutor-end=9}{(}\htmlData{tutor-start=9,tutor-end=10}{L}\htmlData{tutor-start=10,tutor-end=11}{+}\htmlData{tutor-start=11,tutor-end=12}{1}\htmlData{tutor-start=12,tutor-end=13}{)}\htmlData{tutor-start=13,tutor-end=14}{/}\htmlData{tutor-start=14,tutor-end=21}{\alpha }\htmlData{tutor-start=21,tutor-end=28}{\rfloor}$。 2. 确定 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$ 的后继 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{n}\htmlData{tutor-start=4,tutor-end=5}{e}\htmlData{tutor-start=5,tutor-end=6}{x}\htmlData{tutor-start=6,tutor-end=7}{t}}$ 必 $\htmlData{tutor-start=0,tutor-end=4}{\le }\htmlData{tutor-start=4,tutor-end=5}{L}$。 3. 确定 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{n}}$ 的前驱必 $\htmlData{tutor-start=0,tutor-end=1}{>} \htmlData{tutor-start=2,tutor-end=3}{L}$。 4. 由此构建出的周期结构必然包含奇数个元素。具体地，参考解答通过构造不等式链证明了这一点。