2012 中国国家集训队选拔考试（China TST）

books/competition_archive/china_tst/2012_china_tst.pdf · HS-MATH-1024-v2.1-solution-aware

18 题0 个小问/题组

TST 1 / Day 1 · 代数

Complex numbers xi, yi satisfy |xi| = |yi| = 1 for i = 1, 2, . . . , n. Let x = 1 nP nP xi, y = 1 yi n n i=1 i=1 nP |zi| ⩽n. and zi = xyi + yxi −xiyi. Prove that i=1

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题目标签：TST1-DAY1-P1

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该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

TST 1 / Day 1 · 平面几何

Given a scalene triangle ABC. Its incircle touches BC, AC, AB at D, E, F respectvely. Let L, M, N be the symmetric points of D with EF,of E with FD,of F with DE,respectively. Line AL intersects BC at P,line BM intersects CA at Q,line CN intersects AB at R. Prove that P, Q, R are collinear. 2n

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题目标签：TST1-DAY1-P2

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该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

TST 1 / Day 1 · 数论

Let xn = n for all n ∈Z+. Prove there exist infinitely many finite sets A, B of positive integers, satisfying A ∩B = ∅, and Q i∈A = 2012. xi Q xj j∈B

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题目标签：TST1-DAY1-P3

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该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

TST 1 / Day 2 · 平面几何

Given two circles ω1, ω2, S denotes all ∆ABC satisfies that ω1 is the circumcircle of ∆ABC, ω2 is the Aexcircle of ∆ABC , ω2 touches BC, CA, AB at D, E, F. S is not empty, prove that the centroid of ∆DEF is a fixed point.

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题目标签：TST1-DAY2-P1

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该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

TST 1 / Day 2 · 数论

For a positive integer n, denote by τ(n) the number of its positive divisors. For a positive integer n, if τ(m) < τ(n) for all m < n, we call n a good number. Prove that for any positive integer k, there are only finitely many good numbers not divisible by k.

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题目标签：TST1-DAY2-P2

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该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

TST 1 / Day 2 · 代数

n being a given integer, find all functions f : Z →Z, such that for all integers x, y we have f (x + y + f(y)) = f(x) + ny.

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题目标签：TST1-DAY2-P3

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该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

TST 2 / Day 1 · 组合数学

In a simple graph G, we call t pairwise adjacent vertices a t-clique. If a vertex is connected with all other vertices in the graph, we call it a central vertex. Given are two integers n, k such that 3 2 ≤1 2n < k < n. Let G be a graph on n vertices such that (1) G does not contain a (k + 1)-clique; (2) if we add an arbitrary edge to G, that creates a (k + 1)-clique. Find the least possible number of central vertices in G.

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题目标签：TST2-DAY1-P1

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

TST 2 / Day 1 · 组合数学

Prove that there exists a positive real number C with the following property: for any integer n ≥2 and any subset X of the set {1, 2, . . . , n} such that |X| ≥2, there exist x, y, z, w ∈X(not necessarily distinct) such that 0 < |xy −zw| < Cα−4 where α = |X| n .

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题目标签：TST2-DAY1-P2

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

TST 2 / Day 1 · 数论

Let a1 < a2 be two given integers. For any integer n ≥3, let an be the smallest integer which is larger than an−1 and can be uniquely represented as ai+aj, where 1 ≤i < j ≤n−1. Given that there are only a finite number of even numbers in {an}, prove that the sequence {an+1 −an} is eventually periodic, i.e. that there exist positive integers T, N such that for all integers n > N, we have aT+n+1 −aT+n = an+1 −an.

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题目标签：TST2-DAY1-P3

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

TST 2 / Day 2 · 数论

Given an integer n ≥2. Prove that there only exist a finite number of n-tuples of positive integers (a1, a2, . . . , an) which simultaneously satisfy the following three conditions: i=1 gcd(ai, ai+1),where an+1 = a1. - a1 > a2 > . . . > an; - gcd(a1, a2, . . . , an) = 1; - a1 = Pn

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题目标签：TST2-DAY2-P1

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

TST 2 / Day 2 · 代数

Given two integers m, n which are greater than 1. r, s are two given positive real numbers such that r < s. For all aij ≥0 which are not all zeroes,find the maximal value of the expression 1 r (Pn r s ) ij) j=1(Pm i=1 as f = . 1 s (Pm s r ) ij) j=1 ar i=1) Pn

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题目标签：TST2-DAY2-P2

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

TST 2 / Day 2 · 代数

Given an integer n ≥2, a function f : Z →{1, 2, . . . , n} is called good, if for any integer k, 1 ≤k ≤n −1 there exists an integer j(k) such that for every integer m we have f(m + j(k)) ≡f(m + k) −f(m) (mod n + 1). Find the number of good functions.

题解状态：标准答案与规范题解待补充

题目标签：TST2-DAY2-P3

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

TST 3 / Day 1 · 平面几何

In an acute-angled ABC, ∠A > 60◦, H is its orthocenter. M, N are two points on AB, AC respectively, such that ∠HMB = ∠HNC = 60◦. Let O be the circumcenter of triangle HMN. D is a point on the same side with A of BC such that △DBC is an equilateral triangle. Prove that H, O, D are collinear. i=1 bi, we have

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题目标签：TST3-DAY1-P1

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

TST 3 / Day 1 · 数论

Given an integer k ≥2. Prove that there exist k pairwise distinct positive integers a1, a2, . . . , ak such that for any non-negative integers b1, b2, . . . , bk, c1, c2, . . . , ck satisfying a1 ≤bi ≤2ai, i = 1, 2, . . . , k and Qk i=1 bci i < Qk k Y k Y k bci bi. i < i=1 i=1

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题目标签：TST3-DAY1-P2

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

TST 3 / Day 1 · 代数

Find the smallest possible value of a real number c such that for any 2012-degree monic polynomial P(x) = x2012 + a2011x2011 + . . . + a1x + a0 with real coefficients, we can obtain a new polynomial Q(x) by multiplying some of its coefficients by −1 such that every root z of Q(x) satisfies the inequality |Im z| ≤c |Re z| .

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题目标签：TST3-DAY1-P3

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

TST 3 / Day 2 · 组合数学

Given an integer n ≥4. S = {1, 2, . . . , n}. A, B are two subsets of S such that for every pair of (a, b), a ∈A, b ∈B, ab + 1 is a perfect square. Prove that min{|A|, |B|} ≤log2 n.

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题目标签：TST3-DAY2-P1

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

TST 3 / Day 2 · 数论

Find all integers k ≥3 with the following property: There exist integers m, n such that 1 < m < k, 1 < n < k, gcd(m, k) = gcd(n, k) = 1, m + n > k and k | (m −1)(n −1).

题解状态：标准答案与规范题解待补充

题目标签：TST3-DAY2-P2

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

TST 3 / Day 2 · 数学竞赛/待细分

In some squares of a 2012 × 2012 grid there are some beetles, such that no square contain more than one beetle. At one moment, all the beetles fly off the grid and then land on the grid again, also satisfying the condition that there is at most one beetle standing in each square. The vector from the centre of the square from which a beetle B flies to the centre of the square on which it lands is called the translation vector of beetle B. For all possible starting and ending configurations, find the maximum length of the sum of the translation vectors of all beetles.

题解状态：标准答案与规范题解待补充

题目标签：TST3-DAY2-P3

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。