2015 中国国家集训队选拔考试（China TST）

books/competition_archive/china_tst/2015_china_tst.pdf · HS-MATH-1024-v2.1-solution-aware

18 题0 个小问/题组

TST 1 / Day 1 · 平面几何

The circle Γ through A of triangle ABC meets sides AB, AC at E,F respectively, and circumcircle of ABC at P. Prove: Reflection of P across EF is on BC if and only if Γ passes through O (the circumcentre of ABC).

题解状态：标准答案与规范题解待补充

题目标签：TST1-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 1 / Day 1 · 数论

Let a1, a2, a3, · · · be distinct positive integers, and 0 < c < 3 2 . Prove that : There exist infinitely many positive integers k, such that [ak, ak+1] > ck.

题解状态：标准答案与规范题解待补充

题目标签：TST1-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 1 / Day 1 · 数论

Fix positive integers k, n. A candy vending machine has many different colours of candy, where there are 2n candies of each colour. A couple of kids each buys from the vending machine 2 candies of different colours. Given that for any k + 1 kids there are two kids who have at least one colour of candy in common, find the maximum number of kids.

题解状态：标准答案与规范题解待补充

题目标签：TST1-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 1 / Day 2 · 平面几何

Prove that : For each integer n ≥3, there exists the positive integers a1 < a2 < · · · < an , such that for i = 1, 2, · · · , n −2 , With ai, ai+1, ai+2 may be formed as a triangle side length , and the area of the triangle is a positive integer.

题解状态：标准答案与规范题解待补充

题目标签：TST1-DAY2-P4

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 1 / Day 2 · 数论

FIx positive integer n. Prove: For any positive integers a, b, c not exceeding 3n2 +4n, there exist integers x, y, z with absolute value not exceeding 2n and not all 0, such that ax + by + cz = 0

题解状态：标准答案与规范题解待补充

题目标签：TST1-DAY2-P5

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 1 / Day 2 · 数学竞赛/待细分

There are some players in a Ping Pong tournament, where every 2 players play with each other at most once. Given: (1) Each player wins at least a players, and loses to at least b players. (a, b ≥1) (2) For any two players A, B, there exist some players P1, ..., Pk (k ≥2) (where P1 = A,Pk = B), such that Pi wins Pi+1 (i = 1, 2..., k −1). Prove that there exist a + b + 1 distinct players Q1, ...Qa+b+1, such that Qi wins Qi+1 (i = 1, ..., a + b)

题解状态：标准答案与规范题解待补充

题目标签：TST1-DAY2-P6

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

For a positive integer n, and a non empty subset A of {1, 2, ..., 2n}, call A good if the set {u ± v|u, v ∈A} does not contain the set {1, 2, ..., n}. Find the smallest real number c, such that for any positive integer n, and any good subset A of {1, 2, ..., 2n}, |A| ≤cn.

题解状态：标准答案与规范题解待补充

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 · 代数

Let a1, a2, a3, · · · , an be positive real numbers. For the integers n ≥2, prove that 1 n j   Pn Qj 1 n j=1 k=1 ak i=1 ai) ≤n + 1 + (Qn Pn n j     j=1 aj Pn Qj j=1 k=1 ak

题解状态：标准答案与规范题解待补充

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 · 平面几何

Let △ABC be an acute triangle with circumcenter O and centroid G. Let D be the midpoint of BC and E ∈⊙(BC) be a point inside △ABC such that AE ⊥BC. Let F = EG ∩OD and K, L be the point lie on BC such that FK ∥OB, FL ∥OC. Let M ∈AB be a point such that MK ⊥BC and N ∈AC be a point such that NL ⊥BC. Let ω be a circle tangent to OB, OC at B, C, respectively . Prove that ⊙(AMN) is tangent to ω

题解状态：标准答案与规范题解待补充

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 · 代数

Let n be a positive integer, let f1(x), . . . , fn(x) be n bounded real functions, and let a1, . . . , an be n distinct reals. Show that there exists a real number x such that Pn i=1 fi(x) −Pn i=1 fi(x −ai) < 1. c∈C c a∈A a = P b∈B b = P

题解状态：标准答案与规范题解待补充

题目标签：TST2-DAY2-P4

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 · 组合数学

Set S to be a subset of size 68 of {1, 2, ..., 2015}. Prove that there exist 3 pairwise disjoint, non-empty subsets A, B, C such that |A| = |B| = |C| and P

题解状态：标准答案与规范题解待补充

题目标签：TST2-DAY2-P5

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 · 数学竞赛/待细分

Prove that there exist infinitely many integers n such that n2 + 1 is squarefree.

题解状态：标准答案与规范题解待补充

题目标签：TST2-DAY2-P6

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

△ABC is isosceles with AB = AC > BC. Let D be a point in its interior such that DA = DB+DC. Suppose that the perpendicular bisector of AB meets the external angle bisector of ∠ADB at P, and let Q be the intersection of the perpendicular bisector of AC and the external angle bisector of ∠ADC. Prove that B, C, P, Q are concyclic.

题解状态：标准答案与规范题解待补充

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 · 组合数学

Let X be a non-empty and finite set, A1, ..., Ak k subsets of X, satisying: (1) |Ai| ≤3, i = 1, 2, ..., k (2) Any element of X is an element of at least 4 sets among A1, ...., Ak. Show that one can select [3k 7 ] sets from A1, ..., Ak such that their union is X.

题解状态：标准答案与规范题解待补充

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 · 数论

Let a, b be two integers such that their gcd has at least two prime factors. Let S = {x | x ∈ N, x ≡a (mod b)} and call y ∈S irreducible if it cannot be expressed as product of two or more elements of S (not necessarily distinct). Show there exists t such that any element of S can be expressed as product of at most t irreducible elements.

题解状态：标准答案与规范题解待补充

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 · 代数

Let x1, x2, · · · , xn (n ≥2) be a non-decreasing monotonous sequence of positive numbers such that x1, x2 2 , · · · , xn n is a non-increasing monotonous sequence .Prove that Pn i=1 xi ≤n + 1 1 n 2 n√ n! n (Qn i=1 xi)

题解状态：标准答案与规范题解待补充

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 · 组合数学

Let G be the complete graph on 2015 vertices. Each edge of G is dyed red, blue or white. For a subset V of vertices of G, and a pair of vertices (u, v), define L(u, v) = {u, v} ∪{w|w ∈V ∋△uvw has exactly 2 red sides} Prove that, for any choice of V , there exist at least 120 distinct values of L(u, v).

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 · 数论

For all natural numbers n, define f(n) = τ(n!) −τ((n −1)!), where τ(a) denotes the number of positive divisors of a. Prove that there exist infinitely many composite n, such that for all naturals m < n, we have f(m) < f(n).

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。