2021 中国国家集训队选拔考试（China TST）

books/competition_archive/china_tst/2021_china_tst.pdf · HS-MATH-1024-v2.1-solution-aware

24 题0 个小问/题组

TST 1 / Day 1 · 数论

Given positive integers m and n. Let ai,j(1 ≤i ≤m, 1 ≤j ≤n) be non-negative real numbers, such that ai,1 ≥ai,2 ≥· · · ≥ai,n and a1,j ≥a2,j ≥· · · ≥am,j holds for all 1 ≤i ≤m and 1 ≤j ≤n. Denote Xi,j = a1,j + · · · + ai−1,j + ai,j + ai,j−1 + · · · + ai,1, Yi,j = am,j + · · · + ai+1,j + ai,j + ai,j+1 + · · · + ai,n. Prove that m Y n Y m Y n Y Yi,j. Xi,j ≥ i=1 j=1 i=1 j=1

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 · 组合数学

Given positive integers n and k, n > k2 > 4. In a n × n grid, a k-group is a set of k unit squares lying in different rows and different columns. Determine the maximal possible N, such that one can choose N unit squares in the grid and color them, with the following condition holds: in any k-group from the colored N unit squares, there are two squares with the same color, and there are also two squares with different colors.

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

Given positive integer n. Prove that for any integers a1, a2, · · · , an, at least ⌈n(n−6) 19 ⌉numbers from the set {1, 2, · · · , n(n−1) 2 } cannot be represented as ai −aj(1 ≤i, j ≤n).

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 · 数论

Let f(x), g(x) be two polynomials with integer coefficients. It is known that for infinitely many prime p, there exist integer mp such that f(a) ≡g(a + mp) (mod p) holds for all a ∈Z. Prove that there exists a rational number r such that f(x) = g(x + r).

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 · 平面几何

Given a triangle ABC, a circle Ωis tangent to AB, AC at B, C, respectively. Point D is the midpoint of AC, O is the circumcenter of triangle ABC. A circle Γ passing through A, C intersects the minor arc BC on Ωat P, and intersects AB at Q. It is known that the midpoint R of minor arc PQ satisfies that CR ⊥AB. Ray PQ intersects line AC at L, M is the midpoint of AL, N is the midpoint of DR, and X is the projection of M onto ON. Prove that the circumcircle of triangle DNX passes through the center of Γ.

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 · 数论

Given positive integer n and r pairwise distinct primes p1, p2, · · · , pr. Initially, there are (n + 1)r numbers written on the blackboard: pi1 1 pi2 2 · · · pirr (0 ≤i1, i2, · · · , ir ≤n). Alice and Bob play a game by making a move by turns, with Alice going first. In Alice’s round, she erases two numbers a, b (not necessarily different) and write gcd(a, b). In Bob’s round, he erases two numbers a, b (not necessarily different) and write lcm(a, b). The game ends when only one number remains on the blackboard. Determine the minimal possible M such that Alice could guarantee the remaining number no greater than M, regardless of Bob’s move.

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 · 平面几何

A cyclic quadrilateral ABCD has circumcircle Γ, and AB + BC = AD + DC. Let E be the midpoint of arc BCD, and F(̸= C) be the antipode of A wrt Γ. Let I, J, K be the incenter of △ABC, the A-excenter of △ABC, the incenter of △BCD, respectively. Suppose that a point P satisfies △BIC +∼△KPJ. Prove that EK and PF intersect on Γ.

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 · 组合数学

Given positive integers n, k, n ≥2. Find the minimum constant c satisfies the following assertion: For any positive integer m and a kn-regular graph G with m vertices, one could color the vertices of G with n different colors, such that the number of monochrome edges is at most cm.

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 · 数论

Given positive integers a, b, c which are pairwise coprime. Let f(n) denotes the number of the non-negative integer solution (x, y, z) to the equation ax + by + cz = n. Prove that there exists constants α, β, γ ∈R such that for any non-negative integer n, |f(n) − αn2 + βn + γ | < 1 12 (a + b + c) .

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 · 数论

Find all functions f : Z+ →Z+ such that for all positive integers m, n with m ≥n, f(mϕ(n3)) = f(m) · ϕ(n3). Here ϕ(n) denotes the number of positive integers coprime to n and not exceeding n. , where indices are taken modulo 2n + 1. Prove that the

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 · 数论

Let n be a positive integer and a1, a2, . . . a2n+1 be positive reals. For k = 1, 2, . . . , 2n + 1, denote bk = max0≤m≤n 1 2m+1 Pk+m i=k−m ai i=1 ai. number of indices k satisfying bk ≥1 does not exceed 2 P2n+1

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 · 平面几何

Find the smallest positive real constant a, such that for any three points A, B, C on the unit circle, there exists an equilateral triangle PQR with side length a such that all of A, B, C lie on the interior or boundary of △PQR.

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

Given positive integer n ≥5 and a convex polygon P, namely A1A2...An. No diagonals of P are concurrent. Proof that it is possible to choose a point inside every quadrilateral AiAjAkAl(1 ≤ i < j < k < l ≤n) not on diagonals of P, such that the n points chosen are distinct, and any segment connecting these points intersect with some diagonal of P.

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 · 数论

Given distinct positive integer a1, a2, ..., a2020. For n ≥2021, an is the smallest number different from a1, a2, ..., an−1 which doesn’t divide an−2020...an−2an−1. Proof that every number large enough appears in the sequence. n(n−1)

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

Determine the greatest real number C, such that for every positive integer n ≥2, there exists x1, x2, ..., xn ∈[−1, 1], so that Y 1≤i<j≤n (xi −xj) ≥C .

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 · 数学竞赛/待细分

Proof that n X n X n X ⌊n 5ω(m) ≤ 5Ω(m). k ⌋τ(k)2 ≤ m=1 m=1 k=1

题解状态：标准答案与规范题解待补充

题目标签：TST3-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 3 / Day 2 · 数学竞赛/待细分

Determine all f : R →R such that f(xf(y) + y3) = yf(x) + f(y)3

题解状态：标准答案与规范题解待补充

题目标签：TST3-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 3 / Day 2 · 平面几何

Proof that there exist constant λ, so that for any positive integer m(≥2), and any lattice triangle T in the Cartesian coordinate plane, if T contains exactly one m-lattice point in its interior(not containing boundary), then T has area ≤λm3. PS. lattice triangles are triangles whose vertex are lattice points; m-lattice points are lattice points whose both coordinates are divisible by m.

题解状态：标准答案与规范题解待补充

题目标签：TST3-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 4 / Day 1 · 数论

Let n(≥2) be a positive integer. Find the minimum m, so that there exists xij(1 ≤i, j ≤n) satisfying: (1)For every 1 ≤i, j ≤n, xij = max{xi1, xi2, ..., xij} or xij = max{x1j, x2j, ..., xij}. (2)For every 1 ≤i ≤n, there are at most m indices k with xik = max{xi1, xi2, ..., xik}. (3)For every 1 ≤j ≤n, there are at most m indices k with xkj = max{x1j, x2j, ..., xkj}.

题解状态：标准答案与规范题解待补充

题目标签：TST4-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 4 / Day 1 · 平面几何

Let triangleABC(AB < AC) with incenter I circumscribed in ⊙O. Let M, N be midpoint of arc \ BAC and d BC, respectively. D lies on ⊙O so that AD//BC, and E is tangency point of A-excircle of △ABC. Point F is in △ABC so that FI//BC and ∠BAF = ∠EAC. Extend NF to meet ⊙O at G, and extend AG to meet line IF at L. Let line AF and DI meet at K. Proof that ML⊥NK.

题解状态：标准答案与规范题解待补充

题目标签：TST4-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 4 / Day 1 · 数论

Find all positive integer n(≥2) and rational β ∈(0, 1) satisfying the following: There exist positive integers a1, a2, ..., an, such that for any set I ⊆{1, 2, ..., n} which contains at least two elements, X X S( S(ai). ai) = β i∈I i∈I where S(n) denotes sum of digits of decimal representation of n.

题解状态：标准答案与规范题解待补充

题目标签：TST4-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 4 / Day 2 · 数学竞赛/待细分

Suppose x1, x2, ..., x60 ∈[−1, 1] , find the maximum of 60 X x2 i (xi+1 −xi−1), i=1 where xi+60 = xi.

题解状态：标准答案与规范题解待补充

题目标签：TST4-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 4 / Day 2 · 平面几何

Find the smallest real α, such that for any convex polygon P with area 1, there exist a point M in the plane, such that the area of convex hull of P ∪Q is at most α, where Q denotes the image of P under central symmetry with respect to M.

题解状态：标准答案与规范题解待补充

题目标签：TST4-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 4 / Day 2 · 数学竞赛/待细分

Let n(≥2) be an integer. 2n2 contestants participate in a Chinese chess competition, where any two contestant play exactly once. There may be draws. It is known that (1)If A wins B and B wins C, then A wins C. (2)there are at most n3 16 draws. Proof that it is possible to choose n2 contestants and label them Pij(1 ≤i, j ≤n), so that for any i, j, i′, j′ ∈{1, 2, ..., n}, if i < i′, then Pij wins Pi′j′.

题解状态：标准答案与规范题解待补充

题目标签：TST4-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。