2019 中国国家集训队选拔考试（China TST）

books/competition_archive/china_tst/2019_china_tst.pdf · HS-MATH-1024-v2.1-solution-aware

24 题0 个小问/题组

TST 1 / Day 1 · 数学竞赛/待细分

ABCDE is a cyclic pentagon, with circumcentre O. AB = AE = CD. I midpoint of BC. J midpoint of DE. F is the orthocentre of △ABE, and G the centroid of △AIJ.CE intersects BD at H, OG intersects FH at M. Show that AM ⊥CD.

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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

Fix a positive integer n ≥3. Does there exist infinitely many sets S of positive integers {a1, a2, . . . , an, b1, b2, . . . , bn}, such that gcd(a1, a2, . . . , an, b1, b2, . . . , bn) = 1, {ai}n i=1, {bi}n i=1 bi? i=1 ai = Qn i=1 are arithmetic progressions, and Qn i is a fixed value for i=1 MP k

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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 · 平面几何

Find all positive integer n, such that there exists n points P1, . . . , Pn on the unit circle , satisfying the condition that for any point M on the unit circle, Pn a) k = 2018 b) k = 2019.

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

Call a sequence of positive integers {an} good if for any distinct positive integers m, n, one has gcd(m, n) | a2 n and gcd(am, an) | m2 + n2. m + a2 Call a positive integer a to be k-good if there exists a good sequence such that ak = a. Does there exists a k such that there are exactly 2019 k-good positive integers?

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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 · 代数

Determine all functions f : Q →Q such that f(2xy + 1 2) + f(x −y) = 4f(x)f(y) + 1 2 for all x, y ∈Q.

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 · 平面几何

Let k be a positive real. A and B play the following game: at the start, there are 80 zeroes arrange around a circle. Each turn, A increases some of these 80 numbers, such that the total sum added is 1. Next, B selects ten consecutive numbers with the largest sum, and reduces them all to 0. A then wins the game if he/she can ensure that at least one of the number is ≥k at some finite point of time. Determine all k such that A can always win the game.

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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

AB and AC are tangents to a circle ω with center O at B, C respectively. Point P is a variable point on minor arc BC. The tangent at P to ω meets AB, AC at D, E respectively. AO meets BP, CP at U, V respectively. The line through P perpendicular to AB intersects DV at M, and the line through P perpendicular to AC intersects EU at N. Prove that as P varies, MN passes through a fixed point.

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 · 数学竞赛/待细分

Let S be the set of 10-tuples of non-negative integers that have sum 2019. For any tuple in S, if one of the numbers in the tuple is ≥9, then we can subtract 9 from it, and add 1 to the remaining numbers in the tuple. Call thus one operation. If for A, B ∈S we can get from A to B in finitely many operations, then denote A →B. (1) Find the smallest integer k, such that if the minimum number in A, B ∈S respectively are both ≥k, then A →B implies B →A. (2) For the k obtained in (1), how many tuples can we pick from S, such that any two of these tuples A, B that are distinct, A̸ →B.

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 · 代数

Let n be a given even number, a1, a2, · · · , an be non-negative real numbers such that a1 + a2 + · · · + an = 1. Find the maximum possible value of P 1≤i<j≤n min{(i −j)2, (n + i −j)2}aiaj.

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

Does there exist a finite set A of positive integers of at least two elements and an infinite set B of positive integers, such that any two distinct elements in A + B are coprime, and for any coprime positive integers m, n, there exists an element x in A + B satisfying x ≡n (mod m) ? Here A + B = {a + b|a ∈A, b ∈B}.

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 · 平面几何

Let M be the midpoint of BC of triangle ABC. The circle with diameter BC, ω, meets AB, AC at D, E respectively. P lies inside △ABC such that ∠PBA = ∠PAC, ∠PCA = ∠PAB, and 2PM · DE = BC2. Point X lies outside ω such that XM ∥AP, and XB XC = AB AC . Prove that ∠BXC + ∠BAC = 90◦.

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 · 数论

Given coprime positive integers p, q > 1, call all positive integers that cannot be written as px + qy(where x, y are non-negative integers) bad, and define S(p, q) to be the sum of all bad numbers raised to the power of 2019. Prove that there exists a positive integer n, such that for any p, q as described, (p −1)(q −1) divides nS(p, q).

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 · 代数

Given complex numbers x, y, z, with |x|2 + |y|2 + |z|2 = 1. Prove that: |x3 + y3 + z3 −3xyz| ≤1

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

Let S be a set of positive integers, such that n ∈S if and only if X d ≤n d|n,d<n,d∈S Find all positive integers n = 2k · p where k is a non-negative integer and p is an odd prime, such that X d = n d|n,d<n,d∈S

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 · 组合数学

Does there exist a bijection f : N+ →N+, such that there exist a positive integer k, and it’s possible to have each positive integer colored by one of k chosen colors, such that for any x̸ = y , f(x) + y and f(y) + x are not the same color?

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

Find all functions f : R2 →R, such that 1) f(0, x) is non-decreasing ; 2) for any x, y ∈R, f(x, y) = f(y, x) ; 3) for any x, y, z ∈R, (f(x, y) −f(y, z))(f(y, z) −f(z, x))(f(z, x) −f(x, y)) = 0 ; 4) for any x, y, a ∈R, f(x + a, y + a) = f(x, y) + a .

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 · 平面几何

In ∆ABC, AD ⊥BC at D. E, F lie on line AB, such that BD = BE = BF. Let I, J be the incenter and A-excenter. Prove that there exist two points P, Q on the circumcircle of ∆ABC , such that PB = QC, and ∆PEI ∼∆QFJ .

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 · 组合数学

Given positive integers d ≥3, r > 2 and l, with 2d ≤l < rd. Every vertice of the graph G(V, E) is assigned to a positive integer in {1, 2, · · · , l}, such that for any two consecutive vertices in the graph, the integers they are assigned to, respectively, have difference no less than d, and no more than l −d. A proper coloring of the graph is a coloring of the vertices, such that any two consecutive vertices are not the same color. It’s given that there exist a proper subset A of V , such that for G’s any proper coloring with r −1 colors, and for an arbitrary color C, either all numbers in color C appear in A, or none of the numbers in color C appear in A. Show that G has a proper coloring within r −1 colors.

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 · 平面几何

Cyclic quadrilateral ABCD has circumcircle (O). Points M and N are the midpoints of BC and CD, and E and F lie on AB and AD respectively such that EF passes through O and EO = OF. Let EN meet FM at P. Denote S as the circumcenter of △PEF. Line PO intersects AD and BA at Q and R respectively. Suppose OSPC is a parallelogram. Prove that AQ = AR.

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

A graph G(V, E) is triangle-free, but adding any edges to the graph will form a triangle. It’s given that |V | = 2019, |E| > 2018, find the minimum of |E| .

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 · 平面几何

60 points lie on the plane, such that no three points are collinear. Prove that one can divide the points into 20 groups, with 3 points in each group, such that the triangles ( 20 in total) consist of three points in a group have a non-empty intersection.

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 · 组合数学

Prove that there exist a subset A of {1, 2, · · · , 2n} with n elements, such that for any two different non-empty subset of A, the sum of elements of one subset doesn’t divide another’s.

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 · 代数

Find all integer n such that the following property holds: for any positive real numbers a, b, c, x, y, z, with max(a, b, c, x, y, z) = a , a + b + c = x + y + z and abc = xyz, the inequality an + bn + cn ≥xn + yn + zn holds. y is even, and

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 · 数论

Given positive integer n, k such that 2 ≤n < 2k. Prove that there exist a subset A of {0, 1, · · · , n} such that for any x̸ = y ∈A, x k ⌊k 2 ⌋ |A| ≥ 2k · (n + 1)

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。