2022 中国国家集训队选拔考试（China TST）

books/competition_archive/china_tst/2022_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

In a cyclic convex hexagon ABCDEF, AB and DC intersect at G, AF and DE intersect at H. Let M, N be the circumcenters of BCG and EFH, respectively. Prove that the BE, CF and MN are concurrent.

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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

Let p be a prime, A is an infinite set of integers. Prove that there is a subset B of A with 2p − 2 elements, such that the arithmetic mean of any pairwise distinct p elements in B does not belong to A.

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

Let a, b, c, p, q, r be positive integers with p, q, r ≥2. Denote Q = {(x, y, z) ∈Z3 : 0 ≤x ≤a, 0 ≤y ≤b, 0 ≤z ≤c}. Initially, some pieces are put on the each point in Q, with a total of M pieces. Then, one can perform the following three types of operations repeatedly: (1) Remove p pieces on (x, y, z) and place a piece on (x −1, y, z) ; (2) Remove q pieces on (x, y, z) and place a piece on (x, y −1, z) ; (3) Remove r pieces on (x, y, z) and place a piece on (x, y, z −1). Find the smallest positive integer M such that one can always perform a sequence of operations, making a piece placed on (0, 0, 0), no matter how the pieces are distributed initially.

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

Let ABC be an acute triangle with ∠ACB > 2∠ABC. Let I be the incenter of ABC, K is the reflection of I in line BC. Let line BA and KC intersect at D. The line through B parallel to CI intersects the minor arc BC on the circumcircle of ABC at E(E̸ = B). The line through A parallel to BC intersects the line BE at F. Prove that if BF = CE, then FK = AD.

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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 · 平面几何

Let C = {z ∈C : |z| = 1} be the unit circle on the complex plane. Let z1, z2, . . . , z240 ∈C (not necessarily different) be 240 complex numbers, satisfying the following two conditions: (1) For any open arc Γ of length π on C, there are at most 200 of j (1 ≤j ≤240) such that zj ∈Γ. (2) For any open arc γ of length π/3 on C, there are at most 120 of j (1 ≤j ≤240) such that zj ∈γ. Find the maximum of |z1 + z2 + . . . + z240|.

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 · 组合数学

Let m be a positive integer, and A1, A2, . . . , Am (not necessarily different) be m subsets of a finite set A. It is known that for any nonempty subset I of {1, 2 . . . , m}, [ Ai ≥|I| + 1. i∈I Show that the elements of A can be colored black and white, so that each of A1, A2, . . . , Am contains both black and white elements.

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

Find all pairs of positive integers (m, n), such that in a m × n table (with m + 1 horizontal lines and n + 1 vertical lines), a diagonal can be drawn in some unit squares (some unit squares may have no diagonals drawn, but two diagonals cannot be both drawn in a unit square), so that the obtained graph has an Eulerian cycle.

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 · 平面几何

Given a non-right triangle ABC with BC > AC > AB. Two points P1̸ = P2 on the plane satisfy that, for i = 1, 2, if APi, BPi and CPi intersect the circumcircle of the triangle ABC at Di, Ei, and Fi, respectively, then DiEi ⊥DiFi and DiEi = DiFi̸ = 0. Let the line P1P2 intersects the circumcircle of ABC at Q1 and Q2. The Simson lines of Q1, Q2 with respect to ABC intersect at W. Prove that W lies on the nine-point circle of ABC.

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

Let a1, a2, . . . , an be n positive integers that are not divisible by each other, i.e. for any i̸ = j, ai is not divisible by aj. Show that a1 + a2 + · · · + an ≥1.1n2 −2n. Note: A proof of the inequality when n is sufficient large will be awarded points depending on your results.

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

Given a positive integer n, find all n-tuples of real number (x1, x2, . . . , xn) such that 2 X 2 X 2 X · · · f(x1, x2, · · · , xn) = k1x1 + k2x2 + · · · + knxn −1 kn=0 k2=0 k1=0 attains its minimum.

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

Given a positive integer n, let D is the set of positive divisors of n, and let f : D →Z be a function. Prove that the following are equivalent: (a) For any positive divisor m of n, n/d X f(d) . n m/d d|m (b) For any positive divisor k of n, X k f(d). d|k

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

Let m, n be two positive integers with m ≥n ≥2022. Let a1, a2, . . . , an, b1, b2, . . . , bn be 2n real numbers. Prove that the numbers of ordered pairs (i, j) (1 ≤i, j ≤n) such that |ai + bj −ij| ≤m does not exceed 3n√m log n.

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 two circles ω1 and ω2 where ω2 is inside ω1. Show that there exists a point P such that for any line ℓnot passing through P, if ℓintersects circle ω1 at A, B and ℓintersects circle ω2 at C, D, where A, C, D, B lie on ℓin this order, then ∠APC = ∠BPD.

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

Two positive real numbers α, β satisfies that for any positive integers k1, k2, it holds that ⌊k1α⌋̸ = ⌊k2β⌋, where ⌊x⌋denotes the largest integer less than or equal to x. Prove that there exist positive integers m1, m2 such that m1 α + m2 β = 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

Given a positive integer n ≥2. Find all n-tuples of positive integers (a1, a2, . . . , an), such that 1 < a1 ≤a2 ≤a3 ≤· · · ≤an, a1 is odd, and (1) M = 1 2n (a1 −1)a2a3 · · · an is a positive integer; (2) One can pick n-tuples of integers (ki,1, ki,2, . . . , ki,n) for i = 1, 2, . . . , M such that for any 1 ≤i1 < i2 ≤M, there exists j ∈{1, 2, . . . , n} such that ki1,j −ki2,j̸ ≡0, ±1 (mod aj).

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 · 平面几何

Find all positive integer k such that one can find a number of triangles in the Cartesian plane, the centroid of each triangle is a lattice point, the union of these triangles is a square of side length k (the sides of the square are not necessarily parallel to the axis, the vertices of the square are not necessarily lattice points), and the intersection of any two triangles is an empty-set, a common point or a common edge.

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 · 数论

Show that there exist constants c and α > 1 2, such that for any positive integer n, there is a subset A of {1, 2, . . . , n} with cardinality |A| ≥c · nα, and for any x, y ∈A with x̸ = y, the difference x −y is not a perfect square.

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 · 代数

(1) Prove that, on the complex plane, the area of the convex hull of all complex roots of z20 + 63z + 22 = 0 is greater than π. (2) Let a1, a2, . . . , an be complex numbers with sum 1, and k1 < k2 < · · · < kn be odd positive integers. Let ω be a complex number with norm at least 1. Prove that the equation a1zk1 + a2zk2 + · · · + anzkn = w has at least one complex root with norm at most 3n|ω|.

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 · 组合数学

Initially, each unit square of an n × n grid is colored red, yellow or blue. In each round, perform the following operation for every unit square simultaneously: - For a red square, if there is a yellow square that has a common edge with it, then color it yellow. - For a yellow square, if there is a blue square that has a common edge with it, then color it blue. - For a blue square, if there is a red square that has a common edge with it, then color it red. It is known that after several rounds, all unit squares of this n × n grid have the same color. Prove that the grid has became monochromatic no later than the end of the (2n −2)-th round.

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 ABCD be a convex quadrilateral, the incenters of △ABC and △ADC are I, J, respectively. It is known that AC, BD, IJ concurrent at a point P. The line perpendicular to BD through P intersects with the outer angle bisector of ∠BAD and the outer angle bisector ∠BCD at E, F, respectively. Show that PE = PF.

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 · 代数

Find all functions f : R →R such that for any x, y ∈R, the multiset {(f(xf(y)+1), f(yf(x)−1)} is identical to the multiset {xf(f(y)) + 1, yf(f(x)) −1}. Note: The multiset {a, b} is identical to the multiset {c, d} if and only if a = c, b = d or a = d, b = c.

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 · 数论

Find all positive integers a, b, c and prime p satisfying that 2apb = (p + 2)c + 1.

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 · 数论

Let n be a positive integer, x1, x2, . . . , x2n be non-negative real numbers with sum 4. Prove that there exist integer p and q, with 0 ≤q ≤n −1, such that q X n−1 X xp+2i−1 ≤1 and xp+2i ≤1, i=1 i=q+1 where the indices are take modulo 2n. i=q+1 xp+2i = 0. Note: If q = 0, then Pq i=1 xp+2i−1 = 0; if q = n −1, then Pn−1

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 a positive integer n, let D be the set of all positive divisors of n. The subsets A, B of D satisfies that for any a ∈A and b ∈B, it holds that a ∤b and b ∤a. Show that p p p |A| + |B| ≤ |D|.

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。