2023 中国国家集训队选拔考试（China TST）

books/competition_archive/china_tst/2023_china_tst.pdf · HS-MATH-1024-v2.1-solution-aware

24 题0 个小问/题组

TST 1 / Day 1 (March 11, 2023, Chengdu) · 数学竞赛/待细分

Given an integer n ⩾2. Suppose there is a point P inside a convex cyclic 2n-gon A1 . . . A2n satisfying ∠PA1A2 = ∠PA2A3 = . . . = ∠PA2nA1, prove that n Y n Y |A2i−1A2i| = |A2iA2i+1| , i=1 i=1 where A2n+1 = A1.

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题目标签：TST1-DAY1-P1

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TST 1 / Day 1 (March 11, 2023, Chengdu) · 数学竞赛/待细分

n people attend a party. There are no more than n pairs of friends among them. Two people shake hands if and only if they have at least 1 common friend. Given integer m ≥3 such that n ≤m3. Prove that there exists a person A, the number of people that shake hands with A is no more than m −1 times of the number of A’S friends.

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题目标签：TST1-DAY1-P2

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TST 1 / Day 1 (March 11, 2023, Chengdu) · 数论

(1) Let a, b be coprime positive integers. Prove that there exists constants λ and β such that for all integers m, bk ak m−1 X −λm m m k=1 ≤β (2) Prove that there exists N such that for all p > N (where p is a prime number), and any positive integers a, b, c positive integers satisfying p ∤(a+b)(b+c)(c+a), there are at least ⌊p 12⌋ solutions k ∈{1, · · · , p −1} such that ak bk ck + + ≤1 p p p

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题目标签：TST1-DAY1-P3

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TST 1 / Day 2 (March 12, 2023, Chengdu) · 数论

Given m, n ∈N+, define S(m, n) = (a, b) ∈N2 . + | 1 ≤a ≤m, 1 ≤b ≤n, gcd(a, b) = 1 Prove that: for ∀d, r ∈N+, there exists m, n ∈N+, m, n ≥d and |S(m, n)| ≡r (mod d).

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题目标签：TST1-DAY2-P4

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TST 1 / Day 2 (March 12, 2023, Chengdu) · 平面几何

Let △ABC be a triangle, and let P1, · · · , Pn be points inside where no three given points are collinear. Prove that we can partition △ABC into 2n + 1 triangles such that their vertices are among A, B, C, P1, · · · , Pn, and at least n + √n + 1 of them contain at least one of A, B, C.

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题目标签：TST1-DAY2-P5

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TST 1 / Day 2 (March 12, 2023, Chengdu) · 代数

Prove that: (1) In the complex plane, each line (except for the real axis) that crosses the origin has at most one point z, satisfy 1 + z23 z64 ∈R. (2) For any non-zero complex number a and any real number θ, the equation 1 + z23 + az64 = 0 has roots in o n . z ∈C | Re(ze−iθ) ⩾|z| cos π Sθ = Proposed by Yijun Yao

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题目标签：TST1-DAY2-P6

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TST 1 / Day 3 (March 16, 2023, Chengdu) · 数论

Given the integer n ≥2 and a integer a, which is coprime with n. A country has n islands D1, D2, · · · , Dn. For any 1 ≤i̸ = j ≤n, there is a one-way ferry Di to Dj if and only if ij ≡ia (mod n). A tourist can initially fly to any of the islands, and then he can only take a one-way ferry. What is the maximum number of islands he can visit? Created by Zhenhua Qu

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题目标签：TST1-DAY3-P7

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TST 1 / Day 3 (March 16, 2023, Chengdu) · 平面几何

In non-isosceles acute △ABC, AP, BQ, CR is the height of the triangle. A1 is the midpoint of BC, AA1 intersects QR at K, QR intersects a straight line that crosses A and is parallel to BC at point D, the line connecting the midpoint of AH and K intersects DA1 at A2. Similarly define B2, C2. △A2B2C2 is known to be non-degenerate, and its circumscribed circle is ω. Prove that: there are circles ⊙A′, ⊙B′, ⊙C′ tangent to and INSIDE ω satisfying: (1) ⊙A′ is tangent to AB and AC, ⊙B′ is tangent to BC and BA, and ⊙C′ is tangent to CA and CB. (2) A′, B′, C’ are different and collinear. Created by Sihui Zhang

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题目标签：TST1-DAY3-P8

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TST 1 / Day 3 (March 16, 2023, Chengdu) · 数论

Find the largest positive integer m which makes it possible to color several cells of a 70 × 70 table red such that - There are no two red cells satisfying: the two rows in which they are have the same number of red cells, while the two columns in which they are also have the same number of red cells; - There are two rows with exactly m red cells each.

题解状态：标准答案与规范题解待补充

题目标签：TST1-DAY3-P9

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TST 1 / Day 4 (March 17, 2023, Chengdu) · 数学竞赛/待细分

The set of nonempty integers A is said to be ”elegant” if it is for any a ∈A, 1 ≤k ≤2023, b j a b ∈A : = 3k 3k k = 2k. Prove that if the intersection of the integer set S and any ”elegant” set is not empty, then S contains an ”elegant” set. n X n X n X > n2 aijkxiyjzk 3 . j=1 i=1 k=1

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题目标签：TST1-DAY4-P10

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TST 1 / Day 4 (March 17, 2023, Chengdu) · 数学竞赛/待细分

Let n ∈N+. For 1 ≤i, j, k ≤n, aijk ∈{−1, 1}. Prove that: ∃x1, x2, · · · , xn, y1, y2, · · · , yn, z1, z2, · · · , zn ∈ {−1, 1}, satisfy Created by Yu Deng √

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题目标签：TST1-DAY4-P11

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TST 1 / Day 4 (March 17, 2023, Chengdu) · 平面几何

Prove that there exists some positive real number λ such that for any D>1 ∈R, one can always find an acute triangle △ABC in the Cartesian plane such that - A, B, C lie on lattice points; - AB, BC, CA > D; - S△ABC < 3 4 D2 + λ · D4/5.

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题目标签：TST1-DAY4-P12

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TST 2 / Day 1 (March 25, 2023, Hangzhou) · 数论

Does there exists a positive irrational number x, such that there are at most finite positive integers n, satisfy that for any integer 1 ≤k ≤n, {kx} ≥ 1 n+1?

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题目标签：TST2-DAY1-P13

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TST 2 / Day 1 (March 25, 2023, Hangzhou) · 数论

For any nonempty, finite set B and real x, define dB(x) = min b∈B |x −b| (1) Given positive integer m. Find the smallest real number λ (possibly depending on m) such that for any positive integer n and any reals x1, · · · , xn ∈[0, 1], there exists an m-element set B of real numbers satisfying dB(x1) + · · · + dB(xn) ≤λn (2) Given positive integer m and positive real ϵ. Prove that there exists a positive integer n and nonnegative reals x1, · · · , xn, satisfying for any m-element set B of real numbers, we have dB(x1) + · · · + dB(xn) > (1 −ϵ)(x1 + · · · + xn)

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题目标签：TST2-DAY1-P14

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TST 2 / Day 1 (March 25, 2023, Hangzhou) · 平面几何

For a convex quadrilateral ABCD, call a point in the interior of ABCD balanced, if (1) P is not on AC, BD (2) Let AP, BP, CP, DP intersect the boundaries of ABCD at A′, B′, C′, D′, respectively, then AP · PA′ = BP · PB′ = CP · PC′ = DP · PD′ Find the maximum possible number of balanced points.

题解状态：标准答案与规范题解待补充

题目标签：TST2-DAY1-P15

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TST 2 / Day 2 (March 26, 2023, Hangzhou) · 平面几何

Let Γ, Γ1, Γ2 be mutually tangent circles. The three circles are also tangent to a line l. Let Γ, Γ1 be tangent to each other at B1, Γ, Γ2 be tangent to each other at B2, Γ1, Γ2 be tangent to each other at C. Γ, Γ1, Γ2 are tangent to l at A, A1, A2 respectively, where A is between A1, A2. Let D1 = A1C ∩A2B2, D2 = A2C ∩A1B1. Prove that D1D2 is parallel to l.

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题目标签：TST2-DAY2-P16

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TST 2 / Day 2 (March 26, 2023, Hangzhou) · 数学竞赛/待细分

Whether there are integers a1, a2, · · · , that are different from each other, satisfying: (1) For ∀k ∈N+, ak2 > 0 and ak2+k < 0; (2) For ∀n ∈N+, |an+1 −an| ⩽2023√n?

题解状态：标准答案与规范题解待补充

题目标签：TST2-DAY2-P17

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TST 2 / Day 2 (March 26, 2023, Hangzhou) · 数学竞赛/待细分

Find the greatest constant λ such that for any doubly stochastic matrix of order 100, we can pick 150 entries such that if the other 9850 entries were replaced by 0, the sum of entries in each row and each column is at least λ. Note: A doubly stochastic matrix of order n is a n × n matrix, all entries are nonnegative reals, and the sum of entries in each row and column is equal to 1.

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题目标签：TST2-DAY2-P18

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TST 2 / Day 3 (March 29, 2023, Hangzhou) · 平面几何

Let A, B be two fixed points on the unit circle ω, satisfying √ 2 < AB < 2. Let P be a point that can move on the unit circle, and it can move to anywhere on the unit circle satisfying △ABP is acute and AP > AB > BP. Let H be the orthocenter of △ABP and S be a point on the minor arc AP satisfying SH = AH. Let T be a point on the minor arc AB satisfying TB||AP. Let ST ∩BP = Q. Show that (recall P varies) the circle with diameter HQ passes through a fixed point.

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题目标签：TST2-DAY3-P19

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TST 2 / Day 3 (March 29, 2023, Hangzhou) · 数论

Let a, b, d be integers such that |a| ⩾2, d ⩾0 and b ⩾(|a| + 1)d+1. For a real coefficient polynomial f of degree d and integer n, let rn denote the residue of [f(n) · an] mod b. If {rn} is eventually periodic, prove that all the coefficients of f are rational.

题解状态：标准答案与规范题解待补充

题目标签：TST2-DAY3-P20

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TST 2 / Day 3 (March 29, 2023, Hangzhou) · 代数

Given integer n ≥2. Find the minimum value of λ, satisfy that for any real numbers a1, a2, · · · , an and b, n X n X n X p p λ ai |ai|. |ai −b| + v u u tn i=1 i=1 i=1 ⩾

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题目标签：TST2-DAY3-P21

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TST 2 / Day 4 (March 30, 2023, Hangzhou) · 代数

Find all functions f : Z →Z, satisfy that for any integer a, b, c, 2f(a2 + b2 + c2) −2f(ab + bc + ca) = f(a −b)2 + f(b −c)2 + f(c −a)2

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题目标签：TST2-DAY4-P22

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TST 2 / Day 4 (March 30, 2023, Hangzhou) · 数论

Given a prime p and a real number λ ∈(0, 1). Let s and t be positive integers such that s ⩽t < λp 12. S and T are sets of s and t consecutive positive integers respectively, which satisfy |{(x, y) ∈S × T : kx ≡y (mod p)}| ⩾1 + λs. Prove that there exists integers a and b that 1 ⩽a ⩽1 λ, |b| ⩽ t λs and ka ≡b (mod p).

题解状态：标准答案与规范题解待补充

题目标签：TST2-DAY4-P23

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TST 2 / Day 4 (March 30, 2023, Hangzhou) · 数论

Let n be a positive integer. Initially, a 2n × 2n grid has k black cells and the rest white cells. The following two operations are allowed : (1) If a 2 × 2 square has exactly three black cells, the fourth is changed to a black cell; (2) If there are exactly two black cells in a 2×2 square, the black cells are changed to white and white to black. Find the smallest positive integer k such that for any configuration of the 2n × 2n grid with k black cells, all cells can be black after a finite number of operations.

题解状态：标准答案与规范题解待补充

题目标签：TST2-DAY4-P24

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