2024 中国国家集训队选拔考试（China TST）

books/competition_archive/china_tst/2024_china_tst.pdf · HS-MATH-1024-v2.1-solution-aware

24 题0 个小问/题组

TST 1 / Day 1 (March 5, 2024, Beijing) · 组合数学

It is known that each vertex of the convex polyhedron P belongs to three different faces, and each vertex of P can be dyed black and white, so that the two endpoints of each edge of P are different colors. Proof: The interior of each edge of P can be dyed red, yellow, and blue, so that the colors of the three edges connected to each vertex are different, and each face contains two colors of edges. Created by Liang Xiao

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题目标签：TST1-DAY1-P1

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TST 1 / Day 1 (March 5, 2024, Beijing) · 平面几何

In acute triangle △ABC, ∠A > ∠B > ∠C. △AC1B and △CB1A are isosceles triangles such that △AC1B +∼△CB1A. Let lines BB1, CC1 intersects at T. Prove that if all points mentioned above are distinct, ∠ATC isn’t a right angle.

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题目标签：TST1-DAY1-P2

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TST 1 / Day 1 (March 5, 2024, Beijing) · 数论

Given positive integer M. For any n ∈N+, let h(n) be the number of elements in [n] that are coprime to M. Define β := h(M) 3 elements n in [M], satisfy M . Proof: there are at least M q |h(n) −βn| ≤ β · 2ω(M)−3 + 1. Here [n] := {1, 2, . . . , n} for all positive integer n. Proposed by Bin Wang

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题目标签：TST1-DAY1-P3

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TST 1 / Day 2 (March 6, 2024, Beijing) · 数论

Let n be a positive square free integer, S is a subset of [n] := {1, 2, . . . , n} such that |S| ≥n/2. Prove that there exists three elements a, b, c ∈S (can be same), satisfy ab ≡c (mod n). Created by Zhenhua Qu

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题目标签：TST1-DAY2-P4

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TST 1 / Day 2 (March 6, 2024, Beijing) · 数论

Find all functions f : N+ →N+, such that for all positive integer a, b, 2b X f(a + k) = (2b + 1)f(f(a) + b). k=0 Created by Liang Xiao, Yunhao Fu

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题目标签：TST1-DAY2-P5

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TST 1 / Day 2 (March 6, 2024, Beijing) · 平面几何

Let m, n > 2 be integers. A regular n-sided polygon region T on a plane contains a regular msided polygon region with a side length of 1. Prove that any regular m-sided polygon region S on the plane with side length cos π/[m, n] can be translated inside T . In other words, there exists a vector⃗α, such that for each point in S, after translating the vector⃗α at that point, it fall into T . Note: The polygonal area includes both the interior and boundaries. Created by Bin Wang

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题目标签：TST1-DAY2-P6

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TST 1 / Day 3 (March 10, 2024, Beijing) · 数论

For coprime positive integers a, b,denote (a−1 mod b) by the only integer 0 ≤m < b such that am ≡1 (mod b) (1)Prove that for pairwise coprime integers a, b, c, 1 < a < b < c,we have (a−1 mod b) + (b−1 mod c) + (c−1 mod a) > √a. (2)Prove that for any positive integer M,there exists pairwise coprime integers a, b, c, M < a < b < c such that (a−1 mod b) + (b−1 mod c) + (c−1 mod a) < 100√a.

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题目标签：TST1-DAY3-P7

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TST 1 / Day 3 (March 10, 2024, Beijing) · 平面几何

In △ABC, tangents of the circumcircle ⊙O at B, C and at A, B intersects at X, Y respectively. AX cuts BC at D and CY cuts AB at F. Ray DF cuts arc AB of the circumcircle at P. Q, R are on segments AB, AC such that P, Q, R are collinear and QR ∥BO. If PQ2 = PR · QR, find ∠ACB.

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题目标签：TST1-DAY3-P8

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TST 1 / Day 3 (March 10, 2024, Beijing) · 数论

Color the positive integers by four colors c1, c2, c3, c4. (1)Prove that there exists a positive integer n and i, j ∈{1, 2, 3, 4},such that among all the positive divisors of n, the number of divisors with color ci is at least greater than the number of divisors with color cj by 3. (2)Prove that for any positive integer A,there exists a positive integer n and i, j ∈{1, 2, 3, 4},such that among all the positive divisors of n, the number of divisors with color ci is at least greater than the number of divisors with color cj by A.

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题目标签：TST1-DAY3-P9

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TST 1 / Day 4 (March 11, Beijing) · 数论

Let M be a positive integer. f(x) := x3 + ax2 + bx + c ∈Z[x] satisfy |a|, |b|, |c| ≤M. x1, x2 are different roots of f. Prove that |x1 −x2| > 1 M2 + 3M + 1. Created by Jingjun Han

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题目标签：TST1-DAY4-P10

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TST 1 / Day 4 (March 11, Beijing) · 数学竞赛/待细分

There is number 1 on the blackboard initially. The first step is to erase 1 and write two nonnegative reals whose sum is 1. Call the smaller number of the two L2. For integer k ≥2, the k the step is to erase a number on the blackboard arbitrarily and write two nonnegative reals whose sum is the number erased just now. Call the smallest number of the k + 1 on the blackboard Lk+1. Find the maximum of L2 + L3 + · · · + L2024.

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题目标签：TST1-DAY4-P11

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TST 1 / Day 4 (March 11, Beijing) · 数论

Given positive odd number m and integer a. Proof: For any real number c, # x ∈Z ∩[c, c + √m] | x2 ≡a (mod m) ≤2 + log2 m. Proposed by Yinghua Ai

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题目标签：TST1-DAY4-P12

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TST 1 / Day 4 (March 11, Beijing) · 数学竞赛/待细分

For a natural number n, let 2n Cn = 1 n + 1 = (2n)! n!(n + 1)! n be the n-th Catalan number. Prove that for any natural number m, X Ci+jCj+kCk+i = 3 2m + 3C2m+1. i+j+k=m Proposed by Bin Wang

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题目标签：TST1-DAY4-P13

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TST 1 / Day 4 (March 11, Beijing) · 数论

For a positive integer n and a subset S of {1, 2, . . . , n}, let S be ”n-good” if and only if for any x, y ∈S (allowed to be same), if x + y ≤n, then x + y ∈S. Let rn be the smallest real number such that for any positive integer m ≤n, there is always a m-element ”n-good” set, so that the sum of its elements is not more than m · rn. Prove that there exists a real number α such that for any positive integer n, |rn −αn| ≤2024.

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题目标签：TST1-DAY4-P14

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TST 1 / Day 4 (March 11, Beijing) · 数论

n > 1 is an integer. Let real number x > 1 satisfy x101 −nx100 + nx −1 = 0. Prove that for any real 0 < a < b < 1, there exists a positive integer m so that a < {xm} < b. Proposed by Chenjie Yu

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题目标签：TST1-DAY4-P15

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TST 1 / Day 4 (March 11, Beijing) · 数论

m > 1 is an integer such that [2m −√m + 1, 2m] contains a prime. Prove that for any pairwise distinct positive integers a1, a2, . . . , am, there is always 1 ≤i, j ≤m such that ai (ai,aj) ≥m.

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题目标签：TST1-DAY4-P16

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TST 1 / Day 4 (March 11, Beijing) · 数学竞赛/待细分

ABCDE is a convex pentagon with BD = CD = AC, and B, C, D, E are concyclic. If ∠BAC + ∠AED = 180◦and ∠DCA = ∠BDE, prove that AB = DE or AB = 2AE.

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题目标签：TST1-DAY4-P17

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TST 1 / Day 4 (March 11, Beijing) · 数学竞赛/待细分

Let m, n ∈Z≥0, a0, a1, . . . , am, b0, b1, . . . , bn ∈R≥0 For any integer 0 ≤k ≤m + n, define ck := maxi+j=k aibj. Proof m+n X m X n X ai bj. 1 m + n + 1 ck ≥ 1 (m + 1)(n + 1) i=0 j=0 k=0 Created by Yinghua Ai

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题目标签：TST1-DAY4-P18

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TST 1 / Day 4 (March 11, Beijing) · 平面几何

n is a positive integer. An equilateral triangle of side length 3n is split into 9n2 unit equilateral triangles, each colored one of red, yellow, blue, such that each color appears 3n2 times. We call a trapezoid formed by three unit equilateral triangles as a ”standard trapezoid”. If a ”standard trapezoid” contains all three colors, we call it a ”colorful trapezoid”. Find the maximum possible number of ”colorful trapezoids”.

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题目标签：TST1-DAY4-P19

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TST 1 / Day 4 (March 11, Beijing) · 数论

A positive integer is a good number, if its base 10 representation can be split into at least 5 sections, each section with a non-zero digit, and after interpreting each section as a positive integer (omitting leading zero digits), they can be split into two groups, such that each group can be reordered to form a geometric sequence (if a group has 1 or 2 numbers, it is also a geometric sequence), for example 20240327 is a good number, since after splitting it as 2|02|403|2|7, 2|02|2 and 403|7 form two groups of geometric sequences. If a > 1, m > 2, p = 1 + a + a2 + · · · + am is a prime, prove that 10p−1−1 p is a good number. n

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题目标签：TST1-DAY4-P20

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TST 1 / Day 4 (March 11, Beijing) · 代数

Let integer n ≥3, nonnegative real numbers ai,j satisfy ai,j + aj,k ≤ai,k holds for all 1 ≤ i < j < k ≤n. Proof   n2 X . i,j i,j ≥   X 1≤i<j≤n a2 1≤i<j≤n a4 Proposed by Jingjun Han, Dongyi Wei

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题目标签：TST1-DAY4-P21

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TST 1 / Day 4 (March 11, Beijing) · 平面几何

ABC is an isosceles triangle, with AB = AC. D is a moving point such that AD ∥BC, BD > CD. Moving point E is on the arc of BC in circumcircle of ABC not containing A, such that EB < EC. Ray BC contains point F with ∠ADE = ∠DFE. If ray FD intersects ray BA at X, and intersects ray CA at Y , prove that ∠XEY is a fixed angle.

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题目标签：TST1-DAY4-P22

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TST 1 / Day 4 (March 11, Beijing) · 代数

P(z) = anzn + · · · + a1z + z0, with an̸ = 0 is a polynomial with complex coefficients, such that when |z| = 1, |P(z)| ≤1. Prove that for any 0 ≤k ≤n −1, |ak| ≤1 −|an|2. Proposed by Yijun Yao

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题目标签：TST1-DAY4-P23

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TST 1 / Day 4 (March 11, Beijing) · 数学竞赛/待细分

Let N = 102024. S is a square in the Cartesian plane with side length N and the sides parallel to the coordinate axes. Inside there are N points P1, P2, . . . , PN all of which have different x coordinates, and the absolute value of the slope of any connected line between these points is at most 1. Prove that there exists a line l such that at least 2024 of these points is at most distance 1 away from l.

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题目标签：TST1-DAY4-P24

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