2018 中国国家集训队选拔考试（China TST）

books/competition_archive/china_tst/2018_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

Let p, q be positive reals with sum 1. Show that for any n-tuple of reals (y1, y2, ..., yn), there exists an n-tuple of reals (x1, x2, ..., xn) satisfying p · max{xi, xi+1} + q · min{xi, xi+1} = yi for all i = 1, 2, ..., 2017, where x2018 = x1.

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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

A number n is interesting if 2018 divides d(n) (the number of positive divisors of n). Determine all positive integers k such that there exists an infinite arithmetic progression with common difference k whose terms are all interesting.

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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

Circle ω is tangent to sides AB,AC of triangle ABC at D,E respectively, such that D̸ = B, E̸ = C and BD + CE < BC. F,G lies on BC such that BF = BD, CG = CE. Let DG and EF meet at K. L lies on minor arc DE of ω, such that the tangent of L to ω is parallel to BC. Prove that the incenter of △ABC lies on KL.

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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 · 代数

Functions f, g : Z →Z satisfy f(g(x) + y) = g(f(y) + x) for any integers x, y. If f is bounded, prove that g is periodic.

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 · 数论

Given a positive integer k, call n good if among n n n n , , , ..., n at least 0.99n of them are divisible by k. Show that exists some positive integer N such that among 1, 2, ..., N, there are at least 0.99N good numbers.

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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 A1, A2, · · · , Am be m subsets of a set of size n. Prove that !3 m X m X m X . |Ai| |Ai| · |Ai ∩Aj| ≥ 1 mn i=1 j=1 i=1

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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

Given a triangle ABC. D is a moving point on the edge BC. Point E and Point F are on the edge AB and AC, respectively, such that BE = CD and CF = BD. The circumcircle of △BDE and △CDF intersects at another point P other than D. Prove that there exists a fixed point Q, such that the length of QP is constant.

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 · 数论

An integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. For example, 4 can be partitioned in five distinct ways: 4 3 + 1 2 + 2 2 + 1 + 1 1 + 1 + 1 + 1 The number of partitions of n is given by the partition function p (n). So p (4) = 5 . Determine all the positive integers so that p (n) + p (n + 4) = p (n + 2) + p (n + 3).

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

Two positive integers p, q ∈Z+ are given. There is a blackboard with n positive integers written on it. A operation is to choose two same number a, a written on the blackboard, and replace them with a + p, a + q. Determine the smallest n so that such operation can go on infinitely.

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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

Let k, M be positive integers such that k −1 is not squarefree. Prove that there exist a positive real α, such that ⌊α · kn⌋and M are coprime for any positive integer n.

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 positive integers n, k such that n ≥4k, find the minimal value λ = λ(n, k) such that for any positive reals a1, a2, . . . , an, we have n X ≤λ ai q a2 i=1 i+k i + a2 i+1 + · · ·+a2 Where an+i = ai, i = 1, 2, . . . , 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 · 代数

Let M, a, b, r be non-negative integers with a, r ≥2, and suppose there exists a function f : Z →Z satisfying the following conditions: (1) For all n ∈Z, f(r)(n) = an + b where f(r) denotes the composition of r copies of f (2) For all n ≥M, f(n) ≥0 (3) For all n > m > M, n −m|f(n) −f(m) Show that a is a perfect r-th power.

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

Let ω1, ω2 be two non-intersecting circles, with circumcenters O1, O2 respectively, and radii r1, r2 respectively where r1 < r2. Let AB, XY be the two internal common tangents of ω1, ω2, where A, X lie on ω1, B, Y lie on ω2. The circle with diameter AB meets ω1, ω2 at P and Q respectively. If ∠AO1P + ∠BO2Q = 180◦, find the value of PX QY (in terms of r1, r2).

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 · 组合数学

Let G be a simple graph with 100 vertices such that for each vertice u, there exists a vertice v ∈N (u) and N (u) ∩N (v) = ø. Try to find the maximal possible number of edges in G. The N (.) refers to the neighborhood.

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

Prove that there exists a constant C > 0 such that m X H(a1) + H(a2) + · · · + H(am) ≤C iai v u u t i=1 holds for arbitrary positive integer m and any m positive integer a1, a2, · · · , am, where n X H(n) = 1 k. k=1

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

Suppose A1, A2, · · · , An ⊆{1, 2, · · · , 2018} and |Ai| = 2, i = 1, 2, · · · , n, satisfying that Ai + Aj, 1 ≤i ≤j ≤n, are distinct from each other. A + B = {a + b|a ∈A, b ∈B}. Determine the maximal value of n.

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

Let ABC be a triangle with ∠BAC > 90◦, and let O be its circumcenter and ω be its circumcircle. The tangent line of ω at A intersects the tangent line of ω at B and C respectively at point P and Q. Let D, E be the feet of the altitudes from P, Q onto BC, respectively. F, G are two points on PQ different from A, so that A, F, B, E and A, G, C, D are both concyclic. Let M be the midpoint of DE. Prove that DF, OM, EG are concurrent.

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 · 数论

Find all pairs of positive integers (x, y) such that (xy + 1)(xy + x + 2) be 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-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 · 代数

Define the polymonial sequence {fn (x)}n≥1 with f1 (x) = 1, f2n (x) = xfn (x) , f2n+1 (x) = fn (x) + fn+1 (x) , n ≥1. Look for all the rational number a which is a root of certain fn (x) .

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 · 数学竞赛/待细分

There are 32 students in the class with 10 interesting group. Each group contains exactly 16 students. For each couple of students, the square of the number of the groups which are only involved by just one of the two students is defined as their interests −disparity. Define S as the sum of the interests −disparity of all the couples, (= 496) ones in total. Determine the minimal possible value of S.

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

In isosceles △ABC, AB = AC, points D, E, F lie on segments BC, AC, AB such that DE ∥ AB, DF ∥AC. The circumcircle of △ABC ω1 and the circumcircle of △AEF ω2 intersect at A, G. Let DE meet ω2 at K̸ = E. Points L, M lie on ω1, ω2 respectively such that LG ⊥ KG, MG ⊥CG. Let P, Q be the circumcenters of △DGL and △DGM respectively. Prove that A, G, P, Q are concyclic.

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

Let p be a prime and k be a positive integer. Set S contains all positive integers a satisfying 1 ≤a ≤p −1, and there exists positive integer x such that xk ≡a (mod p). Suppose that 3 ≤|S| ≤p −2. Prove that the elements of S, when arranged in increasing order, does not form an arithmetic progression.

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

Suppose the real number λ ∈(0, 1) , and let n be a positive integer. Prove that the modulus of all the roots of the polynomial n n X f (x) = λk(n−k)xk k k=0 are 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 · 代数

Suppose ai, bi, ci, i = 1, 2, · · · , n, are 3n real numbers in the interval [0, 1] . Define S = {(i, j, k) | ai + bj + ck < 1} , T = {(i, j, k) | ai + bj + ck > 2} . Now we know that |S| ≥2018, |T| ≥2018. Try to find the minimal possible value of n.

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。