2016 中国国家集训队选拔考试（China TST）

books/competition_archive/china_tst/2016_china_tst.pdf · HS-MATH-1024-v2.1-solution-aware

18 题0 个小问/题组

TST 1 / Day 1 · 数学竞赛/待细分

ABCDEF is a cyclic hexagon with AB = BC = CD = DE. K is a point on segment AE satisfying ∠BKC = ∠KFE, ∠CKD = ∠KFA. Prove that KC = KF.

题解状态：标准答案与规范题解待补充

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 · 代数

Find the smallest positive number λ , such that for any complex numbers z1, z2, z3 ∈{z ∈ C |z| < 1} ,if z1 + z2 + z3 = 0, then |z1z2 + z2z3 + z3z1|2 + |z1z2z3|2 < λ.

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 · 组合数学

Let n ≥2 be a natural. Define X = {(a1, a2, · · · , an)|ak ∈{0, 1, 2, · · · , k}, k = 1, 2, · · · , n} . For any two elements s = (s1, s2, · · · , sn) ∈X, t = (t1, t2, · · · , tn) ∈X, define s ∨t = (max{s1, t1}, max{s2, t2}, · · · , max{sn, tn}) s ∧t = (min{s1, t1}, min{s2, t2, }, · · · , min{sn, tn}) Find the largest possible size of a proper subset A of X such that for any s, t ∈A, one has s ∨t ∈A, s ∧t ∈A.

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

Let c, d ≥2 be naturals. Let {an} be the sequence satisfying a1 = c, an+1 = ad n + c for n = 1, 2, · · · . Prove that for any n ≥2, there exists a prime number p such that p|an and p̸ |ai for i = 1, 2, · · · n −1.

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 · 平面几何

Refer to the diagram below. Let ABCD be a cyclic quadrilateral with center O. Let the internal angle bisectors of ∠A and ∠C intersect at I and let those of ∠B and ∠D intersect at J. Now extend AB and CD to intersect IJ and P and R respectively and let IJ intersect BC and DA at Q and S respectively. Let the midpoints of PR and QS be M and N respectively. Given that O does not lie on the line IJ, show that OM and ON are perpendicular.

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, n be naturals satisfying n ≥m ≥2 and let S be a set consisting of n naturals. Prove that S has at least 2n−m+1 distinct subsets, each whose sum is divisible by m. (The zero set counts as a subset).

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 · 平面几何

P is a point in the interior of acute triangle ABC. D, E, F are the reflections of P across BC, CA, AB respectively. Rays AP, BP, CP meet the circumcircle of △ABC at L, M, N respectively. Prove that the circumcircles of △PDL, △PEM, △PFN meet at a point T different from P.

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

Find the smallest positive number λ, such that for any 12 points on the plane P1, P2, . . . , P12(can overlap), if the distance between any two of them does not exceed 1, then P 1≤i<j≤12 |PiPj|2 ≤ λ.

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 P be a finite set of primes, A an infinite set of positive integers, where every element of A has a prime factor not in P. Prove that there exist an infinite subset B of A, such that the sum of elements in any finite subset of B has a prime factor not in P.

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

Set positive integer m = 2k · t, where k is a non-negative integer, t is an odd number, and let f(m) = t1−k. Prove that for any positive integer n and for any positive odd number a ≤n, Qn m=1 f(m) is a multiple of a.

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

Does there exist two infinite positive integer sets S, T, such that any positive integer n can be uniquely expressed in the form n = s1t1 + s2t2 + . . . + sktk ,where k is a positive integer dependent on n, s1 < . . . < sk are elements of S, t1, . . . , tk are elements of T?

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 · 平面几何

The diagonals of a cyclic quadrilateral ABCD intersect at P, and there exist a circle Γ tangent to the extensions of AB, BC, AD, DC at X, Y, Z, T respectively. Circle Ωpasses through points A, B, and is externally tangent to circle Γ at S. Prove that SP ⊥ST.

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 · 数学竞赛/待细分

Let n be an integer greater than 1, α is a real, 0 < α < 2, a1, . . . , an, c1, . . . , cn are all positive numbers. For y > 0, let   α ! 1 X f(y) = 1 2 + . cia2 ciaα i i  X ai>y ai≤y If positive number x satisfies x ≥f(y) (for some y), prove that f(x) ≤8 1 α · x.

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

In the coordinate plane the points with both coordinates being rational numbers are called rational points. For any positive integer n, is there a way to use n colours to colour all rational points, every point is coloured one colour, such that any line segment with both endpoints being rational points contains the rational points of every colour?

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 · 平面几何

In cyclic quadrilateral ABCD, AB > BC, AD > DC, I, J are the incenters of △ABC,△ADC respectively. The circle with diameter AC meets segment IB at X, and the extension of JD at Y . Prove that if the four points B, I, J, D are concyclic, then X, Y are the reflections of each other across AC.

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 · 数论

Let a, b, b′, c, m, q be positive integers, where m > 1, q > 1, |b −b′| ≥a. It is given that there exist a positive integer M such that Sq(an + b) ≡Sq(an + b′) + c (mod m) holds for all integers n ≥M. Prove that the above equation is true for all positive integers n. (Here Sq(x) is the sum of digits of x taken in base q).

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 S be a finite set of points on a plane, where no three points are collinear, and the convex hull of S, Ω, is a 2016−gon A1A2 . . . A2016. Every point on S is labelled one of the four numbers ±1, ±2, such that for i = 1, 2, . . . , 1008, the numbers labelled on points Ai and Ai+1008 are the negative of each other. Draw triangles whose vertices are in S, such that any two triangles do not have any common interior points, and the union of these triangles is Ω. Prove that there must exist a triangle, where the numbers labelled on some two of its vertices are the negative of each other.

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 · 平面几何

Find all functions f : R+ →R+ satisfying the following condition: for any three distinct real numbers a, b, c, a triangle can be formed with side lengths a, b, c, if and only if a triangle can be formed with side lengths f(a), f(b), f(c).

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。