2013 中国国家集训队选拔考试（China TST）

books/competition_archive/china_tst/2013_china_tst.pdf · HS-MATH-1024-v2.1-solution-aware

18 题0 个小问/题组

TST 1 / Day 1 March 13th · 平面几何

The quadrilateral ABCD is inscribed in circle ω. F is the intersection point of AC and BD. BA and CD meet at E. Let the projection of F on AB and CD be G and H, respectively. Let M and N be the midpoints of BC and EF, respectively. If the circumcircle of △MNG only meets segment BF at P, and the circumcircle of △MNH only meets segment CF at Q, prove that PQ is parallel to BC.

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题目标签：TST1-DAY1-P1

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TST 1 / Day 1 March 13th · 数论

For the positive integer n, define f(n) = min 2 −m n √ . Let {ni} be a strictly increasing sem∈Z for all i ∈{1, 2, . . .}. Show quence of positive integers. C is a constant such that f(ni) < C n2 i that there exists a real number q > 1 such that ni ⩾qi−1 for all i ∈{1, 2, . . .}.

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题目标签：TST1-DAY1-P2

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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 March 13th · 平面几何

There aren balls numbered 1, 2, · · · , n, respectively. They are painted with 4 colours, red, yellow, blue, and green, according to the following rules: First, randomly line them on a circle. Then let any three clockwise consecutive balls numbered i, j, k, in order. 1) If i > j > k, then the ball j is painted in red; 2) If i < j < k, then the ball j is painted in yellow; 3) If i < j, k < j, then the ball j is painted in blue; 4) If i > j, k > j, then the ball j is painted in green. And now each permutation of the balls determine a painting method. We call two painting methods distinct, if there exists a ball, which is painted with two different colours in that two methods. Find out the number of all distinct painting methods.

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题目标签：TST1-DAY1-P3

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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 March 14th · 代数

Let n and k be two integers which are greater than 1. Let a1, a2, . . . , an, c1, c2, . . . , cm be nonnegative real numbers such that i) a1 ≥a2 ≥. . . ≥an and a1 + a2 + . . . + an = 1; ii) For any integer m ∈{1, 2, . . . , n}, we have that c1 + c2 + . . . + cm ≤mk. Find the maximum of c1ak 1 + c2ak 2 + . . . + cnak n.

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题目标签：TST1-DAY2-P1

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TST 1 / Day 2 March 14th · 平面几何

Let P be a given point inside the triangle ABC. Suppose L, M, N are the midpoints of BC, CA, AB respectively and PL : PM : PN = BC : CA : AB. The extensions of AP, BP, CP meet the circumcircle of ABC at D, E, F respectively. Prove that the circumcentres of APF, APE, BPF, BPD, CPD, CPE are concyclic.

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题目标签：TST1-DAY2-P2

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TST 1 / Day 2 March 14th · 代数

Find all positive real numbers r < 1 such that there exists a set S with the given properties: i) For any real number t, exactly one of t, t + r and t + 1 belongs to S; ii) For any real number t, exactly one of t, t −r and t −1 belongs to S.

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题目标签：TST1-DAY2-P3

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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 3 March 18th · 数论

For a positive integer k ≥2 define Tk = {(x, y) | x, y = 0, 1, . . . , k −1} to be a collection of k2 lattice points on the cartesian coordinate plane. Let d1(k) > d2(k) > · · · be the decreasing sequence of the distinct distances between any two points in Tk. Suppose Si(k) be the number of distances equal to di(k). Prove that for any three positive integers m > n > i we have Si(m) = Si(n).

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题目标签：TST1-DAY3-P1

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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 3 March 18th · 数论

Prove that: there exists a positive constant K, and an integer series {an}, satisfying: (1) 0 < a1 < a2 < · · · < an < · · · ; (2) For any positive integer n, an < 1.01nK; (3) For any finite number of distinct terms in {an}, their sum is not a perfect square.

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题目标签：TST1-DAY3-P2

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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 3 March 18th · 平面几何

Let A be a set consisting of 6 points in the plane. denoted n(A) as the number of the unit circles which meet at least three points of A. Find the maximum of n(A)

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题目标签：TST1-DAY3-P3

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TST 1 / Day 4 March 19th · 数论

For a positive integer N > 1 with unique factorization N = pα1 1 pα2 2 · · · pαk k , we define Ω(N) = α1 + α2 + · · · + αk. Let a1, a2, . . . , an be positive integers and p(x) = (x + a1)(x + a2) · · · (x + an) such that for all positive integers k, Ω(P(k)) is even. Show that n is an even number.

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题目标签：TST1-DAY4-P1

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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 4 March 19th · 数论

Find the greatest positive integer m with the following property: For every permutation a1, a2, · · · , an, · · · of the set of positive integers, there exists positive integers i1 < i2 < · · · < im such that ai1, ai2, · · · , aim is an arithmetic progression with an odd common difference. k

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题目标签：TST1-DAY4-P2

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TST 1 / Day 4 March 19th · 代数

Let n > 1 be an integer and let a0, a1, . . . , an be non-negative real numbers. Definite Sk = Pk ai for k = 0, 1, . . . , n. Prove that i=0 i !2 n−1 X n X ≤4 S2 Sk k −1 1 n n2 45(Sn −S0)2. k=0 k=0

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题目标签：TST1-DAY4-P3

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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 5 March 24th · 数论

Let n ≥2 be an integer. a1, a2, . . . , an are arbitrarily chosen positive integers with (a1, a2, . . . , an) = 1. Let A = a1 + a2 + · · · + an and (A, ai) = di. Let (a2, a3, . . . , an) = D1, (a1, a3, . . . , an) = D2, . . . , (a1, a2, . . . , an−1) = Dn. A −ai nQ Find the minimum of diDi i=1

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题目标签：TST1-DAY5-P1

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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 5 March 24th · 平面几何

The circumcircle of triangle ABC has centre O. P is the midpoint of \ BAC and QP is the diameter. Let I be the incentre of △ABC and let D be the intersection of PI and BC. The circumcircle of △AID and the extension of PA meet at F. The point E lies on the line segment PD such that DE = DQ. Let R, r be the radius of the inscribed circle and circumcircle of △ABC, respectively. Show that if ∠AEF = ∠APE, then sin2 ∠BAC = 2r R

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题目标签：TST1-DAY5-P2

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TST 1 / Day 5 March 24th · 数论

101 people, sitting at a round table in any order, had 1, 2, ..., 101 cards, respectively. A transfer is someone give one card to one of the two people adjacent to him. Find the smallest positive integer k such that there always can through no more than k times transfer, each person hold cards of the same number, regardless of the sitting order.

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题目标签：TST1-DAY5-P3

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TST 1 / Day 6 March 25th · 数论

Let p be a prime number and a, k be positive integers such that pa < k < 2pa. Prove that there exists a positive integer n such that n < p2a, Ck n ≡n ≡k (mod pa). ! !2 n−1 n X n X n Y ≥ a2 + (a2 1 n . bi i i + b2 i )

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题目标签：TST1-DAY6-P1

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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 6 March 25th · 代数

Let k ≥2 be an integer and let a1, a2, · · · , an, b1, b2, · · · , bn be non-negative real numbers. Prove that n n −1 1 n 1 n i=1 i=1 i=1

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题目标签：TST1-DAY6-P2

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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 6 March 25th · 平面几何

A point (x, y) is a lattice point if x, y ∈Z. Let E = {(x, y) : x, y ∈Z}. In the coordinate plane, P and Q are both sets of points in and on the boundary of a convex polygon with vertices on lattice points. Let T = P ∩Q. Prove that if T̸ = ∅and T ∩E = ∅, then T is a non-degenerate convex quadrilateral region.

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题目标签：TST1-DAY6-P3

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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。