2011 中国国家集训队选拔考试（China TST）

books/competition_archive/china_tst/2011_china_tst.pdf · HS-MATH-1024-v2.1-solution-aware

18 题0 个小问/题组

Quiz 1 / Day 1 · 平面几何

In △ABC we have BC > CA > AB. The nine point circle is tangent to the incircle, A-excircle, B-excircle and C-excircle at the points T, TA, TB, TC respectively. Prove that the segments TTB and lines TATC intersect each other. i ≤n3 −n2. i=1 m2

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Quiz 1 / Day 1 · 数学竞赛/待细分

Let S be a set of n points in the plane such that no four points are collinear. Let {d1, d2, · · · , dk} be the set of distances between pairs of distinct points in S, and let mi be the multiplicity of di, i.e. the number of unordered pairs {P, Q} ⊆S with |PQ| = di. Prove that Pk

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题目标签：QUIZ1-DAY1-P2

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Quiz 1 / Day 1 · 数论

A positive integer n is known as an interesting number if n satisfies { n 10k } > n 1010 for all k = 1, 2, . . . 9. Find the number of interesting numbers.

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题目标签：QUIZ1-DAY1-P3

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Quiz 1 / Day 2 · 平面几何

Let one of the intersection points of two circles with centres O1, O2 be P. A common tangent touches the circles at A, B respectively. Let the perpendicular from A to the line BP meet O1O2 at C. Prove that AP ⊥PC.

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Quiz 1 / Day 2 · 数论

Let n be a positive integer and let αn be the number of 1’s within binary representation of n. Show that for all positive integers r, 2n n X k2r. n + k 22n−αn k=−n

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Quiz 1 / Day 2 · 数学竞赛/待细分

For a given integer n ≥2, let a0, a1, . . . , an be integers satisfying 0 = a0 < a1 < . . . < an = 2n−1. Find the smallest possible number of elements in the set {ai + aj | 0 ≤i ≤j ≤n}.

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Quiz 2 / Day 1 · 代数

Let n ≥2 be a given integer. Find all functions f : R →R such that f(x −f(y)) = f(x + yn) + f(f(y) + yn), ∀x, y ∈R.

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Quiz 2 / Day 1 · 数论

Let ℓbe a positive integer, and let m, n be positive integers with m ≥n, such that A1, A2, · · · , Am, B1, · · · , Bm are m + n pairwise distinct subsets of the set {1, 2, · · · , ℓ}. It is known that Ai∆Bj are pairwise distinct, 1 ≤i ≤m, 1 ≤j ≤n, and runs over all nonempty subsets of {1, 2, · · · , ℓ}. Find all possible values of m, n.

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Quiz 2 / Day 1 · 数论

For any positive integer d, prove there are infinitely many positive integers n such that d(n!) −1 is a composite number.

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题目标签：QUIZ2-DAY1-P3

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Quiz 2 / Day 2 · 平面几何

Let AA′, BB′, CC′ be three diameters of the circumcircle of an acute triangle ABC. Let P be an arbitrary point in the interior of △ABC, and let D, E, F be the orthogonal projection of P on BC, CA, AB, respectively. Let X be the point such that D is the midpoint of A′X, let Y be the point such that E is the midpoint of B′Y , and similarly let Z be the point such that F is the midpoint of C′Z. Prove that triangle XY Z is similar to triangle ABC.

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题目标签：QUIZ2-DAY2-P1

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Quiz 2 / Day 2 · 数论

Let {bn}∞ n≥1 be a sequence of positive integers. The sequence {an}∞ n≥1 is defined as follows: a1 is a fixed positive integer and an+1 = abn n + 1, ∀n ≥1. Find all positive integers m ≥3 with the following property: If the sequence {an mod m}∞ n≥1 is eventually periodic, then there exist positive integers q, u, v with 2 ≤q ≤m −1, such that the sequence {bv+ut mod q}∞ t≥1 is purely periodic.

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Quiz 2 / Day 2 · 代数

Let n be a positive integer. Find the largest real number λ such that for all positive real numbers x1, x2, · · · , x2n satisfying the inequality 2n X 2n Y xi, (xi + 2)n ≥ 1 2n i=1 i=1 the following inequality also holds 2n X 2n Y xi. (xi + 1)n ≥λ 1 2n i=1 i=1

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Quiz 3 / Day 1 · 代数

Let n ≥3 be an integer. Find the largest real number M such that for any positive real numbers x1, x2, · · · , xn, there exists an arrangement y1, y2, · · · , yn of real numbers satisfying y2 n X ≥M, i y2 i+2 i+1 −yi+1yi+2 + y2 i=1 where yn+1 = y1, yn+2 = y2.

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Quiz 3 / Day 1 · 数论

Let n > 1 be an integer, and let k be the number of distinct prime divisors of n. Prove that there exists an integer a, 1 < a < n k + 1, such that n | a2 −a.

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Quiz 3 / Day 1 · 组合数学

Let G be a simple graph with 3n2 vertices (n ≥2). It is known that the degree of each vertex of G is not greater than 4n, there exists at least a vertex of degree one, and between any two vertices, there is a path of length ≤3. Prove that the minimum number of edges that G might have is equal to (7n2−3n) 2 .

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Quiz 3 / Day 2 · 平面几何

Let H be the orthocenter of an acute trangle ABC with circumcircle Γ. Let P be a point on the arc BC (not containing A) of Γ, and let M be a point on the arc CA (not containing B) of Γ such that H lies on the segment PM. Let K be another point on Γ such that KM is parallel to the Simson line of P with respect to triangle ABC. Let Q be another point on Γ such that PQ ∥BC. Segments BC and KQ intersect at a point J. Prove that △KJM is an isosceles triangle.

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Quiz 3 / Day 2 · 数论

Let a1, a2, . . . , an, . . . be any permutation of all positive integers. Prove that there exist infinitely many positive integers i such that gcd(ai, ai+1) ≤3 4i.

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题目标签：QUIZ3-DAY2-P2

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Quiz 3 / Day 2 · 代数

Let m and n be positive integers. A sequence of points (A0, A1, . . . , An) on the Cartesian plane is called interesting if Ai are all lattice points, the slopes of OA0, OA1, · · · , OAn are strictly increasing (O is the origin) and the area of triangle OAiAi+1 is equal to 1 2 for i = 0, 1, . . . , n −1. Let (B0, B1, · · · , Bn) be a sequence of points. We may insert a point B between Bi and Bi+1 if −−→ OB = −−→ OBi + −−−−→ OBi+1, and the resulting sequence (B0, B1, . . . , Bi, B, Bi+1, . . . , Bn) is called an extension of the original sequence. Given two interesting sequences (C0, C1, . . . , Cn) and (D0, D1, . . . , Dm), prove that if C0 = D0 and Cn = Dm, then we may perform finitely many extensions on each sequence until the resulting two sequences become identical.

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题目标签：QUIZ3-DAY2-P3

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