2009 中国国家集训队选拔考试（China TST）

books/competition_archive/china_tst/2009_china_tst.pdf · HS-MATH-1024-v2.1-solution-aware

24 题0 个小问/题组

未标注场次 · 平面几何

Let ABC be a triangle. Point D lies on its sideline BC such that ∠CAD = ∠CBA. Circle (O) passing through B, D intersects AB, AD at E, F, respectively. BF meets DE at G.Denote byM the midpoint of AG. Show that CM ⊥AO.

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题目标签：UNLABELED-P1

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未标注场次 · 代数

Given an integer n ≥2, find the maximal constant λ(n) having the following property: if a sequence of real numbers a0, a1, a2, · · · , an satisfies 0 = a0 ≤a1 ≤a2 ≤· · · ≤an, and ai ≥1 i . i=1 a2 2(ai+1 + ai−1), i = 1, 2, · · · , n −1, then (Pn i=1 iai)2 ≥λ(n) Pn

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题目标签：UNLABELED-P2

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未标注场次 · 数论

Prove that for any odd prime number p, the number of positive integer n satisfying p|n! + 1 is less than or equal to cp 2 3 . where c is a constant independent of p.

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题目标签：UNLABELED-P3

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未标注场次 · 代数

Let positive real numbers a, b satisfy b−a > 2. Prove that for any two distinct integers m, n belonging to [a, b), there always exists non-empty set S consisting of certain integers belonging Y x∈S to [ab, (a + 1)(b + 1)) such that mn is square of a rational number. Pn

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题目标签：UNLABELED-P4

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未标注场次 · 组合数学

Let m > 1 be an integer, n is an odd number satisfying 3 ≤n < 2m, number ai,j(i, j ∈N, 1 ≤ i ≤m, 1 ≤j ≤n) satisfies (1) for any 1 ≤j ≤n, a1,j, a2,j, · · · , am,j is a permutation of 1, 2, 3, · · · , m; (2) for any 1 < i ≤m, 1 ≤j ≤n −1, |ai,j −ai,j+1| ≤1 holds. Find the minimal value of M, where M = max1<i<m j=1 ai,j.

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题目标签：UNLABELED-P5

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未标注场次 · 数学竞赛/待细分

Determine whether there exists an arithimethical progression consisting of 40 terms and each of whose terms can be written in the form 2m +3n or not. where m, n are nonnegative integers.

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题目标签：UNLABELED-P6

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Quiz 1 · 平面几何

Given that circle ω is tangent internally to circle Γ at S. ω touches the chord AB of Γ at T. Let O be the center of ω. Point P lies on the line AO. Show that PB ⊥AB if and only if PS ⊥TS.

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题目标签：QUIZ1-P1

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Quiz 1 · 数论

Let n, k be given positive integers satisfying k ≤2n−1. On a table tennis tournament 2n players take part, they play a total of k rounds match, each round is divided into n groups, each group two players match. The two players in different rounds can match on many occasions. Find the greatest positive integer m = f(n, k) such that no matter how the tournament processes, we always find m players each of pair of which didn’t match each other.

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题目标签：QUIZ1-P2

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Quiz 1 · 代数

Let x1, x2, · · · , xm, y1, y2, · · · , yn be positive real numbers. Denote by X = Pm i=1 x, Y = Pn Pn Pn Pm i=1 j=1 i=1 k=1 |xi −xk| j=1 |xi −yj| ≥X2 Pn l=1 |yi −yl| + Y 2 Pm j=1 y. Prove that 2XY Pm

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题目标签：QUIZ1-P3

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Quiz 2 · 数学竞赛/待细分

In convex pentagon ABCDE, denote by AD ∩BE = F, BE ∩CA = G, CA ∩DB = H, DB ∩ EC = I, EC ∩AD = J; AI ∩BE = A′, BJ = B′, CF = C′, DG ∩EC = D′, EH ∩AD = E′. Prove that AB′ B′C · CD′ D′E · EA′ A′B · BC′ C′D · DE′ E′A = 1.

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题目标签：QUIZ2-P1

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Quiz 2 · 组合数学

Find all the pairs of integers (a, b) satisfying ab(a −b)̸ = 0 such that there exists a subset Z0 of set of integers Z, for any integer n, exactly one among three integers n, n + a, n + b belongs to Z0.

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题目标签：QUIZ2-P2

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Quiz 2 · 代数

Consider function f : R →R which satisfies the conditions for any mutually distinct real numbers a, b, c, d satisfying a−b b−c + a−d d−c = 0, f(a), f(b), f(c), f(d) are mutully different and f(a)−f(b) f(b)−f(c) + f(a)−f(d) f(d)−f(c) = 0. Prove that function f is linear

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题目标签：QUIZ2-P3

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Quiz 3 · 组合数学

Let α, β be real numbers satisfying 1 < α < β. Find the greatest positive integer r having the following property: each of positive integers is colored by one of r colors arbitrarily, there always exist two integers x, y having the same color such that α ≤x y ≤β.

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题目标签：QUIZ3-P1

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Quiz 3 · 平面几何

In convex quadrilateral ABCD, CB, DA are external angle bisectors of ∠DCA, ∠CDB, respectively. Points E, F lie on the rays AC, BD respectively such that CEFD is cyclic quadrilateral. Point P lie in the plane of quadrilateral ABCD such that DA, CB are external angle bisectors of ∠PDE, ∠PCF respectively. AD intersects BC at Q. Prove that P lies on AB if and only if Q lies on segment EF.

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题目标签：QUIZ3-P2

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Quiz 3 · 代数

Let f(x) be a n−degree polynomial all of whose coefficients are equal to ±1, and having x = 1 as its m multiple root. If m ≥2k(k ≥2, k ∈N), then n ≥2k+1 −1.

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题目标签：QUIZ3-P3

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Quiz 4 · 平面几何

Given that points D, E lie on the sidelines AB, BC of triangle ABC, respectively, point P is in interior of triangle ABC such that PE = PC and △DEP ∼△PCA. Prove that BP is tangent of the circumcircle of triangle PAD.

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题目标签：QUIZ4-P1

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Quiz 4 · 数论

Find all integers n ≥2 having the following property: for any k integers a1, a2, · · · , ak which aren’t congruent to each other (modulo n), there exists an integer polynomial f(x) such that congruence equation f(x) ≡0(modn) exactly has k roots x ≡a1, a2, · · · , ak(modn).

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题目标签：QUIZ4-P2

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Quiz 4 · 组合数学

Let X be a set containing 2k elements, F is a set of subsets of X consisting of certain k elements such that any one subset of X consisting of k −1 elements is exactly contained in an element of F. Show that k + 1 is a prime number.

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题目标签：QUIZ4-P3

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Quiz 5 · 数论

Let n be a composite. Prove that there exists positive integer m satisfying m|n, m ≤√n, and d(n) ≤d3(m). Where d(k) denotes the number of positive divisors of positive integer k.

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题目标签：QUIZ5-P1

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Quiz 5 · 平面几何

In acute triangle ABC, points P, Q lie on its sidelines AB, AC, respectively. The circumcircle of triangle ABC intersects of triangle APQ at X (different from A). Let Y be the reflection of X in line PQ. Given PX > PB. Prove that S△XPQ > S△Y BC. Where S△XY Z denotes the area of triangle XY Z. a2

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题目标签：QUIZ5-P2

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Quiz 5 · 代数

Let nonnegative real numbers a1, a2, a3, a4 satisfy a1+a2+a3+a4 = 1. Prove that max{P4 1 q i + aiai−1 + a 2. Where for all integers i, ai+4 = ai holds.

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题目标签：QUIZ5-P3

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Quiz 6 · 数论

Let a > b > 1, b is an odd number, let n be a positive integer. If bn|an −1, then ab > 3n n .

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题目标签：QUIZ6-P1

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Quiz 6 · 代数

Find all complex polynomial P(x) such that for any three integers a, b, c satisfying a + b + c̸ = 0, P(a)+P(b)+P(c) a+b+c is an integer.

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题目标签：QUIZ6-P2

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Quiz 6 · 数论

Let (an)n≥1 be a sequence of positive integers satisfying (am, an) = a(m,n) (for all m, n ∈N+). µ( n d|n a Prove that for any n ∈N+, Q d ) d is an integer. where d|n denotes d take all positive divisors of n. Function µ(n) is defined as follows: if n can be divided by square of certain prime number, then µ(1) = 1; µ(n) = 0; if n can be expressed as product of k different prime numbers, then µ(n) = (−1)k.

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题目标签：QUIZ6-P3

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