2008 中国国家集训队选拔考试（China TST）

books/competition_archive/china_tst/2008_china_tst.pdf · HS-MATH-1024-v2.1-solution-aware

24 题0 个小问/题组

TST 1 / Day 1 · 平面几何

Let ABC be a triangle, let AB > AC. Its incircle touches side BC at point E. Point D is the second intersection of the incircle with segment AE (different from E). Point F (different from E) is taken on segment AE such that CE = CF. The ray CF meets BD at point G. Show that CF = FG.

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题目标签：TST1-DAY1-P1

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TST 1 / Day 1 · 数论

The sequence {xn} is defined by x1 = 2, x2 = 12, and xn+2 = 6xn+1 −xn, (n = 1, 2, . . .). Let p be an odd prime number, let q be a prime divisor of xp. Prove that if q̸ = 2, 3, then q ≥2p −1.

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题目标签：TST1-DAY1-P2

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TST 1 / Day 1 · 数论

Suppose that every positve integer has been given one of the colors red, blue,arbitrarily. Prove that there exists an infinite sequence of positive integers a1 < a2 < a3 < · · · < an < · · · , such that inifinite sequence of positive integers a1, a1+a2 2 , a2, a2+a3 2 , a3, a3+a4 2 , · · · has the same color.

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题目标签：TST1-DAY1-P3

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TST 1 / Day 2 · 组合数学

Prove that for arbitary positive integer n ≥4, there exists a permutation of the subsets that contain at least two elements of the set Gn = {1, 2, 3, · · · , n}: P1, P2, · · · , P2n−n−1 such that |Pi ∩Pi+1| = 2, i = 1, 2, · · · , 2n −n −2.

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题目标签：TST1-DAY2-P4

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TST 1 / Day 2 · 数论

For two given positive integers m, n > 1, let aij(i = 1, 2, · · · , n, j = 1, 2, · · · , m) be nonnegative real numbers, not all zero, find the maximum and the minimum values of f, where i=1(Pm j=1(Pn i=1 aij)2 n Pn j=1 aij)2 + m Pm . f = Pm Pm (Pn i=1 i=1 ij j=1 a2 j=1 aij)2 + mn Pn

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题目标签：TST1-DAY2-P5

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TST 1 / Day 2 · 代数

Find the maximal constant M, such that for arbitrary integer n ≥3, there exist two sequences of positive real number a1, a2, · · · , an, and b1, b2, · · · , bn, satisfying (1):Pn k=1 bk = 1, 2bk ≥bk−1 + bk+1, k = 2, 3, · · · , n −1; (2):a2 i=1 aibi, k = 1, 2, 3, · · · , n, an ≡M. k ≤1 + Pk

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题目标签：TST1-DAY2-P6

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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。

Quiz 1 · 平面几何

Let P be an arbitrary point inside triangle ABC, denote by A1 (different from P) the second intersection of line AP with the circumcircle of triangle PBC and define B1, C1 similarly. Prove that ≥8. 1 + 2 · PA PA1 1 + 2 · PB PB1 1 + 2 · PC PC1

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题目标签：QUIZ1-P1

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Quiz 1 · 数论

Let n > 1 be an integer, and n can divide 2φ(n)+3φ(n)+· · ·+nφ(n), let p1, p2, · · · , pk be all distinct prime divisors of n. Show that 1 p1 + 1 p2 + · · · + 1 pk + 1 p1p2···pk is an integer. ( where φ(n) is defined as the number of positive integers ≤n that are relatively prime to n.)

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题目标签：QUIZ1-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。

Quiz 1 · 平面几何

Determine the greatest positive integer n such that in three-dimensional space, there exist n points P1, P2, · · · , Pn, among n points no three points are collinear, and for arbitary 1 ≤i < j < k ≤n, PiPjPk isn’t obtuse triangle.

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题目标签：QUIZ1-P3

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Quiz 2 · 平面几何

Let ABC be a triangle, line l cuts its sides BC, CA, AB at D, E, F, respectively. Denote by O1, O2, O3 the circumcenters of triangle AEF, BFD, CDE, respectively. Prove that the orthocenter of triangle O1O2O3 lies on line l.

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题目标签：QUIZ2-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。

Quiz 2 · 平面几何

In a plane, there is an infinite triangular grid consists of equilateral triangles whose lengths of the sides are equal to 1, call the vertices of the triangles the lattice points, call two lattice points are adjacent if the distance between the two points is equal to 1; A jump game is played by two frogs A, B, ”A jump” is called if the frogs jump from the point which it is lying on to its adjacent point, ” A round jump of A, B” is called if first A jumps and then B by the following rules: Rule (1): A jumps once arbitrarily, then B jumps once in the same direction, or twice in the opposite direction; Rule (2): when A, B sits on adjacent lattice points, they carry out Rule (1) finishing a round jump, or A jumps twice continually, keep adjacent with B every time, and B rests on previous position; If the original positions of A, B are adjacent lattice points, determine whether for A and B,such that the one can exactly land on the original position of the other after a finite round jumps.

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题目标签：QUIZ2-P2

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Quiz 2 · 代数

Let z1, z2, z3 be three complex numbers of moduli less than or equal to 1. w1, w2 are two roots of the equation (z −z1)(z −z2)+(z −z2)(z −z3)+(z −z3)(z −z1) = 0. Prove that, for j = 1, 2, 3, min{|zj −w1|, |zj −w2|} ≤1 holds.

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题目标签：QUIZ2-P3

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Quiz 3 · 平面几何

Let P be the the isogonal conjugate of Q with respect to triangle ABC, and P, Q are in the interior of triangle ABC. Denote by O1, O2, O3 the circumcenters of triangle PBC, PCA, PAB, O′ 1, O′ 2, O′ 3 the circumcenters of triangle QBC, QCA, QAB, O the circumcenter of triangle O1O2O3, O′ the circumcenter of triangle O′ 1O′ 2O′ 3. Prove that OO′ is parallel to PQ.

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题目标签：QUIZ3-P1

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Quiz 3 · 数论

Prove that for arbitary integer n > 16, there exists the set S that contains n positive integers and has the following property:if the subset A of S satisfies for arbitary a, a′ ∈A, a̸ = a′, a+a′ /∈ S holds, then |A| ≤4√n.

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题目标签：QUIZ3-P2

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Quiz 3 · 代数

Let n > m > 1 be odd integers, let f(x) = xn + xm + x + 1. Prove that f(x) can’t be expressed as the product of two polynomials having integer coefficients and positive degrees.

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题目标签：QUIZ3-P3

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Quiz 4 · 平面几何

Given a rectangle ABCD, let AB = b, AD = a(a ≥b), three points X, Y, Z are put inside or on the boundary of the rectangle, arbitrarily. Find the maximum of the minimum of the distances between any two points among the three points. (Denote it by a, b)

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题目标签：QUIZ4-P1

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Quiz 4 · 代数

Let x, y, z be positive real numbers, show that xy x3 + y3 + z3. z + yz x + zx y > 2 3p i=1(1 − 1 2ai ).

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题目标签：QUIZ4-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。

Quiz 4 · 组合数学

Let S be a set that contains n elements. Let A1, A2, · · · , Ak be k distinct subsets of S, where k ≥2, |Ai| = ai ≥1(1 ≤i ≤k). Prove that the number of subsets of S that don’t contain any Ai(1 ≤i ≤k) is greater than or equal to 2n Qk

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题目标签：QUIZ4-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。

Quiz 5 · 平面几何

Let ABC be an acute triangle, let M, N be the midpoints of minor arcs d CA, d AB of the circumcircle of triangle ABC, point D is the midpoint of segment MN, point G lies on minor arc d BC. Denote by I, I1, I2 the incenters of triangle ABC, ABG, ACG respectively.Let P be the second intersection of the circumcircle of triangle GI1I2 with the circumcircle of triangle ABC. Prove that three points D, I, P are collinear.

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题目标签：QUIZ5-P1

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Quiz 5 · 代数

For a given integer n ≥2, determine the necessary and sufficient conditions that real numbers a1, a2, · · · , an, not all zero satisfy such that there exist integers 0 < x1 < x2 < · · · < xn, satisfying a1x1 + a2x2 + · · · + anxn ≥0.

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题目标签：QUIZ5-P2

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Quiz 5 · 数学竞赛/待细分

Let 0 < x1 ≤ x2 2 ≤· · · ≤ xn n , 0 < yn ≤yn−1 ≤· · · ≤y1, Prove that (Pn k=1 xkyk)2 ≤ (Pn k=1(x2 4xkxk−1)yk). where x0 = 0. k −1 k=1 yk)(Pn

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题目标签：QUIZ5-P3

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Quiz 6 · 数学竞赛/待细分

Prove that in a plane, arbitrary n points can be overlapped by discs that the sum of all the diameters is less than n, and the distances between arbitrary two are greater than 1. (where the distances between two discs that have no common points are defined as that the distances between its centers subtract the sum of its radii; the distances between two discs that have common points are zero)

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题目标签：QUIZ6-P1

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Quiz 6 · 代数

Prove that for all n ≥2, there exists n-degree polynomial f(x) = xn + a1xn−1 + · · · + an such that (1) a1, a2, · · · , an all are unequal to 0; (2) f(x) can’t be factorized into the product of two polynomials having integer coefficients and positive degrees; (3) for any integers x, |f(x)| isn’t prime numbers.

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题目标签：QUIZ6-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。

Quiz 6 · 平面几何

Find all positive integers n having the following properties:in two-dimensional Cartesian coordinates, there exists a convex n lattice polygon whose lengths of all sides are odd numbers, and unequal to each other. (where lattice polygon is defined as polygon whose coordinates of all vertices are integers in Cartesian coordinates.)

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题目标签：QUIZ6-P3

解题过程

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