2007 中国国家集训队选拔考试（China TST）

books/competition_archive/china_tst/2007_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

Points A and B lie on the circle with center O. Let point C lies outside the circle; let CS and CT be tangents to the circle. M be the midpoint of minor arc AB of (O). MS, MT intersect AB at points E, F respectively. The lines passing through E, F perpendicular to AB cut OS, OT at X and Y respectively. A line passed through C intersect the circle (O) at P, Q (P lies on segment CQ). Let R be the intersection of MP and AB, and let Z be the circumcentre of triangle PQR. Prove that: X, Y, Z are collinear.

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题目标签：TST1-DAY1-P1

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TST 1 / Day 1 · 数论

A rational number x is called good if it satisfies: x = p q > 1 with p, q being positive integers, gcd(p, q) = 1 and there exists constant numbers α, N such that for any integer n ≥N, |{xn} −α| ≤ 1 2(p + q) Find all the good numbers.

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题目标签：TST1-DAY1-P2

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TST 1 / Day 1 · 平面几何

There are 63 points arbitrarily on the circle C with its diameter being 20. Let S denote the number of triangles whose vertices are three of the 63 points and the length of its sides is no less than 9. Fine the maximum of S.

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题目标签：TST1-DAY1-P3

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TST 1 / Day 2 · 代数

Find all functions f : Q+ 7→Q+ such that: f(x) + f(y) + 2xyf(xy) = f(xy) f(x + y).

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题目标签：TST1-DAY2-P1

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TST 1 / Day 2 · 组合数学

Let x1, . . . , xn be n > 1 real numbers satisfying A = |Pn i=1 xi|̸ = 0 and B = max1≤i<j≤n |xj − xi|̸ = 0. Prove that for any n vectors⃗αi in the plane, there exists a permutation (k1, . . . , kn) of the numbers (1, . . . , n) such that n X xki⃗αi i=1 ≥ AB 2A + B max 1≤i≤n |αi|.

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题目标签：TST1-DAY2-P2

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TST 1 / Day 2 · 数论

Let n be a positive integer, let A be a subset of {1, 2, · · · , n}, satisfying for any two numbers x, y ∈A, the least common multiple of x, y not more than n. Show that |A| ≤1.9√n + 5.

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题目标签：TST1-DAY2-P3

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Quiz 1 · 平面几何

When all vertex angles of a convex polygon are equal, call it equiangular. Prove that p > 2 is a prime number, if and only if the lengths of all sides of equiangular p polygon are rational numbers, it is a regular p polygon.

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题目标签：QUIZ1-P1

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Quiz 1 · 平面几何

Let I be the incenter of triangle ABC. Let M, N be the midpoints of AB, AC, respectively. Points D, E lie on AB, AC respectively such that BD = CE = BC. The line perpendicular to IM through D intersects the line perpendicular to IN through E at P. Prove that AP ⊥BC.

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题目标签：QUIZ1-P2

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Quiz 1 · 代数

Prove that for any positive integer n, there exists only n degree polynomial f(x), satisfying f(0) = 1 and (x + 1)[f(x)]2 −1 is an odd function.

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题目标签：QUIZ1-P3

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Quiz 2 · 数学竞赛/待细分

u, v, w > 0,such that u + v + w + √uvw = 4 prove that p uv v ≥u + v + w w + p vw u + p wu

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题目标签：QUIZ2-P1

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Quiz 2 · 数论

Find all positive integers n such that there exists sequence consisting of 1 and −1 : a1, a2, · · · , an satisfying a1 · 12 + a2 · 22 + · · · + an · n2 = 0.

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题目标签：QUIZ2-P2

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Quiz 2 · 数学竞赛/待细分

Assume there are n ≥3 points in the plane, Prove that there exist three points A, B, C satisfying 1 ≤AB n−1. AC ≤n+1

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题目标签：QUIZ2-P3

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Quiz 3 · 平面几何

Let ABC be a triangle. Circle ω passes through points B and C. Circle ω1 is tangent internally to ω and also to sides AB and AC at T, P, and Q, respectively. Let M be midpoint of arc BC (containing T) of ω. Prove that lines PQ, BC, and MT are concurrent.

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题目标签：QUIZ3-P1

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Quiz 3 · 数学竞赛/待细分

Given an integer k > 1. We call a k−digits decimal integer a1a2 · · · ak is p−monotonic, if for each of integers i satisfying 1 ≤i ≤k −1, when ai is an odd number, ai > ai+1; when ai is an even number, ai < ai+1. Find the number of p−monotonic k−digits integers.

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题目标签：QUIZ3-P2

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Quiz 3 · 数论

Show that there exists a positive integer k such that k · 2n + 1 is composite for all n ∈N0.

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题目标签：QUIZ3-P3

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Quiz 4 · 代数

Let a1, a2, · · · , an be positive real numbers satisfying a1 + a2 + · · · + an = 1. Prove that (a1a2 + a2a3 + · · · + ana1) ≥ n n + 1 a1 a2 2 + a2 + a2 a2 3 + a3 + · · · + an a2 1 + a1

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题目标签：QUIZ4-P1

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Quiz 4 · 代数

After multiplying out and simplifying polynomial (x −1)(x2 −1)(x3 −1) · · · (x2007 −1), getting rid of all terms whose powers are greater than 2007, we acquire a new polynomial f(x). Find its degree and the coefficient of the term having the highest power. Find the degree of f(x) = (1 −x)(1 −x2)...(1 −x2007) (mod x2008).

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题目标签：QUIZ4-P2

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Quiz 4 · 数论

Let n be positive integer, A, B ⊆[0, n] are sets of integers satisfying | A | + | B |≥n + 2. Prove that there exist a ∈A, b ∈B such that a + b is a power of 2.

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题目标签：QUIZ4-P3

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Quiz 5 · 平面几何

Let convex quadrilateral ABCD be inscribed in a circle centers at O. The opposite sides BA, CD meet at H, the diagonals AC, BD meet at G. Let O1, O2 be the circumcenters of triangles AGD, BGC. O1O2 intersects OG at N. The line HG cuts the circumcircles of triangles AGD, BGC at P, Q, respectively. Denote by M the midpoint of PQ. Prove that NO = NM.

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题目标签：QUIZ5-P1

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Quiz 5 · 平面几何

Given n points arbitrarily in the plane P1, P2, . . . , Pn, among them no three points are collinear. Each of Pi (1 ≤i ≤n) is colored red or blue arbitrarily. Let S be the set of triangles having {P1, P2, . . . , Pn} as vertices, and having the following property: for any two segments PiPj and PuPv, the number of triangles having PiPj as side and the number of triangles having PuPv as side are the same in S. Find the least n such that in S there exist two triangles, the vertices of each triangle having the same color.

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题目标签：QUIZ5-P2

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Quiz 5 · 数学竞赛/待细分

Find the smallest constant k such that x √x+y + y √y+z + z √z+x ≤k√x + y + z for all positive x, y, z.

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题目标签：QUIZ5-P3

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Quiz 6 · 数论

Find all the pairs of positive integers (a, b) such that a2 + b −1 is a power of prime number ; a2 + b + 1 can divide b2 −a3 −1, but it can’t divide (a + b −1)2.

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题目标签：QUIZ6-P1

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Quiz 6 · 平面几何

Let ABCD be the inscribed quadrilateral with the circumcircle ω.Let ζ be another circle that internally tangent to ω and to the lines BC and AD at points M, N respectively.Let I1, I2 be the incenters of the △ABC and △ABD.Prove that M, I1, I2, N are collinear.

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题目标签：QUIZ6-P2

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Quiz 6 · 数学竞赛/待细分

Consider a 7 × 7 numbers table aij = (i2 + j)(i + j2), 1 ≤i, j ≤7. When we add arbitrarily each term of an arithmetical progression consisting of 7 integers to corresponding to term of certain row (or column) in turn, call it an operation. Determine whether such that each row of numbers table is an arithmetical progression, after a finite number of operations.

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题目标签：QUIZ6-P3

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