2006 中国国家集训队选拔考试（China TST）

books/competition_archive/china_tst/2006_china_tst.pdf · HS-MATH-1024-v2.1-solution-aware

24 题0 个小问/题组

TST 1 / Day 1 March 19th · 平面几何

ABCD is a trapezoid with AB||CD. There are two circles ω1 and ω2 is the trapezoid such that ω1 is tangent to DA, AB, BC and ω2 is tangent to BC, CD, DA. Let l1 be a line passing through A and tangent to ω2(other than AD), Let l2 be a line passing through C and tangent to ω1 (other than CB). Prove that l1||l2.

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题目标签：TST1-DAY1-P1

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TST 1 / Day 1 March 19th · 数论

Find all positive integer pairs (a, n) such that (a+1)n−an n is an integer.

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题目标签：TST1-DAY1-P2

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TST 1 / Day 1 March 19th · 代数

Given n real numbers a1, a2 . . . an. (n ≥1). Prove that there exists real numbers b1, b2 . . . bn satisfying: (a) For any 1 ≤i ≤n, ai −bi is a positive integer. (b)P 1≤i<j≤n(bi −bj)2 ≤n2−1 12

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题目标签：TST1-DAY1-P3

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TST 1 / Day 2 March 20th · 代数

Two positive valued sequences {an} and {bn} satisfy: (a): a0 = 1 ≥a1, an(bn+1 + bn−1) = an−1bn−1 + an+1bn+1, n ≥1. 3 2 , n ≥1. i=1 bi ≤n (b): Pn Find the general term of {an}.

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题目标签：TST1-DAY2-P1

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TST 1 / Day 2 March 20th · 平面几何

Let ω be the circumcircle of △ABC. P is an interior point of △ABC. A1, B1, C1 are the intersections of AP, BP, CP respectively and A2, B2, C2 are the symmetrical points of A1, B1, C1 with respect to the midpoints of side BC, CA, AB. Show that the circumcircle of △A2B2C2 passes through the orthocentre of △ABC.

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题目标签：TST1-DAY2-P2

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TST 1 / Day 2 March 20th · 代数

Let ai and bi (i = 1, 2, · · · , n) be rational numbers such that for any real number x there is: n X x2 + x + 4 = (aix + b)2 i=1 Find the least possible value of n.

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题目标签：TST1-DAY2-P3

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TST 1 / Day 3 March 22nd · 平面几何

The centre of the circumcircle of quadrilateral ABCD is O and O is not on any of the sides of ABCD. P = AC ∩BD. The circumecentres of △OAB, △OBC, △OCD and △ODA are O1, O2, O3 and O4 respectively. Prove that O1O3, O2O4 and OP are concurrent.

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题目标签：TST1-DAY3-P1

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TST 1 / Day 3 March 22nd · 数学竞赛/待细分

x1, x2, · · · , xn are positive numbers such that Pn ! i=1 xi = 1. Prove that n X ! n X √xi ≤ n2 √n + 1 1 √1 + xi i=1 i=1

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题目标签：TST1-DAY3-P2

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TST 1 / Day 3 March 22nd · 数论

d and n are positive integers such that d | n. The n-number sets (x1, x2, · · · xn) satisfy the following condition: (1) 0 ≤x1 ≤x2 ≤· · · ≤xn ≤n (2) d | (x1 + x2 + · · · xn) Prove that in all the n-number sets that meet the conditions, there are exactly half satisfy xn = n.

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题目标签：TST1-DAY3-P3

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TST 1 / Day 4 March 24th · 平面几何

Let K and M be points on the side AB of a triangle △ABC, and let L and N be points on the side AC. The point K is between M and B, and the point L is between N and C. If BK KM = CL LN , then prove that the orthocentres of the triangles △ABC, △AKL and △AMN lie on one line.

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题目标签：TST1-DAY4-P1

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TST 1 / Day 4 March 24th · 代数

Given three positive real numbers x, y, z such that x + y + z = 1, prove that √ 2 2 . xy √xy+yz + yz √yz+zx + zx √zx+xy ≤

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题目标签：TST1-DAY4-P2

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TST 1 / Day 4 March 24th · 代数

Find all second degree polynomial d(x) = x2 + ax + b with integer coefficients, so that there exists an integer coefficient polynomial p(x) and a non-zero integer coefficient polynomial q(x) that satisfy: (p(x))2 −d(x) (q(x))2 = 1, ∀x ∈R.

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题目标签：TST1-DAY4-P3

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TST 1 / Day 5 March 26th · 数论

Let A be a non-empty subset of the set of all positive integers N∗. If any sufficient big positive integer can be expressed as the sum of 2 elements in A(The two integers do not have to be different), then we call that A is a divalent radical. For x ≥1, let A(x) be the set of all elements in A that do not exceed x, prove that there exist a divalent radical A and a constant number C so that for every x ≥1, there is always |A(x)| ≤C√x.

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题目标签：TST1-DAY5-P1

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TST 1 / Day 5 March 26th · 代数

The function f(n) satisfies f(0) = 0, f(n) = n−f (f(n −1)), n = 1, 2, 3 · · · . Find all polynomials g(x) with real coefficient such that f(n) = [g(n)], n = 0, 1, 2 · · · Where [g(n)] denote the greatest integer that does not exceed g(n).

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题目标签：TST1-DAY5-P2

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TST 1 / Day 5 March 26th · 数论

Given positive integers m and n so there is a chessboard with mn 1 × 1 grids. Colour the grids into red and blue (Grids that have a common side are not the same colour and the grid in the left corner at the bottom is red). Now the diagnol that goes from the left corner at the bottom to the top right corner is coloured into red and blue segments (Every segment has the same colour with the grid that contains it). Find the sum of the length of all the red segments.

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题目标签：TST1-DAY5-P3

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TST 1 / Day 6 March 28th · 数学竞赛/待细分

Let the intersections of ⊙O1 and ⊙O2 be A and B. Point R is on arc AB of ⊙O1 and T is on arc AB on ⊙O2. AR and BR meet ⊙O2 at C and D; AT and BT meet ⊙O1 at Q and P. If PR and TD meet at E and QR and TC meet at F, then prove: AE · BT · BR = BF · AT · AR.

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题目标签：TST1-DAY6-P1

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TST 1 / Day 6 March 28th · 数论

Prove that for any given positive integer m and n, there is always a positive integer k so that 2k −m has at least n different prime divisors.

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题目标签：TST1-DAY6-P2

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TST 1 / Day 6 March 28th · 数论

k and n are positive integers that are greater than 1. N is the set of positive integers. A1, A2, · · · Ak are pairwise not-intersecting subsets of N and A1 ∪A2 ∪· · · ∪Ak = N. Prove that for some i ∈{1, 2, · · · , k}, there exsits infinity many non-factorable n-th degree polynomials so that coefficients of one polynomial are pairwise distinct and all the coeficients are in Ai.

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题目标签：TST1-DAY6-P3

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TST 1 / Day 7 March 31st · 平面几何

H is the orthocentre of △ABC. D, E, F are on the circumcircle of △ABC such that AD ∥ BE ∥CF. S, T, U are the semetrical points of D, E, F with respect to BC, CA, AB. Show that S, T, U, H lie on the same circle.

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题目标签：TST1-DAY7-P1

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TST 1 / Day 7 March 31st · 数论

Given positive integer n, find the biggest real number C which satisfy the condition that if the sum of the reciprocals of a set of integers (They can be the same.) that are greater than 1 is less than C, then we can divide the set of numbers into no more than n groups so that the sum of reciprocals of every group is less than 1.

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题目标签：TST1-DAY7-P2

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TST 1 / Day 7 March 31st · 数论

For a positive integer M, if there exist integers a, b, c and d so that: M ≤a < b ≤c < d ≤M + 49, ad = bc then we call M a GOOD number, if not then M is BAD. Please find the greatest GOOD number and the smallest BAD number.

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题目标签：TST1-DAY7-P3

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TST 1 / Day 8 April 1st · 数论

Let k be an odd number that is greater than or equal to 3. Prove that there exists a kth-degree integer-valued polynomial with non-integer-coefficients that has the following properties: (1) f(0) = 0 and f(1) = 1; and. (2) There exist infinitely many positive integers n so that if the following equation: n = f(x1) + · · · + f(xs), has integer solutions x1, x2, . . . , xs, then s ≥2k −1.

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题目标签：TST1-DAY8-P1

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TST 1 / Day 8 April 1st · 数论

Given positive integers m, a, b, (a, b) = 1. A is a non-empty subset of the set of all positive integers, so that for every positive integer n there is an ∈A and bn ∈A. For all A that satisfy the above condition, find the minimum of the value of |A ∩{1, 2, · · · , m}|

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题目标签：TST1-DAY8-P2

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TST 1 / Day 8 April 1st · 平面几何

△ABC can cover a convex polygon M.Prove that there exsit a triangle which is congruent to △ABC such that it can also cover M and has one side line paralel to or superpose one side line of M.

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题目标签：TST1-DAY8-P3

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