2014 中国国家集训队选拔考试（China TST）

books/competition_archive/china_tst/2014_china_tst.pdf · HS-MATH-1024-v2.1-solution-aware

18 题0 个小问/题组

TST 1 / Day 1 March 12th · 平面几何

ABCD is a cyclic quadrilateral, with diagonals AC, BD perpendicular to each other. Let point F be on side BC, the parallel line EF to AC intersect AB at point E, line FG parallel to BD intersect CD at G. Let the projection of E onto CD be P, projection of F onto DA be Q, projection of G onto AB be R. Prove that QF bisects ∠PQR.

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题目标签：TST1-DAY1-P1

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TST 1 / Day 1 March 12th · 数学竞赛/待细分

Let A be a finite set of positive numbers , B = {a+b c+d|a, b, c, d ∈A}. Show that: |B| ≥2 |A|2 −1, where |X| be the number of elements of the finite set X. (High School Affiliated to Nanjing Normal University )

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题目标签：TST1-DAY1-P2

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TST 1 / Day 1 March 12th · 数论

Let the function f : N∗→N∗such that (1) (f(m), f(n)) ≤(m, n)2014, ∀m, n ∈N∗; (2) n ≤f(n) ≤n + 2014, ∀n ∈N∗ Show that: there exists the positive integers N such that f(n) = n, for each integer n ≥N. (High School Affiliated to Nanjing Normal University )

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题目标签：TST1-DAY1-P3

解题过程

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。

TST 1 / Day 2 March 13th · 代数

For any real numbers sequence {xn} ,suppose that {yn} is a sequence such that: y1 = x1, yn+1 = nP x2 1 2 (n ≥1) . xn+1 −( i ) i=1 Find the smallest positive number λ such that for any real numbers sequence {xn} and all m P m P positive integers m , have 1 x2 λm−iy2 i ≤ i . m i=1 i=1 (High School Affiliated to Nanjing Normal University )

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题目标签：TST1-DAY2-P4

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TST 1 / Day 2 March 13th · 数论

Let a1 < a2 < ... < at be t given positive integers where no three form an arithmetic progression. For k = t, t + 1, ... define ak+1 to be the smallest positive integer larger than ak satisfying the condition that no three of a1, a2, ..., ak+1 form an arithmetic progression. For any x ∈R+ define A(x) to be the number of terms in {ai}i≥1 that are at most x. Show that there exist c > 1 and K > 0 such that A(x) ≥c√x for any x > K.

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题目标签：TST1-DAY2-P5

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TST 1 / Day 2 March 13th · 数论

Let n ≥2 be a positive integer. Fill up a n × n table with the numbers 1, 2, ..., n2 exactly once each. Two cells are termed adjacent if they have a common edge. It is known that for any two adjacent cells, the numbers they contain differ by at most n. Show that there exist a 2 × 2 square of adjacent cells such that the diagonally opposite pairs sum to the same number.

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题目标签：TST1-DAY2-P6

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TST 2 / Day 1 March 17th · 数论

Prove that for any positive integers k and N, k ! 1 N X X (ω(n))k ≤k + 1 N 1 q , n=1 q≤N 1 q is the summation over of prime powers q ≤N (including q = 1). where P q≤N Note: For integer n > 1, ω(n) denotes number of distinct prime factors of n, and ω(1) = 0.

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题目标签：TST2-DAY1-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。

TST 2 / Day 1 March 17th · 数论

Given a fixed positive integer a ≥9. Prove: There exist finitely many positive integers n, satisfying: (1)τ(n) = a (2)n|φ(n) + σ(n) Note: For positive integer n, τ(n) is the number of positive divisors of n, φ(n) is the number of positive integers ≤n and relatively prime with n, σ(n) is the sum of positive divisors of n.

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题目标签：TST2-DAY1-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。

TST 2 / Day 1 March 17th · 数论

A is the set of points of a convex n-gon on a plane. The distinct pairwise distances between any 2 points in A arranged in descending order is d1 > d2 > ... > dm > 0. Let the number of unordered pairs of points in A such that their distance is di be exactly µi, for i = 1, 2, ..., m. Prove: For any positive integer k ≤m, µ1 + µ2 + ... + µk ≤(3k −1)n.

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题目标签：TST2-DAY1-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。

TST 2 / Day 2 March 18th · 平面几何

Given circle O with radius R, the inscribed triangle ABC is an acute scalene triangle, where AB is the largest side. AHA, BHB, CHC are heights on BC, CA, AB. Let D be the symmetric point of HA with respect to HBHC, E be the symmetric point of HB with respect to HAHC. P is the intersection of AD, BE, H is the orthocentre of △ABC. Prove: OP · OH is fixed, and find this value in terms of R. (Edited)

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题目标签：TST2-DAY2-P4

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TST 2 / Day 2 March 18th · 组合数学

Find the smallest positive constant c satisfying: For any simple graph G = G(V, E), if |E| ≥ c|V |, then G contains 2 cycles with no common vertex, and one of them contains a chord. Note: The cycle of graph G(V, E) is a set of distinct vertices v1, v2..., vn ⊆V , vivi+1 ∈E for all 1 ≤i ≤n (n ≥3, vn+1 = v1); a cycle containing a chord is the cycle v1, v2..., vn, such that there exist i, j, 1 < i −j < n −1, satisfying vivj ∈E.

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题目标签：TST2-DAY2-P5

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TST 2 / Day 2 March 18th · 数论

Let k be a fixed even positive integer, N is the product of k distinct primes p1, ..., pk, a, b are two positive integers, a, b ≤N. Denote S1 = {d| d|N, a ≤d ≤b, d has even number of prime factors}, S2 = {d| d|N, a ≤d ≤b, d has odd number of prime factors}, Prove: |S1| −|S2| ≤C k 2 k

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题目标签：TST2-DAY2-P6

解题过程

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。

TST 3 / Day 1 March 23rd · 平面几何

Let the circumcenter of triangle ABC be O. HA is the projection of A onto BC. The extension of AO intersects the circumcircle of BOC at A′. The projections of A′ onto AB, AC are D, E, and OA is the circumcentre of triangle DHAE. Define HB, OB, HC, OC similarly. Prove: HAOA, HBOB, HCOC are concurrent

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题目标签：TST3-DAY1-P1

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TST 3 / Day 1 March 23rd · 平面几何

Let A1A2...A101 be a regular 101-gon, and colour every vertex red or blue. Let N be the number of obtuse triangles satisfying the following: The three vertices of the triangle must be vertices of the 101-gon, both the vertices with acute angles have the same colour, and the vertex with obtuse angle have different colour. (1) Find the largest possible value of N. (2) Find the number of ways to colour the vertices such that maximum N is acheived. (Two colourings a different if for some Ai the colours are different on the two colouring schemes).

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题目标签：TST3-DAY1-P2

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TST 3 / Day 1 March 23rd · 数论

Show that there are no 2-tuples (x, y) of positive integers satisfying the equation (x + 1)(x + 2) · · · (x + 2014) = (y + 1)(y + 2) · · · (y + 4028).

题解状态：标准答案与规范题解待补充

题目标签：TST3-DAY1-P3

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TST 3 / Day 2 March 24th · 数论

Let k be a fixed odd integer, k > 3. Prove: There exist infinitely many positive integers n, such that there are two positive integers d1, d2 satisfying d1, d2 each dividing n2+1 2 , and d1 + d2 = n + k.

题解状态：标准答案与规范题解待补充

题目标签：TST3-DAY2-P4

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TST 3 / Day 2 March 24th · 数学竞赛/待细分

Let n be a given integer which is greater than 1 . Find the greatest constant λ(n) such that for any non-zero complex z1, z2, · · · , zn ,have that n X |zk|2 ≥λ(n) min 1≤k≤n{|zk+1 −zk|2}, k=1 where zn+1 = z1.

题解状态：标准答案与规范题解待补充

题目标签：TST3-DAY2-P5

解题过程

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。

TST 3 / Day 2 March 24th · 数论

For positive integer k > 1, let f(k) be the number of ways of factoring k into product of positive integers greater than 1 (The order of factors are not countered, for example f(12) = 4, as 12 can be factored in these 4 ways: 12, 2 · 6, 3 · 4, 2 · 2 · 3. Prove: If n is a positive integer greater than 1, p is a prime factor of n, then f(n) ≤n p

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题目标签：TST3-DAY2-P6

解题过程

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