2017 中国国家集训队选拔考试（China TST）

books/competition_archive/china_tst/2017_china_tst.pdf · HS-MATH-1024-v2.1-solution-aware

30 题0 个小问/题组

TST 1 / Day 1 · 组合数学

Find out the maximum value of the numbers of edges of a solid regular octahedron that we can see from a point out of the regular octahedron.(We define we can see an edge AB of the regular octahedron from point P outside if and only if the intersection of non degenerate triangle PAB and the solid regular octahedron is exactly edge AB.

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题目标签：TST1-DAY1-P1

解题过程

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TST 1 / Day 1 · 数论

Let x > 1 ,n be positive integer. Prove that {kx} n X n X [kx] < 1 2k −1 k=1 k=1 Where [kx] be the integer part of kx ,{kx} be the decimal part of kx.

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题目标签：TST1-DAY1-P2

解题过程

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TST 1 / Day 1 · 组合数学

Suppose S = {1, 2, 3, ..., 2017},for every subset A of S,define a real number f(A) ≥0 such that: (1) For any A, B ⊂S,f(A∪B)+f(A∩B) ≤f(A)+f(B); (2) For any A ⊂B ⊂S, f(A) ≤f(B); (3) For any k, j ∈S, f({1, 2, . . . , k + 1}) ≥f({1, 2, . . . , k} ∪{j}); (4) For the empty set ∅, f(∅) = 0. Confirm that for any three-element subset T of S,the inequality f(T) ≤27 19f({1, 2, 3}) holds.

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题目标签：TST1-DAY1-P3

解题过程

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TST 1 / Day 2 · 代数

Find out all the integer pairs (m, n) such that there exist two monic polynomials P(x) and Q(x) ,with deg P = m and deg Q = n,satisfy that P(Q(t))̸ = Q(P(t)) holds for any real number t.

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题目标签：TST1-DAY2-P4

解题过程

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TST 1 / Day 2 · 平面几何

In the non-isosceles triangle ABC,D is the midpoint of side BC,E is the midpoint of side CA,F is the midpoint of side AB.The line(different from line BC) that is tangent to the inscribed circle of triangle ABC and passing through point D intersect line EF at X.Define Y, Z similarly.Prove that X, Y, Z are collinear.

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题目标签：TST1-DAY2-P5

解题过程

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TST 1 / Day 2 · 数论

For a given positive integer n and prime number p, find the minimum value of positive integer m that satisfies the following property: for any polynomial f(x) = (x + a1)(x + a2) . . . (x + an) (a1, a2, . . . , an are positive integers), and for any non-negative integer k, there exists a nonnegative integer k′ such that vp(f(k)) < vp(f(k′)) ≤vp(f(k)) + m. Note: for non-zero integer N,vp(N) is the largest non-zero integer t that satisfies pt | N.

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题目标签：TST1-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 2 / Day 1 · 数论

Let n be a positive integer. Let Dn be the set of all divisors of n and let f(n) denote the smallest natural m such that the elements of Dn are pairwise distinct in mod m. Show that there exists a natural N such that for all n ≥N, one has f(n) ≤n0.01.

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题目标签：TST2-DAY1-P1

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 2 / Day 1 · 数学竞赛/待细分

2017 engineers attend a conference. Any two engineers if they converse, converse with each other in either Chinese or English. No two engineers converse with each other more than once. It is known that within any four engineers, there was an even number of conversations and furthermore within this even number of conversations: i) At least one conversation is in Chinese. ii) Either no conversations are in English or the number of English conversations is at least that of Chinese conversations. Show that there exists 673 engineers such that any two of them conversed with each other in Chinese.

题解状态：标准答案与规范题解待补充

题目标签：TST2-DAY1-P2

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 2 / Day 1 · 平面几何

Let ABCD be a quadrilateral and let l be a line. Let l intersect the lines AB, CD, BC, DA, AC, BD at points X, X′, Y, Y ′, Z, Z′ respectively. Given that these six points on l are in the order X, Y, Z, X′, Y ′, Z′, show that the circles with diameter XX′, Y Y ′, ZZ′ are coaxal.

题解状态：标准答案与规范题解待补充

题目标签：TST2-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 2 / Day 2 · 数学竞赛/待细分

An integer n > 1 is given . Find the smallest positive number m satisfying the following conditions for any set {a, b} ⊂{1, 2, · · · , 2n −1} ,there are non-negative integers x, y ( not all zero) such that 2n|ax + by and x + y ≤m.

题解状态：标准答案与规范题解待补充

题目标签：TST2-DAY2-P4

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 2 / Day 2 · 代数

Let ϕ(x) be a cubic polynomial with integer coefficients. Given that ϕ(x) has have 3 distinct real roots u, v, w and u, v, w are not rational number. there are integers a, b, c such that u = av2 + bv + c. Prove that b2 −2b −4ac −7 is a square number .

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题目标签：TST2-DAY2-P5

解题过程

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TST 2 / Day 2 · 组合数学

Let M be a subset of R such that the following conditions are satisfied: a) For any x ∈M, n ∈Z, one has that x + n ∈M. b) For any x ∈M, one has that −x ∈M. c) Both M and R M contain an interval of length larger than 0. For any real x, let M(x) = {n ∈Z+|nx ∈M}. Show that if α, β are reals such that M(α) = M(β), then we must have one of α + β and α −β to be rational.

题解状态：标准答案与规范题解待补充

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 · 数学竞赛/待细分

Let n ≥4 be a natural and let x1, . . . , xn be non-negative reals such that x1 + · · · + xn = 1. Determine the maximum value of x1x2x3 + x2x3x4 + · · · + xnx1x2.

题解状态：标准答案与规范题解待补充

题目标签：TST3-DAY1-P1

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 · 平面几何

Let ABCD be a non-cyclic convex quadrilateral. The feet of perpendiculars from A to BC, BD, CD are P, Q, R respectively, where P, Q lie on segments BC, BD and R lies on CD extended. The feet of perpendiculars from D to AC, BC, AB are X, Y, Z respectively, where X, Y lie on segments AC, BC and Z lies on BA extended. Let the orthocenter of △ABD be H. Prove that the common chord of circumcircles of △PQR and △XY Z bisects BH.

题解状态：标准答案与规范题解待补充

题目标签：TST3-DAY1-P2

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 · 组合数学

Let X be a set of 100 elements. Find the smallest possible n satisfying the following condition: Given a sequence of n subsets of X, A1, A2, . . . , An, there exists 1 ≤i < j < k ≤n such that Ai ⊆Aj ⊆Ak or Ai ⊇Aj ⊇Ak.

题解状态：标准答案与规范题解待补充

题目标签：TST3-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 3 / Day 2 · 数论

Show that there exists a degree 58 monic polynomial P(x) = x58 + a1x57 + · · · + a58 such that P(x) has exactly 29 positive real roots and 29 negative real roots and that log2017 |ai| is a positive integer for all 1 ≤i ≤58.

题解状态：标准答案与规范题解待补充

题目标签：TST3-DAY2-P4

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 · 数论

Show that there exists a positive real C such that for any naturals H, N satisfying H ≥3, N ≥ eCH, for any subset of {1, 2, . . . , N} with size ⌈CHN ln N ⌉, one can find H naturals in it such that the greatest common divisor of any two elements is the greatest common divisor of all H elements.

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 · 组合数学

Every cell of a 2017×2017 grid is colored either black or white, such that every cell has at least one side in common with another cell of the same color. Let V1 be the set of all black cells, V2 be the set of all white cells. For set Vi(i = 1, 2), if two cells share a common side, draw an edge with the centers of the two cells as endpoints, obtaining graphs Gi. If both G1 and G2 are connected paths (no cycles, no splits), prove that the center of the grid is one of the endpoints of G1 or G2.

题解状态：标准答案与规范题解待补充

题目标签：TST3-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 4 / Day 1 · 数学竞赛/待细分

Prove that : 58 X 29 X C58−k C58−2p 4091−2p 2017+kCk 2075−k = p=0 k=0

题解状态：标准答案与规范题解待补充

题目标签：TST4-DAY1-P1

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 4 / Day 1 · 平面几何

In ∆ABC,the excircle of A is tangent to segment BC,line AB and AC at E, D, F respectively.EZ is the diameter of the circle.B1 and C1 are on DF, and BB1 ⊥BC,CC1 ⊥BC.Line ZB1, ZC1 intersect BC at X, Y respectively.Line EZ and line DF intersect at H,ZK is perpendicular to FD at K.If H is the orthocenter of ∆XY Z,prove that:H, K, X, Y are concyclic.

题解状态：标准答案与规范题解待补充

题目标签：TST4-DAY1-P2

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 4 / Day 1 · 数学竞赛/待细分

Find the numbers of ordered array (x1, ..., x100) that satisfies the following conditions: (i)x1, ..., x100 ∈{1, 2, .., 2017}; (ii)2017|x1 + ... + x100; (iii)2017|x2 1 + ... + x2 100.

题解状态：标准答案与规范题解待补充

题目标签：TST4-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 4 / Day 2 · 数学竞赛/待细分

Given integer d > 1, m,prove that there exists integer k > l > 0, such that (22k + d, 22l + d) > m.

题解状态：标准答案与规范题解待补充

题目标签：TST4-DAY2-P4

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 4 / Day 2 · 代数

Given integer m ≥2,x1, ..., xm are non-negative real numbers,prove that: (m −1)m−1(xm 1 + ... + xm m) ≥(x1 + ... + xm)m −mmx1...xm and please find out when the equality holds.

题解状态：标准答案与规范题解待补充

题目标签：TST4-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 4 / Day 2 · 组合数学

A plane has no vertex of a regular dodecahedron on it,try to find out how many edges at most may the plane intersect the regular dodecahedron?

题解状态：标准答案与规范题解待补充

题目标签：TST4-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 5 / 未标注日 · 代数

Given n ≥3. consider a sequence a1, a2, ..., an, if (ai, aj, ak) with i+k=2j (i¡j¡k) and ai + ak̸ = 2aj, we call such a triple a NOT −AP triple. If a sequence has at least one NOT −AP triple, find the least possible number of the NOT −AP triple it contains.

题解状态：标准答案与规范题解待补充

题目标签：TST5-P1

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 5 / 未标注日 · 数学竞赛/待细分

Find the least positive number m such that for any polynimial f(x) with real coefficients, there is a polynimial g(x) with real coefficients (degree not greater than m) such that there exist 2017 distinct number a1, a2, ..., a2017 such that g(ai) = f(ai+1) for i=1,2,...,2017 where indices taken modulo 2017.

题解状态：标准答案与规范题解待补充

题目标签：TST5-P2

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 5 / 未标注日 · 组合数学

For a rational point (x,y), if xy is an integer that divided by 2 but not 3, color (x,y) red, if xy is an integer that divided by 3 but not 2, color (x,y) blue. Determine whether there is a line segment in the plane such that it contains exactly 2017 blue points and 58 red points.

题解状态：标准答案与规范题解待补充

题目标签：TST5-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 5 / 未标注日 · 平面几何

Given a circle with radius 1 and 2 points C, D given on it. Given a constant l with 0 < l ≤2. Moving chord of the circle AB=l and ABCD is a non-degenerated convex quadrilateral. AC and BD intersects at P. Find the loci of the circumcenters of triangles ABP and BCP.

题解状态：标准答案与规范题解待补充

题目标签：TST5-P4

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 5 / 未标注日 · 代数

A(x,y), B(x,y), and C(x,y) are three homogeneous real-coefficient polynomials of x and y with degree 2, 3, and 4 respectively. we know that there is a real-coefficient polinimial R(x,y) such that B(x, y)2 −4A(x, y)C(x, y) = −R(x, y)2. Proof that there exist 2 polynomials F(x,y,z) and G(x,y,z) such that F(x, y, z)2 + G(x, y, z)2 = A(x, y)z2 + B(x, y)z + C(x, y) if for any x, y, z real numbers A(x, y)z2 + B(x, y)z + C(x, y) ≥0

题解状态：标准答案与规范题解待补充

题目标签：TST5-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 5 / 未标注日 · 组合数学

We call a graph with n vertices k −flowing −chromatic if: 1. we can place a chess on each vertex and any two neighboring (connected by an edge) chesses have different colors. 2. we can choose a hamilton cycle v1, v2, · · · , vn, and move the chess on vi to vi+1 with i = 1, 2, · · · , n and vn+1 = v1, such that any two neighboring chess also have different colors. 3. after some action of step 2 we can make all the chess reach each of the n vertices. Let T(G) denote the least number k such that G is k-flowing-chromatic. If such k does not exist, denote T(G)=0. denote χ(G) the chromatic number of G. Find all the positive number m such that there is a graph G with χ(G) ≤m and T(G) ≥2m without a cycle of length small than 2017.

题解状态：标准答案与规范题解待补充

题目标签：TST5-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。