MathDB

2

Part of 2010 China Team Selection Test

Problems(8)

China 2010 quiz1 Problem 2

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9/5/2010
Let ABCDABCD be a convex quadrilateral. Assume line ABAB and CDCD intersect at EE, and BB lies between AA and EE. Assume line ADAD and BCBC intersect at FF, and DD lies between AA and FF. Assume the circumcircles of △BEC\triangle BEC and △CFD\triangle CFD intersect at CC and PP. Prove that ∠BAP=∠CAD\angle BAP=\angle CAD if and only if BD∥EFBD\parallel EF.
geometrycircumcircletrigonometryparallelogramgeometry unsolved
China 2010 quiz2 problem 2

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9/8/2010
Let M={1,2,⋯ ,n}M=\{1,2,\cdots,n\}, each element of MM is colored in either red, blue or yellow. Set A={(x,y,z)∈M×M×M∣x+y+z≡0mod  nA=\{(x,y,z)\in M\times M\times M|x+y+z\equiv 0\mod n, x,y,zx,y,z are of same color},\}, B={(x,y,z)∈M×M×M∣x+y+z≡0mod  n,B=\{(x,y,z)\in M\times M\times M|x+y+z\equiv 0\mod n, x,y,zx,y,z are of pairwise distinct color}.\}. Prove that 2∣A∣≥∣B∣2|A|\geq |B|.
polynomialinequalitiescombinatorics
China 2010 quiz3 problem 2

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9/11/2010
Given positive integer nn, find the largest real number λ=λ(n)\lambda=\lambda(n), such that for any degree nn polynomial with complex coefficients f(x)=anxn+an−1xn−1+⋯+a0f(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+a_0, and any permutation x0,x1,⋯ ,xnx_0,x_1,\cdots,x_n of 0,1,⋯ ,n0,1,\cdots,n, the following inequality holds ∑k=0n∣f(xk)−f(xk+1)∣≥λ∣an∣\sum_{k=0}^n|f(x_k)-f(x_{k+1})|\geq \lambda |a_n|, where xn+1=x0x_{n+1}=x_0.
algebrapolynomialinequalitiesfloor functiontriangle inequalityinequalities unsolved
China 2010 quiz4 problem 2

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9/12/2010
Find all positive real numbers λ\lambda such that for all integers n≥2n\geq 2 and all positive real numbers a1,a2,⋯ ,ana_1,a_2,\cdots,a_n with a1+a2+⋯+an=na_1+a_2+\cdots+a_n=n, the following inequality holds: ∑i=1n1ai−λ∏i=1n1ai≤n−λ\sum_{i=1}^n\frac{1}{a_i}-\lambda\prod_{i=1}^{n}\frac{1}{a_i}\leq n-\lambda.
inequalitiesinequalities unsolved
China 2010 quiz5 Problem 2

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9/13/2010
In a football league, there are n≥6n\geq 6 teams. Each team has a homecourt jersey and a road jersey with different color. When two teams play, the home team always wear homecourt jersey and the road team wear their homecourt jersey if the color is different from the home team's homecourt jersey, or otherwise the road team shall wear their road jersey. It is required that in any two games with 4 different teams, the 4 teams' jerseys have at least 3 different color. Find the least number of color that the nn teams' 2n2n jerseys may use.
combinatorics unsolvedcombinatorics
China 2010 quiz6 Problem 2

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9/14/2010
Prove that there exists a sequence of unbounded positive integers a1≤a2≤a3≤⋯a_1\leq a_2\leq a_3\leq\cdots, such that there exists a positive integer MM with the following property: for any integer n≥Mn\geq M, if n+1n+1 is not prime, then any prime divisor of n!+1n!+1 is greater than n+ann+a_n.
number theoryprime numbersfactorial
China TST 2010, Problem 2

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8/28/2010
Let A={a1,a2,⋯ ,a2010}A=\{a_1,a_2,\cdots,a_{2010}\} and B={b1,b2,⋯ ,b2010}B=\{b_1,b_2,\cdots,b_{2010}\} be two sets of complex numbers. Suppose ∑1≤i<j≤2010(ai+aj)k=∑1≤i<j≤2010(bi+bj)k\sum_{1\leq i<j\leq 2010} (a_i+a_j)^k=\sum_{1\leq i<j\leq 2010}(b_i+b_j)^k holds for every k=1,2,⋯ ,2010k=1,2,\cdots, 2010. Prove that A=BA=B.
algebrapolynomialfunctioncomplex numbersalgebra unsolved
China TST 2010, Problem 5

Source:

8/28/2010
Given integer a1≥2a_1\geq 2. For integer n≥2n\geq 2, define ana_n to be the smallest positive integer which is not coprime to an−1a_{n-1} and not equal to a1,a2,⋯ ,an−1a_1,a_2,\cdots, a_{n-1}. Prove that every positive integer except 1 appears in this sequence {an}\{a_n\}.
inductionnumber theorygreatest common divisoralgebra unsolvedalgebra