MathDB

3

Part of 2008 China Team Selection Test

Problems(7)

Chinese TST 2008 P3

Source:

4/3/2008
Suppose that every positve integer has been given one of the colors red, blue,arbitrarily. Prove that there exists an infinite sequence of positive integers a1<a2<a3<⋯<an<⋯ , a_{1} < a_{2} < a_{3} < \cdots < a_{n} < \cdots, such that inifinite sequence of positive integers a_{1},\frac {a_{1} \plus{} a_{2}}{2},a_{2},\frac {a_{2} \plus{} a_{3}}{2},a_{3},\frac {a_{3} \plus{} a_{4}}{2},\cdots has the same color.
searcharithmetic sequencecombinatorics proposedcombinatoricsRamsey Theory
The greatest integer

Source: Chinese TST

4/5/2008
Determine the greatest positive integer n n such that in three-dimensional space, there exist n points P1,P2,⋯ ,Pn, P_{1},P_{2},\cdots,P_{n}, among n n points no three points are collinear, and for arbitary 1≤i<j<k≤n 1\leq i < j < k\leq n, PiPjPk P_{i}P_{j}P_{k} isn't obtuse triangle.
geometry3D geometrytetrahedroncombinatorics proposedcombinatorics
Complex number

Source: Chinese TST

4/6/2008
Let z1,z2,z3 z_{1},z_{2},z_{3} be three complex numbers of moduli less than or equal to 1 1. w1,w2 w_{1},w_{2} are two roots of the equation (z \minus{} z_{1})(z \minus{} z_{2}) \plus{} (z \minus{} z_{2})(z \minus{} z_{3}) \plus{} (z \minus{} z_{3})(z \minus{} z_{1}) \equal{} 0. Prove that, for j \equal{} 1,2,3, \min\{|z_{j} \minus{} w_{1}|,|z_{j} \minus{} w_{2}|\}\leq 1 holds.
Gaussalgebrapolynomialgeometrycomplex numberscombinatorial geometryalgebra proposed
Polynomial

Source: Chinese TST

4/5/2008
Let n>m>1 n>m>1 be odd integers, let f(x)\equal{}x^n\plus{}x^m\plus{}x\plus{}1. Prove that f(x) f(x) can't be expressed as the product of two polynomials having integer coefficients and positive degrees.
algebrapolynomialalgebra proposed
The number of the subset

Source: Chinese TST

4/9/2008
Let S S be a set that contains n n elements. Let A1,A2,⋯ ,Ak A_{1},A_{2},\cdots,A_{k} be k k distinct subsets of S S, where k\geq 2, |A_{i}| \equal{} a_{i}\geq 1 ( 1\leq i\leq k). Prove that the number of subsets of S S that don't contain any Ai(1≤i≤k) A_{i} (1\leq i\leq k) is greater than or equal to 2^n\prod_{i \equal{} 1}^k(1 \minus{} \frac {1}{2^{a_{i}}}).
probabilitycombinatorics proposedcombinatoricsTSTChina TST
An inequality

Source: Chinese TST

4/6/2008
Let 0 < x_{1}\leq\frac {x_{2}}{2}\leq\cdots\leq\frac {x_{n}}{n}, 0 < y_{n}\leq y_{n \minus{} 1}\leq\cdots\leq y_{1}, Prove that (\sum_{k \equal{} 1}^{n}x_{k}y_{k})^2\leq(\sum_{k \equal{} 1}^{n}y_{k})(\sum_{k \equal{} 1}^{n}(x_{k}^2 \minus{} \frac {1}{4}x_{k}x_{k \minus{} 1})y_{k}). where x_{0} \equal{} 0.
inequalitiesinductionrearrangement inequalityinequalities proposed
Lattice polygon

Source: Chinese TST

4/9/2008
Find all positive integers n n having the following properties:in two-dimensional Cartesian coordinates, there exists a convex n n lattice polygon whose lengths of all sides are odd numbers, and unequal to each other. (where lattice polygon is defined as polygon whose coordinates of all vertices are integers in Cartesian coordinates.)
analytic geometrygeometrymodular arithmetictrapezoidinductiontrigonometrynumber theory proposed