MathDB

2

Part of 2009 China Team Selection Test

Problems(7)

Maximal constant

Source: ChInese TST 2009 P2

4/4/2009
Given an integer n≥2 n\ge 2, find the maximal constant λ(n) \lambda (n) having the following property: if a sequence of real numbers a0,a1,a2,⋯ ,an a_{0},a_{1},a_{2},\cdots,a_{n} satisfies 0 \equal{} a_{0}\le a_{1}\le a_{2}\le \cdots\le a_{n}, and a_{i}\ge\frac {1}{2}(a_{i \plus{} 1} \plus{} a_{i \minus{} 1}),i \equal{} 1,2,\cdots,n \minus{} 1, then (\sum_{i \equal{} 1}^n{ia_{i}})^2\ge \lambda (n)\sum_{i \equal{} 1}^n{a_{i}^2}.
inductionratioLaTeXinequalitiesblogsinequalities proposed
Table tennis match

Source: Chinese TST 2009 1st quiz P2

3/21/2009
Let n,k n,k be given positive integers satisfying k\le 2n \minus{} 1. On a table tennis tournament 2n 2n players take part, they play a total of k k rounds match, each round is divided into n n groups, each group two players match. The two players in different rounds can match on many occasions. Find the greatest positive integer m \equal{} f(n,k) such that no matter how the tournament processes, we always find m m players each of pair of which didn't match each other.
ceiling functionpigeonhole principleinductiongraph theorycombinatorics proposedcombinatorics
Integer set

Source: Chinese TST 2009 2nd quiz P2

3/21/2009
Find all the pairs of integers (a,b) (a,b) satisfying ab(a \minus{} b)\not \equal{} 0 such that there exists a subset Z0 Z_{0} of set of integers Z, Z, for any integer n n, exactly one among three integers n,n \plus{} a,n \plus{} b belongs to Z0 Z_{0}.
modular arithmeticalgebrapolynomialnumber theorycombinatorics proposedcombinatoricsAdditive combinatorics
Areas

Source: Chinese TST 2009 5th P2

4/4/2009
In acute triangle ABC, ABC, points P,Q P,Q lie on its sidelines AB,AC, AB,AC, respectively. The circumcircle of triangle ABC ABC intersects of triangle APQ APQ at X X (different from A A). Let Y Y be the reflection of X X in line PQ. PQ. Given PX>PB. PX>PB. Prove that S△XPQ>S△YBC. S_{\bigtriangleup XPQ}>S_{\bigtriangleup YBC}. Where S△XYZ S_{\bigtriangleup XYZ} denotes the area of triangle XYZ. XYZ.
geometrycircumcirclegeometric transformationreflectionratiotrigonometrygeometry proposed
Necessary and sufficient condition

Source: Chinese TST 2009 3rd quiz P2

3/22/2009
In convex quadrilateral ABCD ABCD, CB,DA CB,DA are external angle bisectors of ∠DCA,∠CDB \angle DCA,\angle CDB, respectively. Points E,F E,F lie on the rays AC,BD AC,BD respectively such that CEFD CEFD is cyclic quadrilateral. Point P P lie in the plane of quadrilateral ABCD ABCD such that DA,CB DA,CB are external angle bisectors of ∠PDE,∠PCF \angle PDE,\angle PCF respectively. AD AD intersects BC BC at Q. Q. Prove that P P lies on AB AB if and only if Q Q lies on segment EF EF.
geometryperpendicular bisectorangle bisectorLaw of Sines
Congruence equation

Source: Chinese TST 2009 4th P2

4/5/2009
Find all integers n≥2 n\ge 2 having the following property: for any k k integers a1,a2,⋯ ,ak a_{1},a_{2},\cdots,a_{k} which aren't congruent to each other (modulo n n), there exists an integer polynomial f(x) f(x) such that congruence equation f(x)≡0(modn) f(x)\equiv 0 (mod n) exactly has k k roots x≡a1,a2,⋯ ,ak(modn). x\equiv a_{1},a_{2},\cdots,a_{k} (mod n).
algebrapolynomialmodular arithmeticnumber theoryprime numbersnumber theory proposed
Complex polynomial

Source: Chinese TST 2009 6th P2

4/4/2009
Find all complex polynomial P(x) P(x) such that for any three integers a,b,c a,b,c satisfying a \plus{} b \plus{} c\not \equal{} 0, \frac{P(a) \plus{} P(b) \plus{} P(c)}{a \plus{} b \plus{} c} is an integer.
algebrapolynomialalgebra proposed