MathDB

1

Part of 1992 China National Olympiad

Problems(2)

China Mathematical Olympiad 1992 problem1

Source: China Mathematical Olympiad 1992 problem1

9/29/2013
Let equation xn+an−1xn−1+an−2xn−2+⋯+a1x+a0=0x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\dots +a_1x+a_0=0 with real coefficients satisfy 0<a0≤a1≤a2≤⋯≤an−1≤10<a_0\le a_1\le a_2\le \dots \le a_{n-1}\le 1. Suppose that λ\lambda (∣λ∣>1|\lambda|>1) is a complex root of the equation, prove that λn+1=1\lambda^{n+1}=1.
inequalitiestriangle inequalityalgebra unsolvedalgebra
China Mathematical Olympiad 1992 problem4

Source: China Mathematical Olympiad 1992 problem4

9/30/2013
A convex quadrilateral ABCDABCD is inscribed in a circle with center OO. The diagonals ACAC, BDBD of ABCDABCD meet at PP. Circumcircles of △ABP\triangle ABP and △CDP\triangle CDP meet at PP and QQ (O,P,QO,P,Q are pairwise distinct). Show that ∠OQP=90∘\angle OQP=90^{\circ}.
geometrycircumcircleparallelogramgeometric transformationreflectiontrapezoidsymmetry