MathDB

6

Part of 2008 239 Open Mathematical Olympiad

Problems(2)

Two variable polynomial and function

Source: 239 2008 S6

7/28/2020
Given a polynomial P(x,y)P(x,y) with real coefficients, suppose that some real function f:R→Rf:\mathbb R \to \mathbb R satisfies P(x,y)=f(x+y)−f(x)−f(y)P(x,y) = f(x+y)-f(x)-f(y)for all x,y∈Rx,y\in\mathbb R. Show that some polynomial qq satisfies P(x,y)=q(x+y)−q(x)−q(y)P(x,y) = q(x+y)-q(x)-q(y)
polynomialalgebrafunction
Three circles and a chord

Source: 239 2008 J6

7/28/2020
ABAB is the chord of the circle SS. Circles S1S_1 and S2S_2 touch the circle SS at points PP and QQ, respectively, and the segment ABAB at point KK. It turned out that ∠PBA=∠QBA\angle{PBA}=\angle{QBA}. Prove that ABAB is the diameter of the circle SS.
geometry