MathDB

3

Part of 2012 Bulgaria National Olympiad

Problems(2)

Winning strategy involving coefficents of a polynomial

Source: Bulgarian National Olympiad 2012 Problem 3

5/21/2012
We are given a real number aa, not equal to 00 or 11. Sacho and Deni play the following game. First is Sasho and then Deni and so on (they take turns). On each turn, a player changes one of the “*” symbols in the equation: ∗x4+∗x3+∗x2+∗x1+∗=0*x^4+*x^3+*x^2+*x^1+*=0 with a number of the type ana^n, where nn is a whole number. Sasho wins if at the end the equation has no real roots, Deni wins otherwise. Determine (in term of aa) who has a winning strategy
algebrapolynomialalgebra proposed
Circumcircle of three points passes through fixed point

Source: Bulgarian National Olympiad 2012 Problem 6

5/21/2012
We are given an acute-angled triangle ABCABC and a random point XX in its interior, different from the centre of the circumcircle kk of the triangle. The lines AX,BXAX,BX and CXCX intersect kk for a second time in the points A1,B1A_1,B_1 and C1C_1 respectively. Let A2,B2A_2,B_2 and C2C_2 be the points that are symmetric of A1,B1A_1,B_1 and C1C_1 in respect to BC,ACBC,AC and ABAB respectively. Prove that the circumcircle of the triangle A2,B2A_2,B_2 and C2C_2 passes through a constant point that does not depend on the choice of XX.
geometrycircumcirclegeometry proposed