MathDB

1

Part of 2017 Azerbaijan Team Selection Test

Problems(2)

Circle Passing Through a Fixed Point

Source: Azerbaijan IMO TST 2017, D2 P1

5/26/2018
Let ABCABC be an acute angled triangle. Points EE and FF are chosen on the sides ACAC and ABAB, respectively, such that BC2=BA×BF+CE×CA.BC^2=BA\times BF+CE\times CA. Prove that for all such EE and FF, circumcircle of the triangle AEFAEF passes through a fixed point different from AA.
geometrycircumcircle
Sequence with Number Theory

Source: Romania 2017 IMO TST 1, problem 3

3/18/2018
Consider the sequence of rational numbers defined by x1=43x_1=\frac{4}{3}, and xn+1=xn2xn2−xn+1x_{n+1}=\frac{x_n^2}{x_n^2-x_n+1}. Show that the nu,erator of the lowest term expression of each sum x1+x2+...+xkx_1+x_2+...+x_k is a perfect square.
Sequencenumber theory