FI-MO 2017

By dividing the integer $m$ by the integer $n, 22$ is the quotient and $5$ the remainder. As the division of the remainder with $n$ continues, the new quotient is $0.4$ and the new remainder is $0.2$ . Find $m$ and $n$ .

Determine $x^2+y^2$ and $x^4+y^4$ , when $x^3+y^3=2$ and $x+y=1$

Consider positive integers $m$ and $n$ for which $m> n$ and the number $22 220 038^m-22 220 038^n$ has are eight zeros at the end. Show that $n> 7$ .

Let $m$ be a positive integer. Two players, Axel and Elina play the game HAUKKU ( $m$ ) proceeds as follows: Axel starts and the players choose integers alternately. Initially, the set of integers is the set of positive divisors of a positive integer $m$ .The player in turn chooses one of the remaining numbers, and removes that number and all of its multiples from the list of selectable numbers. A player who has to choose number $1$ , loses. Show that the beginner player, Axel, has a winning strategy in the HAUKKU ( $m$ ) game for all $m \in Z_{+}$ .

PS. As member Loppukilpailija noted, it should be written $m>1$ , as the statement does not hold for $m = 1$ .

Let $A$ and $B$ be two arbitrary points on the circumference of the circle such that $AB$ is not the diameter of the circle. The tangents to the circle drawn at points $A$ and $B$ meet at $T$ . Next, choose the diameter $XY$ so that the segments $AX$ and $BY$ intersect. Let this be the intersection of $Q$ . Prove that the points $A, B$ , and $Q$ lie on a circle with center $T$ .