ZA-SAMO 2017

Together, the two positive integers $a$ and $b$ have $9$ digits and contain each of the digits $1, 2, 3, 4, 5, 6, 7, 8, 9$ exactly once. For which possible values of $a$ and $b$ is the fraction $a/b$ closest to $1$ ?

Let $ABCD$ be a rectangle with side lengths $AB = CD = 5$ and $BC = AD = 10$ . $W, X, Y, Z$ are points on $AB, BC, CD$ and $DA$ respectively chosen in such a way that $WXYZ$ is a kite, where $\angle ZWX$ is a right angle. Given that $WX = WZ = \sqrt{13}$ and $XY = ZY$ , determine the length of $XY$ .

A representation of $\frac{17}{20}$ as a sum of reciprocals $$\frac{17}{20} = \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_k}$$ is called a calm representation with $k$ terms if the $a_i$ are distinct positive integers and at most one of them is not a power of two.

(a) Find the smallest value of $k$ for which $\frac{17}{20}$ has a calm representation with $k$ terms. (b) Prove that there are infinitely many calm representations of $\frac{17}{20}$ .

Andile and Zandre play a game on a $2017 \times 2017$ board. At the beginning, Andile declares some of the squares forbidden, meaning the nothing may be placed on such a square. After that, they take turns to place coins on the board, with Zandre placing the first coin. It is not allowed to place a coin on a forbidden square or in the same row or column where another coin has already been placed. The player who places the last coin wins the game.

What is the least number of squares Andile needs to declare as forbidden at the beginning to ensure a win? (Assume that both players use an optimal strategy.)

Let $ABC$ be a triangle with circumcircle $\Gamma$ . Let $D$ be a point on segment $BC$ such that $\angle BAD = \angle DAC$ , and let $M$ and $N$ be points on segments $BD$ and $CD$ , respectively, such that $\angle MAD = \angle DAN$ . Let $S, P$ and $Q$ (all different from $A$ ) be the intersections of the rays $AD$ , $AM$ and $AN$ with $\Gamma$ , respectively.

Show that the intersection of $SM$ and $QD$ lies on $\Gamma$ .

Determine all pairs $(P, d)$ of a polynomial $P$ with integer coefficients and an integer $d$ such that the equation $P(x) - P(y) = d$ has infinitely many solutions in integers $x$ and $y$ with $x \neq y$ .