Alice has five real numbers $a < b < c < d < e$. She takes the sum of each pair of numbers and writes down the ten sums. The three smallest sums are $32$, $36$ and $37$, while the two largest sums are $48$ and $51$. Determine $e$.

Let $ABCD$ be a parallelogram with an acute angle at $A$. Let $G$ be the point on the line $AB$, distinct from $B$, such that $CG = CB$. Let $H$ be the point on the line $BC$, distinct from $B$, such that $AB = AH$. Prove that triangle $DGH$ is isosceles.

Find all prime numbers $p$ such that $16p + 1$ is a perfect cube.

Ross wants to play solitaire with his deck of $n$ playing cards, but he’s discovered that the deck is “boxed”: some cards are face up, and others are face down. He wants to turn them all face down again, by repeatedly choosing a block of consecutive cards, removing the block from the deck, turning it over, and replacing it back in the deck at the same point. What is the smallest number of such steps Ross needs in order to guarantee that he can turn all the cards face down again, regardless of how they start out?

Find all pairs $(m, n)$ of positive integers such that the $m \times n$ grid contains exactly $225$ rectangles whose side lengths are odd and whose edges lie on the lines of the grid.

Let $ABCD$ be a quadrilateral. The circumcircle of the triangle $ABC$ intersects the sides $CD$ and $DA$ in the points $P$ and $Q$ respectively, while the circumcircle of $CDA$ intersects the sides $AB$ and $BC$ in the points $R$ and $S$. The lines $BP$ and $BQ$ intersect the line $RS$ in the points $M$ and $N$ respectively. Prove that the points $M, N, P$ and $Q$ lie on the same circle.

Let $a, b, c, d, e$ be distinct positive integers such that $$a^4 + b^4 = c^4 + d^4 = e^5.$$Show that $ac + bd$ is composite.

Find all possible real values for $a, b$ and $c$ such that (a) $a + b + c = 51$, (b) $abc = 4000$, (c) $0 < a \le 10$ and $c \ge 25$.

Let $k$ and $n$ be positive integers, with $k \le n$. A certain class has n students, and among any $k$ of them there is always one that is friends with the other $k- 1$. Find all values of $k$ and $n$ for which there must necessarily be a student who is friends with everyone else in the class.