Given a rectangle $ABCD$ with $AC$ length $e$ and four circles centers $A$, $B$, $C$, $D$ and radii $a$, $b$, $c$, $d$ respectively, satisfying $a + c = b + d \lt e$. Prove you can inscribe a circle inside the quadrilateral whose sides are the two outer common tangents to the circles center $A$ and $C$, and the two outer common tangents to the circles center $B$ and $D$.
a. Given a quadruple $(a, b, c, d)$ of positive reals, transform to the new quadruple $(ab, bc, cd, da)$. Repeat arbitrarily many times. Prove that you can never return to the original quadruple unless $a = b = c = d = 1$.
b. Given $n$ a power of $2$, and an $n$-tuple $(a_1, a_2, \ldots, a_n)$ transform to a new $n$-tuple $(a_1 a_2, a_2 a_3, \ldots, a_{n-1} a_n, a_n a_1)$. If all the members of the original $n$-tuple are $1$ or $-1$, prove that with sufficiently many repetitions you obtain all $1$s.
Given three arbitrary infinite sequences of natural numbers, prove that we can find unequal natural numbers $m$, $n$ such that for each sequence the $m$th member is not less than the $n$th member.
Given an $m \times n$ array of real numbers. You may change the sign of all numbers in a row or of all numbers in a column. Prove that by repeated changes you can obtain an array with all row and column sums non-negative.
Given $n \gt 1$ points, some pairs joined by an edge (an edge never joins a point to itself). Given any two distinct points you can reach one from the other in just one way by moving along edges. Prove that there are $n - 1$ edges.
Given any natural numbers $m$, $n$ and $k$. Prove that we can always find relatively prime natural numbers $r$ and $s$ such that $rm + sn$ is a multiple of $k$.
120 unit squares are arbitrarily arranged in a $20 \times 25$ rectangle (both position and orientation is arbitrary). Prove that it is always possible to place a circle of unit diameter inside the rectangle without intersecting any of the squares.
A and $B$ play the following game with $N$ counters. $A$ divides the counters into $2$ piles, each with at least $2$ counters. Then $B$ divides each pile into $2$ piles, each with at least one counter. $B$ then takes $2$ piles according to a rule which both of them know, and $A$ takes the remaining $2$ piles. Both $A$ and $B$ make their choices in order to end up with as many counters as possible. There are $3$ possibilities for the rule:
$R1$ $B$ takes the biggest heap (or one of them if there is more than one) and the smallest heap (or one of them if there is more than one).
$R2$ $B$ takes the two middling heaps (the two heaps that $A$ would take under $R1$).
$R3$ $B$ has the choice of taking either the biggest and smallest, or the two middling heaps. For each rule, how many counters will $A$ get if both players play optimally?