Find all pairs $(a, b)$ of real numbers with the following property: Given any real numbers $c$ and $d$, if both of the equations $x^2+ax+1=c$ and $x^2+bx+1=d$ have real roots, then the equation $x^2+(a+b)x+1=cd$ has real roots.

Let $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$, $I$ be nine points in space such that $ABCDE$, $ABFGH$, and $GFCDI$ are each regular pentagons with side length $1$. Determine the lengths of the sides of triangle $EHI$.

Call a permutation of the integers $(1,2,\ldots,n)$ $d$-ordered if it does not contains a decreasing subsequence of length $d$. Prove that for every $d=2,3,\ldots,n$, the number of $d$-ordered permutations of $(1,2,\ldots,n)$ is at most $(d-1)^{2n}$.

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that \[f(xy)=\max\{f(x+y),f(x) f(y)\} \] for all real numbers $x$ and $y$.

Find all pairs $(a,b)$ of integers with $a>1$ and $b>1$ such that $a$ divides $b+1$ and $b$ divides $a^3-1$.

Alice and Bob play the following game. It begins with a set of $1000$ $1\times 2$ rectangles. A move consists of choosing two rectangles (a rectangle may consist of one or several $1\times 2$ rectangles combined together) that share a common side length and combining those two rectangles into one rectangle along those sides sharing that common length. The first player who cannot make a move loses. Alice moves first. Describe a winning strategy for Bob.