The Bank of Oslo issues two types of coin: aluminum (denoted $A$) and bronze (denoted $B$). Marianne has $n$ aluminum coins and $n$ bronze coins arranged in a row in some arbitrary initial order. A chain is any subsequence of consecutive coins of the same type. Given a fixed positive integer $k \leq 2n$, Gilberty repeatedly performs the following operation: he identifies the longest chain containing the $k$th coin from the left and moves all coins in that chain to the left end of the row. For example, if $n=4$ and $k=4$, the process starting from the ordering $AABBBABA$ would be $AABBBABA \to BBBAAABA \to AAABBBBA \to BBBBAAAA \to \dotsb$.
Find all pairs $(n,k)$ with $1 \leq k \leq 2n$ such that for every initial ordering, at some moment during the process, the leftmost $n$ coins will all be of the same type.
Find all functions $f \colon \mathbb{R}^+ \to \mathbb{R}^+$ such that for each $x \in \mathbb{R}^+$, there is exactly one $y \in \mathbb{R}^+$ satisfying $$
xf(y)+yf(x) \leq 2.
$$ Here $\mathbb R^+$ denotes the set of positive real numbers.
Let $k$ be a positive integer and let $S$ be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of $S$ around the circle such that the product of any two neighbors is of the form $x^2+x+k$ for some positive integer $x$.
Let $ABCDE$ be a convex pentagon such that $BC=DE$. Assume that there is a point $T$ inside $ABCDE$ with $TB=TD$, $TC=TE$ and $\angle ABT = \angle TEA$. Let line $AB$ intersect lines $CD$ and $CT$ at points $P$ and $Q$, respectively. Assume that the points $P$, $B$, $A$, $Q$ occur on their line in that order. Let line $AE$ intersect $CD$ and $DT$ at points $R$ and $S$, respectively. Assume that the points $R$, $E$, $A$, $S$ occur on their line in that order. Prove that the points $P$, $S$, $Q$, $R$ lie on a circle.
Let $n$ be a positive integer. A *Nordic square* is an $n \times n$ board containing all the integers from $1$ to $n^2$ so that each cell contains exactly one number. An *uphill path* is a sequence of one or more cells such that:
1. the first cell in the sequence is a *valley*, meaning the number written is less than all its orthogonal neighbors,
2. each subsequent cell in the sequence is orthogonally adjacent to the previous cell, and
3. the numbers written in the cells in the sequence are in increasing order.
Find, as a function of $n$, the smallest possible total number of uphill paths in a Nordic square. Two different cells are adjacent precisely when they share a common side. A valley contains a number smaller than the numbers in all its adjacent cells.