We are given a positive integer $r$ and a rectangular board $ABCD$ with dimensions $|AB|=20$, $|BC|=12$. The rectangle is divided into a grid of $20\times12$ unit squares. The following moves are permitted on the board: One can move from one square to another only if the distance between the centers of the two squares is $\sqrt r$. The task is to find a sequence of moves leading from the square corresponding to vertex $A$ to the square corresponding to vertex $B$.
(a) Show that the task cannot be done if $r$ is divisible by $2$ or $3$.
(b) Prove that the task is possible when $r=73$.
(c) Is there a solution when $r=97$?
Let $P$ be a point inside $\triangle ABC$ such that
$$\angle APB-\angle C=\angle APC-\angle B.$$
Let $D,E$ be the incenters of $\triangle APB,\triangle APC$ respectively. Show that $AP$, $BD$, and $CE$ meet in a point.
Let $\mathbb N_0$ denote the set of nonnegative integers. Find all functions $f$ from $\mathbb N_0$ into itself such that
$$f(m+f(n))=f(f(m))+f(n),\qquad\forall m,n\in\mathbb N_0.$$
The positive integers $a$ and $b$ are such that the numbers $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?
Let $ABCDEF$ be a convex hexagon such that $AB$ is parallel to $DE$, $BC$ is parallel to $EF$, and $CD$ is parallel to $AF$. Let $R_A,R_C,R_E$ be the circumradii of triangles $FAB,BCD,DEF$ respectively, and let $P$ denote the perimeter of the hexagon. Prove that
$$R_A+R_C+R_E\ge\frac P2.$$
Let $p,q,n$ be three positive integers with $p+q\lt n$. Let $(x_0,x_1,\ldots,x_n)$ be an $(n+1)$-tuple of integers satisfying the following conditions:
(i) $x_0=x_n=0$.
(ii) For each $i$ with $1\le i\le n$, either $x_i-x_{i-1}=p$ or $x_i-x_{i-1}=-q$.
Show that there exists a pair $(i,j)$ of distinct indices with $(i,j)\ne(0,n)$ such that $x_i=x_j$.