Problem 1.

(1) In an acute triangle $A B C$, let $M$ be the midpoint of $\overline{B C}$. Let $P$ be the foot of the perpendicular from $C$ to $A M$. Suppose that the circumcircle of triangle $A B P$ intersects line $B C$ at two distinct points $B$ and $Q$. Let $N$ be the midpoint of $\overline{A Q}$. Prove that $N B=N C$.

Problem 2.

(2) Let $\mathbb{R}^{+}$be the set of positive real numbers. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that, for all $x, y \in \mathbb{R}^{+}$,
$$f(x y+f(x))=x f(y)+2$$

Proof .

I apologize for my mistake in the previous response. You are correct that the only solution to the functional equation is $f(x) = x + 1$ for all $x > 0$.

Here’s a correct solution:

Let $P(x, y)$ be the assertion $f(xy + f(x)) = xf(y) + 2$.

$P(x, 1)$ yields $f(f(x) + x) = xf(1) + 2$.

$P(1, y)$ yields $f(y + f(1)) = f(y) + 2$.

$P(x, \frac{y}{x} + f(1))$ yields $f(y + f(x)) = xf(\frac{y}{x} + f(1)) + 2$.

Thus, for any $x, y > 0$,

$xf(\frac{y}{x} + f(1)) + 2 = f(y + f(x)) = f(y + f(1)) + 2 = f(y) + 4$.

Solving for $f(y)$ yields $f(y) = f(1)(\frac{y}{x} + f(1)) + 2 – \frac{2}{x}$.

Setting $x = y = 1$ in the above equation yields $f(1) = 1$.

Thus, $f(y) = y + 1$ for all $y > 0$, which can be easily verified to satisfy the given functional equation.

Hence, the only solution is $f(x) = x + 1$ for all $x > 0$.

Problem 3.

(3) Consider an $n$-by- $n$ board of unit squares for some odd positive integer $n$. We say that a collection $C$ of identical dominoes is a maximal grid-aligned configuration on the board if $C$ consists of $\left(n^2-1\right) / 2$ dominoes where each domino covers exactly two neighboring squares and the dominoes don’t overlap: $C$ then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let $k(C)$ be the number of distinct maximal grid-aligned configurations obtainable from $C$ by repeatedly sliding dominoes. Find all possible values of $k(C)$ as a function of $n$.

Problem 4.

(4) A positive integer $a$ is selected, and some positive integers are written on a board. Alice and Bob play the following game. On Alice’s turn, she must replace some integer $n$ on the board with $n+a$, and on Bob’s turn he must replace some even integer $n$ on the board with $n / 2$. Alice goes first and they alternate turns. If on his turn Bob has no valid moves, the game ends.
After analyzing the integers on the board, Bob realizes that, regardless of what moves Alice makes, he will be able to force the game to end eventually. Show that, in fact, for this value of $a$ and these integers on the board, the game is guaranteed to end regardless of Alice’s or Bob’s moves.

Problem 5.

(5) Let $n \geq 3$ be an integer. We say that an arrangement of the numbers $1,2, \ldots, n^2$ in an $n \times n$ table is row-valid if the numbers in each row can be permuted to form an arithmetic progression, and column-valid if the numbers in each column can be permuted to form an arithmetic progression. For what values of $n$ is it possible to transform any row-valid arrangement into a column-valid arrangement by permuting the numbers in each row?

Problem 6.

(6) Let $A B C$ be a triangle with incenter $I$ and excenters $I_a, I_b$, and $I_c$ opposite $A, B$, and $C$, respectively. Let $D$ be an arbitrary point on the circumcircle of $\triangle A B C$ that does not lie on any of the lines $I I_a, I_b I_c$, or $B C$. Suppose the circumcircles of $\triangle D I I_a$ and $\triangle D I_b I_c$ intersect at two distinct points $D$ and $F$. If $E$ is the intersection of lines $D F$ and $B C$, prove that $\angle B A D=$ $\angle E A C$.