NMC 2024

Let $T(a)$ be the sum of digits of $a$. For which positive integers $R$ does there exist a positive integer $n$ such that $\frac{T\left(n^{2}\right)}{T(n)}=R$ ?

Let $\mathcal{Q}_{1}$ be a quadrilateral such that the midpoints of its sides lie on a circle. Prove that there exists a cyclic quadrilateral $\mathcal{Q}_{2}$ with the same sidelengths as $\mathcal{Q}_{1}$, such that two of the angles in $\mathcal{Q}_{2}$ are equal.

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that

$$f(f(x) f(y)+y)=f(x) y+f(y-x+1)$$

for all $x, y \in \mathbb{R}$.

Alice and Bob are playing a game. First, Alice chooses a partition $\mathcal{C}$ of the positive integers into a (not necessarily finite) set of sets, such that each positive integer is in exactly one of the sets in $\mathcal{C}$. Then Bob does the following operation a finite number of times. Choose a set $S \in \mathcal{C}$ not previously chosen, and let $D$ be the set of all positive integers dividing at least one element in $S$. Then add the set $D \backslash S$ (possibly the empty set) to $\mathcal{C}$. Bob wins if there are two equal sets in $\mathcal{C}$ after he has done all his moves, otherwise, Alice wins. Determine which player has a winning strategy.