HMIC 2025

Let $A B C D$ be a convex quadrilateral. Define parabolas $\mathcal{P}_{A}$ $\mathcal{P}_{B}$ $\mathcal{P}_{C}$ , and $\mathcal{P}_{D}$ to have directrices $B D$ $C A$ $D B$ , and $A C$ , and foci $A$ $B$ $C$ , and $D$ , respectively. Prove that no two of these parabolas intersect more than once.

(A parabola with directrix $\ell$ and focus $P$ consists of all points $X$ for which $P X$ equals the distance from $P$ to $\ell$ .)

Find all polynomials $P$ with real coefficients for which there exists a polynomial $Q$ with real coefficients such that for all real $t$ ,

$$\cos (P(t)) = Q(\cos t).$$

Let $ABCD$ be a parallelogram, and let $O$ be a point inside $ABCD$ . Suppose the circumcircles of triangles $OAB$ and $OCD$ intersect at $P \neq O$ , and the circumcircles of triangles $OBC$ and $OAD$ intersect at $Q \neq O$ . Prove $\angle POQ$ equals one of the angles of quadrilateral $ABCD$ .

Determine whether there exist infinitely many pairs of distinct positive integers $m$ and $n$ such that $2^{m} + n$ divides $2^{n} + m$ .

Compute the smallest positive integer $k > 45$ for which there exists a sequence $a_{1}$ , $a_{2}$ , $a_{3}$ , ..., $a_{k - 1}$ of positive integers satisfying the following conditions:

$a_{i} = i$ for all integers $1\leq i\leq 45$ $a_{k - i} = i$ for all integers $1\leq i\leq 45$ , and for any odd integer $1\leq n\leq k - 45$ , the sequence $a_{n}$ , $a_{n + 1}$ , ..., $a_{n + 44}$ is a permutation of $\{1,2,\ldots ,45\}$ .