VN TST 2025

Find all functions $f:\mathbb Q_{>0}\to\mathbb Q_{>0}$ such that, for every $x,y\in\mathbb Q_{>0}$,

$$\frac{f(x)f(y)}{f(xy)}=\frac{(\sqrt{f(x)}+\sqrt{f(y)})^2}{f(x+y)}.$$

Question point values reproduce the supplied source labels; no automatic proof grading is implied.

Let $ABC$ be an acute scalene triangle with orthocenter $H$. Let $D,E,F$ be the reflections of $H$ in $BC,CA,AB$, respectively, and let $A^{\prime},B^{\prime},C^{\prime}$ be the reflections of $A,B,C$ in those same respective lines. Let $S$ be the circumcenter of $A^{\prime} B^{\prime} C^{\prime}$, and $H^{\prime}$ the orthocenter of $DEF$. Let $J$ be the center of the circle through the three perpendicular projections of $H$ onto $B^{\prime} C^{\prime},C^{\prime} A^{\prime},A^{\prime} B^{\prime}$. Prove that $HJ\parallel H^{\prime} S$.

At an applied-mathematics summer camp there are $8m+1$ boys, where $m>5$, and some girls. Each girl knows exactly three boys, and every two boys have exactly one girl whom they both know. Let $n$ be the largest number of girls that can be selected so that each boy knows at most one selected girl. Prove that $n\ge2m+1$.

Find all positive integers $k$ for which there are infinitely many positive integers $n$ such that $\binom{(2025+k)n}{2025n}$ is not divisible by $kn+1$.

An $n\times n$ board has rows and columns numbered $1$ through $n$; the cell in row $i$, column $j$ is denoted $(i,j)$. A subset $A$ of its cells is called good if, for any two cells $(x_1,y),(x_2,y)$ in $A$, none of the cells $(u,v)$ satisfying

$$x_1\lt u\le x_2,\quad v\lt y,$$

or

$$x_1\le u\lt x_2,\quad v>y,$$

belongs to $A$. Find the smallest number of pairwise disjoint good subsets such that every board cell belongs to exactly one subset.

For a prime $p=4k+3$, where $k\in\mathbb Z_{>0}$, consider

$$\begin{aligned}Q(x)={}&px^{2p}-x^{2p-1}+p^2x^{(3p+1)/2}-px^{p+1}\\&+2(p^2+1)x^p-px^{p-1}\\&+p^2x^{(p-1)/2}-x+p.\end{aligned}$$

Find all ordered pairs of polynomials $f(x),g(x)$ with integer coefficients such that $Q(x)=f(x)g(x)$.