Fix positive integers $k$ and $d$.
Prove that for all sufficiently large odd positive integers $n$,
the digits of the base-$2n$ representation of $n^k$ are all greater than $d$.
Proposed by John Berman.
Manual essay; 1 editorial point, not official contest scoring. Automatic answer fields are intentionally empty.
Let $m,n$ be positive integers and let $\mathcal R$ be a $2m\times2n$ grid of unit squares. A domino is a $1\times2$ or $2\times1$ rectangle. An up-right path goes from the lower-left corner to the upper-right corner using exactly $2m+2n$ grid edges. Find the number of these paths dividing $\mathcal R$ into two possibly empty subsets, each tileable with dominoes.
Proposed by Wilbert Chu.
Manual essay; 1 editorial point, not official contest scoring.
Let $n$ be a positive integer and let $a_0\ge a_1\ge\cdots\ge a_n\ge0$ be integers. Prove that $$\sum_{i=0}^n i\binom{a_i}{2}\le\frac12\binom{a_0+a_1+\cdots+a_n}{2}.$$
Proposed by Hung-Hsun Yu.
Manual essay; 1 editorial point, not official contest scoring.
Let $H$ be the orthocenter of an acute triangle $ABC$,
let $F$ be the foot of the altitude from $C$ to $AB$,
and let $P$ be the reflection of $H$ across $BC$.
Suppose that the circumcircle of triangle $AFP$ intersects line $BC$
at two distinct points $X$ and $Y$.
Prove that $CX = CY$.
Proposed by Carl Schildkraut.
Manual essay; 1 editorial point, not official contest scoring. Automatic answer fields are intentionally empty.
Let $S$ be a set of integers such that (i) $\{1,2,\ldots,2025\}\subseteq S$; (ii) if $a,b\in S$ and $\gcd(a,b)=1$, then $ab\in S$; (iii) if $s\in S$ and $s+1$ is composite, then all positive divisors of $s+1$ belong to $S$. Prove that $S$ contains all positive integers.
Proposed by Enrique Treviño.
Manual essay; 1 editorial point, not official contest scoring.