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 $n \gt k \ge 1$ be integers.
Let $P(x) \in \mathbb{R}[x]$ be a polynomial of degree $n$
with no repeated roots and $P(0) \neq 0$.
Suppose that for any real numbers $a_0$, $\dots$, $a_k$
such that the polynomial $a_k x^k + \dots + a_1 x + a_0$ divides $P(x)$,
the product $a_0 a_1 \dots a_k$ is zero.
Prove that $P(x)$ has a nonreal root.
Proposed by Carl Schildkraut.
Manual essay; 1 editorial point, not official contest scoring. Automatic answer fields are intentionally empty.
Alice the architect and Bob the builder play a game.
First, Alice chooses two points $P$ and $Q$ in the plane
and a subset $\mathcal{S}$ of the plane, which are announced to Bob.
Next, Bob marks infinitely many points in the plane, designating each a city.
He may not place two cities within distance at most one unit of each other,
and no three cities he places may be collinear.
Finally, roads are constructed between the cities as follows:
each pair $A$, $B$ of cities is connected with a road along the line segment $AB$
if and only if the following condition holds:
For every city $C$ distinct from $A$ and $B$,
there exists $R\in \mathcal{S}$ such that $\triangle PQR$ is directly similar
to either $\triangle ABC$ or $\triangle BAC$.
Alice wins the game if (i) the resulting roads allow for travel
between any pair of cities via a finite sequence of roads and (ii) no two roads cross.
Otherwise, Bob wins. Determine, with proof, which player has a winning strategy.
Proposed by Carl Schildkraut.
Manual essay; 1 editorial point, not official contest scoring. Automatic answer fields are intentionally empty.
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 $m$ and $n$ be positive integers with $m\geq n$.
There are $m$ cupcakes of different flavors arranged around a circle
and $n$ people who like cupcakes.
Each person assigns a nonnegative real number score to each cupcake,
depending on how much they like the cupcake.
Suppose that for each person $P$, it is possible to partition the circle
of $m$ cupcakes into $n$ groups of consecutive cupcakes
so that the sum of $P$'s scores of the cupcakes in each group is at least $1$.
Prove that it is possible to distribute the $m$ cupcakes to the $n$ people
so that each person $P$ receives cupcakes of total score at least $1$ with respect to $P$.
Proposed by Cheng-Yin Chang and Hung-Hsun Yu.
Manual essay; 1 editorial point, not official contest scoring. Automatic answer fields are intentionally empty.