Annie is playing a game where she starts with a row of positive integers,
written on a blackboard, each of which is a power of $2$.
On each turn, she can erase two adjacent numbers and replace them with a power of $2$
that is greater than either of the erased numbers.
This shortens the row of numbers, and she continues to take turns until only one number remains.
Annie wins the game if the final remaining number is less than $4$ times the sum of the original numbers.
Is it always possible for Annie to win, regardless of the starting row of numbers?
Proposed by Andrew Dittmer.
Manual essay; 1 editorial point, not official contest scoring. Automatic answer fields are intentionally empty.
Let $ABC$ be an acute scalene triangle with no angle equal to $60^\circ$.
Let $\omega$ be the circumcircle of $ABC$.
Let $\Delta_B$ be the equilateral triangle with three vertices on $\omega$,
one of which is $B$.
Let $\ell_B$ be the line through the two vertices of $\Delta_B$ other than $B$.
Let $\Delta_C$ and $\ell_C$ be defined analogously.
Let $Y$ be the intersection of $AC$ and $\ell_B$,
and let $Z$ be the intersection of $AB$ and $\ell_C$.
Let $N$ be the midpoint of minor arc $BC$ on $\omega$.
Let $\mathcal{R}$ be the triangle formed by $\ell_B$, $\ell_C$, and the tangent to $\omega$ through $N$.
Prove that the circumcircle of $AYZ$ and the incircle of $\mathcal{R}$ are tangent.
Proposed by Rogelio Guerrero.
Manual essay; 1 editorial point, not official contest scoring. Automatic answer fields are intentionally empty.
A positive integer $n$ is called solitary if,
for any non-negative integers $a$ and $b$ such that $a+b=n$,
either $a$ or $b$ contains the digit $1$.
Determine, with proof, the number of solitary integers less than $10^{2026}$.
Proposed by Carl Schildkraut.
Manual essay; 1 editorial point, not official contest scoring. Automatic answer fields are intentionally empty.
Let $ABC$ be a triangle.
Points $D$, $E$, and $F$ lie on sides $BC$, $CA$, and $AB$, respectively, such that
$$\angle AFE = \angle BDF = \angle CED.$$
Let $O_A$, $O_B$, and $O_C$ be the circumcenters of triangles $AFE$, $BDF$, and $CED$, respectively.
Let $M$, $N$, and $O$ be the circumcenters of triangles $ABC$, $DEF$, and $O_AO_BO_C$, respectively.
Prove that $OM = ON$.
Proposed by Serena An.
Manual essay; 1 editorial point, not official contest scoring. Automatic answer fields are intentionally empty.
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.
Find all integers $n \geq 3$ such that the following property holds:
if we list the divisors of $n!$ in increasing order
as $1 = d_1 \lt d_2 \lt \dots \lt d_k = n!$, then we have
Let $S_1$, $S_2$, …, $S_{100}$ be finite sets of integers whose intersection is not empty.
For each non-empty $T \subseteq \{S_1, S_2, \ldots, S_{100}\}$,
the size of the intersection of the sets in $T$ is a multiple of $|T|$.
What is the smallest possible number of elements which are in at least $50$ sets?
Proposed by Rishabh Das. One nonofficial editorial point; manual essay grading.
Let $(m,n)$ be positive integers with $n \ge 3$ and draw a regular $n$-gon.
We wish to triangulate this $n$-gon into $n-2$ triangles,
each colored one of $m$ colors, so that each color has an equal sum of areas.
For which $(m,n)$ is such a triangulation and coloring possible?
Proposed by Krit Boonsiriseth. One nonofficial editorial point; manual essay grading.
Let $m$ and $n$ be positive integers.
A circular necklace contains $mn$ beads, each either red or blue.
It turned out that no matter how the necklace was cut into $m$ blocks of $n$
consecutive beads, each block had a distinct number of red beads.
Determine, with proof, all possible values of the ordered pair $(m, n)$.
Proposed by Rishabh Das. One nonofficial editorial point; manual essay grading.
Point $D$ is selected inside acute triangle $ABC$
so that $\angle DAC = \angle ACB$ and $\angle BDC = 90^\circ + \angle BAC$.
Point $E$ is chosen on ray $BD$ so that $AE = EC$.
Let $M$ be the midpoint of $BC$.
Show that line $AB$ is tangent to the circumcircle of triangle $BEM$.
Proposed by Anton Trygub. One nonofficial editorial point; manual essay grading.
Let $n > 2$ be an integer and let $\ell \in \{1, 2,\dots, n\}$.
A collection $A_1$, …, $A_k$ of (not necessarily distinct) subsets of
$\{1, 2,\dots, n\}$ is called $\ell$-large if $|A_i| \ge \ell$ for all $1 \le i \le k$.
Find, in terms of $n$ and $\ell$, the largest real number $c$ such that the inequality
holds for all positive integer $k$,
all nonnegative real numbers $x_1$, $x_2$, …, $x_k$,
and all $\ell$-large collections $A_1$, $A_2$, …, $A_k$
of subsets of $\{1,2,\dots,n\}$.
Proposed by Titu Andreescu, Gabriel Dospinescu. One nonofficial editorial point; manual essay grading.
In an acute triangle $ABC$, let $M$ be the midpoint of $\overline{BC}$.
Let $P$ be the foot of the perpendicular from $C$ to $AM$.
Suppose that the circumcircle of triangle $ABP$
intersects line $BC$ at two distinct points $B$ and $Q$.
Let $N$ be the midpoint of $\overline{AQ}$.
Prove that $NB = NC$.
Proposed by Holden Mui. One nonofficial editorial point; manual essay grading.