Let $ABC$ be an acute triangle inscribed in a circle $\Gamma$. Let $A_1$ be the orthogonal projection of $A$ onto $BC$, so $AA_1$ is an altitude. Let $B_1,C_1$ be the orthogonal projections of $A_1$ onto $AB,AC$, respectively. Point $P$ is such that quadrilateral $AB_1PC_1$ is convex and has the same area as triangle $ABC$. Is it possible that $P$ lies strictly inside $\Gamma$? Justify your answer.
AI-generated diagram of the given triangle and projections:
Let $\alpha,\beta$ be positive real numbers. Emerald makes a trip in the coordinate plane, starting at $(0,0)$. Each minute she moves one unit up or one unit right, always staying in the region $|x-y|\lt 2025$. Upon visiting $(x,y)$, she writes the integer $\lfloor x\alpha+y\beta\rfloor$ there. She writes each nonnegative integer exactly once. Find all pairs $(\alpha,\beta)$ for which such a trip is possible.
Let $P(x)$ be a nonconstant polynomial with integer coefficients such that $P(0)\ne0$. Let $a_1,a_2,a_3,\ldots$ be an infinite sequence of integers such that $P(i-j)$ divides $a_i-a_j$ for all distinct positive integers $i,j$. Prove that the sequence is constant.
Let $n\ge3$ be an integer. There are $n$ cells on a circle, each assigned $0$ or $1$. A rooster stands on one cell and repeatedly does the following:
- On a cell assigned $0$, it changes the number to $1$ and moves to the next cell counterclockwise.
- On a cell assigned $1$, it changes the number to $0$ and moves to the cell after the next cell counterclockwise.
Prove that after sufficiently many operations the following statement holds: if the rooster is on a cell $C$, it will go around the circle exactly three times and stop again at $C$, with every cell assigned the same number as immediately before these three laps.
Consider an infinite sequence $a_1,a_2,\ldots$ of positive integers such that
$$100!\,(a_m+a_{m+1}+\cdots+a_n)$$
is a multiple of $a_{n-m+1}a_{n+m}$ for all positive integers $m\le n$. Prove that the sequence is either bounded or linear.
A sequence is bounded if there is a constant $N$ with $a_n\lt N$ for every positive integer $n$. It is linear if $a_n=n\,a_1$ for every positive integer $n$.
