Team tryout problem set.
Eight olympiad problems across the main contest topics. Work through as many as you can. This set is how we get to know how you think.
Read this first.
- 1.You are not expected to solve all eight. A few complete, well-written solutions tell us far more than eight rushed ones.
- 2.Write full solutions. Explain your reasoning, not just the answer. Partial progress and good ideas count.
- 3.Work on your own. Using the internet to look up a problem defeats the purpose; we want to see your thinking.
- 4.Write up your solutions (typed or neatly handwritten and scanned) and submit them with your application.
Eight problems.
Topics are labeled. Two are marked Challenge, so save them for last.
Find all polynomials $P(x)$ with real coefficients such that
$$P(x)\,P(x+1) = P\!\left(x^2 + x + 1\right) \qquad \text{for all real } x.$$
Hint: think about what the roots of $P$ would have to satisfy.
Prove that for every integer $n$, the number $n^5 - n$ is divisible by $30$.
Each cell of a $3 \times 7$ grid is colored black or white. Prove that there exist two rows and two columns such that the four cells lying at their intersections all have the same color.
Let $a, b, c$ be positive real numbers. Prove that
$$\frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b} \;\ge\; \frac{3}{2}.$$
Let $ABC$ be a triangle with incenter $I$. Line $AI$ meets the circumcircle of $ABC$ again at $M$. Prove that
$$MB = MC = MI.$$
Hint: chase the inscribed angles at $B$ and $C$.
Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that, for all real $x$ and $y$,
$$f(x+y) = f(x) + f(y) \qquad \text{and} \qquad f(xy) = f(x)\,f(y).$$
The numbers $1, 2, 3, \dots, 2026$ are written on a board. In one move you erase two numbers $a$ and $b$ and write their absolute difference $|a - b|$. After $2025$ moves a single number remains. Prove that this number is odd.
Find all integer solutions $(x, y, z)$ of the equation
$$x^2 + y^2 + z^2 = 2xyz.$$
Hint: consider the parity of $x$, $y$, $z$, and what it forces.