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Stage 10 · Mixed

10 problems · Hardest shortlist tier · mathsolympiadprep.com

The answer key prints on its own page at the end.

There are 2022 equally spaced points on a circular track $\gamma$ of circumference 2022. The points are labeled $A_1, A_2, \dots, A_{2022}$ in some order, each label used once. Initially, Bunbun the Bunny begins at $A_1$ . She hops along $\gamma$ from $A_1$ to $A_2$ , then from $A_2$ to $A_3$ , until she reaches $A_{2022}$ , after which she hops back to $A_1$ . When hopping from $P$ to $Q$ , she always hops along the shorter of the two arcs $\text{PQ}$ of $\gamma$ ; if $\text{PQ}$ is a diameter of $\gamma$ , she moves along either semicircle.
Determine the maximal possible sum of the lengths of the 2022 arcs which Bunbun traveled, over all possible labellings of the 2022 points.

USA TST

Geometry Solution and answer checking →
Let $c \geqslant 1$ be an integer. Define a sequence of positive integers by $a_{1}=c$ and
$a_{n+1}=a_{n}^{3}-4 c \cdot a_{n}^{2}+5 c^{2} \cdot a_{n}+c$
for all $n \geqslant 1$ . Prove that for each integer $n \geqslant 2$ there exists a prime number $p$ dividing $a_{n}$ but none of the numbers $a_{1}, \ldots, a_{n-1}$ .

IMO Shortlist

Number theory Solution and answer checking →
Find all polynomials $P(x)$ of odd degree $d$ and with integer coefficients satisfying the following property: for each positive integer $n$ , there exist $n$ positive integers $x_{1}, x_{2}, \ldots, x_{n}$ such that $\frac{1}{2}<\frac{P\left(x_{i}\right)}{P\left(x_{j}\right)}<2$ and $\frac{P\left(x_{i}\right)}{P\left(x_{j}\right)}$ is the $d$ -th power of a rational number for every pair of indices $i$ and $j$ with $1 \leqslant i, j \leqslant n$ .

IMO Shortlist 2016

Algebra Solution and answer checking →
Let $N$ be a positive integer. Geoff and Ceri play a game in which they start by writing the numbers $1, 2, \ldots, N$ on a board. They then take turns to make a move, starting with Geoff. Each move consists of choosing a pair of integers $(k, n)$ , where $k \geqslant 0$ and $n$ is one of the integers on the board, and then erasing every integer $s$ on the board such that $2^{k} \mid n-s$ . The game continues until the board is empty. The player who erases the last integer on the board loses.
Determine all values of $N$ for which Geoff can ensure that he wins, no matter how Ceri plays.
(Indonesia)

IMO Shortlist

Combinatorics Solution and answer checking →
Suppose there are beetles on a chessboard consisting of $2012 \times 2012$ unit squares. Each unit square can accommodate at most one beetle. At a moment, all beetles fly and land on the chessboard again. For a beetle, we call the vector from its flying unit to its landing unit the beetle's "displacement vector". We call the sum of all beetle's "displacement vectors" the "total displacement vectors".
Find the maximum length of "total displacement vector" considering the number of beetles and all possible positions of flying and landing. (posed by Qu Zhenhua)

China National Team Selection Test

Geometry Solution and answer checking →
For every $n \in \mathbb{N}$ let $d(n)$ denote the number of (positive) divisors of $n$ . Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ with the following properties:
(i) $d(f(x))=x$ for all $x \in \mathbb{N}$ ;
(ii) $f(x y)$ divides $(x-1) y^{x y-1} f(x)$ for all $x, y \in \mathbb{N}$ .

International Mathematical Olympiad Spain

Number theory Solution and answer checking →
An integer $n \geqslant 3$ is given. We call an $n$ -tuple of real numbers $(x_{1}, x_{2}, \ldots, x_{n})$ Shiny if for each permutation $y_{1}, y_{2}, \ldots, y_{n}$ of these numbers we have
$\sum_{i=1}^{n-1} y_{i} y_{i+1} = y_{1} y_{2} + y_{2} y_{3} + y_{3} y_{4} + \cdots + y_{n-1} y_{n} \geqslant -1.$
Find the largest constant $K = K(n)$ such that
$\sum_{1 \leqslant i < j \leqslant n} x_{i} x_{j} \geqslant K$
holds for every Shiny $n$ -tuple $(x_{1}, x_{2}, \ldots, x_{n})$ .

International Mathematical Olympiad

Algebra Solution and answer checking →
There are $2010$ students and $100$ classrooms in the Olympiad High School. At the beginning, each of the students is in one of the classrooms. Each minute, as long as not everyone is in the same classroom, somebody walks from one classroom into a different classroom with at least as many students in it (prior to his move). This process will terminate in $M$ minutes. Determine the maximum value of $M$ .

United States — Team Selection Test

Combinatorics Solution and answer checking →
Define the sequence $a_1, a_2, a_3, \dots$ by $a_1 = 1$ and
$a_n = a_{\lfloor n/2 \rfloor} + a_{\lfloor n/3 \rfloor} + \dots + a_{\lfloor n/n \rfloor} + 1$
for $n > 1$ . Prove that there are infinitely many $n$ such that
$a_n \equiv n \pmod{2^{2010}}.$
(This problem was suggested by Gabriel Carroll.)

United States — Team Selection Test 2010

Number theory Solution and answer checking →
Let $\mathcal{P}$ be a regular 2006-gon. A diagonal of $\mathcal{P}$ is called good segment if its endpoints divide the boundary of $\mathcal{P}$ into two parts, each composed of an odd number of sides of $\mathcal{P}$ . The sides of $\mathcal{P}$ are also called good segment.
Suppose $\mathcal{P}$ has been dissected into triangles by 2003 diagonals, no two of which have a common point in the interior of $\mathcal{P}$ . Find the maximum number of isosceles triangles having two good segments that could appear in such a configuration.

International Mathematical Olympiad

Geometry Solution and answer checking →

Answer key — Stage 10 · Mixed

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