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Stage 9 · Mixed

10 problems · IMO P2/P5; hard shortlist · mathsolympiadprep.com

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

Let $n$ be a positive integer. A Nordic square is an $n \times n$ board containing all the integers from $1$ to $n^2$ so that each cell contains exactly one number. Two different cells are considered adjacent if they share a common side. Every cell that is adjacent only to cells containing larger numbers is called a valley. An uphill path is a sequence of one or more cells such that:
(i) the first cell in the sequence is a valley,
(ii) each subsequent cell in the sequence is adjacent to the previous cell, and
(iii) the numbers written in the cells in the sequence are in increasing order.
Find, as a function of $n$ , the smallest possible total number of uphill paths in a Nordic square.
Author: Nikola Petrovi?

International Mathematical Olympiad

Combinatorics Solution and answer checking →
Let $n$ be a positive integer. Let $s: \mathbb{N} \to \{1, \dots, n\}$ be a function such that $n$ divides $m - s(m)$ for all positive integers $m$ . Let $a_0, a_1, a_2, \dots$ be a sequence such that $a_0 = 0$ and
$a_k = a_{k-1} + s(k) \text{ for all } k \ge 1.$
Find all $n$ for which this sequence contains all the residues modulo $(n+1)^2$ .

India — Team Selection Test

Number theory Solution and answer checking →
Let $\omega$ be the circumcircle of a triangle $A B C$ , and let $\Omega_{A}$ be its excircle which is tangent to the segment $B C$ . Let $X$ and $Y$ be the intersection points of $\omega$ and $\Omega_{A}$ . Let $P$ and $Q$ be the projections of $A$ onto the tangent lines to $\Omega_{A}$ at $X$ and $Y$ , respectively. The tangent line at $P$ to the circumcircle of the triangle $A P X$ intersects the tangent line at $Q$ to the circumcircle of the triangle $A Q Y$ at a point $R$ . Prove that $A R \perp B C$ .

IMO Shortlist 2021

Geometry Solution and answer checking →
Let $n$ be a positive integer. Given a sequence $\varepsilon_{1}, \ldots, \varepsilon_{n-1}$ with $\varepsilon_{i}=0$ or $\varepsilon_{i}=1$ for each $i=1, \ldots, n-1$ , the sequences $a_{0}, \ldots, a_{n}$ and $b_{0}, \ldots, b_{n}$ are constructed by the following rules:
$\begin{gathered} a_{0}=b_{0}=1, \quad a_{1}=b_{1}=7 \\ a_{i+1}=\left\{\begin{array}{ll} 2 a_{i-1}+3 a_{i}, & \text{ if } \varepsilon_{i}=0, \\ 3 a_{i-1}+a_{i}, & \text{ if } \varepsilon_{i}=1, \end{array}\right. \quad \text{ for each } i=1, \ldots, n-1, \\ b_{i+1}=\left\{\begin{array}{ll} 2 b_{i-1}+3 b_{i}, & \text{ if } \varepsilon_{n-i}=0, \\ 3 b_{i-1}+b_{i}, & \text{ if } \varepsilon_{n-i}=1, \end{array}\right. \text{ for each } i=1, \ldots, n-1 \end{gathered}$
Prove that $a_{n}=b_{n}$ .

IMO Shortlist

Algebra Solution and answer checking →
Let $n$ and $T$ be positive integers. James has $4 n$ marbles with weights $1,2, \ldots, 4 n$ . He places them on a balance scale, so that both sides have equal weight. Andrew may move a marble from one side of the scale to the other, so that the absolute difference in weights of the two sides remains at most $T$ .
Find, in terms of $n$ , the minimum positive integer $T$ such that Andrew may make a sequence of moves such that each marble ends up on the opposite side of the scale, regardless of how James initially placed the marbles.

IMO Shortlist

Combinatorics Solution and answer checking →
Let $n$ be a positive integer. In this problem, we consider labellings of the squares of a chessboard of size $n \times n$ with the natural numbers from $1$ to $n^2$ such that every number is used exactly once. Given such a labelling, we say a positive integer is a rook product if it is the product of the labels of $n$ squares which have the property that if you place a rook on each of them, no two rooks will attack each other.
(Two rooks are attacking each other, if and only if they are in the same row or column.)
a. Let $n = 8$ . Determine whether there exists a labelling of an $8 \times 8$ chessboard such that the following condition is fulfilled: The difference of any two rook products is always divisible by $65$ .
b. Let $n = 10$ . Determine whether there exists a labelling of a $10 \times 10$ chessboard such that the following condition is fulfilled: The difference of any two rook products is always divisible by $101$ .

Baltic Way Shortlist 2023

Number theory Solution and answer checking →
Let $A = (0,0,0)$ be the origin in the three dimensional coordinate space. The weight of a point is the sum of the absolute values of its coordinates. A point is a primitive lattice point if all its coordinates are integers with their greatest common divisor equal to 1. A square $ABCD$ is called a unbalanced primitive integer square if it has integer side length and the points $B$ and $D$ are primitive lattice points with different weights.
Show that there are infinitely many unbalanced primitive integer squares $AB_iC_iD_i$ such that the plane containing the squares are not parallel to each other.

USA IMO

Geometry Solution and answer checking →
Find the maximum positive number $M$ such that for every $n \in \mathbb{N}^*$ , there are positive numbers $a_1, a_2, \dots, a_n$ and $b_1, b_2, \dots, b_n$ satisfying
$(a) \sum_{k=1}^{n} b_k = 1,\ 2b_k \ge b_{k-1} + b_{k+1},\ k = 2, 3, \dots, n-1,$
$(b) a_k^2 \le 1 + \sum_{i=1}^{k} a_i b_i,\ k = 1, 2, \dots, n,$
$(c) a_n = M.$

China National Team Selection Test

Algebra Solution and answer checking →
A cake has the form of an $n \times n$ square composed of $n^2$ unit squares. Strawberries lie on some of the unit squares so that each row or column contains exactly one strawberry; call this arrangement $\mathcal{A}$ .
Let $\mathcal{B}$ be another such arrangement. Suppose that every grid rectangle with one vertex at the top left corner of the cake contains no fewer strawberries of arrangement $\mathcal{B}$ than of arrangement $\mathcal{A}$ . Prove that arrangement $\mathcal{B}$ can be obtained from $\mathcal{A}$ by performing a number of switches, defined as follows:
A switch consists in selecting a grid rectangle with only two strawberries, situated at its top right corner and bottom left corner, and moving these two strawberries to the other two corners of that rectangle.

IMO Shortlist 2006

Combinatorics Solution and answer checking →
Let $a, b$ be integers, and let $P(x) = a x^{3} + b x$ . For any positive integer $n$ we say that the pair $(a, b)$ is $n$ -good if $n \mid P(m) - P(k)$ implies $n \mid m - k$ for all integers $m, k$ . We say that $(a, b)$ is very good if $(a, b)$ is $n$ -good for infinitely many positive integers $n$ .
a. Find a pair $(a, b)$ which is $51$ -good, but not very good.
b. Show that all $2010$ -good pairs are very good.
(Turkey)

IMO Shortlist

Number theory Solution and answer checking →

Answer key — Stage 9 · Mixed

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