Olympiad problems

2023 Indonesia TST, 1

A number is called [i]Norwegian[/i] if it has three distinct positive divisors whose sum is equal to $2022$. Determine the smallest Norwegian number. (Note: The total number of positive divisors of a Norwegian number is allowed to be larger than $3$.)

2022 Azerbaijan IMO TST, 5

Tags: algebra , combinatorics , IMO Shortlist , AZE IMO TST

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

2022 Germany Team Selection Test, 3

Tags: number theory , IMO Shortlist , AZE IMO TST

Determine all integers $n\geqslant 2$ with the following property: every $n$ pairwise distinct integers whose sum is not divisible by $n$ can be arranged in some order $a_1,a_2,\ldots, a_n$ so that $n$ divides $1\cdot a_1+2\cdot a_2+\cdots+n\cdot a_n.$ [i]Arsenii Nikolaiev, Anton Trygub, Oleksii Masalitin, and Fedir Yudin[/i]

2016 IMO Shortlist, C2

Tags: combinatorics , IMO Shortlist , AZE IMO TST , TST

Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints: [list] [*]each cell contains a distinct divisor; [*]the sums of all rows are equal; and [*]the sums of all columns are equal. [/list]

2023 Belarus Team Selection Test, 3.1

Tags: algebra , IMO Shortlist , AZE IMO TST , TST

Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.

2021 Peru Iberoamerican Team Selection Test, P6

Tags: algebra , combinatorics , IMO Shortlist , AZE IMO TST

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

2022 Thailand TST, 3

Tags: number theory , IMO Shortlist , AZE IMO TST

Determine all integers $n\geqslant 2$ with the following property: every $n$ pairwise distinct integers whose sum is not divisible by $n$ can be arranged in some order $a_1,a_2,\ldots, a_n$ so that $n$ divides $1\cdot a_1+2\cdot a_2+\cdots+n\cdot a_n.$ [i]Arsenii Nikolaiev, Anton Trygub, Oleksii Masalitin, and Fedir Yudin[/i]

2022 Germany Team Selection Test, 3

Tags: number theory , AZE IMO TST

Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers. [i]Proposed by Dorlir Ahmeti, Albania[/i]

2024 Indonesia TST, G

Tags: geometry , AZE IMO TST , IMO Shortlist , TST

Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$. Prove that lines $AD, PM$, and $BC$ are concurrent.

2023 Azerbaijan IMO TST, 3

Tags: number theory , IMO Shortlist , AZE IMO TST

For each $1\leq i\leq 9$ and $T\in\mathbb N$, define $d_i(T)$ to be the total number of times the digit $i$ appears when all the multiples of $1829$ between $1$ and $T$ inclusive are written out in base $10$. Show that there are infinitely many $T\in\mathbb N$ such that there are precisely two distinct values among $d_1(T)$, $d_2(T)$, $\dots$, $d_9(T)$.

Russian TST 2022, P1

Tags: algebra , combinatorics , IMO Shortlist , AZE IMO TST

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

2022 Germany Team Selection Test, 3

Tags: algebra , combinatorics , IMO Shortlist , AZE IMO TST

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

2023 Thailand October Camp, 6

Tags: combinatorics , IMO Shortlist , AZE IMO TST , TST

In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height $0$. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn: [list] [*] The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller. [*] The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter. [/list] We say that a tree is [i]majestic[/i] if its height is at least $10^6$. Determine the largest $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.

2023 Indonesia TST, 1

Tags: algebra , IMO Shortlist , AZE IMO TST , TST

Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.

2023 ISL, C1

Tags: IMO Shortlist , combinatorics , AZE IMO TST

Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A [i]move[/i] consists of the following steps. [list=1] [*]select a $2\times 2$ square in the grid; [*]flip the coins in the top-left and bottom-right unit squares; [*]flip the coin in either the top-right or bottom-left unit square. [/list] Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves. [i]Thanasin Nampaisarn, Thailand[/i]

2022 Saudi Arabia IMO TST, 1

Tags: geometry , IMO Shortlist , AZE IMO TST , Hi

Let $ABCD$ be a parallelogram with $AC=BC.$ A point $P$ is chosen on the extension of ray $AB$ past $B.$ The circumcircle of $ACD$ meets the segment $PD$ again at $Q.$ The circumcircle of triangle $APQ$ meets the segment $PC$ at $R.$ Prove that lines $CD,AQ,BR$ are concurrent.

2023 Belarus Team Selection Test, 1.2

Tags: combinatorics , IMO Shortlist , AZE IMO TST , TST

In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height $0$. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn: [list] [*] The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller. [*] The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter. [/list] We say that a tree is [i]majestic[/i] if its height is at least $10^6$. Determine the largest $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.

2023 ISL, A4

Tags: algebra , Functional inequality , IMO Shortlist , AZE IMO TST , TST

Let $\mathbb R_{>0}$ be the set of positive real numbers. Determine all functions $f \colon \mathbb R_{>0} \to \mathbb R_{>0}$ such that \[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\] for every $x, y \in \mathbb R_{>0}$.

2022 Taiwan TST Round 3, 3

Tags: number theory , IMO Shortlist , AZE IMO TST

Determine all integers $n\geqslant 2$ with the following property: every $n$ pairwise distinct integers whose sum is not divisible by $n$ can be arranged in some order $a_1,a_2,\ldots, a_n$ so that $n$ divides $1\cdot a_1+2\cdot a_2+\cdots+n\cdot a_n.$ [i]Arsenii Nikolaiev, Anton Trygub, Oleksii Masalitin, and Fedir Yudin[/i]

2024 Switzerland - Final Round, 8

Tags: geometry , AZE IMO TST , IMO Shortlist , TST

Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$. Prove that lines $AD, PM$, and $BC$ are concurrent.

2022 Balkan MO Shortlist, C2

Tags: combinatorics , geometry , AZE IMO TST

Alice is drawing a shape on a piece of paper. She starts by placing her pencil at the origin, and then draws line segments of length one, alternating between vertical and horizontal segments. Eventually, her pencil returns to the origin, forming a closed, non-self-intersecting shape. Show that the area of this shape is even if and only if its perimeter is a multiple of eight.

2021 IMO Shortlist, N5

Tags: number theory , AZE IMO TST

Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers. [i]Proposed by Dorlir Ahmeti, Albania[/i]

2022 Bolivia IMO TST, P2

Tags: number theory , AZE IMO TST

Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers. [i]Proposed by Dorlir Ahmeti, Albania[/i]

2023 Estonia Team Selection Test, 6

Tags: combinatorics , IMO Shortlist , AZE IMO TST , TST

In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height $0$. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn: [list] [*] The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller. [*] The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter. [/list] We say that a tree is [i]majestic[/i] if its height is at least $10^6$. Determine the largest $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.

2022 Azerbaijan IMO TST, 1

Tags: combinatorics , geometry , AZE IMO TST

Alice is drawing a shape on a piece of paper. She starts by placing her pencil at the origin, and then draws line segments of length one, alternating between vertical and horizontal segments. Eventually, her pencil returns to the origin, forming a closed, non-self-intersecting shape. Show that the area of this shape is even if and only if its perimeter is a multiple of eight.