Olympiad problems

2005 IMO Shortlist, 4

Find all positive integers $ n$ such that there exists a unique integer $ a$ such that $ 0\leq a < n!$ with the following property: \[ n!\mid a^n \plus{} 1 \] [i]Proposed by Carlos Caicedo, Colombia[/i]

2001 India IMO Training Camp, 1

For any positive integer $n$, show that there exists a polynomial $P(x)$ of degree $n$ with integer coefficients such that $P(0),P(1), \ldots, P(n)$ are all distinct powers of $2$.

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2011 National Olympiad First Round, 14

What is the remainder when $2011^{(2011^{(2011^{(2011^{2011})})})}$ is divided by $19$ ? $\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 1$

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2006 Germany Team Selection Test, 2

Find all positive integers $ n$ such that there exists a unique integer $ a$ such that $ 0\leq a < n!$ with the following property: \[ n!\mid a^n \plus{} 1 \] [i]Proposed by Carlos Caicedo, Colombia[/i]

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1989 IMO Shortlist, 11

Define sequence $ (a_n)$ by $ \sum_{d|n} a_d \equal{} 2^n.$ Show that $ n|a_n.$

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2025 USAJMO, 6

Let $S$ be a set of integers with the following properties: [list] [*] $\{ 1, 2, \dots, 2025 \} \subseteq S$. [*] If $a, b \in S$ and $\gcd(a, b) = 1$, then $ab \in S$. [*] If for some $s \in S$, $s + 1$ is composite, then all positive divisors of $s + 1$ are in $S$. [/list] Prove that $S$ contains all positive integers.

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2009 India National Olympiad, 5

Let $ ABC$ be an acute angled triangle and let $ H$ be its ortho centre. Let $ h_{max}$ denote the largest altitude of the triangle $ ABC$. Prove that: $AH \plus{} BH \plus{} CH\leq2h_{max}$

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2014 IMO Shortlist, A4

Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying \[f\big(f(m)+n\big)+f(m)=f(n)+f(3m)+2014\] for all integers $m$ and $n$. [i]Proposed by Netherlands[/i]

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2017 CMIMC Geometry, 7

Two non-intersecting circles, $\omega$ and $\Omega$, have centers $C_\omega$ and $C_\Omega$ respectively. It is given that the radius of $\Omega$ is strictly larger than the radius of $\omega$. The two common external tangents of $\Omega$ and $\omega$ intersect at a point $P$, and an internal tangent of the two circles intersects the common external tangents at $X$ and $Y$. Suppose that the radius of $\omega$ is $4$, the circumradius of $\triangle PXY$ is $9$, and $XY$ bisects $\overline{PC_\Omega}$. Compute $XY$.

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2024 Austrian MO National Competition, 6

For each prime number $p$, determine the number of residue classes modulo $p$ which can be represented as $a^2+b^2$ modulo $p$, where $a$ and $b$ are arbitrary integers. [i](Daniel Holmes)[/i]

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2019 Tournament Of Towns, 7

On the grid plane all possible broken lines with the following properties are constructed: each of them starts at the point $(0, 0)$, has all its vertices at integer points, each linear segment goes either up or to the right along the grid lines. For each such broken line consider the corresponding [i]worm[/i], the subset of the plane consisting of all the cells that share at least one point with the broken line. Prove that the number of worms that can be divided into dominoes (rectangles $2\times 1$ and $1\times 2$) in exactly $n > 2$ different ways, is equal to the number of positive integers that are less than n and relatively prime to $n$. (Ilke Chanakchi, Ralf Schiffler)

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2000 IMO Shortlist, 1

Determine all positive integers $ n\geq 2$ that satisfy the following condition: for all $ a$ and $ b$ relatively prime to $ n$ we have \[a \equiv b \pmod n\qquad\text{if and only if}\qquad ab\equiv 1 \pmod n.\]

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2009 Balkan MO, 1

Solve the equation \[ 3^x \minus{} 5^y \equal{} z^2.\] in positive integers. [i]Greece[/i]

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2015 China Team Selection Test, 3

Let $a,b$ be two integers such that their gcd has at least two prime factors. Let $S = \{ x \mid x \in \mathbb{N}, x \equiv a \pmod b \} $ and call $ y \in S$ irreducible if it cannot be expressed as product of two or more elements of $S$ (not necessarily distinct). Show there exists $t$ such that any element of $S$ can be expressed as product of at most $t$ irreducible elements.

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2000 National Olympiad First Round, 18

What is the least integer $n\geq 100$ such that $77$ divides $1+2+2^2+2^3+\dots + 2^n$ ? $ \textbf{(A)}\ 101 \qquad\textbf{(B)}\ 105 \qquad\textbf{(C)}\ 111 \qquad\textbf{(D)}\ 119 \qquad\textbf{(E)}\ \text{None} $

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KoMaL A Problems 2022/2023, A. 843

Let $N$ be the set of those positive integers $n$ for which $n\mid k^k-1$ implies $n\mid k-1$ for every positive integer $k$. Prove that if $n_1,n_2\in N$, then their greatest common divisor is also in $N$.

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2013 USA TSTST, 8

Define a function $f: \mathbb N \to \mathbb N$ by $f(1) = 1$, $f(n+1) = f(n) + 2^{f(n)}$ for every positive integer $n$. Prove that $f(1), f(2), \dots, f(3^{2013})$ leave distinct remainders when divided by $3^{2013}$.

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2015 Taiwan TST Round 2, 2

Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying \[f\big(f(m)+n\big)+f(m)=f(n)+f(3m)+2014\] for all integers $m$ and $n$. [i]Proposed by Netherlands[/i]

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2006 India National Olympiad, 1

In a non equilateral triangle $ABC$ the sides $a,b,c$ form an arithmetic progression. Let $I$ be the incentre and $O$ the circumcentre of the triangle $ABC$. Prove that (1) $IO$ is perpendicular to $BI$; (2) If $BI$ meets $AC$ in $K$, and $D$, $E$ are the midpoints of $BC$, $BA$ respectively then $I$ is the circumcentre of triangle $DKE$.

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2012 Thailand Mathematical Olympiad, 3

Let $m, n > 1$ be coprime odd integers. Show that $$\big \lfloor \frac{m^{\phi (n)+1} + n^{\phi (m)+1}}{mn} \rfloor$$ is an even integer, where $\phi$ is Euler’s totient function.

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2011 AMC 10, 23

What is the hundreds digit of $2011^{2011}$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9 $

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1990 IMO, 3

Determine all integers $ n > 1$ such that \[ \frac {2^n \plus{} 1}{n^2} \] is an integer.

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2025 International Zhautykov Olympiad, 3

A pair of positive integers $(x, y)$ is [i] good [/i] if they satisfy $\text{rad}(x) = \text{rad}(y)$ and they do not divide each-other. Given coprime positive integers $a$ and $b$, show that there exist infinitely many $n$ for which there exists a positive integer $m$ such that $(a^n + bm, b^n + am)$ is [i] good[/i]. (Here, $\text{rad}(x)$ denotes the product of $x$'s prime divisors, as usual.)

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2011 Sharygin Geometry Olympiad, 3

Restore the isosceles triangle $ABC$ ($AB = AC$) if the common points $I, M, H$ of bisectors, medians and altitudes respectively are given.

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2023 Hong Kong Team Selection Test, Problem 2

Find the period of the repetend of the fraction $\frac{39}{1428}$ by using [i]binary[/i] numbers, i.e. its binary decimal representation. (Note: When a proper fraction is expressed as a decimal number (of any base), either the decimal number terminates after finite steps, or it is of the form $0.b_1b_2\cdots b_sa_1a_2\cdots a_ka_1a_2\cdots a_ka_1a_2 \cdots a_k \cdots$. Here the repeated sequence $a_1a_2\cdots a_k$ is called the [i]repetend[/i] of the fraction, and the smallest length of the repetend, $k$, is called the [i]period[/i] of the decimal number.)

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