Olympiad problems

2003 Spain Mathematical Olympiad, Problem 2

Does there exist such a finite set of real numbers ${M}$ that has at least two distinct elements and has the property that for two numbers, ${a}$, ${b}$, belonging to ${M}$, the number ${2a - b^2}$ is also an element in ${M}$?

2013 CHMMC (Fall), 4

Tags: number theory

The numbers $25$ and $76$ have the property that when squared in base 10, their squares also end in the same two digits. A positive integer that has at most $3$ digits when expressed in base 21 and also has the property that its base $21$ square ends in the same $3$ digits is called amazing. Find the sum of all amazing numbers. Express your answer in base $21$.

1992 IMO Longlists, 43

Tags: relatively prime , series summation , number theoretic functions , floor function , number theory

Find the number of positive integers $n$ satisfying $\phi(n) | n$ such that \[\sum_{m=1}^{\infty} \left( \left[ \frac nm \right] - \left[\frac{n-1}{m} \right] \right) = 1992\] What is the largest number among them? As usual, $\phi(n)$ is the number of positive integers less than or equal to $n$ and relatively prime to $n.$

2024 CMIMC Algebra and Number Theory, 4

Tags: number theory

For positive integer $n$, let $f(n)$ be the largest integer $k$ such that $k!\leq n$, let $g(n)=n-(f(n))!$, and for $j\geq 1$ let $$g^j(n)=\underbrace{g(\dots(g(n))\dots)}_{\text{$j$ times}}.$$ Find the smallest positive integer $n$ such that $g^{j}(n)> 0$ for all $j<30$ and $g^{30}(n)=0$. [i]Proposed by Connor Gordon[/i]

2016 Iran MO (3rd Round), 1

Tags: polynomial , algebra , number theory

Let $F$ be a subset of the set of positive integers with at least two elements and $P(x)$ be a polynomial with integer coefficients such that for any two distinct elements of $F$ like $a$ and $b$, the following two conditions hold [list] [*] $a+b \in F$, and [*] $\gcd(P(a),P(b))=1$. [/list] Prove that $P(x)$ is a constant polynomial.

2009 Romanian Masters In Mathematics, 1

Tags: least common multiple , number theory , triangle inequality , inequalities , greatest common divisor , floor function , algebra

For $ a_i \in \mathbb{Z}^ \plus{}$, $ i \equal{} 1, \ldots, k$, and $ n \equal{} \sum^k_{i \equal{} 1} a_i$, let $ d \equal{} \gcd(a_1, \ldots, a_k)$ denote the greatest common divisor of $ a_1, \ldots, a_k$. Prove that $ \frac {d} {n} \cdot \frac {n!}{\prod\limits^k_{i \equal{} 1} (a_i!)}$ is an integer. [i]Dan Schwarz, Romania[/i]

PEN D Problems, 7

Tags: congruence , modular arithmetic , number theory

Somebody incorrectly remembered Fermat's little theorem as saying that the congruence $a^{n+1} \equiv a \; \pmod{n}$ holds for all $a$ if $n$ is prime. Describe the set of integers $n$ for which this property is in fact true.

2012 Abels Math Contest (Norwegian MO) Final, 1a

Tags: algebra , number theory

Berit has $11$ twenty kroner coins, $14$ ten kroner coins, and $12$ five kroner coins. An exchange machine can exchange three ten kroner coins into one twenty kroner coin and two five kroner coins, and the reverse. It can also exchange two twenty kroner coins into three ten kroner coins and two five kroner coins, and the reverse. (i) Can Berit get the same number of twenty kroner and ten kroner coins, but no five kroner coins? (ii) Can she get the same number each of twenty kroner, ten kroner, and five kroner coins?

2022 European Mathematical Cup, 2

Tags: divisor , divisibility , number theory , greatest common divisor

We say that a positive integer $n$ is lovely if there exist a positive integer $k$ and (not necessarily distinct) positive integers $d_1$, $d_2$, $\ldots$, $d_k$ such that $n = d_1d_2\cdots d_k$ and $d_i^2 \mid n + d_i$ for $i=1,2,\ldots,k$. a) Are there infinitely many lovely numbers? b) Is there a lovely number, greater than $1$, which is a perfect square of an integer?

2017 Saudi Arabia Pre-TST + Training Tests, 7

Tags: diophantine , diophantine equation , number theory

Find all pairs of integers $(x, y)$ such that $y^3 = 8x^6 + 2x^3 y -y^2$.

2006 Federal Competition For Advanced Students, Part 1, 1

Tags: function , number theory

Let $ n$ be a non-negative integer, which ends written in decimal notation on exactly $ k$ zeros, but which is bigger than $ 10^k$. For a $ n$ is only $ k\equal{}k(n)\geq2$ known. In how many different ways (as a function of $ k\equal{}k(n)\geq2$) can $ n$ be written as difference of two squares of non-negative integers at least?

2011 IMO Shortlist, 6

Tags: polynomial , algebra , divisibility , number theory

Let $P(x)$ and $Q(x)$ be two polynomials with integer coefficients, such that no nonconstant polynomial with rational coefficients divides both $P(x)$ and $Q(x).$ Suppose that for every positive integer $n$ the integers $P(n)$ and $Q(n)$ are positive, and $2^{Q(n)}-1$ divides $3^{P(n)}-1.$ Prove that $Q(x)$ is a constant polynomial. [i]Proposed by Oleksiy Klurman, Ukraine[/i]

1987 IMO Longlists, 69

Tags: polynomial , prime number , quadratic , number theory

Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.[i](IMO Problem 6)[/i] [b][i]Original Formulation[/i][/b] Let $f(x) = x^2 + x + p$, $p \in \mathbb N.$ Prove that if the numbers $f(0), f(1), \cdots , f(\sqrt{p\over 3} )$ are primes, then all the numbers $f(0), f(1), \cdots , f(p - 2)$ are primes. [i]Proposed by Soviet Union. [/i]

2022 Taiwan TST Round 1, N

Tags: number theory

Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.

2008 Finnish National High School Mathematics Competition, 1

Tags: number theory

Foxes, wolves and bears arranged a big rabbit hunt. There were $45$ hunters catching $2008$ rabbits. Every fox caught $59$ rabbits, every wolf $41$ rabbits and every bear $40$ rabbits. How many foxes, wolves and bears were there in the hunting company?

2005 Colombia Team Selection Test, 1

Tags: geometry , 3d geometry , number theory

Let $a,b,c$ be integers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=3$ prove that $abc$ is a perfect cube!

2022 Greece Junior Math Olympiad, 4

Tags: number theory

Find all couples of non-zero integers $(x,y)$ such that, $x^2+y^2$ is a common divisor of $x^5+y$ and $y^5+x$.

2013 Turkey Team Selection Test, 1

Tags: relatively prime , number theory , induction

Let $\phi(n)$ be the number of positive integers less than $n$ that are relatively prime to $n$, where $n$ is a positive integer. Find all pairs of positive integers $(m,n)$ such that \[2^n + (n-\phi(n)-1)! = n^m+1.\]

2018 India PRMO, 12

Tags: algebra , number theory

Determine the number of $8$-tuples $(\epsilon_1, \epsilon_2,...,\epsilon_8)$ such that $\epsilon_1, \epsilon_2, ..., 8 \in \{1,-1\}$ and $\epsilon_1 + 2\epsilon_2 + 3\epsilon_3 +...+ 8\epsilon_8$ is a multiple of $3$.

2012 CHMMC Spring, 1

Tags: combinatorics , number theory

Let $a_k$ be the number of ordered $10$-tuples $(x_1, x_2, ..., x_{10})$ of nonnegative integers such that $$x^2_1+ x^2_2+ ... + x^2_{10} = k.$$ Let $b_k = 0$ if $a_k$ is even and $b_k = 1$ if $a_k$ is odd. Find $\sum^{2012}_{i=1} b_{4i}$.

2018 PUMaC Number Theory B, 7

Tags: number theory

Find the remainder of $$\prod_{n = 2}^{99} (1 - n^2 + n^4)(1 - 2n^2 + n^4)$$ when divided by $101$.

2016 Kosovo National Mathematical Olympiad, 2

Tags: number theory

Evaluate the sum of all three digits number which are not divisible by $13$ .

1979 IMO Longlists, 26

Tags: prime number , factorization , number theory

Let $n$ be a positive integer. If $4^n + 2^n + 1$ is a prime, prove that $n$ is a power of three.

2018 Balkan MO Shortlist, N1

Tags: number theory

For positive integers $m$ and $n$, let $d(m, n)$ be the number of distinct primes that divide both $m$ and $n$. For instance, $d(60, 126) = d(2^2 \cdot 3 \cdot 5, 2 \cdot 3^2 \cdot 7) = 2.$ Does there exist a sequence $(a_n)$ of positive integers such that: [list] [*] $a_1 \geq 2018^{2018};$ [*] $a_m \leq a_n$ whenever $m \leq n$; [*] $d(m, n) = d(a_m, a_n)$ for all positive integers $m\neq n$? [/list] [i](Dominic Yeo, United Kingdom)[/i]

2024 Philippine Math Olympiad, P4

Tags: combinatorics , number theory , function

Let $n$ be a positive integer. Suppose for any $\mathcal{S} \subseteq \{1, 2, \cdots, n\}$, $f(\mathcal{S})$ is the set containing all positive integers at most $n$ that have an odd number of factors in $\mathcal{S}$. How many subsets of $\{1, 2, \cdots, n\}$ can be turned into $\{1\}$ after finitely many (possibly zero) applications of $f$?