Olympiad problems

2001 Singapore Team Selection Test, 1

Let $a, b, c, d$ be four positive integers such that each of them is a difference of two squares of positive integers. Prove that $abcd$ is also a difference of two squares of positive integers.

2024 Nigerian MO Round 3, Problem 2

Tags: number theory

Prove that there exist infinitely many distinct positive integers, $x$ and $y$, such that $$x^3+y^2|x^2+y^3$$

1997 Romania Team Selection Test, 4

Tags: floor function , ceiling function , number theory , prime number , combinatorics , poset

Let $p,q,r$ be distinct prime numbers and let \[A=\{p^aq^br^c\mid 0\le a,b,c\le 5\} \] Find the least $n\in\mathbb{N}$ such that for any $B\subset A$ where $|B|=n$, has elements $x$ and $y$ such that $x$ divides $y$. [i]Ioan Tomescu[/i]

2005 International Zhautykov Olympiad, 3

Tags: vieta , number theory

Find all prime numbers $ p,q < 2005$ such that $ q | p^{2} \plus{} 8$ and $ p|q^{2} \plus{} 8.$

2011 IMO Shortlist, 2

Tags: algebra , polynomial , pigeonhole principle , number theory , combinatorics

Consider a polynomial $P(x) = \prod^9_{j=1}(x+d_j),$ where $d_1, d_2, \ldots d_9$ are nine distinct integers. Prove that there exists an integer $N,$ such that for all integers $x \geq N$ the number $P(x)$ is divisible by a prime number greater than 20. [i]Proposed by Luxembourg[/i]

2014 ELMO Shortlist, 3

Tags: algebra , polynomial , induction , binomial theorem , number theory

Let $t$ and $n$ be fixed integers each at least $2$. Find the largest positive integer $m$ for which there exists a polynomial $P$, of degree $n$ and with rational coefficients, such that the following property holds: exactly one of \[ \frac{P(k)}{t^k} \text{ and } \frac{P(k)}{t^{k+1}} \] is an integer for each $k = 0,1, ..., m$. [i]Proposed by Michael Kural[/i]

2022 Kyiv City MO Round 1, Problem 4

Tags: number theory

In some magic country, there are banknotes only of values $3$, $25$, $80$ hryvnyas. Businessman Victor ate in one restaurant of this country for $2024$ days in a row, and each day (except the first) he spent exactly $1$ hryvnya more than the day before (without any change). Could he have spent exactly $1000000$ banknotes? [i](Proposed by Oleksii Masalitin)[/i]

2011 Belarus Team Selection Test, 2

Tags: number theory , equation , multiplication

Find the least positive integer $n$ for which there exists a set $\{s_1, s_2, \ldots , s_n\}$ consisting of $n$ distinct positive integers such that \[ \left( 1 - \frac{1}{s_1} \right) \left( 1 - \frac{1}{s_2} \right) \cdots \left( 1 - \frac{1}{s_n} \right) = \frac{51}{2010}.\] [i]Proposed by Daniel Brown, Canada[/i]

2010 Balkan MO, 4

Tags: function , greatest common divisor , relatively prime , totient function , number theory

For each integer $n$ ($n \ge 2$), let $f(n)$ denote the sum of all positive integers that are at most $n$ and not relatively prime to $n$. Prove that $f(n+p) \neq f(n)$ for each such $n$ and every prime $p$.

2024 Korea Winter Program Practice Test, Q5

Tags: number theory , polynomial

For each positive integer $n>1$, if $n=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}$($p_i$ are pairwise different prime numbers and $\alpha_i$ are positive integers), define $f(n)$ as $\alpha_1+\alpha_2+\cdots+\alpha_k$. For $n=1$, let $f(1)=0$. Find all pairs of integer polynomials $P(x)$ and $Q(x)$ such that for any positive integer $m$, $f(P(m))=Q(f(m))$ holds.

1998 Switzerland Team Selection Test, 2

Tags: diophantine , diophantine equation , number theory

Find all nonnegative integer solutions $(x,y,z)$ of the equation $\frac{1}{x+2}+\frac{1}{y+2}=\frac{1}{2} +\frac{1}{z+2}$

2017 Romania National Olympiad, 1

Tags: number theory , digit

Consider the set $$M = \left\{\frac{a}{\overline{ba}}+\frac{b}{\overline{ab}} \, | a,b\in\{1,2,3,4,5,6,7,8,9\} \right\}.$$ a) Show that the set $M$ contains no integer. b) Find the smallest and the largest element of $M$

2006 Tournament of Towns, 1

Tags: number theory

Three positive integers $x$ and $y$ are written on the blackboard. Mary records in her notebook the product of any two of them and reduces the third number on the blackboard by $1$. With the new trio of numbers, she repeats the process, and continues until one of the numbers on the blackboard becomes zero. What will be the sum of the numbers in Mary's notebook at that point? (4)

2010 Lithuania National Olympiad, 4

Tags: modular arithmetic , number theory

Decimal digits $a,b,c$ satisfy \[ 37\mid (a0a0\ldots a0b0c0c\ldots 0c)_{10} \] where there are $1001$ a's and $1001$ c's. Prove that $b=a+c$.

2021 Puerto Rico Team Selection Test, 4

Tags: digit , multiple , divisible , number theory

How many numbers $\overline{abcd}$ with different digits satisfy the following property: if we replace the largest digit with the digit $1$ results in a multiple of $30$?

2003 Federal Competition For Advanced Students, Part 1, 1

Tags: number theory

Find all triples of prime numbers $(p, q, r)$ such that $p^q + p^r$ is a perfect square.

2020 China Team Selection Test, 3

Tags: number theory , algebra , least common multiple

For a non-empty finite set $A$ of positive integers, let $\text{lcm}(A)$ denote the least common multiple of elements in $A$, and let $d(A)$ denote the number of prime factors of $\text{lcm}(A)$ (counting multiplicity). Given a finite set $S$ of positive integers, and $$f_S(x)=\sum_{\emptyset \neq A \subset S} \frac{(-1)^{|A|} x^{d(A)}}{\text{lcm}(A)}.$$ Prove that, if $0 \le x \le 2$, then $-1 \le f_S(x) \le 0$.

2023 Argentina National Olympiad, 2

Tags: prime factor , number theory

Find all positive integers $n$ such that all prime factors of $2^n-1$ are less than or equal to $7$.

2019 LIMIT Category B, Problem 7

Tags: diophantine equation , number theory

Find the number of ordered pairs of positive integers for which $$\frac1a+\frac1b=\frac4{2019}$$

1991 Tournament Of Towns, (307) 4

Tags: number theory , digit , recurrence relation

A sequence $a_n$ is determined by the rules $a_0 = 9$ and for any nonnegative $k$, $$a_{k+1}=3a_k^4+4a_k^3.$$ Prove that $a_{10}$ contains more than $1000$ nines in decimal notation. (Yao)

2023 SG Originals, Q6

Tags: functional equation , polynomial , polynomial with integer coeffi , algebra , number theory

$\mathbb{Z}[x]$ represents the set of all polynomials with integer coefficients. Find all functions $f:\mathbb{Z}[x]\rightarrow \mathbb{Z}[x]$ such that for any 2 polynomials $P,Q$ with integer coefficients and integer $r$, the following statement is true. \[P(r)\mid Q(r) \iff f(P)(r)\mid f(Q)(r).\] (We define $a|b$ if and only if $b=za$ for some integer $z$. In particular, $0|0$.) [i]Proposed by the4seasons.[/i]

2025 Malaysian IMO Training Camp, 4

Tags: number theory

For each positive integer $k$, find all positive integer $n$ such that there exists a permutation $a_1,\ldots,a_n$ of $1,2,\ldots,n$ satisfying $$a_1a_2\ldots a_i\equiv i^k \pmod n$$ for each $1\le i\le n$. [i](Proposed by Tan Rui Xuen and Ivan Chan Guan Yu)[/i]

2023 Sinapore MO Open, P3

Tags: number theory

Let $n \geq 2$ be a positive integer. For a positive integer $a$, let $Q_a(x)=x^n+ax$. Let $p$ be a prime and let $S_a=\{b | 0 \leq b \leq p-1, \exists c \in \mathbb {Z}, Q_a(c) \equiv b \pmod p \}$. Show that $\frac{1}{p-1}\sum_{a=1}^{p-1}|S_a|$ is an integer.

2011 Turkey MO (2nd round), 4

Tags: modular arithmetic , number theory , legendre s theorem

$a_{1}=5$ and $a_{n+1}=a_{n}^{3}-2a_{n}^{2}+2$ for all $n\geq1$. $p$ is a prime such that $p=3(mod 4)$ and $p|a_{2011}+1$. Show that $p=3$.

2024 ELMO Shortlist, N7

Tags: number theory , polynomial

For a prime $p$, let $\mathbb{F}_p$ denote the integers modulo $p$, and let $\mathbb{F}_p[x]$ be the set of polynomials with coefficients in $\mathbb{F}_p$. Find all $p$ for which there exists a quartic polynomial $P(x) \in \mathbb{F}_p[x]$ such that for all integers $k$, there exists some integer $\ell$ such that $P(\ell) \equiv k \pmod p$. (Note that there are $p^4(p-1)$ quartic polynomials in $\mathbb{F}_p[x]$ in total.) [i]Aprameya Tripathy[/i]