Olympiad problems

2023 Indonesia TST, 1

Tags: number theory

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 $3$.)

2006 Singapore Senior Math Olympiad, 1

Tags: mutliple , divide , divisible , prime , number theory

Let $a, d$ be integers such that $a,a + d, a+ 2d$ are all prime numbers larger than $3$. Prove that $d$ is a multiple of $6$.

2016 Ecuador Juniors, 1

Tags: digit , number theory

A natural number of five digits is called [i]Ecuadorian [/i]if it satisfies the following conditions: $\bullet$ All its digits are different. $\bullet$ The digit on the far left is equal to the sum of the other four digits. Example: $91350$ is an Ecuadorian number since $9 = 1 + 3 + 5 + 0$, but $54210$ is not since $5 \ne 4 + 2 + 1 + 0$. Find how many Ecuadorian numbers exist.

2013 Iran MO (3rd Round), 3

Tags: modular arithmetic , number theory

Let $p>3$ a prime number. Prove that there exist $x,y \in \mathbb Z$ such that $p = 2x^2 + 3y^2$ if and only if $p \equiv 5, 11 \; (\mod 24)$ (20 points)

2022 Iran-Taiwan Friendly Math Competition, 6

Tags: algebra , number theory , function

Find all completely multipiclative functions $f:\mathbb{Z}\rightarrow \mathbb{Z}_{\geqslant 0}$ such that for any $a,b\in \mathbb{Z}$ and $b\neq 0$, there exist integers $q,r$ such that $$a=bq+r$$ and $$f(r)<f(b)$$ Proposed by Navid Safaei

2010 IMO Shortlist, 1

Tags: equation , number theory , 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]

2022 Abelkonkurransen Finale, 1a

Tags: number theory , perfect square

Determine all positive integers $n$ such that $2022 + 3^n$ is a perfect square.

2015 Postal Coaching, 2

Tags: bounded , number theory

Prove that there exists a real number $C > 1$ with the following property. Whenever $n > 1$ and $a_0 < a_1 < a_2 <\cdots < a_n$ are positive integers such that $\frac{1}{a_0},\frac{1}{a_1} \cdots \frac{1}{a_n}$ form an arithmetic progression, then $a_0 > C^n$.

2010 Hong kong National Olympiad, 3

Tags: number theory , modular arithmetic , floor function

Let $n$ be a positive integer. Let $a$ be an integer such that $\gcd (a,n)=1$. Prove that \[\frac{a^{\phi (n)}-1}{n}=\sum_{i\in R}\frac{1}{ai}\left[\frac{ai}{n}\right]\pmod{n}\] where $R$ is the reduced residue system of $n$ with each element a positive integer at most $n$.

2024 Singapore Junior Maths Olympiad, Q5

Tags: number theory

Find all integer solutions of the equation $$y^2+2y=x^4+20x^3+104x^2+40x+2003.$$ Note: has appeared many times before, see [url=https://artofproblemsolving.com/community/q1_%22x%5E4%2B20x%5E3%2B104x%5E2%22]here[/url]

1981 IMO Shortlist, 12

Tags: algebra , vieta , diophantine equation , polynomial , number theory

Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers in the range $1,2,\ldots,1981$ satisfying $(n^2-mn-m^2)^2=1$.

2012 Irish Math Olympiad, 4

Tags: number theory , prime number

Let $x$ > $1$ be an integer. Prove that $x^5$ + $x$ + $1$ is divisible by at least two distinct prime numbers.

1991 Bulgaria National Olympiad, Problem 3

Tags: number theory

Prove that for every prime number $p\ge5$, (a) $p^3$ divides $\binom{2p}p-2$; (b) $p^3$ divides $\binom{kp}p-k$ for every natural number $k$.

1989 IMO Longlists, 83

Tags: equation , diophantine equation , quadratic , number theory

Let $ a, b \in \mathbb{Z}$ which are not perfect squares. Prove that if \[ x^2 \minus{} ay^2 \minus{} bz^2 \plus{} abw^2 \equal{} 0\] has a nontrivial solution in integers, then so does \[ x^2 \minus{} ay^2 \minus{} bz^2 \equal{} 0.\]

2013 Germany Team Selection Test, 2

Tags: quadratic , number theory

Call admissible a set $A$ of integers that has the following property: If $x,y \in A$ (possibly $x=y$) then $x^2+kxy+y^2 \in A$ for every integer $k$. Determine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers. [i]Proposed by Warut Suksompong, Thailand[/i]

2004 Junior Balkan Team Selection Tests - Romania, 3

Tags: number theory , induction

Let $A$ be a set of positive integers such that a) if $a\in A$, the all the positive divisors of $a$ are also in $A$; b) if $a,b\in A$, with $1<a<b$, then $1+ab \in A$. Prove that if $A$ has at least 3 elements, then $A$ is the set of all positive integers.

2008 IMAR Test, 2

Tags: analytic geometry , number theory , prime number

A point $ P$ of integer coordinates in the Cartesian plane is said [i]visible[/i] if the segment $ OP$ does not contain any other points with integer coordinates (except its ends). Prove that for any $ n\in\mathbb{N}^*$ there exists a visible point $ P_{n}$, at distance larger than $ n$ from any other visible point. [b]Dan Schwarz[/b]

2019 Saint Petersburg Mathematical Olympiad, 1

Tags: number theory , digit , palindrome

A natural number is called a palindrome if it is read in the same way. from left to right and from right to left (in particular, the last digit of the palindrome coincides with the first and therefore not equal to zero). Squares of two different natural numbers have $1001$ digits. Prove that strictly between these squares, there is one palindrome.

2004 Junior Balkan Team Selection Tests - Romania, 4

Tags: number theory

Find all positive integers $n$ for which there exist distinct positive integers $a_1,a_2,\ldots,a_n$ such that \[ \frac 1{a_1} + \frac 2{a_2} + \cdots + \frac n { a_n} = \frac { a_1 + a_2 + \cdots + a_n } n. \]

2018 Romania National Olympiad, 1

Tags: sum of squares , number theory

Find the distinct positive integers $a, b, c,d$, such that the following conditions hold: (1) exactly three of the four numbers are prime numbers; (2) $a^2 + b^2 + c^2 + d^2 = 2018.$

2012 India Regional Mathematical Olympiad, 6

Tags: algebra , polynomial , induction , number theory

Find all positive integers such that $3^{2n}+3n^2+7$ is a perfect square.

2011 Iran Team Selection Test, 4

Tags: number theory , relatively prime

Define a finite set $A$ to be 'good' if it satisfies the following conditions: [list][*][b](a)[/b] For every three disjoint element of $A,$ like $a,b,c$ we have $\gcd(a,b,c)=1;$ [*][b](b)[/b] For every two distinct $b,c\in A,$ there exists an $a\in A,$ distinct from $b,c$ such that $bc$ is divisible by $a.$[/list] Find all good sets.

2018 PUMaC Live Round, 7.2

Tags: number theory , relatively prime

Compute the smallest positive integer $n$ that is a multiple of $29$ with the property that for every positive integer that is relatively prime to $n$, $k^{n}\equiv 1\pmod{n}.$

2023 Chile Junior Math Olympiad, 1

Tags: digit , number theory

Determine the number of three-digit numbers with the following property: The number formed by the first two digits is prime and the number formed by the last two digits is prime.

2011 Akdeniz University MO, 5

Tags: number theory

For all $n \in {\mathbb Z^+}$ we define $$I_n=\{\frac{0}{n},\frac{1}{n},\frac{2}{n},\dotsm,\frac{n-1}{n},\frac{n}{n},\frac{n+1}{n},\dotsm\}$$ infinite cluster. For whichever $x$ and $y$ real number, we say $\mid{x-y}\mid$ is between distance of the $x$ and $y$. [b]a[/b]) For all $n$'s we find a number in $I_n$ such that, the between distance of the this number and $\sqrt 2$ is less than $\frac{1}{2n}$ [b]b[/b]) We find a $n$ such that, between distance of the a number in $I_n$ and $\sqrt 2$ is less than $\frac{1}{2011n}$