This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 15460

2002 Kurschak Competition, 2
Tags: number theory
The Fibonacci sequence is defined as $f_1=f_2=1$, $f_{n+2}=f_{n+1}+f_n$ ($n\in\mathbb{N}$). Suppose that $a$ and $b$ are positive integers such that $\frac ab$ lies between the two fractions $\frac{f_n}{f_{n-1}}$ and $\frac{f_{n+1}}{f_{n}}$. Show that $b\ge f_{n+1}$.

2003 Bulgaria Team Selection Test, 4
Tags: number theory
Is it true that for any permulation $a_1,a_2.....,a_{2002}$ of $1,2....,2002$ there are positive integers $m,n$ of the same parity such that $0<m<n<2003$ and $a_m+a_n=2a_{\frac {m+n}{2}}$

2018 APMO, 5
Tags: polynomial , algebra , number theory
Find all polynomials $P(x)$ with integer coefficients such that for all real numbers $s$ and $t$, if $P(s)$ and $P(t)$ are both integers, then $P(st)$ is also an integer.

1983 IMO Shortlist, 24
Tags: decimal representation , digit , periodic sequence , non-periodical functions , number theory
Let $d_n$ be the last nonzero digit of the decimal representation of $n!$. Prove that $d_n$ is aperiodic; that is, there do not exist $T$ and $n_0$ such that for all $n \geq n_0, d_{n+T} = d_n.$

2011 Bosnia And Herzegovina - Regional Olympiad, 2
Tags: equation , number theory
For positive integers $a$ and $b$ holds $a^3+4a=b^2$. Prove that $a=2t^2$ for some positive integer $t$

1998 Taiwan National Olympiad, 2
Tags: search , function , number theory
Does there exist a solution $(x,y,z,u,v)$ in integers greater than $1998$ to the equation $x^{2}+y^{2}+z^{2}+u^{2}+v^{2}=xyzuv-65$?

1992 Rioplatense Mathematical Olympiad, Level 3, 2
Tags: diophantine equation , diophantine , number theory
Determine the integers $0 \le a \le b \le c \le d$ such that: $$2^n= a^2 + b^2 + c^2 + d^2.$$

2016 European Mathematical Cup, 3
Tags: number theory
Prove that for all positive integers $n$ there exist $n$ distinct, positive rational numbers with sum of their squares equal to $n$. Proposed by Daniyar Aubekerov

2023 Auckland Mathematical Olympiad, 3
Tags: algebra , number theory
Each square on an $8\times 8$ checkers board contains either one or zero checkers. The number of checkers in each row is a multiple of $3$, the number of checkers in each column is a multiple of $5$. Assuming the top left corner of the board is shown below, how many checkers are used in total? [img]https://cdn.artofproblemsolving.com/attachments/0/8/e46929e7ec3fff9be4892ef954ae299e0cb8c7.png[/img]

2019 BMT Spring, 6
Tags: prime number , number theory
Define $ f(n) = \dfrac{n^2 + n}{2} $. Compute the number of positive integers $ n $ such that $ f(n) \leq 1000 $ and $ f(n) $ is the product of two prime numbers.

1981 AMC 12/AHSME, 7
Tags: least common multiple , number theory
How many of the first one hundred positive integers are divisible by all of the numbers $2,3,4,5$? $\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ 4$

2017 Mexico National Olympiad, 2
Tags: average , number theory
A set of $n$ positive integers is said to be [i]balanced[/i] if for each integer $k$ with $1 \leq k \leq n$, the average of any $k$ numbers in the set is an integer. Find the maximum possible sum of the elements of a balanced set, all of whose elements are less than or equal to $2017$.

2014 Online Math Open Problems, 14
Tags: algorithm , euclidean algorithm , greatest common divisor , number theory
What is the greatest common factor of $12345678987654321$ and $12345654321$? [i]Proposed by Evan Chen[/i]

2010 China Girls Math Olympiad, 8
Tags: modular arithmetic , calculus , number theory , integration , inequalities
Determine the least odd number $a > 5$ satisfying the following conditions: There are positive integers $m_1,m_2, n_1, n_2$ such that $a=m_1^2+n_1^2$, $a^2=m_2^2+n_2^2$, and $m_1-n_1=m_2-n_2.$

2020 Bundeswettbewerb Mathematik, 1
Tags: number theory
Show that there are infinitely many perfect squares of the form $50^m-50^n$, but no perfect square of the form $2020^m+2020^n$, where $m$ and $n$ are positive integers.

2000 Hong kong National Olympiad, 3
Tags: number theory
Find all prime numbers $p$ and $q$ such that $\frac{(7^{p}-2^{p})(7^{q}-2^{q})}{pq}$ is an integer.

1999 Akdeniz University MO, 1
Tags: number theory
Prove that, we find infinite numbers such that, this number writeable $1999k+1$ for $k \in {\mathbb N}$ and all digits are equal.

2016 Korea National Olympiad, 7
Tags: number theory
Let $N=2^a p_1^{b_1} p_2^{b_2} \ldots p_k^{b_k}$. Prove that there are $(b_1+1)(b_2+1)\ldots(b_k+1)$ number of $n$s which satisfies these two conditions. $\frac{n(n+1)}{2}\le N$, $N-\frac{n(n+1)}{2}$ is divided by $n$.

2013 Online Math Open Problems, 39
Tags: greatest common divisor , relatively prime , number theory
Find the number of 8-digit base-6 positive integers $(a_1a_2a_3a_4a_5a_6a_7a_8)_6$ (with leading zeros permitted) such that $(a_1a_2\ldots a_8)_6\mid(a_{i+1}a_{i+2}\ldots a_{i+8})_6$ for $i=1,2,\ldots,7$, where indices are taken modulo $8$ (so $a_9=a_1$, $a_{10}=a_2$, and so on). [i]Victor Wang[/i]

2003 South africa National Olympiad, 5
Tags: number theory , algebra , polynomial
Prove that the sum of the squares of two consecutive positive integers cannot be equal to a sum of the fourth powers of two consecutive positive integers.

2021 Argentina National Olympiad, 1
Tags: prime number , number theory
Determine all pairs of prime numbers $p$ and $q$ greater than $1$ and less than $100$, such that the following five numbers: $$p+6,p+10,q+4,q+10,p+q+1,$$ are all prime numbers.

1978 Dutch Mathematical Olympiad, 1
Tags: diophantine equation , diophantine , number theory
Prove that no integer $x$ and $y$ satisfy: $$3x^2 = 9 + y^3.$$

2024 Al-Khwarizmi IJMO, 2
Tags: number theory
For how many $x \in \{1,2,3,\dots, 2024\}$ is it possible that [i]Bekhzod[/i] summed $2024$ non-negative consecutive integers, [i]Ozod[/i] summed $2024+x$ non-negative consecutive integers and they got the same result? [i]Proposed by Marek Maruin, Slovakia[/i]

1985 Iran MO (2nd round), 4
Tags: number theory , floor function
Let $x$ and $y$ be two real numbers. Prove that the equations \[\lfloor x \rfloor + \lfloor y \rfloor =\lfloor x +y \rfloor , \quad \lfloor -x \rfloor + \lfloor -y \rfloor =\lfloor -x-y \rfloor\] Holds if and only if at least one of $x$ or $y$ be integer.

2006 Kyiv Mathematical Festival, 4
Tags: number theory
See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Let $a, b, c, d$ be positive integers and $p$ be prime number such that $a^2+b^2=p$ and $c^2+d^2$ is divisible by $p.$ Prove that there exist positive integers $e$ and $f$ such that $e^2+f^2=\frac{c^2+d^2}{p}.$