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: 916

2018 Regional Olympiad of Mexico Center Zone, 5
Tags: diophantine equation , diophantine , number theory
Find all solutions of the equation $$p ^ 2 + q ^ 2 + 49r ^ 2 = 9k ^ 2-101$$ with $ p$, $q$ and $r$ positive prime numbers and $k$ a positive integer.

2015 Indonesia MO Shortlist, N8
Tags: diophantine equation , infinitely many solutions , number theory
The natural number $n$ is said to be good if there are natural numbers $a$ and $b$ that satisfy $a + b = n$ and $ab | n^2 + n + 1$. (a) Show that there are infinitely many good numbers. (b) Show that if $n$ is a good number, then $7 \nmid n$.

PEN H Problems, 16
Tags: diophantine equation , algebra , function , polynomial
Find all pairs $(a,b)$ of different positive integers that satisfy the equation $W(a)=W(b)$, where $W(x)=x^{4}-3x^{3}+5x^{2}-9x$.

2014 China Team Selection Test, 3
Tags: diophantine equation , number theory
Show that there are no 2-tuples $ (x,y)$ of positive integers satisfying the equation $ (x+1) (x+2)\cdots (x+2014)= (y+1) (y+2)\cdots (y+4028).$

2014 IMAC Arhimede, 3
Tags: diophantine equation , number theory
a) Prove that the equation $2^x + 21^x = y^3$ has no solution in the set of natural numbers. b) Solve the equation $2^x + 21^y = z^2y$ in the set of non-negative integer numbers.

1958 AMC 12/AHSME, 32
Tags: least common multiple , diophantine equation , number theory , modular arithmetic
With $ \$1000$ a rancher is to buy steers at $ \$25$ each and cows at $ \$26$ each. If the number of steers $ s$ and the number of cows $ c$ are both positive integers, then: $ \textbf{(A)}\ \text{this problem has no solution}\qquad\\ \textbf{(B)}\ \text{there are two solutions with }{s}\text{ exceeding }{c}\qquad \\ \textbf{(C)}\ \text{there are two solutions with }{c}\text{ exceeding }{s}\qquad \\ \textbf{(D)}\ \text{there is one solution with }{s}\text{ exceeding }{c}\qquad \\ \textbf{(E)}\ \text{there is one solution with }{c}\text{ exceeding }{s}$

2015 BMT Spring, 9
Tags: diophantine equation , number theory
There exists a unique pair of positive integers $k,n$ such that $k$ is divisible by $6$, and $\sum_{i=1}^ki^2=n^2$. Find $(k,n)$.

2016 Postal Coaching, 5
Tags: diophantine equation , number theory
Find all nonnegative integers $k, n$ which satisfy $2^{2k+1} + 9\cdot 2^k + 5 = n^2.$

2017 Harvard-MIT Mathematics Tournament, 4
Tags: number theory , diophantine equation
Find the number of ordered triples of nonnegative integers $(a, b, c)$ that satisfy \[(ab + 1)(bc + 1)(ca + 1) = 84.\]

2007 Puerto Rico Team Selection Test, 2
Tags: diophantine , diophantine equation , number theory , algebra , system of equations
Find the solutions of positive integers for the system $xy + x + y = 71$ and $x^2y + xy^2 = 880$.

2009 China Girls Math Olympiad, 8
Tags: pell equation , diophantine equation , continued fraction , modular arithmetic , induction , floor function , number theory
For a positive integer $ n,$ $ a_{n}\equal{}n\sqrt{5}\minus{} \lfloor n\sqrt{5}\rfloor$. Compute the maximum value and the minimum value of $ a_{1},a_{2},\ldots ,a_{2009}.$

1987 Czech and Slovak Olympiad III A, 2
Tags: number of solutions , diophantine equation , prime number , number theory
Given a prime $p>3$ and an odd integer $n>0$, show that the equation $$xyz=p^n(x+y+z)$$ has at least $3(n+1)$ different solutions up to symmetry. (That is, if $(x',y',z')$ is a solution and $(x'',y'',z'')$ is a permutation of the previous, they are considered to be the same solution.)

2010 Germany Team Selection Test, 3
Tags: diophantine equation , number theory , induction
Determine all $(m,n) \in \mathbb{Z}^+ \times \mathbb{Z}^+$ which satisfy $3^m-7^n=2.$

1977 Yugoslav Team Selection Test, Problem 2
Tags: diophantine equation , number theory
Determine all $6$-tuples $(p,q,r,x,y,z)$ where $p,q,r$ are prime, and $x,y,z$ natural numbers such that $p^{2x}=q^yr^z+1$.

2023 Dutch BxMO TST, 5
Tags: diophantine equation , modular arithmetic , number theory
Find all pairs of prime numbers $(p,q)$ for which \[2^p = 2^{q-2} + q!.\]

2023 Austrian MO Regional Competition, 4
Tags: gcd , diophantine equation , number theory , greatest common divisor
Determine all pairs $(x, y)$ of positive integers such that for $d = gcd(x, y)$ the equation $$xyd = x + y + d^2$$ holds. [i](Walther Janous)[/i]

KoMaL A Problems 2019/2020, A. 778
Tags: diophantine equation , number theory
Find all square-free integers $d$ for which there exist positive integers $x, y$ and $n$ satisfying $x^2+dy^2=2^n$ Submitted by Kada Williams, Cambridge

PEN H Problems, 77
Tags: diophantine equation , relatively prime , number theory , ratio
Find all pairwise relatively prime positive integers $l, m, n$ such that \[(l+m+n)\left( \frac{1}{l}+\frac{1}{m}+\frac{1}{n}\right)\] is an integer.

2024 Mongolian Mathematical Olympiad, 1
Tags: diophantine equation , number theory , factorial
Find all triples $(a, b, c)$ of positive integers such that $a \leq b$ and \[a!+b!=c^4+2024\] [i]Proposed by Otgonbayar Uuye.[/i]

PEN H Problems, 35
Tags: diophantine equation , algebra , polynomial
Find all cubic polynomials $x^3 +ax^2 +bx+c$ admitting the rational numbers $a$, $b$ and $c$ as roots.

2021 Serbia Team Selection Test, P3
Tags: system of equations , prime number , diophantine equation , number theory
Given is a prime number $p$. Find the number of positive integer solutions $(a, b, c, d)$ of the system of equations $ac+bd = p(a+c)$ and $bc-ad = p(b-d)$.

1982 IMO Longlists, 14
Tags: parametric equation , diophantine equation , geometric progression , polynomial , algebra
Determine all real values of the parameter $a$ for which the equation \[16x^4 -ax^3 + (2a + 17)x^2 -ax + 16 = 0\] has exactly four distinct real roots that form a geometric progression.

2018 India PRMO, 18
Tags: diophantine equation , diophantine , number theory
If $a, b, c \ge 4$ are integers, not all equal, and $4abc = (a+3)(b+3)(c+3)$ then what is the value of $a+b+c$ ?

2013 Bulgaria National Olympiad, 1
Tags: modular arithmetic , diophantine equation , number theory
Find all prime numbers $p,q$, for which $p^{q+1}+q^{p+1}$ is a perfect square. [i]Proposed by P. Boyvalenkov[/i]

2013 Estonia Team Selection Test, 1
Tags: diophantine equation , diophantine , system of equations , number theory
Find all prime numbers $p$ for which one can find a positive integer $m$ and nonnegative integers $a_0,a_1,...,a_m$ less than $p$ such that $$\begin{cases} a_0+a_1p+...+a_{m-1}p^{m-1}+a_{m}p^{m} = 2013 \\ a_0+a_1+...+a_{m-1}+a_{m} = 11\end{cases}$$