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

2019 Serbia Team Selection Test, P5
Tags: number theory , diophantine equation
Solve the equation in nonnegative integers:\\ $2^x=5^y+3$

1997 Estonia National Olympiad, 1
Tags: diophantine equation , diophantine , number theory
Prove that for every integer $n\ge 3$ there are such positives integers $x$ and $y$ such that $2^n = 7x^2 + y^2$

2021 Miklós Schweitzer, 2
Tags: factorial , diophantine equation , number theory
Prove that the equation \[ 2^x + 5^y - 31^z = n! \] has only a finite number of non-negative integer solutions $x,y,z,n$.

2015 India PRMO, 14
Tags: diophantine equation , algebra , number theory
$14.$ If $3^x+2^y=985.$ and $3^x-2^y=473.$ What is the value of $xy ?$

2019 LIMIT Category A, Problem 12
Tags: number theory , diophantine equation , combinatorics
Compute the number of ordered quadruples of positive integers $(a,b,c,d)$ such that $$a!b!c!d!=24!$$$\textbf{(A)}~4$ $\textbf{(B)}~4!$ $\textbf{(C)}~4^4$ $\textbf{(D)}~\text{None of the above}$

2024 Baltic Way, 20
Tags: diophantine equation , discriminant , number theory
Positive integers $a$, $b$ and $c$ satisfy the system of equations \begin{align*} (ab-1)^2&=c(a^2+b^2)+ab+1,\\ a^2+b^2&=c^2+ab. \end{align*} a) Prove that $c+1$ is a perfect square. b) Find all such triples $(a,b,c)$.

2016 Postal Coaching, 2
Tags: prime number , diophantine equation , number theory
Solve the equation for primes $p$ and $q$: $$p^3-q^3=pq^3-1.$$

2017 Istmo Centroamericano MO, 4
Tags: diophantine , diophantine equation , inequalities , number theory , system of equations
Suppose that $a$ and $ b$ are distinct positive integers satisfying $20a + 17b = p$ and $17a + 20b = q$ for certain primes $p$ and $ q$. Determine the minimum value of $p + q$.

1998 Switzerland Team Selection Test, 2
Tags: diophantine , diophantine equation , number theory
Find all nonnegative integer solutions $(x,y,z)$ of the equation $\frac{1}{x+2}+\frac{1}{y+2}=\frac{1}{2} +\frac{1}{z+2}$

KoMaL A Problems 2023/2024, A. 858
Tags: diophantine equation , quadratic residue , number theory
Prove that the only integer solution of the following system of equations is $u=v=x=y=z=0$: $$uv=x^2-5y^2, (u+v)(u+2v)=x^2-5z^2$$

2020 China Team Selection Test, 4
Tags: number theory , diophantine equation
Show that the following equation has finitely many solutions $(t,A,x,y,z)$ in positive integers $$\sqrt{t(1-A^{-2})(1-x^{-2})(1-y^{-2})(1-z^{-2})}=(1+x^{-1})(1+y^{-1})(1+z^{-1})$$

2023 pOMA, 3
Tags: diophantine equation , number theory
Find all positive integers $l$ for which the equation \[ a^3+b^3+ab=(lab+1)(a+b) \] has a solution over positive integers $a,b$.

2005 Austria Beginners' Competition, 1
Tags: number theory , diophantine equation , diophantine
Show that there are no positive integers $a$ und $b$ such that $4a(a + 1) = b(b + 3)$

2020 China Team Selection Test, 4
Tags: diophantine equation , number theory
Show that the following equation has finitely many solutions $(t,A,x,y,z)$ in positive integers $$\sqrt{t(1-A^{-2})(1-x^{-2})(1-y^{-2})(1-z^{-2})}=(1+x^{-1})(1+y^{-1})(1+z^{-1})$$

2020 Jozsef Wildt International Math Competition, W55
Tags: fibonacci number , diophantine equation , fibonacci , number theory
Prove that the equation $$1320x^3=(y_1+y_2+y_3+y_4)(z_1+z_2+z_3+z_4)(t_1+t_2+t_3+t_4+t_5)$$ has infinitely many solutions in the set of Fibonacci numbers. [i]Proposed by Mihály Bencze[/i]

1971 IMO Longlists, 14
Tags: geometry , 3d geometry , diophantine equation , number theory
Note that $8^3 - 7^3 = 169 = 13^2$ and $13 = 2^2 + 3^2.$ Prove that if the difference between two consecutive cubes is a square, then it is the square of the sum of two consecutive squares.

2023 SG Originals, Q2
Tags: diophantine equation , number theory
Find all positive integers $k$ such that there exists positive integers $a, b$ such that \[a^2+4=(k^2-4)b^2.\]

1996 Estonia National Olympiad, 1
Tags: prime , diophantine equation , diophantine , number theory
Let $p$ be a fixed prime. Find all pairs $(x,y)$ of positive numbers satisfying $p(x-y) = xy$.

1988 IMO, 3
Tags: number theory , divisibility , diophantine equation , perfect square , vieta jumping
Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$. Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square.

1994 Abels Math Contest (Norwegian MO), 2b
Tags: diophantine , diophantine equation , number theory
Find all integers $x,y,z$ such that $x^3 +5y^3 = 9z^3$.

2015 IMAR Test, 1
Tags: ro , diophantine equation
Determine all positive integers expressible, for every integer $ n \geq 3 $, in the form \begin{align*} \frac{(a_1 + 1)(a_2 + 1) \ldots (a_n + 1) - 1}{a_1a_2 \ldots a_n}, \end{align*} where $ a_1, a_2, \ldots, a_n $ are pairwise distinct positive integers.

VI Soros Olympiad 1999 - 2000 (Russia), 9.2
Tags: diophantine equation , diophantine , number theory
Find the smallest natural number n such that for all integers $m > n$ there are positive integers $x$ and $y$ for which the equality 1$7x + 23y = m$ holds

2001 Greece JBMO TST, 1
Tags: factorization , diophantine equation , algebra , number theory
a) Factorize $A= x^4+y^4+z^4-2x^2y^2-2y^2z^2-2z^2x^2$ b) Prove that there are no integers $x,y,z$ such that $x^4+y^4+z^4-2x^2y^2-2y^2z^2-2z^2x^2=2000 $

1981 All Soviet Union Mathematical Olympiad, 316
Tags: number theory , diophantine equation , diophantine
Find the natural solutions of the equation $x^3 - y^3 = xy + 61$.

2020 Abels Math Contest (Norwegian MO) Final, 2a
Tags: diophantine , diophantine equation , number theory
Find all natural numbers $k$ such that there exist natural numbers $a_1,a_2,...,a_{k+1}$ with $ a_1!+a_2!+... +a_{k+1}!=k!$ Note that we do not consider $0$ to be a natural number.