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: 436

2014 Hanoi Open Mathematics Competitions, 4
Tags: diophantine equation , prime , diophantine , number theory
If $p$ is a prime number such that there exist positive integers $a$ and $b$ such that $\frac{1}{p}=\frac{1}{a^2}+\frac{1}{b^2}$ then $p$ is (A): $3$, (B): $5$, (C): $11$, (D): $7$, (E) None of the above.

1993 Romania Team Selection Test, 4
Tags: diophantine , number theory
Prove that the equation $ (x\plus{}y)^n\equal{}x^m\plus{}y^m$ has a unique solution in integers with $ x>y>0$ and $ m,n>1$.

2021 Malaysia IMONST 1, 10
Tags: diophantine equation , diophantine , number theory
Determine the number of integer solutions $(x, y, z)$, with $0 \le x, y, z \le 100$, for the equation $$(x - y)^2 + (y + z)^2 = (x + y)^2 + (y - z)^2.$$

2022 Dutch BxMO TST, 3
Tags: diophantine equation , diophantine , number theory
Find all pairs $(p, q)$ of prime numbers such that $$p(p^2 -p - 1) = q(2q + 3).$$

2000 Switzerland Team Selection Test, 7
Tags: diophantine equation , diophantine , number theory
Show that the equation $14x^2 +15y^2 = 7^{2000}$ has no integer solutions.

2020 Puerto Rico Team Selection Test, 2
Tags: system of equations , digit , diophantine equation , diophantine , number theory
The cost of $1000$ grams of chocolate is $x$ dollars and the cost of $1000$ grams of potatoes is $y$ dollars, the numbers $x$ and $y$ are positive integers and have not more than $2$ digits. Mother said to Maria to buy $200$ grams of chocolate and $1000$ grams of potatoes that cost exactly $N$ dollars. Maria got confused and bought $1000$ grams of chocolate and $200$ grams of potatoes that cost exactly $M$ dollars ($M >N$). It turned out that the numbers $M$ and $N$ have no more than two digits and are formed of the same digits but in a different order. Find $x$ and $y$.

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.

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 $x^2y + xy^2 = 880$.

2010 Saudi Arabia BMO TST, 4
Tags: system of equations , system , diophantine , number theory
Find all triples $(x,y, z)$ of integers such that $$\begin{cases} x^2y + y^2z + z^2x= 2010^2 \\ xy^2 + yz^2 + zx^2= -2010 \end{cases}$$

2011 Saudi Arabia Pre-TST, 3.4
Tags: system , diophantine , system of equations , number theory
Find all quadruples $(x,y,z,w)$ of integers satisfying the sys­tem of equations $$x + y + z + w = xy + yz + zx + w^2 - w = xyz - w^3 = - 1$$

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 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 $$\begin{cases} a_0+a_1p+...+a_{m-1}p^{m-1}+a_{m}p^{m} = 2013 \\ a_0+a_1+...+a_{m-1}+a_{m} = 11\end{cases}$$

2005 Austria Beginners' Competition, 2
Tags: inequalities , diophantine , number theory
Determine the number of integer pairs $(x, y)$ such that $(|x| - 2)^2 + (|y| - 2)^2 < 5$ .

2013 Estonia Team Selection Test, 1
Tags: system of equations , diophantine equation , diophantine , 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 $$\begin{cases} a_0+a_1p+...+a_{m-1}p^{m-1}+a_{m}p^{m} = 2013 \\ a_0+a_1+...+a_{m-1}+a_{m} = 11\end{cases}$$

2020 Chile National Olympiad, 4
Tags: diophantine , diophantine equation , system of equations , number theory
Determine all three integers $(x, y, z)$ that are solutions of the system $$x + y -z = 6$$ $$x^3 + y^3 -z^3 = 414$$

1998 Slovenia Team Selection Test, 4
Tags: diophantine , gcd , diophantine equation , number theory
Find all positive integers $x$ and $y$ such that $x+y^2+z^3 = xyz$, where $z$ is the greatest common divisor of $x$ and $y$

2021 Grand Duchy of Lithuania, 4
Tags: diophantine equation , diophantine , number theory
A triplet of positive integers $(x, y, z)$ satisfying $x, y, z > 1$ and $x^3 - yz^3 = 2021$ is called [i]primary [/i] if at least two of the integers $x, y, z$ are prime numbers. a) Find at least one primary triplet. b) Show that there are infinitely many primary triplets.

2024 India Iran Friendly Math Competition, 4
Tags: diophantine , number theory
Prove that there are no integers $x, y, z$ satisfying the equation $$x^2+y^2-z^2=xyz-2.$$ [i]Proposed by Navid Safaei[/i]

2002 Denmark MO - Mohr Contest, 3
Tags: system of equations , diophantine , number theory , algebra
Two positive integers have the sum $2002$. Can $2002$ divide their product?

2010 Contests, 2
Tags: diophantine , diophantine equation , number theory
Determine all triples $(x, y, z)$ of positive integers $x > y > z > 0$, such that $x^2 = y \cdot 2^z + 1$

2015 Saudi Arabia Pre-TST, 1.3
Tags: diophantine equation , diophantine , number theory
Find all integer solutions of the equation $x^2y^5 - 2^x5^y = 2015 + 4xy$. (Malik Talbi)

2016 NZMOC Camp Selection Problems, 7
Tags: diophantine , diophantine equation , number theory
Find all positive integers $n$ for which the equation $$(x^2 + y^2)^n = (xy)^{2016}$$ has positive integer solutions.

2020 LIMIT Category 2, 15
Tags: diophantine equation , diophantine , limit , number theory
How many integer pairs $(x,y)$ satisfies $x^2+y^2=9999(x-y)$?

2019 Hanoi Open Mathematics Competitions, 7
Tags: diophantine equation , diophantine , number theory
Let $p$ and $q$ be odd prime numbers. Assume that there exists a positive integer $n$ such that $pq-1= n^3$. Express $p+q$ in terms of $n$

2016 Regional Olympiad of Mexico Northeast, 1
Tags: diophantine , diophantine equation , number theory
Determine if there is any triple of nonnegative integers, not necessarily different, $(a, b, c)$ such that: $$a^3 + b^3 + c^3 = 2016$$