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

1982 IMO Shortlist, 4
Tags: algebra , polynomial , diophantine equation , parametric equation , geometric progression
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.

1967 IMO Longlists, 44
Tags: polynomial , summation , equation , diophantine equation , algebra
Suppose that $p$ and $q$ are two different positive integers and $x$ is a real number. Form the product $(x+p)(x+q).$ Find the sum $S(x,n) = \sum (x+p)(x+q),$ where $p$ and $q$ take values from 1 to $n.$ Does there exist integer values of $x$ for which $S(x,n) = 0.$

PEN H Problems, 53
Tags: modular arithmetic , diophantine equation
Suppose that $a, b$, and $p$ are integers such that $b \equiv 1 \; \pmod{4}$, $p \equiv 3 \; \pmod{4}$, $p$ is prime, and if $q$ is any prime divisor of $a$ such that $q \equiv 3 \; \pmod{4}$, then $q^{p}\vert a^{2}$ and $p$ does not divide $q-1$ (if $q=p$, then also $q \vert b$). Show that the equation \[x^{2}+4a^{2}= y^{p}-b^{p}\] has no solutions in integers.

1964 Dutch Mathematical Olympiad, 3
Tags: number theory , diophantine , diophantine equation , perfect square
Solve $ (n + 1)(n +10) = q^2$, for certain $q$ and maximum $n$.

2015 Swedish Mathematical Competition, 2
Tags: diophantine , diophantine equation , number theory
Determine all integer solutions to the equation $x^3 + y^3 + 2015 = 0$.

2005 iTest, 34
Tags: diophantine , diophantine equation , perfect square , number theory
If $x$ is the number of solutions to the equation $a^2 + b^2 + c^2 = d^2$ of the form $(a,b,c,d)$ such that $\{a,b,c\}$ are three consecutive square numbers and $d$ is also a square number, find $x$.

PEN H Problems, 18
Tags: diophantine equation
Determine all positive integer solutions $(x, y, z, t)$ of the equation \[(x+y)(y+z)(z+x)=xyzt\] for which $\gcd(x, y)=\gcd(y, z)=\gcd(z, x)=1$.

2012 Cuba MO, 5
Tags: diophantine , diophantine equation , number theory
Find all pairs $(m, n)$ of positive integers such that $m^2 + n^2 =(m + 1)(n + 1).$

I Soros Olympiad 1994-95 (Rus + Ukr), 9.5
Tags: diophantine equation , diophantine , number theory
Find the triplets of natural numbers $(p,q,r)$ that satisfy the equality $$\frac{1}{p}+\frac{q}{q^r -1}=1.$$

2005 Junior Tuymaada Olympiad, 3
Tags: algebra , system of equations , diophantine equation
Tram ticket costs $1$ Tug ($=100$ tugriks). $20$ passengers have only coins in denominations of $2$ and $5$ tugriks, while the conductor has nothing at all. It turned out that all passengers were able to pay the fare and get change. What is the smallest total number of passengers that the tram could have?

2013 Greece JBMO TST, 3
Tags: prime , diophantine equation , number theory
If $p$ is a prime positive integer and $x,y$ are positive integers, find , in terms of $p$, all pairs $(x,y)$ that are solutions of the equation: $p(x-2)=x(y-1)$. (1) If it is also given that $x+y=21$, find all triplets $(x,y,p)$ that are solutions to equation (1).

2017 Israel National Olympiad, 4
Tags: number theory , algebra , diophantine equation
Three rational number $x,p,q$ satisfy $p^2-xq^2$=1. Prove that there are integers $a,b$ such that $p=\frac{a^2+xb^2}{a^2-xb^2}$ and $q=\frac{2ab}{a^2-xb^2}$.

1998 Korea Junior Math Olympiad, 1
Tags: diophantine equation , number theory
Show that there exist no integer solutions $(x, y, z)$ to the equation $$x^3+2y^3+4z^3=9$$

2001 Croatia Team Selection Test, 3
Tags: number theory , diophantine equation , diophantine
Find all solutions of the equation $(a^a)^5 = b^b$ in positive integers.

2013 Dutch BxMO/EGMO TST, 3
Tags: number theory , prime , diophantine equation
Find all triples $(x,n,p)$ of positive integers $x$ and $n$ and primes $p$ for which the following holds $x^3 + 3x + 14 = 2 p^n$

1988 IMO Shortlist, 9
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 Tuymaada Olympiad, 7
Tags: relatively prime , system of equations , algebra , diophantine equation
Prove that there are infinitely many natural numbers $a,b,c,u$ and $v$ with greatest common divisor $1$ satisfying the system of equations: $a+b+c=u+v$ and $a^2+b^2+c^2=u^2+v^2$

2001 Hungary-Israel Binational, 1
Tags: number theory , diophantine equation
Find positive integers $x, y, z$ such that $x > z > 1999 \cdot 2000 \cdot 2001 > y$ and $2000x^{2}+y^{2}= 2001z^{2}.$

2003 Dutch Mathematical Olympiad, 3
Tags: diophantine , consecutive , diophantine equation , number theory
Determine all positive integers$ n$ that can be written as the product of two consecutive integers and as well as the product of four consecutive integers numbers. In the formula: $n = a (a + 1) = b (b + 1) (b + 2) (b + 3)$.

2015 China Team Selection Test, 5
Tags: number theory , diophantine equation
FIx positive integer $n$. Prove: For any positive integers $a,b,c$ not exceeding $3n^2+4n$, there exist integers $x,y,z$ with absolute value not exceeding $2n$ and not all $0$, such that $ax+by+cz=0$

2014 Indonesia MO Shortlist, N1
Tags: number theory , prime , diophantine equation
(a) Let $k$ be an natural number so that the equation $ab + (a + 1) (b + 1) = 2^k$ does not have a positive integer solution $(a, b)$. Show that $k + 1$ is a prime number. (b) Show that there are natural numbers $k$ so that $k + 1$ is prime numbers and equation $ab + (a + 1) (b + 1) = 2^k$ has a positive integer solution $(a, b)$.

PEN H Problems, 52
Tags: diophantine equation
Do there exist two right-angled triangles with integer length sides that have the lengths of exactly two sides in common?

2017 Singapore Junior Math Olympiad, 2
Tags: factorial , diophantine , diophantine equation , number theory
Let $n$ be a positive integer and $a_1,a_2,...,a_{2n}$ be $2n$ distinct integers. Given that the equation $|x-a_1| |x-a_2| ... |x-a_{2n}| =(n!)^2$ has an integer solution $x = m$, find $m$ in terms of $a_1,a_2,...,a_{2n}$

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

1986 Poland - Second Round, 4
Tags: number theory , perfect square , diophantine , diophantine equation
Natural numbers $ x, y, z $ whose greatest common divisor is equal to 1 satisfy the equation $$\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$$ Prove that $ x + y $ is the square of a natural number.