Olympiad problems

1995 Irish Math Olympiad, 2

Determine all integers $ a$ for which the equation $ x^2\plus{}axy\plus{}y^2\equal{}1$ has infinitely many distinct integer solutions $ x,y$.

2015 Indonesia MO Shortlist, N5

Tags: integer , number theory , diophantine equation

Given a prime number $n \ge 5$. Prove that for any natural number $a \le \frac{n}{2} $, we can search for natural number $b \le \frac{n}{2}$ so the number of non-negative integer solutions $(x, y)$ of the equation $ax+by=n$ to be odd*. Clarification: * For example when $n = 7, a = 3$, we can choose$ b = 1$ so that there number of solutions og $3x + y = 7$ to be $3$ (odd), namely: $(0, 7), (1, 4), (2, 1)$

2021 Federal Competition For Advanced Students, P2, 3

Tags: number theory , diophantine equation , diophantine

Find all triples $(a, b, c)$ of natural numbers $a, b$ and $c$, for which $a^{b + 20} (c-1) = c^{b + 21} - 1$ is satisfied. (Walther Janous)

2023 Regional Competition For Advanced Students, 4

Tags: number theory , gcd , greatest common divisor , diophantine equation

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]

PEN H Problems, 53

Tags: diophantine equation , modular arithmetic

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.

2007 Indonesia TST, 3

Tags: inequalities , floor function , diophantine equation , number theory

For each real number $ x$< let $ \lfloor x \rfloor$ be the integer satisfying $ \lfloor x \rfloor \le x < \lfloor x \rfloor \plus{}1$ and let $ \{x\}\equal{}x\minus{}\lfloor x \rfloor$. Let $ c$ be a real number such that \[ \{n\sqrt{3}\}>\dfrac{c}{n\sqrt{3}}\] for all positive integers $ n$. Prove that $ c \le 1$.

2013 Polish MO Finals, 1

Tags: diophantine equation , number theory

Find all solutions of the following equation in integers $x,y: x^4+ y= x^3+ y^2$

PEN H Problems, 90

Tags: diophantine equation

Find all triples of positive integers $(x, y, z)$ such that \[(x+y)(1+xy)= 2^{z}.\]

1980 Polish MO Finals, 2

Tags: number theory , diophantine , diophantine equation

Prove that for every $n$ there exists a solution of the equation $$a^2 +b^2 +c^2 = 3abc$$ in natural numbers $a,b,c$ greater than $n$.

2004 Estonia National Olympiad, 1

Tags: diophantine equation , diophantine , number theory

Find all pairs of integers $(a, b)$ such that $a^2 + ab + b^2 = 1$

1994 Abels Math Contest (Norwegian MO), 2b

Tags: diophantine equation , diophantine , number theory

Find all integers $x,y,z$ such that $x^3 +5y^3 = 9z^3$.

2002 Argentina National Olympiad, 2

Tags: number theory , diophantine equation , diophantine

Determine the smallest positive integer $k$ so that the equation $$2002x+273y=200201+k$$ has integer solutions, and for that value of $k$, find the number of solutions $\left (x,y\right )$ with $x$, $y$ positive integers that have the equation.

PEN H Problems, 39

Tags: diophantine equation , quadratic

Let $A, B, C, D, E$ be integers, $B \neq 0$ and $F=AD^{2}-BCD+B^{2}E \neq 0$. Prove that the number $N$ of pairs of integers $(x, y)$ such that \[Ax^{2}+Bxy+Cx+Dy+E=0,\] satisfies $N \le 2 d( \vert F \vert )$, where $d(n)$ denotes the number of positive divisors of positive integer $n$.

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.

2016 Ecuador Juniors, 2

Tags: diophantine , diophantine equation

Prove that there are no positive integers $x, y$ such that: $(x + 1)^2 + (x + 2)^2 +...+ (x + 9)^2 = y^2$

1990 IMO Longlists, 26

Tags: modular arithmetic , search , diophantine equation , number theory

Prove that there exist infinitely many positive integers $n$ such that the number $\frac{1^2+2^2+\cdots+n^2}{n}$ is a perfect square. Obviously, $1$ is the least integer having this property. Find the next two least integers having this property.

2015 FYROM JBMO Team Selection Test, 1

Tags: number theory , diophantine equation , integer

Solve the equation $x^2+y^4+1=6^z$ in the set of integers.

2005 Taiwan TST Round 1, 1

Tags: diophantine equation , number theory

Prove that there exists infinitely many positive integers $n$ such that $n, n+1$, and $n+2$ can be written as the sum of two perfect squares.

2001 Croatia National Olympiad, Problem 1

Tags: diophantine equation , number theory , preschool

Find all integers $x$ for which $2x^2-x-36$ is the square of a prime number.

2012 Cuba MO, 5

Tags: number theory , diophantine , diophantine equation

Find all pairs $(m, n)$ of positive integers such that $m^2 + n^2 =(m + 1)(n + 1).$

2014 Contests, 3

Tags: number theory , diophantine equation

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.

1976 Czech and Slovak Olympiad III A, 1

Tags: number theory , diophantine equation , perfect square , sum of squares

Determine all integers $x,y,z$ such that \[x^2+y^2=3z^2.\]

1987 USAMO, 1

Tags: algebra , diophantine equation , equation

Determine all solutions in non-zero integers $a$ and $b$ of the equation \[(a^2+b)(a+b^2) = (a-b)^3.\]

2023 Taiwan TST Round 3, 4

Tags: inequalities , am-gm , number theory , diophantine equation

Find all positive integers $a$, $b$ and $c$ such that $ab$ is a square, and \[a+b+c-3\sqrt[3]{abc}=1.\] [i]Proposed by usjl[/i]

2014 NIMO Summer Contest, 15

Tags: diophantine equation

Let $A = (0,0)$, $B=(-1,-1)$, $C=(x,y)$, and $D=(x+1,y)$, where $x > y$ are positive integers. Suppose points $A$, $B$, $C$, $D$ lie on a circle with radius $r$. Denote by $r_1$ and $r_2$ the smallest and second smallest possible values of $r$. Compute $r_1^2 + r_2^2$. [i]Proposed by Lewis Chen[/i]