Olympiad problems

2017 JBMO Shortlist, NT4

Solve in nonnegative integers the equation $5^t + 3^x4^y = z^2$.

2022 Turkey Junior National Olympiad, 3

Tags: diophantine equation , number theory

Let $m, n, a, k$ be positive integers and $k>1$ such that the equality $$5^m+63n+49=a^k$$ holds. Find the minimum value of $k$.

PEN H Problems, 36

Tags: diophantine equation

Prove that the equation $a^2 +b^2 =c^2 +3$ has infinitely many integer solutions $(a, b, c)$.

PEN H Problems, 82

Tags: diophantine equation

Find all triples $(a, b, c)$ of positive integers to the equation \[a! b! = a!+b!+c!.\]

2004 Thailand Mathematical Olympiad, 11

Tags: number theory , diophantine , diophantine equation

Find the number of positive integer solutions to $(x_1 + x_2 + x_3)(y_1 + y_2 + y_3 + y_4) = 91$

1991 AIME Problems, 6

Tags: function , floor function , algebra , number theory , diophantine equation , inequalities , modular arithmetic

Suppose $r$ is a real number for which \[ \left\lfloor r + \frac{19}{100} \right\rfloor + \left\lfloor r + \frac{20}{100} \right\rfloor + \left\lfloor r + \frac{21}{100} \right\rfloor + \cdots + \left\lfloor r + \frac{91}{100} \right\rfloor = 546. \] Find $\lfloor 100r \rfloor$. (For real $x$, $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.)

Oliforum Contest I 2008, 1

Tags: diophantine , diophantine equation , number theory

Let $ p>3$ be a prime. If $ p$ divides $ x$, prove that the equation $ x^2-1=y^p$ does not have positive integer solutions.

2004 Estonia National Olympiad, 1

Tags: number theory , diophantine equation , diophantine

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

1981 Bulgaria National Olympiad, Problem 4

Tags: diophantine equation , number theory

Let $n$ be an odd positive integer. Prove that if the equation $\frac1x+\frac1y=\frac4n$ has a solution in positive integers $x,y$, then $n$ has at least one divisor of the form $4k-1$, $k\in\mathbb N$.

1996 Israel National Olympiad, 1

Tags: diophantine , number theory , diophantine equation

Let $a$ be a prime number and $n > 2$ an integer. Find all integer solutions of the equation $x^n +ay^n = a^2z^n$ .

2012 NZMOC Camp Selection Problems, 6

Tags: diophantine , number theory , diophantine equation

Let $a, b$ and $c$ be positive integers such that $a^{b+c} = b^{c} c$. Prove that b is a divisor of $c$, and that $c$ is of the form $d^b$ for some positive integer $d$.

2024 Bangladesh Mathematical Olympiad, P1

Tags: number theory , diophantine equation , divisibility , factorization

Find all non-negative integers $x, y$ such that\[x^3y+x+y=xy+2xy^2\]

2023 Brazil National Olympiad, 3

Tags: diophantine equation , system of equations , number theory

Let $n$ be a positive integer. Show that there are integers $x_1, x_2, \ldots , x_n$, [i]not all equal[/i], satisfying $$\begin{cases} x_1^2+x_2+x_3+\ldots+x_n=0 \\ x_1+x_2^2+x_3+\ldots+x_n=0 \\ x_1+x_2+x_3^2+\ldots+x_n=0 \\ \vdots \\ x_1+x_2+x_3+\ldots+x_n^2=0 \end{cases}$$ if, and only if, $2n-1$ is not prime.

PEN H Problems, 78

Tags: diophantine equation

Let $x, y$, and $z$ be integers with $z>1$. Show that \[(x+1)^{2}+(x+2)^{2}+\cdots+(x+99)^{2}\neq y^{z}.\]

2013 Swedish Mathematical Competition, 1

Tags: diophantine , diophantine equation , number theory

For $r> 0$ denote by $B_r$ the set of points at distance at most $r$ length units from the origin. If $P_r$ is the set of the points in $B_r$ whit integer coordinates, show that the equation $$xy^3z + 2x^3z^3-3x^5y = 0$$ has an odd number of solutions $(x, y, z)$ in $P_r$.

2013 Korea Junior Math Olympiad, 3

Tags: algebra , sequence , sum , positive integer , diophantine equation , recurrence relation

$\{a_n\}$ is a positive integer sequence such that $a_{i+2} = a_{i+1} +a_i$ (for all $i \ge 1$). For positive integer $n$, define as $$b_n=\frac{1}{a_{2n+1}}\Sigma_{i=1}^{4n-2}a_i$$ Prove that $b_n$ is positive integer.

1905 Eotvos Mathematical Competition, 1

Tags: system of equations , diophantine equation , diophantine , number theory

For given positive integers $n$ and $p$, find neaessary and sufficient conditions for the system of equations $$x + py = n , \\ x + y = p^2$$ to have a solution $(x, y, z)$ of positive integers. Prove also that there is at most one such solution.

1976 Czech and Slovak Olympiad III A, 1

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

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

1997 IMO Shortlist, 6

Tags: number theory , relatively prime , diophantine equation

(a) Let $ n$ be a positive integer. Prove that there exist distinct positive integers $ x, y, z$ such that \[ x^{n\minus{}1} \plus{} y^n \equal{} z^{n\plus{}1}.\] (b) Let $ a, b, c$ be positive integers such that $ a$ and $ b$ are relatively prime and $ c$ is relatively prime either to $ a$ or to $ b.$ Prove that there exist infinitely many triples $ (x, y, z)$ of distinct positive integers $ x, y, z$ such that \[ x^a \plus{} y^b \equal{} z^c.\]

2017 JBMO Shortlist, NT4

2022 Turkey Junior National Olympiad, 3

PEN H Problems, 36

PEN H Problems, 82

2004 Thailand Mathematical Olympiad, 11

1991 AIME Problems, 6

Oliforum Contest I 2008, 1

2004 Estonia National Olympiad, 1

1981 Bulgaria National Olympiad, Problem 4

1996 Israel National Olympiad, 1

2012 NZMOC Camp Selection Problems, 6

2024 Bangladesh Mathematical Olympiad, P1

2023 Brazil National Olympiad, 3

PEN H Problems, 78

2013 Swedish Mathematical Competition, 1

2013 Korea Junior Math Olympiad, 3

1905 Eotvos Mathematical Competition, 1

1976 Czech and Slovak Olympiad III A, 1

1997 IMO Shortlist, 6

2004 VJIMC, Problem 4

2014 Estonia Team Selection Test, 6

2021 Middle European Mathematical Olympiad, 7

2008 Denmark MO - Mohr Contest, 2

1950 Poland - Second Round, 6

2016 India PRMO, 2