Olympiad problems

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Found problems: 916

2011 Saudi Arabia Pre-TST, 3.2

Tags: diophantine , number theory , diophantine equation

Find all pairs of nonnegative integers $(a, b)$ such that $a+2b-b^2=\sqrt{2a+a^2+|2a+1-2b|}$.

2023 Costa Rica - Final Round, 3.6

Tags: digit , diophantine equation , number theory

Given a positive integer $N$, define $u(N)$ as the number obtained by making the ones digit the left-most digit of $N$, that is, taking the last, right-most digit (the ones digit) and moving it leftwards through the digits of $N$ until it becomes the first (left-most) digit; for example, $u(2023) = 3202$. [b](1)[/b] Find a $6$-digit positive integer $N$ such that \[\frac{u(N)}{N} = \frac{23}{35}.\] [b](2)[/b] Prove that there is no positive integer $N$ with less than $6$ digits such that \[\frac{u(N)}{N} = \frac{23}{35}.\]

PEN H Problems, 68

Tags: diophantine equation , ratio , calculus

Consider the system \[x+y=z+u,\] \[2xy=zu.\] Find the greatest value of the real constant $m$ such that $m \le \frac{x}{y}$ for any positive integer solution $(x, y, z, u)$ of the system, with $x \ge y$.

1992 Poland - First Round, 1

Tags: diophantine equation , number theory

Solve the following equation in real numbers: $\frac{(x^2-1)(|x|+1)}{x+sgnx}=[x+1].$

PEN H Problems, 70

Tags: diophantine equation

Show that the equation $\{x^3\}+\{y^3\}=\{z^3\}$ has infinitely many rational non-integer solutions.

1992 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: diophantine equation , diophantine , number theory

Determine the integers $0 \le a \le b \le c \le d$ such that: $$2^n= a^2 + b^2 + c^2 + d^2.$$

1978 Dutch Mathematical Olympiad, 1

Tags: diophantine equation , diophantine , number theory

Prove that no integer $x$ and $y$ satisfy: $$3x^2 = 9 + y^3.$$

2024 Euler Olympiad, Round 2, 1

Tags: euler , diophantine equation , number theory

Find all triples $(a, b,c) $ of positive integers, such that: \[ a! + b! = c!! \] where $(2k)!! = 2 \cdot 4 \cdot \ldots \cdot (2k)$ and $ (2k + 1)!! = 1 \cdot 3 \cdot \ldots \cdot (2k+1).$ [i]Proposed by Stijn Cambie, Belgium [/i]

2018 Ecuador NMO (OMEC), 1

Tags: diophantine , number theory , diophantine equation

Let $a, b$ be integers. Show that the equation $a^2 + b^2 = 26a$ has at least $12$ solutions.

1971 Dutch Mathematical Olympiad, 3

Tags: number theory , diophantine equation , diophantine

Prove that $(0,1)$, $(0, -1)$,$( -1,1)$ and $(-1,-1)$ are the only integer solutions of $$x^2 + x +1 = y^2.$$

2019 Vietnam TST, P4

Tags: diophantine equation , number theory

Find all triplets of positive integers $(x, y, z)$ such that $2^x+1=7^y+2^z$.

PEN H Problems, 43

Tags: diophantine equation

Find all solutions in integers of $x^{3}+2y^{3}=4z^{3}$.

2016 Croatia Team Selection Test, Problem 4

Tags: prime , diophantine equation , prime number , number theory

Find all pairs $(p,q)$ of prime numbers such that $$ p(p^2 - p - 1) = q(2q + 3) .$$

2021 Junior Macedonian Mathematical Olympiad, Problem 3

Tags: diophantine equation , number theory

Find all positive integers $n$ and prime numbers $p$ such that $$17^n \cdot 2^{n^2} - p =(2^{n^2+3}+2^{n^2}-1) \cdot n^2.$$ [i]Authored by Nikola Velov[/i]

PEN H Problems, 57

Tags: diophantine equation , binomial coefficient

Show that the equation ${n \choose k}=m^{l}$ has no integral solution with $l \ge 2$ and $4 \le k \le n-4$.

1989 IMO Longlists, 7

Tags: algebra , polynomial , diophantine equation , rational roots , equation

Prove that $ \forall n > 1, n \in \mathbb{N}$ the equation \[ \sum^n_{k\equal{}1} \frac{x^k}{k!} \plus{} 1 \equal{} 0\] has no rational roots.

2002 Croatia National Olympiad, Problem 3

Tags: number theory , diophantine equation

Find all triples $(x,y,z)$ of natural numbers that verify the equation $$2x^2y^2+2y^2z^2+2z^2x^2-x^4-y^4-z^4=576.$$

2015 China Team Selection Test, 5

Tags: diophantine equation , number theory

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 $ax+by+cz=0$

2015 Saudi Arabia JBMO TST, 1

Tags: number theory , diophantine equation , diophantine

Find all the triples $(x,y,z)$ of positive integers such that $xy+yz+zx-xyz=2015$

PEN H Problems, 13

Tags: 3d geometry , modular arithmetic , diophantine equation , geometry

Find all pairs $(x,y)$ of positive integers that satisfy the equation \[y^{2}=x^{3}+16.\]

VI Soros Olympiad 1999 - 2000 (Russia), 11.4

Tags: diophantine , diophantine equation , number theory

For prime numbers $p$ and $q$, natural numbers $n$, $k$, $r$, the equality $p^{2k}+q^{2n}=r^2$ holds. Prove that the number $r$ is prime.

2015 Costa Rica - Final Round, 4

Tags: number theory , diophantine equation , diophantine

Find all triples $(p,M, z)$ of integers, where $p$ is prime, $m$ is positive and $z$ is negative, that satisfy the equation $$p^3 + pm + 2zm = m^2 + pz + z^2$$

1983 Brazil National Olympiad, 1

Tags: number theory , positive integer , diophantine equation

Show that there are only finitely many solutions to $1/a + 1/b + 1/c = 1/1983$ in positive integers.

2016 Balkan MO Shortlist, N3

Tags: diophantine equation , number theory , diophantine

Find all the integer solutions $(x,y,z)$ of the equation $(x + y + z)^5 = 80xyz(x^2 + y^2 + z^2)$,

2014 Junior Balkan Team Selection Tests - Moldova, 2

Tags: diophantine , diophantine equation , number theory

Determine all pairs of integers $(x, y)$ that satisfy equation $(y - 2) x^2 + (y^2 - 6y + 8) x = y^2 - 5y + 62$.