Olympiad problems

2016 Latvia Baltic Way TST, 17

Can you find five prime numbers $p, q, r, s, t$ such that $p^3+q^3+r^3+s^3 =t^3$?

1989 IMO Shortlist, 4

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

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.

PEN H Problems, 63

Tags: zsigmondy therom , diophantine equation

Show that $\vert 12^m -5^n\vert \ge 7$ for all $m, n \in \mathbb{N}$.

2021 Canadian Mathematical Olympiad Qualification, 6

Tags: diophantine equation , algebra

Show that $(w, x, y, z)=(0,0,0,0)$ is the only integer solution to the equation $$w^{2}+11 x^{2}-8 y^{2}-12 y z-10 z^{2}=0$$

1967 German National Olympiad, 5

Tags: number theory , diophantine , diophantine equation

For each natural number $n$, determine the number $A(n)$ of all integer nonnegative solutions the equation $$5x + 2y + z = 10n.$$

1967 IMO Longlists, 38

Tags: modular arithmetic , number theory , perfect cube , diophantine equation

Does there exist an integer such that its cube is equal to $3n^2 + 3n + 7,$ where $n$ is an integer.

1917 Eotvos Mathematical Competition, 1

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

If $a$ and $b$ are integers and if the solutions of the system of equations $$y - 2x - a = 0$$ $$y^2 - xy + x^2 - b = 0$$ are rational, prove that the solutions are integers.

2021 Ecuador NMO (OMEC), 6

Tags: number theory , diophantine equation

Find all positive integers $a, b, c$ such that $ab+1$ and $c$ are coprimes and: $$a(ba+1)(ca^2+ba+1)=2021^{2021}$$

2013 Costa Rica - Final Round, N1

Tags: number theory , diophantine equation , diophantine

Find all triples $(a, b, p)$ of positive integers, where $p$ is a prime number, such that $a^p - b^p = 2013$.

PEN H Problems, 62

Tags: diophantine equation

Solve the equation $7^x -3^y =4$ in positive integers.

PEN H Problems, 46

Tags: calculus , integration , diophantine equation

Let $a, b, c, d, e, f$ be integers such that $b^{2}-4ac>0$ is not a perfect square and $4acf+bde-ae^{2}-cd^{2}-fb^{2}\neq 0$. Let \[f(x, y)=ax^{2}+bxy+cy^{2}+dx+ey+f\] Suppose that $f(x, y)=0$ has an integral solution. Show that $f(x, y)=0$ has infinitely many integral solutions.

1958 AMC 12/AHSME, 32

Tags: modular arithmetic , least common multiple , number theory , diophantine equation

With $ \$1000$ a rancher is to buy steers at $ \$25$ each and cows at $ \$26$ each. If the number of steers $ s$ and the number of cows $ c$ are both positive integers, then: $ \textbf{(A)}\ \text{this problem has no solution}\qquad\\ \textbf{(B)}\ \text{there are two solutions with }{s}\text{ exceeding }{c}\qquad \\ \textbf{(C)}\ \text{there are two solutions with }{c}\text{ exceeding }{s}\qquad \\ \textbf{(D)}\ \text{there is one solution with }{s}\text{ exceeding }{c}\qquad \\ \textbf{(E)}\ \text{there is one solution with }{c}\text{ exceeding }{s}$

2022 Argentina National Olympiad, 5

Tags: number theory , diophantine equation , diophantine

Find all pairs of positive integers $x,y$ such that $$x^3+y^3=4(x^2y+xy^2-5).$$

1998 USAMTS Problems, 1

Tags: number theory , diophantine equation

Several pairs of positive integers $(m ,n )$ satisfy the condition $19m + 90 + 8n = 1998$. Of these, $(100, 1 )$ is the pair with the smallest value for $n$. Find the pair with the smallest value for $m$.

2018 Korea Junior Math Olympiad, 4

Tags: combinatorics , number theory , diophantine equation

For a positive integer $n$, denote $p(n)$ to be the number of nonnegative integer tuples $(x,y,z,w)$ such that $x+2y+2z+3w=n$. Also, denote $q(n)$ to be the number of nonnegative integer tuples $(a,b,c,d)$ such that (i) $a+b+c+d=n$ (ii) $a \ge b \ge d$ (iii) $a \ge c \ge d$ Prove that for all $n$, $p(n) = q(n)$.

1984 IMO Shortlist, 11

Tags: number theory , equation , polynomial , diophantine equation

Let $n$ be a positive integer and $a_1, a_2, \dots , a_{2n}$ mutually distinct integers. Find all integers $x$ satisfying \[(x - a_1) \cdot (x - a_2) \cdots (x - a_{2n}) = (-1)^n(n!)^2.\]

2014 Indonesia MO Shortlist, N1

Tags: number theory , diophantine equation , prime

(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 $(a, b)$.

2016 Latvia Baltic Way TST, 17

1989 IMO Shortlist, 4

PEN H Problems, 63

2021 Canadian Mathematical Olympiad Qualification, 6

1967 German National Olympiad, 5

1967 IMO Longlists, 38

1917 Eotvos Mathematical Competition, 1

2021 Ecuador NMO (OMEC), 6

2013 Costa Rica - Final Round, N1

PEN H Problems, 62

PEN H Problems, 46

1958 AMC 12/AHSME, 32

2022 Argentina National Olympiad, 5

1998 USAMTS Problems, 1

2018 Korea Junior Math Olympiad, 4

1984 IMO Shortlist, 11

2014 Indonesia MO Shortlist, N1

2002 Junior Balkan Team Selection Tests - Romania, 1

2022 New Zealand MO, 1

2008 Costa Rica - Final Round, 5

2005 Thailand Mathematical Olympiad, 13

2012 Dutch BxMO/EGMO TST, 3

1999 Belarusian National Olympiad, 6

PEN H Problems, 74

2015 Swedish Mathematical Competition, 2