Olympiad problems

2018 Costa Rica - Final Round, N2

Determine all triples $(a, b, c)$ of nonnegative integers that satisfy: $$(c-1) (ab- b -a) = a + b-2$$

2024 Junior Balkan MO, 3

Tags: number theory , diophantine equation , exponent

Find all triples of positive integers $(x, y, z)$ that satisfy the equation $$2020^x + 2^y = 2024^z.$$ [i]Proposed by Ognjen Tešić, Serbia[/i]

1996 Singapore MO Open, 3

Tags: number theory , diophantine equation , diophantine

Let $n$ be a positive integer. Prove that there is no positive integer solution to thxe equation $(x + 2)^n - x^n = 1 + 7^n$.

1996 Singapore MO Open, 4

Tags: diophantine equation , diophantine , number theory

Determine all the solutions of the equation $x^3 + y^3 + z^3 = wx^2y^2z^2$ in natural numbers $x, y, z, w$. Justify your answer

2020 Puerto Rico Team Selection Test, 2

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

The cost of $1000$ grams of chocolate is $x$ dollars and the cost of $1000$ grams of potatoes is $y$ dollars, the numbers $x$ and $y$ are positive integers and have not more than $2$ digits. Mother said to Maria to buy $200$ grams of chocolate and $1000$ grams of potatoes that cost exactly $N$ dollars. Maria got confused and bought $1000$ grams of chocolate and $200$ grams of potatoes that cost exactly $M$ dollars ($M >N$). It turned out that the numbers $M$ and $N$ have no more than two digits and are formed of the same digits but in a different order. Find $x$ and $y$.

2015 Dutch IMO TST, 2

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

Determine all positive integers $n$ for which there exist positive integers $a_1,a_2, ..., a_n$ with $a_1 + 2a_2 + 3a_3 +... + na_n = 6n$ and $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$

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 $4k-1$, $k\in\mathbb N$.

2023 Philippine MO, 1

Tags: algebra , number theory , diophantine equation

Find all ordered pairs $(a, b)$ of positive integers such that $a^2 + b^2 + 25 = 15ab$ and $a^2 + ab + b^2$ is prime.

2022 Dutch BxMO TST, 3

Tags: diophantine equation , diophantine , number theory

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

PEN H Problems, 61

Tags: diophantine equation

Solve the equation $2^x -5 =11^{y}$ in positive integers.

2023 Durer Math Competition Finals, 2

Tags: number theory , diophantine equation

[b]a)[/b] Find all solutions of the equation $p^2+q^2+r^2=pqr$, where $p,q,r$ are positive primes.\\ [b]b)[/b] Show that for every positive integer $N$, there exist three integers $a,b,c\geq N$ with $a^2+b^2+c^2=abc$.

PEN H Problems, 17

Tags: diophantine equation , vieta jumping

Find all positive integers $n$ for which the equation \[a+b+c+d=n \sqrt{abcd}\] has a solution 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 $x^3 + 3x + 14 = 2 p^n$

1985 IMO Longlists, 82

Tags: diophantine equation , algebra , polynomial

Find all cubic polynomials $x^3 +ax^2 +bx+c$ admitting the rational numbers $a$, $b$ and $c$ as roots.

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.

1981 Yugoslav Team Selection Test, Problem 3

Tags: diophantine equation , number theory

Let $a,b$ be nonnegative integers. Prove that $5a>7b$ if and only if there exist nonnegative integers $x,y,z,t$ such that \begin{align*} x+2y+3z+7t&=a,\\ y+2z+5t&=b. \end{align*}

2020 Nigerian MO round 3, #4

Tags: number theory , diophantine equation

let $p$and $q=p+2$ be twin primes. consider the diophantine equation $(+)$ given by $n!+pq^2=(mp)^2$ $m\geq1$, $n\geq1$ i. if $m=p$,find the value of $p$. ii. how many solution quadruple $(p,q,m,n)$ does $(+)$ have ?

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}$

1997 Brazil Team Selection Test, Problem 2

Tags: diophantine equation , number theory

We say that a subset $A$ of $\mathbb N$ is good if for some positive integer $n$, the equation $x-y=n$ admits infinitely many solutions with $x,y\in A$. If $A_1,A_2,\ldots,A_{100}$ are sets whose union is $\mathbb N$, prove that at least one of the $A_i$s is good.

2014 Dutch Mathematical Olympiad, 1

Tags: diophantine equation , diophantine , number theory

Determine all triples $(a,b,c)$, where $a, b$, and $c$ are positive integers that satisfy $a \le b \le c$ and $abc = 2(a + b + c)$.

1971 Dutch Mathematical Olympiad, 3

Tags: number theory , diophantine , diophantine equation

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

2021 Bosnia and Herzegovina Junior BMO TST, 2

Tags: number theory , diophantine equation , diophantine

Let $p, q, r$ be prime numbers and $t, n$ be natural numbers such that $p^2 +qt =(p + t)^n$ and $p^2 + qr = t^4$ . a) Show that $n < 3$. b) Determine all the numbers $p, q, r, t, n$ that satisfy the given conditions.

1986 IMO Shortlist, 3

Tags: diophantine equation , geometry , algebra , pythagorean theorem , number theory

Let $A, B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $ABC$ is an equilateral triangle whose side is $86$ meters long. A boy swam from $A$ directly toward $B$. After covering a distance of $x$ meters, he turned and swam westward, reaching the shore after covering a distance of $y$ meters. If $x$ and $y$ are both positive integers, determine $y.$

2018 India PRMO, 6

Tags: diophantine equation , algebra , number theory , sum , diophantine

Integers $a, b, c$ satisfy $a+b-c=1$ and $a^2+b^2-c^2=-1$. What is the sum of all possible values of $a^2+b^2+c^2$ ?

2005 iTest, 34

Tags: number theory , diophantine , diophantine equation , perfect square

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 $x$.