Olympiad problems

2003 Moldova Team Selection Test, 1

Let $ n\in N^*$. A permutation $ (a_1,a_2,...,a_n)$ of the numbers $ (1,2,...,n)$ is called [i]quadratic [/i] iff at least one of the numbers $ a_1,a_1\plus{}a_2,...,a_1\plus{}a_2\plus{}a\plus{}...\plus{}a_n$ is a perfect square. Find the greatest natural number $ n\leq 2003$, such that every permutation of $ (1,2,...,n)$ is quadratic.

2015 Indonesia MO Shortlist, N5

Tags: diophantine equation , integer , number theory

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: $(0, 7), (1, 4), (2, 1)$

1984 Swedish Mathematical Competition, 5

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

Solve in natural numbers $a,b,c$ the system \[\left\{ \begin{array}{l}a^3 -b^3 -c^3 = 3abc \\ a^2 = 2(a+b+c)\\ \end{array} \right. \]

2014 Thailand TSTST, 2

Tags: diophantine equation , number theory

Prove that the equation $x^8 = n! + 1$ has finitely many solutions in positive integers.

1977 Dutch Mathematical Olympiad, 1

Tags: diophantine equation , number theory , diophantine

Solve, for integers $x$ and $y$ : $$2x^2y = (x+2)^2(y + 1), $$ provided that $(x+2)^2(y + 1)> 1000$.

PEN H Problems, 59

Tags: diophantine equation , modular arithmetic

Solve the equation $28^x =19^y +87^z$, where $x, y, z$ are integers.

PEN H Problems, 48

Tags: search , diophantine equation , modular arithmetic

Solve the equation $x^2 +7=2^n$ in integers.

2008 Thailand Mathematical Olympiad, 7

Tags: diophantine equation , lcm , gcd , diophantine

Two positive integers $m, n$ satisfy the two equations $m^2 + n^2 = 3789$ and $gcd (m, n) + lcm (m, n) = 633$. Compute $m + n$.

2017 Switzerland - Final Round, 7

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

Let $n$ be a natural number such that there are exactly$ 2017$ distinct pairs of natural numbers $(a, b)$, which the equation $$\frac{1}{a}+\frac{1}{b}=\frac{1}{n}$$ fulfilld. Show that $n$ is a perfect square . Remark: $(7, 4) \ne (4, 7)$

1991 Bundeswettbewerb Mathematik, 1

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

Determine all solutions of the equation $4^x + 4^y + 4^z = u^2$ for integers $x,y,z$ and $u$.

2022 Austrian MO Beginners' Competition, 4

Tags: diophantine equation , number theory , diophantine

Determine all prime numbers $p, q$ and $r$ with $p + q^2 = r^4$. [i](Karl Czakler)[/i]

2017 USAJMO, 2

Tags: diophantine equation

Consider the equation \[(3x^3+xy^2)(x^2y+3y^3)=(x-y)^7\] (a) Prove that there are infinitely many pairs $(x,y)$ of positive integers satisfying the equation. (b) Describe all pairs $(x,y)$ of positive integers satisfying the equation.

2015 Bosnia and Herzegovina Junior BMO TST, 1

Tags: equation , diophantine equation , algebra , number theory

Solve equation $x(x+1) = y(y+4)$ where $x$, $y$ are positive integers

2014 Hanoi Open Mathematics Competitions, 4

Tags: diophantine equation , prime , diophantine , number theory

If $p$ is a prime number such that there exist positive integers $a$ and $b$ such that $\frac{1}{p}=\frac{1}{a^2}+\frac{1}{b^2}$ then $p$ is (A): $3$, (B): $5$, (C): $11$, (D): $7$, (E) None of the above.

2012 Ukraine Team Selection Test, 7

Tags: diophantine equation , number theory , relatively prime

Find all pairs of relatively prime integers $(x, y)$ that satisfy equality $2 (x^3 - x) = 5 (y^3 - y)$.

PEN H Problems, 74

Tags: diophantine equation

Find all pairs $(a,b)$ of positive integers that satisfy the equation \[a^{a^{a}}= b^{b}.\]

2007 Indonesia MO, 2

Tags: diophantine equation , relatively prime , number theory

For every positive integer $ n$, $ b(n)$ denote the number of positive divisors of $ n$ and $ p(n)$ denote the sum of all positive divisors of $ n$. For example, $ b(14)\equal{}4$ and $ p(14)\equal{}24$. Let $ k$ be a positive integer greater than $ 1$. (a) Prove that there are infinitely many positive integers $ n$ which satisfy $ b(n)\equal{}k^2\minus{}k\plus{}1$. (b) Prove that there are finitely many positive integers $ n$ which satisfy $ p(n)\equal{}k^2\minus{}k\plus{}1$.

2021 Malaysia IMONST 1, 10

Tags: diophantine equation , diophantine , number theory

Determine the number of integer solutions $(x, y, z)$, with $0 \le x, y, z \le 100$, for the equation $$(x - y)^2 + (y + z)^2 = (x + y)^2 + (y - z)^2.$$

2007 Nordic, 1

Tags: diophantine equation , number theory

Find a solution to the equation $x^2-2x-2007y^2=0$ in positive integers.

2022 Dutch BxMO TST, 3

Tags: diophantine equation , diophantine , number theory

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

2000 Switzerland Team Selection Test, 7

Tags: diophantine equation , diophantine , number theory

Show that the equation $14x^2 +15y^2 = 7^{2000}$ has no integer solutions.

PEN H Problems, 15

Tags: diophantine equation , number theory , factorial

Prove that there are no integers $x$ and $y$ satisfying $x^{2}=y^{5}-4$.

2020 Puerto Rico Team Selection Test, 2

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

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 $y$.

1989 IMO Longlists, 8

Tags: polynomial , equation , root , diophantine equation , algebra

Find the roots $ r_i \in \mathbb{R}$ of the polynomial \[ p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n\] satisfying \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]

2012 Dutch Mathematical Olympiad, 1

Tags: diophantine equation , divisible , number theory

Let $a, b, c$, and $d$ be four distinct integers. Prove that $(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)$ is divisible by $12$.