Olympiad problems

2016 Ecuador Juniors, 2

Prove that there are no positive integers $x, y$ such that: $(x + 1)^2 + (x + 2)^2 +...+ (x + 9)^2 = y^2$

2012 Czech-Polish-Slovak Junior Match, 2

Tags: prime , diophantine , number theory , diophantine equation

Determine all three primes $(a, b, c)$ that satisfied the equality $a^2+ab+b^2=c^2+3$.

2022 Federal Competition For Advanced Students, P1, 4

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

Find all triples $(p, q, r)$ of prime numbers for which $4q - 1$ is a prime number and $$\frac{p + q}{p + r} = r - p$$ holds. [i](Walther Janous)[/i]

2011 China Northern MO, 3

Tags: number theory , diophantine , diophantine equation

Find all positive integer solutions $(x, y, z)$ of the equation $1 + 2^x \cdot 7^y=z^2$.

2017 Istmo Centroamericano MO, 3

Tags: diophantine , number theory , diophantine equation

Find all ordered pairs of integers $(x, y)$ with $y \ge 0$ such that $x^2 + 2xy + y! = 131$.

2013 Junior Balkan Team Selection Tests - Romania, 1

Tags: diophantine , diophantine equation , number theory

Let $n$ be a positive integer. Determine all positive integers $p$ for which there exist positive integers $x_1 < x_2 <...< x_n$ such that $\frac{1}{x_1}+\frac{2}{x_2}+ ... +\frac{n}{x_n}= p$ Irish Mathematical Olympiad

2009 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , diophantine , diophantine equation

Find all non-negative integers $a,b,c,d$ such that $7^a= 4^b + 5^c + 6^d$.

2012 Mathcenter Contest + Longlist, 1

Tags: diophantine equation , diophantine , number theory

Prove without using modulo that there are no integers $a,b,c$ such that $$a^2+b^2-8c = 6$$ [i](Metamorphosis)[/i]

2005 Switzerland - Final Round, 7

Tags: diophantine , number theory , diophantine equation

Let $n\ge 1$ be a natural number. Determine all positive integer solutions of the equation $$7 \cdot 4^n = a^2 + b^2 + c^2 + d^2.$$

1986 ITAMO, 6

Tags: diophantine , diophantine equation , number theory

Show that for any positive integer $n$ there exists an integer $m > 1$ such that $(\sqrt2-1)^n=\sqrt{m}-\sqrt{m-1}$.

1965 Kurschak Competition, 1

Tags: diophantine , number theory , inequalities

What integers $a, b, c$ satisfy $a^2 + b^2 + c^2 + 3 < ab + 3b + 2c$ ?

2021 Junior Balkan Team Selection Tests - Moldova, 7

Tags: diophantine , number theory

Determine all pairs of integer numbers $(a, b)$ that satisfy the relation: $$(a + b)(a + b + 6) = 34 \cdot 3^{|2a-b|}- 7$$

2009 Postal Coaching, 2

Tags: diophantine , number theory , diophantine equation

Solve for prime numbers $p, q, r$ : $$\frac{p}{q} - \frac{4}{r + 1}= 1$$

2017 Regional Olympiad of Mexico Northeast, 3

Tags: diophantine , number theory , diophantine equation

Prove that there is no pair of relatively prime positive integers $(a, b)$ that satisfy the equation $$a^3 + 2017a = b^3 -2017b.$$

2014 Junior Balkan Team Selection Tests - Romania, 2

Tags: number theory , diophantine equation , diophantine

Determine the prime numbers $p$ and $q$ that satisfy the equality: $p^3 + 107 = 2q (17q + 24)$ .

1997 Tournament Of Towns, (548) 2

Tags: number theory , diophantine , diophantine equation

Prove that the equation $x^2 + y^2 - z^2 = 1997$ has infinitely many solutions in integers $x$, $y$ and $z$. (N Vassiliev)

2023-24 IOQM India, 4

Tags: diophantine , number theory , divisibility

Let $x, y$ be positive integers such that $$ x^4=(x-1)\left(y^3-23\right)-1 . $$ Find the maximum possible value of $x+y$.

2010 Hanoi Open Mathematics Competitions, 9

Tags: diophantine , number theory

Let $x,y$ be the positive integers such that $3x^2 +x = 4y^2 + y$. Prove that $x - y$ is a perfect (square).

1965 Swedish Mathematical Competition, 2

Tags: number theory , diophantine equation , diophantine

Find all positive integers m, n such that $m^3 - n^3 = 999$.

2017 QEDMO 15th, 6

Tags: diophantine , diophantine equation , number theory

Find all integers $x,y$ satisfy the $x^3 + y^3 = 3xy$.

1904 Eotvos Mathematical Competition, 2

Tags: diophantine , number theory

If a is a natural number, show that the number of positive integral solutions of the indeterminate equation $$x_1 + 2x_2 + 3x_3 + ... + nx_n = a \ \ (1) $$ is equal to the number of non-negative integral solutions of $$y_1 + 2y_2 + 3y_3 + ... + ny_n = a - \frac{n(n + 1)}{2} \ \ (2)$$ [By a solution of equation (1), we mean a set of numbers $\{x_1, x_2,..., x_n\}$ which satisfies equation (1)].

2022 Dutch IMO TST, 1

Tags: diophantine equation , diophantine , number theory

Determine all positive integers $n \ge 2$ which have a positive divisor $m | n$ satisfying $$n = d^3 + m^3.$$ where $d$ is the smallest divisor of $n$ which is greater than $1$.

2019 Peru EGMO TST, 1

Tags: prime , number theory , diophantine equation , diophantine

Find all the prime numbers $p, q$ and $r$ such that $p^2 + 1 = 74 (q^2 + r^2)$.

2003 Singapore MO Open, 3

Tags: diophantine , number theory , diophantine equation

For any given prime $p$, determine whether the equation $x^2 + y^2 + p^z = 2003$ always has integer solutions in $x, y, z$. Justify your answer

1990 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: diophantine , floor function , algebra

How many positive integer solutions does the equation have $$\left\lfloor\frac{x}{10}\right\rfloor= \left\lfloor\frac{x}{11}\right\rfloor + 1?$$ ($\lfloor x \rfloor$ denotes the integer part of $x$, for example $\lfloor 2\rfloor = 2$, $\lfloor \pi\rfloor = 3$, $\lfloor \sqrt2 \rfloor =1$)