Olympiad problems

2013 Junior Balkan Team Selection Tests - Romania, 1

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]

2017 India IMO Training Camp, 2

Tags: diophantine equation , number theory

Find all positive integers $p,q,r,s>1$ such that $$p!+q!+r!=2^s.$$

2015 Greece National Olympiad, 1

Tags: prime number , number theory , diophantine equation

Find all triplets $(x,y,p)$ of positive integers such that $p$ be a prime number and $\frac{xy^3}{x+y}=p$

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 $$7 \cdot 4^n = a^2 + b^2 + c^2 + d^2.$$

1967 IMO Longlists, 43

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

The equation \[x^5 + 5 \lambda x^4 - x^3 + (\lambda \alpha - 4)x^2 - (8 \lambda + 3)x + \lambda \alpha - 2 = 0\] is given. Determine $\alpha$ so that the given equation has exactly (i) one root or (ii) two roots, respectively, independent from $\lambda.$

PEN H Problems, 62

Tags: diophantine equation

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

2023 Bangladesh Mathematical Olympiad, P3

Tags: factorization , diophantine equation , number theory

Solve the equation for the positive integers: $$(x+2y)^2+2x+5y+9=(y+z)^2$$

1986 ITAMO, 6

Tags: diophantine , diophantine equation , number theory

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

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 $$a^3 + 2017a = b^3 -2017b.$$

2019 Teodor Topan, 1

Tags: number theory , equation , diophantine equation

Solve in the natural numbers the equation $ \log_{6n-19} (n!+1) =2. $ [i]Dragoș Crișan[/i]

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)

2024 CMI B.Sc. Entrance Exam, 5

Tags: diophantine equation , number theory , algebra

Find all solutions for positive integers $(x,y,k,m)$ such that \[ 20x^k+24y^m = 2024\] with $k, m > 1$

2016 Latvia Baltic Way TST, 18

Tags: number theory , diophantine equation , system of equations

Solve the system of equations in integers: $$\begin{cases} a^3=abc+2a+2c \\ b^3=abc-c \\ c^3=abc-a+b \end{cases}$$

PEN A Problems, 2

Tags: arithmetic sequence , diophantine equation , divisibility theory , pell s equation

Find infinitely many triples $(a, b, c)$ of positive integers such that $a$, $b$, $c$ are in arithmetic progression and such that $ab+1$, $bc+1$, and $ca+1$ are perfect squares.