Olympiad problems

2012 Dutch Mathematical Olympiad, 1

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 $12$.

PEN H Problems, 21

Tags: quadratics , LaTeX , Diophantine Equations , APMO

Prove that the equation \[6(6a^{2}+3b^{2}+c^{2}) = 5n^{2}\] has no solutions in integers except $a=b=c=n=0$.

PEN H Problems, 2

Tags: Diophantine Equations

The number $21982145917308330487013369$ is the thirteenth power of a positive integer. Which positive integer?

PEN H Problems, 80

Tags: Diophantine Equations

Prove that if $a, b, c, d$ are integers such that $d=( a+\sqrt[3]{2}b+\sqrt[3]{4}c)^{2}$ then $d$ is a perfect square.

2015 BAMO, 3

Tags: algebra , Diophantine Equations

Let $k$ be a positive integer. Prove that there exist integers $x$ and $y$, neither of which is divisible by $3$, such that $x^2+2y^2 = 3^k$.

1979 IMO Longlists, 44

Tags: algebra , system of equations , Diophantine Equations , IMO , Cauchy-Schwarz inequality

Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $ \sum_{k=1}^{5} kx_{k}=a,$ $ \sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $ \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$

2022 Abelkonkurransen Finale, 1b

Tags: number theory , Diophantine Equations

Find all primes $p$ and positive integers $n$ satisfying \[n \cdot 5^{n-n/p} = p! (p^2+1) + n.\]

2012 IMAC Arhimede, 4

Tags: Diophantine Equations , Diophantine equation , number theory

Solve the following equations in the set of natural numbers: a) $(5+11\sqrt2)^p=(11+5\sqrt2)^q$ b) $1005^x+2011^y=1006^z$

PEN H Problems, 76

Tags: Diophantine Equations

Find all pairs $(m,n)$ of integers that satisfy the equation \[(m-n)^{2}=\frac{4mn}{m+n-1}.\]

PEN H Problems, 75

Tags: Diophantine Equations , pen

Let $a,b$, and $x$ be positive integers such that $x^{a+b}=a^b{b}$. Prove that $a=x$ and $b=x^{x}$.

PEN H Problems, 57

Tags: Diophantine Equations , binomial coefficients

Show that the equation ${n \choose k}=m^{l}$ has no integral solution with $l \ge 2$ and $4 \le k \le n-4$.

2017 German National Olympiad, 6

Tags: number theory , number theory proposed , Germany , Diophantine equation , Diophantine Equations

Prove that there exist infinitely many positive integers $m$ such that there exist $m$ consecutive perfect squares with sum $m^3$. Specify one solution with $m>1$.

1998 Korea - Final Round, 1

Tags: ratio , number theory , relatively prime , Diophantine Equations

Find all pairwise relatively prime positive integers $l, m, n$ such that \[(l+m+n)\left( \frac{1}{l}+\frac{1}{m}+\frac{1}{n}\right)\] is an integer.

PEN H Problems, 18

Tags: Diophantine Equations , pen

Determine all positive integer solutions $(x, y, z, t)$ of the equation \[(x+y)(y+z)(z+x)=xyzt\] for which $\gcd(x, y)=\gcd(y, z)=\gcd(z, x)=1$.

PEN H Problems, 48

Tags: modular arithmetic , search , Diophantine Equations , pen

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

PEN H Problems, 64

Tags: Diophantine Equations

Show that there is no positive integer $k$ for which the equation \[(n-1)!+1=n^{k}\] is true when $n$ is greater than $5$.

1992 IMO Longlists, 36

Tags: algebra , system of equations , Diophantine Equations , polynomial , IMO Shortlist , IMO Longlist

Find all rational solutions of \[a^2 + c^2 + 17(b^2 + d^2) = 21,\]\[ab + cd = 2.\]

2005 Junior Tuymaada Olympiad, 3

Tags: algebra , Diophantine Equations , system of equations

Tram ticket costs $1$ Tug ($=100$ tugriks). $20$ passengers have only coins in denominations of $2$ and $5$ tugriks, while the conductor has nothing at all. It turned out that all passengers were able to pay the fare and get change. What is the smallest total number of passengers that the tram could have?

PEN H Problems, 81

Tags: modular arithmetic , number theory , relatively prime , Diophantine Equations

Find a pair of relatively prime four digit natural numbers $A$ and $B$ such that for all natural numbers $m$ and $n$, $\vert A^m -B^n \vert \ge 400$.

PEN H Problems, 90

Tags: Diophantine Equations

Find all triples of positive integers $(x, y, z)$ such that \[(x+y)(1+xy)= 2^{z}.\]