This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 145

2015 Dutch IMO TST, 2
Tags: system of equations , Diophantine 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 $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$

PEN H Problems, 60
Tags: modular arithmetic , Diophantine Equations
Show that the equation $x^7 + y^7 = {1998}^z$ has no solution in positive integers.

VMEO III 2006, 12.2
Tags: number theory , Diophantine Equations , Diophantine equation , diophantine
Find all positive integers $(m, n)$ that satisfy $$m^2 =\sqrt{n} +\sqrt{2n + 1}.$$

PEN H Problems, 16
Tags: algebra , polynomial , function , Diophantine Equations
Find all pairs $(a,b)$ of different positive integers that satisfy the equation $W(a)=W(b)$, where $W(x)=x^{4}-3x^{3}+5x^{2}-9x$.

2009 Korea Junior Math Olympiad, 8
Tags: number theory , prime numbers , Diophantine Equations
Let a, b, c, d, and e be positive integers. Are there any solutions to $a^2+b^3+c^5+d^7=e^{11}$?

PEN H Problems, 47
Tags: Diophantine Equations
Show that the equation $x^4 +y^4 +4z^4 =1$ has infinitely many rational solutions.

PEN H Problems, 28
Tags: modular arithmetic , number theory , relatively prime , Diophantine Equations
Let $a, b, c$ be positive integers such that $a$ and $b$ are relatively prime and $c$ is relatively prime either to $a$ or $b$. Prove that there exist infinitely many triples $(x, y, z)$ of distinct positive integers such that \[x^{a}+y^{b}= z^{c}.\]

2012 Dutch IMO TST, 3
Tags: Diophantine Equations , diophantine , number theory
Determine all pairs $(x, y)$ of positive integers satisfying $x + y + 1 | 2xy$ and $ x + y - 1 | x^2 + y^2 - 1$.

PEN H Problems, 55
Tags: floor function , Diophantine Equations
Given that \[34! = 95232799cd96041408476186096435ab000000_{(10)},\] determine the digits $a, b, c$, and $d$.

PEN H Problems, 20
Tags: Diophantine Equations
Determine all positive integers $n$ for which the equation \[x^{n}+(2+x)^{n}+(2-x)^{n}= 0\] has an integer as a solution.

PEN H Problems, 22
Tags: symmetry , Diophantine Equations
Find all integers $a,b,c,x,y,z$ such that \[a+b+c=xyz, \; x+y+z=abc, \; a \ge b \ge c \ge 1, \; x \ge y \ge z \ge 1.\]

2012 Dutch IMO TST, 3
Tags: Diophantine Equations , diophantine , number theory
Determine all pairs $(x, y)$ of positive integers satisfying $x + y + 1 | 2xy$ and $ x + y - 1 | x^2 + y^2 - 1$.

2015 IMAR Test, 1
Tags: ro , Diophantine Equations
Determine all positive integers expressible, for every integer $ n \geq 3 $, in the form \begin{align*} \frac{(a_1 + 1)(a_2 + 1) \ldots (a_n + 1) - 1}{a_1a_2 \ldots a_n}, \end{align*} where $ a_1, a_2, \ldots, a_n $ are pairwise distinct positive integers.

2015 Indonesia MO Shortlist, N8
Tags: number theory , Diophantine Equations , infinitely many solutions , Indonesia
The natural number $n$ is said to be good if there are natural numbers $a$ and $b$ that satisfy $a + b = n$ and $ab | n^2 + n + 1$. (a) Show that there are infinitely many good numbers. (b) Show that if $n$ is a good number, then $7 \nmid n$.

PEN H Problems, 63
Tags: Diophantine Equations , zsigmondy therom
Show that $\vert 12^m -5^n\vert \ge 7$ for all $m, n \in \mathbb{N}$.

PEN H Problems, 72
Tags: induction , number theory , relatively prime , absolute value , Diophantine Equations
Find all pairs $(x, y)$ of positive rational numbers such that $x^{y}=y^{x}$.

PEN H Problems, 59
Tags: modular arithmetic , Diophantine Equations
Solve the equation $28^x =19^y +87^z$, where $x, y, z$ are integers.

2016 Abels Math Contest (Norwegian MO) Final, 2a
Tags: Diophantine Equations , diophantine , number theory , system of equations
Find all positive integers $a, b, c, d$ with $a \le b$ and $c \le d$ such that $\begin{cases} a + b = cd \\ c + d = ab \end{cases}$ .

PEN H Problems, 41
Tags: geometry , 3D geometry , number theory , relatively prime , Diophantine Equations
Suppose that $A=1,2,$ or $3$. Let $a$ and $b$ be relatively prime integers such that $a^{2}+Ab^2 =s^3$ for some integer $s$. Then, there are integers $u$ and $v$ such that $s=u^2 +Av^2$, $a =u^3 - 3Avu^2$, and $b=3u^{2}v -Av^3$.

PEN H Problems, 62
Tags: Diophantine Equations , pen
Solve the equation $7^x -3^y =4$ in positive integers.

PEN H Problems, 82
Tags: Diophantine Equations
Find all triples $(a, b, c)$ of positive integers to the equation \[a! b! = a!+b!+c!.\]

PEN H Problems, 5
Tags: Diophantine Equations
Find all pairs $(x, y)$ of rational numbers such that $y^2 =x^3 -3x+2$.

2020 Puerto Rico Team Selection Test, 2
Tags: number theory , systems of equations , Digits , Diophantine Equations , 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 $y$.

PEN H Problems, 85
Tags: inequalities , Diophantine Equations , pen
Find all integer solutions to $2(x^5 +y^5 +1)=5xy(x^2 +y^2 +1)$.

2015 Dutch IMO TST, 2
Tags: system of equations , Diophantine 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 $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$