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: 64

PEN A Problems, 52
Tags: quadratics , modular arithmetic , Divisibility Theory , pen
Let $d$ be any positive integer not equal to 2, 5, or 13. Show that one can find distinct $a$ and $b$ in the set $\{2,5,13,d\}$ such that $ab - 1$ is not a perfect square.

PEN H Problems, 6
Tags: Diophantine Equations , number theory , pen
Show that there are infinitely many pairs $(x, y)$ of rational numbers such that $x^3 +y^3 =9$.

2001 Vietnam Team Selection Test, 3
Tags: algebra unsolved , algebra , number theory , pen
Let a sequence $\{a_n\}$, $n \in \mathbb{N}^{*}$ given, satisfying the condition \[0 < a_{n+1} - a_n \leq 2001\] for all $n \in \mathbb{N}^{*}$ Show that there are infinitely many pairs of positive integers $(p, q)$ such that $p < q$ and $a_p$ is divisor of $a_q$.

PEN H Problems, 33
Tags: geometry , 3D geometry , modular arithmetic , Diophantine Equations , pen
Does there exist an integer such that its cube is equal to $3n^2 +3n+7$, where $n$ is integer?

1993 Iran MO (2nd round), 1
Tags: modular arithmetic , Divisibility Theory , pen
Suppose that $p$ is a prime number and is greater than $3$. Prove that $7^{p}-6^{p}-1$ is divisible by $43$.

PEN S Problems, 6
Tags: algebra , polynomial , induction , complex numbers , Miscellaneous Problems , pen
Suppose that $x$ and $y$ are complex numbers such that \[\frac{x^{n}-y^{n}}{x-y}\] are integers for some four consecutive positive integers $n$. Prove that it is an integer for all positive integers $n$.

PEN A Problems, 73
Tags: number theory , Divisibility Theory , pen
Determine all pairs $(n,p)$ of positive integers such that [list][*] $p$ is a prime, $n>1$, [*] $(p-1)^{n} + 1$ is divisible by $n^{p-1}$. [/list]

PEN A Problems, 42
Tags: modular arithmetic , Divisibility Theory , pen
Suppose that $2^n +1$ is an odd prime for some positive integer $n$. Show that $n$ must be a power of $2$.

PEN E Problems, 41
Tags: limit , trigonometry , pen
Show that $n$ is prime iff $\lim_{r \rightarrow\infty}\,\lim_{s \rightarrow\infty}\,\lim_{t \rightarrow \infty}\,\sum_{u=0}^{s}\left(1-\left(\cos\,\frac{(u!)^{r} \pi}{n} \right)^{2t} \right)=n$ PS : I posted it because it's in the PDF file but not here ...

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 A Problems, 31
Tags: Diophantine equation , Divisibility Theory , pen
Show that there exist infinitely many positive integers $n$ such that $n^{2}+1$ divides $n!$.

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 A Problems, 17
Tags: modular arithmetic , quadratics , number theory , relatively prime , Divisibility Theory , pen
Let $m$ and $n$ be natural numbers such that \[A=\frac{(m+3)^{n}+1}{3m}\] is an integer. Prove that $A$ is odd.