For which real numbers $p$ does the equation $x^{2}+px+3p=0$ have integer solutions ?

Find all positive integers which can be written in the form $(mn+1)/(m+n)$, where $m,n$ are positive integers.

The positive integers $a,b,c$ are less than $99$ and satisfy $a^2+b^2=c^2+99^2$. . Find the minimum and maximum value of $a+b+c$.

Find all pairs of relatively prime numbers ($x, y$) such that $x^2(x + y)$ is divisible by $y^2(y - x)^2$. .

For all positive integers $n$, let $f(n)$ return the smallest positive integer $k$ for which $\tfrac{n}{k}$ is not an integer. For example, $f(6) = 4$ because $1$, $2$, and $3$ all divide $6$ but $4$ does not. Determine the largest possible value of $f(n)$ as $n$ ranges over the set $\{1,2,\ldots, 3000\}$.

Find the smallest multiple of $81$ that only contains the digit $1$. How many $ 1$’s does it contain?

For which integer \( h \), are there infinitely many positive integers \( n \) such that \( \lfloor \sqrt{h^2 + 1} \cdot n \rfloor \) is a perfect square? (Here \( \lfloor x \rfloor \) denotes the integer part of the real number \( x \)?

Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by \[x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,\] has all of its terms relatively prime to $m$. [i]Proposed by Jaroslaw Wroblewski, Poland[/i]

Let $ b$ be an integer greater than $ 5$. For each positive integer $ n$, consider the number \[ x_n = \underbrace{11\cdots1}_{n \minus{} 1}\underbrace{22\cdots2}_{n}5, \] written in base $ b$. Prove that the following condition holds if and only if $ b \equal{} 10$: [i]there exists a positive integer $ M$ such that for any integer $ n$ greater than $ M$, the number $ x_n$ is a perfect square.[/i] [i]Proposed by Laurentiu Panaitopol, Romania[/i]

Find the number of all integers $n$ with $4\le n\le 1023$ which contain no three consecutive equal digits in their binary representations.

Find all pairs of natural numbers $(a, b)$ that fulfill $a^b=(a+b)^a$.

Find all triplets (a, b, c) of positive integers so that $a^2b$, $b^2c$ and $c^2a$ devide $a^3+b^3+c^3$

Find a pair prime numbers $(p, q)$, $p> q$ of , if any, such that $\frac{p^2 - q^2}{4}$ is an odd integer.

Are there integers $ x$, $ y$, not both divisible by $ 5$, such that $ x^2 \plus{} 19y^2 \equal{} 198\cdot 10^{1989}$?

Prove that for any integer $k > 2$, there exist infinitely many positive integers $n$ such that the least common multiple of $n$, $n + 1$,$...$, $n + k - 1$ is greater than the least common multiple of $n + 1$,$n + 2$,$...$, $n + k$.

Prove that for any $n$ there exists a positive integer $k$ such that all the numbers $k\cdot2^s+1~(s=1,\ldots,n)$ are composite.

Which natural numbers can be expressed as the difference of squares of two integers?

Find all primes $p$ such that there exist an integer $n$ and positive integers $k, m$ which satisfies the following. $$ \frac{(mk^2+2)p-(m^2+2k^2)}{mp+2} = n^2$$

Find all four-digit numbers in which the thousands digit is equal to the hundreds digit and the tens digit is equal to the units digit and which are squares of integers.

Let $a$ be a positive integer. Prove that for any pair $(x,y)$ of integer solutions of equation $$x(y^2-2x^2)+x+y+a=0$$ we have: $$|x| \leqslant a+\sqrt{2a^2+2}$$

For all positive integers $n$, show that there exists a positive integer $m$ such that $n$ divides $2^{m} + m$. [i]Proposed by Juhan Aru, Estonia[/i]

[b]p29.[/b] Consider pentagon $ABCDE$. How many paths are there from vertex $A$ to vertex $E$ where no edge is repeated and does not go through $E$. [b]p30.[/b] Let $a_1, a_2, ...$ be a sequence of positive real numbers such that $\sum^{\infty}_{n=1} a_n = 4$. Compute the maximum possible value of $\sum^{\infty}_{n=1}\frac{\sqrt{a_n}}{2^n}$ (assume this always converges). [b]p31.[/b] Define function $f(x) = x^4 + 4$. Let $$P =\prod^{2021}_{k=1} \frac{f(4k - 1)}{f(4k - 3)}.$$ Find the remainder when $P$ is divided by $1000$. [b]p32.[/b] Reduce the following expression to a simplified rational: $\cos^7 \frac{\pi}{9} + \cos^7 \frac{5\pi}{9}+ \cos^7 \frac{7\pi}{9}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n$ be an odd integer and $m=\phi(n)$ be the Euler's totient function. Call a set of residues $T=\{a_1, \cdots, a_k\} \pmod n$ to be [i]good[/i] if $\gcd(a_i, n) > 1$ $\forall i$, and $\gcd(a_i, a_j) = 1, \forall i \neq j$. Define the set $S_n$ consisting of the residues $$\sum_{i=1}^k a_i ^m\pmod{n}$$ over all possible residue sets $T=\{a_1,\cdots,a_k\}$ that is good. Determine $|S_n|$. [i]Proposed by Anzo Teh Zhao Yang[/i]

Prove that the equation $(3x+4y)(4x+5y)=7^z$ doesn't have solution in natural numbers.

We call a pair $(a,b)$ of positive integers, $a<391$, [i]pupusa[/i] if $$\textup{lcm}(a,b)>\textup{lcm}(a,391)$$ Find the minimum value of $b$ across all [i]pupusa[/i] pairs. Fun Fact: OMCC 2017 was held in El Salvador. [i]Pupusa[/i] is their national dish. It is a corn tortilla filled with cheese, meat, etc.