Found problems: 15460
1968 Czech and Slovak Olympiad III A, 2
Show that for any integer $n$ the number \[a_n=\frac{\bigl(2+\sqrt3\bigr)^n-\bigl(2-\sqrt3\bigr)^n}{2\sqrt3}\] is also integer. Determine all integers $n$ such that $a_n$ is divisible by 3.
2000 Tournament Of Towns, 3
The least common multiple of positive integers $a, b, c$ and $d$ is equal to $a + b + c + d$. Prove that $abcd$ is divisible by at least one of $3$ and $5$.
( V Senderov)
1981 IMO Shortlist, 4
Let $\{fn\}$ be the Fibonacci sequence $\{1, 1, 2, 3, 5, \dots.\}. $
(a) Find all pairs $(a, b)$ of real numbers such that for each $n$, $af_n +bf_{n+1}$ is a member of the sequence.
(b) Find all pairs $(u, v)$ of positive real numbers such that for each $n$, $uf_n^2 +vf_{n+1}^2$ is a member of the sequence.
2017 South East Mathematical Olympiad, 7
Let $m$ be a given positive integer. Define $a_k=\frac{(2km)!}{3^{(k-1)m}},k=1,2,\cdots.$ Prove that there are infinite many integers and infinite many non-integers in the sequence $\{a_k\}$.
2021 LMT Spring, B13
Call a $4$-digit number $\overline{a b c d}$ [i]unnoticeable [/i] if $a +c = b +d$ and $\overline{a b c d} +\overline{c d a b}$ is a multiple of $7$. Find the number of unnoticeable numbers.
Note: $a$, $b$, $c$, and $d$ are nonzero distinct digits.
[i]Proposed by Aditya Rao[/i]
2023 Turkey EGMO TST, 2
Find all pairs of $p,q$ prime numbers that satisfy the equation
$$p(p^4+p^2+10q)=q(q^2+3)$$
2018 SG Originals, Q5
Consider a polynomial $P(x,y,z)$ in three variables with integer coefficients such that for any real numbers $a,b,c,$ $$P(a,b,c)=0 \Leftrightarrow a=b=c.$$
Find the largest integer $r$ such that for all such polynomials $P(x,y,z)$ and integers $m,n,$ $$m^r\mid P(n,n+m,n+2m).$$
[i]Proposed by Ma Zhao Yu
1979 IMO Longlists, 5
Describe which positive integers do not belong to the set
\[E = \left\{ \lfloor n+ \sqrt n +\frac 12 \rfloor | n \in \mathbb N\right\}.\]
2025 Harvard-MIT Mathematics Tournament, 9
Let $f$ be the unique polynomial of degree at most $2026$ such that for all $n \in \{1,2, 3, \ldots, 2027\},$ $$f(n)=\begin{cases} 1 & \text{if } $n$ \text{ is a perfect square}, \\
0 & \text{otherwise.}
\end{cases}$$ Suppose that $\tfrac{a}{b}$ is the coefficient of $x^{2025}$ in $f,$ where $a$ and $b$ are integers such that $\gcd(a,b)=1.$ Compute the unique integer $r$ between $0$ and $2026$ (inclusive) such that $a-rb$ is divisible by $2027.$ (Note that $2027$ is prime.)
2023 Azerbaijan IZhO TST, 4
A positive integer $t$ is called a Jane's integer if $t = x^3+y^2$ for some positive integers $x$ and $y$. Prove
that for every integer $n \ge 2$ there exist infinitely many positive integers $m$ such that the set of $n^2$ consecutive
integers $\{m+1,m+2,\dotsc,m+n^2\}$ contains exactly $n + 1$ Jane's integers.
1995 Belarus Team Selection Test, 3
Show that there is no infinite sequence an of natural numbers such that \[a_{a_n}=a_{n+1}a_{n-1}-a_{n}^2\] for all $n\geq 2$
2022 Iberoamerican, 6
Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$, such that $f(a)f(a+b)-ab$ is a perfect square for all $a, b \in \mathbb{N}$.
2004 Romania Team Selection Test, 13
Let $m\geq 2$ be an integer. A positive integer $n$ has the property that for any positive integer $a$ coprime with $n$, we have $a^m - 1\equiv 0 \pmod n$.
Prove that $n \leq 4m(2^m-1)$.
Created by Harazi, modified by Marian Andronache.
2012 Brazil Team Selection Test, 4
Let $p$ be a prime greater than $2$. Prove that there is a prime $q < p$ such that $q^{p-1} - 1$ is not divisible by $p^2$
2012 ELMO Shortlist, 5
Let $n>2$ be a positive integer and let $p$ be a prime. Suppose that the nonzero integers are colored in $n$ colors. Let $a_1,a_2,\ldots,a_{n}$ be integers such that for all $1\le i\le n$, $p^i\nmid a_i$ and $p^{i-1}\mid a_i$. In terms of $n$, $p$, and $\{a_i\}_{i=1}^{n}$, determine if there must exist integers $x_1,x_2,\ldots,x_{n}$ of the same color such that $a_1x_1+a_2x_2+\cdots+a_{n}x_{n}=0$.
[i]Ravi Jagadeesan.[/i]
2011 IFYM, Sozopol, 5
Let $n$, $i$, and $j$ be integers, for which $0<i<j<n$. Is it always true that the binomial coefficients $\binom{n}{i}$ and $\binom{n}{j}$ have a common divisor greater than 1?
2009 All-Russian Olympiad, 5
Given strictly increasing sequence $ a_1<a_2<\dots$ of positive integers such that each its term $ a_k$ is divisible either by 1005 or 1006, but neither term is divisible by $ 97$. Find the least possible value of maximal difference of consecutive terms $ a_{i\plus{}1}\minus{}a_i$.
2012 Serbia JBMO TST, 1
Find all $4$-digit numbers $\overline{abba}$ that are equal to the product of some consecutive prime numbers.
2017 Thailand Mathematical Olympiad, 7
Show that no pairs of integers $(m, n)$ satisfy $2560m^2 + 5m + 6 = n^5$.
.
2014 Contests, 3
We denote the number of positive divisors of a positive integer $m$ by $d(m)$ and the number of distinct prime divisors of $m$ by $\omega(m)$. Let $k$ be a positive integer. Prove that there exist infinitely many positive integers $n$ such that $\omega(n) = k$ and $d(n)$ does not divide $d(a^2+b^2)$ for any positive integers $a, b$ satisfying $a + b = n$.
MathLinks Contest 5th, 5.1
Find all real numbers $a > 1$ such that there exists an integer $k \ge 1$ such that the sequence $\{x_n\}_{n\ge 1}$ formed with the first $k$ digits of the number $\lfloor a^n\rfloor$ is periodical.
2019 Purple Comet Problems, 11
Let $m > n$ be positive integers such that $3(3mn - 2)^2 - 2(3m -3n)^2 = 2019$. Find $3m + n$.
2013 Sharygin Geometry Olympiad, 4
Given a square cardboard of area $\frac{1}{4}$, and a paper triangle of area $\frac{1}{2}$ such that the square of its sidelength is a positive integer. Prove that the triangle can be folded in some ways such that the squace can be placed inside the folded figure so that both of its faces are completely covered with paper.
[i]Proposed by N.Beluhov, Bulgaria[/i]
2020 Thailand TST, 3
Let $a$ and $b$ be two positive integers. Prove that the integer
\[a^2+\left\lceil\frac{4a^2}b\right\rceil\]
is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.)
[i]Russia[/i]
2024 Junior Balkan Team Selection Tests - Moldova, 2
Prove that the number $ \underbrace{88\dots8}_\text{2024\; \textrm{times}}$ is divisible by 2024.