Found problems: 15460
2004 Bulgaria National Olympiad, 5
Let $a,b,c,d$ be positive integers such that the number of pairs $(x,y) \in (0,1)^2$ such that both $ax+by$ and $cx+dy$ are integers is equal with 2004. If $\gcd (a,c)=6$ find $\gcd (b,d)$.
2023 Purple Comet Problems, 4
Positive integer $\underline{a}\,\, \underline{b}\,\, \underline{c}\,\, \underline{d}\,\, \underline{r}\,\, \underline{s}\,\, \underline{t}$ has digits $a$, $b$, $c$, $d$, $r$, $s$, and $t$, in that order, and none of the digits is $0$. The two-digit numbers $\underline{a}\,\, \underline{b}$ , $\underline{b}\,\, \underline{c}$ , $\underline{c}\,\, \underline{d}$ , and $\underline{d}\,\, \underline{r}$ , and the three-digit number $\underline{r}\,\, \underline{s}\,\, \underline{t}$ are all perfect squares. Find $\underline{a}\,\, \underline{b}\,\, \underline{c}\,\, \underline{d}\,\, \underline{r}\,\, \underline{s}\,\, \underline{t}$ .
2002 India IMO Training Camp, 14
Let $p$ be an odd prime and let $a$ be an integer not divisible by $p$. Show that there are $p^2+1$ triples of integers $(x,y,z)$ with $0 \le x,y,z < p$ and such that $(x+y+z)^2 \equiv axyz \pmod p$
2022 Junior Macedonian Mathematical Olympiad, P5
Let $n$ be a positive integer such that $n^5+n^3+2n^2+2n+2$ is a perfect cube. Prove that $2n^2+n+2$ is not a perfect cube.
[i]Proposed by Anastasija Trajanova[/i]
2000 China Second Round Olympiad, 2
Define the sequence $a_1, a_2, \ldots$ and $b_1, b_2, \ldots$ as $a_0=1,a_1=4,a_2=49$ and for $n \geq 0$
$$
\begin{cases}
a_{n+1}=7a_n+6b_n-3, \\
b_{n+1}=8a_n+7b_n-4.
\end{cases}
$$
Prove that for any non-negative integer $n,$ $a_n$ is a perfect square.
2014 Czech-Polish-Slovak Junior Match, 3
Find with all integers $n$ when $|n^3 - 4n^2 + 3n - 35|$ and $|n^2 + 4n + 8|$ are prime numbers.
2024 UMD Math Competition Part I, #24
Let $n\ge3$ be an integer. A regular $n$-gon $P$ is given. We randomly select three distinct vertices of $P$. The probability that these three vertices form an isosceles triangle is $1/m$, where $m$ is an integer. How many such integers $n\le 2024$ are there?
\[\rm a. ~674\qquad \mathrm b. ~675\qquad \mathrm c. ~682 \qquad\mathrm d. ~684\qquad\mathrm e. ~685\]
2016 Mathematical Talent Reward Programme, SAQ: P 3
Prove that for any positive integer $n$ there are $n$ consecutive composite numbers all less than $4^{n+2}$.
2023 ISL, N5
Let $a_1<a_2<a_3<\dots$ be positive integers such that $a_{k+1}$ divides $2(a_1+a_2+\dots+a_k)$ for every $k\geqslant 1$. Suppose that for infinitely many primes $p$, there exists $k$ such that $p$ divides $a_k$. Prove that for every positive integer $n$, there exists $k$ such that $n$ divides $a_k$.
2016 Saudi Arabia Pre-TST, 2.4
Let $n$ be a given positive integer. Prove that there are infinitely many pairs of positive integers $(a, b)$ with $a, b > n$ such that
$$\prod_{i=1}^{2015} (a + i) | b(b + 2016), \prod_{i=1}^{2015}(a + i) \nmid b, \prod_{i=1}^{2015} (a + i)\mid (b + 2016)$$.
2021 OMMock - Mexico National Olympiad Mock Exam, 6
Let $a$ and $b$ be fixed positive integers. We say that a prime $p$ is [i]fun[/i] if there exists a positive integer $n$ satisfying the following conditions:
[list]
[*]$p$ divides $a^{n!} + b$.
[*]$p$ divides $a^{(n + 1)!} + b$.
[*]$p < 2n^2 + 1$.
[/list]
Show that there are finitely many fun primes.
2012 Purple Comet Problems, 29
Let $A=\{1, 3, 5, 7, 9\}$ and $B=\{2, 4, 6, 8, 10\}$. Let $f$ be a randomly chosen function from the set $A\cup B$ into itself. There are relatively prime positive integers $m$ and $n$ such that $\frac{m}{n}$ is the probablity that $f$ is a one-to-one function on $A\cup B$ given that it maps $A$ one-to-one into $A\cup B$ and it maps $B$ one-to-one into $A\cup B$. Find $m+n$.
1997 IMO Shortlist, 17
Find all pairs $ (a,b)$ of positive integers that satisfy the equation: $ a^{b^2} \equal{} b^a$.
2020 SMO, 6
We say that a number is [i]angelic[/i] if it is greater than $10^{100}$ and all of its digits are elements of $\{1,3,5,7,8\}$. Suppose $P$ is a polynomial with nonnegative integer coefficients such that over all positive integers $n$, if $n$ is angelic, then the decimal representation of $P(s(n))$ contains the decimal representation of $s(P(n))$ as a contiguous substring, where $s(n)$ denotes the sum of digits of $n$.
Prove that $P$ is linear and its leading coefficient is $1$ or a power of $10$.
[i]Proposed by Grant Yu[/i]
2012 Estonia Team Selection Test, 1
Prove that for any positive integer $k$ there exist $k$ pairwise distinct integers for which the sum of their squares equals the sum of their cubes.
2006 Croatia Team Selection Test, 4
Find all natural solutions of $3^{x}= 2^{x}y+1.$
2001 Croatia National Olympiad, Problem 1
Find all integers $x$ for which $2x^2-x-36$ is the square of a prime number.
2014 CentroAmerican, 3
A positive integer $n$ is [i]funny[/i] if for all positive divisors $d$ of $n$, $d+2$ is a prime number. Find all funny numbers with the largest possible number of divisors.
2013 Stars Of Mathematics, 1
Prove that for any integers $a,b$, the equation $2abx^4 - a^2x^2 - b^2 - 1 = 0$ has no integer roots.
[i](Dan Schwarz)[/i]
1982 Yugoslav Team Selection Test, Problem 1
Let $p>2$ be a prime number. For $k=1,2,\ldots,p-1$ we denote by $a_k$ the remainder when $k^p$ is divided by $p^2$. Prove that
$$a_1+a_2+\ldots+a_{p-1}=\frac{p^3-p^2}2.$$
2016 Japan MO Preliminary, 9
How many pairs $(a, b)$ for integers $1 \le a, b \le 2015$ which satisfy that $a$ is divisible by $b + 1$ and $2016 - a$ is divisible by $b$.
2016 South East Mathematical Olympiad, 7
Let $A=\{a^3+b^3+c^3-3abc|a,b,c\in\mathbb{N}\}$, $B=\{(a+b-c)(b+c-a)(c+a-b)|a,b,c\in\mathbb{N}\}$, $P=\{n|n\in A\cap B,1\le n\le 2016\}$, find the value of $|P|$.
2016 239 Open Mathematical Olympiad, 1
A natural number $k>1$ is given. The sum of some divisor of $k$ and some divisor of $k - 1$ is equal to $a$,where $a>k + 1$. Prove that at least one of the numbers $a - 1$ or $a + 1$ composite.
2000 IMO Shortlist, 2
For a positive integer $n$, let $d(n)$ be the number of all positive divisors of $n$. Find all positive integers $n$ such that $d(n)^3=4n$.
2012 Online Math Open Problems, 10
A drawer has $5$ pairs of socks. Three socks are chosen at random. If the probability that there is a pair among the three is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, what is $m+n$?
[i]Author: Ray Li[/i]