Found problems: 15460
2013 Dutch BxMO/EGMO TST, 3
Find all triples $(x,n,p)$ of positive integers $x$ and $n$ and primes $p$ for which the following holds $x^3 + 3x + 14 = 2 p^n$
2017 Puerto Rico Team Selection Test, 3
Given are $n$ integers. Prove that at least one of the following conditions applies:
1) One of the numbers is a multiple of $n$.
2) You can choose $k\le n$ numbers whose sum is a multiple of $ n$.
1995 India Regional Mathematical Olympiad, 3
Prove that among any $18$ consecutive three digit numbers there is at least one number which is divisible by the sum of its digits.
2015 Bosnia And Herzegovina - Regional Olympiad, 2
Find all triplets $(p,a,b)$ of positive integers such that $$p=b\sqrt{\frac{a-8b}{a+8b}}$$ is prime
2006 Brazil National Olympiad, 4
A positive integer is [i]bold[/i] iff it has $8$ positive divisors that sum up to $3240$. For example, $2006$ is bold because its $8$ positive divisors, $1$, $2$, $17$, $34$, $59$, $118$, $1003$ and $2006$, sum up to $3240$. Find the smallest positive bold number.
2019 Tournament Of Towns, 4
Consider the following sequence of positive real numbers $\dots<a_{-2}<a_{-1}<a_0<a_1<a_2<\dots$ infinite in both directions. For each positive integer $k$ let $b_k$ be the least integer such that the ratio between the sum of $k$ consecutive terms and the greatest of these $k$ terms is less than or equal to $b_k$(This fact occurs for any sequence of $k$ consecutive numbers). Prove that the sequence $b_1,b_2,b_3,...$ coincides with the sequence $1,2,3,...$ or is eventually constant.
2011 Kyiv Mathematical Festival, 1
Solve the equation $m^{gcd(m,n)} = n^{lcm(m,n)}$ in positive integers, where gcd($m, n$) – greatest common
divisor of $m,n$, and lcm($m, n$) – least common multiple of $m,n$.
2010 China Team Selection Test, 2
Given integer $a_1\geq 2$. For integer $n\geq 2$, define $a_n$ to be the smallest positive integer which is not coprime to $a_{n-1}$ and not equal to $a_1,a_2,\cdots, a_{n-1}$. Prove that every positive integer except 1 appears in this sequence $\{a_n\}$.
Taiwan TST 2015 Round 1, 1
Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\]
[i]Proposed by Titu Andreescu, USA[/i]
2015 Peru IMO TST, 4
Let $n\geq 2$ be an integer. The permutation $a_1,a_2,..., a_n$ of the numbers $1, 2,...,n$ is called [i]quadratic[/i] if $a_ia_{i +1} + 1$ is a perfect square for all $1\leq i \leq n-1.$ The permutation $a_1,a_2,..., a_n$ of the numbers $1, 2,...,n$ is called [i]cubic[/i] if $a_ia_{i + 1} + 1$ is a perfect cube for all $1\leq i \leq n - 1.$
a) Prove that for infinitely many values of $n$ is there at least one quadratic permutation of the numbers $1, 2,...,n.$
b) Prove that for no value of $n$ is there a cubic permutation of the numbers $1, 2,..., n.$
2023 Macedonian Team Selection Test, Problem 1
Let $s(n)$ denote the smallest prime divisor and $d(n)$ denote the number of positive divisors of a positive integer $n>1$. Is it possible to choose $2023$ positive integers $a_{1},a_{2},...,a_{2023}$ with $a_{1}<a_{2}-1<...<a_{2023}-2022$ such that for all $k=1,...,2022$ we have $d(a_{k+1}-a_{k}-1)>2023^{k}$ and $s(a_{k+1}-a_{k}) > 2023^{k}$?
[i]Authored by Nikola Velov[/i]
2014 Contests, 1
A natural number $k$ is such that $k^2 < 2014 < (k +1)^2$. What is the largest prime factor of $k$?
2021 Balkan MO Shortlist, N1
Let $n \geq 2$ be an integer and let \[M=\bigg\{\frac{a_1 + a_2 + ... + a_k}{k}: 1 \le k \le n\text{ and }1 \le a_1 < \ldots < a_k \le n\bigg\}\] be the set of the arithmetic means of the elements of all non-empty subsets of $\{1, 2, ..., n\}$. Find \[\min\{|a - b| : a, b \in M\text{ with } a \neq b\}.\]
1998 Brazil Team Selection Test, Problem 5
Let $p$ be an odd prime integer and $k$ a positive integer not divisible by $p$, $1\le k<2(p+1)$, and let $N=2kp+1$. Prove that the following statements are equivalent:
(i) $N$ is a prime number
(ii) there exists a positive integer $a$, $2\le a<n$, such that $a^{kp}+1$ is divisible by $N$ and $\gcd\left(a^k+1,N\right)=1$.
1996 Denmark MO - Mohr Contest, 4
Regarding a natural number $n$, it is stated that the number $n^2$ has $7$ as the second to last digit. What is the last digit of $n^2$?
2011 VTRMC, Problem 4
Let $m,n$ be positive integers and let $[a]$ denote the residue class$\pmod{mn}$ of the integer $a$ (thus $\{[r]|r\text{ is an integer}\}$ has exactly $mn$ elements). Suppose the set $\{[ar]|r\text{ is an integer}\}$ has exactly $m$ elements. Prove that there is a positive integer $q$ such that $q$ is coprime to $mn$ and $[nq]=[a]$.
2021 Korea Junior Math Olympiad, 2
Let $\{a_n\}$ be a sequence of integers satisfying the following conditions.
[list]
[*] $a_1=2021^{2021}$
[*] $0 \le a_k < k$ for all integers $k \ge 2$
[*] $a_1-a_2+a_3-a_4+ \cdots + (-1)^{k+1}a_k$ is multiple of $k$ for all positive integers $k$.
[/list]
Determine the $2021^{2022}$th term of the sequence $\{a_n\}$.
2002 Tournament Of Towns, 6
In an infinite increasing sequence of positive integers, every term from the $2002^{\text{th}}$ term divides the sum of all preceding terms. Prove that every term starting from some term is equal to the sum of all preceding terms.
Russian TST 2019, P2
Prove that for every odd prime number $p{}$, the following congruence holds \[\sum_{n=1}^{p-1}n^{p-1}\equiv (p-1)!+p\pmod{p^2}.\]
2016 Taiwan TST Round 1, 2
Find all ordered pairs $(a,b)$ of positive integers that satisfy $a>b$ and the equation $(a-b)^{ab}=a^bb^a$.
2012 Moldova Team Selection Test, 10
Let $f:\mathbb{R}\rightarrow\mathbb{R}, f(x,y)=x^2-2y.$ Define the sequences $(a_n)_{n\geq1}$ and $(b_n)_{n\geq1}$ such that $a_{n+1}=f(a_n,b_n), b_{n+1}=f(b_n,a_n).$ If $4a_1-2b_1=7 :$
a) find the smallest $k\in\mathbb{N}$ for which the number $p=2^k\cdot(2^{512}a_9-b_9)$ is an integer.
b) prove that $2^{2^{10}}+2^{2^9}+1$ divides $p.$
2022 New Zealand MO, 5
The sequence $x_1, x_2, x_3, . . .$ is defined by $x_1 = 2022$ and $x_{n+1}= 7x_n + 5$ for all positive integers $n$. Determine the maximum positive integer $m$ such that $$\frac{x_n(x_n - 1)(x_n - 2) . . . (x_n - m + 1)}{m!}$$ is never a multiple of $7$ for any positive integer $n$.
2012 Purple Comet Problems, 28
A bag contains $8$ green candies and $4$ red candies. You randomly select one candy at a time to eat. If you eat five candies, there are relatively prime positive integers $m$ and $n$ so that $\frac{m}{n}$ is the probability that you do not eat a green candy after you eat a red candy. Find $m+n$.
2008 AMC 10, 24
Let $ k\equal{}2008^2\plus{}2^{2008}$. What is the units digit of $ k^2\plus{}2^k$?
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 4 \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ 8$
2021/2022 Tournament of Towns, P7
Let $p$ be a prime number and let $M$ be a convex polygon. Suppose that there are precisely $p$ ways to tile $m$ with equilateral triangles with side $1$ and squares with side $1$. Show there is some side of $M$ of length $p-1$.