Found problems: 15460
1995 Mexico National Olympiad, 1
$N$ students are seated at desks in an $m \times n$ array, where $m, n \ge 3$. Each student shakes hands with the students who are adjacent horizontally, vertically or diagonally. If there are $1020 $handshakes, what is $N$?
1989 All Soviet Union Mathematical Olympiad, 490
A positive integer $n$ has exactly $12$ positive divisors $1 = d_1 < d_2 < d_3 < ... < d_{12} = n$. Let $m = d_4 - 1$. We have $d_m = (d_1 + d_2 + d_4) d_8$. Find $n$.
2000 Mongolian Mathematical Olympiad, Problem 6
Given distinct prime numbers $p_1,\ldots,p_s$ and a positive integer $n$, find the number of positive integers not exceeding $n$ that are divisible by exactly one of the $p_i$.
2014 Switzerland - Final Round, 2
Let $a,b\in\mathbb{N}$ such that :
\[ ab(a-b)\mid a^3+b^3+ab \]
Then show that $\operatorname{lcm}(a,b)$ is a perfect square.
2014 Postal Coaching, 2
Let $A=\{1,2,3,\ldots,40\}$. Find the least positive integer $k$ for which it is possible to partition $A$ into $k$ disjoint subsets with the property that if $a,b,c$ (not necessarily distinct) are in the same subset, then $a\ne b+c$.
2000 Estonia National Olympiad, 1
Find all prime numbers whose sixth power does not give remainder $1$ when dividing by $504$
1986 USAMO, 1
$(\text{a})$ Do there exist 14 consecutive positive integers each of which is divisible by one or more primes $p$ from the interval $2\le p \le 11$?
$(\text{b})$ Do there exist 21 consecutive positive integers each of which is divisible by one or more primes $p$ from the interval $2\le p \le 13$?
2023 Baltic Way, 18
Let $p>7$ be a prime and let $A$ be subset of $\{0,1, \ldots, p-1\}$ with size at least $\frac{p-1}{2}$. Show that for each integer $r$, there exist $a, b, c, d \in A$, not necessarily distinct, such that $ab-cd \equiv r \pmod p$.
2009 Belarus Team Selection Test, 3
Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$.
[i]Proposed by Angelo Di Pasquale, Australia[/i]
2012 Junior Balkan Team Selection Tests - Moldova, 4
Let there be an infinite sequence $ a_{k} $ with $ k\geq 1 $ defined by:
$ a_{k+2} = a_{k} + 14 $ and $ a_{1} = 12 $ , $ a_{2} = 24 $.
[b]a)[/b] Does $2012$ belong to the sequence?
[b]b)[/b] Prove that the sequence doesn't contain perfect squares.
KoMaL A Problems 2018/2019, A. 734
For an arbitrary positive integer $m$, not divisible by $3$, consider the permutation $x \mapsto 3x \pmod{m}$ on the set $\{ 1,2,\dotsc ,m-1\}$. This permutation can be decomposed into disjointed cycles; for instance, for $m=10$ the cycles are $(1\mapsto 3\to 9,\mapsto 7,\mapsto 1)$, $(2\mapsto 6\mapsto 8\mapsto 4\mapsto 2)$ and $(5\mapsto 5)$. For which integers $m$ is the number of cycles odd?
1971 IMO Shortlist, 10
Prove that we can find an infinite set of positive integers of the from $2^n-3$ (where $n$ is a positive integer) every pair of which are relatively prime.
VMEO IV 2015, 12.4
We call the [i]tribi [/i] of a positive integer $k$ (denoted $T(k)$) the number of all pairs $11$ in the binary representation of $k$. e.g $$T(1)=T(2)=0,\, T(3)=1, \,T(4)=T(5)=0,\,T(6)=1,\,T(7)=2.$$
Calculate $S_n=\sum_{k=1}^{2^n}T(K)$.
2003 All-Russian Olympiad Regional Round, 9.4
Two players take turns writing on the board in a row from left to right arbitrary numbers. The player loses, after whose move one or more several digits written in a row form a number divisible by $11$. Which player will win if played correctly?
1997 Polish MO Finals, 1
The positive integers $x_1, x_2, ... , x_7$ satisfy $x_6 = 144$, $x_{n+3} = x_{n+2}(x_{n+1}+x_n)$ for $n = 1, 2, 3, 4$. Find $x_7$.
1995 Mexico National Olympiad, 4
Find $26$ elements of $\{1, 2, 3, ... , 40\}$ such that the product of two of them is never a square.
Show that one cannot find $27$ such elements.
2003 Iran MO (3rd Round), 7
$f_{1},f_{2},\dots,f_{n}$ are polynomials with integer coefficients. Prove there exist a reducible $g(x)$ with integer coefficients that $f_{1}+g,f_{2}+g,\dots,f_{n}+g$ are irreducible.
2003 Federal Math Competition of S&M, Problem 4
Let $S$ be the subset of $N$($N$ is the set of all natural numbers) satisfying:
i)Among each $2003$ consecutive natural numbers there exist at least one contained in $S$;
ii)If $n \in S$ and $n>1$ then $[\frac{n}{2}] \in S$
Prove that:$S=N$
I hope it hasn't posted before. :lol: :lol:
2008 VJIMC, Problem 4
The numbers of the set $\{1,2,\ldots,n\}$ are colored with $6$ colors. Let
$$S:=\{(x,y,z)\in\{1,2,\ldots,n\}^3:x+y+z\equiv0\pmod n\text{ and }x,y,z\text{ have the same color}\}$$and
$$D:=\{(x,y,z)\in\{1,2,\ldots,n\}^3:x+y+z\equiv0\pmod n\text{ and }x,y,z\text{ have three different colors}\}.$$Prove that
$$|D|\le2|S|+\frac{n^2}2.$$
1996 Singapore Senior Math Olympiad, 3
Prove that for any positive even integer $n$ larger than $38$, $n$ can be written as $a\times b+c\times d$ where $a, b, c, d$ are odd integers larger than $1$.
2015 Iran Team Selection Test, 2
Assume that $a_1, a_2, a_3$ are three given positive integers consider the following sequence:
$a_{n+1}=\text{lcm}[a_n, a_{n-1}]-\text{lcm}[a_{n-1}, a_{n-2}]$ for $n\ge 3$
Prove that there exist a positive integer $k$ such that $k\le a_3+4$ and $a_k\le 0$.
($[a, b]$ means the least positive integer such that$ a\mid[a,b], b\mid[a, b]$ also because $\text{lcm}[a, b]$ takes only nonzero integers this sequence is defined until we find a zero number in the sequence)
2021 Azerbaijan EGMO TST, 1
p is a prime number, k is a positive integer
Find all (p, k): $k!=(p^3-1)(p^3-p)(p^3-p^2)$
2021 Romania National Olympiad, 3
Given is an positive integer $a>2$
a) Prove that there exists positive integer $n$ different from $1$, which is not a prime, such that $a^n=1(mod n)$
b) Prove that if $p$ is the smallest positive integer, different from $1$, such that $a^p=1(mod p)$, then $p$ is a prime.
c) There does not exist positive integer $n$, different from $1$, such that $2^n=1(mod n)$
2021 HMNT, 4
The sum of the digits of the time $19$ minutes ago is two less than the sum of the digits of the time right now. Find the sum of the digits of the time in $19$ minutes. (Here, we use a standard $12$-hour clock of the form $hh:mm$.)
1982 All Soviet Union Mathematical Olympiad, 342
What minimal number of numbers from the set $\{1,2,...,1982\}$ should be deleted to provide the property:
[i]none of the remained numbers equals to the product of two other remained numbers[/i]?