Found problems: 85335
2016 Brazil Team Selection Test, 2
For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2016$ good partitions.
PS. [url=https://artofproblemsolving.com/community/c6h1268855p6622233]2015 ISL C3 [/url] has 2015 instead of 2016
2012 All-Russian Olympiad, 2
Does there exist natural numbers $a,b,c$ all greater than $10^{10}$ such that their product is divisible by each of these numbers increased by $2012$?
2006 May Olympiad, 5
With $28$ points, a “triangular grid” of equal sides is formed, as shown in the figure.
One operation consists of choosing three points that are the vertices of an equilateral triangle and removing these three points from the grid. If after performing several of these operations there is only one point left, in what positions can that point remain?
Give all the possibilities and indicate in each case the operations carried out.
Justify why the remaining point cannot be in another position.
[img]https://cdn.artofproblemsolving.com/attachments/f/c/1cedfe0e1c5086b77151538265f8e253e93d2e.gif[/img]
2020 Peru Cono Sur TST., P5
Find the smallest positive integer $n$ such that for any $n$ distinct real numbers $b_1, b_2,\ldots ,b_n$ in the interval $[ 1, 1000 ]$ there always exist $b_i$ and $b_j$ such that:
$$0<b_i-b_j<1+3\sqrt[3]{b_ib_j}$$
2025 All-Russian Olympiad, 10.7
A competition consists of $25$ sports, each awarding one gold medal to a winner. $25$ athletes participate, each in all $25$ sports. There are also $25$ experts, each of whom must predict the number of gold medals each athlete will win. In each prediction, the medal counts must be non-negative integers summing to $25$. An expert is called competent if they correctly guess the number of gold medals for at least one athlete. What is the maximum number \( k \) such that the experts can make their predictions so that at least \( k \) of them are guaranteed to be competent regardless of the outcome? \\
2022 Czech-Austrian-Polish-Slovak Match, 6
Consider 26 letters $A,..., Z$. A string is a finite sequence consisting of those letters. We say that a string $s$ is nice if it contains each of the 26 letters at least once, and each permutation of letters $A,..., Z$ occurs in $s$ as a subsequences the same number of times. Prove that:
(a) There exists a nice string.
(b) Any nice string contains at least $2022$ letters.
1927 Eotvos Mathematical Competition, 1
Let the integers $a, b, c, d$ be relatively prime to $$m = ad - bc.$$
Prove that the pairs of integers $(x,y)$ for which $ax+by$ is a multiple of $m$ are identical with those for which $cx + dy$ is a multiple of $m$.
2007 Vietnam Team Selection Test, 1
Given two sets $A, B$ of positive real numbers such that: $|A| = |B| =n$; $A \neq B$ and $S(A)=S(B)$, where $|X|$ is the number of elements and $S(X)$ is the sum of all elements in set $X$. Prove that we can fill in each unit square of a $n\times n$ square with positive numbers and some zeros such that:
a) the set of the sum of all numbers in each row equals $A$;
b) the set of the sum of all numbers in each column equals $A$.
c) there are at least $(n-1)^{2}+k$ zero numbers in the $n\times n$ array with $k=|A \cap B|$.
2015 Korea National Olympiad, 3
A positive integer $n$ is given. If there exists sets $F_1, F_2, \cdots F_m$ satisfying the following conditions, prove that $m \le n$. (For sets $A, B$, $|A|$ is the number of elements of $A$. $A-B$ is the set of elements that are in $A$ but not $B$. $\text{min}(x,y)$ is the number that is not larger than the other.)
(i): For all $1 \le i \le m$, $F_i \subseteq \{1,2,\cdots,n\}$
(ii): For all $1 \le i < j \le m$, $\text{min}(|F_i-F_j|,|F_j-F_i|) = 1$
2004 Irish Math Olympiad, 1
1. (a) For which positive integers n, does 2n divide the sum of the first n positive
integers?
(b) Determine, with proof, those positive integers n (if any) which have the
property that 2n + 1 divides the sum of the first n positive integers.
Estonia Open Junior - geometry, 2020.1.5
A circle $c$ with center $A$ passes through the vertices $B$ and $E$ of a regular pentagon $ABCDE$. The line $BC$ intersects the circle $c$ for second time at point $F$. Prove that the lines $DE$ and $EF$ are perpendicular.
2014 IMAC Arhimede, 5
Let $p$ be a prime number. The natural numbers $m$ and $n$ are written in the system with the base $p$ as $n = a_0 + a_1p +...+ a_kp^k$ and $m = b_0 + b_1p +..+ b_kp^k$. Prove that
$${n \choose m} \equiv \prod_{i=0}^{k}{a_i \choose b_i} (mod p)$$
1986 Bundeswettbewerb Mathematik, 4
Given the finite set $M$ with $m$ elements and $1986$ further sets $M_1,M_2,M_3,...,M_{1986}$, each of which contains more than $\frac{m}{2}$ elements from $M$ . Show that no more than ten elements need to be marked in order for any set $M_i$ ($i =1, 2, 3,..., 1986$) contains at least one marked element.
2009 Abels Math Contest (Norwegian MO) Final, 1b
Show that the sum of three consecutive perfect cubes can always be written as the difference between two perfect squares.
2008 Greece Team Selection Test, 2
In a village $X_0$ there are $80$ tourists who are about to visit $5$ nearby villages $X_1,X_2,X_3,X_4,X_5$.Each of them has chosen to visit only one of them.However,there are cases when the visit in a village forces the visitor to visit other villages among $X_1,X_2,X_3,X_4,X_5$.Each tourist visits only the village he has chosen and the villages he is forced to.If $X_1,X_2,X_3,X_4,X_5$ are totally visited by $40,60,65,70,75$ tourists respectively,then find how many tourists had chosen each one of them and determine all the ordered pairs $(X_i,X_j):i,j\in \{1,2,3,4,5\}$ which are such that,the visit in $X_i$ forces the visitor to visit $X_j$ as well.
1997 IMC, 3
Let $A,B \in \mathbb{R}^{n\times n}$ with $A^2+B^2=AB$. Prove that if $BA-AB$ is invertible then $3|n$.
2006 Kyiv Mathematical Festival, 2
See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url]
2006 equilateral triangles are located in the square with side 1. The sum of their perimeters is equal to 300. Prove that at least three of them have a common point.
1982 Poland - Second Round, 3
Prove that for every natural number $ n \geq 2 $ the inequality holds
$$
\log_n 2 \cdot \log_n 4 \cdot \log_n 6 \ldots \log_n (2n - 2) \leq 1.$$
1984 IMO Longlists, 33
Let $ d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $ n$ vertices (where $ n>3$). Let $ p$ be its perimeter. Prove that:
\[ n\minus{}3<{2d\over p}<\Bigl[{n\over2}\Bigr]\cdot\Bigl[{n\plus{}1\over 2}\Bigr]\minus{}2,\]
where $ [x]$ denotes the greatest integer not exceeding $ x$.
1998 National Olympiad First Round, 10
Let $ p$ and $ q$ be two consecutive terms of the sequence of odd primes. The number of positive divisor of $ p \plus{} q$, at least
$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6$
2002 District Olympiad, 2
Let $ ABCD $ be an inscriptible quadrilateral and $ M $ be a point on its circumcircle, distinct from its vertices. Let $ H_1,H_2,H_3,H_4 $ be the orthocenters of $ MAB,MBC, MCD, $ respectively, $ MDA, $ and $ E,F, $ the midpoints of the segments $ AB, $ respectivley, $ CD. $ Prove that:
[b]a)[/b] $ H_1H_2H_3H_4 $ is a parallelogram.
[b]b)[/b] $ H_1H_3=2\cdot EF. $
2007 Junior Tuymaada Olympiad, 4
An acute-angle non-isosceles triangle $ ABC $ is given. The point $ H $ is its orthocenter, the points $ O $ and $ I $ are the centers of its circumscribed and inscribed circles, respectively. The circumcircle of the triangle $ OIH $ passes through the vertex $ A $. Prove that one of the angles of the triangle is $ 60^\circ $.
2021 LMT Fall, 12
Let $x, y$, and $z$ be three not necessarily real numbers that satisfy the following system of equations:
$x^3 -4 = (2y +1)^2$
$y^3 -4 = (2z +1)^2$
$z^3 -4 = (2x +1)^2$.
Find the greatest possible real value of $(x -1)(y -1)(z -1)$.
2005 Balkan MO, 1
Let $ABC$ be an acute-angled triangle whose inscribed circle touches $AB$ and $AC$ at $D$ and $E$ respectively. Let $X$ and $Y$ be the points of intersection of the bisectors of the angles $\angle ACB$ and $\angle ABC$ with the line $DE$ and let $Z$ be the midpoint of $BC$. Prove that the triangle $XYZ$ is equilateral if and only if $\angle A = 60^\circ$.
2014 India PRMO, 6
What is the smallest possible natural number $n$ for which the equation $x^2 -nx + 2014 = 0$ has integer roots?