Found problems: 85335
2005 Baltic Way, 10
Let $m = 30030$ and let $M$ be the set of its positive divisors which have exactly $2$ prime factors. Determine the smallest positive integer $n$ with the following property: for any choice of $n$ numbers from $M$, there exist 3 numbers $a$, $b$, $c$ among them satisfying $abc=m$.
PEN B Problems, 5
Let $p$ be an odd prime. If $g_{1}, \cdots, g_{\phi(p-1)}$ are the primitive roots $\pmod{p}$ in the range $1<g \le p-1$, prove that \[\sum_{i=1}^{\phi(p-1)}g_{i}\equiv \mu(p-1) \pmod{p}.\]
2017 Romania National Olympiad, 2
Let be a square $ ABCD, $ a point $ E $ on $ AB, $ a point $ N $ on $ CD, $ points $ F,M $ on $ BC, $ name $
P $ the intersection of $ AN $ with $ DE, $ and name $ Q $ the intersection of $ AM $ with $ EF. $ If the triangles $ AMN $ and $ DEF $ are equilateral, prove that $ PQ=FM. $
2018 Czech-Polish-Slovak Match, 6
We say that a positive integer $n$ is [i]fantastic[/i] if there exist positive rational numbers $a$ and $b$ such that
$$ n = a + \frac 1a + b + \frac 1b.$$
[b](a)[/b] Prove that there exist infinitely many prime numbers $p$ such that no multiple of $p$ is fantastic.
[b](b)[/b] Prove that there exist infinitely many prime numbers $p$ such that some multiple of $p$ is fantastic.
[i]Proposed by Walther Janous, Austria[/i]
2019 Brazil Team Selection Test, 4
Let $p \geq 7$ be a prime number and $$S = \bigg\{jp+1 : 1 \leq j \leq \frac{p-5}{2}\bigg\}.$$ Prove that at least one element of $S$ can be written as $x^2+y^2$, where $x, y$ are integers.
2011 Saint Petersburg Mathematical Olympiad, 6
There is infinite sequence of composite numbers $a_1,a_2,...,$ where $a_{n+1}=a_n-p_n+\frac{a_n}{p_n}$ ; $p_n$ is smallest prime divisor of $a_n$. It is known, that $37|a_n$ for every $n$.
Find possible values of $a_1$
2023 Assam Mathematics Olympiad, 12
In quadrilateral $ABCD$, $AD || BC$, diagonals $AC$ and $BD$ are perpendicular to each other, $X$ and $Y$ are mid-points of $AB$ and $CD$ respectively. Prove that $AB + CD \geq AD + BC$.
2024 Bosnia and Herzegovina Junior BMO TST, 4.
Let $m$ and $n$ be natural numbers. Every one of the $m*n$ squares of the $m*n$ board is colored either black or white, so that no 2 neighbouring squares are the same color(the board is colored like in chess").In one step we can pick 2 neighbouring squares and change their colors like this:
[b]- [/b]a white square becomes black;
[b]-[/b]a black square becomes blue;
[b]-[/b]a blue square becomes white.
For which $m$ and $n$ can we ,in a finite sequence of these steps, switch the starting colors from white to black and vice versa.
2015 BAMO, 2
Members of a parliament participate in various committees. Each committee consists of at least $2$ people, and it is known that every two committees have at least one member in common. Prove that it is possible to give each member a colored hat (hats are available in black, white or red) so that every committee contains at least $2$ members with different hat colors.
2012 Vietnam National Olympiad, 1
Define a sequence $\{x_n\}$ as: $\left\{\begin{aligned}& x_1=3 \\ & x_n=\frac{n+2}{3n}(x_{n-1}+2)\ \ \text{for} \ n\geq 2.\end{aligned}\right.$
Prove that this sequence has a finite limit as $n\to+\infty.$ Also determine the limit.
2006 AMC 10, 5
A 2 x 3 rectangle and a 3 x 4 rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square?
$ \textbf{(A) } 16 \qquad \textbf{(B) } 25 \qquad \textbf{(C) } 36 \qquad \textbf{(D) } 49 \qquad \textbf{(E) } 64$
2010 Indonesia TST, 1
find all pairs of relatively prime natural numbers $ (m,n) $ in such a way that there exists non constant polynomial f satisfying \[ gcd(a+b+1, mf(a)+nf(b) > 1 \]
for every natural numbers $ a $ and $ b $
1970 Vietnam National Olympiad, 2
Let $N=1890*1930*1970$, find the number of divisors of N which are not divisors of $45$
2013 Kosovo National Mathematical Olympiad, 4
Calculate:
$\sqrt{3\sqrt{5\sqrt{3\sqrt{5...}}}}$
CIME I 2018, 12
Define a permutation of the set $\{1,2,3,...,n\}$ to be $\textit{sortable}$ if upon cancelling an appropriate term of such permutation, the remaining $n-1$ terms are in increasing order. If $f(n)$ is the number of sortable permutations of $\{1,2,3,...,n\}$, find the remainder when $$\sum\limits_{i=1}^{2018} (-1)^i \cdot f(i)$$ is divided by $1000$. Note that the empty set is considered sortable.
[i]Proposed by [b]FedeX333X[/b][/i]
2014 Dutch IMO TST, 2
Let $\triangle ABC$ be a triangle. Let $M$ be the midpoint of $BC$ and let $D$ be a point on the interior of side $AB$. The intersection of $AM$ and $CD$ is called $E$. Suppose that $|AD|=|DE|$. Prove that $|AB|=|CE|$.
2016 Miklós Schweitzer, 2
Let $K=(V,E)$ be a finite, simple, complete graph. Let $d$ be a positive integer. Let $\phi:E\to \mathbb{R}^d$ be a map from the edge set to Euclidean space, such that the preimage of any point in the range defines a connected graph on the entire vertex set $V$, and the points assigned to the edges of any triangle in $K$ are collinear. Show that the range of $\phi$ is contained in a line.
2018 Benelux, 1
(a) Determine the minimal value of
$\displaystyle\left(x+\dfrac{1}{y}\right)\left(x+\dfrac{1}{y}-2018\right)+\left(y+\dfrac{1}{x}\right)\left(y+\dfrac{1}{x}-2018\right), $
where $x$ and $y$ vary over the positive reals.
(b) Determine the minimal value of
$\displaystyle\left(x+\dfrac{1}{y}\right)\left(x+\dfrac{1}{y}+2018\right)+\left(y+\dfrac{1}{x}\right)\left(y+\dfrac{1}{x}+2018\right), $
where $x$ and $y$ vary over the positive reals.
2011 International Zhautykov Olympiad, 1
Given is trapezoid $ABCD$, $M$ and $N$ being the midpoints of the bases of $AD$ and $BC$, respectively.
a) Prove that the trapezoid is isosceles if it is known that the intersection point of perpendicular bisectors of the lateral sides belongs to the segment $MN$.
b) Does the statement of point a) remain true if it is only known that the intersection point of perpendicular bisectors of the lateral sides belongs to the line $MN$?
2005 India IMO Training Camp, 3
A merida path of order $2n$ is a lattice path in the first quadrant of $xy$- plane joining $(0,0)$ to $(2n,0)$ using three kinds of steps $U=(1,1)$, $D= (1,-1)$ and $L= (2,0)$, i.e. $U$ joins $x,y)$ to $(x+1,y+1)$ etc... An ascent in a merida path is a maximal string of consecutive steps of the form $U$. If $S(n,k)$ denotes the number of merdia paths of order $2n$ with exactly $k$ ascents, compute $S(n,1)$ and $S(n,n-1)$.
2021 CMIMC, 2.4
What is the $101$st smallest integer which can represented in the form $3^a+3^b+3^c$, where $a,b,$ and $c$ are integers?
[i]Proposed by Dilhan Salgado[/i]
2020 Australian Maths Olympiad, 7
A $\emph{tetromino tile}$ is a tile that can be formed by gluing together four unit square tiles, edge to edge. For each positive integer $\emph{n}$, consider a bathroom whose floor is in the shape of a $2\times2 n$ rectangle. Let $T_n$ be the number of ways to tile this bathroom floor with tetromino tiles. For example, $T_2 = 4$ since there are four ways to tile a $2\times4$ rectangular bathroom floor with tetromino tiles, as shown below.
[click for diagram]
Prove that each of the numbers $T_1, T_2, T_3, ...$ is a perfect square.
2003 Bosnia and Herzegovina Team Selection Test, 3
Prove that for every positive integer $n$ holds:
$(n-1)^n+2n^n \leq (n+1)^{n} \leq 2(n-1)^n+2n^{n}$
2001 China Western Mathematical Olympiad, 1
The sequence $ \{x_n\}$ satisfies $ x_1 \equal{} \frac {1}{2}, x_{n \plus{} 1} \equal{} x_n \plus{} \frac {x_n^2}{n^2}$. Prove that $ x_{2001} < 1001$.
II Soros Olympiad 1995 - 96 (Russia), 9.9
There are $5$ ingots weighing $1$, $2$, $3$, $4$ and $5$ kg with an unknown copper content that varies in different ingots. Each ingot must be divided into $5$ parts and $5$ new ingots of the same mass of $1$, $2$, $3$, $4$ and $5$ kg must be made. This requires that the percentage of copper in all pieces be the same, regardless of what it was in the original pieces. What parts should each piece be divided into?