Found problems: 85335
2009 India IMO Training Camp, 3
Let $ a,b$ be two distinct odd natural numbers.Define a Sequence $ { < a_n > }_{n\ge 0}$ like following:
$ a_1 \equal{} a \\
a_2 \equal{} b \\
a_n \equal{} \text{largest odd divisor of }(a_{n \minus{} 1} \plus{} a_{n \minus{} 2})$.
Prove that there exists a natural number $ N$ such that $ a_n \equal{} gcd(a,b) \forall n\ge N$.
2013 EGMO, 3
Let $n$ be a positive integer.
(a) Prove that there exists a set $S$ of $6n$ pairwise different positive integers, such that the least common multiple of any two elements of $S$ is no larger than $32n^2$.
(b) Prove that every set $T$ of $6n$ pairwise different positive integers contains two elements the least common multiple of which is larger than $9n^2$.
III Soros Olympiad 1996 - 97 (Russia), 9.3
Draw the set of projections of a square given on a plane onto all possible lines passing through a given point $O$ of the plane lying outside the square.
1978 Polish MO Finals, 5
For a given real number $a$, define the sequence $(a_n)$ by $a_1 = a$ and
$$a_{n+1} =\begin{cases}
\dfrac12 \left(a_n -\dfrac{1}{a_n}\right) \,\,\, if \,\,\, a_n \ne 0, \\
0 \,\,\, if \,\,\, a_n = 0 \end{cases}$$
Prove that the sequence $(a_n)$ contains infinitely many nonpositive terms.
2011 China Team Selection Test, 1
Let $H$ be the orthocenter of an acute trangle $ABC$ with circumcircle $\Gamma$. Let $P$ be a point on the arc $BC$ (not containing $A$) of $\Gamma$, and let $M$ be a point on the arc $CA$ (not containing $B$) of $\Gamma$ such that $H$ lies on the segment $PM$. Let $K$ be another point on $\Gamma$ such that $KM$ is parallel to the Simson line of $P$ with respect to triangle $ABC$. Let $Q$ be another point on $\Gamma$ such that $PQ \parallel BC$. Segments $BC$ and $KQ$ intersect at a point $J$. Prove that $\triangle KJM$ is an isosceles triangle.
2002 Turkey MO (2nd round), 2
Two circles are externally tangent to each other at a point $A$ and internally tangent to a third circle $\Gamma$ at points $B$ and $C.$ Let $D$ be the midpoint of the secant of $\Gamma$ which is tangent to the smaller circles at $A.$ Show that $A$ is the incenter of the triangle $BCD$ if the centers of the circles are not collinear.
1996 Baltic Way, 6
Let $a,b,c,d$ be positive integers such that $ab\equal{}cd$. Prove that $a\plus{}b\plus{}c\plus{}d$ is a composite number.
May Olympiad L1 - geometry, 2000.2
Let $ABC$ be a right triangle in $A$ , whose leg measures $1$ cm. The bisector of the angle $BAC$ cuts the hypotenuse in $R$, the perpendicular to $AR$ on $R$ , cuts the side $AB$ at its midpoint. Find the measurement of the side $AB$ .
2020 Abels Math Contest (Norwegian MO) Final, 1a
In how many ways can the circles be coloured using three colours, so that no two circles connected by a line segment have the same colour?
[img]https://cdn.artofproblemsolving.com/attachments/3/2/e2bd61786aa4269593233311e85204cff071ec.png[/img]
2012 Kazakhstan National Olympiad, 3
The sequence $a_{n}$ defined as follows: $a_{1}=4, a_{2}=17$ and for any $k\geq1$ true equalities
$a_{2k+1}=a_{2}+a_{4}+...+a_{2k}+(k+1)(2^{2k+3}-1)$
$a_{2k+2}=(2^{2k+2}+1)a_{1}+(2^{2k+3}+1)a_{3}+...+(2^{3k+1}+1)a_{2k-1}+k$
Find the smallest $m$ such that $(a_{1}+...a_{m})^{2012^{2012}}-1$ divided $2^{2012^{2012}}$
2022 Israel TST, 3
A class has 30 students. To celebrate 'Tu BiShvat' each student chose some dried fruits out of $n$ different kinds. Say two students are friends if they both chose from the same type of fruit. Find the minimal $n$ so that it is possible that each student has exactly \(6\) friends.
2010 Contests, 1
Solve in the real numbers $x, y, z$ a system of the equations:
\[
\begin{cases}
x^2 - (y+z+yz)x + (y+z)yz = 0 \\
y^2 - (z + x + zx)y + (z+x)zx = 0 \\
z^2 - (x+y+xy)z + (x+y)xy = 0. \\
\end{cases}
\]
2011 Princeton University Math Competition, A3 / B6
Shirley has a magical machine. If she inputs a positive even integer $n$, the machine will output $n/2$, but if she inputs a positive odd integer $m$, the machine will output $m+3$. The machine keeps going by automatically using its output as a new input, stopping immediately before it obtains a number already processed. Shirley wants to create the longest possible output sequence possible with initial input at most $100$. What number should she input?
2004 IMO Shortlist, 3
Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying
\[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\]
for any two positive integers $ m$ and $ n$.
[i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers:
$ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$.
By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$).
[i]Proposed by Mohsen Jamali, Iran[/i]
2023 Costa Rica - Final Round, 3.2
Find all ordered pairs of positive integers $(r, s)$ for which there are exactly $35$ ordered pairs of positive integers $(a, b)$ such that the least common multiple of $a$ and $b$ is $2^r \cdot 3^s$.
2005 Sharygin Geometry Olympiad, 7
Two circles with radii $1$ and $2$ have a common center at the point $O$. The vertex $A$ of the regular triangle $ABC$ lies on the larger circle, and the middpoint of the base $CD$ lies on the smaller one. What can the angle $BOC$ be equal to?
2017 Kosovo Team Selection Test, 4
For every $n \in \mathbb{N}_{0}$, prove that
$\sum_{k=0}^{\left[\frac{n}{2} \right]}{2}^{n-2k} \binom{n}{2k}=\frac{3^{n}+1}{2}$
2001 South africa National Olympiad, 4
$n$ red and $n$ blue points on a plane are given so that no three of the $2n$ points are collinear. Prove that it is always possible to split up the points into $n$ pairs, with one red and one blue point in each pair, so that no two of the $n$ line segments which connect the two members of a pair intersect.
2013 Hanoi Open Mathematics Competitions, 7
Let $ABC$ be a triangle with $\angle A = 90^o, \angle B = 60^o$ and $BC = 1$ cm. Draw outside of $\vartriangle ABC$ three equilateral triangles $ABD,ACE$ and $BCF$. Determine the area of $\vartriangle DEF$.
2006 AIME Problems, 1
In quadrilateral $ABCD, \angle B$ is a right angle, diagonal $\overline{AC}$ is perpendicular to $\overline{CD},$ $AB=18, BC=21,$ and $CD=14.$ Find the perimeter of $ABCD$.
2006 Cezar Ivănescu, 3
[b]a)[/b] Given two positive reals $ x,y, $ prove that $ \min\left( x,1/x+y,1/y \right)\le\sqrt 2. $ and determine when equality holds.
[b]b)[/b] Find all triplets of real numbers $ (a,b,c) $ having the property that for every triplet of real numbers $ (x,y,z) , $ the following equality holds:
$$ |ax+by+cz|+|bx+cy+az|+|cx+ay+bz|=|x|+|y|+|z| $$
2018 Peru IMO TST, 6
Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.
2011 239 Open Mathematical Olympiad, 2
There are $100$ people in the group. Is it possible that for each pair of people exist at least $50$ others, so every in that group knows exactly one person from the pair?
1951 Moscow Mathematical Olympiad, 204
* Given several numbers each of which is less than $1951$ and the least common multiple of any two of which is greater than $1951$. Prove that the sum of their reciprocals is less than $2$.
1997 Moldova Team Selection Test, 5
Let $P(x)\in\mathbb{Z}[x]$ with deg $P=2015$. Let $Q(x)=(P(x))^2-9$. Prove that: the number of distinct roots of $Q(x)$ can not bigger than $2015$