Found problems: 85335
2003 Iran MO (3rd Round), 24
$ A,B$ are fixed points. Variable line $ l$ passes through the fixed point $ C$. There are two circles passing through $ A,B$ and tangent to $ l$ at $ M,N$. Prove that circumcircle of $ AMN$ passes through a fixed point.
2017 Iran MO (3rd round), 3
$30$ volleyball teams have participated in a league. Any two teams have played a match with each other exactly once. At the end of the league, a match is called [b]unusual[/b] if at the end of the league, the winner of the match have a smaller amount of wins than the loser of the match. A team is called [b]astonishing[/b] if all its matches are [b]unusual[/b] matches.
Find the maximum number of [b]astonishing[/b] teams.
2009 Tuymaada Olympiad, 2
An arrangement of chips in the squares of $ n\times n$ table is called [i]sparse[/i] if every $ 2\times 2$ square contains at most 3 chips. Serge put chips in some squares of the table (one in a square) and obtained a sparse arrangement. He noted however that if any chip is moved to any free square then the arrangement is no more sparce. For what $ n$ is this possible?
[i]Proposed by S. Berlov[/i]
1969 IMO Longlists, 24
$(GBR 1)$ The polynomial $P(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k$, where $a_0,\cdots, a_k$ are integers, is said to be divisible by an integer $m$ if $P(x)$ is a multiple of $m$ for every integral value of $x$. Show that if $P(x)$ is divisible by $m$, then $a_0 \cdot k!$ is a multiple of $m$. Also prove that if $a, k,m$ are positive integers such that $ak!$ is a multiple of $m$, then a polynomial $P(x)$ with leading term $ax^k$can be found that is divisible by $m.$
LMT Speed Rounds, 2010.14
On the team round, an LMT team of six students wishes to divide itself into two distinct groups of three, one group to work on part $1,$ and one group to work on part $2.$ In addition, a captain of each group is designated. In how many ways can this be done?
2013 Kazakhstan National Olympiad, 2
Let in triangle $ABC$ incircle touches sides $AB,BC,CA$ at $C_1,A_1,B_1$ respectively. Let $\frac {2}{CA_1}=\frac {1}{BC_1}+\frac {1}{AC_1}$ .Prove that if $X$ is intersection of incircle and $CC_1$ then $3CX=CC_1$
2017 Ukrainian Geometry Olympiad, 4
Let $ABCD$ be a parallelogram and $P$ be an arbitrary point of the circumcircle of $\Delta ABD$, different from the vertices. Line $PA$ intersects the line $CD$ at point $Q$. Let $O$ be the center of the circumcircle $\Delta PCQ$. Prove that $\angle ADO = 90^o$.
2016 Saudi Arabia GMO TST, 2
Let $a, b$ be given two real number with $a \ne 0$. Find all polynomials $P$ with real coefficients such that
$x P(x - a) = (x - b)P(x)$ for all $x\in R$
2012 NIMO Summer Contest, 3
Let
\[
S = \sum_{i = 1}^{2012} i!.
\]
The tens and units digits of $S$ (in decimal notation) are $a$ and $b$, respectively. Compute $10a + b$.
[i]Proposed by Lewis Chen[/i]
2018 China Team Selection Test, 3
Circle $\omega$ is tangent to sides $AB$,$AC$ of triangle $ABC$ at $D$,$E$ respectively, such that $D\neq B$, $E\neq C$ and $BD+CE<BC$. $F$,$G$ lies on $BC$ such that $BF=BD$, $CG=CE$. Let $DG$ and $EF$ meet at $K$. $L$ lies on minor arc $DE$ of $\omega$, such that the tangent of $L$ to $\omega$ is parallel to $BC$. Prove that the incenter of $\triangle ABC$ lies on $KL$.
1993 IMO Shortlist, 7
Let $A$, $B$, $C$, $D$ be four points in the plane, with $C$ and $D$ on the same side of the line $AB$, such that $AC \cdot BD = AD \cdot BC$ and $\angle ADB = 90^{\circ}+\angle ACB$. Find the ratio
\[\frac{AB \cdot CD}{AC \cdot BD}, \]
and prove that the circumcircles of the triangles $ACD$ and $BCD$ are orthogonal. (Intersecting circles are said to be orthogonal if at either common point their tangents are perpendicuar. Thus, proving that the circumcircles of the triangles $ACD$ and $BCD$ are orthogonal is equivalent to proving that the tangents to the circumcircles of the triangles $ACD$ and $BCD$ at the point $C$ are perpendicular.)
2000 Junior Balkan Team Selection Tests - Romania, 1
Solve in natural the equation
$9^x-3^x=y^4+2y^3+y^2+2y$
_____________________________
Azerbaijan Land of the Fire :lol:
2010 Danube Mathematical Olympiad, 3
All sides and diagonals of a convex $n$-gon, $n\ge 3$, are coloured one of two colours. Show that there exist $\left[\frac{n+1}{3}\right]$ pairwise disjoint monochromatic segments.
[i](Two segments are disjoint if they do not share an endpoint or an interior point).[/i]
2020 BMT Fall, 5
Call a positive integer [i]prime-simple[/i] if it can be expressed as the sum of the squares of two distinct prime numbers. How many positive integers less than or equal to $100$ are prime-simple?
ICMC 2, 5
For continuously differentiable function \(f : [0, 1] \to\mathbb{R}\) with \(f (1/2) = 0\), show that
\[\left(\int_0^1 f(x)\mathrm{d}x\right)^2\leq \frac{1}{4}\int_0^1\left(f'(x)\right)^2\mathrm{d}x\]
2016 AMC 8, 9
What is the sum of the distinct prime integer divisors of $2016$?
$\textbf{(A) }9\qquad\textbf{(B) }12\qquad\textbf{(C) }16\qquad\textbf{(D) }49\qquad \textbf{(E) }63$
2012 Sharygin Geometry Olympiad, 5
Let $ABC$ be an isosceles right-angled triangle. Point $D$ is chosen on the prolongation of the hypothenuse $AB$ beyond point $A$ so that $AB = 2AD$. Points $M$ and $N$ on side $AC$ satisfy the relation $AM = NC$. Point $K$ is chosen on the prolongation of $CB$ beyond point $B$ so that $CN = BK$. Determine the angle between lines $NK$ and $DM$.
(M.Kungozhin)
2004 Regional Olympiad - Republic of Srpska, 4
Set $S=\{1,2,...,n\}$ is firstly divided on $m$ disjoint nonempty subsets, and then on $m^2$ disjoint nonempty subsets. Prove that some $m$ elements of set $S$ were after first division in same set, and after the second division were in $m$ different sets
1994 AMC 8, 20
Let $W,X,Y$ and $Z$ be four different digits selected from the set
$\{ 1,2,3,4,5,6,7,8,9\}.$
If the sum $\dfrac{W}{X} + \dfrac{Y}{Z}$ is to be as small as possible, then $\dfrac{W}{X} + \dfrac{Y}{Z}$ must equal
$\text{(A)}\ \dfrac{2}{17} \qquad \text{(B)}\ \dfrac{3}{17} \qquad \text{(C)}\ \dfrac{17}{72} \qquad \text{(D)}\ \dfrac{25}{72} \qquad \text{(E)}\ \dfrac{13}{36}$
2023 Brazil Cono Sur TST, 4
Let $p$ be a prime number. Determine all positive integers $a$ such that the sequence $(a_n)_{n\geq 0}$ defined by $a_0=a$ and $a_{n+1}=pa_n-(p-1)\lfloor \sqrt[p]{a_ n} \rfloor^p$, for every $n\geq0$, is eventually constant.
2002 AMC 10, 24
Riders on a Ferris wheel travel in a circle in a vertical plane. A particular wheel has radius $ 20$ feet and revolves at the constant rate of one revolution per minute. How many seconds does it take a rider to travel from the bottom of the wheel to a point $ 10$ vertical feet above the bottom?
$ \textbf{(A)}\ 5 \qquad
\textbf{(B)}\ 6 \qquad
\textbf{(C)}\ 7.5 \qquad
\textbf{(D)}\ 10 \qquad
\textbf{(E)}\ 15$
2020 Israel National Olympiad, 7
Let $P$ be a point inside a triangle $ABC$, $d_a$, $d_b$ and $d_c$ be distances from $P$ to the lines $BC$, $AC$ and $AB$ respectively, $R$ be a radius of the circumcircle and $r$ be a radius of the inscribed circle for $\Delta ABC.$ Prove that:
$$\sqrt{d_a}+\sqrt{d_b}+\sqrt{d_c}\leq\sqrt{2R+5r}.$$
MathLinks Contest 6th, 1.2
Let $ABCD$ be a rectangle of center $O$ in the plane $\alpha$, and let $V \notin\alpha$ be a point in space such that $V O \perp \alpha$. Let $A' \in (V A)$, $B'\in (V B)$, $C'\in (V C)$, $D'\in (V D)$ be four points, and let $M$ and $N$ be the midpoints of the segments $A'C'$ and $B'D'$. .Prove that $MN \parallel \alpha$ if and only if $V , A', B', C', D'$ all lie on a sphere.
2020 Caucasus Mathematical Olympiad, 1
By one magic nut, Wicked Witch can either turn a flea into a beetle or a spider into a bug; while by one magic acorn, she can either turn a flea into a spider or a beetle into a bug. In the evening Wicked Witch had spent 20 magic nuts and 23 magic acorns. By these actions, the number of beetles increased by 5. Determine what was the change in the number of spiders. (Find all possible answers and prove that the other answers are impossible.)
2008 Pan African, 1
Let $x$ and $y$ be two positive reals. Prove that $xy\le\frac{x^{n+2}+y^{n+2}}{x^n+y^n}$ for all non-negative integers $n$.