Found problems: 85335
2019 Latvia Baltic Way TST, 10
Let $\triangle ABC$ be an acute angled triangle with orthocenter $H$ and let $M$ be a midpoint of $BC$. Circle with diameter $AH$ is $\omega_1$ and circle with center $M$ is $\omega_2$. If $\omega_2$ is tangent to circumcircle of $\triangle ABC$, then prove that circles $\omega_1$ and $\omega_2$ are tangent to each other.
Estonia Open Junior - geometry, 2012.2.3
Two circles $c$ and $c'$ with centers $O$ and $O'$ lie completely outside each other. Points $A, B$, and $C$ lie on the circle $c$ and points $A', B'$, and $C$ lie on the circle $c'$ so that segment $AB\parallel A'B'$, $BC \parallel B'C'$, and $\angle ABC = \angle A'B'C'$. The lines $AA', BB$', and $CC'$ are all different and intersect in one point $P$, which does not coincide with any of the vertices of the triangles $ABC$ or $A'B'C'$. Prove that $\angle AOB = \angle A'O'B'$.
1957 Moscow Mathematical Olympiad, 369
Represent $1957$ as the sum of $12$ positive integer summands $a_1, a_2, ... , a_{12}$ for which the number $a_1! \cdot a_2! \cdot a_3! \cdot ... \cdot a_{12}!$ is minimal.
2004 IMO, 3
Define a "hook" to be a figure made up of six unit squares as shown below in the picture, or any of the figures obtained by applying rotations and reflections to this figure.
[asy]
unitsize(0.5 cm);
draw((0,0)--(1,0));
draw((0,1)--(1,1));
draw((2,1)--(3,1));
draw((0,2)--(3,2));
draw((0,3)--(3,3));
draw((0,0)--(0,3));
draw((1,0)--(1,3));
draw((2,1)--(2,3));
draw((3,1)--(3,3));
[/asy]
Determine all $ m\times n$ rectangles that can be covered without gaps and without overlaps with hooks such that
- the rectangle is covered without gaps and without overlaps
- no part of a hook covers area outside the rectangle.
2018 Brazil Team Selection Test, 3
In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.
2013 HMNT, 3
Let $ABC$ be a triangle with $AB = 5$, $BC = 4$, and $CA = 3$. Initially, there is an ant at each vertex. The ants start walking at a rate of $1$ unit per second, in the direction $A \to B \to C \to A$ (so the ant starting at $A$ moves along ray $\overrightarrow{AB}$, etc.). For a positive real number $t$ less than$ 3$, let $A(t)$ be the area of the triangle whose vertices are the positions of the ants after $t$ seconds have elapsed. For what positive real number $t$ less than $3$ is $A(t)$ minimized?
1958 AMC 12/AHSME, 6
The arithmetic mean between $ \frac {x \plus{} a}{x}$ and $ \frac {x \minus{} a}{x}$, when $ x \not \equal{} 0$, is:
$ \textbf{(A)}\ {2}\text{, if }{a \not \equal{} 0}\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ {1}\text{, only if }{a \equal{} 0}\qquad \textbf{(D)}\ \frac {a}{x}\qquad \textbf{(E)}\ x$
1964 AMC 12/AHSME, 30
If $(7+4\sqrt{3})x^2+(2+\sqrt{3})x-2=0$, the larger root minus the smaller root is:
$ \textbf{(A)}\ -2+3\sqrt{3}\qquad\textbf{(B)}\ 2-\sqrt{3}\qquad\textbf{(C)}\ 6+3\sqrt{3}\qquad\textbf{(D)}\ 6-3\sqrt{3}\qquad\textbf{(E)}\ 3\sqrt{3}+2 $
2022 Turkey Junior National Olympiad, 3
Let $m, n, a, k$ be positive integers and $k>1$ such that the equality $$5^m+63n+49=a^k$$
holds. Find the minimum value of $k$.
2012 Today's Calculation Of Integral, 808
For a constant $c$, a sequence $a_n$ is defined by $a_n=\int_c^1 nx^{n-1}\left(\ln \left(\frac{1}{x}\right)\right)^n dx\ (n=1,\ 2,\ 3,\ \cdots).$
Find $\lim_{n\to\infty} a_n$.
2014 ELMO Shortlist, 11
Let $p$ be a prime satisfying $p^2\mid 2^{p-1}-1$, and let $n$ be a positive integer. Define
\[ f(x) = \frac{(x-1)^{p^n}-(x^{p^n}-1)}{p(x-1)}. \]
Find the largest positive integer $N$ such that there exist polynomials $g(x)$, $h(x)$ with integer coefficients and an integer $r$ satisfying $f(x) = (x-r)^N g(x) + p \cdot h(x)$.
[i]Proposed by Victor Wang[/i]
1916 Eotvos Mathematical Competition, 1
If $ a$ and $b$ are positive numbers, prove that the equation
$$\frac{1}{x}+\frac{1}{x - a}+\frac{1}{x+ b}= 0$$
has two rea] roots, one between $ a/3$ and $2a/3$, and one between $-2b/3$ and $-b/3$.
2013 IMO Shortlist, C1
Let $n$ be an positive integer. Find the smallest integer $k$ with the following property; Given any real numbers $a_1 , \cdots , a_d $ such that $a_1 + a_2 + \cdots + a_d = n$ and $0 \le a_i \le 1$ for $i=1,2,\cdots ,d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most $1$.
2022 Germany Team Selection Test, 1
Given a triangle $ABC$ and three circles $x$, $y$ and $z$ such that $A \in y \cap z$, $B \in z \cap x$ and $C \in x \cap y$.
The circle $x$ intersects the line $AC$ at the points $X_b$ and $C$, and intersects the line $AB$ at the points $X_c$ and $B$.
The circle $y$ intersects the line $BA$ at the points $Y_c$ and $A$, and intersects the line $BC$ at the points $Y_a$ and $C$.
The circle $z$ intersects the line $CB$ at the points $Z_a$ and $B$, and intersects the line $CA$ at the points $Z_b$ and $A$.
(Yes, these definitions have the symmetries you would expect.)
Prove that the perpendicular bisectors of the segments $Y_a Z_a$, $Z_b X_b$ and $X_c Y_c$ concur.
2016 Junior Balkan Team Selection Tests - Romania, 3
ABCD=cyclic quadrilateral,$AC\cap BD=X$
AA'$\perp $BD,A'$\in$BD
CC'$\perp $BD,C'$\in$BD
BB'$\perp $AC,B'$\in$AC
DD'$\perp $AC,D'$\in$AC
Prove that:
a)Prove that perpendiculars from midpoints of the sides to the opposite sides are concurrent.The point is called Mathot Point
b)A',B',C',D' are concyclic
c)If O'=circumcenter of (A'B'C') prove that O'=midpoint of the line that connects the orthocente of triangle XAB and XCD
d)O' is the Mathot Point
1993 India Regional Mathematical Olympiad, 6
If $a,b,c,d$ are four positive reals such that $abcd= 1$ , prove that $(1+a) (1+b) (1 +c ) (1 +d ) \geq 16.$
2002 China Girls Math Olympiad, 3
Find all positive integers $ k$ such that for any positive numbers $ a, b$ and $ c$ satisfying the inequality
\[ k(ab \plus{} bc \plus{} ca) > 5(a^2 \plus{} b^2 \plus{} c^2),\]
there must exist a triangle with $ a, b$ and $ c$ as the length of its three sides respectively.
2016 Junior Regional Olympiad - FBH, 2
Find set of positive integers divisible with $8$ which sum of digits is $7$ and product is $6$
Geometry Mathley 2011-12, 9.2
Let $ABDE, BCFZ$ and $CAKL$ be three arbitrary rectangles constructed outside a triangle $ABC$. Let $EF$ meet $ZK$ at $M$, and $N$ be the intersection of the lines through $F,Z$ perpendicular to $FL,ZD$. Prove that $A,M,N$ are collinear.
Kostas Vittas
2018 Belarusian National Olympiad, 10.6
The vertices of the convex quadrilateral $ABCD$ lie on the parabola $y=x^2$. It is known that $ABCD$ is cyclic and $AC$ is a diameter of its circumcircle. Let $M$ and $N$ be the midpoints of the diagonals of $AC$ and $BD$ respectively. Find the length of the projection of the segment $MN$ on the axis $Oy$.
2020-2021 OMMC, 14
There exist positive integers $N, M$ such that $N$'s remainders modulo the four integers $6, 36,$ $216,$ and $M$ form an increasing nonzero geometric sequence in that order. Find the smallest possible value of $M$.
2017 Israel Oral Olympiad, 6
What is the maximal number of vertices of a convex polyhedron whose each face is either a regular triangle or a square?
Ukraine Correspondence MO - geometry, 2019.11
Let $O$ be the center of the circle circumscribed around the acute triangle $ABC$, and let $N$ be the midpoint of the arc $ABC$ of this circle. On the sides $AB$ and $BC$ mark points $D$ and $E$ respectively, such that the point $O$ lies on the segment $DE$. The lines $DN$ and $BC$ intersect at the point $P$, and the lines $EN$ and $AB$ intersect at the point $Q$. Prove that $PQ \perp AC$.
2020 Online Math Open Problems, 17
Let $ABC$ be a triangle with $AB=11,BC=12,$ and $CA=13$, let $M$ and $N$ be the midpoints of sides $CA$ and $AB$, respectively, and let the incircle touch sides $CA$ and $AB$ at points $X$ and $Y$, respectively. Suppose that $R,S,$ and $T$ are the midpoints of line segments $MN,BX,$ and $CY$, respectively. Then $\sin\angle{SRT}=\frac{k\sqrt{m}}{n}$, where $k,m,$ and $n$ are positive integers such that $\gcd(k,n)=1$ and $m$ is not divisible by the square of any prime. Compute $100k+10m+n$.
[i]Proposed by Tristan Shin[/i]
1975 IMO, 5
Can there be drawn on a circle of radius $1$ a number of $1975$ distinct points, so that the distance (measured on the chord) between any two points (from the considered points) is a rational number?