Found problems: 25757
2018 China Northern MO, 5
A right triangle has the property that it's sides are pairwise relatively prime positive integers and that the ratio of it's area to it's perimeter is a perfect square. Find the minimum possible area of this triangle.
LMT Team Rounds 2010-20, B11
$\vartriangle ABC$ is an isosceles triangle with $AB = AC$. Let $M$ be the midpoint of $BC$ and $E$ be the point on AC such that $AE :CE = 5 : 3$. Let $X$ be the intersection of $BE$ and $AM$. Given that the area of $\vartriangle CM X$ is $15$, find the area of $\vartriangle ABC$.
2022 VIASM Summer Challenge, Problem 4
Given a triangle $ABC$ inscribed in $(O)$. Choose points $M,N,P$ on the sides $AB,BC,CA$ such that $AMNP$ is a parallelogram. The segment $CM$ intersects $NP$ at $E$; the segment $BP$ intersects $NM$ at $F$; and the segment $BE$ intersects $CF$ at $D.$
a) Prove that: $A,D,N$ are collinear.
b) Let $I,J$ be the circumcenters of $\triangle MBF, \triangle PCE,$ respectively.
Prove that: $OD$ passes through the midpoint of $IJ.$
2012 Czech-Polish-Slovak Match, 3
Let $ABCD$ be a cyclic quadrilateral with circumcircle $\omega$. Let $I, J$ and $K$ be the incentres of the triangles $ABC, ACD$ and $ABD$ respectively. Let $E$ be the midpoint of the arc $DB$ of circle $\omega$ containing the point $A$. The line $EK$ intersects again the circle $\omega$ at point $F$ $(F \neq E)$. Prove that the points $C, F, I, J$ lie on a circle.
2014 Balkan MO Shortlist, G4
Let $A_0B_0C_0$ be a triangle with area equal to $\sqrt 2$. We consider the excenters $A_1$,$B_1$ and $C_1$ then we consider the excenters ,say $A_2,B_2$ and $C_2$,of the triangle $A_1B_1C_1$. By continuing this procedure ,examine if it is possible to arrive to a triangle $A_nB_nC_n$ with all coordinates rational.
2008 Postal Coaching, 1
In triangle $ABC,\angle B > \angle C, T$ is the midpoint of arc $BAC$ of the circumcicle of $ABC$, and $I$ is the incentre of $ABC$. Let $E$ be point such that $\angle AEI = 90^0$ and $AE$ is parallel to $BC$. If $TE$ intersects the circumcircle of $ABC$ at $P(\ne T)$ and $\angle B = \angle IPB$, determine $\angle A$.
2015 Kazakhstan National Olympiad, 6
The quadrilateral $ABCD$ has an incircle of diameter $d$ which touches $BC$ at $K$ and touches $DA$ at $L$. Is it always true that the harmonic mean of $AB$ and $CD$ is equal to $KL$ if and only if the geometric mean of $AB$ and $CD$ is equal to $d$?
1965 All Russian Mathematical Olympiad, 058
A circle is circumscribed around the triangle $ABC$. Chords, from the midpoint of the arc $AC$ to the midpoints of the arcs $AB$ and $BC$, intersect sides $[AB]$ and $[BC]$ in the points $D$ and $E$. Prove that $(DE)$ is parallel to $(AC)$ and passes through the centre of the inscribed circle.
2017 Switzerland - Final Round, 8
Let $ABC$ be an isosceles triangle with vertex $A$ and $AB> BC$. Let $k$ be the circle with center $A$ passsing through $B$ and $C$. Let $H$ be the second intersection of $k$ with the altitude of the triangle $ABC$ through $B$. Further let $G$ be the second intersection of $k$ with the median through $B$ in triangle $ABC$. Let $X$ be the intersection of the lines $AC$ and $GH$. Show that $C$ is the midpoint of $AX$.
2014 Kurschak Competition, 3
Let $K$ be a closed convex polygonal region, and let $X$ be a point in the plane of $K$. Show that there exists a finite sequence of reflections in the sides of $K$, such that $K$ contains the image of $X$ after these reflections.
2022 Harvard-MIT Mathematics Tournament, 7
Point $P$ is located inside a square $ABCD$ of side length $10$. Let $O_1$, $O_2$, $O_3$, $O_4$ be the circumcenters of $P AB$, $P BC$, $P CD$, and $P DA$, respectively. Given that $P A+P B +P C +P D = 23\sqrt2$ and the area of $O_1O_2O_3O_4$ is $50$, the second largest of the lengths $O_1O_2$, $O_2O_3$, $O_3O_4$, $O_4O_1$ can be written as $\sqrt{\frac{a}{b}}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.
2008 Stars Of Mathematics, 3
Consider a convex quadrilateral, and the incircles of the triangles determined by one of its diagonals. Prove that the tangency points of the incircles with the diagonal are symmetrical with respect to the midpoint of the diagonal if and only if the line of the incenters passes through the crossing point of the diagonals.
[i]Dan Schwarz[/i]
2014 AIME Problems, 8
Circle $C$ with radius $2$ has diameter $\overline{AB}$. Circle $D$ is internally tangent to circle $C$ at $A$. Circle $E$ is internally tangent to circle $C,$ externally tangent to circle $D,$ and tangent to $\overline{AB}$. The radius of circle $D$ is three times the radius of circle $E$ and can be written in the form $\sqrt{m} - n,$ where $m$ and $n$ are positive integers. Find $m+n$.
2022 Bundeswettbewerb Mathematik, 4
Some points in the plane are either colored red or blue. The distance between two points of the opposite color is at most 1.
Prove that there exists a circle with diameter $\sqrt{2}$ such that no two points outside of this circle have same color. It is enough to prove this claim for a finite number of colored points.
III Soros Olympiad 1996 - 97 (Russia), 10.5
Two circles intersect at two points $A$ and $B$. The radii of these circles are equal to $R$ and $r$, respectively; the angle between the radii going to the points of intersection is equal to $a$. A chord $KM$ of length $b$ is taken in a circle of radius $r$. Straight lines $KA$, $KB$, $MA$ and $MB$ intersect the other circle for second time at four points. Find the area of the quadrilateral with vertices at these points.
1990 IMO Longlists, 91
Quadrilateral $ABCD$ has an inscribed circle with center $O$. Knowing that $AB = CD$, and $M, K$ are the midpoints of $BC, AD$ respectively. Prove that $OM = OK.$
2004 Iran MO (3rd Round), 7
Suppose $F$ is a polygon with lattice vertices and sides parralell to x-axis and y-axis.Suppose $S(F),P(F)$ are area and perimeter of $F$.
Find the smallest k that:
$S(F) \leq k.P(F)^2$
2000 Iran MO (3rd Round), 2
Isosceles triangles $A_3A_1O_2$ and $A_1A_2O_3$ are constructed on the sides of
a triangle $A_1A_2A_3$ as the bases, outside the triangle. Let $O_1$ be a point
outside $\Delta A_1A_2A_3$ such that
$\angle O_1A_3A_2 =\frac 12\angle A_1O_3A_2$ and $\angle O_1A_2A_3 =\frac 12\angle A_1O_2A_3$.
Prove that $A_1O_1\perp O_2O_3$, and if $T$ is the projection of $O_1$ onto $A_2A_3$,
then $\frac{A_1O_1}{O_2O_3} = 2\frac{O_1T}{A_2A_3}$.
2022 3rd Memorial "Aleksandar Blazhevski-Cane", P4
Let $ABC$ be an acute triangle with incircle $\omega$, incenter $I$, and $A$-excircle $\omega_{a}$. Let $\omega$ and $\omega_{a}$ meet $BC$ at $X$ and $Y$, respectively. Let $Z$ be the intersection point of $AY$ and $\omega$ which is closer to $A$. The point $H$ is the foot of the altitude from $A$. Show that $HZ$, $IY$ and $AX$ are concurrent.
[i]Proposed by Nikola Velov[/i]
V Soros Olympiad 1998 - 99 (Russia), 11.6
Cut the $10$ cm $x 20$ cm rectangle into two pieces with one straight cut so that they can be placed inside the $19.4$ cm diameter circle without intersecting.
2006 Harvard-MIT Mathematics Tournament, 4
Let $ABC$ be a triangle such that $AB=2$, $CA=3$, and $BC=4$. A semicircle with its diameter on $BC$ is tangent to $AB$ and $AC$. Compute the area of the semicircle.
1987 Tournament Of Towns, (157) 1
From vertex $A$ in square $ABCD$ (of side length $1$ ) two lines are drawn , one intersecting side $BC$ and the other intersecting side $CD$. The angle between these lines is $\theta$. From vertices $B$ and $D$ we construct perpendiculars to each of these lines . Find the area of the quadrilateral whose vertices are the four feet of these perpendiculars.
1982 Czech and Slovak Olympiad III A, 3
In the plane with coordinates $x,y$, find an example of a convex set $M$ that contains infinitely many lattice points (i.e. points with integer coordinates), but at the same time only finitely many lattice points from $M$ lie on each line in that plane.
Kvant 2024, M2813
The quadrilateral $ABCD$ is described around a circle centered on $I$. Let the diagonals $AC$ and $BD$ intersect at point $E$. The perpendicular bisectors to the segments $AC$ and $BD$ intersect at the point $P$ lying inside the triangle $BEC$. The circumscribed circles of the triangles $APC$ and $BPD$ intersect at points $P$ and $Q$. Prove that $I$ lies on the line $PQ$.
[i] Proposed by Tran Quang Hung [/i]
2011 Indonesia MO, 5
[asy]
draw((0,1)--(4,1)--(4,2)--(0,2)--cycle);
draw((2,0)--(3,0)--(3,3)--(2,3)--cycle);
draw((1,1)--(1,2));
label("1",(0.5,1.5));
label("2",(1.5,1.5));
label("32",(2.5,1.5));
label("16",(3.5,1.5));
label("8",(2.5,0.5));
label("6",(2.5,2.5));
[/asy]
The image above is a net of a unit cube. Let $n$ be a positive integer, and let $2n$ such cubes are placed to build a $1 \times 2 \times n$ cuboid which is placed on a floor. Let $S$ be the sum of all numbers on the block visible (not facing the floor). Find the minimum value of $n$ such that there exists such cuboid and its placement on the floor so $S > 2011$.