Found problems: 25757
Denmark (Mohr) - geometry, 2012.1
Inside a circle with radius $6$ lie four smaller circles with centres $A,B,C$ and $D$. The circles touch each other as shown. The point where the circles with centres $A$ and $C$ touch each other is the centre of the big circle. Calculate the area of quadrilateral $ABCD$.
[img]https://1.bp.blogspot.com/-FFsiOOdcjao/XzT_oJYuQAI/AAAAAAAAMVk/PpyUNpDBeEIESMsiElbexKOFMoCXRVaZwCLcBGAsYHQ/s0/2012%2BMohr%2Bp1.png[/img]
Indonesia MO Shortlist - geometry, g10
Given two circles with one of the centers of the circle is on the other circle. The two circles intersect at two points $C$ and $D$. The line through $D$ intersects the two circles again at $A$ and $ B$. Let $H$ be the midpoint of the arc $AC$ that does not contain $D$ and the segment $HD$ intersects circle that does not contain $H$ at point $E$. Show that $E$ is the center of the incircle of the triangle $ACD$.
2010 Romania Team Selection Test, 2
Let $ABC$ be a scalene triangle, let $I$ be its incentre, and let $A_1$, $B_1$ and $C_1$ be the points of contact of the excircles with the sides $BC$, $CA$ and $AB$, respectively. Prove that the circumcircles of the triangles $AIA_1$, $BIB_1$ and $CIC_1$ have a common point different from $I$.
[i]Cezar Lupu & Vlad Matei[/i]
2012 HMNT, 9
Triangle $ABC$ satisfies $\angle B > \angle C$. Let $M$ be the midpoint of $BC$, and let the perpendicular bisector of $BC$ meet the circumcircle of $\vartriangle ABC$ at a point $D$ such that points $A$, $D$, $C$, and $B$ appear on the circle in that order. Given that $\angle ADM = 68^o$ and $\angle DAC = 64^o$ , find $\angle B$.
1990 Swedish Mathematical Competition, 2
The points $A_1, A_2,.. , A_{2n}$ are equally spaced in that order along a straight line with $A_1A_2 = k$. $P$ is chosen to minimise $\sum PA_i$. Find the minimum.
2010 Ukraine Team Selection Test, 2
Let $ABCD$ be a quadrilateral inscribled in a circle with the center $O, P$ be the point of intersection of the diagonals $AC$ and $BD$, $BC\nparallel AD$. Rays $AB$ and $DC$ intersect at the point $E$. The circle with center $I$ inscribed in the triangle $EBC$ touches $BC$ at point $T_1$. The $E$-excircle with center $J$ in the triangle $EAD$ touches the side $AD$ at the point T$_2$. Line $IT_1$ and $JT_2$ intersect at $Q$. Prove that the points $O, P$, and $Q$ lie on a straight line.
2006 Estonia National Olympiad, 4
Let O be the circumcentre of an acute triangle ABC and let A′, B′ and C′ be the
circumcentres of triangles BCO, CAO and ABO, respectively. Prove that the area of triangle ABC does not exceed the area of triangle A′B′C′.
2018 India PRMO, 17
Triangles $ABC$ and $DEF$ are such that $\angle A = \angle D, AB = DE = 17, BC = EF = 10$ and $AC - DF = 12$. What is $AC + DF$?
2022 Sharygin Geometry Olympiad, 13
Eight points in a general position are given in the plane. The areas of all $56$ triangles with vertices at these points are written in a row. Prove that it is possible to insert the symbols "$+$" and "$-$" between them in such a way
that the obtained sum is equal to zero.
2016 Ecuador Juniors, 3
Let $P_1P_2 . . . P_{2016 }$ be a cyclic polygon of $2016$ sides. Let $K$ be a point inside the polygon and let $M$ be the midpoint of the segment $P_{1000}P_{2000}$. Knowing that $KP_1 = KP_{2011} = 2016$ and $KM$ is perpendicular to $P_{1000}P_{2000}$, find the length of segment $KP_{2016}$.
2023 Macedonian Team Selection Test, Problem 2
Let $ABC$ be an acute triangle such that $AB<AC$ and $AB<BC$. Let $P$ be a point on the segment $BC$ such that $\angle APB = \angle BAC$. The tangent to the circumcircle of triangle $ABC$ at $A$ meets the circumcircle of triangle $APB$ at $Q \neq A$. Let $Q'$ be the reflection of $Q$ with respect to the midpoint of $AB$. The line $PQ$ meets the segment $AQ'$ at $S$. Prove that
$$\frac{1}{AB}+\frac{1}{AC} > \frac{1}{CS}.$$
[i]Authored by Nikola Velov[/i]
1987 Traian Lălescu, 1.4
Through a given point inside a circle, construct two perpendicular chords such that the sum of their lengths would be:
[b]a)[/b] maximum.
[b]b)[/b] minimum.
2018 Stanford Mathematics Tournament, 6
In $\vartriangle AB$C, $AB = 3$, $AC = 6,$ and $D$ is drawn on $BC$ such that $AD$ is the angle bisector of $\angle BAC$. $D$ is reflected across $AB$ to a point $E$, and suppose that $AC$ and $BE$ are parallel. Compute $CE$.
2014 Contests, 2
Consider two circles of radius one, and let $O$ and $O'$ denote their centers. Point $M$ is selected on either circle. If $OO' = 2014$, what is the largest possible area of triangle $OMO'$?
[i]Proposed by Evan Chen[/i]
1982 National High School Mathematics League, 11
Length of edges of regular triangle $ABC$ are $4$, $D\in BC,E\in CA,F\in AB$, satisfying: $|AE|=|BF|=|CD|=1$. $BE\cap CF=R, CF\cap AD=Q, AD\cap BE=S$. $P$ is a point inside $\triangle RQS$ or on its sides. Note that $x=d(P,BC),y=d(P,CA),z=d(P,AB)$.
[b](a)[/b] $xyz$ get its minumum value when $P=R$ (or$Q,S$).
[b](b)[/b] Calculate the minumum value of $xyz$.
2012 Sharygin Geometry Olympiad, 2
We say that a point inside a triangle is good if the lengths of the cevians passing through this point are inversely proportional to the respective side lengths. Find all the triangles for which the number of good points is maximal.
(A.Zaslavsky, B.Frenkin)
2018 South East Mathematical Olympiad, 3
Let $O$ be the circumcenter of acute $\triangle ABC$($AB<AC$), the angle bisector of $\angle BAC$ meets $BC$ at $T$ and $M$ is the midpoint of $AT$. Point $P$ lies inside $\triangle ABC$ such that $PB\perp PC$. $D,E$ distinct from $P$ lies on the perpendicular to $AP$ through $P$ such that $BD=BP, CE=CP$. If $AO$ bisects segment $DE$, prove that $AO$ is tangent to the circumcircle of $\triangle AMP$.
1970 IMO Shortlist, 6
In the triangle $ABC$ let $B'$ and $C'$ be the midpoints of the sides $AC$ and $AB$ respectively and $H$ the foot of the altitude passing through the vertex $A$. Prove that the circumcircles of the triangles $AB'C'$,$BC'H$, and $B'CH$ have a common point $I$ and that the line $HI$ passes through the midpoint of the segment $B'C'.$
2011 Irish Math Olympiad, 4
The incircle $\mathcal{C}_1$ of triangle $ABC$ touches the sides $AB$ and $AC$ at the points $D$ and $E$, respectively. The incircle $\mathcal{C}_2$ of the triangle $ADE$ touches the sides $AB$ and $AC$ at the points $P$ and $Q$, and intersects the circle $\mathcal{C}_1$ at the points $M$ and $n$. Prove that
(a) the center of the circle $\mathcal{C}_2$ lies on the circle $\mathcal{C}_1$.
(b) the four points $M,N,P,Q$ in appropriate order form a rectangle if and only if twice the radius of $\mathcal{C}_1$ is three times the radius of $\mathcal{C}_2$.
2014 Purple Comet Problems, 7
Inside the $7\times8$ rectangle below, one point is chosen a distance $\sqrt2$ from the left side and a distance $\sqrt7$ from the bottom side. The line segments from that point to the four vertices of the rectangle are drawn. Find the area of the shaded region.
[asy]
import graph;
size(4cm);
pair A = (0,0);
pair B = (9,0);
pair C = (9,7);
pair D = (0,7);
pair P = (1.5,3);
draw(A--B--C--D--cycle,linewidth(1.5));
filldraw(A--B--P--cycle,rgb(.76,.76,.76),linewidth(1.5));
filldraw(C--D--P--cycle,rgb(.76,.76,.76),linewidth(1.5));
[/asy]
1949 Moscow Mathematical Olympiad, 163
Prove that if opposite sides of a hexagon are parallel and the diagonals connecting opposite vertices have equal lengths, a circle can be circumscribed around the hexagon.
2013 Online Math Open Problems, 21
Let $ABC$ be a triangle with $AB = 5$, $AC = 8$, and $BC = 7$. Let $D$ be on side $AC$ such that $AD = 5$ and $CD = 3$. Let $I$ be the incenter of triangle $ABC$ and $E$ be the intersection of the perpendicular bisectors of $\overline{ID}$ and $\overline{BC}$. Suppose $DE = \frac{a\sqrt{b}}{c}$ where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. Find $a+b+c$.
[i]Proposed by Ray Li[/i]
2018 Moldova EGMO TST, 3
Let $\triangle ABC $ be an acute triangle.$O$ denote its circumcenter.Points $D$,$E$,$F$ are the midpoints of the sides $BC$,$CA$,and $AB$.Let $M$ be a point on the side $BC$ . $ AM \cap EF = \big\{ N \big\} $ . $ON \cap \big( ODM \big) = \big\{ P \big\} $ Prove that $M'$ lie on $\big(DEF\big)$ where $M'$ is the symmetrical point of $M$ thought the midpoint of $DP$.
2015 Moldova Team Selection Test, 3
Consider an acute triangle $ABC$, points $E,F$ are the feet of the perpendiculars from $B$ and $C$ in $\triangle ABC$. Points $I$ and $J$ are the projections of points $F,E$ on the line $BC$, points $K,L$ are on sides $AB,AC$ respectively such that $IK \parallel AC$ and $JL \parallel AB$. Prove that the lines $IE$,$JF$,$KL$ are concurrent.
2017 NIMO Problems, 3
Let $ABCD$ be a cyclic quadrilateral with circumradius $100\sqrt{3}$ and $AC=300$. If $\angle DBC = 15^{\circ}$, then find $AD^2$.
[i]Proposed by Anand Iyer[/i]