Found problems: 25757
2005 Slovenia National Olympiad, Problem 3
Let $T$ be a point inside a square $ABCD$. The lines $TA,TB,TC,TD$ meet the circumcircle of $ABCD$ again at $A',B',C',D'$, respectively. Prove that $A'B'\cdot C'D'=A'D'\cdot B'C'$.
2014 Sharygin Geometry Olympiad, 20
A quadrilateral $KLMN$ is given. A circle with center $O$ meets its side $KL$ at points $A$ and $A_1$, side $LM$ at points $B$ and $B_1$, etc. Prove that if the circumcircles of triangles $KDA, LAB, MBC$ and $NCD$ concur at point $P$, then
a) the circumcircles of triangles $KD_1A_1, LA_1B_1, MB_1C_1$ and $NC1D1$ also concur at some point $Q$;
b) point $O$ lies on the perpendicular bisector to $PQ$.
2023 Thailand October Camp, 6
Let $ABC$ be an acute triangle with orthocenter $H$. Let $D$ and $E$ be feet of the altitudes from $B$ and $C$ respectively. Let $M$ be the midpoint of segment $AH$ and $F$ be the intersection point of $AH$ and $DE$. Furthermore, let $P$ and $Q$ be the points
inside triangle $ADE$ so that $P$ is an intersection of $CM$ and the circumcircle of $DFH$, and $Q$ is an intersection of $BM$ and the circumcircle of $EFH$. Prove that the intersection of lines $DQ$ and $EP$ lies on segment $AH$.
2010 Oral Moscow Geometry Olympiad, 6
Perpendicular bisectors of the sides $BC$ and $AC$ of an acute-angled triangle $ABC$ intersect lines $AC$ and $BC$ at points $M$ and $N$. Let point $C$ move along the circumscribed circle of triangle $ABC$, remaining in the same half-plane relative to $AB$ (while points $A$ and $B$ are fixed). Prove that line $MN$ touches a fixed circle.
2002 All-Russian Olympiad, 2
The diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$ meet at $O$. The circumcircles of triangles $AOB$ and $COD$ intersect again at $K$. Point $L$ is such that the triangles $BLC$ and $AKD$ are similar and equally oriented. Prove that if the quadrilateral $BLCK$ is convex, then it is tangent [has an incircle].
1974 AMC 12/AHSME, 19
In the adjoining figure $ABCD$ is a square and $CMN$ is an equilateral triangle. If the area of $ABCD$ is one square inch, then the area of $CMN$ in square inches is
[asy]
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);
draw((.82,0)--(1,1)--(0,.76)--cycle);
label("A", (0,0), S);
label("B", (1,0), S);
label("C", (1,1), N);
label("D", (0,1), N);
label("M", (0,.76), W);
label("N", (.82,0), S);
[/asy]
$ \textbf{(A)}\ 2\sqrt{3}-3 \qquad\textbf{(B)}\ 1-\frac{\sqrt{3}}{3} \qquad\textbf{(C)}\ \frac{\sqrt{3}}{4} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{3} \qquad\textbf{(E)}\ 4-2\sqrt{3} $
1962 IMO, 5
On the circle $K$ there are given three distinct points $A,B,C$. Construct (using only a straightedge and a compass) a fourth point $D$ on $K$ such that a circle can be inscribed in the quadrilateral thus obtained.
1974 Chisinau City MO, 76
Altitude $AH$ and median $AM$ of the triangle $ABC$ satisfy the relation: $\angle ABM = \angle CBH$. Prove that triangle $ABC$ is isosceles or right-angled.
2005 Iran MO (3rd Round), 3
Prove that in acute-angled traingle ABC if $r$ is inradius and $R$ is radius of circumcircle then: \[a^2+b^2+c^2\geq 4(R+r)^2\]
Champions Tournament Seniors - geometry, 2016.3
Let $t$ be a line passing through the vertex $A$ of the equilateral $ABC$, parallel to the side $BC$. On the side $AC$ arbitrarily mark the point $D$. Bisector of the angle $ABD$ intersects the line $t$at the point $E$. Prove that $BD=CD+AE$.
1986 IMO Longlists, 76
Let $A, B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $ABC$ is an equilateral triangle whose side is $86$ meters long. A boy swam from $A$ directly toward $B$. After covering a distance of $x$ meters, he turned and swam westward, reaching the shore after covering a distance of $y$ meters. If $x$ and $y$ are both positive integers, determine $y.$
2021 JBMO TST - Turkey, 1
In an acute-angled triangle $ABC$, the circle with diameter $[AB]$ intersects the altitude drawn from vertex $C$ at a point $D$ and the circle with diameter $[AC]$ intersects the altitude drawn from vertex $B$ at a point $E$. Let the lines $BD$ and $CE$ intersect at $F$. Prove that $$AF\perp DE$$
1999 Turkey MO (2nd round), 2
Problem-2:
Given a circle with center $O$, the two tangent lines from a point $S$ outside the circle touch the circle at points $P$ and $Q$. Line $SO$ intersects the circle at $A$ and $B$, with $B$ closer to $S$. Let $X$ be an interior point of minor arc $PB$, and let line $OS$ intersect lines $QX$ and $PX$ at $C$ and $D$, respectively. Prove that
$\frac{1}{\left| AC \right|}+\frac{1}{\left| AD \right|}=\frac{2}{\left| AB \right|}$.
2005 USA Team Selection Test, 1
Let $n$ be an integer greater than $1$. For a positive integer $m$, let $S_{m}= \{ 1,2,\ldots, mn\}$. Suppose that there exists a $2n$-element set $T$ such that
(a) each element of $T$ is an $m$-element subset of $S_{m}$;
(b) each pair of elements of $T$ shares at most one common element;
and
(c) each element of $S_{m}$ is contained in exactly two elements of $T$.
Determine the maximum possible value of $m$ in terms of $n$.
2009 Hanoi Open Mathematics Competitions, 10
Prove that $d^2+(a-b)^2<c^2$ ,where $d$ is diameter of the inscribed circle of $\vartriangle ABC$
2010 Denmark MO - Mohr Contest, 5
An equilateral triangle $ABC$ is given. With $BC$ as diameter, a semicircle is drawn outside the triangle. On the semicircle, points $D$ and $E$ are chosen such that the arc lengths $BD, DE$ and $EC$ are equal. Prove that the line segments $AD$ and $AE$ divide the side $BC$ into three equal parts.
[img]https://1.bp.blogspot.com/-hQQV-Of96Ls/XzXCZjCledI/AAAAAAAAMV0/SwXa4mtEEm04onYbFGZiTc5NSpkoyvJLwCLcBGAsYHQ/s0/2010%2BMohr%2Bp5.png[/img]
2013 Sharygin Geometry Olympiad, 1
Let $ABC$ be an isosceles triangle with $AB = BC$. Point $E$ lies on the side $AB$, and $ED$ is the perpendicular from $E$ to $BC$. It is known that $AE = DE$. Find $\angle DAC$.
VI Soros Olympiad 1999 - 2000 (Russia), 10.2
$37$ points are arbitrarily marked on the plane. Prove that among them there must be either two points at a distance greater than $6$, or two points at a distance less than $1.5$.
1988 AMC 12/AHSME, 6
A figure is an equiangular parallelogram if and only if it is a
$ \textbf{(A)}\ \text{rectangle}\qquad\textbf{(B)}\ \text{regular polygon}\qquad\textbf{(C)}\ \text{rhombus}\qquad\textbf{(D)}\ \text{square}\qquad\textbf{(E)}\ \text{trapezoid} $
2000 Mediterranean Mathematics Olympiad, 4
Let $P,Q,R,S$ be the midpoints of the sides $BC,CD,DA,AB$ of a convex quadrilateral, respectively. Prove that
\[4(AP^2+BQ^2+CR^2+DS^2)\le 5(AB^2+BC^2+CD^2+DA^2)\]
2020 Dutch IMO TST, 2
Given is a triangle $ABC$ with its circumscribed circle and $| AC | <| AB |$. On the short arc $AC$, there is a variable point $D\ne A$. Let $E$ be the reflection of $A$ wrt the inner bisector of $\angle BDC$. Prove that the line $DE$ passes through a fixed point, regardless of point $D$.
Ukraine Correspondence MO - geometry, 2015.8
On the sides $BC, AC$ and $AB$ of the equilateral triangle $ABC$ mark the points $D, E$ and $F$ so that $\angle AEF = \angle FDB$ and $\angle AFE = \angle EDC$. Prove that $DA$ is the bisector of the angle $EDF$.
2003 National Olympiad First Round, 9
How many integer triangles are there with inradius $1$?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ \text{Infinite}
$
OIFMAT III 2013, 6
The acute triangle $ABC$ is inscribed in a circle with center $O$. Let $D$ be the intersection of the bisector of angle $BAC$ with segment $BC$ and $ P$ the intersection point of $AB$ with the perpendicular on $OA$ passing through $D$. Show that $AC = AP$.
2018 South East Mathematical Olympiad, 6
In the isosceles triangle $ABC$ with $AB=AC$, the center of $\odot O$ is the midpoint of the side $BC$, and $AB,AC$ are tangent to the circle at points $E,F$ respectively. Point $G$ is on $\odot O$ with $\angle AGE = 90^{\circ}$. A tangent line of $\odot O$ passes through $G$, and meets $AC$ at $K$. Prove that line $BK$ bisects $EF$.