Found problems: 25757
1992 Balkan MO, 3
Let $D$, $E$, $F$ be points on the sides $BC$, $CA$, $AB$ respectively of a triangle $ABC$ (distinct from the vertices). If the quadrilateral $AFDE$ is cyclic, prove that \[ \frac{ 4 \mathcal A[DEF] }{\mathcal A[ABC] } \leq \left( \frac{EF}{AD} \right)^2 . \]
[i]Greece[/i]
1967 Swedish Mathematical Competition, 2
You are given a ruler with two parallel straight edges a distance $d$ apart. It may be used
(1) to draw the line through two points,
(2) given two points a distance $\ge d$ apart, to draw two parallel lines, one through each point,
(3) to draw a line parallel to a given line, a distance d away.
One can also (4) choose an arbitrary point in the plane,
and (5) choose an arbitrary point on a line.
Show how to construct :
(A) the bisector of a given angle, and
(B) the perpendicular to the midpoint of a given line segment.
2016 Balkan MO Shortlist, G1
Let $ABCD$ be a quadrilateral ,circumscribed about a circle. Let $M$ be a point on the side $AB$. Let $I_{1}$,$I_{2}$ and $I_{3}$ be the incentres of triangles $AMD$, $CMD$ and $BMC$ respectively. Prove that $I_{1}I_{2}I_{3}M$ is circumscribed.
1971 Canada National Olympiad, 8
A regular pentagon is inscribed in a circle of radius $r$. $P$ is any point inside the pentagon. Perpendiculars are dropped from $P$ to the sides, or the sides produced, of the pentagon.
a) Prove that the sum of the lengths of these perpendiculars is constant.
b) Express this constant in terms of the radius $r$.
1998 Iran MO (3rd Round), 2
Let $ M$ and $ N$ be two points inside triangle $ ABC$ such that
\[ \angle MAB \equal{} \angle NAC\quad \mbox{and}\quad \angle MBA \equal{} \angle NBC.
\]
Prove that
\[ \frac {AM \cdot AN}{AB \cdot AC} \plus{} \frac {BM \cdot BN}{BA \cdot BC} \plus{} \frac {CM \cdot CN}{CA \cdot CB} \equal{} 1.
\]
2008 Sharygin Geometry Olympiad, 8
(B.Frenkin, A.Zaslavsky) A convex quadrilateral was drawn on the blackboard. Boris marked the centers of four excircles each touching one side of the quadrilateral and the extensions of two adjacent sides. After this, Alexey erased the quadrilateral. Can Boris define its perimeter?
2021 Latvia Baltic Way TST, P9
Pentagon $ABCDE$ with $CD\parallel BE$ is inscribed in circle $\omega$. Tangent to $\omega$ through $B$ intersects line $AC$ at $F$ in a way that $A$ lies between $C$ and $F$. Lines $BD$ and $AE$ intersect at $G$. Prove that $FG$ is tangent to the circumcircle of $\triangle ADG$.
2020 Junior Balkan Team Selection Tests - Moldova, 11
Let $\triangle ABC$ be an acute triangle. The bisector of $\angle ACB$ intersects side $AB$ in $D$. The circumcircle of triangle $ADC$ intersects side $BC$ in $C$ and $E$ with $C \neq E$. The line parallel to $AE$ which passes through $B$ intersects line $CD$ in $F$. Prove that the triangle $\triangle AFB$ is isosceles.
2013 NIMO Problems, 4
Let $a,b,c$ be the answers to problems $4$, $5$, and $6$, respectively. In $\triangle ABC$, the measures of $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, $c$ in degrees, respectively. Let $D$ and $E$ be points on segments $AB$ and $AC$ with $\frac{AD}{BD} = \frac{AE}{CE} = 2013$. A point $P$ is selected in the interior of $\triangle ADE$, with barycentric coordinates $(x,y,z)$ with respect to $\triangle ABC$ (here $x+y+z=1$). Lines $BP$ and $CP$ meet line $DE$ at $B_1$ and $C_1$, respectively. Suppose that the radical axis of the circumcircles of $\triangle PDC_1$ and $\triangle PEB_1$ pass through point $A$. Find $100x$.
[i]Proposed by Evan Chen[/i]
2019 Tournament Of Towns, 5
The point $M$ inside a convex quadrilateral $ABCD$ is equidistant from the lines $AB$ and $CD$ and is equidistant from the lines $BC$ and $AD$. The area of $ABCD$ occurred to be equal to $MA\cdot MC +MB \cdot MD$. Prove that the quadrilateral $ABCD$ is
a) tangential (circumscribed),
b) cyclic (inscribed).
(Nairi Sedrakyan)
1967 IMO Shortlist, 1
$A_0B_0C_0$ and $A_1B_1C_1$ are acute-angled triangles. Describe, and prove, how to construct the triangle $ABC$ with the largest possible area which is circumscribed about $A_0B_0C_0$ (so $BC$ contains $B_0, CA$ contains $B_0$, and $AB$ contains $C_0$) and similar to $A_1B_1C_1.$
2021 Czech and Slovak Olympiad III A, 6
An acute triangle $ABC$ is given. Let us denote $X$ for each of its inner points $X_a, X_b, X_c$ its images in axial symmetries sequentially along the lines $BC, CA, AB$. Prove that all $X_aX_bX_c$ triangles have a common interior point.
(Josef Tkadlec)
1997 Israel Grosman Mathematical Olympiad, 2
Is there a planar polygon whose vertices have integer coordinates and whose area is $1/2$, such that this polygon is
(a) a triangle with at least two sides longer than $1000$?
(b) a triangle whose sides are all longer than $1000$?
(c) a quadrangle?
Estonia Open Senior - geometry, 2015.2.5
The triangle $K_2$ has as its vertices the feet of the altitudes of a non-right triangle $K_1$. Find all possibilities for the sizes of the angles of $K_1$ for which the triangles $K_1$ and $K_2$ are similar.
2016 IberoAmerican, 3
Let $ABC$ be an acute triangle and $\Gamma$ its circumcircle. The lines tangent to $\Gamma$ through $B$ and $C$ meet at $P$. Let $M$ be a point on the arc $AC$ that does not contain $B$ such that $M \neq A$ and $M \neq C$, and $K$ be the point where the lines $BC$ and $AM$ meet. Let $R$ be the point symmetrical to $P$ with respect to the line $AM$ and $Q$ the point of intersection of lines $RA$ and $PM$. Let $J$ be the midpoint of $BC$ and $L$ be the intersection point of the line $PJ$ and the line through $A$ parallel to $PR$. Prove that $L, J, A, Q,$ and $K$ all lie on a circle.
2010 HMNT, 5
Circle $O$ has chord $AB$. A circle is tangent to $O$ at $T$ and tangent to$ AB$ at $X$ such that $AX = 2XB$. What is $\frac{AT}{BT}$ ?
2002 Tournament Of Towns, 5
An acute triangle was dissected by a straight cut into two pieces which are not necessarily triangles. Then one of the pieces were dissected by a straight cut into two pieces and so on. After a few dissections it turns out the pieces were all triangles. Is it possible they were all obtuse?
1998 Iran MO (3rd Round), 2
Let $ABCDEF$ be a convex hexagon such that $AB = BC, CD = DE$ and $EF = FA$. Prove that
\[\frac{AB}{BE}+\frac{CD}{AD}+\frac{EF}{CF} \geq \frac{3}{2}.\]
2005 Germany Team Selection Test, 2
Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$.
[i]Proposed by Hojoo Lee, Korea[/i]
1985 IMO Longlists, 87
Determine the radius of a sphere $S$ that passes through the centroids of each face of a given tetrahedron $T$ inscribed in a unit sphere with center $O$. Also, determine the distance from $O$ to the center of $S$ as a function of the edges of $T.$
Kyiv City MO 1984-93 - geometry, 1990.10.5
A circle centered at a point $(0, 1)$ on the coordinate plane intersects the parabola $y = x^2$ at four points: $A, B, C, D.$ Find the largest possible value of the area of the quadrilateral $ABCD$.
2018 PUMaC Team Round, 14
Find the sum of the positive integer solutions to the equation $\left\lfloor\sqrt[3]{x}\right\rfloor+\left\lfloor\sqrt[4]{x}\right\rfloor=4.$
2000 Manhattan Mathematical Olympiad, 4
Three rectangles, each of area $6$ square inches, are placed inside a $4$ in. by $4$ in. square. Prove that, no matter how the three rectangles are shaped and arranged, (for example, like in the picture below), one can find two of them which have a common area of at least $2/3$ square inches.
2019 ITAMO, 5
Let $ABC$ be an acute angled triangle$.$ Let $D$ be the foot of the internal angle bisector of $\angle BAC$ and let $M$ be the midpoint of $AD.$ Let $X$ be a point on segment $BM$ such that $\angle MXA=\angle DAC.$ Prove that $AX$ is perpendicular to $XC.$
2003 Oral Moscow Geometry Olympiad, 4
In triangle $ABC$, $M$ is the point of intersection of the medians, $O$ is the center of the inscribed circle, $A', B', C'$ are the touchpoints with the sides $BC, CA, AB$, respectively. Prove that if $CA'= AB$, then $OM$ and $AB$ are perpendicular.
PS. There is a a typo