Found problems: 25757
1994 Tournament Of Towns, (419) 7
Consider an arbitrary “figure” $F$ (non convex polygon). A chord of $F$ is defined to be a segment which lies entirely within $ F$ and whose ends are on its boundary.
(a) Does there always exist a chord of $F$ that divides its area in half?
(b) Prove that for any $F$ there exists a chord such that the area of each of the two parts of $F$ is not less than $ 1/3$ of the area of $F$.
(c) Can the number $1/3$ in (b) be changed to a greater one?
(V Proizvolov)
MathLinks Contest 7th, 1.1
Given is an acute triangle $ ABC$ and the points $ A_1,B_1,C_1$, that are the feet of its altitudes from $ A,B,C$ respectively. A circle passes through $ A_1$ and $ B_1$ and touches the smaller arc $ AB$ of the circumcircle of $ ABC$ in point $ C_2$. Points $ A_2$ and $ B_2$ are defined analogously.
Prove that the lines $ A_1A_2$, $ B_1B_2$, $ C_1C_2$ have a common point, which lies on the Euler line of $ ABC$.
1995 Mexico National Olympiad, 3
$A, B, C, D$ are consecutive vertices of a regular $7$-gon. $AL$ and $AM$ are tangents to the circle center $C$ radius $CB$. $N$ is the intersection point of $AC$ and $BD$. Show that $L, M, N$ are collinear.
2019 AIME Problems, 7
Triangle $ABC$ has side lengths $AB=120$, $BC=220$, and $AC=180$. Lines $\ell_{A}$, $\ell_{B}$, and $\ell_{C}$ are drawn parallel to $\overline{BC}$, $\overline{AC}$, and $\overline{AB}$, respectively, such that the intersection of $\ell_{A}$, $\ell_{B}$, and $\ell_{C}$ with the interior of $\triangle ABC$ are segments of length $55$, $45$, and $15$, respectively. Find the perimeter of the triangle whose sides lie on $\ell_{A}$, $\ell_{B}$, and $\ell_{C}$.
2020 Purple Comet Problems, 4
The gure below shows a large circle with area $120$ containing a circle with half of the radius of the large circle and six circles with a quarter of the radius of the large circle. Find the area of the shaded region.
[img]https://cdn.artofproblemsolving.com/attachments/7/9/064a05feb9bd67896c079a5141bf7556d7165b.png[/img]
Ukrainian TYM Qualifying - geometry, IV.7
Let $ABCD$ be the quadrilateral whose area is the largest among the quadrilaterals with given sides $a, b, c, d$, and let $PORS$ be the quadrilateral inscribed in $ABCD$ with the smallest perimeter. Find this perimeter.
2010 Contests, 1a
The point $P$ lies on the edge $AB$ of a quadrilateral $ABCD$.
The angles $BAD, ABC$ and $CPD$ are right, and $AB = BC + AD$.
Show that $BC = BP$ or $AD = BP$.
I Soros Olympiad 1994-95 (Rus + Ukr), 11.2
Given a rectangle $ABCD$ with $AB> BC$. On the side $CD$, take a point $L$ such that $BL$ and $AC$ are perpendicular. Let $K$ be the intersection point of segments $BL$ and $AC$. It is known that segments $AL$. and $DK$ are perpendicular. Find $\angle ACB.$
1980 IMO, 11
A triangle $(ABC)$ and a point $D$ in its plane satisfy the relations
\[\frac{BC}{AD}=\frac{CA}{BD}=\frac{AB}{CD}=\sqrt{3}.\]
Prove that $(ABC)$ is equilateral and $D$ is its center.
2012 Indonesia TST, 3
Given a cyclic quadrilateral $ABCD$ with the circumcenter $O$, with $BC$ and $AD$ not parallel. Let $P$ be the intersection of $AC$ and $BD$. Let $E$ be the intersection of the rays $AB$ and $DC$. Let $I$ be the incenter of $EBC$ and the incircle of $EBC$ touches $BC$ at $T_1$. Let $J$ be the excenter of $EAD$ that touches $AD$ and the excircle of $EAD$ that touches $AD$ touches $AD$ at $T_2$. Let $Q$ be the intersection between $IT_1$ and $JT_2$. Prove that $O,P,Q$ are collinear.
2003 Romania National Olympiad, 1
Find the locus of the points $ M $ that are situated on the plane where a rhombus $ ABCD $ lies, and satisfy:
$$ MA\cdot MC+MB\cdot MD=AB^2 $$
[i]Ovidiu Pop[/i]
2013 Costa Rica - Final Round, G3
Let $ABCD$ be a rectangle with center $O$ such that $\angle DAC = 60^o$. Bisector of $\angle DAC$ cuts a $DC$ at $S$, $OS$ and $AD$ intersect at $L$, $BL$ and $AC$ intersect at $M$. Prove that $SM \parallel CL$.
2024 Brazil National Olympiad, 3
Let \( n \geq 3 \) be a positive integer. In a convex polygon with \( n \) sides, all the internal bisectors of its \( n \) internal angles are drawn. Determine, as a function of \( n \), the smallest possible number of distinct lines determined by these bisectors.
2007 District Olympiad, 2
All $ 2n\ge 2 $ squares of a $ 2\times n $ rectangle are colored with three colors. We say that a color has a [i]cut[/i] if there is some column (from all $ n $) that has both squares colored with it. Determine:
[b]a)[/b] the number of colorings that have no cuts.
[b]b)[/b] the number of colorings that have a single cut.
2008 Tuymaada Olympiad, 6
Let $ ABCD$ be an isosceles trapezoid with $ AD \parallel BC$. Its diagonals $ AC$ and $ BD$ intersect at point $ M$. Points $ X$ and $ Y$ on the segment $ AB$ are such that $ AX \equal{} AM$, $ BY \equal{} BM$. Let $ Z$ be the midpoint of $ XY$ and $ N$ is the point of intersection of the segments $ XD$ and $ YC$. Prove that the line $ ZN$ is parallel to the bases of the trapezoid.
[i]Author: A. Akopyan, A. Myakishev[/i]
1985 IMO Longlists, 9
A polyhedron has $12$ faces and is such that:
[b][i](i)[/i][/b] all faces are isosceles triangles,
[b][i](ii)[/i][/b] all edges have length either $x$ or $y$,
[b][i](iii)[/i][/b] at each vertex either $3$ or $6$ edges meet, and
[b][i](iv)[/i][/b] all dihedral angles are equal.
Find the ratio $x/y.$
2023 Junior Balkan Team Selection Tests - Moldova, 9
Let $ AD $, $ BE $ and $ CF $ be the altitudes of $ \Delta ABC $. The points $ P, \, \, Q, \, \, R $ and $ S $ are the feet of the perpendiculars drawn from the point $ D $ on the segments $ BA $, $ BE $, $ CF $ and $ CA $, respectively. Prove that the points $ P, \, \, Q, \, \, R $ and $ S $ are collinear.
2009 Costa Rica - Final Round, 6
Let $ \Delta ABC$ with incircle $ \Gamma$, let $ D, E$ and $ F$ the tangency points of $ \Gamma$ with sides $ BC, AC$ and $ AB$, respectively and let $ P$ the intersection point of $ AD$ with $ \Gamma$.
$ a)$ Prove that $ BC, EF$ and the straight line tangent to $ \Gamma$ for $ P$ concur at a point $ A'$.
$ b)$ Define $ B'$ and $ C'$ in an anologous way than $ A'$. Prove that $ A'\minus{}B'\minus{}C'$
2022 Belarusian National Olympiad, 11.6
The incircle of a right-angled triangle $ABC$ touches hypotenus $AB$ at $P$, $BC$ and $AC$ at $R$ and $Q$ respectively. $C_1$ and $C_2$ are reflections of $C$ in $PQ$ and $PR$.
Find the angle $C_1IC_2$, where $I$ is the incenter of $ABC$.
2011 District Olympiad, 3
Let $ABCA'B'C'$ a right triangular prism with the bases equilateral triangles. A plane $\alpha$ containing point $A$ intersects the rays $BB'$ and $CC'$ at points E and $F$, so that $S_ {ABE} + S_{ACF} = S_{AEF}$. Determine the measure of the angle formed by the plane $(AEF)$ with the plane $(BCC')$.
2013 Lusophon Mathematical Olympiad, 6
Consider a triangle $ABC$. Let $S$ be a circumference in the interior of the triangle that is tangent to the sides $BC$, $CA$, $AB$ at the points $D$, $E$, $F$ respectively. In the exterior of the triangle we draw three circumferences $S_A$, $S_B$, $S_C$. The circumference $S_A$ is tangent to $BC$ at $L$ and to the prolongation of the lines $AB$, $AC$ at the points $M$, $N$ respectively. The circumference $S_B$ is tangent to $AC$ at $E$ and to the prolongation of the line $BC$ at $P$. The circumference $S_C$ is tangent to $AB$ at $F$ and to the prolongation of the line $BC$ at $Q$. Show that the lines $EP$, $FQ$ and $AL$ meet at a point of the circumference $S$.
1898 Eotvos Mathematical Competition, 3
Let $A, B, C, D$ be four given points on a straight line $e$. Construct a square such that two of its parallel sides (or their extensions) go through $A$ and $B$ respectively, and the other two sides (and their extensions) go through $C$ and $D$ respectively.
2020 Turkey Team Selection Test, 6
In a triangle $\triangle ABC$, $D$ and $E$ are respectively on $AB$ and $AC$ such that $DE\parallel BC$. $P$ is the intersection of $BE$ and $CD$. $M$ is the second intersection of $(APD)$ and $(BCD)$ , $N$ is the second intersection of $(APE)$ and $(BCE)$. $w$ is the circle passing through $M$ and $N$ and tangent to $BC$. Prove that the lines tangent to $w$ at $M$ and $N$ intersect on $AP$.
1998 AMC 12/AHSME, 18
A right circular cone of volume $ A$, a right circular cylinder of volume $ M$, and a sphere of volume $ C$ all have the same radius, and the common height of the cone and the cylinder is equal to the diameter of the sphere. Then
$ \textbf{(A)}\ A \minus{} M \plus{} C \equal{} 0 \qquad \textbf{(B)}\ A \plus{} M \equal{} C \qquad \textbf{(C)}\ 2A \equal{} M \plus{} C$
$ \textbf{(D)}\ A^2 \minus{} M^2 \plus{} C^2 \equal{} 0 \qquad \textbf{(E)}\ 2A \plus{} 2M \equal{} 3C$
2009 Junior Balkan Team Selection Tests - Moldova, 3
Let $ABC$ be a triangle with $\angle BCA=20.$ Let points $D\in(BC), F\in(AC)$ be such that $CD=DF=FB=BA.$ Find $\angle ADF.$