Found problems: 25757
Brazil L2 Finals (OBM) - geometry, 2008.3
Let $P$ be a convex pentagon with all sides equal. Prove that if two of the angles of $P$ add to $180^o$, then it is possible to cover the plane with $P$, without overlaps.
2019 BAMO, B
In the figure below, parallelograms $ABCD$ and $BFEC$ have areas $1234$ cm$^2$ and $2804$ cm$^2$, respectively.
Points $M$ and $N$ are chosen on sides $AD$ and $FE$, respectively, so that segment $MN$ passes through $B$.
Find the area of $\vartriangle MNC$.
[img]https://cdn.artofproblemsolving.com/attachments/b/6/8b57b632191bdb3a27ab7c59e2376dab23950b.png[/img]
2024 Al-Khwarizmi IJMO, 1
We have triangle $ABC$ with area $S$. In one step we can move only one vertex at a time so that the area of the triangle during movement remains constant. Prove that we can move this triangle into any other arbitrary triangle $DEF$ with area $S$.
[i]Proposed by Marek Maruin, Slovakia[/i]
1997 IMO Shortlist, 20
A quick solution:
Let R be the foot of the perpend. from X to BC. Let's assume Q and R are in the interior of the segms AC and BC (respectively) and P in the ext of AD. P, R, Q are colinear (Simson's thm). PQ tangent to circle XRD iff XRQ=XDR iff Pi-XCA=XDR iff XBA=XDR=XDC=ADB iff XBC+ABC=ADB=DAC+ACB iff XAC+ABC=DAC+ACD iff ABC=ACD=ACB iff AB=AC. It's the same for all the other cases.
1999 Junior Balkan MO, 4
Let $ABC$ be a triangle with $AB=AC$. Also, let $D\in[BC]$ be a point such that $BC>BD>DC>0$, and let $\mathcal{C}_1,\mathcal{C}_2$ be the circumcircles of the triangles $ABD$ and $ADC$ respectively. Let $BB'$ and $CC'$ be diameters in the two circles, and let $M$ be the midpoint of $B'C'$. Prove that the area of the triangle $MBC$ is constant (i.e. it does not depend on the choice of the point $D$).
[i]Greece[/i]
2017 Estonia Team Selection Test, 4
Let $ABC$ be an isosceles triangle with apex $A$ and altitude $AD$. On $AB$, choose a point $F$ distinct from $B$ such that $CF$ is tangent to the incircle of $ABD$. Suppose that $\vartriangle BCF$ is isosceles. Show that those conditions uniquely determine:
a) which vertex of $BCF$ is its apex,
b) the size of $\angle BAC$
2022 Sharygin Geometry Olympiad, 8.6
Two circles meeting at points $A, B$ and a point $O$ lying outside them are given. Using a compass and a ruler construct a ray with origin $O$ meeting the first circle at point $C$ and the second one at point $D$ in such a way that the ratio $OC : OD$ be maximal.
2017 Purple Comet Problems, 21
The diagram below shows a large circle. Six congruent medium-sized circles are each internally tangent to the large circle and tangent to two neighboring medium-sized circles. Three congruent small circles are mutually tangent to one another and are each tangent to two medium-sized circles as shown. The ratio of the area of the large circle to the area of one of the small circles can be written as $m+\sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.
[img]https://cdn.artofproblemsolving.com/attachments/a/4/fcffd7ee6e8d3da0641525e7a987d13ce05496.png[/img]
2013 Portugal MO, 6
In each side of a regular polygon with $n$ sides, we choose a point different from the vertices and we obtain a new polygon of $n$ sides. For which values of $n$ can we obtain a polygon such that the internal angles are all equal but the polygon isn't regular?
2023 China Second Round, 1
Let $A,B$ be two fixed points on a plane and $\Omega$ a fixed semicircle arc with diameter $AB$. Let $T$ be another fixed point on $\Omega$, and $\omega$ a fixed circle that passes through $A$ and $T$ and has its center in $\Delta ABT$. Let $P$ be a moving point on the arc $TB$ (endpoints excluded), and $C,D$ be two moving points on $\omega$ such that $C$ lies on segment $AP$, $C,D$ lies on different sides of line $AB$ and $CD\ \bot \ AB$. Denote the circumcenter of $\Delta CDP$ of $K$. Prove that
(i) $K$ lies on the circumcircle of $\Delta TDP$.
(ii) $K$ is a fixed point.
2006 Germany Team Selection Test, 2
The lengths of the altitudes of a triangle are positive integers, and the length of the radius of the incircle is a prime number.
Find the lengths of the sides of the triangle.
2013 IFYM, Sozopol, 8
Let $K$ be a point on the angle bisector, such that $\angle BKL=\angle KBL=30^\circ$. The lines $AB$ and $CK$ intersect in point $M$ and lines $AC$ and $BK$ intersect in point $N$. Determine $\angle AMN$.
1967 IMO Shortlist, 3
The $n$ points $P_1,P_2, \ldots, P_n$ are placed inside or on the boundary of a disk of radius 1 in such a way that the minimum distance $D_n$ between any two of these points has its largest possible value $D_n.$ Calculate $D_n$ for $n = 2$ to 7. and justify your answer.
2018 Rioplatense Mathematical Olympiad, Level 3, 2
Let $P$ be a point outside a circumference $\Gamma$, and let $PA$ be one of the tangents from $P$ to $\Gamma$. Line $l$ passes through $P$ and intersects $\Gamma$ at $B$ and $C$, with $B$ between $P$ and $C$. Let $D$ be the symmetric of $B$ with respect to $P$. Let $\omega_1$ and $\omega_2$ be the circles circumscribed to the triangles $DAC$ and $PAB$ respectively. $\omega_1$ and $\omega _2$ intersect at $E \neq A$. Line $EB$ cuts back to $\omega _1 $ in $F$. Prove that $CF = AB$.
2013 AMC 12/AHSME, 24
Let $ABC$ be a triangle where $M$ is the midpoint of $\overline{AC}$, and $\overline{CN}$ is the angle bisector of $\angle ACB$ with $N$ on $\overline{AB}$. Let $X$ be the intersection of the median $\overline{BM}$ and the bisector $\overline{CN}$. In addition $\bigtriangleup BXN$ is equilateral and $AC=2$. What is $BN^2$?
$\textbf{(A)}\ \frac{10-6\sqrt{2}}{7} \qquad\textbf{(B)}\ \frac{2}{9} \qquad\textbf{(C)}\ \frac{5\sqrt{2} - 3\sqrt{3}}{8} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{6} \qquad\textbf{(E)}\ \frac{3\sqrt{3} - 4}{5}$.
2011 Iran MO (2nd Round), 3
The line $l$ intersects the extension of $AB$ in $D$ ($D$ is nearer to $B$ than $A$) and the extension of $AC$ in $E$ ($E$ is nearer to $C$ than $A$) of triangle $ABC$. Suppose that reflection of line $l$ to perpendicular bisector of side $BC$ intersects the mentioned extensions in $D'$ and $E'$ respectively. Prove that if $BD+CE=DE$, then $BD'+CE'=D'E'$.
2005 Tournament of Towns, 4
Two ants crawl along the perimeter of a polygonal table, so that the distance between them is always $10$ cm. Each side of the table is more than $1$ meter long. At the initial moment both ants are on the same side of the table.
(a) [i](2 points)[/i] Suppose that the table is a convex polygon. Is it always true that both ants can visit each point on the perimeter?
(b) [i](4 points)[/i] Is it always true (this time without assumption of convexity) that each point on the perimeter can be visited by at least one ant?
2020-21 KVS IOQM India, 4
Let $ABC$ be an isosceles triangle with $AB=AC$ and incentre $I$. If $AI=3$ and the distance from $I$ to $BC$ is $2$, what is the square of length on $BC$?
Kyiv City MO Seniors 2003+ geometry, 2013.10.4
The two circles ${{w} _ {1}}, \, \, {{w} _ {2}}$ touch externally at the point $Q$. The common external tangent of these circles is tangent to ${{w} _ {1}}$ at the point $B$, $BA$ is the diameter of this circle. A tangent to the circle ${{w} _ {2}} $ is drawn through the point $A$, which touches this circle at the point $C$, such that the points $B$ and $C$ lie in one half-plane relative to the line $AQ$. Prove that the circle ${{w} _ {1}}$ bisects the segment $C $.
(Igor Nagel)
2011 Turkey MO (2nd round), 2
Let $ABC$ be a triangle $D\in[BC]$ (different than $A$ and $B$).$E$ is the midpoint of $[CD]$. $F\in[AC]$ such that $\widehat{FEC}=90$ and $|AF|.|BC|=|AC|.|EC|.$ Circumcircle of $ADC$ intersect $[AB]$ at $G$ different than $A$.Prove that tangent to circumcircle of $AGF$ at $F$ is touch circumcircle of $BGE$ too.
2017 IMO, 4
Let $R$ and $S$ be different points on a circle $\Omega$ such that $RS$ is not a diameter. Let $\ell$ be the tangent line to $\Omega$ at $R$. Point $T$ is such that $S$ is the midpoint of the line segment $RT$. Point $J$ is chosen on the shorter arc $RS$ of $\Omega$ so that the circumcircle $\Gamma$ of triangle $JST$ intersects $\ell$ at two distinct points. Let $A$ be the common point of $\Gamma$ and $\ell$ that is closer to $R$. Line $AJ$ meets $\Omega$ again at $K$. Prove that the line $KT$ is tangent to $\Gamma$.
[i]Proposed by Charles Leytem, Luxembourg[/i]
2021 Poland - Second Round, 5
Find the largest positive integer $n$ with the following property: there are rectangles $A_1, ... , A_n$ and $B_1,... , B_n,$ on the plane , each with sides parallel to the axis of the coordinate system, such that the rectangles $A_i$ and $B_i$ are disjoint for all $i \in \{1,..., n\}$, but the rectangles $A_i$ and $B_j$ have a common point for all $i, j \in \{1,..., n\}$, $i \ne j$.
[i]Note: By points belonging to a rectangle we mean all points lying either in its interior, or on any of its sides, including its vertices [/i]
2020 Purple Comet Problems, 18
In isosceles $\vartriangle ABC, AB = AC, \angle BAC$ is obtuse, and points $E$ and $F$ lie on sides $AB$ and $AC$, respectively, so that $AE = 10, AF = 15$. The area of $\vartriangle AEF$ is $60$, and the area of quadrilateral $BEFC$ is $102$. Find $BC$.
2011 ELMO Shortlist, 5
Given positive reals $x,y,z$ such that $xy+yz+zx=1$, show that
\[\sum_{\text{cyc}}\sqrt{(xy+kx+ky)(xz+kx+kz)}\ge k^2,\]where $k=2+\sqrt{3}$.
[i]Victor Wang.[/i]
2012 AMC 8, 25
A square with area 4 is inscribed in a square with area 5, with one vertex of the smaller square on each side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length $a$, and the other of length $b$. What is the value of $ab$ ?
[asy]
draw((0,2)--(2,2)--(2,0)--(0,0)--cycle);
draw((0,0.3)--(0.3,2)--(2,1.7)--(1.7,0)--cycle);
label("$a$",(-0.1,0.15));
label("$b$",(-0.1,1.15));
[/asy]
$\textbf{(A)}\hspace{.05in}\dfrac15 \qquad \textbf{(B)}\hspace{.05in}\dfrac25 \qquad \textbf{(C)}\hspace{.05in}\dfrac12 \qquad \textbf{(D)}\hspace{.05in}1 \qquad \textbf{(E)}\hspace{.05in}4 $