Found problems: 25757
2022 MMATHS, 2
Triangle $ABC$ has $AB = 3$, $BC = 4$, and $CA = 5$. Points $D$, $E$, $F$, $G$, $H$, and $I$ are the reflections of $A$ over $B$, $B$ over $A$, $B$ over $C$, $C$ over $B$, $C$ over $A$, and $A$ over $C$, respectively. Find the area of hexagon $EFIDGH$.
2023 AIME, 12
Let $\triangle ABC$ be an equilateral triangle with side length $55$. Points $D$, $E$, and $F$ lie on sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$, respectively, with $BD=7$, $CE=30$, and $AF=40$. A unique point $P$ inside $\triangle ABC$ has the property that \[\measuredangle AEP=\measuredangle BFP=\measuredangle CDP.\] Find $\tan^{2}\left(\measuredangle AEP\right)$.
Kyiv City MO 1984-93 - geometry, 1991.10.2
In an acute-angled triangle $ABC$ on the sides $AB$, $BC$, $AC$, the points $C_1$, $A_1$, and $B_1$ are marked such that the segments $AA_1$, $BB_1$, $CC_1$ intersect at some point $O$ and the angles $AA_1C$, $BB_1A$, $CC_1B$ are equal. Prove that $AA_1$, $BB_1$, and $CC_1$ are the altitudes of the triangle.
2021 Sharygin Geometry Olympiad, 9.2
A cyclic pentagon is given. Prove that the ratio of its area to the sum of the diagonals is not greater than the quarter of the circumradius.
1999 Harvard-MIT Mathematics Tournament, 2
A ladder is leaning against a house with its lower end $15$ feet from the house. When the lower end is pulled $9$ feet farther from the house, the upper end slides $13$ feet down. How long is the ladder (in feet)?
2004 Korea - Final Round, 1
An isosceles triangle with $AB=AC$ has an inscribed circle $O$, which touches its sides $BC,CA,AB$ at $K,L,M$ respectively. The lines $OL$ and $KM$ intersect at $N$; the lines $BN$ and $CA$ intersect at $Q$. Let $P$ be the foot of the perpendicular from $A$ on $BQ$. Suppose that $BP=AP+2\cdot PQ$. Then, what values can the ratio $\frac{AB}{BC}$ assume?
2017 HMNT, 4
Triangle $ABC$ has $AB=10$, $BC=17$, and $CA=21$. Point $P$ lies on the circle with diameter $AB$. What is the greatest possible area of $APC$?
2012 China Team Selection Test, 2
Given a scalene triangle $ABC$. Its incircle touches $BC,AC,AB$ at $D,E,F$ respectvely. Let $L,M,N$ be the symmetric points of $D$ with $EF$,of $E$ with $FD$,of $F$ with $DE$,respectively. Line $AL$ intersects $BC$ at $P$,line $BM$ intersects $CA$ at $Q$,line $CN$ intersects $AB$ at $R$. Prove that $P,Q,R$ are collinear.
1952 Miklós Schweitzer, 2
Is it possible to find three conics in the plane such that any straight line in the plane intersects at least two of the conics and through any point of the plane pass tangents to at least two of them?
2013 HMIC, 3
Triangle $ABC$ is inscribed in a circle $\omega$ such that $\angle A = 60^o$ and $\angle B = 75^o$. Let the bisector of angle $A$ meet $BC$ and $\omega$ at $E$ and $D$, respectively. Let the reflections of $A$ across $D$ and $C$ be $D'$ and $C'$ , respectively. If the tangent to $\omega$ at $A$ meets line $BC$ at $P$, and the circumcircle of $APD'$ meets line $AC$ at $F \ne A$, prove that the circumcircle of $C'FE$ is tangent to $BC$ at $E$.
2005 Denmark MO - Mohr Contest, 1
This figure is cut out from a sheet of paper. Folding the sides upwards along the dashed lines, one gets a (non-equilateral) pyramid with a square base. Calculate the area of the base.
[img]https://1.bp.blogspot.com/-lPpfHqfMMRY/XzcBIiF-n2I/AAAAAAAAMW8/nPs_mLe5C8srcxNz45Wg-_SqHlRAsAmigCLcBGAsYHQ/s0/2005%2BMohr%2Bp1.png[/img]
2021 Dutch BxMO TST, 1
Given is a cyclic quadrilateral $ABCD$ with $|AB| = |BC|$. Point $E$ is on the arc $CD$ where $A$ and $B$ are not on. Let $P$ be the intersection point of $BE$ and $CD$ , let $Q$ be the intersection point of $AE$ and $BD$ . Prove that $PQ \parallel AC$.
2017 Federal Competition For Advanced Students, 2
Let $ABCDE$ be a regular pentagon with center $M$. A point $P$ (different from $M$) is chosen on the line
segment $MD$. The circumcircle of $ABP$ intersects the line segment $AE$ in $A$ and $Q$ and
the line through $P$ perpendicular to $CD$ in $P$ and $R$.
Prove that $AR$ and $QR$ have same length.
[i]proposed by Stephan Wagner[/i]
2011 Math Prize For Girls Problems, 7
If $z$ is a complex number such that
\[
z + z^{-1} = \sqrt{3},
\]
what is the value of
\[
z^{2010} + z^{-2010} \, ?
\]
1999 Vietnam Team Selection Test, 3
Let a convex polygon $H$ be given. Show that for every real number $a \in (0, 1)$ there exist 6 distinct points on the sides of $H$, denoted by $A_1, A_2, \ldots, A_6$ clockwise, satisfying the conditions:
[b]I.[/b] $(A_1A_2) = (A_5A_4) = a \cdot (A_6A_3)$.
[b]II.[/b] Lines $A_1A_2, A_5A_4$ are equidistant from $A_6A_3$.
(By $(AB)$ we denote vector $AB$)
2023 ELMO Shortlist, G5
Let \(ABC\) be an acute triangle with circumcircle \(\omega\). Let \(P\) be a variable point on the arc \(BC\) of \(\omega\) not containing \(A\). Squares \(BPDE\) and \(PCFG\) are constructed such that \(A\), \(D\), \(E\) lie on the same side of line \(BP\) and \(A\), \(F\), \(G\) lie on the same side of line \(CP\). Let \(H\) be the intersection of lines \(DE\) and \(FG\). Show that as \(P\) varies, \(H\) lies on a fixed circle.
[i]Proposed by Karthik Vedula[/i]
2017 239 Open Mathematical Olympiad, 2
Inside the circle $\omega$ through points $A, B$ point $C$ is chosen. An arbitrary point $X$ is selected on the segment $BC$. The ray $AX$ cuts the circle in $Y$. Prove that all circles $CXY$ pass through a two fixed points that is they intersect and are coaxial, independent of the position of $X$.
1971 IMO Longlists, 1
The points $S(i, j)$ with integer Cartesian coordinates $0 < i \leq n, 0 < j \leq m, m \leq n$, form a lattice. Find the number of:
[b](a)[/b] rectangles with vertices on the lattice and sides parallel to the coordinate axes;
[b](b)[/b] squares with vertices on the lattice and sides parallel to the coordinate axes;
[b](c)[/b] squares in total, with vertices on the lattice.
2020-21 KVS IOQM India, 2
If $ABCD$ is a rectangle and $P$ is a point inside it such that $AP=33, BP=16, DP=63$.
Find $CP$.
2010 Gheorghe Vranceanu, 1
Let $ A_1,B_1,C_1 $ be the middlepoints of the sides of a triangle $ ABC $ and let $ A_2,B_2,C_2 $ be on the middle of the paths $ CAB,ABC,BCA, $ respectively. Prove that $ A_1A_2,B_1B_2,C_1C_2 $ are concurrent.
2015 Belarus Team Selection Test, 3
Consider a fixed circle $\Gamma$ with three fixed points $A, B,$ and $C$ on it. Also, let us fix a real number $\lambda \in(0,1)$. For a variable point $P \not\in\{A, B, C\}$ on $\Gamma$, let $M$ be the point on the segment $CP$ such that $CM =\lambda\cdot CP$ . Let $Q$ be the second point of intersection of the circumcircles of the triangles $AMP$ and $BMC$. Prove that as $P$ varies, the point $Q$ lies on a fixed circle.
[i]Proposed by Jack Edward Smith, UK[/i]
2000 Harvard-MIT Mathematics Tournament, 12
At a dance, Abhinav starts from point $(a, 0)$ and moves along the negative $x$ direction with speed $v_a$, while Pei-Hsin starts from $(0,6)$ and glides in the negative $y$-direction with speed $v_b$. What is the distance of closest approach between the two?
2010 Oral Moscow Geometry Olympiad, 5
All edges of a regular right pyramid are equal to $1$, and all vertices lie on the side surface of a (infinite) right circular cylinder of radius $R$. Find all possible values of $R$.
2011 IMAR Test, 2
The area of a convex polygon in the plane is equally shared by the four standard quadrants, and all non-zero lattice points lie outside the polygon. Show that the area of the polygon is less than $4$.
2007 Argentina National Olympiad, 6
Julián chooses $2007$ points of the plane between which there are no $3$ aligned, and draw with red all the segments that join two of those points. Next, Roberto draws several lines. Its objective is for each red segment to be cut inside by (at least) one of the lines. Determine the minor $\ell$ lines such that, no matter how Julián chooses the $2007$ points, with the properly chosen $\ell$ lines, Roberto will achieve his objective with certainty.