Found problems: 25757
2022 Durer Math Competition Finals, 11
In rectangle $ABCD$, diagonal $AC$ is met by the angle bisector from $B$ at $B'$ and the angle bisector from $D$ at $D'$. Diagonal $BD$ is met by the angle bisector from $A$ at $A'$ and the angle bisector from $C$ at $C'$. The area of quadrilateral $A'B'C'D'$ is $\frac{9}{16}$ the area of rectangle $ABCD$. What is the ratio of the longer side and shorter side of rectangle $ABCD$?
2014 Contests, 4
Let $n$ be a positive integer. A 4-by-$n$ rectangle is divided into $4n$ unit squares in the usual way. Each unit square is colored black or white. Suppose that every white unit square shares an edge with at least one black unit square. Prove that there are at least $n$ black unit squares.
2023 ELMO Shortlist, G8
Convex quadrilaterals \(ABCD\), \(A_1B_1C_1D_1\), and \(A_2B_2C_2D_2\) are similar with vertices in order. Points \(A\), \(A_1\), \(B_2\), \(B\) are collinear in order, points \(B\), \(B_1\), \(C_2\), \(C\) are collinear in order, points \(C\), \(C_1\), \(D_2\), \(D\) are collinear in order, and points \(D\), \(D_1\), \(A_2\), \(A\) are collinear in order. Diagonals \(AC\) and \(BD\) intersect at \(P\), diagonals \(A_1C_1\) and \(B_1D_1\) intersect at \(P_1\), and diagonals \(A_2C_2\) and \(B_2D_2\) intersect at \(P_2\). Prove that points \(P\), \(P_1\), and \(P_2\) are collinear.
[i]Proposed by Holden Mui[/i]
2009 Flanders Math Olympiad, 3
Consider a line segment $[AB]$ with midpoint $M$ and perpendicular bisector $m$. For each point$ X \ne M$ on m consider we are the intersection point $Y$ of the line $BX$ with the bisector from the angle $\angle BAX$. As $X$ approaches $M$, then approaches $Y$ to a point of $[AB]$. Which?
[img]https://cdn.artofproblemsolving.com/attachments/a/3/17d72a23011a9ec22deb20184717cc9c020a2b.png[/img]
[hide=original wording]Beschouw een lijnstuk [AB] met midden M en middelloodlijn m. Voor elk punt X 6= M op m beschouwenwe het snijpunt Y van de rechte BX met de bissectrice van de hoek < BAX . Als X tot M nadert, dan nadert
Y tot een punt van [AB]. Welk? [/hide]
2009 Thailand Mathematical Olympiad, 9
In triangle $\vartriangle ABC, D$ and $E$ are midpoints of the sides $BC$ and $AC$, respectively. Lines $AD$ and $BE$ are drawn intersecting at $P$. It turns out that $\angle CAD = 15^o$ and $\angle APB = 60^o$. What is the value of $AB/BC$ ?
1999 Kurschak Competition, 2
Given a triangle on the plane, construct inside the triangle the point $P$ for which the centroid of the triangle formed by the three projections of $P$ onto the sides of the triangle happens to be $P$.
2024 Silk Road, 2
Let $ABCD$ be a non-isosceles trapezoid with $AB \parallel CD$. A circle through $A$ and $B$ meets $AD$, $BC$ at $E, F$. The segments $AF, BE$ meet at $G$. The circumcircles of $\triangle ADG$ and $\triangle BCG$ meet at $H$. Show that if $GD=GC$, $H$ is the orthocenter of $\triangle ABG$.
2006 China Team Selection Test, 1
$H$ is the orthocentre of $\triangle{ABC}$. $D$, $E$, $F$ are on the circumcircle of $\triangle{ABC}$ such that $AD \parallel BE \parallel CF$. $S$, $T$, $U$ are the semetrical points of $D$, $E$, $F$ with respect to $BC$, $CA$, $AB$. Show that $S, T, U, H$ lie on the same circle.
2004 Harvard-MIT Mathematics Tournament, 8
If $x$ and $y$ are real numbers with $(x+y)^4=x-y$, what is the maximum possible value of $y$?
2012 Argentina National Olympiad Level 2, 3
Let $ABC$ be a triangle with $\angle A= 105^\circ$ and $\angle B= 45^\circ$. Let $L$ be a point on side $BC$ such that $AL$ is the bisector of angle $\angle BAC$ and let $M$ be the midpoint of side $AC$. Suppose that lines $AL$ and $BM$ intersect at point $P$. Calculate the ratio $\dfrac{AP}{AL}$.
2018 Pan African, 4
Given a triangle $ABC$, let $D$ be the intersection of the line through $A$ perpendicular to $AB$, and the line through $B$ perpendicular to $BC$. Let $P$ be a point inside the triangle. Show that $DAPB$ is cyclic if and only if $\angle BAP = \angle CBP$.
2019 ELMO Shortlist, G6
Let $ABC$ be an acute scalene triangle and let $P$ be a point in the plane. For any point $Q\neq A,B,C$, define $T_A$ to be the unique point such that $\triangle T_ABP \sim \triangle T_AQC$ and $\triangle T_ABP, \triangle T_AQC$ are oriented in the same direction (clockwise or counterclockwise). Similarly define $T_B, T_C$.
a) Find all $P$ such that there exists a point $Q$ with $T_A,T_B,T_C$ all lying on the circumcircle of $\triangle ABC$. Call such a pair $(P,Q)$ a [i]tasty pair[/i] with respect to $\triangle ABC$.
b) Keeping the notations from a), determine if there exists a tasty pair which is also tasty with respect to $\triangle T_AT_BT_C$.
[i]Proposed by Vincent Huang[/i]
1975 All Soviet Union Mathematical Olympiad, 209
Denote the midpoints of the convex hexagon $A_1A_2A_3A_4A_5A_6$ diagonals $A_6A_2$, $A_1A_3$, $A_2A_4$, $A_3A_5$, $A_4A_6$, $A_5A_1$ as $B_1, B_2, B_3, B_4, B_5, B_6$ respectively. Prove that if the hexagon $B_1B_2B_3B_4B_5B_6$ is convex, than its area equals to the quarter of the initial hexagon.
2018 BMT Spring, 6
A triangle $T$ has all integer side lengths and at most one of its side lengths is greater than ten. What is the largest possible area of $T$ ?
1957 Moscow Mathematical Olympiad, 371
Given quadrilateral $ABCD$ and the directions of its sides. Inscribe a rectangle in $ABCD$.
2019 Iranian Geometry Olympiad, 3
Three circles $\omega_1$, $\omega_2$ and $\omega_3$ pass through one common point, say $P$. The tangent line to $\omega_1$ at $P$ intersects $\omega_2$ and $\omega_3$ for the second time at points $P_{1,2}$ and $P_{1,3}$, respectively. Points $P_{2,1}$, $P_{2,3}$, $P_{3,1}$ and $P_{3,2}$ are similarly defined. Prove that the perpendicular bisector of segments $P_{1,2}P_{1,3}$, $P_{2,1}P_{2,3}$ and $P_{3,1}P_{3,2}$ are concurrent.
[i]Proposed by Mahdi Etesamifard[/i]
2016 Puerto Rico Team Selection Test, 3
Let $ABCD$ be a cyclic quadrilateral. Let $ P$ be the intersection of the lines $BC$ and $AD$. Line $AC$ cuts the circumscribed circle of the triangle $BDP$ in $S$ and $T$, with $S$ between $ A$ and $C$. The line $BD$ intersects the circumscribed circle of the triangle $ACP$ in $U$ and $V$, with $U$ between $ B$ and $D$. Prove that $PS = PT = PU = PV$.
2012 Math Prize For Girls Problems, 16
Say that a complex number $z$ is [i]three-presentable[/i] if there is a complex number $w$ of absolute value $3$ such that $z = w - \frac{1}{w}$. Let $T$ be the set of all three-presentable complex numbers. The set $T$ forms a closed curve in the complex plane. What is the area inside $T$?
VI Soros Olympiad 1999 - 2000 (Russia), 10.5
Two different points $A$ and $B$ have been marked on the circle $\omega$. We consider all points $X$ of the circle $\omega$, different from $A$ and $B$. Let $Y$ be the middpoint of the chord $AX$ and $Z$ be the projection of point $A$ on the line $BX$. Prove that all straight lines $YZ$ pass through a certain fixed point that does not depend on the choice of point $X$.
2021 Princeton University Math Competition, 15
Let $\vartriangle ABC$ be an acute triangle with angles $\angle BAC = 70^o$, $\angle ABC = 60^o$, let $D, E$ be the feet of perpendiculars from $B, C$ to $AC$, $AB$, respectively, and let $H$ be the orthocenter of $ABC$. Let $F$ be a point on the shorter arc $AB$ of circumcircle of $ABC$ satisfying $\angle F AB = 10^o$ and let $G$ be the foot of perpendicular from $H$ to $AF$. If $I = BF \cap EG$ and $J = CF \cap DG$, compute the angle $\angle GIJ$.
2014 Sharygin Geometry Olympiad, 8
Let $M$ be the midpoint of the chord $AB$ of a circle centered at $O$. Point $K$ is symmetric to $M$ with respect to $O$, and point $P$ is chosen arbitrarily on the circle. Let $Q$ be the intersection of the line perpendicular to $AB$ through $A$ and the line perpendicular to $PK$ through $P$. Let $H$ be the projection of $P$ onto $AB$. Prove that $QB$ bisects $PH$.
(Tran Quang Hung)
2012 Sharygin Geometry Olympiad, 2
In a triangle $ABC$ the bisectors $BB'$ and $CC'$ are drawn. After that, the whole picture except the points $A, B'$, and $C'$ is erased. Restore the triangle using a compass and a ruler.
(A.Karlyuchenko)
2001 Stanford Mathematics Tournament, 7
The median to a 10 cm side of a triangle has length 9 cm and is perpendicular to a second median of the triangle. Find the exact value in centimeters of the length of the third median.
2001 Federal Math Competition of S&M, Problem 4
Parallelogram $ABCD$ is the base of a pyramid $SABCD$. Planes determined by triangles $ASC$ and $BSD$ are mutually perpendicular. Find the area of the side $ASD$, if areas of sides $ASB,BSC$ and $CSD$ are equal to $x,y$ and $z$, respectively.
1980 IMO Shortlist, 8
Three points $A,B,C$ are such that $B \in ]AC[$. On the side of $AC$ we draw the three semicircles with diameters $[AB], [BC]$ and $[AC]$. The common interior tangent at $B$ to the first two semi-circles meets the third circle in $E$. Let $U$ and $V$ be the points of contact of the common exterior tangent to the first two semi-circles. Denote the area of the triangle $ABC$ as $S(ABC)$. Evaluate the ratio $R=\frac{S(EUV)}{S(EAC)}$ as a function of $r_1 = \frac{AB}{2}$ and $r_2 = \frac{BC}{2}$.