Found problems: 25757
2019 Sharygin Geometry Olympiad, 5
Let $A, B, C$ and $D$ be four points in general position, and $\omega$ be a circle passing through $B$ and $C$. A point $P$ moves along $\omega$. Let $Q$ be the common point of circles $\odot (ABP)$ and $\odot (PCD)$ distinct from $P$. Find the locus of points $Q$.
2017 Yasinsky Geometry Olympiad, 4
Diagonals of trapezium $ABCD$ are mutually perpendicular and the midline of the trapezium is $5$. Find the length of the segment that connects the midpoints of the bases of the trapezium.
2007 Iran MO (3rd Round), 7
A ring is the area between two circles with the same center, and width of a ring is the difference between the radii of two circles.
[img]http://i18.tinypic.com/6cdmvi8.png[/img]
a) Can we put uncountable disjoint rings of width 1(not necessarily same) in the space such that each two of them can not be separated.
[img]http://i19.tinypic.com/4qgx30j.png[/img]
b) What's the answer if 1 is replaced with 0?
2024 Regional Olympiad of Mexico West, 4
Let $\triangle ABC$ be a triangle and $\omega$ its circumcircle. The tangent to $\omega$ through $B$ cuts the parallel to $BC$ through $A$ at $P$. The line $CP$ cuts the circumcircle of $\triangle ABP$ again in $Q$ and line $AQ$ cuts $\omega$ at $R$. Prove that $BQCR$ is parallelogram if and only if $AC=BC$.
2021 Science ON Juniors, 3
Circles $\omega_1$ and $\omega_2$ are externally tangent to each other at $P$. A random line $\ell$ cuts $\omega_1$ at $A$ and $C$ and $\omega_2$ at $B$ and $D$ (points $A,C,B,D$ are in this order on $\ell$). Line $AP$ meets $\omega_2$ again at $E$ and line $BP$ meets $\omega_1$ again at $F$. Prove that the radical axis of circles $(PCD)$ and $(PEF)$ is parallel to $\ell$.
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[i](Vlad Robu)[/i]
2008 Singapore Team Selection Test, 1
In triangle $ABC$, $D$ is a point on $AB$ and $E$ is a point on $AC$ such that $BE$ and $CD$ are bisectors of $\angle B$ and $\angle C$ respectively. Let $Q,M$ and $N$ be the feet of perpendiculars from the midpoint $P$ of $DE$ onto $BC,AB$ and $AC$, respectively. Prove that $PQ=PM+PN$.
2019 BMT Spring, 14
A regular hexagon has positive integer side length. A laser is emitted from one of the hexagon’s corners, and is reflected off the edges of the hexagon until it hits another corner. Let $a$ be the distance that the laser travels. What is the smallest possible value of $a^2$ such that $a > 2019$?
You need not simplify/compute exponents.
2024 Kurschak Competition, 1
The quadrilateral $ABCD$ is divided into cyclic quadrilaterals with pairwise disjoint interiors. None of the vertices of the cyclic quadrilaterals in the decomposition is an interior point of a side of any cyclic quadrilateral in the decomposition or of a side of the quadrilateral $ABCD$. Prove that $ABCD$ is also a cyclic quadrilateral.
IV Soros Olympiad 1997 - 98 (Russia), 9.6
A chord is drawn through the intersection point of the diagonals of an inscribed quadrilateral. It is known that the parts of this chord located outside the quadrilateral have lengths equal to $\frac13$ and $\frac14$ of this chord. In what ratio is this chord divided by the intersection point of the diagonals of the quadrilateral?
1967 IMO Shortlist, 1
Prove that a tetrahedron with just one edge length greater than $1$ has volume at most $ \frac{1}{8}.$
2018 IFYM, Sozopol, 7
On the sides $AC$ and $AB$ of an acute $\triangle ABC$ are chosen points $M$ and $N$ respectively. Point $P$ is an intersection point of the segments $BM$ and $CN$ and point $Q$ is an inner point for the quadrilateral $ANPM$, for which $\angle BQC = 90^\circ$ and $\angle BQP = \angle BMQ$. If the quadrilateral $ANPM$ is inscribed in a circle, prove that $\angle QNC = \angle PQC$.
2011 India Regional Mathematical Olympiad, 1
Let $ABC$ be an acute angled scalene triangle with circumcentre $O$ and orthocentre $H.$ If $M$ is the midpoint of $BC,$ then show that $AO$ and $HM$ intersect on the circumcircle of $ABC.$
Durer Math Competition CD 1st Round - geometry, 2008.C3
Given the squares $ABCD$ and $DEFG$, whose only common point is $D$. Let the midpoints of segments $AG$, $GE$, $EC$, and $CA$ be $H, I, J$, and $K$ respectively . Prove that $HIJK$ is a square.
[img]https://cdn.artofproblemsolving.com/attachments/f/d/c3313e5bbf581977a74ea2b114d14950e38605.png[/img]
2023 Tuymaada Olympiad, 6
In the plane $n$ segments with lengths $a_1, a_2, \dots , a_n$ are drawn. Every ray beginning at the point $O$ meets at least one of the segments. Let $h_i$ be the distance from $O$ to the $i$-th segment (not the line!) Prove the inequality
\[\frac{a_1}{h_1}+\frac{a_2}{h_2} + \ldots + \frac{a_i}{h_i} \geqslant 2 \pi.\]
2006 AMC 10, 7
The $ 8\times 18$ rectangle $ ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $ y$?
[asy] unitsize(2mm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=4; draw((0,4)--(18,4)--(18,-4)--(0,-4)--cycle); draw((6,4)--(6,0)--(12,0)--(12,-4)); label("$D$",(0,4),NW); label("$C$",(18,4),NE); label("$B$",(18,-4),SE); label("$A$",(0,-4),SW); label("$y$",(9,1)); [/asy]$ \textbf{(A) } 6\qquad \textbf{(B) } 7\qquad \textbf{(C) } 8\qquad \textbf{(D) } 9\qquad \textbf{(E) } 10$
2002 Austrian-Polish Competition, 2
Let $P_{1}P_{2}\dots P_{2n}$ be a convex polygon with an even number of corners. Prove that there exists a diagonal $P_{i}P_{j}$ which is not parallel to any side of the polygon.
2020 CHMMC Winter (2020-21), 1
Triangle $ABC$ has circumcircle $\Omega$. Chord $XY$ of $\Omega$ intersects segment $AC$ at point $E$ and segment $AB$ at point $F$ such that $E$ lies between $X$ and $F$. Suppose that $A$ bisects arc $\widehat{XY}$. Given that $EC = 7, FB = 10, AF = 8$, and $YF - XE = 2$, find the perimeter of triangle $ABC$.
1974 Dutch Mathematical Olympiad, 1
A convex quadrilateral with area $1$ is divided into four quadrilaterals divided by connecting the midpoints of the opposite sides. Prove that each of those four quadrilaterals has area $< \frac38$.
2004 Baltic Way, 6
A positive integer is written on each of the six faces of a cube. For each vertex of the cube we compute the product of the numbers on the three adjacent faces. The sum of these products is $1001$. What is the sum of the six numbers on the faces?
2006 China Team Selection Test, 1
The centre of the circumcircle of quadrilateral $ABCD$ is $O$ and $O$ is not on any of the sides of $ABCD$. $P=AC \cap BD$. The circumecentres of $\triangle{OAB}$, $\triangle{OBC}$, $\triangle{OCD}$ and $\triangle{ODA}$ are $O_1$, $O_2$, $O_3$ and $O_4$ respectively.
Prove that $O_1O_3$, $O_2O_4$ and $OP$ are concurrent.
2009 Balkan MO Shortlist, G1
In the triangle $ABC, \angle BAC$ is acute, the angle bisector of $\angle BAC$ meets $BC$ at $D, K$ is the foot of the perpendicular from $B$ to $AC$, and $\angle ADB = 45^o$. Point $P$ lies between $K$ and $C$ such that $\angle KDP = 30^o$. Point $Q$ lies on the ray $DP$ such that $DQ = DK$. The perpendicular at $P$ to $AC$ meets $KD$ at $L$. Prove that $PL^2 = DQ \cdot PQ$.
1974 Chisinau City MO, 78
Each point of the sphere of radius $r\ge 1$ is colored in one of $n$ colors ($n \ge 2$), and for each color there is a point on the sphere colored in this color. Prove that there are points $A_i$, $B_i$, $i= 1, ..., n$ on the sphere such that the colors of the points $A_1, ..., A_n$ are pairwise different and the color of the point $B_i$ at a distance of $1$ from $A_i$ is different from the color of the point $A_1, i= 1, ..., n$
1997 Hungary-Israel Binational, 3
Can a closed disk can be decomposed into a union of two congruent parts having no common point?
2019 Baltic Way, 15
Let $n \geq 4$, and consider a (not necessarily convex) polygon $P_1P_2\hdots P_n$ in the plane. Suppose that, for each $P_k$, there is a unique vertex $Q_k\ne P_k$ among $P_1,\hdots, P_n$ that lies closest to it. The polygon is then said to be [i]hostile[/i] if $Q_k\ne P_{k\pm 1}$ for all $k$ (where $P_0 = P_n$, $P_{n+1} = P_1$).
(a) Prove that no hostile polygon is convex.
(b) Find all $n \geq 4$ for which there exists a hostile $n$-gon.
2004 Germany Team Selection Test, 2
Let $n \geq 5$ be a given integer. Determine the greatest integer $k$ for which there exists a polygon with $n$ vertices (convex or not, with non-selfintersecting boundary) having $k$ internal right angles.
[i]Proposed by Juozas Juvencijus Macys, Lithuania[/i]