Found problems: 25757
IV Soros Olympiad 1997 - 98 (Russia), 10.2
Let $M $be the point of intersection of the diagonals of the inscribed quadrilateral $ABCD$. Prove that if $AB = AM,$ then a line passing through $M$ perpendicular to $AD$ passes through the midpoint of the arc $BC$.
2005 Tournament of Towns, 5
Prove that if a regular icosahedron and a regular dodecahedron have a common circumsphere, then they have a common insphere.
[i](7 points)[/i]
2007 ITest, 38
Find the largest positive integer that is equal to the cube of the sum of its digits.
2009 Saint Petersburg Mathematical Olympiad, 3
Streets of Moscow are some circles (rings) with common center $O$ and some straight lines from center $O$ to external ring. Point $A,B$ - two crossroads on external ring. Three friends want to move from $A$ to $B$. Dima goes by external ring, Kostya goes from $A$ to $O$ then to $B$. Sergey says, that there is another way, that is shortest. Prove, that he is wrong.
2021 JHMT HS, 3
Let $ABCDEF$ be a convex hexagon such that $AB=CD=EF=20, \ BC=DE=FA=21,$ and $\angle A=\angle C=\angle E=90^{\circ}.$ The area of $ABCDEF$ can then be expressed in the form $a+\tfrac{b\sqrt{c}}{d},$ where $a,\ b,\ c,$ and $d$ are positive integers, $b$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d.$
2005 Serbia Team Selection Test, 2
A convex angle $xOy$ and a point $M$ inside it are given in the plane. Prove that there is a unique point $P$ in the plane with the following property:
- For any line $l$ through $M$, meeting the rays $x$ and $y$ (or their extensions) at $X$ and $Y$, the angle $XPY$ is not obtuse.
2009 Brazil Team Selection Test, 4
There is given a convex quadrilateral $ ABCD$. Prove that there exists a point $ P$ inside the quadrilateral such that
\[
\angle PAB \plus{} \angle PDC \equal{} \angle PBC \plus{} \angle PAD \equal{} \angle PCD \plus{} \angle PBA \equal{} \angle PDA \plus{} \angle PCB = 90^{\circ}
\]
if and only if the diagonals $ AC$ and $ BD$ are perpendicular.
[i]Proposed by Dusan Djukic, Serbia[/i]
2017 Junior Balkan Team Selection Tests - Moldova, Problem 7
Given is an acute triangle $ABC$ and the median $AM.$ Draw $BH\perp AC.$ The line which goes through $A$ and is perpendicular to $AM$ intersects $BH$ at $E.$ On the opposite ray of the ray $AE$ choose $F$ such that $AE=AF.$ Prove that $CF\perp AB.$
2023 Bulgaria JBMO TST, 3
Let $ABC$ be a non-isosceles triangle with circumcircle $k$, incenter $I$ and $C$-excenter $I_C$. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of arc $\widehat{ACB}$ on $k$. Prove that $\angle IMI_C + \angle INI_C = 180^{\circ}$.
2019 Peru IMO TST, 3
Let $I,\ O$ and $\Gamma$ be the incenter, circumcenter and the circumcircle of triangle $ABC$, respectively. Line $AI$ meets $\Gamma$ at $M$ $(M\neq A)$. The circumference $\omega$ is tangent internally to $\Gamma$ at $T$, and is tangent to the lines $AB$ and $AC$. The tangents through $A$ and $T$ to $\Gamma$ intersect at $P$. Lines $PI$ and $TM$ meet at $Q$. Prove that the lines $QA$ and $MO$ meet at a point on $\Gamma$.
VMEO III 2006, 11.2
Given a triangle $ABC$, incircle $(I)$ touches $BC,CA,AB$ at $D,E,F$ respectively. Let $M$ be a point inside $ABC$. Prove that $M$ lie on $(I)$ if and only if one number among $\sqrt{AE\cdot S_{BMC}},\sqrt{BF\cdot S_{CMA}},\sqrt{CD\cdot S_{AMB}}$ is sum of two remaining numbers ($S_{ABC}$ denotes the area of triangle $ABC$)
2024 AMC 10, 22
Let $\mathcal K$ be the kite formed by joining two right triangles with legs $1$ and $\sqrt3$ along a common hypotenuse. Eight copies of $\mathcal K$ are used to form the polygon shown below. What is the area of triangle $\Delta ABC$?
[img]https://cdn.artofproblemsolving.com/attachments/1/3/03abbd4df2932f4a1d16a34c2b9e15b683dedb.png[/img]
$\textbf{(A) }2+3\sqrt3\qquad\textbf{(B) }\dfrac92\sqrt3\qquad\textbf{(C) }\dfrac{10+8\sqrt3}{3}\qquad\textbf{(D) }8\qquad\textbf{(E) }5\sqrt3$
Champions Tournament Seniors - geometry, 2017.4
Let $AD$ be the bisector of triangle $ABC$. Circle $\omega$ passes through the vertex $A$ and touches the side $BC$ at point $D$. This circle intersects the sides $AC$ and $AB$ for the second time at points $M$ and $N$ respectively. Lines $BM$ and $CN$ intersect the circle for the second time $\omega$ at points $P$ and $Q$, respectively. Lines $AP$ and $AQ$ intersect side $BC$ at points $K$ and $L$, respectively. Prove that $KL=\frac12 BC$
2010 Tournament Of Towns, 7
A square is divided into congruent rectangles with sides of integer lengths. A rectangle is important if it has at least one point in common with a given diagonal of the square. Prove that this diagonal bisects the total area of the important rectangles
1991 IberoAmerican, 1
Each vertex of a cube is assigned an 1 or a -1, and each face is assigned the product of the numbers assigned to its vertices. Determine the possible values the sum of these 14 numbers can attain.
2012 District Olympiad, 4
Consider a tetrahedron $ABCD$ in which $AD \perp BC$ and $AC \perp BD$. We denote by $E$ and $F$ the projections of point $B$ on the lines $AD$ and $AC$, respectively. If $M$ and $N$ are the midpoints of the segments $[AB]$ and $[CD]$, respectively, show that $MN \perp EF$
1997 Taiwan National Olympiad, 2
Given a line segment $AB$ in the plane, find all possible points $C$ such that in the triangle $ABC$, the altitude from $A$ and the median from $B$ have the same length.
2021 Science ON all problems, 2
In triangle $ABC$, we have $\angle ABC=\angle ACB=44^o$. Point $M$ is in its interior such that $\angle MBC=16^o$ and $\angle MCB=30^o$. Prove that $\angle MAC=\angle MBC$.
[i] (Andra Elena Mircea)[/i]
2016 CHMMC (Fall), 15
In a $5 \times 5$ grid of squares, how many nonintersecting pairs rectangles of rectangles are there? (Note sharing a vertex or edge still means the rectangles intersect.)
2013 AIME Problems, 10
Given a circle of radius $\sqrt{13}$, let $A$ be a point at a distance $4 + \sqrt{13}$ from the center $O$ of the circle. Let $B$ be the point on the circle nearest to point $A$. A line passing through the point $A$ intersects the circle at points $K$ and $L$. The maximum possible area for $\triangle BKL$ can be written in the form $\tfrac{a-b\sqrt{c}}{d}$, where $a$, $b$, $c$, and $d$ are positive integers, $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.
2007 Sharygin Geometry Olympiad, 15
In a triangle $ABC$, let $AA', BB'$ and $CC'$ be the bisectors. Suppose $A'B' \cap CC' =P$ and $A'C' \cap BB'= Q$. Prove that $\angle PAC = \angle QAB$.
2020 Vietnam Team Selection Test, 6
In the scalene acute triangle $ABC$, $O$ is the circumcenter. $AD, BE, CF$ are three altitudes. And $H$ is the orthocenter. Let $G$ be the reflection point of $O$ through $BC$. Draw the diameter $EK$ in $\odot (GHE)$, and the diameter $FL$ in $\odot (GHF)$.
a) If $AK, AL$ and $DE, DF$ intersect at $U, V$ respectively, prove that $UV\parallel EF$.
b) Suppose $S$ is the intersection of the two tangents of the circumscribed circle of $\triangle ABC$ at $B$ and $C$. $T$ is the intersection of $DS$ and $HG$. And $M,N$ are the projection of $H$ on $TE,TF$ respectively. Prove that $M,N,E,F$ are concyclic.
2010 ITAMO, 4
In a trapezium $ABCD$, the sides $AB$ and $CD$ are parallel and the angles $\angle ABC$ and $\angle BAD$ are acute. Show that it is possible to divide the triangle $ABC$ into 4 disjoint triangle $X_1. . . , X_4$ and the triangle $ABD$ into 4 disjoint triangles $Y_1,. . . , Y_4$ such that the triangles $X_i$ and $Y_i$ are congruent for all $i$.
2007 Princeton University Math Competition, 7
A set of points $P_i$ [i]covers[/i] a polygon if for every point in the polygon, a line can be drawn inside the polygon to at least one $P_i$. Points $A_1, A_2, \cdots, A_n$ in the plane form a $2007$-gon, not necessarily convex. Find the minimum value of $n$ such that for any such polygon, we can pick $n$ points inside it that cover the polygon.
1986 All Soviet Union Mathematical Olympiad, 431
Given two points inside a convex dodecagon (twelve sides) situated $10$ cm far from each other. Prove that the difference between the sum of distances, from the point to all the vertices, is less than $1$ m for those points.