Found problems: 25757
2007 Harvard-MIT Mathematics Tournament, 30
$ABCD$ is a cyclic quadrilateral in which $AB=3$, $BC=5$, $CD=6$, and $AD=10$. $M$, $I$, and $T$ are the feet of the perpendiculars from $D$ to lines $AB$, $AC$, and $BC$ respectively. Determine the value of $MI/IT$.
2015 Saudi Arabia IMO TST, 1
Let $ABC$ be an acute-angled triangle inscribed in the circle $(O)$, $H$ the foot of the altitude of $ABC$ at $A$ and $P$ a point inside $ABC$ lying on the bisector of $\angle BAC$. The circle of diameter $AP$ cuts $(O)$ again at $G$. Let $L$ be the projection of $P$ on $AH$. Prove that if $GL$ bisects $HP$ then $P$ is the incenter of the triangle $ABC$.
Lê Phúc Lữ
2015 NIMO Problems, 6
Let $\triangle ABC$ be a triangle with $BC = 4, CA= 5, AB= 6$, and let $O$ be the circumcenter of $\triangle ABC$. Let $O_b$ and $O_c$ be the reflections of $O$ about lines $CA$ and $AB$ respectively. Suppose $BO_b$ and $CO_c$ intersect at $T$, and let $M$ be the midpoint of $BC$. Given that $MT^2 = \frac{p}{q}$ for some coprime positive integers $p$ and $q$, find $p+q$.
[i]Proposed by Sreejato Bhattacharya[/i]
2018 Brazil Undergrad MO, 12
Let $ABC$ be an equilateral triangle. $A $ point $P$ is chosen at random within this triangle. What is the probability that the sum of the distances from point $P$ to the sides of triangle $ABC$ are measures of the sides of a triangle?
1971 IMO, 1
All faces of the tetrahedron $ABCD$ are acute-angled. Take a point $X$ in the interior of the segment $AB$, and similarly $Y$ in $BC, Z$ in $CD$ and $T$ in $AD$.
[b]a.)[/b] If $\angle DAB+\angle BCD\ne\angle CDA+\angle ABC$, then prove none of the closed paths $XYZTX$ has minimal length;
[b]b.)[/b] If $\angle DAB+\angle BCD=\angle CDA+\angle ABC$, then there are infinitely many shortest paths $XYZTX$, each with length $2AC\sin k$, where $2k=\angle BAC+\angle CAD+\angle DAB$.
Ukrainian From Tasks to Tasks - geometry, 2016.8
Let $ABCD$ be a convex quadrilateral. It is known that $S_{ABD} = 7$, $S_{BCD}= 5$ and $S_{ABC}= 3$. Inside the quadrilateral mark the point $X$ so that $ABCX$ is a parallelogram. Find $S_{ADX}$ and $S_{BDX}$.
Geometry Mathley 2011-12, 9.1
Let $ABC$ be a triangle with $(O), (I)$ being the circumcircle, and incircle respectively. Let $(I)$ touch $BC,CA$, and $AB$ at $A_0, B_0, C_0$ let $BC,CA$, and $AB$ intersect $B_0C_0, C_0A_0, A_0Bv$ at $A_1, B_1$, and $C_1$ respectively. Prove that $OI$ passes through the orthocenter of four triangles $A_0B_0C_0, A_0B_1C_1, B_0C_1A_1,C_0A_1B_1$.
Nguyễn Minh Hà
2019 Estonia Team Selection Test, 11
Given a circle $\omega$ with radius $1$. Let $T$ be a set of triangles good, if the following conditions apply:
(a) the circumcircle of each triangle in the set $T$ is $\omega$;
(b) The interior of any two triangles in the set $T$ has no common point.
Find all positive real numbers $t$, for which for each positive integer $n$ there is a good set of $n$ triangles, where the perimeter of each triangle is greater than $t$.
2020 Ukrainian Geometry Olympiad - December, 2
On a circle noted $n$ points. It turned out that among the triangles with vertices in these points exactly half of the acute. Find all values $n$ in which this is possible.
2001 Swedish Mathematical Competition, 3
Show that if $b = \frac{a+c}{2}$ in the triangle $ABC$, then $\cos (A-C) + 4 \cos B = 3$.
2011 Uzbekistan National Olympiad, 2
Let triangle ABC with $ AB=c$ $AC=b$ $BC=a$ $R$ circumradius, $p$ half peremetr of $ABC$.
I f $\frac{acosA+bcosB+ccosC}{asinA+bsinB+csinC}=\frac{p}{9R}$ then find all value of $cosA$.
2017 Romania Team Selection Test, P1
Let $ABCD$ be a trapezium, $AD\parallel BC$, and let $E,F$ be points on the sides$AB$ and $CD$, respectively. The circumcircle of $AEF$ meets $AD$ again at $A_1$, and the circumcircle of $CEF$ meets $BC$ again at $C_1$. Prove that $A_1C_1,BD,EF$ are concurrent.
2012 AMC 12/AHSME, 20
A trapezoid has side lengths $3, 5, 7,$ and $11$. The sum of all the possible areas of the trapezoid can be written in the form of $r_1 \sqrt{n_1} + r_2 \sqrt{n_2} + r_3$, where $r_1, r_2,$ and $r_3$ are rational numbers and $n_1$ and $n_2$ are positive integers not divisible by the square of a prime. What is the greatest integer less than or equal to
\[r_1 + r_2 + r_3 + n_1 + n_2?\]
$ \textbf{(A)}\ 57\qquad\textbf{(B)}\ 59\qquad\textbf{(C)}\ 61\qquad\textbf{(D)}\ 63\qquad\textbf{(E)}\ 65 $
2008 Indonesia TST, 1
Let $ABCD$ be a square with side $20$ and $T_1, T_2, ..., T_{2000}$ are points in $ABCD$ such that no $3$ points in the set $S = \{A, B, C, D, T_1, T_2, ..., T_{2000}\}$ are collinear. Prove that there exists a triangle with vertices in $S$, such that the area is less than $1/10$.
2022 USAJMO, 4
Let $ABCD$ be a rhombus, and let $K$ and $L$ be points such that $K$ lies inside the rhombus, $L$ lies outside the rhombus, and $KA = KB = LC = LD$. Prove that there exist points $X$ and $Y$ on lines $AC$ and $BD$ such that $KXLY$ is also a rhombus.
[i]Proposed by Ankan Bhattacharya[/i]
2019 Caucasus Mathematical Olympiad, 6
In a triangle $ABC$ with $\angle BAC = 90^{\circ}$ let $BL$ be the bisector, $L\in AC$. Let $D$ be a point symmetrical to $A$ with respect to $BL$. Let $M$ be the circumcenter of $ADC$. Prove that $CM$, $DL$, and $AB$ are concurrent.
2013 Turkey Junior National Olympiad, 3
Let $ABC$ be a triangle such that $AC>AB.$ A circle tangent to the sides $AB$ and $AC$ at $D$ and $E$ respectively, intersects the circumcircle of $ABC$ at $K$ and $L$. Let $X$ and $Y$ be points on the sides $AB$ and $AC$ respectively, satisfying
\[ \frac{AX}{AB}=\frac{CE}{BD+CE} \quad \text{and} \quad \frac{AY}{AC}=\frac{BD}{BD+CE} \]
Show that the lines $XY, BC$ and $KL$ are concurrent.
2010 Sharygin Geometry Olympiad, 2
Bisectors $AA_1$ and $BB_1$ of a right triangle $ABC \ (\angle C=90^\circ )$ meet at a point $I.$ Let $O$ be the circumcenter of triangle $CA_1B_1.$ Prove that $OI \perp AB.$
2006 China Team Selection Test, 1
$ABCD$ is a trapezoid with $AB || CD$. There are two circles $\omega_1$ and $\omega_2$ is the trapezoid such that $\omega_1$ is tangent to $DA$, $AB$, $BC$ and $\omega_2$ is tangent to $BC$, $CD$, $DA$. Let $l_1$ be a line passing through $A$ and tangent to $\omega_2$(other than $AD$), Let $l_2$ be a line passing through $C$ and tangent to $\omega_1$ (other than $CB$).
Prove that $l_1 || l_2$.
2006 All-Russian Olympiad Regional Round, 9.6
In an acute triangle $ABC$, the angle bisector$AD$ and altitude $BE$ are drawn. Prove that angle $CED$ is greater than $45^o$.
2009 Estonia Team Selection Test, 3
Find all natural numbers $n$ for which there exists a convex polyhedron satisfying the following conditions:
(i) Each face is a regular polygon.
(ii) Among the faces, there are polygons with at most two different numbers of edges.
(iii) There are two faces with common edge that are both $n$-gons.
2012 HMNT, 9
Consider triangle $ABC$ where $BC = 7$, $CA = 8$, and $AB = 9$. $D$ and $E$ are the midpoints of $BC$ and $CA$, respectively, and $AD$ and $BE$ meet at $G$. The reflection of $G$ across $D$ is $G'$, and $G'E$ meets $CG$ at $P$. Find the length $PG$.
2017-IMOC, G2
Given two acute triangles $\vartriangle ABC, \vartriangle DEF$. If $AB \ge DE, BC \ge EF$ and $CA \ge FD$, show that the area of $\vartriangle ABC$ is not less than the area of $\vartriangle DEF$
1991 AMC 12/AHSME, 24
The graph, $G$ of $y = \log_{10}x$ is rotated $90^{\circ}$ counter-clockwise about the origin to obtain a new graph $G'$. Which of the following is an equation for $G'$?
$ \textbf{(A)}\ y = \log_{10}\left(\frac{x + 90}{9}\right)\qquad\textbf{(B)}\ y = \log_{x}10\qquad\textbf{(C)}\ y = \frac{1}{x + 1}\qquad\textbf{(D)}\ y = 10^{-x}\qquad\textbf{(E)}\ y = 10^{x} $
2006 All-Russian Olympiad Regional Round, 10.1
Natural numbers from $1$ to $200$ were divided into $50$ sets. Prove that one of them contains three numbers that are the lengths of the sides some triangle.