Found problems: 25757
Indonesia MO Shortlist - geometry, g11
Given triangle $ABC$ and point $P$ on the circumcircle of triangle $ABC$. Suppose the line $CP$ intersects line $AB$ at point $E$ and line $BP$ intersect line $AC$ at point $F$. Suppose also the perpendicular bisector of $AB$ intersects $AC$ at point $K$ and the perpendicular bisector of $AC$ intersects $AB$ at point $J$. Prove that $$\left( \frac{CE}{BF}\right)^2= \frac{AJ \cdot JE }{ AK \cdot KF}$$
2013 India IMO Training Camp, 2
In a triangle $ABC$ with $B = 90^\circ$, $D$ is a point on the segment $BC$ such that the inradii of triangles $ABD$ and $ADC$ are equal. If $\widehat{ADB} = \varphi$ then prove that $\tan^2 (\varphi/2) = \tan (C/2)$.
1953 Polish MO Finals, 3
Through each vertex of a tetrahedron with a given volume $ V $, a plane is drawn parallel to the opposite face of the tetrahedron. Calculate the volume of the tetrahedron formed by these planes.
2022 Oral Moscow Geometry Olympiad, 1
Given an isosceles trapezoid $ABCD$. The bisector of angle $B$ intersects the base $AD$ at point $L$. Prove that the center of the circle circumscribed around triangle $BLD$ lies on the circle circumscribed around the trapezoid.
(Yu. Blinkov)
1974 AMC 12/AHSME, 19
In the adjoining figure $ABCD$ is a square and $CMN$ is an equilateral triangle. If the area of $ABCD$ is one square inch, then the area of $CMN$ in square inches is
[asy]
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);
draw((.82,0)--(1,1)--(0,.76)--cycle);
label("A", (0,0), S);
label("B", (1,0), S);
label("C", (1,1), N);
label("D", (0,1), N);
label("M", (0,.76), W);
label("N", (.82,0), S);
[/asy]
$ \textbf{(A)}\ 2\sqrt{3}-3 \qquad\textbf{(B)}\ 1-\frac{\sqrt{3}}{3} \qquad\textbf{(C)}\ \frac{\sqrt{3}}{4} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{3} \qquad\textbf{(E)}\ 4-2\sqrt{3} $
LMT Speed Rounds, 14
In obtuse triangle $ABC$ with $AB = 7$, $BC = 20$, and $C A = 15$, let point $D$ be the foot of the altitude from $C$ to line $AB$. Evaluate $[ACD]+[BCD]$. (Note that $[XY Z]$ means the area of triangle $XY Z$.)
[i]Proposed by Jonathan Liu[/i]
1956 AMC 12/AHSME, 45
A wheel with a rubber tire has an outside diameter of $ 25$ in. When the radius has been decreased a quarter of an inch, the number of revolutions in one mile will:
$ \textbf{(A)}\ \text{be increased about }2\% \qquad\textbf{(B)}\ \text{be increased about }1\%$
$ \textbf{(C)}\ \text{be increased about }20\% \qquad\textbf{(D)}\ \text{be increased about }\frac {1}{2}\% \qquad\textbf{(E)}\ \text{remain the same}$
2022 MMATHS, 10
Suppose that $A_1A_2A_3$ is a triangle with $A_1A_2 = 16$ and $A_1A_3 = A_2A_3 = 10$. For each integer $n \ge 4$, set An to be the circumcenter of triangle $A_{n-1}A_{n-2}A_{n-3}$. There exists a unique point $Z$ lying in the interiors of the circumcircles of triangles $A_kA_{k+1}A_{k+2}$ for all integers $k \ge 1$. If $ZA^2_1+ ZA^2_2+ ZA^2_3+ ZA^2_4$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd(a, b) = 1$, find $a + b$.
1988 Iran MO (2nd round), 2
In a cyclic quadrilateral $ABCD$, let $I,J$ be the midpoints of diagonals $AC, BD$ respectively and let $O$ be the center of the circle inscribed in $ABCD.$ Prove that $I, J$ and $O$ are collinear.
1997 AIME Problems, 2
The nine horizontal and nine vertical lines on an $8\times8$ checkerboard form $r$ rectangles, of which $s$ are squares. The number $s/r$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
2021 All-Russian Olympiad, 6
In tetrahedron $ABCS$ no two edges have equal length. Point $A'$ in plane $BCS$ is symmetric to $S$ with respect to the perpendicular bisector of $BC$. Points $B'$ and $C'$ are defined analagously. Prove that planes $ABC, AB'C', A'BC'$ abd $A'B'C$ share a common point.
2017 Iran Team Selection Test, 1
$ABCD$ is a trapezoid with $AB \parallel CD$. The diagonals intersect at $P$. Let $\omega _1$ be a circle passing through $B$ and tangent to $AC$ at $A$. Let $\omega _2$ be a circle passing through $C$ and tangent to $BD$ at $D$. $\omega _3$ is the circumcircle of triangle $BPC$.
Prove that the common chord of circles $\omega _1,\omega _3$ and the common chord of circles $\omega _2, \omega _3$ intersect each other on $AD$.
[i]Proposed by Kasra Ahmadi[/i]
1951 Polish MO Finals, 6
Given a circle and a segment $ MN $. Find a point $ C $ on the circle such that the triangle $ ABC $, where $ A $ and $ B $ are the intersection points of the lines $ MC $ and $ NC $ with the circle, is similar to the triangle $ MNC $.
1991 Kurschak Competition, 2
A convex polyhedron has two triangle and three quadrilateral faces. Connect every vertex of one of the triangle faces with the intersection point of the diagonals in the quadrilateral face opposite to it. Show that the resulting three lines are concurrent.
2008 Grigore Moisil Intercounty, 2
Given a convex quadrilateral $ ABCD, $ find the locus of points $ X $ that verify the qualities:
$$ XA^2+XB^2+CD^2=XB^2+XC^2+DA^2=XC^2+XD^2+AB^2=XD^2+XA^2+BC^2 $$
[i]Maria Pop[/i]
2004 India IMO Training Camp, 1
Let $ABC$ be a triangle and $I$ its incentre. Let $\varrho_1$ and $\varrho_2$ be the inradii of triangles $IAB$ and $IAC$ respectively.
(a) Show that there exists a function $f: ( 0, \pi ) \mapsto \mathbb{R}$ such that \[ \frac{ \varrho_1}{ \varrho_2} = \frac{f(C)}{f(B)} \] where $B = \angle ABC$ and $C = \angle BCA$
(b) Prove that \[ 2 ( \sqrt{2} -1 ) < \frac{ \varrho_1} { \varrho_2} < \frac{ 1 + \sqrt{2}}{2} \]
2000 French Mathematical Olympiad, Exercise 2
Let $A,B,C$ be three distinct points in space, $(A)$ the sphere with center $A$ and radius $r$. Let $E$ be the set of numbers $R>0$ for which there is a sphere $(H)$ with center $H$ and radius $R$ such that $B$ and $C$ are outside the sphere, and the points of the sphere $(A)$ are strictly inside it.
(a) Suppose that $B$ and $C$ are on a line with $A$ and strictly outside $(A)$. Show that $E$ is nonempty and bounded, and determine its supremum in terms of the given data.
(b) Find a necessary and sufficient condition for $E$ to be nonempty and bounded
(c) Given $r$, compute the smallest possible supremum of $E$, if it exists.
2013 Argentina National Olympiad Level 2, 2
Let $ABC$ be a right triangle. It is known that there are points $D$ on the side $AC$ and $E$ on the side $BC$ such that $AB = AD = BE$ and $BD$ is perpendicular to $DE$. Calculate the ratios $\frac{AB}{BC}$ and $\frac{BC}{CA}$.
1993 AMC 8, 18
The rectangle shown has length $AC=32$, width $AE=20$, and $B$ and $F$ are midpoints of $\overline{AC}$ and $\overline{AE}$, respectively. The area of quadrilateral $ABDF$ is
[asy]
pair A,B,C,D,EE,F;
A = (0,20); B = (16,20); C = (32,20); D = (32,0); EE = (0,0); F = (0,10);
draw(A--C--D--EE--cycle);
draw(B--D--F);
dot(A); dot(B); dot(C); dot(D); dot(EE); dot(F);
label("$A$",A,NW);
label("$B$",B,N);
label("$C$",C,NE);
label("$D$",D,SE);
label("$E$",EE,SW);
label("$F$",F,W);
[/asy]
$\text{(A)}\ 320 \qquad \text{(B)}\ 325 \qquad \text{(C)}\ 330 \qquad \text{(D)}\ 335 \qquad \text{(E)}\ 340$
2023 Korea Junior Math Olympiad, 8
A red equilateral triangle $T$ with side length $1$ is drawn on a plane. For a positive real $c$, we place three blue equilateral triangle shaped paper with side length $c$ on a plane to cover $T$ completely. Find the minimum value of $c$. As shown in the picture, it doesn't matter if the blue papers overlap each other or stick out from $T$. Folding or tearing the paper is not allowed.
2017 Sharygin Geometry Olympiad, 4
Alex dissects a paper triangle into two triangles. Each minute after this he dissects one of obtained triangles into two triangles. After some time (at least one hour) it appeared that all obtained triangles were congruent. Find all initial triangles for which this is possible.
2011 Korea National Olympiad, 2
Let $ABC$ be a triangle and its incircle meets $BC, AC, AB$ at $D, E$ and $F$ respectively. Let point $ P $ on the incircle and inside $ \triangle AEF $. Let $ X=PB \cap DF , Y=PC \cap DE, Q=EX \cap FY $. Prove that the points $ A$ and $Q$ lies on $DP$ simultaneously or located opposite sides from $DP$.
2022-23 IOQM India, 23
In a triangle $ABC$, the median $AD$ divides $\angle{BAC}$ in the ratio $1:2$. Extend $AD$ to $E$ such that $EB$ is perpendicular $AB$. Given that $BE=3,BA=4$, find the integer nearest to $BC^2$.
2008 Junior Balkan MO, 2
The vertices $ A$ and $ B$ of an equilateral triangle $ ABC$ lie on a circle $k$ of radius $1$, and the vertex $ C$ is in the interior of the circle $ k$. A point $ D$, different from $ B$, lies on $ k$ so that $ AD\equal{}AB$. The line $ DC$ intersects $ k$ for the second time at point $ E$. Find the length of the line segment $ CE$.
1960 IMO Shortlist, 4
Construct triangle $ABC$, given $h_a$, $h_b$ (the altitudes from $A$ and $B$), and $m_a$, the median from vertex $A$.