Found problems: 25757
2020 Abels Math Contest (Norwegian MO) Final, 4b
The triangle $ABC$ has a right angle at $A$. The centre of the circumcircle is called $O$, and the base point of the normal from $O$ to $AC$ is called $D$. The point $E$ lies on $AO$ with $AE = AD$. The angle bisector of $\angle CAO$ meets $CE$ in $Q$. The lines $BE$ and $OQ$ intersect in $F$. Show that the lines $CF$ and $OE$ are parallel.
2000 Rioplatense Mathematical Olympiad, Level 3, 2
In a triangle $ABC$, points $D, E$ and $F$ are considered on the sides $BC, CA$ and $AB$ respectively, such that the areas of the triangles $AFE, BFD$ and $CDE$ are equal. Prove that $$\frac{(DEF) }{ (ABC)} \ge \frac{1}{4}$$
Note: $(XYZ)$ is the area of triangle $XYZ$.
Swiss NMO - geometry, 2010.9
Let $ k$ and $ k'$ two concentric circles centered at $ O$, with $ k'$ being larger than $ k$. A line through $ O$ intersects $ k$ at $ A$ and $ k'$ at $ B$ such that $ O$ seperates $ A$ and $ B$. Another line through $ O$ intersects $ k$ at $ E$ and $ k'$ at $ F$ such that $ E$ separates $ O$ and $ F$.
Show that the circumcircle of $ \triangle{OAE}$ and the circles with diametres $ AB$ and $ EF$ have a common point.
2002 May Olympiad, 3
In a triangle $ABC$, right in $A$ and isosceles, let $D$ be a point on the side $AC$ ($A \ne D \ne C$) and $E$ be the point on the extension of $BA$ such that the triangle $ADE$ is isosceles. Let $P$ be the midpoint of segment $BD$, $R$ be the midpoint of the segment $CE$ and $Q$ the intersection point of $ED$ and $BC$. Prove that the quadrilateral $ARQP$ is a square.
1951 AMC 12/AHSME, 49
The medians of a right triangle which are drawn from the vertices of the acute angles are $ 5$ and $ \sqrt {40}$. The value of the hypotenuse is:
$ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 2\sqrt {40} \qquad\textbf{(C)}\ \sqrt {13} \qquad\textbf{(D)}\ 2\sqrt {13} \qquad\textbf{(E)}\ \text{none of these}$
1988 Balkan MO, 1
Let $ABC$ be a triangle and let $M,N,P$ be points on the line $BC$ such that $AM,AN,AP$ are the altitude, the angle bisector and the median of the triangle, respectively. It is known that
$\frac{[AMP]}{[ABC]}=\frac{1}{4}$ and $\frac{[ANP]}{[ABC]}=1-\frac{\sqrt{3}}{2}$.
Find the angles of triangle $ABC$.
1972 IMO Shortlist, 7
Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.
2015 Latvia Baltic Way TST, 7
Two circle $\Gamma_1$ and $\Gamma_2$ intersect at points $A$ and $B$, point $P$ is not on the line $AB$. Line $AP$ intersects again $\Gamma_1$ and $\Gamma_2$ at points $K$ and $L$ respectively, line $BP$ intersects again $\Gamma_1$ and $\Gamma_2$ at points $M$ and $N$ respectively and all the points mentioned so far are different. The centers of the circles circumscribed around the triangles $KMP$ and $LNP$ are $O_1$ and $O_2$ respectively. Prove that $O_1O_2$ is perpendicular to $AB$.
1995 All-Russian Olympiad, 2
A chord $CD$ of a circle with center $O$ is perpendicular to a diameter $AB$. A chord $AE$ bisects the radius $OC$. Show that the line $DE$ bisects the chord $BC$
[i]V. Gordon[/i]
1957 AMC 12/AHSME, 6
An open box is constructed by starting with a rectangular sheet of metal $ 10$ in. by $ 14$ in. and cutting a square of side $ x$ inches from each corner. The resulting projections are folded up and the seams welded. The volume of the resulting box is:
$ \textbf{(A)}\ 140x \minus{} 48x^2 \plus{} 4x^3 \qquad \textbf{(B)}\ 140x \plus{} 48x^2 \plus{} 4x^3\qquad \\\textbf{(C)}\ 140x \plus{} 24x^2 \plus{} x^3\qquad \textbf{(D)}\ 140x \minus{} 24x^2 \plus{} x^3\qquad \textbf{(E)}\ \text{none of these}$
2013 Chile TST Ibero, 3
The incircle of triangle $\triangle ABC$ touches $AC$ and $BC$ at $E$ and $D$ respectively. The excircle corresponding to $A$ touches the extensions of $BC$ at $A_1$, $CA$ at $B_1$, and $AB$ at $C_1$. Let $DE \cap A_1B_1 = L$. Prove that $L$ belongs to the circumcircle of triangle $\triangle A_1B_1C_1$.
2021 Yasinsky Geometry Olympiad, 2
In the triangle $ABC$, it is known that $AB = BC = 20$ cm, and $AC = 24$ cm. The point $M$ lies on the side $BC$ and is equidistant from sides $AB$ and $AC$. Find this distance.
(Alexander Shkolny)
2019 Thailand TST, 1
Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.
2021 BMT, 23
Shivani has a single square with vertices labeled $ABCD$. She is able to perform the following transformations:
$\bullet$ She does nothing to the square.
$\bullet$ She rotates the square by $90$, $180$, or $270$ degrees.
$\bullet$ She reflects the square over one of its four lines of symmetry.
For the first three timesteps, Shivani only performs reflections or does nothing. Then for the next three timesteps, she only performs rotations or does nothing. She ends up back in the square’s original configuration. Compute the number of distinct ways she could have achieved this.
1939 Moscow Mathematical Olympiad, 047
Prove that for any triangle the bisector lies between the median and the height drawn from the same vertex.
1976 Poland - Second Round, 6
Six points are placed on the plane such that each three of them are the vertices of a triangle with sides of different lengths. Prove that the shortest side of one of these triangles is also the longest side of another of them.
1983 AMC 12/AHSME, 19
Point $D$ is on side $CB$ of triangle $ABC$. If \[ \angle{CAD} = \angle{DAB} = 60^\circ,\quad AC = 3\quad\mbox{ and }\quad AB = 6, \] then the length of $AD$ is
$\text{(A)} \ 2 \qquad \text{(B)} \ 2.5 \qquad \text{(C)} \ 3 \qquad \text{(D)} \ 3.5 \qquad \text{(E)} \ 4$
1984 Kurschak Competition, 2
$A_1B_1A_2$, $B_1A_2B_2$, $A_2B_2A_3$,...,$B_{13}A_{14}B_{14}$, $A_{14}B_{14}A_1$ and $B_{14}A_1B_1$ are equilateral rigid plates that can be folded along the edges $A_1B_1$,$B_1A_2$, ..., $A_{14}B_{14}$ and $B_{14}A_1$ respectively. Can they be folded so that all $28$ plates lie in the same plane?
2013 NIMO Summer Contest, 6
Let $ABC$ and $DEF$ be two triangles, such that $AB=DE=20$, $BC=EF=13$, and $\angle A = \angle D$. If $AC-DF=10$, determine the area of $\triangle ABC$.
[i]Proposed by Lewis Chen[/i]
2008 Sharygin Geometry Olympiad, 5
(A.Zaslavsky) Given two triangles $ ABC$, $ A'B'C'$. Denote by $ \alpha$ the angle between the altitude and the median from vertex $ A$ of triangle $ ABC$. Angles $ \beta$, $ \gamma$, $ \alpha'$, $ \beta'$, $ \gamma'$ are defined similarly. It is known that $ \alpha \equal{} \alpha'$, $ \beta \equal{} \beta'$, $ \gamma \equal{} \gamma'$. Can we conclude that the triangles are similar?
2009 Math Prize For Girls Problems, 13
The figure below shows a right triangle $ \triangle ABC$.
[asy]unitsize(15);
pair A = (0, 4);
pair B = (0, 0);
pair C = (4, 0);
draw(A -- B -- C -- cycle);
pair D = (2, 0);
real p = 7 - 3sqrt(3);
real q = 4sqrt(3) - 6;
pair E = p + (4 - p)*I;
pair F = q*I;
draw(D -- E -- F -- cycle);
label("$A$", A, N);
label("$B$", B, S);
label("$C$", C, S);
label("$D$", D, S);
label("$E$", E, NE);
label("$F$", F, W);[/asy]
The legs $ \overline{AB}$ and $ \overline{BC}$ each have length $ 4$. An equilateral triangle $ \triangle DEF$ is inscribed in $ \triangle ABC$ as shown. Point $ D$ is the midpoint of $ \overline{BC}$. What is the area of $ \triangle DEF$?
1983 IMO Longlists, 3
[b](a)[/b] Given a tetrahedron $ABCD$ and its four altitudes (i.e., lines through each vertex, perpendicular to the opposite face), assume that the altitude dropped from $D$ passes through the orthocenter $H_4$ of $\triangle ABC$. Prove that this altitude $DH_4$ intersects all the other three altitudes.
[b](b)[/b] If we further know that a second altitude, say the one from vertex A to the face $BCD$, also passes through the orthocenter $H_1$ of $\triangle BCD$, then prove that all four altitudes are concurrent and each one passes through the orthocenter of the respective triangle.
2015 AMC 10, 17
A line that passes through the origin intersects both the line $x=1$ and the line $y=1+\frac{\sqrt{3}}{3}x$. The three lines create an equilateral triangle. What is the perimeter of the triangle?
$ \textbf{(A) }2\sqrt{6}\qquad\textbf{(B) }2+2\sqrt{3}\qquad\textbf{(C) }6\qquad\textbf{(D) }3+2\sqrt{3}\qquad\textbf{(E) }6+\frac{\sqrt{3}}{3} $
1990 French Mathematical Olympiad, Problem 5
In a triangle $ABC$, $\Gamma$ denotes the excircle corresponding to $A$, $A',B',C'$ are the points of tangency of $\Gamma$ with $BC,CA,AB$ respectively, and $S(ABC)$ denotes the region of the plane determined by segments $AB',AC'$ and the arc $C'A'B'$ of $\Gamma$.
Prove that there is a triangle $ABC$ of a given perimeter $p$ for which the area of $S(ABC)$ is maximal. For this triangle, give an approximate measure of the angle at $A$.
2012 France Team Selection Test, 2
Let $ABC$ be an acute-angled triangle with $AB\not= AC$. Let $\Gamma$ be the circumcircle, $H$ the orthocentre and $O$ the centre of $\Gamma$. $M$ is the midpoint of $BC$. The line $AM$ meets $\Gamma$ again at $N$ and the circle with diameter $AM$ crosses $\Gamma$ again at $P$. Prove that the lines $AP,BC,OH$ are concurrent if and only if $AH=HN$.