Found problems: 1049
A rhombus is given with one diagonal twice the length of the other diagonal. Express the side of the rhombus is terms of $K$, where $K$ is the area of the rhombus in square inches.
${{ \textbf{(A)}\ \sqrt{K} \qquad\textbf{(B)}\ \frac{1}{2}\sqrt{2K} \qquad\textbf{(C)}\ \frac{1}{3}\sqrt{3K} \qquad\textbf{(D)}\ \frac{1}{4}\sqrt{4K} }\qquad\textbf{(E)}\ \text{None of these are correct} } $
It is known that $\triangle ABC$ is an acute triangle. Let $C'$ be the foott of the perpendicular from $C$ to $AB$, and $D$, $E$ two distinct points on $CC'$. The feet of the perpendiculars from $D$ to $AC$ and $BC$ are $F$ and $G$, respectively. Show that if $DGEF$ is a parallelogram then $ABC$ is isosceles.
A circle with radius $1$ is inscribed in a square and circumscribed about another square as shown. Which fraction is closest to the ratio of the circle's shaded area to the area between the two squares?
[asy]
filldraw((-1,-1)--(-1,1)--(1,1)--(1,-1)--cycle,mediumgray,black);
filldraw(Circle((0,0),1), mediumgray,black);
filldraw((-1,0)--(0,1)--(1,0)--(0,-1)--cycle,white,black);[/asy]
$ \textbf{(A)}\ \frac{1}2\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ \frac{3}2\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ \frac{5}2 $
In a triangle $ABC$ , let $P$ be a point on its circumscribed circle (on the arc $AC$ that does not contain $B$). Let $H,H_1,H_2$ and $H_3$ be the orthocenters of triangles $ABC, BCP, ACP$ and $ABP$, respectively. Let $L = PB \cap AC$ and $J = HH_2 \cap H_1H_3$. If $M$ and $N$ are the midpoints of $JH$ and $LP$, respectively, prove that $MN$ and $JL$ intersect at their midpoint.
Let ABCD be a parallelogram, and P a point such that
$2 PDA=ABP$ and
$2 PAD=PCD$
Show that $AB=BP=CP$
Given four non-coplanar points, is it always possible to find a plane such that the orthogonal projections of the points onto the plane are the vertices of a parallelogram? How many such planes are there in general?
Let $ABC$ be triangle with $H$ is the orthocenter and $I$ is incenter. Denote $A_{1}, A_{2}, B_{1}, B_{2}, C_{1}, C_{2}$ be the points on the rays $AB, AC, BC, CA, CB$, respectively such that $$AA_{1} = AA_{2} = BC, BB_{1} = BB_{2} = CA, CC_{1} = CC_{2} = AB.$$ Suppose that $B_{1}B_{2}$ cuts $C_{1}C_{2}$ at $A'$, $C_{1}C_{2}$ cuts $A_{1}A_{2}$ at $B'$ and $A_{1}A_{2}$ cuts $B_{1}B_{2}$ at $C'$.
a) Prove that area of triangle $A'B'C'$ is smaller than or equal to the area of triangle $ABC$.
b) Let $J$ be circumcenter of triangle $A'B'C'$. $AJ$ cuts $BC$ at $R$, $BJ$ cuts $CA$ at $S$ and $CJ$ cuts $AB$ at $T$. Suppose that $(AST), (BTR), (CRS)$ intersect at $K$. Prove that if triangle $ABC$ is not isosceles then $HIJK$ is a parallelogram.
What is the ratio of the area of the shaded square to the area of the large square? (The figure is drawn to scale)
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draw((0,0)--(0,4)--(4,4)--(4,0)--cycle);
draw((0,0)--(4,4));
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draw((1,1)--(2,0)--(4,2));
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$ \text{(A)}\ \frac{1}{6}\qquad\text{(B)}\ \frac{1}{7}\qquad\text{(C)}\ \frac{1}{8}\qquad\text{(D)}\ \frac{1}{12}\qquad\text{(E)}\ \frac{1}{16} $
Let $ABCD$ be a square. Let $E, Z$ be points on the sides $AB, CD$ of the square respectively, such that $DE\parallel BZ$. Assume that the triangles $\triangle EAD, \triangle ZCB$ and the parallelogram $BEDZ$ have the same area.
If the distance between the parallel lines $DE$ and $BZ$ is equal to $1$, determine the area of the square.
Let $P$ and $P^{\prime }$ be two parallelograms with equal area, and let their sidelengths be $a,$ $b$ and $a^{\prime },$ $b^{\prime }.$ Assume that $a^{\prime }\leq a\leq b\leq b^{\prime },$ and moreover, it is possible to place the segment $b^{\prime }$ such that it completely lies in the interior of the parallelogram $P.$
Show that the parallelogram $P$ can be partitioned into four polygons such that these four polygons can be composed again to form the parallelogram $%
P^{\prime }.$
Let $ABC$ be a triangle. A circle passing through $A$ and $B$ intersects segments $AC$ and $BC$ at $D$ and $E$, respectively. Lines $AB$ and $DE$ intersect at $F$, while lines $BD$ and $CF$ intersect at $M$. Prove that $MF = MC$ if and only if $MB\cdot MD = MC^2$.
Let $ABCD$ be a convex quadrilateral with $AB$ is not parallel to $CD$. Circle $\omega_1$ with center $O_1$ passes through $A$ and $B$, and touches segment $CD$ at $P$. Circle $\omega_2$ with center $O_2$ passes through $C$ and $D$, and touches segment $AB$ at $Q$. Let $E$ and $F$ be the intersection of circles $\omega_1$ and $\omega_2$. Prove that $EF$ bisects segment $PQ$ if and only if $BC$ is parallel to $AD$.
Three vertices of parallelogram $PQRS$ are $P(-3,-2)$, $Q(1,-5)$, $R(9,1)$ with $P$ and $R$ diagonally opposite. The sum of the coordinates of vertex $S$ is:
$\textbf{(A)}\ 13 \qquad
\textbf{(B)}\ 12 \qquad
\textbf{(C)}\ 11 \qquad
\textbf{(D)}\ 10 \qquad
\textbf{(E)}\ 9$
Let $w$-circumcircle of triangle $ABC$ with an obtuse angle $C$ and $C '$symmetric point of point $C$ with respect to $AB$. $M$ midpoint of $AB$. $C'M$ intersects $w$ at $N$ ($C '$ between $M$ and $N$). Let $BC'$ second crossing point $w$ in $F$, and $AC'$ again crosses the $w$ at point $E$. $K$-midpoint $EF$. Prove that the lines $AB, CN$ and$ KC'$are concurrent.
[color=blue][size=150]PERU TST IMO - 2006[/size]
Saturday, may 20.[/color]
[b]Question 04[/b]
In an actue-angled triangle $ABC$ draws up: its circumcircle $w$
with center $O$, the circumcircle $w_1$ of the triangle $AOC$ and
the diameter $OQ$ of $w_1$. The points are chosen $M$ and $N$ on
the straight lines $AQ$ and $AC$, respectively, in such a way that
the quadrilateral $AMBN$ is a parallelogram.
Prove that the intersection point of the straight lines $MN$ and $BQ$ belongs the
circumference $w_1.$
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[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=88513]Spanish version[/url]
$\text{\LaTeX}{}$ed by carlosbr
Let $\Gamma$ be a circunference and $O$ its center. $AE$ is a diameter of $\Gamma$ and $B$ the midpoint of one of the arcs $AE$ of $\Gamma$. The point $D \ne E$ in on the segment $OE$. The point $C$ is such that the quadrilateral $ABCD$ is a parallelogram, with $AB$ parallel to $CD$ and $BC$ parallel to $AD$. The lines $EB$ and $CD$ meets at point $F$. The line $OF$ cuts the minor arc $EB$ of $\Gamma$ at $I$.
Prove that the line $EI$ is the angle bissector of $\angle BEC$.
In a given tedrahedron $ ABCD$ let $ K$ and $ L$ be the centres of edges $ AB$ and $ CD$ respectively. Prove that every plane that contains the line $ KL$ divides the tedrahedron into two parts of equal volume.
Given a parallelogram $ABCD$, join $A$ to the midpoints $E$ and $F$ of the opposite sides $BC$ and $CD$. $AE$ and $AF$ intersect the diagonal $BD$ in $M$ and $N$. Prove that $M$ and $N$ divide $BD$ into three equal parts.
Let $ABC$ be an acute triangle with $AB \neq AC$. Let $D$ be the foot of the altitude from $A$ and $\omega$ the circumcircle of the triangle. Let $\omega_1$ be the circle tangent to $AD$, $BD$ and $\omega$. Let $\omega_2$ be the circle tangent to $AD$, $CD$ and $\omega$. Let $\ell$ be the interior common tangent to both $\omega_1$ and $\omega_2$, different from $AD$.
Prove that $\ell$ passes through the midpoint of $BC$ if and only if $2BC = AB + AC$.
Consider a parallelogram $ABCD$ where $AB>AD$. Let $X$ and $Y$, distinct from $B$, be points on the ray $BD^\rightarrow$ such that $CX=CB$ and $AY=AB$. Prove that $DX=DY$. Note: The notation $BD^\rightarrow$ denotes the ray originating from point $B$ passing through point $D$.
Three circles $\mathcal{K}_1$, $\mathcal{K}_2$, $\mathcal{K}_3$ of radii $R_1,R_2,R_3$ respectively, pass through the point $O$ and intersect two by two in $A,B,C$. The point $O$ lies inside the triangle $ABC$.
Let $A_1,B_1,C_1$ be the intersection points of the lines $AO,BO,CO$ with the sides $BC,CA,AB$ of the triangle $ABC$. Let $ \alpha = \frac {OA_1}{AA_1} $, $ \beta= \frac {OB_1}{BB_1} $ and $ \gamma = \frac {OC_1}{CC_1} $ and let $R$ be the circumradius of the triangle $ABC$. Prove that
\[ \alpha R_1 + \beta R_2 + \gamma R_3 \geq R. \]
On the sides $ AB,BC,CD,DA $ of the parallelogram $ ABCD, $ consider the points $ M,N,P, $ respectively, $ Q, $ such that $ \overrightarrow{MN} +\overrightarrow{QP} =\overrightarrow{AC} . $ Show that $ \overrightarrow{PN} +\overrightarrow{QM} = \overrightarrow{DB} . $
Let $A$ and $B$ be two distinct points on the circle $\Gamma$, not diametrically opposite. The point $P$, distinct from $A$ and $B$, varies on $\Gamma$. Find the locus of the orthocentre of triangle $ABP$.
Show that the four perpendiculars dropped from the midpoints of the sides of a cyclic quadrilateral to the respective opposite sides are concurrent.
[b]Note by Darij:[/b] A [i]cyclic quadrilateral [/i]is a quadrilateral inscribed in a circle.
Consider a convex quadrilateral, and the incircles of the triangles determined by one of its diagonals. Prove that the tangency points of the incircles with the diagonal are symmetrical with respect to the midpoint of the diagonal if and only if the line of the incenters passes through the crossing point of the diagonals.
[i]Dan Schwarz[/i]