Found problems: 1049
Let $ABC$ be a triangle. The incircle of $ABC$ touches the sides $AB$ and $AC$ at the points $Z$ and $Y$, respectively. Let $G$ be the point where the lines $BY$ and $CZ$ meet, and let $R$ and $S$ be points such that the two quadrilaterals $BCYR$ and $BCSZ$ are parallelogram.
Prove that $GR=GS$.
[i]Proposed by Hossein Karke Abadi, Iran[/i]
On a non-rhombus parallelogram $ ABCD, $ the vertex $ B $ is projected on $ AC $ in the point $ E. $ The perpendicular on $ BD $ thru $ E $ intersects the lines $ BC $ and $ AB $ in $ F $ and $ G, $ respectively. Show that $ EF=EG $ if and only if $ \angle ABC=90^{\circ } . $
[i]Mircea Becheanu[/i]
Let $ABCD$ be a parallelogram and $M,N$ be points in the plane such that $C \in (AM)$ and $D \in (BN)$. Lines $NA,NC$ meet lines $MB,MD$ at points $E,F,G,H$. Show that points $E,F,G,H$ lie on a circle if and only if $ABCD$ is a rhombus.
Let $M$ be an arbitrary point on side $BC$ of triangle $ABC$. $W$ is a circle which is tangent to $AB$ and $BM$ at $T$ and $K$ and is tangent to circumcircle of $AMC$ at $P$. Prove that if $TK||AM$, circumcircles of $APT$ and $KPC$ are tangent together.
Let $ABCD$ and $A^{\prime }B^{\prime}C^{\prime }D^{\prime }$ be two arbitrary parallelograms in the space, and let $M,$ $N,$ $P,$ $Q$ be points dividing the segments $AA^{\prime },$ $BB^{\prime },$ $CC^{\prime },$ $DD^{\prime }$ in equal ratios.
[b]a.)[/b] Prove that the quadrilateral $MNPQ$ is a parallelogram.
[b]b.)[/b] What is the locus of the center of the parallelogram $MNPQ,$ when the point $M$ moves on the segment $AA^{\prime }$ ?
(Consecutive vertices of the parallelograms are labelled in alphabetical order.
It is given isosceles triangle $ABC$ with $AB = AC$. $AD$ is diameter of circumcircle of triangle $ABC$. On the side $BC$ is chosen point $E$. On the sides $AC, AB$ there are points $F, G$ respectively such that $AFEG$ is parallelogram. Prove that $DE$ is perpendicular to $FG$.
Consider the triangle $ABC$. Let $P, Q$ inside the angle $A$ such that $\angle BAP=\angle CAQ$ and $PBQC$ is a parallelogram. Show that $\angle ABP=\angle ACP.$
Let $ABCD$ be a parallelogram. The circle with diameter $AC$ intersects the line $BD$ at points $P$ and $Q$. The perpendicular to the line $AC$ passing through the point $C$ intersects the lines $AB$ and $AD$ at points $X$ and $Y$, respectively. Prove that the points $P,Q,X$ and $Y$ lie on the same circle.
The altitudes of a parallelogram are greater than $1$. Does this yield that the unit square may be covered by this parallelogram?
Let $ABC$ be an isosceles triangle with $AB = AC$. Point $D$ lies on side $AC$ such that $BD$ is the angle bisector of $\angle ABC$. Point $E$ lies on side $BC$ between $B$ and $C$ such that $BE = CD$. Prove that $DE$ is parallel to $AB$.
Let $r,R$ and $r_a$ be the radii of the incircle, circumcircle and A-excircle of the triangle $ABC$ with $AC>AB$, respectively. $I,O$ and $J_A$ are the centers of these circles, respectively. Let incircle touches the $BC$ at $D$, for a point $E \in (BD)$ the condition $A(IEJ_A)=2A(IEO)$ holds.
Prove that
\[ED=AC-AB \iff R=2r+r_a.\]
Consider a triangle $ABC$. For every point M $\in BC$ ,we define $N \in CA$ and $P \in AB$ such that $APMN$ is a parallelogram. Let $O$ be the intersection of $BN$ and $CP$. Find $M \in BC$ such that $\angle PMO=\angle OMN$
Given a circle $K$, find the locus of vertices $A$ of parallelograms $ABCD$ with diagonals $AC \leq BD$, such that $BD$ is inside $K$.
A triangle has sides with lengths $a,b,c$ such that
\[ a^2 + b^2 = 5c^2 \]
Prove that medians to the sides of lengths $a$ and $b$ are perpendicular.
Prove that in any parallelepiped the sum of the lengths of the edges is less than or equal to twice the sum of the lengths of the four diagonals.
Call an $n$-gon to be [i]lattice[/i] if its vertices are lattice points. Prove that inside every lattice convex pentagon there exists a lattice point.
$\textbf{Bake Sale}$
Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies di ffer, as shown.
$\circ$ Art's cookies are trapezoids:
[asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8));
draw(origin--(5,0)--(5,3)--(2,3)--cycle);
draw(rightanglemark((5,3), (5,0), origin));
label("5 in", (2.5,0), S);
label("3 in", (5,1.5), E);
label("3 in", (3.5,3), N);[/asy]
$\circ$ Roger's cookies are rectangles:
[asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8));
draw(origin--(4,0)--(4,2)--(0,2)--cycle);
draw(rightanglemark((4,2), (4,0), origin));
draw(rightanglemark((0,2), origin, (4,0)));
label("4 in", (2,0), S);
label("2 in", (4,1), E);[/asy]
$\circ$ Paul's cookies are parallelograms:
[asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8));
draw(origin--(3,0)--(2.5,2)--(-0.5,2)--cycle);
draw((2.5,2)--(2.5,0), dashed);
draw(rightanglemark((2.5,2),(2.5,0), origin));
label("3 in", (1.5,0), S);
label("2 in", (2.5,1), W);[/asy]
$\circ$ Trisha's cookies are triangles:
[asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8));
draw(origin--(3,0)--(3,4)--cycle);
draw(rightanglemark((3,4),(3,0), origin));
label("3 in", (1.5,0), S);
label("4 in", (3,2), E);[/asy]
Each friend uses the same amount of dough, and Art makes exactly 12 cookies. Art's cookies sell for 60 cents each. To earn the same amount from a single batch, how much should one of Roger's cookies cost in cents?
$ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 75\qquad\textbf{(E)}\ 90$
Let $ ABC$ be a triangle, $ D$ be the midpoint of $ BC$, $ E$ be a point on segment $ AC$ such that $ BE\equal{}2AD$ and $ F$ is the intersection point of $ AD$ with $ BE$. If $ \angle DAC\equal{}60^{\circ}$, find the measure of the angle $ FEA$.
Let $OFT$ and $NOT$ be two similar triangles (with the same orientation) and let $FANO$ be a parallelogram. Show that
\[\vert OF\vert \cdot \vert ON\vert=\vert OA\vert \cdot \vert OT\vert.\]
Let $ O$ be the circumcenter of an acute triangle $ ABC$ and let $ k$ be the circle with center $ P$ that is tangent to $ O$ at $ A$ and tangent to side $ BC$ at $ D$. Circle $ k$ meets $ AB$ and $ AC$ again at $ E$ and $ F$ respectively. The lines $ OP$ and $ EP$ meet $ k$ again at $ I$ and $ G$. Lines $ BO$ and $ IG$ intersect at $ H$. Prove that $ \frac{{DF}^2}{AF}\equal{}GH$.
Consider a triangle $\Delta ABC$ and three points $D,E,F$ such that: $B$ and $E$ are on different side of the line $AC$, $C$ and $D$ are on different sides of $AB$, $A$ and $F$ are on the same side of the line $BC$. Also $\Delta ADB \sim \Delta CEA \sim \Delta CFB$. Let $M$ be the middle point of $AF$. Prove that:
a)$\Delta BDF \sim \Delta FEC$.
b) $M$ is the middle point of $DE$.
[i]Dan Branzei[/i]
In an acute angled triangle $ABC$, $\angle A = 30^{\circ}$, $H$ is the orthocenter, and $M$ is the midpoint of $BC$. On the line $HM$, take a point $T$ such that $HM = MT$. Show that $AT = 2 BC$.
Consider an $ABCD$ parallelogram with $\overline{AD}$ $=$ $\overline{BD}$. Point E lies in segment $\overline{BD}$ in such a way that $\overline{AE}$ $=$ $\overline{DE}$. The extension of line $\overline{AE}$ cuts segment $\overline{BC}$ and $F$. if line $\overline{DF}$ is the bisector of the $\angle CED$. Find the value of the $\angle ABD$
$\textbf{4.1.}$ Point $E$ lies in segment $\overline{BD}$ means that exits a point $E$ in the segment $\overline{BD}$ in other words lies refers to the same thing found
[i]Proposed by Alicia Smith, Francisco Morazan[/i]
Given a non-isoceles triangle $ABC$ inscribes a circle $(O,R)$ (center $O$, radius $R$). Consider a varying line $l$ such that $l\perp OA$ and $l$ always intersects the rays $AB,AC$ and these intersectional points are called $M,N$. Suppose that the lines $BN$ and $CM$ intersect, and if the intersectional point is called $K$ then the lines $AK$ and $BC$ intersect.
$1$, Assume that $P$ is the intersectional point of $AK$ and $BC$. Show that the circumcircle of the triangle $MNP$ is always through a fixed point.
$2$, Assume that $H$ is the orthocentre of the triangle $AMN$. Denote $BC=a$, and $d$ is the distance between $A$ and the line $HK$. Prove that $d\leq\sqrt{4R^2-a^2}$ and the equality occurs iff the line $l$ is through the intersectional point of two lines $AO$ and $BC$.
A $k\times \ell$ 'parallelogram' is drawn on a paper with hexagonal cells (it consists of $k$ horizontal rows of $\ell$ cells each). In this parallelogram a set of non-intersecting sides of hexagons is chosen; it divides all the vertices into pairs.
Juniors) How many vertical sides can there be in this set?
Seniors) How many ways are there to do that?
[asy]
size(120);
defaultpen(linewidth(0.8));
path hex = dir(30)--dir(90)--dir(150)--dir(210)--dir(270)--dir(330)--cycle;
for(int i=0;i<=3;i=i+1)
{
for(int j=0;j<=2;j=j+1)
{
real shiftx=j*sqrt(3)/2+i*sqrt(3),shifty=j*3/2;
draw(shift(shiftx,shifty)*hex);
}
}
[/asy]
[i](T. Doslic)[/i]