A rhombus has area $36$ and the longer diagonal is twice as long as the shorter diagonal. What is the perimeter of the rhombus?

In triangle $ABC$, $D$ is a variable point on line $BC$. Points $E,F$ are on segments $AC, AB$ respectively such that $BF=BD$ and $CD=CE$. Circles $(AEF)$ and $(ABC)$ meet again at $S$. Lines $EF$ and $BC$ meet at $P$ and circles $(PDS)$ and $(AEF)$ meet again at $Q$. Prove that, as $D$ varies, isogonal conjugate of $Q$ with respect to triangle $ ABC$ lies on a fixed circle. [i]Proposed by Serdar Bozdag[/i]

Determine all pairs $(h,s)$ of positive integers with the following property: If one draws $h$ horizontal lines and another $s$ lines which satisfy (i) they are not horizontal, (ii) no two of them are parallel, (iii) no three of the $h + s$ lines are concurrent, then the number of regions formed by these $h + s$ lines is 1992.

The line parallel to side $AC$ of a right triangle $ABC$ $(\angle C=90^\circ)$ intersects sides $AB$ and $BC$ at $M$ and $N$, respectively, so that the $CN / BN = AC / BC = 2$. Let $O$ be the intersection point of the segments $AN$ and $CM$ and $K$ be a point on the segment $ON$ such that $MO + OK = KN$. The perpendicular line to $AN$ at point $K$ and the bisector of triangle $ABC$ of $\angle B$ meet at point $T$. Find the angle $\angle MTB$.

A triangle has side lengths $a, b, c$ and angles $A, B, C$ as usual (with $b$ opposite $B$ etc). Show that if $$a(1 - 2 \cos A) + b(1 - 2 \cos B) + c(1 - 2 \cos C) = 0$$ then the triangle is equilateral.

Let $ ABC$ be a triangle with circum-circle $ \Gamma$. Let $ M$ be a point in the interior of triangle $ ABC$ which is also on the bisector of $ \angle A$. Let $ AM, BM, CM$ meet $ \Gamma$ in $ A_{1}, B_{1}, C_{1}$ respectively. Suppose $ P$ is the point of intersection of $ A_{1}C_{1}$ with $ AB$; and $ Q$ is the point of intersection of $ A_{1}B_{1}$ with $ AC$. Prove that $ PQ$ is parallel to $ BC$.

In triangle $ABC$, sides $CB$ and $CA$ are equal to $a$ and $b$, respectively. The bisector of the angle $ACB$ intersects the side $AB$ at the point $K$ and the circumscribed circle of the triangle at point $M$. The circumscribed circle of the triangle $AMK$ intersects for second time the straight line $CA$ at the point $P$. Find the length of the segment $AP$.

$9.$ What is the greatest possible perimeter of a right-angled triangle with integer side lengths if one of the sides has length $12 ?$

Point $P$ is taken inside the square $ABCD$ such that $BP + DP=25$, $CP - AP = 15$ and $\angle$[b]ABP =[/b] $\angle$[b]ADP[/b]. What is the radius of the circumcircle of $ABCD$?

In triangle $ABC$, points $M_1, M_2$ and $M_3$ are midpoints of sides $AB$, $BC$ and $AC$, respectively, while points $H_1, H_2$ and $H_3$ are bases of altitudes drawn from $C$, $A$ and $B$, respectively. Prove that one can construct a triangle from segments $H_1M_2, H_2M_3$ and $H_3M_1$. [i](3 points)[/i]

A plastic snap-together cube has a protruding snap on one side and receptacle holes on the other five sides as shown. What is the smallest number of these cubes that can be snapped together so that only receptacle holes are showing? [asy] draw((0,0)--(4,0)--(4,4)--(0,4)--cycle); draw(circle((2,2),1)); draw((4,0)--(6,1)--(6,5)--(4,4)); draw((6,5)--(2,5)--(0,4)); draw(ellipse((5,2.5),0.5,1)); fill(ellipse((3,4.5),1,0.25),black); fill((2,4.5)--(2,5.25)--(4,5.25)--(4,4.5)--cycle,black); fill(ellipse((3,5.25),1,0.25),black); [/asy] $\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 8$

$D$ is the midpoint of the side $BC$ of triangle $ABC$. $E$ and $F$ are points on $CA$ and $AB$ respectively, such that $BE$ is perpendicular to $CA$ and $CF$ is perpendicular to $AB$. If $DEF$ is an equilateral triangle, does it follow that $ABC$ is also equilateral?

In triangle $ABC$, let $A'$ is the symmetrical of $A$ with respect to the circumcenter $O$ of $ABC$. Prove that: [b]a)[/b] The sum of the squares of the tangents segments drawn from $A$ and $A'$ to the incircle of $ABC$ equals $$4R^2-4Rr-2r^2$$ where $R$ and $r$ are the radii of the circumscribed and inscribed circles of $ABC$ respectively. [b]b)[/b] The circle with center $A'$ and radius $A'I$ intersects the circumcircle of $ABC$ in a point $L$ such that $$AL=\sqrt{ AB.AC}$$ where $I$ is the centre of the inscribed circle of $ABC$.

Let $ABC$ be a triangle and let $M$ be the midpoint of the side $BC$ . The circle with radius $MA$ centered in $M$ meets the lines $AB$ and $AC$ again at $B^{'}$ and $C^{'}$, respectively , and the tangents to this circle at $B^{'}$ and $C^{'}$ meet at $D$ . Show that the perpendicular bisector of the segment $BC$ bisects the segment $AD$.

Prove that inside any acute-angled triangle, there exists a point $P$ such that the feet of the perpendiculars dropped from $P$ to the sides of the triangle are the vertices of an equilateral triangle. (NB Vassiliev)

Height of a circular truncated cone is $8$. Center of sphere $O_1$ with a radius of $2$ is on the axis of the circular truncated cone. Sphere $O_1$ is tangent to the top surface and the flank. We can put another sphere $O_2$, satisfying that sphere $O_2$ with a radius of $3$ have only one common point with sphere $O_1$, bottom surface and the flank. Besides $O_2$, how many spheres can we put inside the circular truncated cone? $\text{(A)}1\qquad\text{(B)}2\qquad\text{(C)}3\qquad\text{(D)}4$

Let us choose arbitrarily $n$ vertices of a regular $2n$-gon and color them red. The remaining vertices are colored blue. We arrange all red-red distances into a non-decreasing sequence and do the same with the blue-blue distances. Prove that the sequences are equal.

Let $E$ and $F$ be the midpoints of the lines $AB$ and $DA$ of a square $ABCD$, respectively and let $G$ be the intersection of $DE$ with $CF$. Find the aspect ratio of sidelengths of the triangle $EGC$, $| EG | : | GC | : | CE |$.

Let $D$ be the midpoint of the base $BC$ of an isosceles triangle $ABC,$ $E$ be the point at the side $AC$ such that $\angle CDE=60^\circ,$ and $M$ be the midpoint of $DE.$ Prove that $\angle AME=\angle BMD.$

The altitudes of a triangle have integer length and its inradius is a prime number. Find all possible values of the sides of the triangle.

The midpoint of the edge $SA$ of the triangular pyramid of $SABC$ has equal distances from all the vertices of the pyramid. Let $SH$ be the height of the pyramid. Prove that $BA^2 + BH^2 = C A^2 + CH^2$.

A man gets lost in a large forest, the boundary of which is a straight line. (We can assume that the forest fills the half-plane.) It is known that the distance from a person to Granina forest does not exceed $2$ km. a) Suggest a path along which he will certainly be able to get out of the forest after walking no more than $14$ km. (Of course, a person does not know in which direction the border of the forest is, BUT he has the opportunity to move along any pre-selected curve. It is believed that a person left the forest as soon as he reached its border, while the border of the forest is invisible to him, no matter how close he would have approached it.) b) Find a path with the same property and length no more than $13$ km.

In $\vartriangle ABC$, two straight lines drawn from an arbitrary point $D$ on $AB$ are parallel to $AC$ , $BC$ and intersect $BC$ , $AC$ at $F$ , $G$, respectively. Prove that the sum of the circumferences of the circles circumscribed around $\vartriangle ADG$ and $\vartriangle BDF$ is equal to the circumference of the circle circumscribed around $\vartriangle ABC$.

In the adjoining figure $AB$ and $BC$ are adjacent sides of square $ABCD$; $M$ is the midpoint of $AB$; $N$ is the midpoint of $BC$; and $AN$ and $CM$ intersect at $O$. The ratio of the area of $AOCD$ to the area of $ABCD$ is [asy] draw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)--(2,1)--(2,2)--(1,0)); label("A", (0,0), S); label("B", (2,0), S); label("C", (2,2), N); label("D", (0,2), N); label("M", (1,0), S); label("N", (2,1), E); label("O", (1.2, .8)); [/asy] $ \textbf{(A)}\ \frac{5}{6} \qquad\textbf{(B)}\ \frac{3}{4} \qquad\textbf{(C)}\ \frac{2}{3} \qquad\textbf{(D)}\ \frac{\sqrt{3}}{2} \qquad\textbf{(E)}\ \frac{(\sqrt{3}-1)}{2} $

Let $I$ be the incentre and $O$ the circumcentre of a non-equilateral triangle $ABC$. Prove that $\angle AIO \le 90^{\circ}$ if and only if $2BC\le AB+AC$.