In isosceles triangle $ABC$ sides $AB$ and $BC$ have length $125$ while side $AC$ has length $150$. Point $D$ is the midpoint of side $AC$. $E$ is on side $BC$ so that $BC$ and $DE$ are perpendicular. Similarly, $F$ is on side $AB$ so that $AB$ and $DF$ are perpendicular. Find the area of triangle $DEF$.

Laura won the local math olympiad and was awarded a "magical" ruler. With it, she can draw (as usual) lines in the plane, and she can also measure segments and replicate them anywhere in the plane; but she can also divide a segment into as many equal parts as she wishes; for instance, she can divide any segment into $17$ equal parts. Laura drew a parallelogram $ABCD$ and decided to try out her magical ruler; with it, she found the midpoint $M$ of side $CD$, and she extended $CB$ beyond $B$ to point $N$ so that segments $CB$ and $BN$ were equal in length. Unfortunately, her mischievous little brother came along and erased everything on Laura's picture except for points $A, M$, and $N$. Using Laura's magical ruler, help her reconstruct the original parallelogram $ABCD$: write down the steps that she needs to follow and prove why this will lead to reconstructing the original parallelogram $ABCD$.

We will say that two pyramids touch each other by faces if they have no common interior points and if the intersection of a face of one of them with a face of the other is either a triangle or a polygon. Is it possible to place $8$ tetrahedra in such a way that every two of them touch each other by faces? (A Andjans, Riga)

The quadrilateral $ABCD$ with perpendicular diagonals is inscribed in the circle with center $O$, the points $M,N$ are the midpoints of $[BC]$ and $[CD]$ respectively. Find the ratio of areas of the figures $OMCN$ and $ABCD$

Prove that if the two altitudes of a tetrahedron intersect, then the other two atltitudes intersect also.

Points $A, B, C$ are vertices of an equilateral triangle inscribed in a circle. Point $D$ lies on the shorter arc $\overarc {AB}$ . Prove that $AD + BD = DC$.

The quadrilateral $ABCD$ has perpendicular diagonals that meet at $O$. The incenters of triangles $ABC, BCD, CDA, DAB$ form a quadrilateral with perimeter $P$. Show that the sum of the inradii of the triangles $AOB, BOC, COD, DOA$ is less than or equal to $\frac{P} {2}$.

A quadrilateral $ABCD$ is inscribed in a circle. Suppose that $|DA| =|BC|= 2$ and$ |AB| = 4$. Let $E $ be the intersection point of lines $BC$ and $DA$. Suppose that $\angle AEB = 60^o$ and that $|CD| <|AB|$. Calculate the radius of the circle.

Segments $AB$, $BC$ and $CA$ are, respectively, diagonals of squares $K_1$, $K_2$, $K3$. Prove that if triangle $ABC$ is acute, then it completely covered by squares $K_1$, $K_2$ and $K_3$.

Let $ABC$ be an acute triangle with $AB<AC$. $E,F$ are foots of the altitudes drawn from $B,C$ respectively. Let $M$ be the midpoint of segment $BC$. The tangent at $A$ to the circumcircle of $ABC$ cuts $BC$ in $P$ and $EF$ cuts the parallel to $BC$ from $A$ at $Q$. Prove that $PQ$ is perpendicular to $AM$.

Triangle $ABC$ in the plane $\Pi$ is called [i]good [/i] if it has the following property: For any point $D$ in space outside the plane $\Pi$, it is possible to construct a triangle with sides of lengths $CD,BD,AD$. Find all good triangles

A semicircle has diameter $AB$ and center $S$,with a point $M$ on the circumference.$U,V$ are the incircles of sectors $ASM$ and $BSM$.Prove that circles $U,V$ can be seperated by a line perpendicular to $AB$.

Given convex quadrilateral $ABCD$. We construct the similar triangles $APB, BQC, CRD, DSA$ outside $ABCD$ so that $\angle PAB = \angle QBC = \angle RCD = \angle SDA, \angle PBA = \angle QCB = \angle RDC = \angle SAD$. Prove that if $PQRS$ is a parallelogram, so is $ABCD$.

Two circles $ \Omega_{1}$ and $ \Omega_{2}$ are externally tangent to each other at a point $ I$, and both of these circles are tangent to a third circle $ \Omega$ which encloses the two circles $ \Omega_{1}$ and $ \Omega_{2}$. The common tangent to the two circles $ \Omega_{1}$ and $ \Omega_{2}$ at the point $ I$ meets the circle $ \Omega$ at a point $ A$. One common tangent to the circles $ \Omega_{1}$ and $ \Omega_{2}$ which doesn't pass through $ I$ meets the circle $ \Omega$ at the points $ B$ and $ C$ such that the points $ A$ and $ I$ lie on the same side of the line $ BC$. Prove that the point $ I$ is the incenter of triangle $ ABC$. [i]Alternative formulation.[/i] Two circles touch externally at a point $ I$. The two circles lie inside a large circle and both touch it. The chord $ BC$ of the large circle touches both smaller circles (not at $ I$). The common tangent to the two smaller circles at the point $ I$ meets the large circle at a point $ A$, where the points $ A$ and $ I$ are on the same side of the chord $ BC$. Show that the point $ I$ is the incenter of triangle $ ABC$.

Consider an acute-angled triangle $ABC$ and its circumcircle. Let $D$ be a point on the arc $AB$ which does not include point $C$ and let $A_1$ and $B_1$ be points on the lines $DA$ and $DB$, respectively, such that $CA_1 \perp DA$ and $CB_1 \perp DB$. Prove that $|AB| \ge |A_1B_1|$.

Let $U$ be the center of the circle inscribed in the triangle $ABC$, $O_1$, $O_2$ and $O_3$ the centers of the circles circumscribed by the triangles $BCU$, $CAU$ and $ABU$ respectively. Prove that the circles circumscribed around the triangles $ABC$ and $O_1O_2O_3$ have the same center.

Let $ABC$ be a triangle and let $D\in (BC)$ be the foot of the $A$- altitude. The circle $w$ with the diameter $[AD]$ meet again the lines $AB$ , $AC$ in the points $K\in (AB)$ , $L\in (AC)$ respectively. Denote the meetpoint $M$ of the tangents to the circle $w$ in the points $K$ , $L$ . Prove that the ray $[AM$ is the $A$-median in $\triangle ABC$ ([b][u]Serbia[/u][/b]).

The figure below shows a barn in the shape of two congruent pentagonal prisms that intersect at right angles and have a common center. The ends of the prisms are made of a 12 foot by 7 foot rectangle surmounted by an isosceles triangle with sides 10 feet, 10 feet, and 12 feet. Each prism is 30 feet long. Find the volume of the barn in cubic feet. [center][img]https://snag.gy/Ox9CUp.jpg[/img][/center]

If the altitude of an equilateral triangle is $ \sqrt {6}$, then the area is: $ \textbf{(A)}\ 2\sqrt {2} \qquad\textbf{(B)}\ 2\sqrt {3} \qquad\textbf{(C)}\ 3\sqrt {3} \qquad\textbf{(D)}\ 6\sqrt {2} \qquad\textbf{(E)}\ 12$

The locus of the midpoint of a line segment that is drawn from a given external point $ P$ to a given circle with center $ O$ and radius $ r$, is: $ \textbf{(A)}\ \text{a straight line perpendicular to }\overline{PO} \\ \textbf{(B)}\ \text{a straight line parallel to } \overline{PO} \\ \textbf{(C)}\ \text{a circle with center }P\text{ and radius }r \\ \textbf{(D)}\ \text{a circle with center at the midpoint of }\overline{PO}\text{ and radius }2r \\ \textbf{(E)}\ \text{a circle with center at the midpoint }\overline{PO}\text{ and radius }\frac{1}{2}r$

A diagonal of a convex hexagon is called [i]long[/i] if it decomposes the hexagon into two quadrangles. Each pair of [i]long[/i] diagonals decomposes the hexagon into two triangles and two quadrangles. Given is a hexagon with the property, that for each decomposition by two [i]long[/i] diagonals the resulting triangles are both isosceles with the side of the hexagon as base. Show that the hexagon has a circumcircle.

Let $ABC$ be an acute triangle with altitude $AD$ ($D \in BC$). The line through $C$ parallel to $AB$ meets the perpendicular bisector of $AD$ at $G$. Show that $AC = BC$ if and only if $\angle AGC = 90^{\circ}$.

In equilateral triangle $ABC$ with side length $2$, let the parabola with focus $A$ and directrix $BC$ intersect sides $AB$ and $AC$ at $A_1$ and $A_2$, respectively. Similarly, let the parabola with focus $B$ and directrix $CA$ intersect sides $BC$ and $BA$ at $B_1$ and $B_2$, respectively. Finally, let the parabola with focus $C$ and directrix $AB$ intersect sides $CA$ and $C_B$ at $C_1$ and $C_2$, respectively. Find the perimeter of the triangle formed by lines $A_1A_2$, $B_1B_2$, $C_1C_2$.

Point $P$ lies in the interior of rectangle $ABCD$ such that $AP + CP = 27$, $BP - DP = 17$, and $\angle DAP \cong \angle DCP$. Compute the area of rectangle $ABCD$. [i]Proposed by Aaron Lin[/i]

Let $P$ and $Q$ be fixed points in the Euclidean plane. Consider another point $O_0$. Define $O_{i+1}$ as the center of the unique circle passing through $O_i$, $P$ and $Q$. (Assume that $O_i,P,Q$ are never collinear.) How many possible positions of $O_0$ satisfy that $O_{2021}=O_{0}$? [i]Proposed by Fei Peng[/i]