In $ \triangle{ABC}$, we have $ AB\equal{}1$ and $ AC\equal{}2$. Side $ BC$ and the median from $ A$ to $ BC$ have the same length. What is $ BC$? $ \textbf{(A)}\ \frac{1\plus{}\sqrt2}{2} \qquad \textbf{(B)}\ \frac{1\plus{}\sqrt3}{2} \qquad \textbf{(C)}\ \sqrt2 \qquad \textbf{(D)}\ \frac{3}{2} \qquad \textbf{(E)}\ \sqrt3$

Let $ABCD$ be a square, point $E$ be the midpoint of the side $BC$. On the side $AB$ mark a point $F$ such that $FE \perp DE$. Prove that $AF + BE = DF$. (Ercole Suppa, Italy)

Let $ABC$ be an equilateral triangle. Let $P$ and $S$ be points on $AB$ and $AC$, respectively, and let $Q$ and $R$ be points on $BC$ such that $PQRS$ is a rectangle. If $PQ = \sqrt3 PS$ and the area of $PQRS$ is $28\sqrt3$, what is the length of $PC$?

Let $ABC$ be a triangle and $D$ the midpoint of the side $BC$. Define points $E$ and $F$ on $AC$ and $B$, respectively, such that $DE=DF$ and $\angle EDF =\angle BAC$. Prove that $$DE\geq \frac {AB+AC} 4.$$

Let the circles $k_1$ and $k_2$ intersect at two points $A$ and $B$, and let $t$ be a common tangent of $k_1$ and $k_2$ that touches $k_1$ and $k_2$ at $M$ and $N$ respectively. If $t\perp AM$ and $MN=2AM$, evaluate the angle $NMB$.

In a convex trapezoid $ABCD$, side $AD$ is twice the length of the other sides. Let $E$ and $F$ be points on segments $AC$ and $BD$, respectively, such that $\angle BEC = 70^\circ$ and $\angle BFC = 80^\circ$. Determine the ratio of the areas of quadrilaterals $BEFC$ and $ABCD$. [i]Proposed by Zaza Meliqidze, Georgia [/i]

A regular pentagon of side length $1$ is given. Determine the smallest $r$ for which the pentagon can be covered by five discs of radius $r$ and justify your answer.

Let $P_1$ be a convex polyhedron with vertices $A_1,A_2,\ldots,A_9$. Let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves $A_1$ to $A_i$. Prove that at least two of the polyhedra $P_1,P_2,\ldots,P_9$ have an interior point in common.

We are given a finite collection of segments in the plane, of total length 1. Prove that there exists a line $ l$ such that the sum of the lengths of the projections of the given segments to the line $ l$ is less than $ \frac{2}{\pi}.$

In triangle $ABC$ ($AB\not= AC$), the incircle is tangent to the sides of $BC$ ,$CA$ , $AB$ at $D$ ,$E$, $F$ respectively. Let $AD$ meet the incircle again at point $P$, let $EF$ and the line passing through the point $P$ and perpendicular to $AD$ intersect at $Q$. Let $AQ$ intersect $DE$ at $X$ and $DF$ at $Y$. Prove that $AX=AY$. [i](tatari/nightmare)[/i]

The sides of the triangle $ABC$ lie on the sides of the angle $MAN$. Construct a triangle $ABC$ if the point $O$ of the intersection of its medians is given.

Is it possible to inscribe an $N$-gon in a circle so that all the lengths of its sides are different and all its angles (in degrees) are integer, where a) $N = 19$, b) $N = 20$ ? Mikhail Malkin

Let $ABC$ be a triangle with $AB = 12$, $BC = 5$, $AC = 13$. Let$ D$ and $E$ be the feet of the internal and external angle bisectors from $B$, respectively. (The external angle bisector from $B$ bisects the angle between $BC$ and the extension of $AB$.) Let $\omega$ be the circumcircle of $\vartriangle BDE$, extend $AB$ so that it intersects $\omega$ again at $F$. Extend $F C$ to meet $\omega$ again at $X$, and extend $AX$ to meet $\omega$ again at $G$. Find $F G$.

Given the equilateral triangle $ABC$. It is known that the radius of the inscribed circle is in this triangle is equal to $1$. The rectangle $ABDE$ is such that point $C$ belongs to its side $DE$. Find the radius of the circle circumscribed around the rectangle $ABDE$.

A toilet paper roll is a cylinder of radius $8$ and height $6$ with a hole in the shape of a cylinder of radius $2$ and the same height. That is, the bases of the roll are annuli with inner radius $2$ and outer radius $8$. Compute the surface area of the roll.

Consider the dark square in an array of unit squares, part of which is shown. The first ring of squares around this center square contains $ 8$ unit squares. The second ring contains $ 16$ unit squares. If we continue this process, the number of unit squares in the $ 100^\text{th}$ ring is $ \textbf{(A)}\ 396 \qquad \textbf{(B)}\ 404 \qquad \textbf{(C)}\ 800 \qquad \textbf{(D)}\ 10,\!000 \qquad \textbf{(E)}\ 10,\!404$ [asy]unitsize(3mm); defaultpen(linewidth(1pt)); fill((2,2)--(2,7)--(7,7)--(7,2)--cycle, mediumgray); fill((3,3)--(6,3)--(6,6)--(3,6)--cycle, gray); fill((4,4)--(5,4)--(5,5)--(4,5)--cycle, black); for(real i=0; i<=9; ++i) { draw((i,0)--(i,9)); draw((0,i)--(9,i)); }[/asy]

On the plane we mark $n$ ($n > 2$) straight lines passing through one point $O$ in such a way that for any two of them there is a marked straight line that bisects one of the pairs of vertical angles, formed by these straight lines. Prove that the drawn straight lines divide full angle into equal parts.

Find all positive integers $n \geq 3$, for which it is possible to draw $n$ chords on a circle, with their $2n$ endpoints being pairwise distinct, such that each chords intersects exactly $k$ others for: (a) $k=n-2$, (b) $k=n-3$.

The vertices $ A$ and $ B$ of an equilateral triangle $ ABC$ lie on a circle $k$ of radius $1$, and the vertex $ C$ is in the interior of the circle $ k$. A point $ D$, different from $ B$, lies on $ k$ so that $ AD\equal{}AB$. The line $ DC$ intersects $ k$ for the second time at point $ E$. Find the length of the line segment $ CE$.

Let \( n \geq 3 \) be an integer. For every two adjacent vertices \( A \) and \( B \) of a convex \( n \)-gon, we find a vertex \( C \) such that the angle \( \angle ACB \) is the largest, and write down the measure in degrees. Find the smallest possible value of the sum of the written \( n \) numbers.

Let $ABCD$ be a parallelogram. Let $M$ be a point on the side $AB$ and $N$ be a point on the side $BC$ such that the segments $AM$ and $CN$ have equal lengths and are non-zero. The lines $AN$ and $CM$ meet at $Q$. Prove that the line $DQ$ is the bisector of the angle $\measuredangle ADC$. [i]Alternative formulation.[/i] Let $ABCD$ be a parallelogram. Let $M$ and $N$ be points on the sides $AB$ and $BC$, respectively, such that $AM=CN\neq 0$. The lines $AN$ and $CM$ intersect at a point $Q$. Prove that the point $Q$ lies on the bisector of the angle $\measuredangle ADC$.

let $\omega_1$ be a circle with $O_1$ as its center , let $\omega_2$ be a circle passing through $O_1$ with center $O_2$ let $A$ be one of the intersection of $\omega_1$ and $\omega_2$ let $x$ be a line tangent line to $\omega_1$ passing from $A$ let $\omega_3$ be a circle passing through $O_1,O_2$ with its center on the line $x$ and intersect $\omega_2$ at $P$ (not $O_1$) prove that the reflection of $P$ through $x$ is on $\omega_1$

A right triangle is drawn on the plane. How to use only a compass to mark two points, such that the distance between them is equal to the diameter of the circle inscribed in this triangle?

The points $A_0, A_1, A_2, A_3, A_4$ divide a unit circle (circle of radius $1$) into five equal parts. Prove that the chords $A_0, A_1, A_0, A_2$ satisfy $$(A_0A_1 \cdot A_0A_2)^2= 5$$

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.