Three cyclists ride along a circular road with radius $1$ km counterclockwise. Their velocities are constant and different. Does there necessarily exist (in a sufficiently long time) a moment when all the three distances between cyclists are greater than $1$ km? by V. Protasov

Let $\mathcal C$ be a circle of radius $1$. (a) Determine the triangles $ABC$ inscribed in $\mathcal C$ for which $AB^2+BC^2+CA^2$ is maximal. (b) Determine the quadrilaterals $ABCD$ inscribed in $\mathcal C$ for which $AB^2+AC^2+AD^2+BC^2+BD^2+CD^2$ is maximal.

Point $D$ lies on side $AB$ of triangle $ABC$, and $$\frac{AD}{DC} = \frac{AB}{BC}.$$ Prove that angle $C$ is obtuse. (Sergey Berlov)

Let $O$ be a point in the plane of the triangle $ABC$. A circle $\mathcal{C}$ which passes through $O$ intersects the second time the lines $OA,OB,OC$ in $P,Q,R$ respectively. The circle $\mathcal{C}$ also intersects for the second time the circumcircles of the triangles $BOC$, $COA$ and $AOB$ respectively in $K,L,M$. Prove that the lines $PK,QL$ and $RM$ are concurrent.

Consider a circle centered at $O$ with radius $r$ and a line $\ell$ not passing through $O$. A grasshopper is jumping to and fro between the points of the circle and the line, the length of each jump being $r$. Prove that there are at most $8$ points for the grasshopper to reach.

Let $ABCDEF$ be a hexagon inscribed in a circle (with vertices in that order) with $\angle B + \angle C > 180^o$ and $\angle E + \angle F > 180^o$. Let the lines $AB$ and $CD$ intersect at $X$ and the lines $AF$ and $DE$ intersect at $S$. Let $XY$ and $ST$ be the diameters of the circumcircles of $\vartriangle BCX$ and $\vartriangle EFS$ respectively. If $U$ is the intersection point of the lines $BX$ and $ES$ and $V$ is the intersection point of the lines $BY$ and $ET,$ prove that the lines $UV, XY$ and $ST$ are all parallel.

In a triangle $ABC$ with $\angle ACB = 90^{\circ}$ $D$ is the foot of the altitude of $C$. Let $E$ and $F$ be the reflections of $D$ with respect to $AC$ and $BC$. Let $O_1$ and $O_2$ be the circumcenters of $\triangle {ECB}$ and $\triangle {FCA}$. Show that: $$2O_1O_2=AB$$

In $ \triangle ABC$, $ AB \equal{} BC$, and $ BD$ is an altitude. Point $ E$ is on the extension of $ \overline{AC}$ such that $ BE \equal{} 10$. The values of $ \tan CBE$, $ \tan DBE$, and $ \tan ABE$ form a geometric progression, and the values of $ \cot DBE$, $ \cot CBE$, $ \cot DBC$ form an arithmetic progression. What is the area of $ \triangle ABC$? [asy]unitsize(3mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair D=(0,0), C=(3,0), A=(-3,0), B=(0, 8), Ep=(6,0); draw(A--B--Ep--cycle); draw(D--B--C); label("$A$",A,S); label("$D$",D,S); label("$C$",C,S); label("$E$",Ep,S); label("$B$",B,N);[/asy]$ \textbf{(A)}\ 16 \qquad \textbf{(B)}\ \frac {50}{3} \qquad \textbf{(C)}\ 10\sqrt3 \qquad \textbf{(D)}\ 8\sqrt5 \qquad \textbf{(E)}\ 18$

Let $ABCD$ be a parallelogram, let $X$ and $Y$ in the segments $AB$ and $CD$, respectively. The segments $AY$ and $DX$ intersects in $P$ and the segments $BY$ and $DX$ intersects in $Q$, show that the line $PQ$ passes in a fixed point(independent of the positions of the points $X$ and $Y$). I guess that the fixed point is the midpoint of $BD$.

Triangle $ABC$ has side lengths $AB=120$, $BC=220$, and $AC=180$. Lines $\ell_{A}$, $\ell_{B}$, and $\ell_{C}$ are drawn parallel to $\overline{BC}$, $\overline{AC}$, and $\overline{AB}$, respectively, such that the intersection of $\ell_{A}$, $\ell_{B}$, and $\ell_{C}$ with the interior of $\triangle ABC$ are segments of length $55$, $45$, and $15$, respectively. Find the perimeter of the triangle whose sides lie on $\ell_{A}$, $\ell_{B}$, and $\ell_{C}$.

In triangle $\vartriangle ABC$, $D$ is the midpoint of $BC$. Points $E$ and $F$ are chosen on side $AC$ so that $AF = F E = EC$. Let $AD$ intersect $BE$ and $BF$ and $G$ and $H$, respectively. Find the ratio of the areas of $\vartriangle BGH$ and $\vartriangle ABC$.

Let $P$ be a point on the circumcircle of a triangle $A_{1}A_{2}A_{3}$, and let $H$ be the orthocenter of the triangle. The feet $B_{1},B_{2},B_{3}$ of the perpendiculars from $P$ to $A_{2}A_{3},A_{3}A_{1},A_{1}A_{2}$ lie on a line. Prove that this line bisects the segment $PH$.

$ABCD$ is a square such that $AB$ lies on the line $y=x+4$ and points $C$ and $D$ lie on the graph of parabola $y^2=x$. Compute the sum of all possible areas of $ABCD$.

If an integer $a > 1$ is given such that $\left(a-1\right)^3+a^3+\left(a+1\right)^3$ is the cube of an integer, then show that $4\mid a$.

Two circles intersect at points $M$ and $N$. Let $A$ be a point on the first circle, distinct from $M,N$. The lines $AM$ and $AN$ meet the second circle again at $B$ and $C$, respectively. Prove that the tangent to the first circle at $A$ is parallel to $BC$.

Let $\omega$ be a circle in the plane and $A,B$ two points lying on it. We denote by $M$ the midpoint of $AB$ and let $P \ne M$ be a new point on $AB$. Build circles $\gamma$ and $\delta$ tangent to $AB$ at $P$ and to $\omega$ at $C$, respectively $D$. Consider $E$ to be the point diametrically opposed to $D$ in $\omega$. Prove that the circumcenter of $\triangle BMC$ lies on the line $BE$.

Consider a rectangular parallelepiped $ABCDA'B'C'D'$ in which we denote $AB=a,\ BC=b,\ AA'=c$. Let $DE\perp AC,\ DF\perp A'C,\ E\in AC,\ F \in A'C$ and $C'P\perp B'D',\ C'Q\perp BD',\ P\in B'D',\ Q\in BD'$. Prove that the planes $(DEF)$ and $(C'PQ)$ are perpendicular if and only if $a^2+c^2=b^2$. [i]Sorin Peligrad[/i]

Let $\vartriangle ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. Let $D$, $E$, and $F$ be the midpoints of $AB$, $BC$, and $CA$ respectively. Imagine cutting $\vartriangle ABC$ out of paper and then folding $\vartriangle AFD$ up along $FD$, folding $\vartriangle BED$ up along $DE$, and folding $\vartriangle CEF$ up along $EF$ until $A$, $B$, and $C$ coincide at a point $G$. The volume of the tetrahedron formed by vertices $D$, $E$, $F$, and $G$ can be expressed as $\frac{p\sqrt{q}}{r}$ , where $p$, $q$, and $r$ are positive integers, $p$ and $r$ are relatively prime, and $q$ is square-free. Find $p + q + r$.

A convex polygon is circumscribed around a circle. Points of contact of its sides with the circle form a polygon with the same set of angles (the order of angles may differ). Is it true that the polygon is regular?

Let $ABCD$ be a cyclic quadrilateral with perpendicular diagonals and circumcenter $O$. Let $g$ be the line obtained by reflection of the diagonal $AC$ along the angle bisector of $\angle BAD$. Prove that the point $O$ lies on the line $g$. [i]Proposed by Theresia Eisenkölbl[/i]

Three 12 cm $\times$ 12 cm squares are each cut into two pieces $A$ and $B$, as shown in the first figure below, by joining the midpoints of two adjacent sides. These six pieces are then attached to a regular hexagon, as shown in the second figure, so as to fold into a polyhedron. What is the volume (in $\text{cm}^3$) of this polyhedron? [asy] defaultpen(fontsize(10)); size(250); draw(shift(0, sqrt(3)+1)*scale(2)*rotate(45)*polygon(4)); draw(shift(-sqrt(3)*(sqrt(3)+1)/2, -(sqrt(3)+1)/2)*scale(2)*rotate(165)*polygon(4)); draw(shift(sqrt(3)*(sqrt(3)+1)/2, -(sqrt(3)+1)/2)*scale(2)*rotate(285)*polygon(4)); filldraw(scale(2)*polygon(6), white, black); pair X=(2,0)+sqrt(2)*dir(75), Y=(-2,0)+sqrt(2)*dir(105), Z=(2*dir(300))+sqrt(2)*dir(225); pair[] roots={2*dir(0), 2*dir(60), 2*dir(120), 2*dir(180), 2*dir(240), 2*dir(300)}; draw(roots[0]--X--roots[1]); label("$B$", centroid(roots[0],X,roots[1])); draw(roots[2]--Y--roots[3]); label("$B$", centroid(roots[2],Y,roots[3])); draw(roots[4]--Z--roots[5]); label("$B$", centroid(roots[4],Z,roots[5])); label("$A$", (1+sqrt(3))*dir(90)); label("$A$", (1+sqrt(3))*dir(210)); label("$A$", (1+sqrt(3))*dir(330)); draw(shift(-10,0)*scale(2)*polygon(4)); draw((sqrt(2)-10,0)--(-10,sqrt(2))); label("$A$", (-10,0)); label("$B$", centroid((sqrt(2)-10,0),(-10,sqrt(2)),(sqrt(2)-10, sqrt(2))));[/asy]

Two equilateral triangles $ABC$ and $DEF$, each with side length $1$, are drawn in $2$ parallel planes such that when one plane is projected onto the other, the vertices of the triangles form a regular hexagon $AF BDCE$. Line segments $AE$, $AF$, $BF$, $BD$, $CD,$ and $CE$ are drawn, and suppose that each of these segments also has length $1$. Compute the volume of the resulting solid that is formed.

On sides $AB, BC$ and $CA$ of triangle $ABC$ are located points $P, M$ and $K$, respectively, so that $AM, BK$ and $CP$ intersect in one point and the sum of the vectors $\overrightarrow{AM}, \overrightarrow{BK}$ and $\overrightarrow{CP}$ equals $ \overrightarrow{0}$. Prove that $K, M$ and $P$ are midpoints of the sides of triangle $ABC$ on which they are located.

Triangle $ABC_0$ has a right angle at $C_0$. Its side lengths are pairwise relatively prime positive integers, and its perimeter is $p$. Let $C_1$ be the foot of the altitude to $\overline{AB}$, and for $n\geq 2$, let $C_n$ be the foot of the altitude to $\overline{C_{n-2}B}$ in $\triangle C_{n-2}C_{n-1}B$. The sum $\sum\limits_{n=1}^{\infty}C_{n-1}C_n = 6p$. Find $p$.

Let $D$, $E$, $F$ be the midpoints of the arcs $BC$, $CA$, $AB$ on the circumcircle of a triangle $ABC$ not containing the points $A$, $B$, $C$, respectively. Let the line $DE$ meets $BC$ and $CA$ at $G$ and $H$, and let $M$ be the midpoint of the segment $GH$. Let the line $FD$ meet $BC$ and $AB$ at $K$ and $J$, and let $N$ be the midpoint of the segment $KJ$. a) Find the angles of triangle $DMN$; b) Prove that if $P$ is the point of intersection of the lines $AD$ and $EF$, then the circumcenter of triangle $DMN$ lies on the circumcircle of triangle $PMN$.