The distance between any two of six given points in the plane is at least $1$. Prove that the distance between some two points is at least $\sqrt{\frac{5+\sqrt5}{2}}$

Determine the locus of points, for whom the ratio of the distances to two given points has a constant value.

Line $ l_2$ intersects line $ l_1$ and line $ l_3$ is parallel to $ l_1$. The three lines are distinct and lie in a plane. The number of points equidistant from all three lines is: $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$

Prove that one can find a $n_{0} \in \mathbb{N}$ such that $\forall m \geq n_{0}$, there exist three positive integers $a$, $b$ , $c$ such that (i) $m^3 < a < b < c < (m+1)^3$; (ii) $abc$ is the cube of an integer.

Let $S$ be a circle with centre $O$. A chord $AB$, not a diameter, divides $S$ into two regions $R_1$ and $R_2$ such that $O$ belongs to $R_2$. Let $S_1$ be a circle with centre in $R_1$, touching $AB$ at $X$ and $S$ internally. Let $S_2$ be a circle with centre in $R_2$, touching $AB$ at $Y$, the circle $S$ internally and passing through the centre of $S$. The point $X$ lies on the diameter passing through the centre of $S_2$ and $\angle YXO=30^o$. If the radius of $S_2$ is $100 $ then what is the radius of $S_1$?

Find the side length of the largest square that can be inscribed in the unit cube.

Let $\vartriangle ABC$ be a triangle with $AB = 4$ and $AC = \frac72$ . Let $\omega$ denote the $A$-excircle of $\vartriangle ABC$. Let $\omega$ touch lines $AB$, $AC$ at the points $D$, $E$, respectively. Let $\Omega$ denote the circumcircle of $\vartriangle ADE$. Consider the line $\ell$ parallel to $BC$ such that $\ell$ is tangent to $\omega$ at a point $F$ and such that $\ell$ does not intersect $\Omega$. Let $\ell$ intersect lines $AB$, $AC$ at the points $X$, $Y$ , respectively, with $XY = 18$ and $AX = 16$. Let the perpendicular bisector of $XY$ meet the circumcircle of $\vartriangle AXY$ at $P$, $Q$, where the distance from $P$ to $F$ is smaller than the distance from $Q$ to$ F$. Let ray $\overrightarrow {PF}$ meet $\Omega$ for the first time at the point $Z$. If $PZ^2 = \frac{m}{n}$ for relatively prime positive integers $m$, $n$, find $m + n$.

Given an $ABC$ acute triangle with $O$ the center of the circumscribed circle. Suppose that $\omega$ is a circle that is tangent to the line $AO$ at point $A$ and also tangent to the line $BC$. Prove that $\omega$ is also tangent to the circumcircle of the triangle $BOC$.

Let $k_1, k_2$ and $k_3$ be three circles with centers $O_1, O_2$ and $O_3$ respectively, such that no center is inside of the other two circles. Circles $k_1$ and $k_2$ intersect at $A$ and $P$, circles $k_1$ and $k_3$ intersect and $C$ and $P$, circles $k_2$ and $k_3$ intersect at $B$ and $P$. Let $X$ be a point on $k_1$ such that the line $XA$ intersects $k_2$ at $Y$ and the line $XC$ intersects $k_3$ at $Z$, such that $Y$ is nor inside $k_1$ nor inside $k_3$ and $Z$ is nor inside $k_1$ nor inside $k_2$. a) Prove that $\triangle XYZ$ is simular to $\triangle O_1O_2O_3$ b) Prove that the $P_{\triangle XYZ} \le 4P_{\triangle O_1O_2O_3}$. Is it possible to reach equation?$ *Note: $P$ denotes the area of a triangle*

Denote $X,Y$ two convex polygons, such that $X$ is contained inside $Y$. Denote $S (X)$, $P (X)$, $S (Y)$, $P (Y)$ the area and perimeter of the first and second polygons, respectively. Prove that $$ \frac{S(X)}{P(X)}<2 \frac{S(Y)}{P(Y)}.$$

Let $\Gamma_1$ be a circle and $P$ a point outside of $\Gamma_1$. The tangents from $P$ to $\Gamma_1$ touch the circle at $A$ and $B$. Let $M$ be the midpoint of $PA$ and $\Gamma_2$ the circle through $P$, $A$ and $B$. Line $BM$ cuts $\Gamma_2$ at $C$, line $CA$ cuts $\Gamma_1$ at $D$, segment $DB$ cuts $\Gamma_2$ at $E$ and line $PE$ cuts $\Gamma_1$ at $F$, with $E$ in segment $PF$. Prove lines $AF$, $BP$, and $CE$ are concurrent.

In $\triangle ABC$, $\angle A=30^{\circ}$, $BC=13$. Given $2$ circles $\gamma_1, \gamma_2$ ith radius $r_1,r_2$ contain $A$ and touch $BC$ at $B$ and $C$ respectively. Find $r_1r_2$.

Let $E_1, E_2$, and $E_3$ be three mutually intersecting ellipses, all in the same plane. Their foci are respectively $F_2, F_3; F_3, F_1$; and $F_1, F_2$. The three foci are not on a straight line. Prove that the common chords of each pair of ellipses are concurrent.

In a convex pentagon $ABCDE$, the area of the triangles $ABC, ABD, ACD$ and $ADE$ are equal and have the value $F$. What is the area of the triangle $BCE$ ?

$\triangle ABC$ satisfies $\angle A=60^{\circ}$. Call its circumcenter and orthocenter $O, H$, respectively. Let $M$ be a point on the segment $BH$, then choose a point $N$ on the line $CH$ such that $H$ lies between $C, N$, and $\overline{BM}=\overline{CN}$. Find all possible value of \[\frac{\overline{MH}+\overline{NH}}{\overline{OH}}\]

Let $E$ be an ellipse where the length of the major axis is $26$, the length of the minor axis is $24$, and the foci are at points $R$ and $S$. Let $A$ and $B$ be points on the ellipse such that $RASB$ forms a non-degenerate quadrilateral, lines $RA$ and $SB$ intersect at $P$ with segment $PR$ containing $A$, and lines $RB$ and $AS$ intersect at Q with segment $QR$ containing $B$. Given that $RA = AS$, $AP = 26$, the perimeter of the non-degenerate quadrilateral $RP SQ$ is $m +\sqrt{n}$, where $m$ and $n$ are integers. Compute $m + n$.

Point $P$ is located inside a square $ABCD$ of side length $10$. Let $O_1$, $O_2$, $O_3$, $O_4$ be the circumcenters of $P AB$, $P BC$, $P CD$, and $P DA$, respectively. Given that $P A+P B +P C +P D = 23\sqrt2$ and the area of $O_1O_2O_3O_4$ is $50$, the second largest of the lengths $O_1O_2$, $O_2O_3$, $O_3O_4$, $O_4O_1$ can be written as $\sqrt{\frac{a}{b}}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.

A cylinder with length $\ell$ has a radius of $6$ meters, and three spheres with radii $3, 4$, and $5$ meters are placed inside the cylinder. If the spheres are packed into the cylinder such that $\ell$ is minimized, determine the length $\ell$.

All the vertices of a convex pentagon are on lattice points. Prove that the area of the pentagon is at least $\frac{5}{2}$. [i]Bogdan Enescu[/i]

Convex quadrilateral $ABCD$ is described on a circle $\omega$, and is not a trapezius inscribed in a circle. Let the tangency points of $\omega$ and $AB, BC, CD, DA$ be $K, L, M, N$ respectively. A circle with a center $I_K$, different from $\omega$ is tangent to the segement $AB$ and lines $AD, BC$. A circle with center $I_L$, different from $\omega$ is tangent to segment $BC$ and lines $AB, CD$. A circle with center $I_M$, different from $\omega$ is tangent to segment $CD$ and lines $AD, BC$. A circle with center $I_N$, different from $\omega$ is tangent to segment $AD$ and lines $AB, CD$. Prove that the lines $I_KK, I_LL, I_MM, I_NN$ are concurrent.

Let $ ABC$, $ l$ and $ P$ be arbitrary triangle, line and point. $ A',B',C'$ are reflections of $ A,B,C$ in point $ P$. $ A''$ is a point on $ B'C'$ such that $ AA''\parallel l$. $ B'',C''$ are defined similarly. Prove that $ A'',B'',C''$ are collinear.

Consider two quadrilaterals $ABCD$ and $A'B'C'D'$ in an affine Euclidian plane such that $AB = A'B', BC = B'C', CD = C'D'$, and $DA = D'A'$. Prove that the following two statements are true: [b](a)[/b] If the diagonals $BD$ and $AC$ are mutually perpendicular, then the diagonals $B'D'$ and $A'C'$ are also mutually perpendicular. [b](b)[/b] If the perpendicular bisector of $BD$ intersects $AC$ at $M$, and that of $B'D'$ intersects $A'C'$ at $M'$, then $\frac{\overline{MA}}{\overline{MC}}=\frac{\overline{M'A'}}{\overline{M'C'}}$ (if $MC = 0$ then $M'C' = 0$).

Inside the triangle $ABC $, the point $P $ is selected so that $BC = AP $ and $\angle APC = 180 {} ^ \circ - \angle ABC $. On the side $AB $ there is a point $K $, for which $AK = KB + PC $. Prove that $\angle AKC = 90 {} ^ \circ $. (Danilo Hilko)

Let $ P $ be a point on the side $ AB $ of the triangle $ ABC. $ The parallels through $ P $ of the medians $ AA_1,BB_1 $ intersect $ BC,AC $ at $ R,Q, $ respectively. Show that $ P, $ the middlepoint of $ RQ $ and the centroid of $ ABC $ are collinear.

In trapezoid $ABCD$, diagonal $AC$ is the bisector of angle $A$. Point $K$ is the midpoint of diagonal $AC$. It is known that $DC = DK$. Find the ratio of the bases $AD: BC$.