What is the minimum possible perimeter of a triangle two of whose sides are along the x- and y-axes and such that the third contains the point $(1,2)$?

Let $\Gamma_{1}$ be a circle. $AB$ is a diameter, $\ell$ is the tangent at $B$, and $M$ is a point on $\Gamma_{1}$ other than $A$. $\Gamma_{2}$ is a circle tangent to $\ell$, and also to $\Gamma_{1}$ at $M$. a) Determine the point of tangency $P$ of $\ell$ and $\Gamma_{2}$ and find the locus of the center of $\Gamma_{2}$ as $M$ varies. b) Show that there exists a circle that is always orthogonal to $\Gamma_{2}$, regardless of the position of $M$.

Let $P$ be a point in the interior of a triangle $ABC$ and let the rays $\overrightarrow{AP}, \overrightarrow{BP}$ and $\overrightarrow{CP}$ intersect the sides $BC, CA$ and $AB$ in $A_1,B_1$ and $C_1$, respectively. Let $D$ be the foot of the perpendicular from $A_1$ to $B_1C_1$. Show that \[\frac{CD}{BD}=\frac{B_1C}{BC_1} \cdot \frac{C_1A}{AB_1}.\]

Let $M_n=\{(u,v) \in S^n \times S^n: u \cdot v=0\}$, where $n \ge 2$, and $u \cdot v$ is the Euclidean inner product of $u$ and $v$. Suppose that the topology of $M_n$ is induces from $S^n \times S^n$. (1) Prove that $M_n$ is a connected regular submanifold of $S^n \times S^n$. (2) $M_n$ is Lie Group if and only if $n=2$.

Find out the maximum possible area of the triangle $ABC$ whose medians have lengths satisfying inequalities $m_a \le 2, m_b \le 3, m_c \le 4$.

In equilateral $\triangle ABC$ let points $D$ and $E$ trisect $\overline{BC}$. Then $\sin \left( \angle DAE \right)$ can be expressed in the form $\tfrac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is an integer that is not divisible by the square of any prime. Find $a+b+c$.

Minimal distance of a finite set of different points in space is length of the shortest segment, whose both ends belong to this set and segment has length greater than $0$. a) Prove there exist set of $8$ points on sphere with radius $R$, whose minimal distance is greater than $1,15R$. b) Does there exist set of $8$ points on sphere with radius $R$, whose minimal distance is greater than $1,2R$?

Prove that the sum of an odd number of vectors of length 1, of common origin $O$ and all situated in the same semi-plane determined by a straight line which goes through $O,$ is at least 1.

In triangle $ABC$, $AB > BC > CA$, $AB=6$, $\angle{B}-\angle{C}=90^o$. The incircle touches $BC$ at $E$ and $EF$ is a diameter of the incircle. Radical $AF$ intersect $BC$ at $D$. $DE$ equals to the circumradius of $\triangle{ABC}$. Find $BC$ and $AC$.

Consider a triangle $ABC$ with $BC = 3$. Choose a point $D$ on $BC$ such that $BD = 2$. Find the value of \[AB^2 + 2AC^2 - 3AD^2.\]

In $\mathbb{R}^3$, let there be a cube $Q$ and a sequence of other cubes, all of which are homothetic to $Q$ with coefficients of homothety that are each smaller than $1$. Prove that if this sequence of homothetic cubes completely fills $Q$, the sum of their coefficients of homothety is not less than $4$.

$ABCD$ is cyclic quadrilateral and $O$ is the center of its circumcircle. Suppose that $AD \cap BC = E$ and $AC \cap BD = F$. Circle $\omega$ is tanget to line $AC$ and $BD$. $PQ$ is a diameter of $\omega$ that $F$ is orthocenter of $EPQ$. Prove that line $OE$ is passing through center of $\omega$ [i]Proposed by Mahdi Etesami Fard [/i]

In a triangle $ABC$, the altitude from $A$ meets the circumcircle again at $T$ . Let $O$ be the circumcenter. The lines $OA$ and $OT$ intersect the side $BC$ at $Q$ and $M$, respectively. Prove that \[\frac{S_{AQC}}{S_{CMT}} = \biggl( \frac{ \sin B}{\cos C} \biggr)^2 .\]

Equilateral $\triangle ABC$ with side length $14$ is rotated about its center by angle $\theta$, where $0 < \theta < 60^{\circ}$, to form $\triangle DEF$. The area of hexagon $ADBECF$ is $91\sqrt{3}$. What is $\tan\theta$? [asy] defaultpen(fontsize(13)); size(200); pair O=(0,0),A=dir(225),B=dir(-15),C=dir(105),D=rotate(38.21,O)*A,E=rotate(38.21,O)*B,F=rotate(38.21,O)*C; draw(A--B--C--A,gray+0.4);draw(D--E--F--D,gray+0.4); draw(A--D--B--E--C--F--A,black+0.9); dot(O); dot("$A$",A,dir(A)); dot("$B$",B,dir(B)); dot("$C$",C,dir(C)); dot("$D$",D,dir(D)); dot("$E$",E,dir(E)); dot("$F$",F,dir(F)); [/asy] $\textbf{(A)}~\displaystyle\frac{3}{4}\qquad\textbf{(B)}~\displaystyle\frac{5\sqrt{3}}{11}\qquad\textbf{(C)}~\displaystyle\frac{4}{5}\qquad\textbf{(D)}~\displaystyle\frac{11}{13}\qquad\textbf{(E)}~\displaystyle\frac{7\sqrt{3}}{13}$

An angle is given in a plane. Using only a compass, one must find out $(a)$ if this angle is acute. Find the minimal number of circles one must draw to be sure. $(b)$ if this angle equals $31^{\circ}$.(One may draw as many circles as one needs).

Let $\Omega$ and $O$ be the circumcircle and the circumcentre of an acute-angled triangle $ABC$ with $AB > BC$. The angle bisector of $\angle ABC$ intersects $\Omega$ at $M \ne B$. Let $\Gamma$ be the circle with diameter $BM$. The angle bisectors of $\angle AOB$ and $\angle BOC$ intersect $\Gamma$ at points $P$ and $Q,$ respectively. The point $R$ is chosen on the line $P Q$ so that $BR = MR$. Prove that $BR\parallel AC$. (Here we always assume that an angle bisector is a ray.) [i]Proposed by Sergey Berlov, Russia[/i]

(a) Let the incenter of $\triangle ABC$ be $I$. We connect $I$ other $3$ vertices and divide $\triangle ABC$ into $3$ small triangles which has area $2,3$ and $4$. Find the area of the inscribed circle of $\triangle ABC$. (b) Let $ABCD$ be a parallelogram. Point $E,F$ is on $AB,BC$ respectively. If $[AED]=7,[EBF]=3,[CDF]=6$, then find $[DEF].$ (Here $[XYZ]$ denotes the area of $XYZ$)

Consider $ \rho$ a semicircle of diameter $ AB$. A parallel to $ AB$ cuts the semicircle at $ C, D$ such that $ AD$ separates $ B, C$. The parallel at $ AD$ through $ C$ intersects the semicircle the second time at $ E$. Let $ F$ be the intersection point of the lines $ BE$ and $ CD$. The parallel through $ F$ at $ AD$ cuts $ AB$ in $ P$. Prove that $ PC$ is tangent to $ \rho$. [i]Author: Cosmin Pohoata[/i]

Circle $\omega$ lies inside the circle $\Omega$ and touches it internally at point $P$. Point $S$ is taken on $\omega$ and the tangent to $\omega$ is drawn through it. This tangent meets $\Omega$ at points $A$ and $B$. Let $I$ be the centre of $\omega$. Find the locus of circumcentres of triangles $AIB$.

Let $ABCD$ be a trapezoid with $AB\parallel CD$. The bisectors of $\angle CDA$ and $\angle DAB$ meet at $E$, the bisectors of $\angle ABC$ and $\angle BCD$ meet at $F$, the bisectors of $\angle BCD$ and $\angle CDA$ meet at $G$, and the bisectors of $\angle DAB$ and $\angle ABC$ meet at $H$. Quadrilaterals $EABF$ and $EDCF$ have areas $24$ and $36$, respectively, and triangle $ABH$ has area $25$. Find the area of triangle $CDG$.

Note that $8^3 - 7^3 = 169 = 13^2$ and $13 = 2^2 + 3^2.$ Prove that if the difference between two consecutive cubes is a square, then it is the square of the sum of two consecutive squares.

Let $ABC$ be a triangle, and let $BCDE$, $CAFG$, $ABHI$ be squares that do not overlap the triangle with centers $X$, $Y$, $Z$ respectively. Given that $AX=6$, $BY=7$, and $CA=8$, find the area of triangle $XYZ$.

Let be a convex polygon with $n >5$ vertices and area $1$. Prove that there exists a convex hexagon inside the given polygon with area at least $\dfrac{3}{4}$

There are $10$ points on a circle and all possible segments are drawn on the which two of these points are the endpoints. Determine the probability that selecting two segments randomly, they intersect at some point (it could be on the circumference).

Let $ P$ be the the isogonal conjugate of $ Q$ with respect to triangle $ ABC$, and $ P,Q$ are in the interior of triangle $ ABC$. Denote by $ O_{1},O_{2},O_{3}$ the circumcenters of triangle $ PBC,PCA,PAB$, $ O'_{1},O'_{2},O'_{3}$ the circumcenters of triangle $ QBC,QCA,QAB$, $ O$ the circumcenter of triangle $ O_{1}O_{2}O_{3}$, $ O'$ the circumcenter of triangle $ O'_{1}O'_{2}O'_{3}$. Prove that $ OO'$ is parallel to $ PQ$.