Let $t$ be positive number. Draw two tangent lines to the palabola $y=x^{2}$ from the point $(t,-1).$ Denote the area of the region bounded by these tangent lines and the parabola by $S(t).$ Find the minimum value of $\frac{S(t)}{\sqrt{t}}.$

Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.

Let $\{R_i\}_{1\leq i\leq n}$ be a family of disjoint closed rectangular surfaces with total area 4 such that their projections of the $Ox$ axis is an interval. Prove that there exist a triangle with vertices in $\displaystyle \bigcup_{i=1}^n R_i$ which has an area of at least 1. [Thanks Grobber for the correction]

The polygon enclosed by the solid lines in the figure consists of $ 4$ congruent squares joined edge-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing? [asy]unitsize(10mm); defaultpen(fontsize(10pt)); pen finedashed=linetype("4 4"); filldraw((1,1)--(2,1)--(2,2)--(4,2)--(4,3)--(1,3)--cycle,grey,black+linewidth(.8pt)); draw((0,1)--(0,3)--(1,3)--(1,4)--(4,4)--(4,3)-- (5,3)--(5,2)--(4,2)--(4,1)--(2,1)--(2,0)--(1,0)--(1,1)--cycle,finedashed); draw((0,2)--(2,2)--(2,4),finedashed); draw((3,1)--(3,4),finedashed); label("$1$",(1.5,0.5)); draw(circle((1.5,0.5),.17)); label("$2$",(2.5,1.5)); draw(circle((2.5,1.5),.17)); label("$3$",(3.5,1.5)); draw(circle((3.5,1.5),.17)); label("$4$",(4.5,2.5)); draw(circle((4.5,2.5),.17)); label("$5$",(3.5,3.5)); draw(circle((3.5,3.5),.17)); label("$6$",(2.5,3.5)); draw(circle((2.5,3.5),.17)); label("$7$",(1.5,3.5)); draw(circle((1.5,3.5),.17)); label("$8$",(0.5,2.5)); draw(circle((0.5,2.5),.17)); label("$9$",(0.5,1.5)); draw(circle((0.5,1.5),.17));[/asy] $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$

Let $ABC$ be a triangle, and $M$ the midpoint of its side $BC$. Let $\gamma$ be the incircle of triangle $ABC$. The median $AM$ of triangle $ABC$ intersects the incircle $\gamma$ at two points $K$ and $L$. Let the lines passing through $K$ and $L$, parallel to $BC$, intersect the incircle $\gamma$ again in two points $X$ and $Y$. Let the lines $AX$ and $AY$ intersect $BC$ again at the points $P$ and $Q$. Prove that $BP = CQ$.

$ABCD$ is a square of center $O$. On the sides $DC$ and $AD$ the equilateral triangles DAF and DCE have been constructed. Decide if the area of the $EDF$ triangle is greater, less or equal to the area of the $DOC$ triangle. [img]https://4.bp.blogspot.com/-o0lhdRfRxl0/XNYtJgpJMmI/AAAAAAAAKKg/lmj7KofAJosBZBJcLNH0JKjW3o17CEMkACK4BGAYYCw/s1600/may4_2.gif[/img]

Alice and Bob play the following "point guessing game." First, Alice marks an equilateral triangle $ABC$ and a point $D$ on segment $BC$ satisfying $BD=3$ and $CD=5$. Then, Alice chooses a point $P$ on line $AD$ and challenges Bob to mark a point $Q\neq P$ on line $AD$ such that $\frac{BQ}{QC}=\frac{BP}{PC}$. Alice wins if and only if Bob is unable to choose such a point. If Alice wins, what are the possible values of $\frac{BP}{PC}$ for the $P$ she chose?

$\triangle ABC$ has semiperimeter $s$ and area $F$ . A square $P QRS$ with side length $x$ is inscribed in $ABC$ with $P$ and $Q$ on $BC$, $R$ on $AC$, and $S$ on $AB$. Similarly, $y$ and $z$ are the sides of squares two vertices of which lie on $AC$ and $AB$, respectively. Prove that \[\frac 1x +\frac 1y + \frac 1z \le \frac{s(2+\sqrt3)}{2F}\]

We consider two circles $k_1(M_1, r_1)$ and $k_2(M_2, r_2)$ with $z = M_1M_2 > r_1+r_2$ and a common outer tangent with the tangent points $P_1$ and $P2$ (that is, they lie on the same side of the connecting line $M_1M_2$). We now change the radii so that their sum is $r_1+r_2 = c$ remains constant. What set of points does the midpoint of the tangent segment $P_1P_2$ run through, when $r_1$ varies from $0$ to $c$?

Let scalene triangle $ABC$ have circumcenter $O$ and incenter $I$. Its incircle $\omega$ is tangent to sides $BC,CA,$ and $AB$ at $D,E,$ and $F$, respectively. Let $P$ be the foot of the altitude from $D$ to $EF$, and let line $DP$ intersect $\omega$ again at $Q \ne D$. The line $OI$ intersects the altitude from $A$ to$ BC$ at $T$. Given that $OI \|BC,$ show that $PQ=PT$.

Let $ABC$ be a triangle with circumcircle $\Gamma$. $D$ is the midpoint of arc $BC$ (this arc does not contain $A$). $E$ is the common point of $BC$ and the perpendicular bisector of $BD$. $F$ is the common point of $AC$ and the parallel to $AB$ containing $D$. $G$ is the common point of $EF$ and $AB$. $H$ is the common point of $GD$ and $AC$. Show that $GAH$ is isosceles.

Let $A_1A_2A_3A_4$ be a non-cyclic quadrilateral. Let $O_1$ and $r_1$ be the circumcentre and the circumradius of the triangle $A_2A_3A_4$. Define $O_2,O_3,O_4$ and $r_2,r_3,r_4$ in a similar way. Prove that \[\frac{1}{O_1A_1^2-r_1^2}+\frac{1}{O_2A_2^2-r_2^2}+\frac{1}{O_3A_3^2-r_3^2}+\frac{1}{O_4A_4^2-r_4^2}=0.\] [i]Proposed by Alexey Gladkich, Israel[/i]

The sides of $\triangle ABC$ measure 11,20, and 21 units. We fold it along $PQ,QR,RP$ where $P,Q,R$ are the midpoints of its sides until $A,B,C$ coincide. What is the volume of the resulting tetrahedron?

Let $ABC$ be a right triangle with hypotenuse $BC$. The tangent to the circumcircle of triangle $ABC$ at $A$ intersects the line $BC$ at $T$. The points $D$ and $E$ are chosen so that $AD = BD, AE = CE,$ and $\angle CBD = \angle BCE < 90^{\circ}$. Prove that $D, E,$ and $T$ are collinear. Proposed by [i]Nikola Velov, Macedonia[/i]

A right-angled triangle has perimeter $60$ and the altitude of the hypotenuse has a length $12$. Determine the lengths of the sides.

Let the convex quadrilateral $ABCD$ with $AD = BC$ ¸and $\angle A + \angle B = 120^o$. Take a point $P$ in the plane so that the line $CD$ separates the points $A$ and $P$, and the $DCP$ triangle is equilateral. Show that the triangle $ABP$ is equilateral. It is the true statement for a non-convex quadrilateral?

Let $ ABC $ be a triangle with $ AB=AC $ and chose such that $ \angle BAC <120^{\circ } . $ On the altitude of $ ABC $ from $ A, $ consider the point $ O $ so that $ \angle BOC =120^{\circ } , $ and an arbitrary point $ M\neq O $ in the interior of $ ABC. $ Show that $ MA+MB+MC>OA+OB+OC. $ [i]Gheorghe Bucur[/i]

3-There is given a trapezoid $ ABCD$ in the plane with $ BC\parallel{}AD$.We know that the angle bisectors of the angles of the trapezoid are concurrent at $ O$.Let $ T$ be the intersection of the diagonals $ AC,BD$.Let $ Q$ be on $ CD$ such that $ \angle OQD \equal{} 90^\circ$.Prove that if the circumcircle of the triangle $ OTQ$ intersects $ CD$ again at $ P$ then $ TP\parallel{}AD$.

Let $ABC$ be an acute-angled triangle, $\Gamma$ its circumcircle and $M$ the midpoint of $BC$. Let $N$ be a point in the arc $BC$ of $\Gamma$ not containing $A$ such that $\angle NAC= \angle BAM$. Let $R$ be the midpoint of $AM$, $S$ the midpoint of $AN$ and $T$ the foot of the altitude through $A$. Prove that $R$, $S$ and $T$ are collinear.

On the circle $O,$ six points $A,B,C,D,E,F$ are on the circle counterclockwise. $BD$ is the diameter of the circle and it is perpendicular to $CF.$ Also, lines $CF, BE, AD$ is concurrent. Let $M$ be the foot of altitude from $B$ to $AC$ and let $N$ be the foot of altitude from $D$ to $CE.$ Prove that the area of $\triangle MNC$ is less than half the area of $\square ACEF.$

All of the polygons in the figure below are regular. Prove that $ABCD$ is an isosceles trapezoid. [img]https://cdn.artofproblemsolving.com/attachments/e/a/3f4de32becf4a90bf0f0b002fb4d8e724e8844.png[/img] [i]Proposed by Mahdi Etesamifard - Iran[/i]

Let $O$ is a point in a plane $P$ and let $[OX,[OY,[OZ$ is distinct ray in $P$. Prove that, if $A \in [OX$ , $B \in [OY$ and $C \in [OZ$ points such that $\triangle OAB$ , $\triangle OBC$ and $\triangle OCA$ 's perimeter is 2, there is only one $(A,B,C)$ triple

Two circles $\Gamma_1$ and $\Gamma_2$are given with centres $O_1$ and $O_2$ and common exterior tangents $\ell_1$ and $\ell_2$. The line $\ell_1$ intersects $\Gamma_1$ in $A$ and $\Gamma_2$ in $B$. Let $X$ be a point on segment $O_1O_2$, not lying on $\Gamma_1$ or $\Gamma_2$. The segment $AX$ intersects $\Gamma_1$ in $Y \ne A$ and the segment $BX$ intersects $\Gamma_2$ in $Z \ne B$. Prove that the line through $Y$ tangent to $\Gamma_1$ and the line through $Z$ tangent to $\Gamma_2$ intersect each other on $\ell_2$.

A acute triangle $\triangle ABC$ has circumcenter $O$. The circumcircle of $OAB$, called $O_1$, and the circumcircle of $OAC$, called $O_2$, meets $BC$ again at $D ( \not=B )$ and $E ( \not= C )$ respectively. The perpendicular bisector of $BC$ hits $AC$ again at $F$. Prove that the circumcenter of $\triangle ADE$ lies on $AC$ if and only if the centers of $O_1, O_2$ and $F$ are colinear.

Prove that if in a tetrahedron $ ABCD $ opposite edges are equal, i.e. $ AB = CD $, $ AC = BD $, $ AD = BC $, then the lines passing through the midpoints of opposite edges are mutually perpendicular and are the axes of symmetry of the tetrahedron.