In every $1\times1$ square of an $m\times n$ table we have drawn one of two diagonals. Prove that there is a path including these diagonals either from left side to the right side, or from the upper side to the lower side.

In parallelogram $ABCD$, point $K$ is marked on diagonal $AC$. Circle $s_1$ passes through point $K$ and touches lines $AB$ and $AD$ ($s_1$ intersects the diagonal $AC$ for the second time on the segment $AK$). Circle $s_2$ passes through point $K$ and touches lines $CB$ and $CD$ ($s_2$ intersects for the second time diagonal $AC$ on segment $KC$). Prove that for all positions of the point $K$ on the diagonal $AC$, the straight lines connecting the centers of circles $ s_1$ and $s_2$, will be parallel to each other.

In an acute triangle $ ABC$, let $ D$ be a point on $ \left[AC\right]$ and $ E$ be a point on $ \left[AB\right]$ such that $ \angle ADB\equal{}\angle AEC\equal{}90{}^\circ$. If perimeter of triangle $ AED$ is 9, circumradius of $ AED$ is $ \frac{9}{5}$ and perimeter of triangle $ ABC$ is 15, then $ \left|BC\right|$ is $\textbf{(A)}\ 5 \qquad\textbf{(B)}\ \frac{24}{5} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ \frac{48}{5}$

Let $\mathcal{B}$ be the set of rectangular boxes that have volume $23$ and surface area $54$. Suppose $r$ is the least possible radius of a sphere that can fit any element of $\mathcal{B}$ inside it. Then $r^{2}$ can be expressed as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$. [i]Proposed by David Monk, United Kingdom[/i]

Let $P$ be an arbitrary point inside triangle $ABC$, denote by $A_{1}$ (different from $P$) the second intersection of line $AP$ with the circumcircle of triangle $PBC$ and define $B_{1},C_{1}$ similarly. Prove that $\left(1 \plus{} 2\cdot\frac {PA}{PA_{1}}\right)\left(1 \plus{} 2\cdot\frac {PB}{PB_{1}}\right)\left(1 \plus{} 2\cdot\frac {PC}{PC_{1}}\right)\geq 8$.

In a circle $C$ with centre $O$ and radius $r$, let $C_1$, $C_2$ be two circles with centres $O_1$, $O_2$ and radii $r_1$, $r_2$ respectively, so that each circle $C_i$ is internally tangent to $C$ at $A_i$ and so that $C_1$, $C_2$ are externally tangent to each other at $A$. Prove that the three lines $OA$, $O_1 A_2$, and $O_2 A_1$ are concurrent.

Let $A_1A_2A_3A_4A_5$ be a convex, cyclic pentagon with $\angle A_i + \angle A_{i+1} >180^{\circ}$ for all $i \in \{1,2,3,4,5\}$ (all indices modulo $5$ in the problem). Define $B_i$ as the intersection of lines $A_{i-1}A_i$ and $A_{i+1}A_{i+2}$, forming a star. The circumcircles of triangles $A_{i-1}B_{i-1}A_i$ and $A_iB_iA_{i+1}$ meet again at $C_i \neq A_i$, and the circumcircles of triangles $B_{i-1}A_iB_i$ and $B_iA_{i+1}B_{i+1}$ meet again at $D_i \neq B_i$. Prove that the ten lines $A_iC_i, B_iD_i$, $i \in \{1,2,3,4,5\}$, have a common point.

$ABCDE$ is a convex pentagon with $AB = CD = EA = 1$, $\angle ABC = \angle DEA = 90^o$, and $BC + DE = 1$. What is the area of the pentagon?

Prove that for an arbitrary triangle $ABC$ : $sin \frac{A}{2} sin \frac{B}{2} sin \frac{C}{2} < \frac{1}{4}$.

Given triangle $ABC$, squares $ABEF, BCGH, CAIJ$ are constructed externally on side $AB, BC, CA$, respectively. Let $AH \cap BJ = P_1$, $BJ \cap CF = Q_1$, $CF \cap AH = R_1$, $AG \cap CE = P_2$, $BI \cap AG = Q_2$, $CE \cap BI = R_2$. Prove that triangle $P_1 Q_1 R_1$ is congruent to triangle $P_2 Q_2 R_2$.

Let $ABC$ be a triangle and $D, E$ and $F$ the foots of heights from $A, B$ and $C$ respectively. Let $D_1$ be such a point on $EF$, that $DF = D_1 E$ where $E$ is between $D_1$ and $F$. Similarly, let $D_2$ be such a point on $EF$, that $DE = D_2 F$ where $F$ is between $E$ and $D_2$. Let the bisector of $DD_1$ intersect $AB$ at $P$ and let the bisector of $DD_2$ intersect $AC$ at $Q$. Prove that, $PQ$ bisects $BC$.

The bisectors of the angles $\sphericalangle CAB$ and $\sphericalangle BCA$ intersect the circumcircle of $ABC$ in $P$ and $Q$ respectively. These bisectors intersect each other in point $I$. Prove that $PQ \perp BI$.

Circles $ C_1$ and $ C_2$ with centers at $ O_1$ and $ O_2$ respectively meet at points $ A$ and $ B$. The radii $ O_1B$ and $ O_2B$ meet $ C_1$ and $ C_2$ at $ F$ and$ E$. The line through $ B$ parallel to $ EF$ intersects $ C_1$ again at $ M$ and $ C_2$ again at $ N$. Prove that $ MN \equal{} AE \plus{} AF$.

The altitudes $AD$ and $BE$ of an acute triangle $ABC$ intersect at point $H$. Let $F$ be the intersection of the line $AB$ and the line that is parallel to the side BC and goes through the circumcenter of $ABC$. Let $M$ be the midpoint of the segment $AH$. Prove that $\angle CMF = 90^o$

The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with $ 9$ trapezoids, let $ x$ be the angle measure in degrees of the larger interior angle of the trapezoid. What is $ x$? [asy]unitsize(4mm); defaultpen(linewidth(.8pt)); int i; real r=5, R=6; path t=r*dir(0)--r*dir(20)--R*dir(20)--R*dir(0); for(i=0; i<9; ++i) { draw(rotate(20*i)*t); } draw((-r,0)--(R+1,0)); draw((-R,0)--(-R-1,0));[/asy]$ \textbf{(A)}\ 100 \qquad \textbf{(B)}\ 102 \qquad \textbf{(C)}\ 104 \qquad \textbf{(D)}\ 106 \qquad \textbf{(E)}\ 108$

In a circular tower with an internal diameter of $ 2$ m, there is a spiral staircase with a height of $ 6$ m. The height of each stair step is $ 0.15$ m. In the horizontal projection, the steps form adjacent circular sections with an angle of $ 18^\circ $. The narrower ends of the steps are mounted in a round pillar with a diameter of $ 0.64$ m, the axis of which coincides with the axis of the tower. Calculate the greatest length of a straight rod that can be moved up these stairs from the bottom to the top (do not take into account the thickness of the rod or the thickness of the boards from which the stairs are made).

Let $C$ is at a circle. $[AB]$ is a diameter this circle. $D$ is a point at $[AB]$. Perpendicular from $C$ to $[AB]$'s foot on the $[AB]$ is $E$, perpendicular from $A$ to $[CD]$'s foot on the $[CD]$ is $F$. Prove that, $$[DC][FC]=[BD][EA]$$

$P_1,P_2,\ldots,P_{100}$ are $100$ points on the plane, no three of them are collinear. For each three points, call their triangle [b]clockwise[/b] if the increasing order of them is in clockwise order. Can the number of [b]clockwise[/b] triangles be exactly $2017$? [i]Proposed by Morteza Saghafian[/i]

Given a triangle $ABC$. Let $O$ be the circumcenter of this triangle $ABC$. Let $H$, $K$, $L$ be the feet of the altitudes of triangle $ABC$ from the vertices $A$, $B$, $C$, respectively. Denote by $A_{0}$, $B_{0}$, $C_{0}$ the midpoints of these altitudes $AH$, $BK$, $CL$, respectively. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$, $CA$, $AB$ at the points $D$, $E$, $F$, respectively. Prove that the four lines $A_{0}D$, $B_{0}E$, $C_{0}F$ and $OI$ are concurrent. (When the point $O$ concides with $I$, we consider the line $OI$ as an arbitrary line passing through $O$.)

Consider a circle with center $O$ and radius $R,$ and let $A$ and $B$ be two points in the plane of this circle. [b]a.)[/b] Draw a chord $CD$ of the circle such that $CD$ is parallel to $AB,$ and the point of the intersection $P$ of the lines $AC$ and $BD$ lies on the circle. [b]b.)[/b] Show that generally, one gets two possible points $P$ ($P_{1}$ and $P_{2}$) satisfying the condition of the above problem, and compute the distance between these two points, if the lengths $OA=a,$ $OB=b$ and $AB=d$ are given.

From the foot of one altitude of the acute triangle, perpendiculars are drawn on the other two sides, that meet the other sides at $P$ and $Q$. Show that the length of $PQ$ does not depend on which of the three altitudes is selected.

Let $ABCD$ be a cyclic qudrilaterlal such that $AB.BC=2.CD.DA$ Prove that $8.BD^2 \leq 9.AC^2$

A triangle $ABC$ is given. Find all the segments $XY$ that lie inside the triangle such that $XY$ and five of the segments $XA,XB, XC, YA,YB,YC$ divide the triangle $ABC$ into $5$ regions with equal areas. Furthermore, prove that all the segments $XY$ have a common point.

A right rectangular prism has integer side lengths $a$, $b$, and $c$. If $\text{lcm}(a,b)=72$, $\text{lcm}(a,c)=24$, and $\text{lcm}(b,c)=18$, what is the sum of the minimum and maximum possible volumes of the prism? [i]Proposed by Deyuan Li and Andrew Milas[/i]