Let $ ABC$ be a tringle and let $ P$ be an interior point such that $ \angle BPC \equal{} 90 ,\angle BAP \equal{} \angle BCP$.Let $ M,N$ be the mid points of $ AC,BC$ respectively.Suppose $ BP \equal{} 2PM$.Prove that $ A,P,N$ are collinear.

Let's take a $ABCD$ rectangle of paper; the side $AB$ measures $5$ cm and the side $BC$ measures $9$ cm. We do three folds: 1.We take the $AB$ side on the $BC$ side and call $P$ the point on the $BC$ side that coincides with $A$. A right trapezoid $BCDQ$ is then formed. 2. We fold so that $B$ and $Q$ coincide. A $5$-sided polygon $RPCDQ$ is formed. 3. We fold again by matching $D$ with $C$ and $Q$ with $P$. A new right trapezoid $RPCS$. After making these folds, we make a cut perpendicular to $SC$ by its midpoint $T$, obtaining the right trapezoid $RUTS$. Calculate the area of the figure that appears as we unfold the last trapezoid $RUTS$.

A cylinder radius $12$ and a cylinder radius $36$ are held tangent to each other with a tight band. The length of the band is $m\sqrt{k}+n\pi$ where $m$, $k$, and $n$ are positive integers, and $k$ is not divisible by the square of any prime. Find $m + k + n$. [asy] size(150); real t=0.3; void cyl(pair x, real r, real h) { pair xx=(x.x,t*x.y); path B=ellipse(xx,r,t*r), T=ellipse((x.x,t*x.y+h),r,t*r), S=xx+(r,0)--xx+(r,h)--(xx+(-r,h))--xx-(r,0); unfill(S--cycle); draw(S); unfill(B); draw(B); unfill(T); draw(T); } real h=8, R=3,r=1.2; pair X=(0,0), Y=(R+r)*dir(-50); cyl(X,R,h); draw(shift((0,5))*yscale(t)*arc(X,R,180,360)); cyl(Y,r,h); void str (pair x, pair y, real R, real r, real h, real w) { real u=(angle(y-x)+asin((R-r)/(R+r)))*180/pi+270; path P=yscale(t)*(arc(x,R,180,u)--arc(y,r,u,360)); path Q=shift((0,h))*P--shift((0,h+w))*reverse(P)--cycle; fill(Q,grey);draw(Q); } str(X,Y,R,r,3.5,1.5);[/asy]

Let $m$ and $n$ be positive integers, and let $\mathcal R$ be a $2m\times{2n}$ grid of unit squares. A [i]domino[/i] is a $1\times2{}$ or $2\times{1}$ rectangle. A subset $S$ of grid squares in $\mathcal R$ is [i]domino-tileable[/i] if dominoes can be placed to cover every square of $S$ exactly once with no domino extending outside of $S$. [i]Note[/i]: The empty set is domino tileable. An [i]up-right path[/i] is a path from the lower-left corner of $\mathcal R$ to the upper-right corner of $\mathcal R$ formed by exactly $2m+2n$ edges of the grid squares. Determine, with proof, in terms of $m$ and $n$, the number of up-right paths that divide $\mathcal R$ into two domino-tileable subsets.

There are three rectangles in the following figure. The lengths of some segments are shown. Find the length of the segment $XY$ . [img]https://2.bp.blogspot.com/-x7GQfMFHzAQ/W6K57utTEkI/AAAAAAAAJFQ/1-5WhhuerMEJwDnWB09sTemNLdJX7_OOQCK4BGAYYCw/s320/igo%2B2018%2Bintermediate%2Bp1.png[/img] Proposed by Hirad Aalipanah

In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a [i]box[/i]. Two boxes [i]intersect[/i] if they have a common point in their interior or on their boundary. Find the largest $ n$ for which there exist $ n$ boxes $ B_1$, $ \ldots$, $ B_n$ such that $ B_i$ and $ B_j$ intersect if and only if $ i\not\equiv j\pm 1\pmod n$. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

In a billiard with shape of a rectangle $ABCD$ with $AB=2013$ and $AD=1000$, a ball is launched along the line of the bisector of $\angle BAD$. Supposing that the ball is reflected on the sides with the same angle at the impact point as the angle shot , examine if it shall ever reach at vertex B.

Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ are perpendicular. Let $O$ be the circumcenter of $ABCD$, $K$ the intersection of the diagonals, $ L\neq O $ the intersection of the circles circumscribed to $OAC$ and $OBD$, and $G$ the intersection of the diagonals of the quadrilateral whose vertices are the midpoints of the sides of $ABCD$. Prove that $O, K, L$ and $G$ are collinear

Let $ABCD$ be a rectangle. A line $r$ moves parallel to $AB$ and intersects diagonal $AC$ , forming two triangles opposite the vertex, inside the rectangle. Prove that the sum of the areas of these triangles is minimal when $r$ passes through the midpoint of segment $AD$ .

Given a rectangle $A_1A_2A_3A_4$. Four circles with $A_i$ as their centres have their radiuses $r_1, r_2, r_3, r_4$; and $r_1+r_3=r_2+r_4<d$, where d is a diagonal of the rectangle. Two pairs of the outer common tangents to {the first and the third} and {the second and the fourth} circumferences make a quadrangle. Prove that you can inscribe a circle into that quadrangle.

For an integer $m\geq 1$, we consider partitions of a $2^m\times 2^m$ chessboard into rectangles consisting of cells of chessboard, in which each of the $2^m$ cells along one diagonal forms a separate rectangle of side length $1$. Determine the smallest possible sum of rectangle perimeters in such a partition. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

At each vertex of a $4\times5$ rectangle there is a house. Find the path of the minimum length connecting all these houses.

Let $\triangle ABC$ with $\angle BAC=120 ^\circ$. Let $D, E, F$ points on sides $BC, CA, AB$, respectively, such that $AD, BE, CF$ are angle bisectors on $\triangle ABC$. \\ Prove that $\triangle ABC$ is isosceles if and only if $\triangle DEF$ is right-angled isosceles.

As in the picture below, the rectangle on the left hand side has been divided into four parts by line segments which are parallel to a side of the rectangle. The areas of the small rectangles are $A,B,C$ and $D$. Similarly, the small rectangles on the right hand side have areas $A^\prime,B^\prime,C^\prime$ and $D^\prime$. It is known that $A\leq A^\prime$, $B\leq B^\prime$, $C\leq C^\prime$ but $D\leq B^\prime$. [asy] import graph; size(12cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.3,xmax=12.32,ymin=-10.68,ymax=6.3; draw((0,3)--(0,0)); draw((3,0)--(0,0)); draw((3,0)--(3,3)); draw((0,3)--(3,3)); draw((2,0)--(2,3)); draw((0,2)--(3,2)); label("$A$",(0.86,2.72),SE*lsf); label("$B$",(2.38,2.7),SE*lsf); label("$C$",(2.3,1.1),SE*lsf); label("$D$",(0.82,1.14),SE*lsf); draw((5,2)--(11,2)); draw((5,2)--(5,0)); draw((11,0)--(5,0)); draw((11,2)--(11,0)); draw((8,0)--(8,2)); draw((5,1)--(11,1)); label("$A'$",(6.28,1.8),SE*lsf); label("$B'$",(9.44,1.82),SE*lsf); label("$C'$",(9.4,0.8),SE*lsf); label("$D'$",(6.3,0.86),SE*lsf); dot((0,3),linewidth(1pt)+ds); dot((0,0),linewidth(1pt)+ds); dot((3,0),linewidth(1pt)+ds); dot((3,3),linewidth(1pt)+ds); dot((2,0),linewidth(1pt)+ds); dot((2,3),linewidth(1pt)+ds); dot((0,2),linewidth(1pt)+ds); dot((3,2),linewidth(1pt)+ds); dot((5,0),linewidth(1pt)+ds); dot((5,2),linewidth(1pt)+ds); dot((11,0),linewidth(1pt)+ds); dot((11,2),linewidth(1pt)+ds); dot((8,0),linewidth(1pt)+ds); dot((8,2),linewidth(1pt)+ds); dot((5,1),linewidth(1pt)+ds); dot((11,1),linewidth(1pt)+ds); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy] Prove that the big rectangle on the left hand side has area smaller or equal to the area of the big rectangle on the right hand side, i.e. $A+B+C+D\leq A^\prime+B^\prime+C^\prime+D^\prime$.

(1) Let $ f(x)$ be a continuous function defined on $ [a,\ b]$, it is known that there exists some $ c$ such that \[ \int_a^b f(x)\ dx \equal{} (b \minus{} a)f(c)\ (a < c < b)\] Explain the fact by using graph. Note that you don't need to prove the statement. (2) Let $ f(x) \equal{} a_0 \plus{} a_1x \plus{} a_2x^2 \plus{} \cdots\cdots \plus{} a_nx^n$, Prove that there exists $ \theta$ such that \[ f(\sin \theta) \equal{} a_0 \plus{} \frac {a_1}{2} \plus{} \frac {a_3}{3} \plus{} \cdots\cdots \plus{} \frac {a_n}{n \plus{} 1},\ 0 < \theta < \frac {\pi}{2}.\]