$\textbf{Bake Sale}$ Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown. $\circ$ Art's cookies are trapezoids: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(5,0)--(5,3)--(2,3)--cycle); draw(rightanglemark((5,3), (5,0), origin)); label("5 in", (2.5,0), S); label("3 in", (5,1.5), E); label("3 in", (3.5,3), N);[/asy] $\circ$ Roger's cookies are rectangles: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(4,0)--(4,2)--(0,2)--cycle); draw(rightanglemark((4,2), (4,0), origin)); draw(rightanglemark((0,2), origin, (4,0))); label("4 in", (2,0), S); label("2 in", (4,1), E);[/asy] $\circ$ Paul's cookies are parallelograms: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(2.5,2)--(-0.5,2)--cycle); draw((2.5,2)--(2.5,0), dashed); draw(rightanglemark((2.5,2),(2.5,0), origin)); label("3 in", (1.5,0), S); label("2 in", (2.5,1), W);[/asy] $\circ$ Trisha's cookies are triangles: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(3,4)--cycle); draw(rightanglemark((3,4),(3,0), origin)); label("3 in", (1.5,0), S); label("4 in", (3,2), E);[/asy] Each friend uses the same amount of dough, and Art makes exactly 12 cookies. How many cookies will be in one batch of Trisha's cookies? $ \textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 24$

Let $E$ be a point inside the rhombus $ABCD$ such that $|AE|=|EB|$, $m(\widehat{EAB})=12^\circ$, and $m(\widehat{DAE})=72^\circ$. What is $m(\widehat{CDE})$ in degrees? $ \textbf{(A)}\ 64 \qquad\textbf{(B)}\ 66 \qquad\textbf{(C)}\ 68 \qquad\textbf{(D)}\ 70 \qquad\textbf{(E)}\ 72 $

Let $ABC$ be a triangle with $AB = AC$. Let $M$ be the midpoint of $BC$. Let the circles with diameters $AC$ and $BM$ intersect at points $M$ and $P$. Let $MP$ intersect $AB$ at $Q$. Let $R$ be a point on $AP$ such that $QR \parallel BP$. Prove that $CP$ bisects $\angle RCB$.

Several non-intersecting diagonals divide a convex polygon into triangles. At each vertex of the polygon the number of triangles adjacent to it is written. Is it possible to reconstruct all the diagonals using these numbers if the diagonals are erased?

Let $ABC$ be an acute-angled triangle with $AC > AB$, let $O$ be its circumcentre, and let $D$ be a point on the segment $BC$. The line through $D$ perpendicular to $BC$ intersects the lines $AO, AC,$ and $AB$ at $W, X,$ and $Y,$ respectively. The circumcircles of triangles $AXY$ and $ABC$ intersect again at $Z \ne A$. Prove that if $W \ne D$ and $OW = OD,$ then $DZ$ is tangent to the circle $AXY.$

Two satellites are orbiting the earth in the equatorial plane at an altitude $h$ above the surface. The distance between the satellites is always $d$, the diameter of the earth. For which $h$ is there always a point on the equator at which the two satellites subtend an angle of $90^\circ$?

The incircle of $\vartriangle ABC$ touches the sides $BC,CA,AB$ at $D,E,F$ respectively. A circle through $A$ and $B$ encloses $\vartriangle ABC$ and intersects the line $DE$ at points $P$ and $Q$. Prove that the midpoint of $AB$ lies on the circumircle of $\vartriangle PQF$.

Let $\triangle ABC$ be an acute triangle with circumcenter $O$ such that $AB<AC$, let $Q$ be the intersection of the external bisector of $\angle A$ with $BC$, and let $P$ be a point in the interior of $\triangle ABC$ such that $\triangle BPA$ is similar to $\triangle APC$. Show that $\angle QPA + \angle OQB = 90^{\circ}$. [i]Alex Zhu.[/i]

If $T(n)$ is the numbers of triangles with integers sizes(not congruent with each other) with it´s perimeter is equal to $n$, prove that: $$T(2012)<T(2015)$$ $$T(2013)=T(2016)$$

Three cevians divided the triangle into six triangles, the areas of which are marked in the figure. 1) Prove that $S_1 \cdot S_2 \cdot S_3 =Q_1 \cdot Q_2 \cdot Q_3$. 2) Determine whether it is true that if $S_1 = S_2 = S_3$, then $Q_1 = Q_2 = Q_3$. [img]https://cdn.artofproblemsolving.com/attachments/c/d/3e847223b24f783551373e612283e10e477e62.png[/img]

Chord $ \left[AD\right]$ is perpendicular to the diameter $ \left[BC\right]$ of a circle. Let $ E$ and $ F$ be the midpoints of the arcs $ AC$ and $ CD$, respectively. If $ AD\bigcap BE\equal{}\left\{G\right\}$, $ AF\bigcap BC\equal{}\left\{H\right\}$ and $ m(AC)\equal{}\alpha$, find the measure of angle $ BHC$ in terms of $ \alpha$. $\textbf{(A)}\ 90{}^\circ \minus{}\frac{\alpha }{2} \qquad\textbf{(B)}\ 60{}^\circ \minus{}\frac{\alpha }{3} \qquad\textbf{(C)}\ \alpha \minus{}30{}^\circ \\ \qquad\textbf{(D)}\ 15{}^\circ \plus{}\frac{\alpha }{2} \qquad\textbf{(E)}\ \frac{180{}^\circ \minus{}2\alpha }{3}$

A right parallelepiped (i.e. a parallelepiped one of whose edges is perpendicular to a face) is given. Its vertices have integral coordinates, and no other points with integral coordinates lie on its faces or edges. Prove that the volume of this parallelepiped is a sum of three perfect squares. [i]Proposed by A. Golovanov[/i]

[hide=E stands for Euclid, L stands for Lobachevsky]they had two problem sets under those two names[/hide] [b]E1.[/b] What is the perimeter of a rectangle if its area is $24$ and one side length is $6$? [b]E2.[/b] John moves 3 miles south, then $2$ miles west, then $7$ miles north, and then $5$ miles east. What is the length of the shortest path, in miles, from John's current position to his original position? [b]E3.[/b] An equilateral triangle $ABC$ is drawn with side length $2$. The midpoints of sides $AB$, $BC$, and $CA$ are constructed, and are connected to form a triangle. What is the perimeter of the newly formed triangle? [b]E4.[/b] Let triangle $ABC$ have sides $AB = 74$ and $AC = 5$. What is the sum of all possible integral side lengths of BC? [b]E5.[/b] What is the area of quadrilateral $ABCD$ on the coordinate plane with $A(1, 0)$, $B(0, 1)$, $C(1, 3)$, and $D(5, 2)$? [b]E6 / L1.[/b] Let $ABCD$ be a square with side length $30$. A circle centered at the center of $ABCD$ with diameter $34$ is drawn. Let $E$ and $F$ be the points at which the circle intersects side $AB$. What is $EF$? [b]E7 / L2.[/b] What is the area of the quadrilateral bounded by $|2x| + |3y| = 6$? [b]E8.[/b] A circle $O$ with radius $2$ has a regular hexagon inscribed in it. Upon the sides of the hexagon, equilateral triangles of side length $2$ are erected outwards. Find the area of the union of these triangles and circle $O$. [b]L3.[/b] Right triangle $ABC$ has hypotenuse $AB$. Altitude $CD$ divides $AB$ into segments $AD$ and $DB$, with $AD = 20$ and $DB = 16$. What is the area of triangle $ABC$? [b]L4.[/b] Circle $O$ has chord $AB$. Extend $AB$ past $B$ to a point $C$. A ray from $C$ is drawn, and this ray intersects circle $O$. Let point $D$ be the point of intersection of the ray and the circle that is closest to point $C$. Given $AB = 20$, $BC = 16$, and $OA = \frac{201}{6}$ , find the longest possible length of $CD$. [b]L5.[/b] Consider a circular cone with vertex $A$. The cone's height is $4$ and the radius of its base is $3$. Inscribe a sphere inside the cone. Find the ratio of the volume of the cone to the volume of the sphere. [b]L6.[/b] A disk of radius $\frac12$ is randomly placed on the coordinate plane. What is the probability that it contains a lattice point (point with integer coordinates)? [b]L7.[/b] Let $ABC$ be an equilateral triangle of side length $2$. Let $D$ be the midpoint of $BC$, and let $P$ be a variable point on $AC$. By moving $P$ along $AC$, what is the minimum perimeter of triangle $BDP$? [b]L8.[/b] Let $ABCD$ be a rectangle with $AB = 8$ and $BC = 9$. Let $DEFG$ be a rhombus, where $G$ is on line $BC$ and $A$ is on line $EF$. If $m\angle EFG = 30^o, what is $DE$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given an acute $\vartriangle ABC$ with orthocenter $H$. Let $M_a$ be the midpoint of $BC. M_aH$ intersects the circumcircle of $\vartriangle ABC$ at $X_a$ and $AX_a$ intersects $BC$ at $Y_a$. Define $Y_b, Y_c$ in a similar way. Prove that $Y_a, Y_b,Y_c$ are collinear. [img]https://2.bp.blogspot.com/-yjISBHtRa0s/XnSKTrhhczI/AAAAAAAALds/e_rvs9glp60L1DastlvT0pRFyP7GnJnCwCK4BGAYYCw/s320/imoc2017%2Bg4.png[/img]

Let $P$ be a polygon that is convex and symmetric to some point $O$. Prove that for some parallelogram $R$ satisfying $P\subset R$ we have \[\frac{|R|}{|P|}\leq \sqrt 2\] where $|R|$ and $|P|$ denote the area of the sets $R$ and $P$, respectively. [i]Proposed by Witold Szczechla, Poland[/i]

Three squares of area $4, 9$ and $36$ are inscribed in the triangle as shown in the figure. Find the area of the big triangle [img]https://cdn.artofproblemsolving.com/attachments/9/7/3e904a9c78307e1be169ec0b95b1d3d24c1aa2.png[/img]

A circle passes through the points $A,C$ of triangle $ABC$ intersects with the sides $AB,BC$ at points $D,E$ respectively. Let $ \frac{BD}{CE}=\frac{3}{2}$, $BE=4$, $AD=5$ and $AC=2\sqrt{7} $. Find the angle $ \angle BDC$.

Let $O$ be the circumcenter of triangle $ABC$ with $|AB|=5$, $|BC|=6$, $|AC|=7$. Let $A_1$, $B_1$, $C_1$ be the reflections of $O$ over the lines $BC$, $AC$, $AB$, respectively. What is the distance between $A$ and the circumcenter of triangle $A_1B_1C_1$? $ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ \sqrt {29} \qquad\textbf{(C)}\ \dfrac {19}{2\sqrt 6} \qquad\textbf{(D)}\ \dfrac {35}{4\sqrt 6} \qquad\textbf{(E)}\ \sqrt {\dfrac {35}3} $

Consider a circle with an arc \(AB\) and a point \(C\) on this arc. Let \(D\) be the midpoint of arc \(BC\) and \(M\) the midpoint of chord \(AD\). Suppose the tangent lines to the circle at point \(D\) intersect the ray \(AC\) at point \(K\). Prove that the areas of triangle \(MBD\) and quadrilateral \(MCKD\) are equal if and only if the measure of arc \(AB\) is \(180^\circ\). [i]Proposed by Irakli Shalibashvili, Georgia [/i]

Let $\Gamma$ be a circle with center $A$, radius $1$ and diameter $BX$. Let $\Omega$ be a circle with center $C$, radius $1$ and diameter $DY $, where $X$ and $Y$ are on the same side of $AC$. $\Gamma$ meets $\Omega$ at two points, one of which is $Z$. The lines tangent to $\Gamma$ and $\Omega$ that pass through $Z$ cut out a sector of the plane containing no part of either circle and with angle $60^\circ$. If $\angle XY C = \angle CAB$ and $\angle XCD = 90^\circ$, then the length of $XY$ can be written in the form $\tfrac{\sqrt a+\sqrt b}{c}$ for integers $a, b, c$ where $\gcd(a, b, c) = 1$. Find $a + b + c$.

Let $\triangle ABC$ be connected to the circle $\Gamma$. The angular bisector of $\angle BAC$ intersects $BC$ to $D$. Straight line $BP$ intersects $AC$ to $E$, and straight line $CP$ intersects $AB$ to $F$. Let the tangent of the circle $\Gamma$ at point $A$ intersect the line $EF$ at the point $Q$. Proof: $PQ\parallel BC$.

Let $P_1$ be a regular $n$-gon, where $n\in\mathbb{N}$. We construct $P_2$ as the regular $n$-gon whose vertices are the midpoints of the edges of $P_1$. Continuing analogously, we obtain regular $n$-gons $P_3,P_4,\ldots ,P_m$. For $m\ge n^2-n+1$, find the maximum number $k$ such that for any colouring of vertices of $P_1,\ldots ,P_m$ in $k$ colours there exists an isosceles trapezium $ABCD$ whose vertices $A,B,C,D$ have the same colour. [i]Radu Ignat[/i]

Let $M$, $N$ and $P$ be three non-collinear points. Construct using straight edge and compass a triangle for which $M$ and $N$ are the midpoints of two of its sides, and $P$ is its orthocenter.

Triangle $ABC$ is inscribed in circle $O$. Tangent $PD$ is drawn from $A$, $D$ is on ray $BC$, $P$ is on ray $DA$. Line $PU$ ($U \in BD$) intersects circle $O$ at $Q$, $T$, and intersect $AB$ and $AC$ at $R$ and $S$ respectively. Prove that if $QR=ST$, then $PQ=UT$.

Let $\vartriangle ABC$ be an equilateral triangle with side $1$. The points $P$, $Q$, $R$ are chosen on the sides of the segments $AB$, $BC$, $AC$ respectively in such a way that $$\frac{AP}{PB}=\frac{BQ}{QC}=\frac{CR}{RA}=\frac25.$$ Find the area of triangle $PQR$. [img]https://cdn.artofproblemsolving.com/attachments/8/4/6184d66bd3ae23db29a93eeef241c46ae0ad44.png[/img]