Tamara has three rows of two 6-feet by 2-feet flower beds in her garden. The beds are separated and also surrounded by 1-foot-wide walkways, as shown on the diagram. What is the total area of the walkways, in square feet? [asy] unitsize(0.7cm); path p1 = (0,0)--(15,0)--(15,10)--(0,10)--cycle; fill(p1,lightgray); draw(p1); for (int i = 1; i <= 8; i += 7) { for (int j = 1; j <= 7; j += 3 ) { path p2 = (i,j)--(i+6,j)--(i+6,j+2)--(i,j+2)--cycle; draw(p2); fill(p2,white); } } draw((0,8)--(1,8),Arrows); label("1",(0.5,8),S); draw((7,8)--(8,8),Arrows); label("1",(7.5,8),S); draw((14,8)--(15,8),Arrows); label("1",(14.5,8),S); draw((11,0)--(11,1),Arrows); label("1",(11,0.5),W); draw((11,3)--(11,4),Arrows); label("1",(11,3.5),W); draw((11,6)--(11,7),Arrows); label("1",(11,6.5),W); draw((11,9)--(11,10),Arrows); label("1",(11,9.5),W); label("6",(4,1),N); label("2",(1,2),E); [/asy] $\textbf{(A) }72 \qquad \textbf{(B) }78 \qquad \textbf{(C) }90 \qquad \textbf{(D) }120 \qquad \textbf{(E) }150 $

Let $ABCD$ be a rectangle and let $E \in (BD)$ such that $m( \angle DAE) =15^o$. Let $F \in AB$ such that $EF \perp AB$. It is known that $EF=\frac12 AB$ and $AD = a$. Find the measure of the angle $\angle EAC$ and the length of the segment $(EC)$.

Let $ABCD$ be an inscribed quadrilateral in a circle $c(O,R)$ (of circle $O$ and radius $R$). With centers the vertices $A,B,C,D$, we consider the circles $C_{A},C_{B},C_{C},C_{D}$ respectively, that do not intersect to each other . Circle $C_{A}$ intersects the sides of the quadrilateral at points $A_{1} , A_{2}$ , circle $C_{B}$ intersects the sides of the quadrilateral at points $B_{1} , B_{2}$ , circle $C_{C}$ at points $C_{1} , C_{2}$ and circle $C_{D}$ at points $C_{1} , C_{2}$ . Prove that the quadrilateral defined by lines $A_{1}A_{2} , B_{1}B_{2} , C_{1}C_{2} , D_{1}D_{2}$ is cyclic.

On grid paper, mark three nodes so that in the triangle they formed, the sum of the two smallest medians equals to half-perimeter.

Given an acute triangle $ABC$. Points $B'$ and $C'$ are symmetrical, respectively, to vertices $B$ and $ C$ wrt straight lines $AC$ and $AB$. Let $P$ be the intersection point of the circumcircles of triangles $ABB'$ and $ACC'$, different from $A$. Prove that the center of the circumcircle of triangle $ABC$ lies on line $PA$.

A convex quadrilateral is inscribed in a circle of radius $1$. Prove that the difference between its perimeter and the sum of the lengths of its diagonals is greater than zero and less than $2.$

We are given an acute triangle $ABC$, and inside it a point $P$, which is not on any of the heights $AA_1$, $BB_1$, $CC_1$. The rays $AP$, $BP$, $CP$ intersect the circumcircle of $ABC$ at points $A_2$, $B_2$, $C_2$. Prove that the circles $AA_1A_2$, $BB_1B_2$ and $CC_1C_2$ are concurrent.

In triangle $ABC$, let the circumcenter, incenter, and orthocenter be $O$, $I$, and $H$ respectively. Segments $AO$, $AI$, and $AH$ intersect the circumcircle of triangle $ABC$ at $D$, $E$, and $F$. $CD$ intersects $AE$ at $M$ and $CE$ intersects $AF$ at $N$. Prove that $MN$ is parallel to $BC$.

Acute triangle $ABC$ is inscribed in circle $\omega$. Let $H$ and $O$ denote its orthocenter and circumcenter, respectively. Let $M$ and $N$ be the midpoints of sides $AB$ and $AC$, respectively. Rays $MH$ and $NH$ meet $\omega$ at $P$ and $Q$, respectively. Lines $MN$ and $PQ$ meet at $R$. Prove that $OA\perp RA$.

In a triangle $ABC$ with $AC\ne BC$, $M$ is the midpoint of $AB$ and $\angle A=\alpha$, $\angle B=\beta$, $\angle ACM=\varphi$ and $\angle BSM=\Psi$. Prove that $$\frac{\sin\alpha\sin\beta}{\sin(\alpha-\beta)}=\frac{\sin\varphi\sin\Psi}{\sin(\varphi-\Psi)}.$$

The altitudes $AN$ and $BM$ are drawn in triangle $ABC$. Prove that the perpendicular bisector to the segment $NM$ divides the segment $AB$ in half.

$8$ points lie on the faces of unit cube and form another cube. What can be length of edge of this cube?

Consider a fixed triangle $ABC$ such that $AB=AC.$ Let $M$ be the midpoint of $BC.$ Let $P$ be a variable point inside $\triangle ABC,$ such that $\angle PBC=\angle PCA.$ Prove that the sum of the measures of $\angle BPM$ and $\angle APC$ is constant.

Consider a square of side length $12$ centimeters. Irina draws another square that has $8$ centimeters more of perimeter than the original square. What is the area of the square drawn by Irina?

Let $L$ be the line with slope $\tfrac{5}{12}$ that contains the point $A=(24,-1)$, and let $M$ be the line perpendicular to line $L$ that contains the point $B=(5,6)$. The original coordinate axes are erased, and line $L$ is made the $x$-axis, and line $M$ the $y$-axis. In the new coordinate system, point $A$ is on the positive $x$-axis, and point $B$ is on the positive $y$-axis. The point $P$ with coordinates $(-14,27)$ in the original system has coordinates $(\alpha,\beta)$ in the new coordinate system. Find $\alpha+\beta$.

Consider an acute triangle $ABC$ of area $S$. Let $CD \perp AB$ ($D \in AB$), $DM \perp AC$ ($M \in AC$) and $DN \perp BC$ ($N \in BC$). Denote by $H_1$ and $H_2$ the orthocentres of the triangles $MNC$, respectively $MND$. Find the area of the quadrilateral $AH_1BH_2$ in terms of $S$.

On a circle with diameter $AB$, lie points $C$ and $D$. $XY$ is the diameter passing through the midpoint $K$ of the chord $CD$. Point $M$ is the projection of point $X$ onto line $AC$, and point $N$ is the projection of point $Y$ on line $BD$. Prove that points $M, N$ and $K$ are collinear. (A. Zaslavsky)

Rectangle $R_0$ has sides of lengths $3$ and $4$. Rectangles $R_1$, $R_2$, and $R_3$ are formed such that: $\bullet$ all four rectangles share a common vertex $P$, $\bullet$ for each $n = 1, 2, 3$, one side of $R_n$ is a diagonal of $R_{n-1}$, $\bullet$ for each $n = 1, 2, 3$, the opposite side of $R_n$ passes through a vertex of $R_{n-1}$ such that the center of $R_n$ is located counterclockwise of the center of $R_{n-1}$ with respect to $P$. Compute the total area covered by the union of the four rectangles. [img]https://cdn.artofproblemsolving.com/attachments/3/1/e9edd39e60e4a4defdb127b93b19ab0d0f443c.png[/img]

Let \( ABC \) be a triangle and \( D \) the foot of the altitude from \( A \). Let \( M \) be a point such that \( MB = MC \). Let \( E \) and \( F \) be the intersections of the circumcircle of \( BMD \) and \( CMD \) with \( AD \). Let \( G \) and \( H \) be the intersections of \( MB \) and \( MC \) with \( AD \). Prove that \( EG = FH \).

A circle and a square are drawn on the plane so that they overlap. Together, the two shapes cover an area of $329$ square units. The area common to both shapes is $101$ square units. The area of the circle is $234$ square units. What is the perimeter of the square in units? $\mathrm{(A) \ } 14 \qquad \mathrm{(B) \ } 48 \qquad \mathrm {(C) \ } 56 \qquad \mathrm{(D) \ } 64 \qquad \mathrm{(E) \ } 196$

[b]p1.[/b] The following figure represents a rectangular piece of paper $ABCD$ whose dimensions are $4$ inches by $3$ inches. When the paper is folded along the line segment $EF$, the corners $A$ and $C$ coincide. (a) Find the length of segment $EF$. (b) Extend $AD$ and $EF$ so they meet at $G$. Find the area of the triangle $\vartriangle AEG$. [img]https://cdn.artofproblemsolving.com/attachments/d/4/e8844fd37b3b8163f62fcda1300c8d63221f51.png[/img] [b]p2.[/b] (a) Let $p$ be a prime number. If $a, b, c$, and $d$ are distinct integers such that the equation $(x -a)(x - b)(x - c)(x - d) - p^2 = 0$ has an integer solution $r$, show that $(r - a) + (r - b) + (r - c) + (r - d) = 0$. (b) Show that $r$ must be a double root of the equation $(x - a)(x - b)(x - c)(x - d) - p^2 = 0$. [b]p3.[/b] If $\sin x + \sin y + \sin z = 0$ and $\cos x + \cos y + \cos z = 0$, prove the following statements. (a) $\cos (x - y) = -\frac12$ (b) $\cos (\theta - x) + \cos(\theta - y) + \cos (\theta - z) = 0$, for any angle $\theta$. (c) $\sin^2 x + \sin^2 y + \sin^2 z =\frac32$ [b]p4.[/b] Let $|A|$ denote the number of elements in the set $A$. (a) Construct an infinite collection $\{A_i\}$ of infinite subsets of the set of natural numbers such that $|A_i \cap A_j | = 0$ for $i \ne j$. (b) Construct an infinite collection $\{B_i\}$ of infinite subsets of the set of natural numbers such that $|B_i \cap B_j |$ gives a distinct integer for every pair of $i$ and $j$, $i \ne j$. [b]p5.[/b] Consider the equation $x^4 + y^4 = z^5$. (a) Show that the equation has a solution where $x, y$, and $z$ are positive integers. (b) Show that the equation has infinitely many solutions where $x, y$, and $z$ are positive integers. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCD$ be the trapezium with bases $AB$ and $CD$, such that $\sphericalangle ABC = 90^{\circ}$. The bisector of angle $BAD$ intersects the segment $BC$ in the point $P$. Show that if $\sphericalangle APD = 45^{\circ}$, then area of quadrilateral $APCD$ is equal to the area of the triangle $ABP$.

Let $ABC$ be a triangle with circumcircle $\Gamma$ and circumcentre $O$. The circle with centre on the line $AB$ and passing through the points $A$ and $O$ intersects $\Gamma$ again in $D$. Similarly, the circle with centre on the line $AC$ and passing through the points $A$ and $O$ intersects $\Gamma$ again in $E$. Prove that $BD$ is parallel with $CE$.

Let $R$ be the set of all rectangles centered at the origin and with perimeter $1$ (the center of a rectangle is the intersection point of its two diagonals). Let $S$ be a region that contains all of the rectangles in $R$ (region $A$ contains region $B$, if $B$ is completely inside of $A$). The minimum possible area of $S$ has the form $\pi a$, where $a$ is a real number. Find $1/a$.

Let $X$ be an arbitrary point on side $BC$ of triangle $ABC$. Triangle $T$ is formed by the angle bisectors of the angles $\angle ABC$, $\angle ACB$ and $\angle AXC$. Prove that the circle circumscribed around the triangle $T$, passes through the vertex $A$. (Dmytro Prokopenko)