Let $CD$ be a chord of a circle $\Gamma_1$ and $AB$ a diameter of $\Gamma_1$ perpendicular to $CD$ at $N$ with $AN > NB$. A circle $\Gamma_2$ centered at $C$ with radius $CN$ intersects $\Gamma_1$ at points $P$ and $Q$. The line $PQ$ intersects $CD$ at $M$ and $AC$ at $K$; and the extension of $NK$ meets $\Gamma_2$ at $L$. Prove that $PQ$ is perpendicular to $AL$

With a roller shear, rectangular sheets of $1420$ mm wide should be made, namely with a width of $500$ mm and a total length of $1000$ m as well as a width of $300$ mm and a total length of $1800$ m can be cut. So far it has been based on the attached drawing cut, in which the gray area represents the waste, which is quite large. A socialist brigade proposes cutting in such a way that waste is significantly reduced becomes. a) What percentage is the waste if cutting continues as before? b) How does the brigade have to cut so that the waste is as small as possible and what is the total length of the starting sheets is required in this case? c) What percentage is the waste now? [img]https://cdn.artofproblemsolving.com/attachments/f/8/c6c88b79abb5d34674bf54524ae1731985c3f7.png[/img]

Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron, each of whose edges measures 1 meter. A bug, starting from vertex $A$, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let $p = n/729$ be the probability that the bug is at vertex $A$ when it has crawled exactly 7 meters. Find the value of $n$.

To each side of the regular $p$-gon of side length $1$ there is attached a $1 \times k$ rectangle, partitioned into $k$ unit cells, where $k$ and $p$ are given positive integers and p an odd prime. Let $P$ be the resulting nonconvex star-like polygonal figure consisting of $kp + 1$ regions ($kp$ unit cells and the $p$-gon). Each region is to be colored in one of three colors, adjacent regions having different colors. Furthermore, it is required that the colored figure should not have a symmetry axis. In how many ways can this be done?

You are given a quadrilateral $ABCD$. It is known that $\angle BAC = 30^o$, $\angle D = 150^o$ and, in addition, $AB = BD$. Prove that $AC$ is the bisector of angle $C$.

Circles centered at $ A$ and $ B$ each have radius $ 2$, as shown. Point $ O$ is the midpoint of $ \overline{AB}$, and $ OA \equal{} 2\sqrt {2}$. Segments $ OC$ and $ OD$ are tangent to the circles centered at $ A$ and $ B$, respectively, and $ EF$ is a common tangent. What is the area of the shaded region $ ECODF$? [asy]unitsize(6mm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=3; pair O=(0,0); pair A=(-2*sqrt(2),0); pair B=(2*sqrt(2),0); pair G=shift(0,2)*A; pair F=shift(0,2)*B; pair C=shift(A)*scale(2)*dir(45); pair D=shift(B)*scale(2)*dir(135); pair X=A+2*dir(-60); pair Y=B+2*dir(240); path P=C--O--D--Arc(B,2,135,90)--G--Arc(A,2,90,45)--cycle; fill(P,gray); draw(Circle(A,2)); draw(Circle(B,2)); dot(A); label("$A$",A,W); dot(B); label("$B$",B,E); dot(C); label("$C$",C,W); dot(D); label("$D$",D,E); dot(G); label("$E$",G,N); dot(F); label("$F$",F,N); dot(O); label("$O$",O,S); draw(G--F); draw(C--O--D); draw(A--B); draw(A--X); draw(B--Y); label("$2$",midpoint(A--X),SW); label("$2$",midpoint(B--Y),SE);[/asy]$ \textbf{(A)}\ \frac {8\sqrt {2}}{3}\qquad \textbf{(B)}\ 8\sqrt {2} \minus{} 4 \minus{} \pi \qquad \textbf{(C)}\ 4\sqrt {2}$ $ \textbf{(D)}\ 4\sqrt {2} \plus{} \frac {\pi}{8}\qquad \textbf{(E)}\ 8\sqrt {2} \minus{} 2 \minus{} \frac {\pi}{2}$

Prove that an arbitrary triangle can be cut into three polygons, one of which must be an obtuse triangle, so that they can then be folded into a rectangle. (Turning over parts is possible).

Triangle $ABC$ is isosceles ($AB=AC$). From $A$, we draw a line $\ell$ parallel to $BC$. $P,Q$ are on perpendicular bisectors of $AB,AC$ such that $PQ\perp BC$. $M,N$ are points on $\ell$ such that angles $\angle APM$ and $\angle AQN$ are $\frac\pi2$. Prove that \[\frac{1}{AM}+\frac1{AN}\leq\frac2{AB}\]

Let $ABC$ be a triangle with centroid $G$. Let the circumcircle of triangle $AGB$ intersect the line $BC$ in $X$ different from $B$; and the circucircle of triangle $AGC$ intersect the line $BC$ in $Y$ different from $C$. Prove that $G$ is the centroid of triangle $AXY$.

Let $n\ge2$. For any two $n$-vectors $\vec x=(x_1,\ldots,x_n)$ and $\vec y=(y_1,\ldots,y_n)$, we define $$f\left(\vec x,\vec y\right)=x_1\overline{y_1}-\sum_{i=2}^nx_i\overline{y_i}.$$Prove that if $f\left(\vec x,\vec x\right)\ge0$, and $f\left(\vec y,\vec y\right)\ge0$, then $\left|f\left(\vec x,\vec y\right)\right|^2\ge f\left(\vec x,\vec x\right)f\left(\vec y,\vec y\right)$.

Consider two circles of radius one, and let $O$ and $O'$ denote their centers. Point $M$ is selected on either circle. If $OO' = 2014$, what is the largest possible area of triangle $OMO'$? [i]Proposed by Evan Chen[/i]

Given $\vartriangle ABC$ and a point $P$ inside that triangle. The parallelograms $CPBL$, $APCM$ and $BPAN$ are constructed. Prove that $AL$, $BM$ and $CN$ pass through one point $S$, and that $S$ is the midpoint of $AL$, $BM$ and $CN$.

To fold a paper airplane, Austin starts with a square paper $F OLD$ with side length $2$. First, he folds corners $L$ and $D$ to the square’s center. Then, he folds corner $F$ to corner $O$. What is the longest distance between two corners of the resulting figure?

Describe all closed bounded figures $\Phi$ in the plane any two points of which are connectable by a semicircle lying in $\Phi$.

Let $ABC$ be a triangle with $\angle C = 90^\circ$ and $CA \neq CB$. Let $CH$ be an altitude and $CL$ be an interior angle bisector. Show that for $X \neq C$ on the line $CL$, we have $\angle XAC \neq \angle XBC$. Also show that for $Y \neq C$ on the line $CH$ we have $\angle YAC \neq \angle YBC$. [i]Bulgaria[/i]

Let $P$ be an arbitrary point on side $BC$ of triangle $ABC$. Let $K$ be the incenter of triangle $PAB$. Let the incircle of triangle $PAC$ touch $BC$ at $F$. Point $G$ on $CK$ is such that $FG // PK$. Find the locus of $G$.

Given a quadrilateral $ ABCD $ inscribed in a circle. The images of the points $ A $ and $ C $ in symmetry with respect to the line $ BD $ are the points $ A' $ and $ C' $, respectively, and the images of the points $ B $ and $ D $ in symmetry with respect to the line $ AC $ are the points $ B'$ and $D'$ respectively. Prove that the points $ A' $, $ B' $, $ C' $, $ D' $ lie on the circle.

Three parallel lines $L_1, L_2, L_2$ are drawn in the plane such that the perpendicular distance between $L_1$ and $L_2$ is $3$ and the perpendicular distance between lines $L_2$ and $L_3$ is also $3$. A square $ABCD$ is constructed such that $A$ lies on $L_1$, $B$ lies on $L_3$ and $C$ lies on $L_2$. Find the area of the square.

Let $ABC$ be a triangle with $\angle B \ge 2\angle C$. Denote by $D$ the foot of the altitude from $A$ and by $M$ be the midpoint of $BC$. Prove that $DM \ge \frac{AB}{2}$.

Let $ABC$ be an acute triangle such that $AC<BC$ and $\omega$ its circumcircle. $M$ is the midpoint of $BC$. Points $F$ and $E$ are chosen in $AB$ and $BC$, respectively, such that $AC=CF$ and $EB=EF$. The line $AM$ intersects $\omega$ in $D\neq A$. The line $DE$ intersects the line $FM$ in $G$. Prove that $G$ lies on $\omega$.

Let $BE$ and $CF$ be altitudes in triangle $ABC$. If $AE = 24$, $EC = 60$, and $BF = 31$, determine $AF$.

A line passes through $ A(1,1)$ and $ B(100,1000)$. How many other points with integer coordinates are on the line and strictly between $ A$ and $ B$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 9$

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 $ k$ be the inscribed circle of non-isosceles triangle $ \triangle ABC$, which center is $ S$. Circle $ k$ touches sides $ BC,CA,AB$ in points $ P,Q,R$ respectively. Line $ QR$ intersects $ BC$ in point $ M$. Let a circle which contains points $ B$ and $ C$ touch $ k$ in point $ N$. Circumscribed circle of $ \triangle MNP$ intersects line $ AP$ in point $ L$, different from $ P$. Prove that points $ S,L$ and $ M$ are collinear.

Let $ ABC$ be a triangle, let $ AB > AC$. Its incircle touches side $ BC$ at point $ E$. Point $ D$ is the second intersection of the incircle with segment $ AE$ (different from $ E$). Point $ F$ (different from $ E$) is taken on segment $ AE$ such that $ CE \equal{} CF$. The ray $ CF$ meets $ BD$ at point $ G$. Show that $ CF \equal{} FG$.