The side $AC$ of an acute triangle $ABC$ is the diameter of the circle $c_1$ and side $BC$ is the diameter of the circle $c_2$. Let $E$ be the foot of the altitude drawn from the vertex $B$ of the triangle and $F$ the foot of the altitude drawn from the vertex $A$. In addition, let $L$ and $N$ be the points of intersection of the line $BE$ with the circle $c_1$ (the point $L$ lies on the segment $BE$) and the points of intersection of $K$ and $M$ of line $AF$ with circle $c_2$ (point $K$ is in section $AF$). Prove that $K LM N$ is a cyclic quadrilateral.

Let $y = mx + b$ be the image when the line $x - 3y + 11 = 0$ is reflected across the x-axis. The value of $m + b$ is $ \textbf{(A)}\ -6\qquad\textbf{(B)}\ -5\qquad\textbf{(C)}\ -4\qquad\textbf{(D)}\ -3\qquad\textbf{(E)}\ -2 $

Four rays spread out from point $O$ in a $3$-dimensional space in a way that the angle between every two rays is $a$. Find $\cos a$.

$BE$ is the altitude of acute triangle $ABC$. The line $\ell$ touches the circumscribed circle of the triangle $ABC$ at point $B$. A perpendicular $CF$ is drawn from $C$ on line $\ell$. Prove that the lines $EF$ and $AB$ are parallel.

Inside a circle, point $K$ is taken such that the ray drawn from $K$ through the centre $O$ of the circle and the chord perpendicular to this ray passing through $K$ divide the circle into three pieces with equal area. Let $L$ be one of the endpoints of the chord mentioned. Does the inequality $\angle KOL < 75^o$ hold?

Let $ P$ be the the isogonal conjugate of $ Q$ with respect to triangle $ ABC$, and $ P,Q$ are in the interior of triangle $ ABC$. Denote by $ O_{1},O_{2},O_{3}$ the circumcenters of triangle $ PBC,PCA,PAB$, $ O'_{1},O'_{2},O'_{3}$ the circumcenters of triangle $ QBC,QCA,QAB$, $ O$ the circumcenter of triangle $ O_{1}O_{2}O_{3}$, $ O'$ the circumcenter of triangle $ O'_{1}O'_{2}O'_{3}$. Prove that $ OO'$ is parallel to $ PQ$.

$20$ points with no three collinear are given. How many obtuse triangles can be formed by these points? $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 2{{10}\choose{3}} \qquad \textbf{(D)}\ 3{{10}\choose{3}} \qquad \textbf{(E)}\ {{20}\choose{3}}$

Sohom constructs a square $BERK$ of side length $10$. Darlnim adds points $T$, $O$, $W$, and $N$, which are the midpoints of $\overline{BE}$, $\overline{ER}$, $\overline{RK}$, and $\overline{KB}$, respectively. Lastly, Sylvia constructs square $CALI$ whose edges contain the vertices of $BERK$, such that $\overline{CA}$ is parallel to $\overline{BO}$. Compute the area of $CALI$. [img]https://cdn.artofproblemsolving.com/attachments/0/9/0fda0c273bb73b85f3b1bc73661126630152b3.png[/img]

Given a circle $C$ and a diameter $AB$ in it, mark a point $P$ on $AB$ different from the ends. In one of the two arcs determined by $AB$ choose the points $M$ and $N$ such that $\angle APM = 60 ^ \circ = \angle BPN$. The segments $MP$ and $NP$ are drawn to obtain three curvilinear triangles; $APM $, $MPN$ and $NPB$ (the sides of the curvilinear triangle $APM$ are the segments $AP$ and $PM$ and the arc $AM$). In each curvilinear triangle a circle is inscribed, that is, a circle is built tangent to the three sides. Show that the sum of the radii of the three inscribed circles is less than or equal to the radius of $C$.

Given a triangle $ABC$ for which $\angle BAC \neq 90^{\circ}$, let $B_1, C_1$ be variable points on $AB,AC$, respectively. Let $B_2,C_2$ be the points on line $BC$ such that a spiral similarity centered at $A$ maps $B_1C_1$ to $C_2B_2$. Denote the circumcircle of $AB_1C_1$ by $\omega$. Show that if $B_1B_2$ and $C_1C_2$ concur on $\omega$ at a point distinct from $B_1$ and $C_1$, then $\omega$ passes through a fixed point other than $A$. [i]Proposed by Max Jiang[/i]

Consider a parallelogram $[ABCD]$ such that $\angle DAB$ is an acute angle. Let $G$ be a point in line $AB$ different from $B$ such that $\overline{BC}=\overline{GC}$, and let $H$ be a point in line $BC$ different from $B$ such that $\overline{AB}=\overline{AH}$. Prove that triangle $[GDH]$ is isosceles.

Let $F$ be the correspondence associating with every point $P = (x, y)$ the point $P' = (x', y')$ such that \[ x'= ax + b,\qquad y'= ay + 2b. \qquad (1)\] Show that if $a \neq 1$, all lines $PP'$ are concurrent. Find the equation of the set of points corresponding to $P = (1, 1)$ for $b = a^2$. Show that the composition of two mappings of type $(1)$ is of the same type.

10.5 Let $BB'$, $CC'$ be the altitudes of an acute triangle $ABC$. Two circles through $A$ and $C'$ are tangent to $BC$ at points $P$ and $Q$. Prove that $A, B', P, Q$ are concyclic.

Pick a point $C$ on a semicircle with diameter $AB$. Let $P$ and $Q$ be two points on segment $AB$ such that $AP= AC$ and $BQ= BC$. The point $O$ is the center of the circumscribed circle of triangle $CPQ$ and point $H$ is the orthocenter of triangle $CPQ$ . Prove that for all posible locations of point $C$, the line $OH$ is passing through a fixed point. (Mykhailo Sydorenko)

[b]p1.[/b] Chris goes to Matt's Hamburger Shop to buy a hamburger. Each hamburger must contain exactly one bread, one lettuce, one cheese, one protein, and at least one condiment. There are two kinds of bread, two kinds of lettuce, three kinds of cheese, three kinds of protein, and six different condiments: ketchup, mayo, mustard, dill pickles, jalape~nos, and Matt's Magical Sunshine Sauce. How many different hamburgers can Chris make? [b]p2.[/b] The degree measures of the interior angles in convex pentagon $NICKY$ are all integers and form an increasing arithmetic sequence in some order. What is the smallest possible degree measure of the pentagon's smallest angle? [b]p3.[/b] Daniel thinks of a two-digit positive integer $x$. He swaps its two digits and gets a number $y$ that is less than $x$. If $5$ divides $x-y$ and $7$ divides $x+y$, find all possible two-digit numbers Daniel could have in mind. [b]p4.[/b] At the Lio Orympics, a target in archery consists of ten concentric circles. The radii of the circles are $1$, $2$, $3$, $...$, $9$, and $10$ respectively. Hitting the innermost circle scores the archer $10$ points, the next ring is worth $9$ points, the next ring is worth 8 points, all the way to the outermost ring, which is worth $1$ point. If a beginner archer has an equal probability of hitting any point on the target and never misses the target, what is the probability that his total score after making two shots is even? [b]p5.[/b] Let $F(x) = x^2 + 2x - 35$ and $G(x) = x^2 + 10x + 14$. Find all distinct real roots of $F(G(x)) = 0$. [b]p6.[/b] One day while driving, Ivan noticed a curious property on his car's digital clock. The sum of the digits of the current hour equaled the sum of the digits of the current minute. (Ivan's car clock shows $24$-hour time; that is, the hour ranges from $0$ to $23$, and the minute ranges from $0$ to $59$.) For how many possible times of the day could Ivan have observed this property? [b]p7.[/b] Qi Qi has a set $Q$ of all lattice points in the coordinate plane whose $x$- and $y$-coordinates are between $1$ and $7$ inclusive. She wishes to color $7$ points of the set blue and the rest white so that each row or column contains exactly $1$ blue point and no blue point lies on or below the line $x + y = 5$. In how many ways can she color the points? [b]p8.[/b] A piece of paper is in the shape of an equilateral triangle $ABC$ with side length $12$. Points $A_B$ and $B_A$ lie on segment $AB$, such that $AA_B = 3$, and $BB_A = 3$. Define points $B_C$ and $C_B$ on segment $BC$ and points $C_A$ and $A_C$ on segment $CA$ similarly. Point $A_1$ is the intersection of $A_CB_C$ and $A_BC_B$. Define $B_1$ and $C_1$ similarly. The three rhombi - $AA_BA_1A_C$,$BB_CB_1B_A$, $CC_AC_1C_B$ - are cut from triangle $ABC$, and the paper is folded along segments $A_1B_1$, $B_1C_1$, $C_1A_1$, to form a tray without a top. What is the volume of this tray? [b]p9.[/b] Define $\{x\}$ as the fractional part of $x$. Let $S$ be the set of points $(x, y)$ in the Cartesian coordinate plane such that $x + \{x\} \le y$, $x \ge 0$, and $y \le 100$. Find the area of $S$. [b]p10.[/b] Nicky likes dolls. He has $10$ toy chairs in a row, and he wants to put some indistinguishable dolls on some of these chairs. (A chair can hold only one doll.) He doesn't want his dolls to get lonely, so he wants each doll sitting on a chair to be adjacent to at least one other doll. How many ways are there for him to put any number (possibly none) of dolls on the chairs? Two ways are considered distinct if and only if there is a chair that has a doll in one way but does not have one in the other. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On the plane is give a broken line $ABCD$ in which $AB = BC = CD = 1$, and $AD$ is not equal to $1$. The positions of $B$ and $C$ are fixed but $A$ and $D$ change their positions in turn according to the following rule (preserving the distance rules given): the point $A$ is reflected with respect to the line $BD$, then $D$ is reflected with respect to the line $AC$ (in which $A$ occupies its new position), then $A$ is reflected with respect to the line $BD$ ($D$ occupying its new position), $D$ is reflected with respect to the line $AC$, and so on. Prove that after several steps $A$ and $D$ coincide with their initial positions. (M Kontzewich)

A semicircle of diameter $ 1$ sits at the top of a semicircle of diameter $ 2$, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of this lune. [asy]unitsize(2.5cm); defaultpen(fontsize(10pt)+linewidth(.8pt)); filldraw(Circle((0,.866),.5),grey,black); label("1",(0,.866),S); filldraw(Circle((0,0),1),white,black); draw((-.5,.866)--(.5,.866),linetype("4 4")); clip((-1,0)--(1,0)--(1,2)--(-1,2)--cycle); draw((-1,0)--(1,0)); label("2",(0,0),S);[/asy]$ \textbf{(A)}\ \frac {1}{6}\pi \minus{} \frac {\sqrt {3}}{4} \qquad \textbf{(B)}\ \frac {\sqrt {3}}{4} \minus{} \frac {1}{12}\pi \qquad \textbf{(C)}\ \frac {\sqrt {3}}{4} \minus{} \frac {1}{24}\pi\qquad\textbf{(D)}\ \frac {\sqrt {3}}{4} \plus{} \frac {1}{24}\pi$ $ \textbf{(E)}\ \frac {\sqrt {3}}{4} \plus{} \frac {1}{12}\pi$

In a triangle $ABC,$ the internal and external bisectors of the angle $A$ intersect the line $BC$ at $D$ and $E$ respectively. The line $AC$ meets the circle with diameter $DE$ again at $F.$ The tangent line to the circle $ABF$ at $A$ meets the circle with diameter $DE$ again at $G.$ Show that $AF = AG.$

In triangle $ABC$ ($AB > BC$), $K$ and $M$ are the midpoints of sides $AB$ and $AC$, $O$ is the point of intersection of the angle bisectors. Let $P$ be the intersection point of lines $KM$ and $CO$, and the point $Q$ is such that $QP \perp KM$ and $QM \parallel BO$. Prove that $QO \perp AC$.

A circle and a square have the same perimeter. Then: $ \textbf{(A)}\ \text{their areas are equal} \qquad\textbf{(B)}\ \text{the area of the circle is the greater}$ $ \textbf{(C)}\ \text{the area of the square is the greater}$ $ \textbf{(D)}\ \text{the area of the circle is } \pi \text{ times the area of the square} \\ \qquad\textbf{(E)}\ \text{none of these}$

Prove the inequality \[(-a+b+c)^2(a-b+c)^2(a+b-c)^2 \geq (-a^2+b^2+c^2)(a^2-b^2+c^2)(a^2+b^2-c^2)\] for all real numbers $a, b, c.$

1. Let $ABC$ be an acute-angled triangle with $AB\neq AC$. The circle with diameter $BC$ intersects the sides $AB$ and $AC$ at $M$ and $N$ respectively. Denote by $O$ the midpoint of the side $BC$. The bisectors of the angles $\angle BAC$ and $\angle MON$ intersect at $R$. Prove that the circumcircles of the triangles $BMR$ and $CNR$ have a common point lying on the side $BC$.

$ABCD$ is inscribed in unit circle $\Gamma$. Let $\Omega_1$, $\Omega_2$ be the circumcircles of $\vartriangle ABD$, $\vartriangle CBD$ respectively. Circles $\Omega_1$, $\Omega_2$ are tangent to segment $BD$ at $M$,$N$ respectively. Line A$M$ intersects $\Gamma$, $\Omega_1$ again at points $X_1$,$X_2$ respectively (different from $A$, $M$). Let $\omega_1$ be the circle passing through $X_1$, $X_2$ and tangent to $\Omega_1$. Line $CN$ intersects $\Gamma$, $\Omega_2$ again at points $Y_1$, $Y_2$ respectively (different from $C$, $N$). Let $\omega_2$ be the circle passing through $Y_1$, $Y_2$ and tangent to $\Omega_2$. Circles $\Omega_1$,$\Omega_2$, $\omega_1$, $\omega_2$ have radii $R_1$, $R_2$, $r_1$, $r_2$ respectively. Prove that $$r_1+r_2-R_1-R_2=1.$$ [img]https://cdn.artofproblemsolving.com/attachments/1/5/70471f2419fadc4b2183f5fe74f0c7a2e69ed4.png[/img] [url=https://www.geogebra.org/m/vxx8ghww]geogebra file[/url]

$ABC$ is a triangle. The feet of the perpendiculars from $B$ and $C$ to the angle bisector at $A$ are $K, L$ respectively. $N$ is the midpoint of $BC$, and $AM$ is an altitude. Show that $K,L,N,M$ are concyclic.

In triangle $ABC$ $\omega$ is incircle and $\omega_1$,$\omega_2$,$\omega_3$ is tangents to $\omega$ and two sides of $ABC$. $r, r_1, r_2, r_3$ is radius of $\omega, \omega_1, \omega_2, \omega_3$. Prove that $\sqrt{r_1 r_2}+\sqrt{r_2 r_3}+\sqrt{r_3 r_1}=r$