Let $M$ be the midpoint of the side $AB$ of a triangle $ABC$. A circle through point $C$ that has a point of tangency to the line $AB$ at point $A$ and a circle through point $C$ that has a point of tangency to the line $AB$ at point $B$ intersect the second time at point $N$. Prove that $|CM|^2 + |CN|^2 - |MN|^2 = |CA|^2 + |CB|^2 - |AB|^2$.

In the non-isosceles triangle $ABC$ consider the points $X$ on $[AB]$ and $Y$ on $[AC]$ such that $[BX]=[CY]$, $M$ and $N$ are the midpoints of the segments $[BC]$, respectively $[XY]$, and the straight lines $XY$ and $BC$ meet in $K$. Prove that the circumcircle of triangle $KMN$ contains a point, different from $M$ , which is independent of the position of the points $X$ and $Y$.

In an acute-angled triangle \(ABC\), \(H\) be the orthocenter of it and \(D\) be any point on the side \(BC\). The points \(E, F\) are on the segments \(AB, AC\), respectively, such that the points \(A, B, D, F\) and \(A, C, D, E\) are cyclic. The segments \(BF\) and \(CE\) intersect at \(P.\) \(L\) is a point on \(HA\) such that \(LC\) is tangent to the circumcircle of triangle \(PBC\) at \(C.\) \(BH\) and \(CP\) intersect at \(X\). Prove that the points \(D, X, \) and \(L\) lie on the same line. [i]Proposed by Theoklitos Parayiou, Cyprus [/i]

Let $ABC$ be an acute triangle.The lines $l_1$ and $l_2$ are perpendicular to $AB$ at the points $A$ and $B$, respectively.The perpendicular lines from the midpoint $M$ of $AB$ to the lines $AC$ and $BC$ intersect $l_1$ and $l_2$ at the points $E$ and $F$, respectively.If $D$ is the intersection point of the lines $EF$ and $MC$, prove that \[\angle ADB = \angle EMF.\]

A non-isosceles triangle $A_{1}A_{2}A_{3}$ has sides $a_{1}$, $a_{2}$, $a_{3}$ with the side $a_{i}$ lying opposite to the vertex $A_{i}$. Let $M_{i}$ be the midpoint of the side $a_{i}$, and let $T_{i}$ be the point where the inscribed circle of triangle $A_{1}A_{2}A_{3}$ touches the side $a_{i}$. Denote by $S_{i}$ the reflection of the point $T_{i}$ in the interior angle bisector of the angle $A_{i}$. Prove that the lines $M_{1}S_{1}$, $M_{2}S_{2}$ and $M_{3}S_{3}$ are concurrent.

Let $\triangle ABC$ be an acute scalene triangle with incenter $I$ and incircle $\omega$. Two points $X$ and $Y$ are chosen on minor arcs $AB$ and $AC$, respectively, of the circumcircle of triangle $\triangle ABC$ such that $XY$ is tangent to $\omega$ at $P$ and $\overline{XY}\perp \overline{AI}$. Let $\omega$ be tangent to sides $AC$ and $AB$ at $E$ and $F$, respectively. Denote the intersection of lines $XF$ and $YE$ as $T$. Prove that if the circumcircles of triangles $\triangle TEF$ and $\triangle ABC$ are tangent at some point $Q$, then lines $PQ$, $XE$, and $YF$ are concurrent. [i]Proposed by Andrew Wen[/i]

The gure below shows a large circle with area $120$ containing a circle with half of the radius of the large circle and six circles with a quarter of the radius of the large circle. Find the area of the shaded region. [img]https://cdn.artofproblemsolving.com/attachments/7/9/064a05feb9bd67896c079a5141bf7556d7165b.png[/img]

Let $\triangle ABC$ be an equilateral triangle with side length $55$. Points $D$, $E$, and $F$ lie on sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$, respectively, with $BD=7$, $CE=30$, and $AF=40$. A unique point $P$ inside $\triangle ABC$ has the property that \[\measuredangle AEP=\measuredangle BFP=\measuredangle CDP.\] Find $\tan^{2}\left(\measuredangle AEP\right)$.

From among $2^{1/2},$ $3^{1/3},$ $8^{1/8},$ $9^{1/9}$ those which have the greatest and the next to the greatest values, in that order, are \[ \begin{array}{rlrlrlrl} \hbox {(A)}& 3^{1/3},\ 2^{1/2} \quad & \hbox {(B)}& 3^{1/3},\ 8^{1/8} \quad & \hbox {(C)}& 3^{1/3},\ 9^{1/9} \quad & \hbox {(D)}& 8^{1/8},\ 9^{1/9} \\ \hbox {(E)}& \multicolumn{3}{l}{\hbox{None of these}} \end{array} \]

Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\frac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ = 48$. Let $R$ be the midpoint of $\overline{PQ}$. What is $AR+BR+CR+DR$? $ \textbf{(A)}\ 180 \qquad\textbf{(B)}\ 184 \qquad\textbf{(C)}\ 188 \qquad\textbf{(D)}\ 192\qquad\textbf{(E)}\ 196 $

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 $S$.

Let $ABCD$ be a convex cyclic quadrilateral with diagonals $AC$ and $BD$. Each of the four vertixes are reflected across the diagonal on which the do not lie. (a) Investigate when the four points thus obtained lie on a straight line and give as simple an equivalent condition as possible to the cyclic quadrilateral $ABCD$ for it. (b) Show that in all other cases the four points thus obtained lie on one circle. (Theresia Eisenkölbl)

$ABCD$ is a square with side length 20. A light beam is radiated from $A$ and intersects sides $BC,CD,DA$ respectively and reaches the midpoint of side $AB$. What is the length of the path that the beam has taken? [img]https://s8.uupload.ir/files/photo14908575660_2r3g.jpg[/img] [i]Proposed by Mahdi Etesamifard - Iran[/i]

Let $A$ and $B$ be two distinct points on the circle $\Gamma$, not diametrically opposite. The point $P$, distinct from $A$ and $B$, varies on $\Gamma$. Find the locus of the orthocentre of triangle $ABP$.

Prove that any triangle can be divided into $22$ triangles, each of which has an angle of $22^o$, and another $23$ triangles, each of which has an angle of $23^o$.

Let $ABC$ and $A'B'C'$ be any two coplanar triangles. Let $L$ be a point such that $AL || BC, A'L || B'C'$ , and $M,N$ similarly defined. The line $BC$ meets $B'C'$ at $P$, and similarly defined are $Q$ and $R$. Prove that $PL, QM, RN$ are concurrent.

[b]p1.[/b] Nicky is studying biology and has a tank of $17$ lizards. In one day, he can either remove $5$ lizards or add $2$ lizards to his tank. What is the minimum number of days necessary for Nicky to get rid of all of the lizards from his tank? [b]p2.[/b] What is the maximum number of spheres with radius $1$ that can fit into a sphere with radius $2$? [b]p3.[/b] A positive integer $x$ is sunny if $3x$ has more digits than $x$. If all sunny numbers are written in increasing order, what is the $50$th number written? [b]p4.[/b] Quadrilateral $ABCD$ satisfies $AB = 4$, $BC = 5$, $DA = 4$, $\angle DAB = 60^o$, and $\angle ABC = 150^o$. Find the area of $ABCD$. [b]p5. [/b]Totoro wants to cut a $3$ meter long bar of mixed metals into two parts with equal monetary value. The left meter is bronze, worth $10$ zoty per meter, the middle meter is silver, worth $25$ zoty per meter, and the right meter is gold, worth $40$ zoty per meter. How far, in meters, from the left should Totoro make the cut? [b]p6.[/b] If the numbers $x_1, x_2, x_3, x_4$, and $x5$ are a permutation of the numbers $1, 2, 3, 4$, and $5$, compute the maximum possible value of $$|x_1 - x_2| + |x_2 - x_3| + |x_3 - x_4| + |x_4 - x_5|.$$ [b]p7.[/b] In a $3 \times 4$ grid of $12$ squares, find the number of paths from the top left corner to the bottom right corner that satisfy the following two properties: $\bullet$ The path passes through each square exactly once. $\bullet$ Consecutive squares share a side. Two paths are considered distinct if and only if the order in which the twelve squares are visited is different. For instance, in the diagram below, the two paths drawn are considered the same. [img]https://cdn.artofproblemsolving.com/attachments/7/a/bb3471bbde1a8f58a61d9dd69c8527cfece05a.png[/img] [b]p8.[/b] Scott, Demi, and Alex are writing a computer program that is $25$ ines long. Since they are working together on one computer, only one person may type at a time. To encourage collaboration, no person can type two lines in a row, and everyone must type something. If Scott takes $10$ seconds to type one line, Demi takes $15$ seconds, and Alex takes $20$ seconds, at least how long, in seconds, will it take them to finish the program? [b]p9.[/b] A hand of four cards of the form $(c, c, c + 1, c + 1)$ is called a tractor. Vinjai has a deck consisting of four of each of the numbers $7$, $8$, $9$ and $10$. If Vinjai shuffles and draws four cards from his deck, compute the probability that they form a tractor. [b]p10. [/b]The parabola $y = 2x^2$ is the wall of a fortress. Totoro is located at $(0, 4)$ and fires a cannonball in a straight line at the closest point on the wall. Compute the y-coordinate of the point on the wall that the cannonball hits. [b]p11. [/b]How many ways are there to color the squares of a $10$ by $10$ grid with black and white such that in each row and each column there are exactly two black squares and between the two black squares in a given row or column there are exactly [b]4[/b] white squares? Two configurations that are the same under rotations or reflections are considered different. [b]p12.[/b] In rectangle $ABCD$, points $E$ and $F$ are on sides $AB$ and $CD$, respectively, such that $AE = CF > AD$ and $\angle CED = 90^o$. Lines $AF, BF, CE$ and $DE$ enclose a rectangle whose area is $24\%$ of the area of $ABCD$. Compute $\frac{BF}{CE}$ . [b]p13.[/b] Link cuts trees in order to complete a quest. He must cut $3$ Fenwick trees, $3$ Splay trees and $3$ KD trees. If he must also cut 3 trees of the same type in a row at some point during his quest, in how many ways can he cut the trees and complete the quest? (Trees of the same type are indistinguishable.) [b]p14.[/b] Find all ordered pairs (a, b) of positive integers such that $\sqrt{64a + b^2} + 8 = 8\sqrt{a} + b$. [b]p15.[/b] Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle BCD = 108^o$, $\angle CDE = 168^o$ and $AB =BC = CD = DE$. Find the measure of $\angle AEB$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCD$ be a convex quadrilateral with $\angle ABC>90$, $CDA>90$ and $\angle DAB=\angle BCD$. Denote by $E$ and $F$ the reflections of $A$ in lines $BC$ and $CD$, respectively. Suppose that the segments $AE$ and $AF$ meet the line $BD$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $BEK$ and $DFL$ are tangent to each other. $\emph{Slovakia}$

Triangle $ABC$ has $AB=10$, $BC=17$, and $CA=21$. Point $P$ lies on the circle with diameter $AB$. What is the greatest possible area of $APC$?

Angle bisectors $AD$ and $CE$ of triangle $ABC$ intersect at point $O$. A line symmetrical to $ AB$ with respect to $CE$ intersects the line symmetric $BC$ with respect to $AD$, at point $K$. Prove that $KO \perp AC$.

Two circles $K_1$ and $K_2$ of different radii intersect at two points $A$ and $B$, let $C$ and $D$ be two points on $K_1$ and $K_2$, respectively, such that $A$ is the midpoint of the segment $CD$. The extension of $DB$ meets $K_1$ at another point $E$, the extension of $CB$ meets $K_2$ at another point $F$. Let $l_1$ and $l_2$ be the perpendicular bisectors of $CD$ and $EF$, respectively. i) Show that $l_1$ and $l_2$ have a unique common point (denoted by $P$). ii) Prove that the lengths of $CA$, $AP$ and $PE$ are the side lengths of a right triangle.

A square $ABCD$ and point $E$ are drawn in a plane such that lengths $DE < BE$ and $\vartriangle ACE$ is equilateral. Compute $\angle BAE$.

If five geometric means are inserted between 8 and 5832, the fifth term in the geometric series: $\textbf{(A)}\ 648 \qquad \textbf{(B)}\ 832 \qquad \textbf{(C)}\ 1168 \qquad \textbf{(D)}\ 1944 \qquad \textbf{(E)}\ \text{None of these}$

The functions $ f(x) ,\ g(x)$ satify that $ f(x) \equal{} \frac {x^3}{2} \plus{} 1 \minus{} x\int_0^x g(t)\ dt,\ g(x) \equal{} x \minus{} \int_0^1 f(t)\ dt$. Let $ l_1,\ l_2$ be the tangent lines of the curve $ y \equal{} f(x)$, which pass through the point $ (a,\ g(a))$ on the curve $ y \equal{} g(x)$. Find the minimum area of the figure bounded by the tangent tlines $ l_1,\ l_2$ and the curve $ y \equal{} f(x)$ .

Each of the squares of an $8 \times 8$ board can be colored white or black. Find the number of colorings of the board such that every $2 \times 2$ square contains exactly 2 black squares and 2 white squares.