In triangle $ABC$, $M\in(BC)$, $\frac{BM}{BC}=\alpha$, $N\in(CA)$, $\frac{CN}{CA}=\beta$, $P\in(AB)$, $\frac{AP}{AB}=\gamma$. Let $AM\cap BN=\{D\}$, $BN\cap CP=\{E\}$, $CP\cap AM=\{F\}$. Prove that $S_{DEF}=S_{BMD}+S_{CNE}+S_{APF}$ iff $\alpha+\beta+\gamma=1$.

Right triangle $ABC$ is inscribed in circle $W$. $\angle{CAB}=65$ degrees, and $\angle{CBA}=25$ degrees. The median from $C$ to $AB$ intersects $W$ and line $D$. Line $l_1$ is drawn tangent to $W$ at $A$. Line $l_2$ is drawn tangent to $W$ at $D$. The lines $l_1$ and $l_2$ intersect at $P$ Determine $\angle{APD}$

[size=130]In $\triangle ABC$, $\angle A=60^\circ$. $\odot I$ is the incircle of $\triangle ABC$. $\odot I$ is tangent to sides $AB$, $AC$ at $D$, $E$, respectively. Line $DE$ intersects line $BI$ and $CI$ at $F$, $G$ respectively. Prove that [/size]$FG=\frac{BC}{2}$.

A cow lives on a cubic planet of side length $12$. It is tied on a leash $12$ units long that is staked at the center of one of the faces of the cube. The total surface area that the cow can graze is $A \pi+B( \sqrt3 -1)$. Find $A + B$.

Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, $CA = 15$. Company $XYZ$ wants to locate their base at the point $P$ in the plane minimizing the total distance to their workers, who are located at vertices $A$, $B$, and $C$. There are $1$, $5$, and $4$ workers at $A$, $B$, and $C$, respectively. Find the minimum possible total distance Company $XYZ$'s workers have to travel to get to $P$.

Determine the maximal area triangle such that all of its vertices satisfy $\frac{x^2}{9} + \frac{y^2}{16} = 1$.

$X_1, X_2, . . . , X_{2015}$ are $2015$ points in the plane such that for all $1 \le i, j \le 2015$, the line segment $X_iX_{i+1} = X_jX_{j+1}$ and angle $\angle X_iX_{i+1}X_{i+2} = \angle X_jX_{j+1}X_{j+2}$ (with cyclic indices such that $X_{2016} = X_1$ and $X_{2017} = X_2$). Given fixed $X_1$ and $X_2$, determine the number of possible locations for $X_3$.

$ D$ is a point on the edge $ BC$ of triangle $ ABC$ such that $ AD\equal{}\frac{BD^2}{AB\plus{}AD}\equal{}\frac{CD^2}{AC\plus{}AD}$. $ E$ is a point such that $ D$ is on $ [AE]$ and $ CD\equal{}\frac{DE^2}{CD\plus{}CE}$. Prove that $ AE\equal{}AB\plus{}AC$.

Let $ABC$ be a triangle with circumcircle $\Gamma$. Let $\omega_0$ be a circle tangent to chord $AB$ and arc $ACB$. For each $i = 1, 2$, let $\omega_i$ be a circle tangent to $AB$ at $T_i$ , to $\omega_0$ at $S_i$ , and to arc $ACB$. Suppose $\omega_1 \ne \omega_2$. Prove that there is a circle passing through $S_1, S_2, T_1$, and $T_2$, and tangent to $\Gamma$ if and only if $\angle ACB = 90^o$. .

For a convex quadrilateral $ABCD$, call a point in the interior of $ABCD$ [b]balanced[/b], if (1) $P$ is not on $AC,BD$ (2) Let $AP,BP,CP,DP$ intersect the boundaries of $ABCD$ at $A', B', C', D'$, respectively, then $$AP \cdot PA' = BP \cdot PB' = CP \cdot PC' = DP \cdot PD'$$ Find the maximum possible number of balanced points.

A circle passes through the three vertices of an isosceles triangle that has two sides of length $ 3$ and a base of length $ 2$. What is the area of this circle? $ \textbf{(A)}\ 2\pi\qquad \textbf{(B)}\ \frac {5}{2}\pi\qquad \textbf{(C)}\ \frac {81}{32}\pi\qquad \textbf{(D)}\ 3\pi\qquad \textbf{(E)}\ \frac {7}{2}\pi$

There are $\frac{k(k+1)}{2}+1$ points on the planes, some are connected by disjoint segments ( also point can not lies on segment, that connects two other points). It is true, that plane is divided to some parallelograms and one infinite region. What maximum number of segments can be drawn ? [i] A.Kupavski, A. Polyanski[/i]

[u]Round 1[/u] [b]p1.[/b] Alex is on a week-long mining quest. Each morning, she mines at least $1$ and at most $10$ diamonds and adds them to her treasure chest (which already contains some diamonds). Every night she counts the total number of diamonds in her collection and finds that it is divisible by either $22$ or $25$. Show that she miscounted. [b]p2.[/b] Hermione set out a row of $11$ Bertie Bott’s Every Flavor Beans for Ron to try. There are $5$ chocolateflavored beans that Ron likes and $6$ beans flavored like earwax, which he finds disgusting. All beans look the same, and Hermione tells Ron that a chocolate bean always has another chocolate bean next to it. What is the smallest number of beans that Ron must taste to guarantee he finds a chocolate one? [b]p3.[/b] There are $101$ pirates on a pirate ship: the captain and $100$ crew. Each pirate, including the captain, starts with $1$ gold coin. The captain makes proposals for redistributing the coins, and the crew vote on these proposals. The captain does not vote. For every proposal, each crew member greedily votes “yes” if he gains coins as a result of the proposal, “no” if he loses coins, and passes otherwise. If strictly more crew members vote “yes” than “no,” the proposal takes effect. The captain can make any number of proposals, one after the other. What is the largest number of coins the captain can accumulate? [b]p4.[/b] There are $100$ food trucks in a circle and $10$ gnomes who sample their menus. For the first course, all the gnomes eat at different trucks. For each course after the first, gnome #$1$ moves $1$ truck left or right and eats there; gnome #$2$ moves $2$ trucks left or right and eats there; ... gnome #$10$ moves $10$ trucks left or right and eats there. All gnomes move at the same time. After some number of courses, each food truck had served at least one gnome. Show that at least one gnome ate at some food truck twice. [b]p5.[/b] The town of Lumenville has $100$ houses and is preparing for the math festival. The Tesla wiring company lays lengths of power wire in straight lines between the houses so that power flows between any two houses, possibly by passing through other houses.The Edison lighting company hangs strings of lights in straight lines between pairs of houses so that each house is connected by a string to exactly one other. Show that however the houses are arranged, the Edison company can always hang their strings of lights so that the total length of the strings is no more than the total length of the power wires the Tesla company used. [img]https://cdn.artofproblemsolving.com/attachments/9/2/763de9f4138b4dc552247e9316175036c649b6.png[/img] [u]Round 2[/u] [b]p6.[/b] What is the largest number of zeros that could appear at the end of $1^n + 2^n + 3^n + 4^n$, where n can be any positive integer? [b]p7.[/b] A tennis academy has $2023$ members. For every group of 1011 people, there is a person outside of the group who played a match against everyone in it. Show there is someone who has played against all $2022$ other members. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Suppose you have $15$ circles of radius $1$. Compute the side length of the smallest equilateral triangle that could possibly contain all the circles, if you are free to arrange them in any shape, provided they don’t overlap.

Let triangle $ABC$ be a given acute scalene triangle with altitudes $BE$ and $CF$. Let $D$ be the point where the incircle of $\,\triangle ABC$ touches side $BC$. The circumcircle of $\triangle BDE$ meets line $AB$ again at point $K$, the circumcircle of $\triangle CDF$ meets line $AC$ again at point $L$. The circumcircle of $\triangle BDE$ and $\triangle CDF$ meet line $KL$ again at $X$ and $Y$, respectively. Prove that the incenter of $\triangle DXY$ lies on the incircle of $\,\triangle ABC$. [i]Proposed by Luu Dong, Vietnam[/i]

In the figure, triangle $ADE$ is produced from triangle $ABC$ by a rotation by $90^o$ about the point $A$. If angle $D$ is $60^o$ and angle $E$ is $40^o$, how large is then angle $u$? [img]https://1.bp.blogspot.com/-6Fq2WUcP-IA/Xzb9G7-H8jI/AAAAAAAAMWY/hfMEAQIsfTYVTdpd1Hfx15QPxHmfDLEkgCLcBGAsYHQ/s0/2009%2BMohr%2Bp1.png[/img]

Let $I,H,O$ be the incenter, centroid, and circumcenter of the nonisosceles triangle $ABC$. Prove that $AI \parallel HO$ if and only if $\angle BAC =120^{\circ}$.

In the plane of the triangle $ ABC $ a straight line moves which intersects the sides $ AC $ and $ BC $ in such points $ D $ and $ E $ that $ AD = BE $. Find the locus of the midpoint $ M $ of the segment $ DE $.

A cube lies on the plane. After being rolled a few times (over its edges), it is brought back to its initial location with the same face up. Could the top face have been rotated by 90 degrees? [i](5 points)[/i]

Let $ABC$ be an acute-angled triangle with $AB=AC$, let $D$ be the foot of perpendicular, of the point $C$, to the line $AB$ and the point $M$ is the midpoint of $AC$. Finally, the point $E$ is the second intersection of the line $BC$ and the circumcircle of $\triangle CDM$. Prove that the lines $AE, BM$ and $CD$ are concurrents if and only if $CE=CM$.

Which regular polygon can be obtained (and how) by cutting a cube with a plane ?

The vertices of $\Delta ABC$ lie on the graphics of the function $f(x)=x^2$ and its centroid is $M(1,7)$. Determine the greatest possible value of the area of $\Delta ABC$.

Square $ABCD$ has center $O$, $AB=900$, $E$ and $F$ are on $AB$ with $AE<BF$ and $E$ between $A$ and $F$, $m\angle EOF =45^\circ$, and $EF=400$. Given that $BF=p+q\sqrt{r}$, wherer $p,q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r$.

Cut the $9 \times 10$ grid rectangle along the grid lines into several squares so that there are exactly two of them with odd sidelengths.

Let $ABCD$ be a cyclic quadrilateral with all distinct sides that has an inscribed circle. The incircle of $ABCD$ has center $I$ and is tangent to $AB$, $BC$, $CD$, and $DA$ at points $W$, $X$, $Y$, and $Z$, respectively. Let $K$ be the intersection of the lines $WX$ and $YZ$. Prove that $KI$ is tangent to the circumcircle of triangle $AIC$.