Let $P_1P_2\ldots P_n$ be a convex polygon in the plane. We assume that for any arbitrary choice of vertices $P_i,P_j$ there exists a vertex in the polygon $P_k$ distinct from $P_i,P_j$ such that $\angle P_iP_kP_j=60^{\circ}$. Show that $n=3$. [i]Radu Todor[/i]

Let $a,b,c$ be the sides and $\alpha,\beta,\gamma$ be the corresponding angles of a triangle. Prove the equality $$\left(\frac bc+\frac cb\right)\cos\alpha+\left(\frac ca+\frac ac\right)\cos\beta+\left(\frac ab+\frac ba\right)\cos\gamma=3.$$

The extension of the bisector of angle $A$ of triangle $ABC$ intersects with the circumscribed circle of this triangle at point $W$. A straight line is drawn through $W$, which is parallel to side $AB$ and intersects sides $BC$ and $AC$ , at points $N$ and $K$, respectively. Prove that the line $AW$ is tangent to the circumscribed circle of $\vartriangle CNW$. (Sergey Yakovlev)

Four points are chosen at random on the surface of a sphere. What is the probability that the center of the sphere lies inside the tetrahedron whose vertices are at the four points?

Line $\ell$ is perpendicular to one of the medians of the triangle. The median perpendiculars to the sides of this triangle intersect the line $\ell$ at three points. Prove that one of them is the midpoint of the segment formed by the other two.

Circles $C_{1}$ and $C_{2}$ are tangent to each other at $K$ and are tangent to circle $C$ at $M$ and $N$. External tangent of $C_{1}$ and $C_{2}$ intersect $C$ at $A$ and $B$. $AK$ and $BK$ intersect with circle $C$ at $E$ and $F$ respectively. If AB is diameter of $C$, prove that $EF$ and $MN$ and $OK$ are concurrent. ($O$ is center of circle $C$.)

A scalene triangle $ABC$ is inscribed in a circle $\Gamma$. The bisector of angle $A$ meets $BC$ at $E$. Let $M$ be the midpoint of the arc $BAC$. The line $ME$ intersects $\Gamma$ again at $D$. Show that the circumcentre of triangle $AED$ coincides with the intersection point of the tangent to $\Gamma$ at $D$ and the line $BC$.

Given triangle ABC with incenter I. Let P,Q be point on circumcircle such that $\angle API=\angle CPI$ and $\angle BQI=\angle CQI$.Prove that $BP,AQ$ and $OI$ are concurrent.

Let $ABC$ be a triangle with $AB<AC$. There are two points $X$ and $Y$ on the angle bisector of $B\widehat AC$ such that $X$ is between $A$ and $Y$ and $BX$ is parallel to $CY$. Let $Z$ be the reflection of $X$ with respect to $BC$. Line $YZ$ cuts line $BC$ at point $P$. If line $BY$ cuts line $CX$ at point $K$, prove that $KA=KP$.

A circumcircle of triangle $ABC$ meets the sides $AD$ and $CD$ of a parallelogram $ABCD$ at points $K$ and $L$ respectively. Let $M$ be the midpoint of arc $KL$ not containing $B$. Prove that $DM \perp AC$. by E.Bakaev

Let $ABCD$ be a parallelogram, such that the point $M$ is the midpoint of the side $CD$ and lies on the bisector of the angle $\angle BAD$. Prove that $\angle AMB = 90^o$.

We draw two lines $(\ell_1) , (\ell_2)$ through the orthocenter $H$ of the triangle $ABC$ such that each one is dividing the triangle into two figures of equal area and equal perimeters. Find the angles of the triangle.

Figure 1 is called a "stack map." The numbers tell how many cubes are stacked in each position. Fig. 2 shows these cubes, and Fig. 3 shows the view of the stacked cubes as seen from the front. Which of the following is the front view for the stack map in Fig. 4? [asy] unitsize(24); draw((0,0)--(2,0)--(2,2)--(0,2)--cycle); draw((1,0)--(1,2)); draw((0,1)--(2,1)); draw((5,0)--(7,0)--(7,1)--(20/3,4/3)--(20/3,13/3)--(19/3,14/3)--(16/3,14/3)--(16/3,11/3)--(13/3,11/3)--(13/3,2/3)--cycle); draw((20/3,13/3)--(17/3,13/3)--(17/3,10/3)--(14/3,10/3)--(14/3,1/3)); draw((20/3,10/3)--(17/3,10/3)--(17/3,7/3)--(20/3,7/3)); draw((17/3,7/3)--(14/3,7/3)); draw((7,1)--(6,1)--(6,2)--(5,2)--(5,0)); draw((5,1)--(6,1)--(6,0)); draw((20/3,4/3)--(6,4/3)); draw((17/3,13/3)--(16/3,14/3)); draw((17/3,10/3)--(16/3,11/3)); draw((14/3,10/3)--(13/3,11/3)); draw((5,2)--(13/3,8/3)); draw((5,1)--(13/3,5/3)); draw((6,2)--(17/3,7/3)); draw((9,0)--(11,0)--(11,4)--(10,4)--(10,3)--(9,3)--cycle); draw((11,3)--(10,3)--(10,0)); draw((11,2)--(9,2)); draw((11,1)--(9,1)); draw((13,0)--(16,0)--(16,2)--(13,2)--cycle); draw((13,1)--(16,1)); draw((14,0)--(14,2)); draw((15,0)--(15,2)); label("Figure 1",(1,0),S); label("Figure 2",(17/3,0),S); label("Figure 3",(10,0),S); label("Figure 4",(14.5,0),S); label("$1$",(1.5,.2),N); label("$2$",(.5,.2),N); label("$3$",(.5,1.2),N); label("$4$",(1.5,1.2),N); label("$1$",(13.5,.2),N); label("$3$",(14.5,.2),N); label("$1$",(15.5,.2),N); label("$2$",(13.5,1.2),N); label("$2$",(14.5,1.2),N); label("$4$",(15.5,1.2),N);[/asy] [asy] unitsize(18); draw((0,0)--(3,0)--(3,2)--(1,2)--(1,4)--(0,4)--cycle); draw((0,3)--(1,3)); draw((0,2)--(1,2)--(1,0)); draw((0,1)--(3,1)); draw((2,0)--(2,2)); draw((5,0)--(8,0)--(8,4)--(7,4)--(7,3)--(6,3)--(6,2)--(5,2)--cycle); draw((8,3)--(7,3)--(7,0)); draw((8,2)--(6,2)--(6,0)); draw((8,1)--(5,1)); draw((10,0)--(12,0)--(12,4)--(11,4)--(11,3)--(10,3)--cycle); draw((12,3)--(11,3)--(11,0)); draw((12,2)--(10,2)); draw((12,1)--(10,1)); draw((14,0)--(17,0)--(17,4)--(16,4)--(16,2)--(14,2)--cycle); draw((17,3)--(16,3)); draw((17,2)--(16,2)--(16,0)); draw((17,1)--(14,1)); draw((15,0)--(15,2)); draw((19,0)--(22,0)--(22,4)--(20,4)--(20,1)--(19,1)--cycle); draw((22,3)--(20,3)); draw((22,2)--(20,2)); draw((22,1)--(20,1)--(20,0)); draw((21,0)--(21,4)); label("(A)",(1.5,0),S); label("(B)",(6.5,0),S); label("(C)",(11,0),S); label("(D)",(15.5,0),S); label("(E)",(20.5,0),S);[/asy]

Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let $ B$ be the total area of the blue triangles, $ W$ the total area of the white squares, and $ R$ the area of the red square. Which of the following is correct? [asy]unitsize(3mm); fill((-4,-4)--(-4,4)--(4,4)--(4,-4)--cycle,blue); fill((-2,-2)--(-2,2)--(2,2)--(2,-2)--cycle,red); path onewhite=(-3,3)--(-2,4)--(-1,3)--(-2,2)--(-3,3)--(-1,3)--(0,4)--(1,3)--(0,2)--(-1,3)--(1,3)--(2,4)--(3,3)--(2,2)--(1,3)--cycle; path divider=(-2,2)--(-3,3)--cycle; fill(onewhite,white); fill(rotate(90)*onewhite,white); fill(rotate(180)*onewhite,white); fill(rotate(270)*onewhite,white);[/asy] $ \textbf{(A)}\ B \equal{} W \qquad \textbf{(B)}\ W \equal{} R \qquad \textbf{(C)}\ B \equal{} R \qquad \textbf{(D)}\ 3B \equal{} 2R \qquad \textbf{(E)}\ 2R \equal{} W$

Let $ABC$ be a triangle with $\angle A = 90^o$ and let $D$ be an orthogonal projection of $A$ onto $BC$. The midpoints of $AD$ and $AC$ are called $E$ and $F$, respectively. Let $M$ be the circumcentre of $\vartriangle BEF$. Prove that $AC\parallel BM$.

Given is a non-isosceles triangle $ABC$ with $\angle ABC=60^{\circ}$, and in its interior, a point $T$ is selected such that $\angle ATC= \angle BTC=\angle BTA=120^{\circ}$. Let $M$ the intersection point of the medians in $ABC$. Let $TM$ intersect $(ATC)$ at $K$. Find $TM/MK$.

A line $d$ is called [i]faithful[/i] to triangle $ABC$ if $d$ be in plane of triangle $ABC$ and the reflections of $d$ over the sides of $ABC$ be concurrent. Prove that for any two triangles with acute angles lying in the same plane, either there exists exactly one [i]faithful[/i] line to both of them, or there exist infinitely [i]faithful[/i] lines to them.

Triangle $ABC$ is equilateral with $AB=1$. Points $E$ and $G$ are on $\overline{AC}$ and points $D$ and $F$ are on $\overline{AB}$ such that both $\overline{DE}$ and $\overline{FG}$ are parallel to $\overline{BC}$. Furthermore, triangle $ADE$ and trapezoids $DFGE$ and $FBCG$ all have the same perimeter. What is $DE+FG$? [asy] size(180); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); real s=1/2,m=5/6,l=1; pair A=origin,B=(l,0),C=rotate(60)*l,D=(s,0),E=rotate(60)*s,F=m,G=rotate(60)*m; draw(A--B--C--cycle^^D--E^^F--G); dot(A^^B^^C^^D^^E^^F^^G); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$D$",D,S); label("$E$",E,NW); label("$F$",F,S); label("$G$",G,NW); [/asy] $\textbf{(A) }1\qquad \textbf{(B) }\dfrac{3}{2}\qquad \textbf{(C) }\dfrac{21}{13}\qquad \textbf{(D) }\dfrac{13}{8}\qquad \textbf{(E) }\dfrac{5}{3}\qquad$

Let $ABCD$ be a cyclic quadrilateral inscribed in circle $O$, where $AC\perp BD$. $M$ be the midpoint of arc $ADC$. Circle $(DOM)$ intersect $DA,DC$ at $E,F$. Prove that $BE=BF$.

[b]p1.[/b] Let $D(n)$ denote the number of positive factors of the integer $n$. For example, $D(6) = 4$ , since the factors of $6$ are $1, 2, 3$ , and $6$ . Note that $D(n) = 2$ if and only if $n$ is a prime number. (a) Describe the set of all solutions to the equation $D(n) = 5$ . (b) Describe the set of all solutions to the equation $D(n) = 6$ . (c) Find the smallest $n$ such that $D(n) = 21$ . [b]p2.[/b] At a party with $n$ married couples present (and no one else), various people shook hands with various other people. Assume that no one shook hands with his or her spouse, and no one shook hands with the same person more than once. At the end of the evening Mr. Jones asked everyone else, including his wife, how many hands he or she had shaken. To his surprise, he got a different answer from each person. Determine the number of hands that Mr. Jones shook that evening, (a) if $n = 2$ . (b) if $n = 3$ . (c) if $n$ is an arbitrary positive integer (the answer may depend on $n$). [b]p3.[/b] Let $n$ be a positive integer. A square is divided into triangles in the following way. A line is drawn from one corner of the square to each of $n$ points along each of the opposite two sides, forming $2n + 2$ nonoverlapping triangles, one of which has a vertex at the opposite corner and the other $2n + 1$ of which have a vertex at the original corner. The figure shows the situation for $n = 2$ . Assume that each of the $2n + 1$ triangles with a vertex in the original corner has area $1$. Determine the area of the square, (a) if $n = 1$ . (b) if $n$ is an arbitrary positive integer (the answer may depend on $n$). [img]https://cdn.artofproblemsolving.com/attachments/1/1/62a54011163cc76cc8d74c73ac9f74420e1b37.png[/img] [b]p4.[/b] Arthur and Betty play a game with the following rules. Initially there are one or more piles of stones, each pile containing one or more stones. A legal move consists either of removing one or more stones from one of the piles, or, if there are at least two piles, combining two piles into one (but not removing any stones). Arthur goes first, and play alternates until a player cannot make a legal move; the player who cannot move loses. (a) Determine who will win the game if initially there are two piles, each with one stone, assuming that both players play optimally. (b) Determine who will win the game if initially there are two piles, each with $n$ stones, assuming that both players play optimally; $n$ is a positive integer, and the answer may depend on $n$ . (c) Determine who will win the game if initially there are $n$ piles, each with one stone, assuming that both players play optimally; $n$ is a positive integer, and the answer may depend on $n$ . [b]p5.[/b] Suppose $x$ and $y$ are real numbers such that $0 < x < y$. Define a sequence$ A_0 , A_1 , A_2, A_3, ...$ by-setting $A_0 = x$ , $A_1 = y$ , and then $A_n= |A_{n-1}| - A_{n-2}$ for each $n \ge 2$ (recall that $|A_{n-1}|$ means the absolute value of $A_{n-1}$ ). (a) Find all possible values for $A_6$ in terms of $x$ and $y$ . (b) Find values of $x$ and $y$ so that $A_{1987} = 1987$ and $A_{1988} = -1988$ (simultaneously). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$\triangle ABC$, $AB>AC$. the incircle $I$ of $\triangle ABC$ meet $BC$ at point $D$, $AD$ meet $I$ again at $E$. $EP$ is a tangent of $I$, and $EP$ meet the extension line of $BC$ at $P$. $CF\parallel PE$, $CF\cap AD=F$. the line $BF$ meet $I$ at $M,N$, point $M$ is on the line segment $BF$, the line segment $PM$ meet $I$ again at $Q$. Show that $\angle ENP=\angle ENQ$

Let $X$ be an interior point of a rectangle $ABCD$. Let the bisectors of $\angle DAX$ and $\angle CBX$ intersect in $P$. A point $Q$ satisfies $\angle QAP=\angle QBP=90^\circ$. Show that $PX=QX$.

Given an acute triangle $ABC$. with $H$ as its orthocenter, lines $\ell_1$ and $\ell_2$ go through $H$ and are perpendicular to each other. Line $\ell_1$ cuts $BC$ and the extension of $AB$ on $D$ and $Z$ respectively. Whereas line $\ell_2$ cuts $BC$ and the extension of $AC$ on $E$ and $X$ respectively. If the line through $D$ and parallel to $AC$ and the line through $E$ parallel to $AB$ intersects at $Y$, prove that $X,Y,Z$ are collinear.

G. H. Hardy once went to visit Srinivasa Ramanujan in the hospital, and he started the conversation with: “I came here in taxi-cab number $1729$. That number seems dull to me, which I hope isn’t a bad omen.” “Nonsense,” said Ramanujan. “The number isn’t dull at all. It’s quite interesting. It’s the smallest number that can be expressed as the sum of two cubes in two different ways.” Ramujan had immediately seen that $1729=12^3+1^3=10^3+9^3$. What is the smallest positive integer representable as the sum of the cubes of [i]three[/i] positive integers in two different ways?

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