Given three circumferences of the same radius in a plane. a) All three are crossing in one point $K$. Consider three arcs $AK,CK,EK$ : the $A,C,E$ are the points of the circumferences intersection and the arcs are taken in the clockwise direction. Every arc is inside one circle, outside the second and on the border of the third one. Prove that the sum of the arcs is $180$ degrees. b) Consider the case, when the three circles give a curvilinear triangle $BDF$ as their intersection (instead of one point $K$). The arcs are taken in the clockwise direction. Every arc is inside one circle, outside the second and on the border of the third one. Prove that the sum of the $AB, CD$ and $EF$ arcs is $180$ degrees.

For a triangle $ ABC,$ let $ k$ be its circumcircle with radius $ r.$ The bisectors of the inner angles $ A, B,$ and $ C$ of the triangle intersect respectively the circle $ k$ again at points $ A', B',$ and $ C'.$ Prove the inequality \[ 16Q^3 \geq 27 r^4 P,\] where $ Q$ and $ P$ are the areas of the triangles $ A'B'C'$ and $ABC$ respectively.

The points $P$ and $Q$ on the side $AC$ of the non-isosceles $\Delta ABC$ are such that $\angle ABP=\angle QBC<\frac{1}{2}\angle ABC$. The angle bisectors of $\angle A$ and $\angle C$ intersect the segment $BP$ in points $K$ and $L$ and the segment $BQ$ in points $M$ and $N$, respectively. Prove that $AC$,$KN$, and $LM$ are concurrent.

Let $P_1 , P_2 , ...$ be arbitrary points and A be a connected compact set in the plane with a diameter greater than 4. Show that for some point P in A , $\overline {PP_1} \cdot \overline {PP_2} \cdots \overline {PP_n}>1$. Furthermore, prove that this is no longer necessarily true for compact sets of diameter 4.

Let $S$ be the set of circles in the coordinate plane that are tangent to each of the three circles with equations $x^{2}+y^{2}=4$, $x^{2}+y^{2}=64$, and $(x-5)^{2}+y^{2}=3$. What is the sum of the areas of all circles in $S$? $\textbf{(A)}~48\pi\qquad\textbf{(B)}~68\pi\qquad\textbf{(C)}~96\pi\qquad\textbf{(D)}~102\pi\qquad\textbf{(E)}~136\pi\qquad$

Two circles with centres $A$ and $B$ lie inside an angle. They touch each other and both sides of the angle. Prove that the circle with the diameter $AB$ touches both sides of the angle. (V. Prasolov)

Let $D$ be an interior point on the side $BC$ of an acute-angled triangle $ABC$. Let the circumcircle of triangle $ADB$ intersect $AC$ again at $E(\ne A)$ and the circumcircle of triangle $ADC$ intersect $AB$ again at $F(\ne A)$. Let $AD$, $BE$, and $CF$ intersect the circumcircle of triangle $ABC$ again at $D_1(\ne A)$, $E_1(\ne B)$ and $F_1(\ne C)$, respectively. Let $I$ and $I_1$ be the incentres of triangles $DEF$ and $D_1E_1F_1$, respectively. Prove that $E,F, I, I_1$ are concyclic.

A convex $n{}$-gon with $n > 4$ is such that if a diagonal cuts a triangle from it then this triangle is isosceles. Prove that there are at least 2 equal sides among any 4 sides of the $n{}$-gon. [i]Maxim Didin[/i]

Given a point $E$ on the diameter $AC$ of the certain circle. Draw a chord $BD$ to maximise the area of the quadrangle $ABCD$.

Let $ ABCDE$ be a convex pentagon. If $ P$, $ Q$, $ R$ and $ S$ are the respective centroids of the triangles $ ABE$, $ BCE$, $ CDE$ and $ DAE$, show that $ PQRS$ is a parallelogram and its area is $ 2/9$ of that of $ ABCD$.

Prove that the sum of the lengths of the polyhedron edges exceeds its tripled diameter (distance between two farest vertices).

Angle $A$ in triangle $ABC$ is equal to $120^o$. Prove that the distance from the center of the circumscribed circle to the orthocenter is equal to $AB + AC$. (V. Protasov)

Let $ABC$ be a triangle such that \[ \left( \cot \dfrac{A}{2} \right)^2 + \left( 2\cot \dfrac{B}{2} \right)^2 + \left( 3\cot \dfrac{C}{2} \right)^2 = \left( \dfrac{6s}{7r} \right)^2, \] where $s$ and $r$ denote its semiperimeter and its inradius, respectively. Prove that triangle $ABC$ is similar to a triangle $T$ whose side lengths are all positive integers with no common divisors and determine these integers.

A big $ L$ is formed as shown. What is its area? [asy]unitsize(4mm); defaultpen(linewidth(.8pt)+fontsize(12pt)); draw((0,0)--(5,0)--(5,2)--(2,2)--(2,8)--(0,8)--cycle); label("5",(2.5,0),S); label("2",(5,1),E); label("2",(1,8),N); label("8",(0,4),W);[/asy]$ \textbf{(A)}\ 22 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 26 \qquad \textbf{(D)}\ 28 \qquad \textbf{(E)}\ 30$

Hi guys, I was just reading over old posts that I made last year ( :P ) and saw how much the level of Getting Started became harder. To encourage more people from posting, I decided to start a Problem of the Day. This is how I'll conduct this: 1. In each post (not including this one since it has rules, etc) everyday, I'll post the problem. I may post another thread after it to give hints though. 2. Level of problem.. This is VERY important. All problems in this thread will be all AHSME or problems similar to this level. No AIME. Some AHSME problems, however, that involve tough insight or skills will not be posted. The chosen problems will be usually ones that everyone can solve after working. Calculators are allowed when you solve problems but it is NOT necessary. 3. Response.. All you have to do is simply solve the problem and post the solution. There is no credit given or taken away if you get the problem wrong. This isn't like other threads where the number of problems you get right or not matters. As for posting, post your solutions here in this thread. Do NOT PM me. Also, here are some more restrictions when posting solutions: A. No single answer post. It doesn't matter if you put hide and say "Answer is ###..." If you don't put explanation, it simply means you cheated off from some other people. I've seen several posts that went like "I know the answer" and simply post the letter. What is the purpose of even posting then? Huh? B. Do NOT go back to the previous problem(s). This causes too much confusion. C. You're FREE to give hints and post different idea, way or answer in some cases in problems. If you see someone did wrong or you don't understand what they did, post here. That's what this thread is for. 4. Main purpose.. This is for anyone who visits this forum to enjoy math. I rememeber when I first came into this forum, I was poor at math compared to other people. But I kindly got help from many people such as JBL, joml88, tokenadult, and many other people that would take too much time to type. Perhaps without them, I wouldn't be even a moderator in this forum now. This site clearly made me to enjoy math more and more and I'd like to do the same thing. That's about the rule.. Have fun problem solving! Next post will contain the Day 1 Problem. You can post the solutions until I post one. :D

Let $ABCD$ be a trapezoid ($AD//BC$) with $\angle A=\angle B= 90^o$ and $AD<BC$. Let $E$ be the intersection point of the non parallel sides $AB$ and $CD$, $Z$ be the symmetric point of $A$ wrt line $BC$ and $M$ be the midpoint of $EZ$. If it is given than line $CM$ is perpendicular on line $DZ$, then prove that line $ZC$ is perpendicular on line $EC$.

Let $ m,n$ be two natural nonzero numbers and sets $ A \equal{} \{ 1,2,...,n\}, B \equal{} \{1,2,...,m\}$. We say that subset $ S$ of Cartesian product $ A \times B$ has property $ (j)$ if $ (a \minus{} x)(b \minus{} y)\le 0$ for each pairs $ (a,b),(x,y) \in S$. Prove that every set $ S$ with propery $ (j)$ has at most $ m \plus{} n \minus{} 1$ elements. [color=#FF0000]The statement was edited, in order to reflect the actual problem asked. The sign of the inequality was inadvertently reversed into $ (a \minus{} x)(b \minus{} y)\ge 0$, and that accounts for the following two posts.[/color]

Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. A line passing through point $B$ intersects $\omega_1$ for the second time at point $C$ and $\omega_2$ at point $D$. The line $AC$ intersects circle $\omega_2$ for the second time at point $F$, and the line $AD$ intersects the circle $\omega_1$ for the second time at point $E$ . Let point $O$ be the center of the circle circumscribed around $\vartriangle AEF$. Prove that $OB \perp CD$.

A positive integer $n$ is [i]nice[/i] if there is a positive integer $m$ with exactly four positive divisors (including $1$ and $m$) such that the sum of the four divisors is equal to $n$. How many numers in the set $\{2010, 2011, 2012,\ldots,2019\}$ are nice? ${ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D}}\ 4\qquad\textbf{(E)}\ 5 $

Let $ABCD$ be a convex quadrilateral. $AC$ and $BD$ meet at $P$, with $\angle APD=60^{\circ}$. Let $E,F,G$, and $H$ be the midpoints of $AB,BC,CD$ and $DA$ respectively. Find the greatest positive real number $k$ for which \[EG+3HF\ge kd+(1-k)s \] where $s$ is the semi-perimeter of the quadrilateral $ABCD$ and $d$ is the sum of the lengths of its diagonals. When does the equality hold?

What is the maximum area of a triangle that can be inscribed in an ellipse with semi-axes $a$ and $b$? $\text{(A) }ab\frac{3\sqrt{3}}{4}\qquad\text{(B) }ab\qquad\text{(C) }ab\sqrt{2}\qquad\text{(D) }\left(a+b\right)\frac{3\sqrt{3}}{4}\qquad\text{(E) }\left(a+b\right)\sqrt{2}$

Let $ ABC$ be a triangle, all of whose angles are greater than $ 45^{\circ}$ and smaller than $ 90^{\circ}$. (a) Prove that one can fit three squares inside $ ABC$ in such a way that: (i) the three squares are equal (ii) the three squares have common vertex $ K$ inside the triangle (iii) any two squares have no common point but $ K$ (iv) each square has two opposite vertices onthe boundary of $ ABC$, while all the other points of the square are inside $ ABC$. (b) Let $ P$ be the center of the square which has $ AB$ as a side and is outside $ ABC$. Let $ r_{C}$ be the line symmetric to $ CK$ with respect to the bisector of $ \angle BCA$. Prove that $ P$ lies on $ r_{C}$.

Given is a parallelogram $XYZT$, and the variable points $A, B, C, D$ lie on the sides $XY, XT, TZ, ZY$ respectively, so that $ABCD$ is cyclic with circumcenter $O$, $AC \parallel XT$, and $BD \parallel XY$. Let $P$ be the intersection point of the lines $AD$ and $BC$, and let $Q$ be the intersection of the lines $AB$ and $CD$. Prove that the circle $(POQ)$ passes through a fixed point.

Let $k$ be the circle and $A$ and $B$ points on circle which are not diametrically opposite. On minor arc $AB$ lies point arbitrary point $C$. Let $D$, $E$ and $F$ be foots of perpendiculars from $C$ on chord $AB$ and tangents of circle $k$ in points $A$ and $B$. Prove that $CD= \sqrt {CE \cdot CF}$

Let $ABC$ be an acute scalene triangle and $K$ be a point inside it that belongs to the bisector of the angle $\angle ABC$. Let$ P$ be the point where the line $AK$ intersects the line perpendicular to $AB$ that passes through $B$, and let $Q$ be the point where the line $CK$ intersects the line perpendicular to $CB$ that passes through $B$. Let $L$ be the foot of the perpendicular drawn from $K$ on the line $AC$. Prove that if $P Q$ is perpendicular to $BL$, then $K$ is the incenter of $ABC$.