4-Point $ P$ is taken on the segment $ BC$ of the scalene triangle $ ABC$ such that $ AP \neq AB,AP \neq AC$.$ l_1,l_2$ are the incenters of triangles $ ABP,ACP$ respectively. circles $ W_1,W_2$ are drawn centered at $ l_1,l_2$ and with radius equal to $ l_1P,l_2P$,respectively. $ W_1,W_2$ intersects at $ P$ and $ Q$. $ W_1$ intersects $ AB$ and $ BC$ at $ Y_1( \mbox{the intersection closer to B})$ and $ X_1$,respectively. $ W_2$ intersects $ AC$ and $ BC$ at $ Y_2(\mbox{the intersection closer to C})$ and $ X_2$,respectively.PROVE THE CONCURRENCY OF $ PQ,X_1Y_1,X_2Y_2$.

Find all functions $f: R \to R$ that satisfy the condition $(f (x) - f (y)) (u - v) = (f (u) - f (v)) (x -y)$ for arbitrary real $x, y, u, v$ such that $x + y = u + v$.

Given a set $S$ of positive real numbers, let $$\Sigma (S) = \Bigg\{ \sum_{x \in A} x : \emptyset \neq A \subset S \Bigg\}.$$ be the set of all the sums of elements of non-empty subsets of $S$. Find the least constant $L> 0$ with the following property: for every integer greater than $1$ and every set $S$ of $n$ positive real numbers, it is possible partition $\Sigma(S)$ into $n$ subsets $\Sigma_1,\ldots, \Sigma_n$ so that the ratio between the largest and smallest element of each $\Sigma_i$ is at most $L$.

Complex numbers $a,b,w,x,y,z,p$ satisfy \begin{align*} \frac{(x-w)\lvert a-w \rvert}{(a-w)\lvert x-w \rvert}&=\text{(cyclic variants)};\\ \frac{(z-w)\lvert b-w \rvert}{(b-w)\lvert z-w \rvert}&=\text{(cyclic variants)};\\ p &= \frac{\sum_{\text{cyc}} \frac w{\lvert p-w \rvert}}{\sum_{\text{cyc}}\frac1{\lvert p-w \rvert}}; \end{align*} where cyclic sums, equations, etc. are wrt $w,x,y,z$. Prove that there exists a real $k$ such that \[\sum_{\text{cyc}} \frac{(x-w)(a-w)}{\lvert x-w\rvert (p-w)} =k\sum_{\text{cyc}} \frac{(z-w)(b-w)}{\lvert z-w\rvert(p-w)}.\] [i]Neal Yan[/i]

Let be the sequence of sets $ \left(\left\{ A\in\mathcal{M}_2\left(\mathbb{R} \right) | A^{n+1} =2007^nA\right\}\right)_{n\ge 1} . $ [b]a)[/b] Prove that each term of the above sequence hasn't a finite cardinal. [b]b)[/b] Determine the intersection of the fourth element of the above sequence with the $ 2007\text{th} $ element. [i]Gheorghe Iurea[/i] [hide=Note]Similar with [url]https://artofproblemsolving.com/community/c7h1928039p13233629[/url].[/hide]

Consider the sequence of six real numbers 60, 10, 100, 150, 30, and $x$. The average (arithmetic mean) of this sequence is equal to the median of the sequence. What is the sum of all the possible values of $x$? (The median of a sequence of six real numbers is the average of the two middle numbers after all the numbers have been arranged in increasing order.)

A triangle with integral sides has perimeter $8$. The area of the triangle is $\textbf{(A) }2\sqrt{2}\qquad \textbf{(B) }\dfrac{16}{9}\sqrt{3}\qquad \textbf{(C) }2\sqrt{3}\qquad \textbf{(D) }4\qquad \textbf{(E) }4\sqrt{2}$

Let $k$ be a circle with diameter $AB$. Let $C$ be a point on the straight line $AB$, so that $B$ between $A$ and $C$ lies. Let $T$ be a point on $k$ such that $CT$ is a tangent to $k$. Let $l$ be the parallel to $CT$ through $A$ and $D$ the intersection of $l$ and the perpendicular to $AB$ through $T$. Show that the line $DB$ bisects segment $CT$.

RyNo, a little rhinoceros, has $17$ scratch marks on its body. Some are horizontal and the rest are vertical. Some are on the left side and the rest are on the right side. If RyNo rubs one side of its body against a tree, two scratch marks, either both horizontal or both vertical, will disappear from that side. However, at the same time, two new scratch marks, one horizontal and one vertical, will appear on the other side. If there are less than two horizontal and less than two vertical scratch marks on the side being rubbed, then nothing happens. If RyNo continues to rub its body against trees, is it possible that at some point in time, the numbers of horizontal and vertical scratch marks have interchanged on each side of its body?

For any integer $n\geq 2$, we compute the integer $h(n)$ by applying the following procedure to its decimal representation. Let $r$ be the rightmost digit of $n$. [list][*]If $r=0$, then the decimal representation of $h(n)$ results from the decimal representation of $n$ by removing this rightmost digit $0$. [*]If $1\leq r \leq 9$ we split the decimal representation of $n$ into a maximal right part $R$ that solely consists of digits not less than $r$ and into a left part $L$ that either is empty or ends with a digit strictly smaller than $r$. Then the decimal representation of $h(n)$ consists of the decimal representation of $L$, followed by two copies of the decimal representation of $R-1$. For instance, for the number $17,151,345,543$, we will have $L=17,151$, $R=345,543$ and $h(n)=17,151,345,542,345,542$.[/list] Prove that, starting with an arbitrary integer $n\geq 2$, iterated application of $h$ produces the integer $1$ after finitely many steps. [i]Proposed by Gerhard Woeginger, Austria[/i]

Let $ABC$ be a triangle. The incircle of $ABC$ touches $BC,AC,AB$ at $D,E,F$, respectively. Each pair of the incircles of triangles $AEF, BDF,CED$ has two pair of common external tangents, one of them being one of the sides of $ABC$. Show that the other three tangents divide triangle $DEF$ into three triangles and three parallelograms.

A line is called $good$ if it bisects perimeter and area of a figure at the same time.Prove that: [i]a)[/i] all of the good lines in a triangle concur. [i]b)[/i] all of the good lines in a regular polygon concur too.

Given $4$ vectors $a,b,c,d$ in the plane, such that $a+b+c+d=0$. Prove the following inequality: $$|a|+|b|+|c|+|d| \ge |a+d|+|b+d|+|c+d|$$

The diagram below contains eight line segments, all the same length. Each of the angles formed by the intersections of two segments is either a right angle or a $45$ degree angle. If the outside square has area $1000$, find the largest integer less than or equal to the area of the inside square. [asy] size(130); real r = sqrt(2)/2; defaultpen(linewidth(0.8)); draw(unitsquare^^(r,0)--(0,r)^^(1-r,0)--(1,r)^^(r,1)--(0,1-r)^^(1-r,1)--(1,1-r)); [/asy]

There are 15 people at a party; each person has 10 friends. To greet each other each person hugs all their friends. How many hugs are exchanged at this party?

Consider the equation $x^4 + ax^3 + bx^2 + ax + 1 = 0$ with real coefficients $a, b$. Determine the number of distinct real roots and their multiplicities for various values of $a$ and $b$. Display your result graphically in the $(a, b)$ plane.

In an aquarium there are nine small fish. The aquarium is cube shaped with a side length of two meters and is completely filled with water. Show that it is always possible to find two small fish with a distance of less than $\sqrt3$ meters.

Let $ y$ be the function of $ x$ satisfying the differential equation $ y'' \minus{} y \equal{} 2\sin x$. (1) Let $ y \equal{} e^xu \minus{} \sin x$, find the differential equation with which the function $ u$ with respect to $ x$ satisfies. (2) If $ y(0) \equal{} 3,\ y'(0) \equal{} 0$, then determine $ y$.

Let $a_1,\ldots,a_n,b_1,\ldots,b_n$ be positive numbers. Show that \[\sqrt{\left(a_1+\cdots+a_n\right)\left(b_1+\cdots+b_n\right)}\ge\sqrt{a_1b_1}+\cdots+\sqrt{a_nb_n}\] and prove that equality holds if and only if \[\frac{a_1}{b_1}=\cdots=\frac{a_n}{b_n}.\]

A circle of radius $ 2$ is inscribed in a semicircle, as shown. The area inside the semicircle but outside the circle is shaded. What fraction of the semicircle's area is shaded? [asy]unitsize(6mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; filldraw(Arc((0,0),4,0,180)--cycle,gray,black); filldraw(Circle((0,2),2),white,black); dot((0,2)); draw((0,2)--((0,2)+2*dir(60))); label("$2$",midpoint((0,2)--((0,2)+2*dir(60))),SE);[/asy]$ \textbf{(A)}\ \frac{1}{2}\qquad \textbf{(B)}\ \frac{\pi}{6}\qquad \textbf{(C)}\ \frac{2}{\pi}\qquad \textbf{(D)}\ \frac{2}{3}\qquad \textbf{(E)}\ \frac{3}{\pi}$

Points $ K$, $ L$, $ M$, and $ N$ lie in the plane of the square $ ABCD$ so that $ AKB$, $ BLC$, $ CMD$, and $ DNA$ are equilateral triangles. If $ ABCD$ has an area of $ 16$, find the area of $ KLMN$. [asy]unitsize(2cm); defaultpen(fontsize(8)+linewidth(0.8)); pair A=(-0.5,0.5), B=(0.5,0.5), C=(0.5,-0.5), D=(-0.5,-0.5); pair K=(0,1.366), L=(1.366,0), M=(0,-1.366), N=(-1.366,0); draw(A--N--K--A--B--K--L--B--C--L--M--C--D--M--N--D--A); label("$A$",A,SE); label("$B$",B,SW); label("$C$",C,NW); label("$D$",D,NE); label("$K$",K,NNW); label("$L$",L,E); label("$M$",M,S); label("$N$",N,W);[/asy] $ \textbf{(A)}\ 32 \qquad \textbf{(B)}\ 16 \plus{} 16\sqrt {3} \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 32 \plus{} 16\sqrt {3} \qquad \textbf{(E)}\ 64$

Six regular octagons each with sides of length 2 are placed in a two-by-three array and inscribed in a square as shown. The area of the square can be written in the form $m+n\sqrt{2}$ where $m$ and $n$ are positive integers. Find $m+n$. [asy] size(175); pair A=dir(-45/2); real h=A.x; path P=A; for(int k=1;k<8;++k) P=P--rotate(45*k)*A; P=P--cycle; for(real x=-2*h;x<3*h;x+=2*h) for(real y:new real[]{-h,h} ) draw (shift((x,y))*P); pair C=dir(45/2)+(2*h,h), X=(C.x+C.y,0), Y=(0,C.x+C.y); draw(X--Y--(-X)--(-Y)--cycle);[/asy]

On the bisector of angle $A$ of triangle $ABC$, points $D$ and $F$ are taken inside the triangle so that $\angle DBC = \angle FBA$. Prove that: a) $\angle DCB = \angle FCA$ b) a circle passing through $B$ and $F$ and tangent to the segment $BC$ is tangle to the circumscribed circle of the triangle $ABC$.

A stack of $2000$ cards is labelled with the integers from $1$ to $2000,$ with different integers on different cards. The cards in the stack are not in numerical order. The top card is removed from the stack and placed on the table, and the next card is moved to the bottom of the stack. The new top card is removed from the stack and placed on the table, to the right of the card already there, and the next card in the stack is moved to the bottom of the stack. The process - placing the top card to the right of the cards already on the table and moving the next card in the stack to the bottom of the stack - is repeated until all cards are on the table. It is found that, reading from left to right, the labels on the cards are now in ascending order: $1,2,3,\ldots,1999,2000.$ In the original stack of cards, how many cards were above the card labelled 1999?

Find a positive integer $n$ with 1000 digits, all distinct from zero, with the following property: it's possible to group the digits of $n$ into 500 pairs in such a way that if the two digits of each pair are multiplied and then add the 500 products, it results a number $m$ that is a divisor of $n$.