The incircle of a non-isosceles triangle $ABC$ touches $AB$ at point $C'$. The circle with diameter $BC'$ meets the incircle and the bisector of angle $B$ again at points $A_1$ and $A_2$ respectively. The circle with diameter $AC'$ meets the incircle and the bisector of angle $A$ again at points $B_1$ and $B_2$ respectively. Prove that lines $AB, A_1B_1, A_2B_2$ concur. (E. H. Garsia)

You are given an unlimited supply of red, blue, and yellow cards to form a hand. Each card has a point value and your score is the sum of the point values of those cards. The point values are as follows: the value of each red card is 1, the value of each blue card is equal to twice the number of red cards, and the value of each yellow card is equal to three times the number of blue cards. What is the maximum score you can get with fifteen cards?

Let $ABC$ be a triangle . Let point $X$ be in the triangle and $AX$ intersects $BC$ in $Y$ . Draw the perpendiculars $YP,YQ,YR,YS$ to lines $CA,CX,BX,BA$ respectively. Find the necessary and sufficient condition for $X$ such that $PQRS$ be cyclic .

$x_1,...,x_n$ are non-negative reals and $n \geq 3$. Prove that at least one of the following inequalities is true: \[ \sum_{i=1} ^n \frac{x_i}{x_{i+1}+x_{i+2}} \geq \frac{n}{2}, \] \[ \sum_{i=1} ^n \frac{x_i}{x_{i-1}+x_{i-2}} \geq \frac{n}{2} . \]

If $ABCD$ is a rectangle and $P$ is a point inside it such that $AP=33, BP=16, DP=63$. Find $CP$.

A point $P$ lies on the internal angle bisector of $\angle BAC$ of a triangle $\triangle ABC$. Point $D$ is the midpoint of $BC$ and $PD$ meets the external angle bisector of $\angle BAC$ at point $E$. If $F$ is the point such that $PAEF$ is a rectangle then prove that $PF$ bisects $\angle BFC$ internally or externally.

Imagine a regular a $2015$-gon with edge length $2$. At each vertex, draw a unit circle centered at that vertex and color the circle’s circumference orange. Now, another unit circle $S$ is placed inside the polygon such that it is externally tangent to two adjacent circles centered at the vertices. This circle $S$ is allowed to roll freely in the interior of the polygon as long as it remains externally tangent to the vertex circles. As it rolls, $S$ turns the color of any point it touches into black. After it rolls completely around the interior of the polygon, the total length of the black lengths can be expressed in the form $\tfrac{p\pi}{q}$ for positive integers $p, q$ satisfying $\gcd(p, q) = 1$. What is $p + q$?

Let $a,b,c$ be real number greater than $1$. Let \[S=\log_a {bc}+\log_b {ca}+\log_c {ab}\] Find the minimum possible value of $S$.

In the $x$-$y$ plane, for the origin $ O$, given an isosceles triangle $ OAB$ with $ AO \equal{} AB$ such that $ A$ is on the first quadrant and $ B$ is on the $ x$ axis. Denote the area by $ s$. Find the area of the common part of the traingle and the region expressed by the inequality $ xy\leq 1$ to give the area as the function of $ s$.

Find the maximum value of $M =\frac{x}{2x + y} +\frac{y}{2y + z}+\frac{z}{2z + x}$ , $x,y, z > 0$

Three identical square sheets of paper each with side length $6{ }$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\circ$ about its center and the top sheet is rotated clockwise $60^\circ$ about its center, resulting in the $24$-sided polygon shown in the figure below. The area of this polygon can be expressed in the form $a-b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. What is $a+b+c?$ [asy] size(160); defaultpen(linewidth(1.1)); path square = (1,1)--(1,-1)--(-1,-1)--(-1,1)--cycle; filldraw(square,white); filldraw(rotate(30)*square,white); filldraw(rotate(60)*square,white); dot((0,0),linewidth(7)); [/asy] $\textbf{(A)}\: 75\qquad\textbf{(B)} \: 93\qquad\textbf{(C)} \: 96\qquad\textbf{(D)} \: 129\qquad\textbf{(E)} \: 147$

Let $ A_1,B_1,C_1 $ be the middlepoints of the sides of a triangle $ ABC $ and let $ A_2,B_2,C_2 $ be on the middle of the paths $ CAB,ABC,BCA, $ respectively. Prove that $ A_1A_2,B_1B_2,C_1C_2 $ are concurrent.

Let $k > 1, n > 2018$ be positive integers, and let $n$ be odd. The nonzero rational numbers $x_1,x_2,\ldots,x_n$ are not all equal and satisfy $$x_1+\frac{k}{x_2}=x_2+\frac{k}{x_3}=x_3+\frac{k}{x_4}=\ldots=x_{n-1}+\frac{k}{x_n}=x_n+\frac{k}{x_1}$$ Find: a) the product $x_1 x_2 \ldots x_n$ as a function of $k$ and $n$ b) the least value of $k$, such that there exist $n,x_1,x_2,\ldots,x_n$ satisfying the given conditions.

The names of $8$ people are written on slips of paper and placed in a hat. Each of the $8$ people then randomly draw a piece of paper (without replacement). Then, the people are formed into groups satisfying the following requirements: (i) Each person is in the same group as the person who drew his piece of paper. (ii)There are as many groups as possible while still satisfying condition (i). On average, how many groups will there be? (There might be “groups” of only one person.)

Determine the (real) domain of a function $$y=\sqrt{1-\frac{x}{4}|x|+\sqrt{1-\frac{x}{2}|x|\,}\,}-\sqrt{1-\frac{x}{4}|x|-\sqrt{1-\frac{x}{2}|x|\,}\,}$$ and draw its graph.

Consider a fixed circle $\Gamma$ with three fixed points $A, B,$ and $C$ on it. Also, let us fix a real number $\lambda \in(0,1)$. For a variable point $P \not\in\{A, B, C\}$ on $\Gamma$, let $M$ be the point on the segment $CP$ such that $CM =\lambda\cdot CP$ . Let $Q$ be the second point of intersection of the circumcircles of the triangles $AMP$ and $BMC$. Prove that as $P$ varies, the point $Q$ lies on a fixed circle. [i]Proposed by Jack Edward Smith, UK[/i]

The first four terms in an arithmetic sequence are $ x \plus{} y$, $ x \minus{} y$, $ xy$, and $ x/y$, in that order. What is the fifth term? $ \textbf{(A)}\ \minus{}\frac{15}{8} \qquad \textbf{(B)}\ \minus{}\frac{6}{5} \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ \frac{27}{20} \qquad \textbf{(E)}\ \frac{123}{40}$

Torus $\mathcal T$ is the surface produced by revolving a circle with radius $3$ around an axis in the plane a distance $6$ from the center of the circle. When a sphere of radius $11$ rests inside $\mathcal T$, it is internally tangent to $\mathcal T$ along a circle with radius $r_{i}$, and when it rests outside $\mathcal T$, it is externally tangent along a circle with radius $r_{o}$. The difference $r_{i}-r_{o}=\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Do there exists many infinitely points like $(x_1,y_1),(x_2,y_2),...$ such that for any sequences like {$b_1,b_2,...$} of real numbers there exists a polynomial $P(x,y)\in R[x,y]$ such that we have for all $i$ : $P(x_{i},y_{i})=b_{i}$

At a dance, Abhinav starts from point $(a, 0)$ and moves along the negative $x$ direction with speed $v_a$, while Pei-Hsin starts from $(0,6)$ and glides in the negative $y$-direction with speed $v_b$. What is the distance of closest approach between the two?

There are 100 blue lines drawn on the plane, among which there are no parallel lines and no three of which pass through one point. The intersection points of the blue lines are marked in red. Could it happen that the distance between any two red dots lying on the same blue line is equal to an integer? [i]From the folklore[/i]

A triangle is divided into nine smaller triangles, where counters with the number zero are placed at each of the ten vertices. A [i]movement[/i] consists of choosing one of the nine triangles and applying the same operation to the three counters of that triangle: adding a unit or subtracting a unit. The figure illustrates a possible [i]movement[/i]. We shall call the integer number n [i]dominant [/i] if it is possible, after a few moves, to obtain a configuration in which the counter numbers are all consecutive and the largest of these numbers is $n$. Determine all [i]dominant [/i] numbers. [img]https://cdn.artofproblemsolving.com/attachments/7/3/731160e6e9a2b3292a31c4555d4adbc7028596.png[/img]

In the coordinate plane, consider points $A = (0, 0)$, $B = (11, 0)$, and $C = (18, 0)$. Line $\ell_A$ has slope 1 and passes through $A$. Line $\ell_B$ is vertical and passes through $B$. Line $\ell_C$ has slope $-1$ and passes through $C$. The three lines $\ell_A$, $\ell_B$, and $\ell_C$ begin rotating clockwise about points $A$, $B$, and $C$, respectively. They rotate at the same angular rate. At any given time, the three lines form a triangle. Determine the largest possible area of such a triangle.

Let $f(x,y)$ be a continuous function on the square $$S=\{(x,y):0\leq x\leq 1, 0\leq y\leq 1\}.$$ For each point $(a,b)$ in the interior of $S$, let $S_{(a,b)}$ be the largest square that is contained in $S$, is centered at $(a,b)$, and has sides parallel to those of $S$. If the double integral $\int \int f(x,y) dx dy$ is zero when taken over each square $S_{(a,b)}$, must $f(x,y)$ be identically zero on $S$?

Let $ABC$ be an acute triangle. Let $D, E,$ and $F$ be the feet of altitudes from $A, B,$ and $C$ to sides $BC, CA,$ and $AB$, respectively, and let $Q$ be the foot of altitude from A to line $EF$ . Given that $AQ = 20, BC = 15,$ and $AD = 24$, compute the perimeter of triangle $DEF.$