Let $A B C$ be a triangle in the $x y$ plane, where $B$ is at the origin $(0,0)$. Let $B C$ be produced to $D$ such that $B C: C D=1: 1, C A$ be produced to $E$ such that $C A: A E=1: 2$ and $A B$ be produced to $F$ such that $A B: B F=1: 3$. Let $G(32,24)$ be the centroid of the triangle $A B C$ and $K$ be the centroid of the triangle $D E F$. Find the length $G K$.

For all integers $x,y$, a non-negative integer $f(x,y)$ is written on the point $(x,y)$ on the coordinate plane. Initially, $f(0,0) = 4$ and the value written on all remaining points is $0$. For integers $n, m$ that satisfies $f(n,m) \ge 2$, define '[color=#9a00ff]Seehang[/color]' as the act of reducing $f(n,m)$ by $1$, selecting 3 of $f(n,m+1), f(n,m-1), f(n+1,m), f(n-1,m)$ and increasing them by 1. Prove that after a finite number of '[color=#0f0][color=#9a00ff]Seehang[/color][/color]'s, it cannot be $f(n,m)\le 1$ for all integers $n,m$.

Let $c$ be a real number, and let $z_1, z_2$ be the two complex numbers satisfying the quadratic $z^2 - cz + 10 = 0$. Points $z_1, z_2, \frac{1}{z_1}$, and $\frac{1}{z_2}$ are the vertices of a (convex) quadrilateral $Q$ in the complex plane. When the area of $Q$ obtains its maximum value, $c$ is the closest to which of the following? $\textbf{(A)}~4.5\qquad\textbf{(B)}~5\qquad\textbf{(C)}~5.5\qquad\textbf{(D)}~6\qquad\textbf{(E)}~6.5$

A girls' camp is located $ 300$ rods from a straight road. On this road, a boys' camp is located $ 500$ rods from the girls' camp. It is desired to build a canteen on the road which shall be exactly the same distance from each camp. The distance of the canteen from each of the camps is: $ \textbf{(A)}\ 400\text{ rods} \qquad\textbf{(B)}\ 250\text{ rods} \qquad\textbf{(C)}\ 87.5\text{ rods} \qquad\textbf{(D)}\ 200\text{ rods}\\ \textbf{(E)}\ \text{none of these}$

Let \(n\) be a positive integer and consider an \(n\times n\) square grid. For \(1\le k\le n\), a [i]python[/i] of length \(k\) is a snake that occupies \(k\) consecutive cells in a single row, and no other cells. Similarly, an [i]anaconda[/i] of length \(k\) is a snake that occupies \(k\) consecutive cells in a single column, and no other cells. The grid contains at least one python or anaconda, and it satisfies the following properties: [list] [*]No cell is occupied by multiple snakes. [*]If a cell in the grid is immediately to the left or immediately to the right of a python, then that cell must be occupied by an anaconda. [*]If a cell in the grid is immediately to above or immediately below an anaconda, then that cell must be occupied by a python. [/list] Prove that the sum of the squares of the lengths of the snakes is at least \(n^2\). [i]Proposed by Linus Tang[/i]

The trapezoid $ABCD$ of bases $AB$ and $CD$, has $\angle A = 90^o, AB = 6, CD = 3$ and $AD = 4$. Let $E, G, H$ be the circumcenters of triangles $ABC, ACD, ABD$, respectively. Find the area of the triangle $EGH$.

Given a cube of unit side. Let $A$ and $B$ be two opposite vertex. Determine the radius of the sphere, with center inside the cube, tangent to the three faces of the cube with common point $A$ and tangent to the three sides with common point $B$.

[b]p1.[/b] What is the smallest positive integer divisible by $20$, $12$, and $13$? [b]p2.[/b] Two circles of radius $5$ are placed in the plane such that their centers are $7$ units apart. What is the largest possible distance between a point on one circle and a point on the other? [b]p3.[/b] In a magic square, all the numbers in the rows, columns, and diagonals sum to the same value. How many $2\times 2$ magic squares containing the integers $\{1, 2, 3, 4\}$ are there? [b]p4.[/b] Ethan's sock drawer contains two pairs of white socks and one pair of red socks. Ethan picks two socks at random. What is the probability that he picks two white socks? [b]p5.[/b] The sum of the time on a digital clock is the sum of the digits displayed on the screen. For example, the sum of the time at $10:23$ would be $6$. Assuming the clock is a $12$ hour clock, what is the greatest possible positive difference between the sum of the time at some time and the sum of the time one minute later? [b]p6.[/b] Given the expression $1 \div 2 \div 3 \div 4$, what is the largest possible resulting value if one were to place parentheses $()$ somewhere in the expression? [b]p7.[/b] At a convention, there are many astronomers, astrophysicists, and cosmologists. At $first$, all the astronomers and astrophysicists arrive. At this point, $\frac35$ of the people in the room are astronomers. Then, all the cosmologists come, so now, $30\%$ of the people in the room are astrophysicists. What fraction of the scientists are cosmologists? [b]p8.[/b] At $10:00$ AM, a minuteman starts walking down a $1200$-step stationary escalator at $40$ steps per minute. Halfway down, the escalator starts moving up at a constant speed, while the minuteman continues to walk in the same direction and at the same pace that he was going before. At $10:55$ AM, the minuteman arrives back at the top. At what speed is the escalator going up, in steps per minute? [b]p9.[/b] Given that $x_1 = 57$, $x_2 = 68$, and $x_3 = 32$, let $x_n = x_{n-1} -x_{n-2} +x_{n-3}$ for $n \ge 4$. Find $x_{2013}$. [b]p10.[/b] Two squares are put side by side such that one vertex of the larger one coincides with a vertex of the smaller one. The smallest rectangle that contains both squares is drawn. If the area of the rectangle is $60$ and the area of the smaller square is $24$, what is the length of the diagonal of the rectangle? [b]p11.[/b] On a dield trip, $2$ professors, $4$ girls, and $4$ boys are walking to the forest to gather data on butterflies. They must walk in a line with following restrictions: one adult must be the first person in the line and one adult must be the last person in the line, the boys must be in alphabetical order from front to back, and the girls must also be in alphabetical order from front to back. How many such possible lines are there, if each person has a distinct name? [b]p12.[/b] Flatland is the rectangle with vertices $A, B, C$, and $D$, which are located at $(0, 0)$, $(0, 5)$, $(5, 5)$, and $(5, 0)$, respectively. The citizens put an exact map of Flatland on the rectangular region with vertices $(1, 2)$, $(1, 3)$, $(2, 3)$, and $(2, 2)$ in such a way so that the location of $A$ on the map lies on the point $(1, 2)$ of Flatland, the location of $B$ on the map lies on the point $(1, 3)$ of Flatland, the location of C on the map lies on the point $(2, 3)$ of Flatland, and the location of D on the map lies on the point $(2, 2)$ of Flatland. Which point on the coordinate plane is thesame point on the map as where it actually is on Flatland? [b]p13.[/b] $S$ is a collection of integers such that any integer $x$ that is present in $S$ is present exactly $x$ times. Given that all the integers from $1$ through $22$ inclusive are present in $S$ and no others are, what is the average value of the elements in $S$? [b]p14.[/b] In rectangle $PQRS$ with $PQ < QR$, the angle bisector of $\angle SPQ$ intersects $\overline{SQ}$ at point $T$ and $\overline{QR }$ at $U$. If $PT : TU = 3 : 1$, what is the ratio of the area of triangle $PTS$ to the area of rectangle $PQRS$? [b]p15.[/b] For a function $f(x) = Ax^2 + Bx + C$, $f(A) = f(B)$ and $A + 6 = B$. Find all possible values of $B$. [b]p16.[/b] Let $\alpha$ be the sum of the integers relatively prime to $98$ and less than $98$ and $\beta$ be the sum of the integers not relatively prime to $98$ and less than $98$. What is the value of $\frac{\alpha}{\beta}$ ? [b]p17.[/b] What is the value of the series $\frac{1}{3} + \frac{3}{9} + \frac{6}{27} + \frac{10}{81} + \frac{15}{243} + ...$? [b]p18.[/b] A bug starts at $(0, 0)$ and moves along lattice points restricted to $(i, j)$, where $0 \le i, j \le 2$. Given that the bug moves $1$ unit each second, how many different paths can the bug take such that it ends at $(2, 2)$ after $8$ seconds? [b]p19.[/b] Let $f(n)$ be the sum of the digits of $n$. How many different values of $n < 2013$ are there such that $f(f(f(n))) \ne f(f(n))$ and $f(f(f(n))) < 10$? [b]p20.[/b] Let $A$ and $B$ be points such that $\overline{AB} = 14$ and let $\omega_1$ and $\omega_2$ be circles centered at $A$ and $B$ with radii $13$ and $15$, respectively. Let $C$ be a point on $\omega_1$ and $D$ be a point on $\omega_2$ such that $\overline{CD}$ is a common external tangent to $\omega_1$ and $\omega_2$. Let $P$ be the intersection point of the two circles that is closer to $\overline{CD}$. If $M$ is the midpoint of $\overline{CD}$, what is the length of segment $\overline{PM}$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A [i]move[/i] consists of the following steps. [list=1] [*]select a $2\times 2$ square in the grid; [*]flip the coins in the top-left and bottom-right unit squares; [*]flip the coin in either the top-right or bottom-left unit square. [/list] Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves. [i]Thanasin Nampaisarn, Thailand[/i]

Let $\triangle{ABC}$ be an isoceles triangle with $\angle A=90^{\circ}$. There exists a point $P$ inside $\triangle{ABC}$ such that $\angle PAB=\angle PBC=\angle PCA$ and $AP=10$. Find the area of $\triangle{ABC}$.

Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$-axis. What is the least positive integer $n$ such that performing the sequence of transformations transformations $T_1, T_2, T_3, \dots, T_n$ returns the point $(1,0)$ back to itself? $\textbf{(A) } 359 \qquad \textbf{(B) } 360\qquad \textbf{(C) } 719 \qquad \textbf{(D) } 720 \qquad \textbf{(E) } 721$

On the coordinate plane, Let $C$ be the graph of $y=(\ln x)^2\ (x>0)$ and for $\alpha >0$, denote $L(\alpha)$ be the tangent line of $C$ at the point $(\alpha ,\ (\ln \alpha)^2).$ (1) Draw the graph. (2) Let $n(\alpha)$ be the number of the intersection points of $C$ and $L(\alpha)$. Find $n(\alpha)$. (3) For $0<\alpha <1$, let $S(\alpha)$ be the area of the region bounded by $C,\ L(\alpha)$ and the $x$-axis. Find $S(\alpha)$. 2010 Tokyo Institute of Technology entrance exam, Second Exam.

Let $ABC$ be an equilateral triangle. From the vertex $A$ we draw a ray towards the interior of the triangle such that the ray reaches one of the sides of the triangle. When the ray reaches a side, it then bounces off following the law of reflection, that is, if it arrives with a directed angle $\alpha$, it leaves with a directed angle $180^{\circ}-\alpha$. After $n$ bounces, the ray returns to $A$ without ever landing on any of the other two vertices. Find all possible values of $n$.

In a plane a circle with radius $R$ and center $w$ and a line $\Lambda$ are given. The distance between $w$ and $\Lambda$ is $d, d > R$. The points $M$ and $N$ are chosen on $\Lambda$ in such a way that the circle with diameter $MN$ is externally tangent to the given circle. Show that there exists a point $A$ in the plane such that all the segments $MN$ are seen in a constant angle from $A.$