Consider the shape formed from taking equilateral triangle $ABC$ with side length $6$ and tracing out the arc $BC$ with center $A$. Set the shape down on line $l$ so that segment $AB$ is perpendicular to $l$, and $B$ touches $l$. Beginning from arc $BC$ touching $l$, we roll $ABC$ along $l$ until both points $A$ and $C$ are on the line. The area traced out by the roll can be written in the form $n\pi$, where $n$ is an integer. Find $n$.

Find the number of ways to completely cover a $2 \times 10$ rectangular grid of unit squares with $2 \times 1$ rectangles $R$ and $\sqrt{2}$ - $\sqrt{2}$ - $2$ triangles $T$ such that the following all hold: [list] [*] a placement of $R$ must have all of its sides parallel to the grid lines, [*] a placement of $T$ must have its longest side parallel to a grid line, [*] the tiles are non-overlapping, and [*] no tile extends outside the boundary of the grid. [/list] (The figure below shows an example of such a tiling; consider tilings that differ by reflections to be distinct.) [asy] unitsize(1cm); fill((0,0)--(10,0)--(10,2)--(0,2)--cycle, grey); draw((0,0)--(10,0)--(10,2)--(0,2)--cycle); draw((1,0)--(1,2)); draw((1,2)--(3,0)); draw((1,0)--(3,2)); draw((3,2)--(5,0)); draw((3,0)--(5,2)); draw((2,1)--(4,1)); draw((5,0)--(5,2)); draw((7,0)--(7,2)); draw((5,1)--(7,1)); draw((8,0)--(8,2)); draw((8,0)--(10,2)); draw((8,2)--(10,0)); [/asy]

[u]Round 6[/u] [b]p16.[/b] Given that $x$ and $y$ are positive real numbers such that $x^3+y = 20$, the maximum possible value of $x + y$ can be written as $\frac{a\sqrt{b}}{c}$ +d where $a$, $b$, $c$, and $d$ are positive integers such that $gcd(a,c) = 1$ and $b$ is square-free. Find $a +b +c +d$. [b]p17.[/b] In $\vartriangle DRK$ , $DR = 13$, $DK = 14$, and $RK = 15$. Let $E$ be the intersection of the altitudes of $\vartriangle DRK$. Find the value of $\lfloor DE +RE +KE \rfloor$. [b]p18.[/b] Subaru the frog lives on lily pad $1$. There is a line of lily pads, numbered $2$, $3$, $4$, $5$, $6$, and $7$. Every minute, Subaru jumps from his current lily pad to a lily pad whose number is either $1$ or $2$ greater, chosen at random from valid possibilities. There are alligators on lily pads $2$ and $5$. If Subaru lands on an alligator, he dies and time rewinds back to when he was on lily pad number $1$. Find the expected number of jumps it takes Subaru to reach pad $7$. [u]Round 7[/u] This set has problems whose answers depend on one another. [b]p19.[/b] Let $B$ be the answer to Problem $20$ and let $C$ be the answer to Problem $21$. Given that $$f (x) = x^3-Bx-C = (x-r )(x-s)(x-t )$$ where $r$, $s$, and $t$ are complex numbers, find the value of $r^2+s^2+t^2$. [b]p20.[/b] Let $A$ be the answer to Problem $19$ and let $C$ be the answer to Problem $21$. Circles $\omega_1$ and $\omega_2$ meet at points $X$ and $Y$ . Let point $P \ne Y$ be the point on $\omega_1$ such that $PY$ is tangent to $\omega_2$, and let point $Q \ne Y$ be the point on $\omega_2$ such that $QY$ is tangent to $\omega_1$. Given that $PX = A$ and $QX =C$, find $XY$ . [b]p21.[/b] Let $A$ be the answer to Problem $19$ and let $B$ be the answer to Problem $20$. Given that the positive difference between the number of positive integer factors of $A^B$ and the number of positive integer factors of $B^A$ is $D$, and given that the answer to this problem is an odd prime, find $\frac{D}{B}-40$. [u]Round 8[/u] [b]p22.[/b] Let $v_p (n)$ for a prime $p$ and positive integer $n$ output the greatest nonnegative integer $x$ such that $p^x$ divides $n$. Find $$\sum^{50}_{i=1}\sum^{i}_{p=1} { v_p (i )+1 \choose 2},$$ where the inner summation only sums over primes $p$ between $1$ and $i$ . [b]p23.[/b] Let $a$, $b$, and $c$ be positive real solutions to the following equations. $$\frac{2b^2 +2c^2 -a^2}{4}= 25$$ $$\frac{2c^2 +2a^2 -b^2}{4}= 49$$ $$\frac{2a^2 +2b^2 -c^2}{4}= 64$$ The area of a triangle with side lengths $a$, $b$, and $c$ can be written as $\frac{x\sqrt{y}}{z}$ where $x$ and $z$ are relatively prime positive integers and $y$ is square-free. Find $x + y +z$. [b]p24.[/b] Alan, Jiji, Ina, Ryan, and Gavin want to meet up. However, none of them know when to go, so they each pick a random $1$ hour period from $5$ AM to $11$ AM to meet up at Alan’s house. Find the probability that there exists a time when all of them are at the house at one time. [b]Round 9 [/b] [b]p25.[/b] Let $n$ be the number of registered participantsin this $LMT$. Estimate the number of digits of $\left[ {n \choose 2} \right]$ in base $10$. If your answer is $A$ and the correct answer is $C$, then your score will be $$\left \lfloor \max \left( 0,20 - \left| \ln \left( \frac{A}{C}\right) \cdot 5 \right|\right| \right \rfloor.$$ [b]p26.[/b] Let $\gamma$ be theminimum value of $x^x$ over all real numbers $x$. Estimate $\lfloor 10000\gamma \rfloor$. If your answer is $A$ and the correct answer is $C$, then your score will be $$\left \lfloor \max \left( 0,20 - \left| \ln \left( \frac{A}{C}\right) \cdot 5 \right|\right| \right \rfloor.$$ [b]p27.[/b] Let $$E = \log_{13} 1+log_{13}2+log_{13}3+...+log_{13}513513.$$ Estimate $\lfloor E \rfloor$. If your answer is $A$ and the correct answer is $C$, your score will be $$\left \lfloor \max \left( 0,20 - \left| \ln \left( \frac{A}{C}\right) \cdot 5 \right|\right| \right \rfloor.$$ PS. You should use hide for answers. Rounds 1-5 have been posted [url=https://artofproblemsolving.com/community/c3h3167127p28823220]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCD$ be a tangential quadrilateral with $BC> BA$. The point $P$ is on the segment $BC$, such that $BP = BA$ . Show that the bisector of $\angle BCD$, the perpendicular on line $BC$ through $P$ and the perpendicular on $BD$ through $A$, intersect at one point.

A $101\times 101$ square grid is given with rows and columns numbered in order from $1$ to $101$. Each square that is contained in both an even-numbered row and an even-numbered column is cut out. A small section of the grid is shown below, with the cut-out squares in black. Compute the maximum number of $L$-triominoes (pictured below) that can be placed in the grid so that each $L$-triomino lies entirely inside the grid and no two overlap. Each $L$-triomino may be placed in the orientation pictured below, or rotated by $90^o$, $180^o$, or $270^o$. [img]https://cdn.artofproblemsolving.com/attachments/2/5/016d4e823e3df4b83556a49f7e612d40e3deba.png[/img]

Before the FIFA world cup, the football coach of F country want to test seven players $A_1, A_2, \cdots, A_7$. He asks them to join in three training matches (90 minutes each), and everyone must appear in each match at least once. Suppose that at any moment during a match, one and only one of them enters the field, and the total time (measured in minutes) on the field for $A_1, A_2, A_3, A_4$ are multiples of $7$ and the total time for$A_5, A_6, A_7$ are multiples of $13$. If the number of substitutions of players during each match is not limited, find the number of different cases. Note: If and only if the total time of a certian player is different, then the case is considered different.

What is the maximal number of crosses than can fit in a $10\times 11$ board without overlapping? Is this problem well-known? [asy] size(4.58cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -3.18, xmax = 1.4, ymin = -0.22, ymax = 3.38; /* image dimensions */ /* draw figures */ draw((-3.,2.)--(1.,2.)); draw((-2.,3.)--(-2.,0.)); draw((-2.,0.)--(-1.,0.)); draw((-1.,0.)--(-1.,3.)); draw((-1.,3.)--(-2.,3.)); draw((-3.,1.)--(1.,1.)); draw((1.,1.)--(1.,2.)); draw((-3.,2.)--(-3.,1.)); draw((0.,2.)--(0.,1.)); draw((-1.,2.)--(-1.,1.)); draw((-2.,2.)--(-2.,1.)); /* dots and labels */ clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]

Let $ \triangle ABC $ be equilateral. On the side $ AB $ points $ C_{1} $ and $ C_{2} $, on the side $ AC $ points $ B_{1} $ and $ B_{2} $ are chosen, and on the side $ BC $ points $ A_{1} $ and $ A_{2} $ are chosen. The following condition is given : $ A_{1}A_{2} $ = $ B_{1}B_{2} $ = $ C_{1}C_{2} $. Let the intersection lines $ A_{2}B_{1}$ and $ B_{2}C_{1} $, $ B_{2}C_{1} $ and $ C_{2}A_{1} $ and $ C_{2}A_{1} $ and $ A_{2}B_{1} $ are $ E $, $ F $, and $ G $ respectively. Show that the triangle formed by $ B_{1}A_{2} $, $ A_{1}C_{2} $ and $ C_{1}B_{2} $ is similar to $ \triangle EFG $.

Let $a_0,b_0,c_0,a,b,c$ be integers such that $\gcd(a_0,b_0,c_0)=\gcd(a,b,c)=1$. Prove that there exists a positive integer $n$ and integers $a_1,a_2,\ldots,a_n=a,b_1,b_2,\ldots,b_n=b,c_1,c_2,\ldots,c_n=c$ such that for all $1\le i\le n$, $a_{i-1}a_i+b_{i-1}b_i+c_{i-1}c_i=1$. [i]Proposed by Michael Ren.[/i]

Solve the system of equations \[(x+\sqrt{x^2+1})(y+\sqrt{y^2+1})=1,\]\[y+\frac{y}{\sqrt{x^2-1}}+\frac{35}{12}=0.\]

Let $D$ be a point inside an acute triangle $ABC$, such that $\angle BAD = \angle DBC$ and $\angle DAC = \angle BCD$. Let $P$ be a point on the circumcircle of the triangle $ADB$. Suppose $P$ are itself outside the triangle $ABC$. A line through $P$ intersects the ray $BA$ in $X$ and ray $CA$ in $Y$, so that $\angle XPB = \angle PDB$. Show that $BY$ and $CX$ intersect on $AD$.

The cubic equation $x^3 + ax^2 + bx + c = 0$ has three real roots. Show that $a^2-3b\geq 0$, and that $\sqrt{a^2-3b}$ is less than or equal to the difference between the largest and smallest roots.

Given a positive integer $n$ and a prime $q$, prove that the number $n^q+(\frac{n-1}{2})^2$ can't be a power of $q$.

Sum of all roots of the equation $$cos^{100} x + a_1 cos^{99} x + a_2cos^{98} x +... + a_99 cos x+ a_{100} = 0$$, in interval $\left[\pi, \frac{3\pi}{2} \right]$, is equal to $21\pi$, and the sum of all roots of the equation $$sin^{100} x + a_1 sin^{99} x + a_2sin ^{98} x +... + a_99sin x+ a_{100} = 0$$, in the same interval, is equal to $24\pi $. How many roots does the first equation have on the segment $\left[ \frac{\pi}{2}, \pi\right]$?

Let $a_1, a_2, ..., a_{2012}$ be pairwise distinct integers. Show that the equation $(x -a_1)(x - a_2)...(x - a_{2012}) = (1006!)^2$ has at most one integral solution.

Let $ ABC$ be a triangle with centroid $ G$ and $ A_1,B_1,C_1$ midpoints of the sides $ BC,CA,AB$. A paralel through $ A_1$ to $ BB_1$ intersects $ B_1C_1$ at $ F$. Prove that triangles $ ABC$ and $ FA_1A$ are similar if and only if quadrilateral $ AB_1GC_1$ is cyclic.

Let $a,b,c$ be integers satisfying $a+b+c=1$ and $ab+bc+ca<abc$. Show that $ab+bc+ca<2abc$.

Let $A = \{1,2,3,4,5,6,7,8\}$, $B = \{9,10,11,12,13,14,15,16\}$ and $C =\{17,18,19,20,21,22,23,24\}$. Find the number of triples $(x, y, z)$ such that $x \in A, y \in B, z \in C $ and $x + y + z = 36$.

Grid the plane forming an infinite board. In each cell of this board, there is a lamp, initially turned off. A permitted operation consists of selecting a square of \(3\times 3\), \(4\times 4\), or \(5\times 5\) cells and changing the state of all lamps in that square (those that are off become on, and those that are on become off). (a) Prove that for any finite set of lamps, it is possible to achieve, through a finite sequence of permitted operations, that those are the only lamps turned on on the board. (b) Prove that if in a sequence of permitted operations only two out of the three square sizes are used, then it is impossible to achieve that at the end the only lamps turned on on the board are those in a \(2\times 2\) square.

How many lattice points $P$ in or on the circle $x^2+y^2=25$ have the property that there exists a unique line with rational slope through $P$ that divides the circle into two parts with equal areas? [i]Proposed by Nathan Ramesh

Let $m\ge 2$ be an integer, $A$ a finite set of integers (not necessarily positive) and $B_1,B_2,...,B_m$ subsets of $A$. Suppose that, for every $k=1,2,...,m$, the sum of the elements of $B_k$ is $m^k$. Prove that $A$ contains at least $\dfrac{m}{2}$ elements.

Prove that $$(1 + x)(1 + x^2)(1 + x^4)(1 + x^8) ... (1 + x^{2^{k-1}} ) = 1 + x + x^2 + x^3 +... + x^{2^{k-1}}$$

Among the points corresponding to number $1,2,...,2n$ on the real line, $n$ are colored in blue and $n$ in red. Let $a_1,a_2,...,a_n$ be the blue points and $b_1,b_2,...,b_n$ be the red points. Prove that the sum $\mid a_1-b_1\mid+...+\mid a_n-b_n\mid$ does not depend on coloring , and compute its value. :roll:

At least $3$ players take part in a tennis tournament. Each participant plays exactly one match against each other participant. After the tournament has ended, we find out that each player has won at least one match. (There are no ties in tennis). Show that in the tournament, there was at least one trio of players $A,B,C$ such that $A$ beat $B$, $B$ beat $C$, and $C$ beat $A$.

Define $ f: \mathbb{R}\to\mathbb{R}$ by \[ f(x)\equal{}\begin{cases}x&\text{if }x\le e\\ xf(\ln x)&\text{if }x>e\end{cases}\] Does $ \displaystyle\sum_{n\equal{}1}^{\infty}\frac1{f(n)}$ converge?