300 students participate on the international math olympiad. Every student speaks in exactly two of the official languages of the olympiad and every language is spoken by 100 people (it is known that students speak only the official languages). Prove that the students can be sited on a circular table, such that no two neighbors spoke the same language.

A circumscribed quadrilateral $ABCD$ is given. $E$ and $F$ are the intersection points of opposite sides of the $ABCD$. It turned out that the radii of the inscribed circles of the triangles $AEF$ and $CEF$ are equal. Prove that $AC \bot BD$.

Let $n$ be a positive integer. Let $a,b,x$ be real numbers, with $a \neq b$ and let $M_n$ denote the $2n x 2n $ matrix whose $(i,j)$ entry $m_{ij}$ is given by $m_{ij}=x$ if $i=j$, $m_{ij}=a$ if $i \not= j$ and $i+j$ is even, $m_{ij}=b$ if $i \not= j$ and $i+j$ is odd. For example $ M_2=\begin{vmatrix}x& b& a & b\\ b& x & b &a\\ a & b& x & b\\ b & a & b & x \end{vmatrix}$. Express $\lim_{x\to\ 0} \frac{ det M_n}{ (x-a)^{(2n-2)} }$ as a polynomial in $a,b $ and $n$ . P.S. How write in latex $m_{ij}=...$ with symbol for the system (because is multiform function?)

Let $\Lambda$ denote the set of points $(x,y)$ in 2D space with integer coordinates such that $0\leq x\leq 4$ and $0\leq y\leq 2$. That is, \[ \Lambda=\{ (x,y) \in \mathbb{Z}^2: 0\leq x\leq 4, \ 0\leq y\leq 2 \}. \] Find the number of ways to connect points of $\Lambda$ with segments of length $\sqrt{2}$ or $\sqrt{5}$ such that the interior of any unit square with vertices in $\Lambda$ contains part of exactly one segment; an example is shown below (connections that differ by reflections are distinct). [asy] unitsize(1cm); dot((0,0)); dot((1,0)); dot((2,0)); dot((3,0)); dot((4,0)); dot((0,1)); dot((1,1)); dot((2,1)); dot((3,1)); dot((4,1)); dot((0,2)); dot((1,2)); dot((2,2)); dot((3,2)); dot((4,2)); draw((0,0)--(1,1)); draw((0,2)--(2,1)); draw((1,1)--(2,0)); draw((2,0)--(3,2)); draw((3,1)--(4,2)); draw((3,0)--(4,1)); [/asy]

[u]Round 1[/u] [b]p1.[/b] How many powers of $2$ are greater than $3$ but less than $2013$? [b]p2.[/b] What number is equal to six greater than three times the answer to this question? [b]p3.[/b] Surya Cup-a-tea-raju goes to Starbucks Coffee to sip coffee out of a styrofoam cup. The cup is a cylinder, open on one end, with base radius $3$ centimeters and height $10$ centimeters. What is the exterior surface area of the styrofoam cup? [u]Round 2[/u] [b]p4.[/b] Andrew has two $6$-foot-length sticks that he wishes to make into two of the sides of the entrance to his fort, with the ground being the third side. If he wants to make his entrance in the shape of a triangle, what is the largest area that he can make the entrance? [b]p5.[/b] Ethan and Devin met a fairy who told them “if you have less than $15$ dollars, I will give you cake”. If both had integral amounts of dollars, and Devin had 5 more dollars than Ethan, but only Ethan got cake, how many different amounts of money could Ethan have had? [b]p6.[/b] If $2012^x = 2013$, for what value of $a$, in terms of $x$, is it true that $2012^a = 2013^2$? [u]Round 3[/u] [b]p7.[/b] Find the ordered triple $(L, M, T)$ of positive integers that makes the following equation true: $$1 + \dfrac{1}{L + \dfrac{1}{M+\dfrac{1}{T}}}=\frac{79}{43}.$$ [b]p8.[/b] Jonathan would like to start a banana plantation so he is saving up to buy an acre of land, which costs $\$600,000$. He deposits $\$300,000$ in the bank, which gives $20\%$ interest compounded at the end of each year. At this rate, how many years will Jonathan have to wait until he can buy the acre of land? [b]p9.[/b] Arul and Ethan went swimming at their town pool and started to swim laps to see who was in better shape. After one hour of swimming at their own paces, Ethan completed $32$ more laps than Arul. However, after that, Ethan got tired and swam at half his original speed while Arul’s speed didn’t change. After one more hour, Arul swam a total of $320$ laps. How many laps did Ethan swim after two hours? [u]Round 4[/u] [b]p10.[/b] A right triangle with a side length of $6$ and a hypotenuse of 10 has circles of radius $1$ centered at each vertex. What is the area of the space inside the triangle but outside all three circles? [b]p11.[/b] In isosceles trapezoid $ABCD$, $\overline{AB} \parallel\overline{CD}$ and the lengths of $\overline{AB}$ and $\overline{CD}$ are $2$ and $6$, respectively. Let the diagonals of the trapezoid intersect at point $E$. If the distance from $E$ to $\overline{CD}$ is $9$, what is the area of triangle $ABE$? [b]p12.[/b] If $144$ unit cubes are glued together to form a rectangular prism and the perimeter of the base is $54$ units, what is the height? PS. You should use hide for answers. Rounds 6-8 are [url=https://artofproblemsolving.com/community/c3h3136014p28427163]here[/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3137069p28442224]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We form a decimal code of $21$ digits. the code may start with $0$. Determine the probability that the fragment $0123456789$ appears in the code.

Prove that $n$ ($\ge 4$) points of the plane are vertices of a convex $n$-gon if and only if any $4$ of them are vertices of a convex quadrilateral.

In how many ways can we place the numbers from $1$ to $100$ in a $2\times 50$ rectangle (divided into $100$ unit squares) so that any two consecutive numbers are always placed in squares with a common side?

An equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M\in\overline{AB}$, $Q\in\overline{AC}$, and $N,P\in\overline{BC}$. Let $S$ be the intersection of $\overleftrightarrow{MN}$ and $\overleftrightarrow{PQ}$. Denote by $\ell$ the angle bisector of $\angle MSQ$. Prove that $\overline{OI}$ is parallel to $\ell$, where $O$ is the circumcenter of triangle $ABC$, and $I$ is the incenter of triangle $ABC$.

[u]Round 5[/u] [b]p13.[/b] Points $A, B, C$, and $D$ lie in a plane with $AB = 6$, $BC = 5$, and $CD = 5$, and $AB$ is perpendicular to $BC$. Point E lies on line $AD$ such that $D \ne E$, $AE = 3$ and $CE = 5$. Find $DE$. [b]p14.[/b] How many ordered pairs of integers $(x,y)$ are solutions to $x^2y = 36 + y$? [b]p15.[/b] Chicken nuggets come in boxes of two sizes, $a$ nuggets per box and $b$ nuggets per box. We know that $899$ nuggets is the largest number of nuggets we cannot obtain with some combination of $a$-sized boxes and $b$-sized boxes. How many different pairs $(a, b)$ are there with $a < b$? [u]Round 6[/u] [b]p16.[/b] You are playing a game with coins with your friends Alice and Bob. When all three of you flip your respective coins, the majority side wins. For example, if Alice, Bob, and you flip Heads, Tails, Heads in that order, then you win. If Alice, Bob, and you flip Heads, Heads, Tails in that order, then you lose. Notice that more than one person will “win.” Alice and Bob design their coins as follows: a value $p$ is chosen randomly and uniformly between $0$ and $1$. Alice then makes a biased coin that lands on heads with probability $p$, and Bob makes a biased coin that lands on heads with probability $1 -p$. You design your own biased coin to maximize your chance of winning without knowing $p$. What is the probability that you win? [b]p17.[/b] There are $N$ distinct students, numbered from $1$ to $N$. Each student has exactly one hat: $y$ students have yellow hats, $b$ have blue hats, and $r$ have red hats, where $y + b + r = N$ and $y, b, r > 0$. The students stand in a line such that all the $r$ people with red hats stand in front of all the $b$ people with blue hats. Anyone wearing red is standing in front of everyone wearing blue. The $y$ people with yellow hats can stand anywhere in the line. The number of ways for the students to stand in a line is $2016$. What is $100y + 10b + r$? [b]p18.[/b] Let P be a point in rectangle $ABCD$ such that $\angle APC = 135^o$ and $\angle BPD = 150^o$. Suppose furthermore that the distance from P to $AC$ is $18$. Find the distance from $P$ to $BD$. [u]Round 7 [/u] [b]p19.[/b] Let triangle $ABC$ be an isosceles triangle with $|AB| = |AC|$. Let $D$ and $E$ lie on $AB$ and $AC$, respectively. Suppose $|AD| = |BC| = |EC|$ and triangle $ADE$ is isosceles. Find the sum of all possible values of $\angle BAC$ in radians. Write your answer in the form $2 arcsin \left( \frac{a}{b}\right) + \frac{c}{d} \pi$, where $\frac{a}{b}$ and $\frac{c}{d}$ are in lowest terms, $-1 \le \frac{a}{b} \le 1$, and $-1 \le \frac{c}{d} \le 1$. [b]p20.[/b] Kevin is playing a game in which he aims to maximize his score. In the $n^{th}$ round, for $n \ge 1$, a real number between $0$ and $\frac{1}{3^n}$ is randomly generated. At each round, Kevin can either choose to have the randomly generated number from that round as his score and end the game, or he can choose to pass on the number and continue to the next round. Once Kevin passes on a number, he CANNOT claim that number as his score. Kevin may continue playing for as many rounds as he wishes. If Kevin plays optimally, the expected value of his score is $a + b\sqrt{c}$ where $a, b$, and $c$ are integers and $c$ is positive and not divisible by any positive perfect square other than $1$. What is $100a + 10b + c$? [b]p21.[/b] Lisa the ladybug (a dimensionless ladybug) lives on the coordinate plane. She begins at the origin and walks along the grid, at each step moving either right or up one unit. The path she takes ends up at $(2016, 2017)$. Define the “area” of a path as the area below the path and above the $x$-axis. The sum of areas over all paths that Lisa can take can be represented as as $a \cdot {{4033} \choose {2016}}$ . What is the remainder when $a$ is divided by $1000$? PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2782871p24446475]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let I be the incenter of triangle $ABC$ and let $k$ be a circle through the points $A$ and $B$. The circle intersects * the line $AI$ in points $A$ and $P$ * the line $BI$ in points $B$ and $Q$ * the line $AC$ in points $A$ and $R$ * the line $BC$ in points $B$ and $S$ with none of the points $A,B,P,Q,R$ and $S$ coinciding and such that $R$ and $S$ are interior points of the line segments $AC$ and $BC$, respectively. Prove that the lines $PS$, $QR$, and $CI$ meet in a single point. (Stephan Wagner)

Let $x$ and $y$ be real numbers with $x>y$ such that $x^2y^2+x^2+y^2+2xy=40$ and $xy+x+y=8$. Find the value of $x$.

A palindrome is a word that reads the same backwards as forwards, such as “eye”, “race car”, and “qwertyytrewq”. How many letters are in the smallest palindrome containing the letters b, o, g, t, r, and o, not necessarily in that order and not necessarily adjacent?

Prove that if $ p$ is a prime number, then $ 7p+3^{p}-4$ is not a perfect square.

Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as \begin{align*} (x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z) \end{align*} with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.

In acute triangle $ ABC$, show that: $ \sin^3{A}\cos^2{(B \minus{} C)} \plus{} \sin^3{B}\cos^2{(C \minus{} A)} \plus{} \sin^3{C}\cos^2{(A \minus{} B)} \leq 3\sin{A} \sin{B} \sin{C}$ and find out when the equality holds.