[u]Round 1 [/u] [b]p1. [/b]Twelve people, some are knights and some are knaves, are sitting around a table. Knaves always lie and knights always tell the truth. At some point they start up a conversation. The first person says, “There are no knights around this table.” The second says, “There is at most one knight at this table.” The third – “There are at most two knights at the table.” And so on until the 12th says, “There are at most eleven knights at the table.” How many knights are at the table? Justify your answer. [b]p2.[/b] Show that in the sequence $10017$, $100117$, $1001117$, $...$ all numbers are divisible by $53$. [b]p3.[/b] Harry and Draco have three wands: a bamboo wand, a willow wand, and a cherry wand, all of the same length. They must perform a spell wherein they take turns picking a wand and breaking it into three parts – first Harry, then Draco, then Harry again. But in order for the spell to work, Harry has to make sure it is possible to form three triangles out of the pieces of the wands, where each triangle has a piece from each wand. How should he break the wands to ensure the success of the spell? [b]p4.[/b] A $2\times 2\times 2$ cube has $4$ equal squares on each face. The squares that share a side are called neighbors (thus, each square has $4$ neighbors – see picture). Is it possible to write an integer in each square in such a way that the sum of each number with its $4$ neighbors is equal to $13$? If yes, show how. If no, explain why not. [img]https://cdn.artofproblemsolving.com/attachments/8/4/0f7457f40be40398dee806d125ba26780f9d3a.png[/img] [b]p5.[/b] Two girls are playing a game. The first player writes the letters $A$ or $B$ in a row, left to right, adding one letter on her turn. The second player switches any two letters after each move by the first player (the letters do not have to be adjacent), or does nothing, which also counts as a move. The game is over when each player has made $2011$ moves. Can the second player plan her moves so that the resulting letters form a palindrome? (A palindrome is a sequence that reads the same forward and backwards, e.g. $AABABAA$.) [u]Round 2 [/u] [b]p6.[/b] A red square is placed on a table. $2010$ white squares, each the same size as the red square, are then placed on the table in such a way that the red square is fully covered and the sides of every white square are parallel to the sides of the red square. Is it always possible to remove one of the white squares so the red square remains completely covered? [b]p7.[/b] A computer starts with a given positive integer to which it randomly adds either $54$ or $77$ every second and prints the resulting sum after each addition. For example, if the computer is given the number $1$, then a possible output could be: $1$, $55$, $109$, $186$, $…$ Show that after finitely many seconds the computer will print a number whose last two digits are the same. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] Compute $$\sqrt{(\sqrt{63} +\sqrt{112} +\sqrt{175})(-\sqrt{63} +\sqrt{112} +\sqrt{175})(\sqrt{63}-\sqrt{112} +\sqrt{175})(\sqrt{63} +\sqrt{112} -\sqrt{175})}$$ [b]p2.[/b] Consider the set $S = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. How many distinct $3$-element subsets are there such that the sum of the elements in each subset is divisible by $3$? [b]p3.[/b] Let $a^2$ and $b^2$ be two integers. Consider the triangle with one vertex at the origin, and the other two at the intersections of the circle $x^2 + y^2 = a^2 + b^2$ with the graph $ay = b|x|$. If the area of the triangle is numerically equal to the radius of the circle, what is this area? [b]p4.[/b] Suppose $f(x) = x^3 + x - 1$ has roots $a$, $b$ and $c$. What is $$\frac{a^3}{1-a}+\frac{b^3}{1-b}+\frac{c^3}{1-c} ?$$ [b]p5.[/b] Lisa has a $2D$ rectangular box that is $48$ units long and $126$ units wide. She shines a laser beam into the box through one of the corners such that the beam is at a $45^o$ angle with respect to the sides of the box. Whenever the laser beam hits a side of the box, it is reflected perfectly, again at a $45^o$ angle. Compute the distance the laser beam travels until it hits one of the four corners of the box. [b]p6.[/b] How many ways can we form a group with an odd number of members (plural) from $99$ people total? Express your answer in the form $a^b + c$, where $a$, $b$, and $c$ are integers, and $a$ is prime. [b]p7.[/b] Let $$S = \log_2 9 \log_3 16 \log_4 25 ...\log_{999} 1000000.$$ Compute the greatest integer less than or equal to $\log_2 S$. [b]p8.[/b] A prison, housing exactly four hundred prisoners in four hundred cells numbered $1$-$400$, has a really messed-up warden. One night, when all the prisoners are asleep and all of their doors are locked, the warden toggles the locks on all of their doors (that is, if the door is locked, he unlocks the door, and if the door is unlocked, he locks it again), starting at door $1$ and ending at door $400$. The warden then toggles the lock on every other door starting at door $2$ ($2$, $4$, $6$, etc). After he has toggled the lock on every other door, the warden then toggles every third door (doors $3$, $6$, $9$, etc.), then every fourth door, etc., finishing by toggling every $400$th door (consisting of only the $400$th door). He then collapses in exhaustion. Compute the number of prisoners who go free (that is, the number of unlocked doors) when they wake up the next morning. [b]p9.[/b] Let $A$ and $B$ be fixed points on a $2$-dimensional plane with distance $AB = 1$. An ant walks on a straight line from point $A$ to some point $C$ on the same plane and finds that the distance from itself to $B$ always decreases at any time during this walk. Compute the area of the locus of points where point $C$ could possibly be located. [b]p10.[/b] A robot starts in the bottom left corner of a $4 \times 4$ grid of squares. How many ways can it travel to each square exactly once and then return to its start if it is only allowed to move to an adjacent (not diagonal) square at each step? [b]p11.[/b] Assuming real values for $p$, $q$, $r$, and $s$, the equation $$x^4 + px^3 + qx^2 + rx + s$$ has four non-real roots. The sum of two of these roots is $4 + 7i$, and the product of the other two roots is $3 - 4i$. Find $q$. [b]p12.[/b] A cube is inscribed in a right circular cone such that one face of the cube lies on the base of the cone. If the ratio of the height of the cone to the radius of the cone is $2 : 1$, what fraction of the cone's volume does the cube take up? Express your answer in simplest radical form. [b]p13.[/b] Let $$y =\dfrac{1}{1 +\dfrac{1}{9 +\dfrac{1}{5 +\dfrac{1}{9 +\dfrac{1}{5 +...}}}}}$$ If $y$ can be represented as $\frac{a\sqrt{b} + c}{d}$, where $b$ is not divisible by the square of any prime, and the greatest common divisor of $a$ and $d$ is $1$, find the sum $a + b + c + d$. [b]p14.[/b] Alice wants to paint each face of an octahedron either red or blue. She can paint any number of faces a particular color, including zero. Compute the number of ways in which she can do this. Two ways of painting the octahedron are considered the same if you can rotate the octahedron to get from one to the other. [b]p15.[/b] Find $n$ in the equation $$133^5 + 110^5 + 84^5 + 27^5 = n^5,$$ where $n$ is an integer less than $170$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Show that for any real numbers $a_{3},a_{4},...,a_{85}$, not all the roots of the equation $a_{85}x^{85}+a_{84}x^{84}+...+a_{3}x^{3}+3x^{2}+2x+1=0$ are real.

Let $a,b,c$ be positive real numbers such that $a+b+c=10$ and $ab+bc+ca=25$. Let $m=\min\{ab,bc,ca\}$. Find the largest possible value of $m$.

Find all prime numbers $p$, for which there exist $x, y \in \mathbb{Q}^+$ and $n \in \mathbb{N}$, satisfying $x+y+\frac{p}{x}+\frac{p}{y}=3n$.

Let be a number $ n\ge 2, $ a binary funcion $ b:\mathbb{Z}\rightarrow\mathbb{Z}_2, $ and $ \frac{n^3+5n}{6} $ consecutive integers. Show that among these consecutive integers there are $ n $ of them, namely, $ b_1,b_2,\ldots ,b_n, $ that have the properties: $ \text{(i)} b\left( b_1\right) =b\left( b_2\right) =\cdots =b\left( b_n\right) $ $ \text{(ii)} 1\le b_2-b_1\le b_3-b_2\le \cdots\le b_n-b_{n-1} $

Define the sequences $(a_n),(b_n)$ by \begin{align*} & a_n, b_n > 0, \forall n\in\mathbb{N_+} \\ & a_{n+1} = a_n - \frac{1}{1+\sum_{i=1}^n\frac{1}{a_i}} \\ & b_{n+1} = b_n + \frac{1}{1+\sum_{i=1}^n\frac{1}{b_i}} \end{align*} 1) If $a_{100}b_{100} = a_{101}b_{101}$, find the value of $a_1-b_1$; 2) If $a_{100} = b_{99}$, determine which is larger between $a_{100}+b_{100}$ and $a_{101}+b_{101}$.

Find all functions $f:\mathbb{R}\to\mathbb{R}$ for which \[ x(f(x+1)-f(x)) = f(x), \] for all $x\in\mathbb{R}$ and \[ | f(x) - f(y) | \leq |x-y| , \] for all $x,y\in\mathbb{R}$. [i]Mihai Piticari[/i]

Vanya has chosen two positive numbers, x and y. He wrote the numbers x+y, x-y, x/y, and xy, and has shown these numbers to Petya. However, he didn't say which of the numbers was obtained from which operation. Show that Petya can uniquely recover x and y.

Call a polynomial $f(x)$ [i]excellent[/i] if its coefficients are all in [0, 1) and $f(x)$ is an integer for all integers $x$. a) Compute the number of excellent polynomials with degree at most 3. b) Compute the number of excellent polynomials with degree at most $n$, in terms of $n$. c) Find the minimum $n\ge3$ for which there exists an excellent polynomial of the form $\frac{1}{n!}x^n+g(x)$, where $g(x)$ is a polynomial of degree at most $n-3$.

If the remainder is $2013$ when a polynomial with coefficients from the set $\{0,1,2,3,4,5\}$ is divided by $x-6$, what is the least possible value of the coefficient of $x$ in this polynomial? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 1 $

The function $g$, with domain and real numbers, fulfills the following: $\bullet$ $g (x) \le x$, for all real $x$ $\bullet$ $g (x + y) \le g (x) + g (y)$ for all real $x,y$ Find $g (1990)$.

Given $5n$ real numbers $r_i, s_i, t_i, u_i, v_i \geq 1 (1 \leq i \leq n)$, let $R = \frac {1}{n} \sum_{i=1}^{n} r_i$, $S = \frac {1}{n} \sum_{i=1}^{n} s_i$, $T = \frac {1}{n} \sum_{i=1}^{n} t_i$, $U = \frac {1}{n} \sum_{i=1}^{n} u_i$, $V = \frac {1}{n} \sum_{i=1}^{n} v_i$. Prove that $\prod_{i=1}^{n}\frac {r_i s_i t_i u_i v_i + 1}{r_i s_i t_i u_i v_i - 1} \geq \left(\frac {RSTUV +1}{RSTUV - 1}\right)^n$.

Show that for any $n\geq 2$, $2^{2^n}+1$ ends with 7

Let $f(x) = ax^3 + bx^2 + cx + d$ be a polynomial with real coefficients. Given that $f(x)$ has three real positive roots and that $f(0) < 0$, prove that $2b^3+ 9a^2 d - 7abc \le 0$.

In quadrilateral $ABCD$, $BC=8$, $CD=12$, $AD=10$, and $m\angle A= m\angle B = 60^\circ$. Given that $AB=p + \sqrt{q}$, where $p$ and $q$ are positive integers, find $p+q$.

Let $\alpha \in (1, +\infty)$ be a real number, and let $P(x) \in \mathbb{R}[x]$ be a monic polynomial with degree $24$, such that (i) $P(0) = 1$. (ii) $P(x)$ has exactly $24$ positive real roots that are all less than or equal to $\alpha$. Show that $|P(1)| \le \left( \frac{19}{5}\right)^5 (\alpha-1)^{24}$.

Find all the real values of $ \lambda$ for which the system of equations $ x\plus{}y\plus{}z\plus{}v\equal{}0$ and $ \left(xy\plus{}yz\plus{}zv\right)\plus{}\lambda\left(xz\plus{}xv\plus{}yv\right)\equal{}0$, has a unique real solution.

Real numbers $a,b,c,d$ are given. Solve the system of equations (unknowns $x,y,z,u)$\[ x^{2}-yz-zu-yu=a\] \[ y^{2}-zu-ux-xz=b\] \[ z^{2}-ux-xy-yu=c\] \[ u^{2}-xy-yz-zx=d\]

Bristy wants to build a special set $A$. She starts with $A=\{0, 42\}$. At any step, she can add an integer $x$ to the set $A$ if it is a root of a polynomial which uses the already existing integers in $A$ as coefficients. She keeps doing this, adding more and more numbers to $A$. After she eventually runs out of numbers to add to $A$, how many numbers will be in $A$?

Determine all functions $f : Z\to Z$, where $Z$ is the set of integers, such that $$f(m + f(f(n))) = -f(f(m + 1)) - n$$ for all integers $m$ and $n$.

For all $x,y,z$ positive real numbers, find the all $c$ positive real numbers that providing $$\frac{x^3y+y^3z+z^3x}{x+y+z}+\frac{4c}{xyz}\ge2c+2$$

Let $n\geq 2$ be an integer and let $a_1, a_2, \ldots, a_n$ be positive real numbers with sum $1$. Prove that $$\sum_{k=1}^n \frac{a_k}{1-a_k}(a_1+a_2+\cdots+a_{k-1})^2 < \frac{1}{3}.$$

Let $a,b,c$ be real numbers $a + b +c = 0$. Show that [list=a] [*] $\displaystyle \frac{a^2 + b^2 + c^2}{2} \cdot \frac{a^3 + b^3 + c^3}{3} = \frac{a^5 + b^5 + c^5}{5}$. [*] $\displaystyle \frac{a^2 + b^2 + c^2}{2} \cdot \frac{a^5 + b^5 + c^5}{5} = \frac{a^7 + b^7 + c^7}{7}$. [/list] [I]Folklore[/I]

Find all positive integers $n$ for which there exist real numbers $x_1, x_2,. . . , x_n$ satisfying all of the following conditions: (i) $-1 <x_i <1,$ for all $1\leq i \leq n.$ (ii) $ x_1 + x_2 + ... + x_n = 0.$ (iii) $\sqrt{1 - x_1^2} +\sqrt{1 - x^2_2} + ... +\sqrt{1 - x^2_n} = 1.$