Given a quadratic trinomial $p(x)$ with integer coefficients such that $p(x)$ is not divisible by $3$ for all integers $x$. Prove that there exist polynomials $f(x)$ and $h(x)$ with integer coefficients such that $$ p(x)\cdot f(x)+3h(x)=x^6+x^4+x^2+1. $$ [i](I. Gorodnin)[/i]

Call a polynomial $p(x)$ with positive integer roots [i]corrupt[/i] if there exists an integer that cannot be expressed as a sum of (not necessarily positive) multiples of its roots. The polynomial $A(x)$ is monic, corrupt, and has distinct roots. As well, $A(0)$ has $7$ positive divisors. Find the least possible value of $|A(1)|$.

Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\] for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$. Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.

The sequence $\{a_n\}$ is defined as follows: \begin{align*} a_n = \sqrt{1 + \left(1 + \frac 1n \right)^2} + \sqrt{1 + \left(1 - \frac 1n \right)^2}. \end{align*} What is the value of the expression given below? \begin{align*} \frac{4}{a_1} + \frac{4}{a_2} + \dots + \frac{4}{a_{96}}.\end{align*} $\textbf{(A)}\ \sqrt{18241} \qquad\textbf{(B)}\ \sqrt{18625} - 1 \qquad\textbf{(C)}\ \sqrt{18625} \qquad\textbf{(D)}\ \sqrt{19013} - 1\qquad\textbf{(E)}\ \sqrt{19013}$

Polynomial P(x) with integer coefficient is called [i]cube-presented[/i] if it can be represented as sum of several cube of polynomials with integer coefficients. Examples: $3x + 3x^2$ is cube-represented because $3x + 3x^2 = (x + 1)^3 +(-x)^3 + (-1)^3$. a) Is $3x^2$ a cube-represented polynomial? b). How many quadratic polynomial P(x) with integer coefficients belong to the set $\{1,2, 3, ...,2017\}$ which is cube-represented?

Find all homogeneous linear recursive sequences such that there is a $ T$ such that $ a_n\equal{}a_{n\plus{}T}$ for each $ n$.

[u]Round 1[/u] [b]p1.[/b] Alice and Bob played $25$ games of rock-paper-scissors. Alice played rock $12$ times, scissors $6$ times, and paper $7$ times. Bob played rock $13$ times, scissors $9$ times, and paper $3$ times. If there were no ties, who won the most games? (Remember, in each game each player picks one of rock, paper, or scissors. Rock beats scissors, scissors beat paper, and paper beats rock. If they choose the same object, the result is a tie.) [b]p2.[/b] On the planet Vulcan there are eight big volcanoes and six small volcanoes. Big volcanoes erupt every three years and small volcanoes erupt every two years. In the past five years, there were $30$ eruptions. How many volcanoes could erupt this year? [b]p3.[/b] A tangle is a sequence of digits constructed by picking a number $N\ge 0$ and writing the integers from $0$ to $N$ in some order, with no spaces. For example, $010123459876$ is a tangle with $N = 10$. A palindromic sequence reads the same forward or backward, such as $878$ or $6226$. The shortest palindromic tangle is $0$. How long is the second-shortest palindromic tangle? [b]p4.[/b] Balls numbered $1$ to $N$ have been randomly arranged in a long input tube that feeds into the upper left square of an $8 \times 8$ board. An empty exit tube leads out of the lower right square of the board. Your goal is to arrange the balls in order from $1$ to $N$ in the exit tube. As a move, you may 1. move the next ball in line from the input tube into the upper left square of the board, 2. move a ball already on the board to an adjacent square to its right or below, or 3. move a ball from the lower right square into the exit tube. No square may ever hold more than one ball. What is the largest number $N$ for which you can achieve your goal, no matter how the balls are initially arranged? You can see the order of the balls in the input tube before you start. [img]https://cdn.artofproblemsolving.com/attachments/1/8/bbce92750b01052db82d58b96584a36fb5ca5b.png[/img] [b]p5.[/b] A $2018 \times 2018$ board is covered by non-overlapping $2 \times 1$ dominoes, with each domino covering two squares of the board. From a given square, a robot takes one step to the other square of the domino it is on and then takes one more step in the same direction. Could the robot continue moving this way forever without falling off the board? [img]https://cdn.artofproblemsolving.com/attachments/9/c/da86ca4ff0300eca8e625dff891ed1769d44a8.png[/img] [u]Round 2[/u] [b]p6.[/b] Seventeen teams participated in a soccer tournament where a win is worth $1$ point, a tie is worth $0$ points, and a loss is worth $-1$ point. Each team played each other team exactly once. At least $\frac34$ of all games ended in a tie. Show that there must be two teams with the same number of points at the end of the tournament. [b]p7.[/b] The city of Old Haven is known for having a large number of secret societies. Any person may be a member of multiple societies. A secret society is called influential if its membership includes at least half the population of Old Haven. Today, there are $2018$ influential secret societies. Show that it is possible to form a council of at most $11$ people such that each influential secret society has at least one member on the council. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Denote by $a_n$ the greatest number that is not divisible by $3$ and that divides $n$. Consider the sequence $s_0 = 0, s_n = a_1 +a_2+\cdots+a_n, n \in \mathbb N$. Denote by $A(n)$ the number of all sums $s_k \ (0 \leq k \leq 3^n, k \in \mathbb N_0)$ that are divisible by $3$. Prove the formula \[A(n) = 3^{n-1} + 2 \cdot 3^{(n/2)-1} \cos \left(\frac{n\pi}{6}\right), \qquad n\in \mathbb N_0.\]

Let $p = 2^{16}+1$ be an odd prime. Define $H_n = 1+ \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n}$. Compute the remainder when \[ (p-1)! \sum_{n = 1}^{p-1} H_n \cdot 4^n \cdot \binom{2p-2n}{p-n} \] is divided by $p$. [i]Proposed by Yang Liu[/i]

Let $A$ be a set of functions $f : R\to R$. For all $f_1, f_2 \in A$ there exists a $f_3 \in A$ such that $f_1(f_2(y) - x)+ 2x = f_3(x + y)$ for all $x, y \in R$. Prove that for all $f \in A$, we have $f(x - f(x))= 0$ for all $x \in R$.

The polynomials $k_n(x_1, \ldots, x_n)$, where $n$ is a non-negative integer, satisfy the following conditions \[k_0=1\] \[k_1(x_1)=x_1\] \[k_n(x_1, \ldots, x_n) = x_nk_{n-1}(x_1, \ldots , x_{n-1}) + (x_n^2+x_{n-1}^2)k_{n-2}(x_1,\ldots,x_{n-2})\] Prove that for each non-negative $n$ we have $k_n(x_1,\ldots,x_n)=k_n(x_n,\ldots,x_1)$.

Let $a,b,c,d >0$ and $abcd=1$. Prove that: \[ \frac{1}{(1+a)^2}+\frac{1}{(1+b)^2}+\frac{1}{(1+c)^2}+\frac{1}{(1+d)^2} \geq 1 \]

Let be two real numbers $ x,y, $ and a natural number $ n_0 $ such that $ \{ n_0x \} = \{ n_0y \} $ and $ \{ (n_0+1)x \} = \{ (n_0+1)y \} ,$ where $ \{\} $ denotes the fractional part. Show that $ \{ nx \} =\{ ny \} , $ for any natural number $ n. $ [i]Ovidiu Pop[/i]

Let $S$ be a real number. It is known that however we choose several numbers from the interval $(0, 1]$ with sum equal to $S$, these numbers can be separated into two subsets with the following property: The sum of the numbers in one of the subsets doesn’t exceed 1 and the sum of the numbers in the other subset doesn’t exceed 5. Find the greatest possible value of $S$.

Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that $$\frac{1}{a\sqrt{c^2 + 1}} + \frac{1}{b\sqrt{a^2 + 1}} + \frac{1}{c\sqrt{b^2+1}} > 2.$$

Find all positive integer(s) $n$ such that $n^2+32n+8$ is a perfect square.

Calculate $$\sum_{n=1}^{1989}\frac{1}{\sqrt{n+\sqrt{n^2-1}}}$$

For some positive integer $n,$ Elmo writes down the equation \[x_1+x_2+\dots+x_n=x_1+x_2+\dots+x_n.\] Elmo inserts at least one $f$ to the left side of the equation and adds parentheses to create a valid functional equation. For example, if $n=3,$ Elmo could have created the equation \[f(x_1+f(f(x_2)+x_3))=x_1+x_2+x_3.\] Cookie Monster comes up with a function $f: \mathbb{Q}\to\mathbb{Q}$ which is a solution to Elmo's functional equation. (In other words, Elmo's equation is satisfied for all choices of $x_1,\dots,x_n\in\mathbb{Q})$. Is it possible that there is no integer $k$ (possibly depending on $f$) such that $f^k(x)=x$ for all $x$? [i]Srinivas Arun[/i]

What is the sum of the solutions to the equation $\sqrt[4]x =\displaystyle \frac{12}{7-\sqrt[4]x}$?

Today was the 5th Kettering Olympiad - and here are the problems, which are very good intermediate problems. 1. Find all real $x$ so that $(1+x^2)(1+x^4)=4x^3$ 2. Mark and John play a game. They have $100$ pebbles on a table. They take turns taking at least one at at most eight pebbles away. The person to claim the last pebble wins. Mark goes first. Can you find a way for Mark to always win? What about John? 3. Prove that $\sin x + \sin 3x + \sin 5x + ... + \sin 11 x = (1-\cos 12 x)/(2 \sin x)$ 4. Mark has $7$ pieces of paper. He takes some of them and splits each into $7$ pieces of paper. He repeats this process some number of times. He then tells John he has $2000$ pieces of paper. John tells him he is wrong. Why is John right? 5. In a triangle $ABC$, the altitude, angle bisector, and median split angle $A$ into four equal angles. Find the angles of $ABC.$ 6. There are $100$ cities. There exist airlines connecting pairs of cities. a) Find the minimal number of airlines such that with at most $k$ plane changes, one can go from any city to any other city. b) Given that there are $4852$ airlines, show that, given any schematic, one can go from any city to any other city.

Prove that the equation \[x^3 - 3 \tan\frac{\pi}{12} x^2 - 3x + \tan\frac{\pi}{12}= 0\] has one root $x_1 = \tan \frac{\pi}{36}$, and find the other roots.

Let $n$ be a positive integer. Assume that $n$ numbers are to be chosen from the table $\begin{array}{cccc}0 & 1 & \cdots & n-1\\ n & n+1 & \cdots & 2n-1\\ \vdots & \vdots & \ddots & \vdots\\(n-1)n & (n-1)n+1 & \cdots & n^2-1\end{array} $ with no two of them from the same row or the same column. Find the maximal value of the product of these $n$ numbers.

Prove that for $a, b, c \in [0;1]$, $$(1-a)(1+ab)(1+ac)(1-abc) \leq (1+a)(1-ab)(1-ac)(1+abc).$$

Among all sets of real numbers $\{ x_1 , x_2 , ... , x_{20} \}$ from the open interval $(0, 1 )$ such that $$x_1x_2...x_{20}= ( 1 - x_1 ) ( 1 -x_2 ) ... (1 - x_{20} )$$ find the one for which $x_1 x_2... x_{20}$ is maximal. (A Cherniatiev)

Let the sequence $ \{a_n\}_{n\geq 1}$ be defined by $ a_1 \equal{} 20$, $ a_2 \equal{} 30$ and $ a_{n \plus{} 2} \equal{} 3a_{n \plus{} 1} \minus{} a_n$ for all $ n\geq 1$. Find all positive integers $ n$ such that $ 1 \plus{} 5a_n a_{n \plus{} 1}$ is a perfect square.