Find all polynomials $P$ with integer coefficients, for which there exists a number $N$, such that for every natural number $n \geq N$, all prime divisors of $n+2^{\lfloor \sqrt{n} \rfloor}$ are also divisors of $P(n)$.

Find all functions $ f: \mathbb{R}\to\mathbb{R}$ such that we have $f(x + y) = f(x) + f(y) + f(xy)$ for all $ x,y\in \mathbb{R}$

Assume $x_1,x_2,..,x_5$ are positive numbers such that $x_1 < x_2$ and $x_3,x_4, x_5$ are all greater than $x_2$. Prove that if $a > 0$, then $$\frac{1}{(x_1 +x_3)^a}+ \frac{1}{(x_2 +x_4)^a}+ \frac{1}{(x_2 +x_5)^a} <\frac{1}{(x_1 +x_2)^a}+ \frac{1}{(x_2 +x_3)^a}+ \frac{1}{(x_4 +x_5)^a}$$

Given arbitrary complex numbers $w_1,w_2,\ldots,w_n$, show that there exists a positive integer $k\le 2n+1$ for which $\text{Re} (w_1^k+w_2^k+\cdots+w_n^k)\ge 0$.

[u]Round 1[/u] [b]p1.[/b] At a fortune-telling exam, $13$ witches are sitting in a circle. To pass the exam, a witch must correctly predict, for everybody except herself and her two neighbors, whether they will pass or fail. Each witch predicts that each of the $10$ witches she is asked about will fail. How many witches could pass? [b]p2.[/b] Out of $152$ coins, $7$ are counterfeit. All counterfeit coins have the same weight, and all real coins have the same weight, but counterfeit coins are lighter than real coins. How can you find $19$ real coins if you are allowed to use a balance scale three times? [b]p3.[/b] The digits of a number $N$ increase from left to right. What could the sum of the digits of $9 \times N$ be? [b]p4.[/b] The sides and diagonals of a pentagon are colored either blue or red. You can choose three vertices and flip the colors of all three lines that join them. Can every possible coloring be turned all blue by a sequence of such moves? [img]https://cdn.artofproblemsolving.com/attachments/5/a/644aa7dd995681fc1c813b41269f904283997b.png[/img] [b]p5.[/b] You have $100$ pancakes, one with a single blueberry, one with two blueberries, one with three blueberries, and so on. The pancakes are stacked in a random order. Count the number of blueberries in the top pancake and call that number $N$. Pick up the stack of the top $N$ pancakes and flip it upside down. Prove that if you repeat this counting-and-flipping process, the pancake with one blueberry will eventually end up at the top of the stack. [u]Round 2[/u] [b]p6.[/b] A circus owner will arrange $100$ fleas on a long string of beads, each flea on her own bead. Once arranged, the fleas start jumping using the following rules. Every second, each flea chooses the closest bead occupied by one or more of the other fleas, and then all fleas jump simultaneously to their chosen beads. If there are two places where a flea could jump, she jumps to the right. At the start, the circus owner arranged the fleas so that, after some time, they all gather on just two beads. What is the shortest amount of time it could take for this to happen? [b]p7.[/b] The faraway land of Noetheria has $2016$ cities. There is a nonstop flight between every pair of cities. The price of a nonstop ticket is the same in both directions, but flights between different pairs of cities have different prices. Prove that you can plan a route of $2015$ consecutive flights so that each flight is cheaper than the previous one. It is permissible to visit the same city several times along the way. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the smallest positive integer value of $N$ such that field $K=\mathbb{Q}(\sqrt{N},\ \sqrt{i+1})$, where $i=\sqrt{-1}$, is Galois extension on $\mathbb{Q}$, then find the Galois group $Gal(K/\mathbb{Q}).$

If $3a=1+\sqrt 2$, what is the largest integer not exceeding $9a^4-6a^3+8a^2-6a+9$? $ \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 9 \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ \text{None of the preceding} $

Given distinct real numbers $a_1,a_2,...,a_n$, find the minimum value of the function $$y = |x-a_1|+|x-a_2|+...+|x-a_n|, \,\,\, x \in R.$$

Find all real parameters $q$ for which there is a $p\in[0,1]$ such that the equation $$x^4+2px^3+(2p^2-p)x^2+(p-1)p^2x+q=0$$has four real roots.

[u]Round 1[/u] [b]p1.[/b] This year, the Mathathon consists of $7$ rounds, each with $3$ problems. Another math test, Aspartaime, consists of $3$ rounds, each with $5$ problems. How many more problems are on the Mathathon than on Aspartaime? [b]p2.[/b] Let the solutions to $x^3 + 7x^2 - 242x - 2016 = 0 $be $a, b$, and $c$. Find $a^2 + b^2 + c^2$. (You might find it helpful to know that the roots are all rational.) [b]p3.[/b] For triangle $ABC$, you are given $AB = 8$ and $\angle A = 30^o$ . You are told that $BC$ will be chosen from amongst the integers from $1$ to $10$, inclusive, each with equal probability. What is the probability that once the side length $BC$ is chosen there is exactly one possible triangle $ABC$? [u]Round 2 [/u] [b]p4.[/b] It’s raining! You want to keep your cat warm and dry, so you want to put socks, rain boots, and plastic bags on your cat’s four paws. Note that for each paw, you must put the sock on before the boot, and the boot before the plastic bag. Also, the items on one paw do not affect the items you can put on another paw. How many different orders are there for you to put all twelve items of rain footwear on your cat? [b]p5.[/b] Let $a$ be the square root of the least positive multiple of $2016$ that is a square. Let $b$ be the cube root of the least positive multiple of $2016$ that is a cube. What is $ a - b$? [b]p6.[/b] Hypersomnia Cookies sells cookies in boxes of $6, 9$ or $10$. You can only buy cookies in whole boxes. What is the largest number of cookies you cannot exactly buy? (For example, you couldn’t buy $8$ cookies.) [u]Round 3 [/u] [b]p7.[/b] There is a store that sells each of the $26$ letters. All letters of the same type cost the same amount (i.e. any ‘a’ costs the same as any other ‘a’), but different letters may or may not cost different amounts. For example, the cost of spelling “trade” is the same as the cost of spelling “tread,” even though the cost of using a ‘t’ may be different from the cost of an ‘r.’ If the letters to spell out $1$ cost $\$1001$, the letters to spell out $2$ cost $\$1010$, and the letters to spell out $11$ cost $\$2015$, how much do the letters to spell out $12$ cost? [b]p8.[/b] There is a square $ABCD$ with a point $P$ inside. Given that $PA = 6$, $PB = 9$, $PC = 8$. Calculate $PD$. [b]p9.[/b] How many ordered pairs of positive integers $(x, y)$ are solutions to $x^2 - y^2 = 2016$? [u]Round 4 [/u] [b]p10.[/b] Given a triangle with side lengths $5, 6$ and $7$, calculate the sum of the three heights of the triangle. [b]p11. [/b]There are $6$ people in a room. Each person simultaneously points at a random person in the room that is not him/herself. What is the probability that each person is pointing at someone who is pointing back to them? [b]p12.[/b] Find all $x$ such that $\sum_{i=0}^{\infty} ix^i =\frac34$. PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2782837p24446063]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine all positive real numbers $x$ for which $$\left [x\right ]+\left [\sqrt{1996x}\right ]=1996$$ is verified Clarification:The brackets indicate the integer part of the number they enclose.

For a positive integer $n$ consider any partition of the set $\{ 1,2,\ldots ,2n \}$ into $n$ two-element subsets $P_1,P_2\ldots,P_n$. In each subset $P_i$, let $p_i$ be the product of the two numbers in $P_i$. Prove that \[\frac{1}{p_1}+\frac{1}{p_2}+\ldots + \frac{1}{p_n}<1 \]

Let $p, q, r$ be positive real numbers, not all equal, such that some two of the equations \begin{eqnarray*} px^2 + 2qx + r &=& 0 \\ qx^2 + 2rx + p &=& 0 \\ rx^2 + 2px + q &=& 0 . \\ \end{eqnarray*} have a common root, say $\alpha$. Prove that $a)$ $\alpha$ is real and negative; $b)$ the remaining third quadratic equation has non-real roots.

The function $f: \mathbb{R}\rightarrow\mathbb{R}$ satisfies $f(\textrm{cot}x)=\sin2x+\cos2x$, for any $x\in(0,\pi)$. Find the minimum and maximum value of $g: [-1;1]\rightarrow\mathbb{R}$, $g(x)=f(x)\cdot f(1-x)$.

Find the smallest possible value of the expression \[\left\lfloor\frac{a+b+c}{d}\right\rfloor+\left\lfloor\frac{b+c+d}{a}\right\rfloor+\left\lfloor\frac{c+d+a}{b}\right\rfloor+\left\lfloor\frac{d+a+b}{c}\right\rfloor\] in which $a,~ b,~ c$, and $d$ vary over the set of positive integers. (Here $\lfloor x\rfloor$ denotes the biggest integer which is smaller than or equal to $x$.)

Show that there are infintely many pairs $(a,b)$ of relatively prime integers (not necessarily positive) such that both the equations \begin{eqnarray*} x^2 +ax +b &=& 0 \\ x^2 + 2ax + b &=& 0 \\ \end{eqnarray*} have integer roots.

Let $k > 1{}$ be a real number, $n\geqslant 3$ be an integer, and $x_1 \geqslant x_2\geqslant\cdots\geqslant x_n$ be positive real numbers. Prove that \[\frac{x_1+kx_2}{x_2+x_3}+\frac{x_2+kx_3}{x_3+x_4}+\cdots+\frac{x_n+kx_1}{x_1+x_2}\geqslant\frac{n(k+1)}{2}.\][i]Ilija Jovcheski[/i]

Do there exist polynomials $p(x), q(x), r(x)$ whose coefficients are positive integers such that $p(x) = (x^{2}-3x+3) q(x)$ and $q(x) = (\frac{x^{2}}{20}-\frac{x}{15}+\frac{1}{12}) r(x)$?

Let $a, b, c$ be real numbers such that $a + b + c = 0$. Given that $a^3 + b^3 + c^3 \neq 0$, $a^2 + b^2 + c^2 \neq 0$, determine all possible values for: $$\frac{a^5 + b^5 + c^5}{(a^3 + b^3 + c^3)(a^2 + b^2 + c^2)}$$

For some real number $r$, the polynomial $8x^3-4x^2-42x+45$ is divisible by $(x-r)^2$. Which of the following numbers is closest to $r$? $\text{(A)} \ 1.22 \qquad \text{(B)} \ 1.32 \qquad \text{(C)} \ 1.42 \qquad \text{(D)} \ 1.52 \qquad \text{(E)} \ 1.62$

Consider those functions $ f: \mathbb{N} \mapsto \mathbb{N}$ which satisfy the condition \[ f(m \plus{} n) \geq f(m) \plus{} f(f(n)) \minus{} 1 \] for all $ m,n \in \mathbb{N}.$ Find all possible values of $ f(2007).$ [i]Author: Nikolai Nikolov, Bulgaria[/i]

Find all real numbers $p$ such that the equation $$\sqrt{x^2-5p^2}=px-1$$ has a root $x=3$. Then, solve the equation for the determined values of $p$.

Let $P(x)$ be a polynomial with degree $n$ and real coefficients. For all $0 \le y \le 1$ holds $\mid p(y) \mid \le 1$. Prove that $p(-\frac{1}{n}) \le 2^{n+1} -1$

Solve over the real numbers the equation $$3^{\log_5(5x-10)}-2=5^{-1+\log_3x}.$$

On a blackboard the product $log_{( )}[ ]\times\dots\times log_{( )}[ ]$ is written (there are 50 logarithms in the product). Donald has $100$ cards: $[2], [3],\dots, [51]$ and $(52),\dots,(101)$. He is replacing each $()$ with some card of form $(x)$ and each $[]$ with some card of form $[y]$. Find the difference between largest and smallest values Donald can achieve.