For any positive integer $n$, we define $$S_n=\sum_{k=1}^n \frac{2^k}{k^2}.$$ Prove that there are no polynomials $P,Q$ with real coefficients such that for any positive integer $n$, we have $\frac{S_{n+1}}{S_n}=\frac{P(n)}{Q(n)}$.

Determine all functions $f:\mathbb{N}\to \mathbb{N}$ such that for all $a, b \in \mathbb{N}$ the following conditions hold: $(i)$ $f(f(a)+b) \mid b^a-1$; $(ii)$ $f(f(a))\geq f(a)-1$.

Let $P(x)=ax^3+bx^2+cx+d$ be a polynomial with real coefficients such that \[\min\{d,b+d\}> \max\{|{c}|,|{a+c}|\}\] Prove that $P(x)$ do not have a real root in $[-1,1]$.

The distance between cities $A$ and $B$ is $30$ km. A bus departed from $A$, which makes a stop every $5$ km for $2$ minutes. The bus moves between stops at a speed of $80$ km / h. Simultaneously with the departure of the bus from $A$, a cyclist leaves $B$ to meet it, traveling at a speed of $27$ km / h. How far from $A$ will the cyclist meet the bus?

Shirley has a magical machine. If she inputs a positive even integer $n$, the machine will output $n/2$, but if she inputs a positive odd integer $m$, the machine will output $m+3$. The machine keeps going by automatically using its output as a new input, stopping immediately before it obtains a number already processed. Shirley wants to create the longest possible output sequence possible with initial input at most $100$. What number should she input?

Let $W,A,B$ be fixed real numbers with $W>0$. Prove that the following statements are equivalent. [list] [*] For all $x, y, z\ge 0$ satisfying $x+y\le z+W, x+z\le y+W, y+z\le x+W$ we have $Axyz+B\ge x^2+y^2+z^2$. [*] $B\ge W^2$ and $AW^3+B\ge 3W^2$. [/list] [i]Proposed by Ákos Somogyi, London[/i]

Let $ a,b,c $ be three positive numbers. Show that the equation $$ a^x+b^x=c^x $$ has, at most, one real solution.

Let \[f = X^{n}+a_{n-1}X^{n-1}+\ldots+a_{1}X+a_{0}\] be an integer polynomial of degree $n \geq 3$ such that $a_{k}+a_{n-k}$ is even for all $k \in \overline{1,n-1}$ and $a_{0}$ is even. Suppose that $f = gh$, where $g,h$ are integer polynomials and $\deg g \leq \deg h$ and all the coefficients of $h$ are odd. Prove that $f$ has an integer root.

Find all real values of the parameter $a$ for which the system of equations \[x^4 = yz - x^2 + a,\] \[y^4 = zx - y^2 + a,\] \[z^4 = xy - z^2 + a,\] has at most one real solution.

Let $\dots, a_{-1}, a_0, a_1, a_2, \dots$ be a sequence of positive integers satisfying the folloring relations: $a_n = 0$ for $n < 0$, $a_0 = 1$, and for $n \ge 1$, \[a_n = a_{n - 1} + 2(n - 1)a_{n - 2} + 9(n - 1)(n - 2)a_{n - 3} + 8(n - 1)(n - 2)(n - 3)a_{n - 4}.\] Compute \[\sum_{n \ge 0} \frac{10^n a_n}{n!}.\]

[b]p1.[/b] Let $a_0, a_1,...,a_n$ be such that $a_n \ne 0$ and $$(1 + x + x^3)^{341}(1 + 2x + x^2 + 2x^3 + 2x^4 + x^6)^{342} =\sum^n_{i=0}a_ix^i,$$ Find the number of odd numbers in the sequence a0; a1; : : : an. [b]p2.[/b] Let $F_0 = 1$, $F_1 = 1$ and F$_k = F_{k-1} + F_{k-2}$. Let $P(x) =\sum^{99}_{k=0} x^{F_k}$ . The remainder when $P(x)$ is divided by $x^3 - 1$ can be expressed as $ax^2 + bx + c$. Find $2a + b$. [b]p3.[/b] Let $a_n$ be the number of permutations of the numbers $S = \{1, 2,...,n\}$ such that for all $k$ with $1 \le k \le n$, the sum of $k$ and the number in the $k$th position of the permutation is a power of $2$. Compute $a_{2^0} + a_{2^1} +... + a_{2^{20}}$ . [b]p4.[/b] Three identical balls are painted white and black, so that half of each sphere is a white hemisphere, and the other half is a black one. The three balls are placed on a plane surface, each with a random orientation, so that each ball has a point of contact with the other two. What is the probability that at at least one point of contact between two of the balls, both balls are the same color? [b]p5.[/b] Compute the greatest positive integer $n$ such that there exists an odd integer $a$, for which $\frac{a^{2^n}-1}{4^{4^4}}$ is not an integer. [b]p6.[/b] You are blind and cannot feel the difference between a coin that is heads up or tails up. There are $100$ coins in front of you and are told that exactly $10$ of them are heads up. On the back of this paper, explain how you can split the otherwise indistinguishable coins into two groups so that both groups have the same number of heads. [b]p7.[/b] On the back of this page, write the best math pun you can think of. You’ll get a point if we chuckle. [b]p8.[/b] Pick an integer between $1$ and $10$. If you pick $k$, and $n$ total teams pick $k$, then you’ll receive $\frac{k}{10n}$ points. [b]p9.[/b] There are four prisoners in a dungeon. Tomorrow, they will be separated into a group of three in one room, and the other in a room by himself. Each will be given a hat to wear that is either black or white – two will be given white and two black. None of them will be able to communicate with each other and none will see his or her own hat color. The group of three is lined up, so that the one in the back can see the other two, the second can see the first, but the first cannot see the others. If anyone is certain of their hat color, then they immediately shout that they know it to the rest of the group. If they can secretly prove it to the guard, they are saved. They only say something if they’re sure. Which person is sure to survive? [b]p10.[/b] Down the road, there are $10$ prisoners in a dungeon. Tomorrow they will be lined up in a single room and each given a black or white hat – this time they don’t know how many of each. The person in the back can see everyone’s hat besides his own, and similarly everyone else can only see the hats of the people in front of them. The person in the back will shout out a guess for his hat color and will be saved if and only if he is right. Then the person in front of him will have to guess, and this will continue until everyone has the opportunity to be saved. Each person can only say his or her guess of “white” or “black” when their turn comes, and no other signals may be made. If they have the night before receiving the hats to try to devise some sort of code, how many people at a minimum can be saved with the most optimal code? Describe the code on the back of this paper for full points. [b]p11.[/b] A few of the problems on this mixer contest were taken from last year’s event. One of them had fewer than $5$ correct answers, and most of the answers given were the same incorrect answer. Half a point will be given if you can guess the number of the problem on this test that corresponds to last year’s question, and another $.5$ points will be given if you can guess the very common incorrect answer. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a_1,a_2,\ldots,a_n$ ($n\geq 2$) be nonnegative real numbers whose sum is $\frac{n}{2}$. For every $i=1,\ldots,n$ define $$b_i=a_i+a_ia_{i+1}+a_ia_{i+1}a_{i+2}+\cdots+ a_ia_{i+1}\cdots a_{i+n-2}+2a_ia_{i+1}\cdots a_{i+n-1}$$ where $a_{j+n}=a_j$ for every $j$. Prove that $b_i\geq 1$ holds for at least one index $i$.

Zeros of a fourth-degree polynomial $f (x)$ form an arithmetic progression. Prove that the zeros of $f '(x)$ also form an arithmetic progression.

Find the complex numbers $x,y,z$,with $\mid x\mid=\mid y\mid=\mid z\mid$,knowing that $x+y+z$ and $x^{3}+y^{3}+z^{3}$ are be real numbers.

In the coordinate plane,the graps of functions $y=sin x$ and $y=tan x$ are drawn, along with the coordinate axes. Using compass and ruler, construct a line tangent to the graph of sine at a point above the axis, $Ox$, as well at a point below that axis (the line can also meet the graph at several other points)

Find all polynomials $P$ such that $P(x) + \binom{2018}{2}P(x+2)+...+\binom{2018}{2106}P(x+2016)+P(x+2018)=$ $=\binom{2018}{1}P(x+1)+\binom{2018}{3}P(x+3)+...+\binom{2018}{2105}P(x+2015)+\binom{2018}{2107}P(x+2017)$ for all real numbers $x$.

Does there exist a sequence $ F(1), F(2), F(3), \ldots$ of non-negative integers that simultaneously satisfies the following three conditions? [b](a)[/b] Each of the integers $ 0, 1, 2, \ldots$ occurs in the sequence. [b](b)[/b] Each positive integer occurs in the sequence infinitely often. [b](c)[/b] For any $ n \geq 2,$ \[ F(F(n^{163})) \equal{} F(F(n)) \plus{} F(F(361)). \]

Let $p$ be a prime number. Find all polynomials $P$ with integer coefficients with the following properties: $(a)$ $P(x)>x$ for all positive integers $x$. $(b)$ The sequence defined by $p_0:=p$, $p_{n+1}:=P(p_n)$ for all positive integers $n$, satisfies the property that for all positive integers $m$ there exists some $l\geq 0$ such that $m\mid p_l$.

[b]p1. [/b]The longest diagonal of a regular hexagon is 12 inches long. What is the area of the hexagon, in square inches? [b]p2.[/b] When Al and Bob play a game, either Al wins, Bob wins, or they tie. The probability that Al does not win is $\frac23$ , and the probability that Bob does not win is $\frac34$ . What is the probability that they tie? [b]p3.[/b] Find the sum of the $a + b$ values over all pairs of integers $(a, b)$ such that $1 \le a < b \le 10$. That is, compute the sum $$(1 + 2) + (1 + 3) + (1 + 4) + ...+ (2 + 3) + (2 + 4) + ...+ (9 + 10).$$ [b]p4.[/b] A $3 \times 11$ cm rectangular box has one vertex at the origin, and the other vertices are above the $x$-axis. Its sides lie on the lines $y = x$ and $y = -x$. What is the $y$-coordinate of the highest point on the box, in centimeters? [b]p5.[/b] Six blocks are stacked on top of each other to create a pyramid, as shown below. Carl removes blocks one at a time from the pyramid, until all the blocks have been removed. He never removes a block until all the blocks that rest on top of it have been removed. In how many different orders can Carl remove the blocks? [img]https://cdn.artofproblemsolving.com/attachments/b/e/9694d92eeb70b4066b1717fedfbfc601631ced.png[/img] [b]p6.[/b] Suppose that a right triangle has sides of lengths $\sqrt{a + b\sqrt{3}}$,$\sqrt{3 + 2\sqrt{3}}$, and $\sqrt{4 + 5\sqrt{3}}$, where $a, b$ are positive integers. Find all possible ordered pairs $(a, b)$. [b]p7.[/b] Farmer Chong Gu glues together $4$ equilateral triangles of side length $ 1$ such that their edges coincide. He then drives in a stake at each vertex of the original triangles and puts a rubber band around all the stakes. Find the minimum possible length of the rubber band. [b]p8.[/b] Compute the number of ordered pairs $(a, b)$ of positive integers less than or equal to $100$, such that a $b -1$ is a multiple of $4$. [b]p9.[/b] In triangle $ABC$, $\angle C = 90^o$. Point $P$ lies on segment $BC$ and is not $B$ or $C$. Point $I$ lies on segment $AP$. If $\angle BIP = \angle PBI = \angle CAB = m^o$ for some positive integer $m$, find the sum of all possible values of $m$. [b]p10.[/b] Bob has $2$ identical red coins and $2$ identical blue coins, as well as $4$ distinguishable buckets. He places some, but not necessarily all, of the coins into the buckets such that no bucket contains two coins of the same color, and at least one bucket is not empty. In how many ways can he do this? [b]p11.[/b] Albert takes a $4 \times 4$ checkerboard and paints all the squares white. Afterward, he wants to paint some of the square black, such that each square shares an edge with an odd number of black squares. Help him out by drawing one possible configuration in which this holds. (Note: the answer sheet contains a $4 \times 4$ grid.) [b]p12.[/b] Let $S$ be the set of points $(x, y)$ with $0 \le x \le 5$, $0 \le y \le 5$ where $x$ and $y$ are integers. Let $T$ be the set of all points in the plane that are the midpoints of two distinct points in $S$. Let $U$ be the set of all points in the plane that are the midpoints of two distinct points in $T$. How many distinct points are in $U$? (Note: The points in $T$ and $U$ do not necessarily have integer coordinates.) [b]p13.[/b] In how many ways can one express $6036$ as the sum of at least two (not necessarily positive) consecutive integers? [b]p14.[/b] Let $a, b, c, d, e, f$ be integers (not necessarily distinct) between $-100$ and $100$, inclusive, such that $a + b + c + d + e + f = 100$. Let $M$ and $m$ be the maximum and minimum possible values, respectively, of $$abc + bcd + cde + def + ef a + f ab + ace + bdf.$$ Find $\frac{M}{m}$. [b]p15.[/b] In quadrilateral $ABCD$, diagonal $AC$ bisects diagonal $BD$. Given that $AB = 20$, $BC = 15$, $CD = 13$, $AC = 25$, find $DA$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On a chessboard a square is chosen . The sum of the squares of distances from its centre to the centre of all black squares is designated by $a$ and to the centre of all white squares by $b$. Prove that $a = b$. (A. Andj ans, Riga)

The sequence $x_1,x_2, ...$ is defined by the following equations: $$x_1=19, \ \ x_2=97, \ \ x_{n+2} =x_n - \frac{1}{x_{n+1}}$$ for $n \ge 1$. Prove that there exists a positive integer $k$ such that $x_k=0$ and find $k$. (A Berzinsh)

Consider two functions $f , \,g \,:\mathbb{R} \to \mathbb{R}$ such that from some $a>0$ holds $g(x)=f(x+a)$ for any $x \in \mathbb{R}$. If $f$ is even and $g$ is odd, prove that both functions are periodic.

Let $A=(a_{jk})$ be a $10\times 10$ array of positive real numbers such that the sum of numbers in row as well as in each column is $1$. Show that there exists $j<k$ and $l<m$ such that \[a_{jl}a_{km}+a_{jm}a_{kl}\ge \frac{1}{50}\]

Find all finite sets $M \subset R, |M| \ge 2$, satisfying the following condition: [i]for all $a, b \in M, a \ne b$, the number $a^3 - \frac{4}{9}b$ also belongs to $M$. [/i] (I. Voronovich)

Let $a, b, c$ be the side lengths of a triangle. Prove that $2 (a^3 + b^3 + c^3) < (a + b + c) (a^2 + b^2 + c^2) \le 3 (a^3 + b^3 + c^3)$