A polynomial $p(x) = \sum_{j=1}^{2n-1} a_j x^j$ with real coefficients is called [i]mountainous[/i] if $n \ge 2$ and there exists a real number such that the polynomial's coefficients satisfy $a_1=1, a_{j+1}-a_j=k$ for $1 \le j \le n-1,$ and $a_{j+1}-a_j=-k$ for $n \le j \le 2n-2;$ we call $k$ the [i]step size[/i] of $p(x).$ A real number $k$ is called [i]good[/i] if there exists a mountainous polynomial $p(x)$ with step size $k$ such that $p(-3)=0.$ Let $S$ be the sum of all good numbers $k$ satisfying $k \ge 5$ or $k \le 3.$ If $S=\tfrac{b}{c}$ for relatively prime positive integers $b,c,$ find $b+c.$

Let $ X$ be the set of all positive integers greater than or equal to $ 8$ and let $ f: X\rightarrow X$ be a function such that $ f(x\plus{}y)\equal{}f(xy)$ for all $ x\ge 4, y\ge 4 .$ if $ f(8)\equal{}9$, determine $ f(9) .$

Polynomials $u_i(x) = a_ix+b_i$ ($a_i,b_i \in R$, $ i = 1,2,3$) satisfy $$u_1(x)^n +u_2(x)^n = u_3(x)^n$$ for some integer $n \ge 2.$ Prove that there exist real numbers $A$,$B$,$c_1$,$c_2$,$c_3$ such that $u_i(x) = c_i(Ax+B)$ for $i = 1,2,3$.

Let $ p(x)\equal{}a_0 \plus{}a_1 x\plus{}...\plus{}a_n x^n$ be a polynomial with nonnegative real coefficients. Suppose that $ p(4)\equal{}2$ and $ p(16)\equal{}8$. Prove that $ p(8) \le 4$ and find all such $ p$ with $ p(8)\equal{}4$.

Let $r, s, t, u$ be the distinct roots of the polynomial $x^4 + 2x^3 + 3x^2 + 3x + 5$. For $n \ge 1$, define $s_n = r^n + s^n + t^n + u^n$ and $t_n = s_1 + s_2 + ...+ s_n$. Compute $t_4 + 2t_3 + 3t_2 + 3t_1 + 5$.

Find a non-zero polynomial $f(x, y)$ such that $f(\lfloor 3t \rfloor, \lfloor 5t \rfloor) = 0$ for all real numbers $t$.

Determine all pairs of positive real numbers $(a, b)$ with $a > b$ that satisfy the following equations: $a\sqrt{a}+ b\sqrt{b} = 134$ and $a\sqrt{b}+ b\sqrt{a} = 126$.

[b]p1.[/b] Twelve people, three of whom are in the Mafia and one of whom is a police inspector, randomly sit around a circular table. What is the probability that the inspector ends up sitting next to at least one of the Mafia? [b]p2.[/b] Of the positive integers between $1$ and $1000$, inclusive, how many of them contain neither the digit “$4$” nor the digit “$7$”? [b]p3.[/b] You are really bored one day and decide to invent a variation of chess. In your variation, you create a new piece called the “krook,” which, on any given turn, can move either one square up or down, or one square left or right. If you have a krook at the bottom-left corner of the chessboard, how many different ways can the krook reach the top-right corner of the chessboard in exactly $17$ moves? [b]p4.[/b] Let $p$ be a prime number. What is the smallest positive integer that has exactly $p$ different positive integer divisors? Write your answer as a formula in terms of $p$. [b]p5.[/b] You make the square $\{(x, y)| - 5 \le x \le 5, -5 \le y \le 5\}$ into a dartboard as follows: (i) If a player throws a dart and its distance from the origin is less than one unit, then the player gets $10$ points. (ii) If a player throws a dart and its distance from the origin is between one and three units, inclusive, then the player gets awarded a number of points equal to the number of the quadrant that the dart landed on. (The player receives no points for a dart that lands on the coordinate axes in this case.) (iii) If a player throws a dart and its distance from the origin is greater than three units, then the player gets $0$ points. If a person throws three darts and each hits the board randomly (i.e with uniform distribution), what is the expected value of the score that they will receive? [b]p6.[/b] Teddy works at Please Forget Meat, a contemporary vegetarian pizza chain in the city of Gridtown, as a deliveryman. Please Forget Meat (PFM) has two convenient locations, marked with “$X$” and “$Y$ ” on the street map of Gridtown shown below. Teddy, who is currently at $X$, needs to deliver an eggplant pizza to $\nabla$ en route to $Y$ , where he is urgently needed. There is currently construction taking place at $A$, $B$, and $C$, so those three intersections will be completely impassable. How many ways can Teddy get from $X$ to $Y$ while staying on the roads (Traffic tickets are expensive!), not taking paths that are longer than necessary (Gas is expensive!), and that let him pass through $\nabla$ (Losing a job is expensive!)? [img]https://cdn.artofproblemsolving.com/attachments/e/0/d4952e923dc97596ad354ed770e80f979740bc.png[/img] [b]p7.[/b] $x, y$, and $z$ are positive real numbers that satisfy the following three equations: $$x +\frac{1}{y}= 4 \,\,\,\,\, y +\frac{1}{z}= 1\,\,\,\,\, z +\frac{1}{x}=\frac73.$$ Compute $xyz$. [b]p8.[/b] Alan, Ben, and Catherine will all start working at the Duke University Math Department on January $1$st, $2009$. Alan’s work schedule is on a four-day cycle; he starts by working for three days and then takes one day off. Ben’s work schedule is on a seven-day cycle; he starts by working for five days and then takes two days off. Catherine’s work schedule is on a ten-day cycle; she starts by working for seven days and then takes three days off. On how many days in $2009$ will none of the three be working? [b]p9.[/b] $x$ and $y$ are complex numbers such that $x^3 + y^3 = -16$ and $(x + y)^2 = xy$. What is the value of $|x + y|$? [b]p10.[/b] Call a four-digit number “well-meaning” if (1) its second digit is the mean of its first and its third digits and (2) its third digit is the mean of its second and fourth digits. How many well-meaning four-digit numbers are there? (For a four-digit number, its first digit is its thousands [leftmost] digit and its fourth digit is its units [rightmost] digit. Also, four-digit numbers cannot have “$0$” as their first digit.) [b]p11.[/b] Suppose that $\theta$ is a real number such that $\sum^{\infty}{k=2} \sin \left(2^k\theta \right)$ is well-defined and equal to the real number $a$. Compute: $$\sum^{\infty}{k=0} \left(\cot^3 \left(2^k\theta \right)-\cot \left(2^k\theta \right) \right) \sin^4 \left(2^k\theta \right).$$ Write your answer as a formula in terms of $a$. [b]p12.[/b] You have $13$ loaded coins; the probability that they come up as heads are $\cos\left( \frac{0\pi}{24 }\right)$,$ \cos\left( \frac{1\pi}{24 }\right)$, $\cos\left( \frac{2\pi}{24 }\right)$, $...$, $\cos\left( \frac{11\pi}{24 }\right)$ and $\cos\left( \frac{12\pi}{24 }\right)$, respectively. You throw all $13$ of these coins in the air at once. What is the probability that an even number of them come up as heads? [b]p13.[/b] Three married couples sit down on a long bench together in random order. What is the probability that none of the husbands sit next to their respective wives? [b]p14.[/b] What is the smallest positive integer that has at least $25$ different positive divisors? [b]p15.[/b] Let $A_1$ be any three-element set, $A_2 = \{\emptyset\}$, and $A_3 = \emptyset$. For each $i \in \{1, 2, 3\}$, let: (i) $B_i = \{\emptyset,A_i\}$, (ii) $C_i$ be the set of all subsets of $B_i$, (iii) $D_i = B_i \cup C_i$, and (iv) $k_i$ be the number of different elements in $D_i$. Compute $k_1k_2k_3$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Show that \[ \cos x^2 + \cos y^2 - \cos xy < 3 \] for reals $x$, $y$.

Let $x, y, z$ be real numbers from interval $(0, 1)$. Prove that $$\frac{1}{x(1-y)}+\frac{1}{y(1-x)}+\frac{1}{z(1-x)}\ge \frac{3}{xyz+(1-x)(1-y)(1-z)}$$

Write down all real numbers $(x, y)$ satisfying two conditions: $x^{2018} + y^2 = 2$, and $x^2 + y^{2018} = 2$.

Let $\mathcal{M}=\mathbb{Q}[x,y,z]$ be the set of three-variable polynomials with rational coefficients. Prove that for any non-zero polynomial $P\in \mathcal{M}$ there exists non-zero polynomials $Q,R\in \mathcal{M}$ such that \[ R(x^2y,y^2z,z^2x) = P(x,y,z)Q(x,y,z). \]

Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?

a) The real number $a$ and the continuous function $f : [a, \infty) \rightarrow [a, \infty)$ are such that $|f(x)-f(y)| < |x–y|$ for every two different $x, y \in [a, \infty)$. Is it always true that the equation $f(x)=x$ has only one solution in the interval $[a, \infty)$? b) The real numbers $a$ and $b$ and the continuous function $f : [a, b] \rightarrow [a, b]$ are such that $|f(x)-f(y)| < |x–y|$, for every two different $x, y \in [a, b]$. Is it always true that the equation $f(x)=x$ has only one solution in the interval $[a, b]$?

[b]p1.[/b] (a) Put the numbers $1$ to $6$ on the circle in such way that for any five consecutive numbers the sum of first three (clockwise) is larger than the sum of remaining two. (b) Can you arrange these numbers so it works both clockwise and counterclockwise. [b]p2.[/b] A girl has a box with $1000$ candies. Outside the box there is an infinite number of chocolates and muffins. A girl may replace: $\bullet$ two candies in the box with one chocolate bar, $\bullet$ two muffins in the box with one chocolate bar, $\bullet$ two chocolate bars in the box with one candy and one muffin, $\bullet$ one candy and one chocolate bar in the box with one muffin, $\bullet$ one muffin and one chocolate bar in the box with one candy. Is it possible that after some time it remains only one object in the box? [b]p3.[/b] Find any integer solution of the puzzle: $WE+ST+RO+NG=128$ (different letters mean different digits between $1$ and $9$). [b]p4.[/b] Two consecutive three‐digit positive integer numbers are written one after the other one. Show that the six‐digit number that is obtained is not divisible by $1001$. [b]p5.[/b] There are $9$ straight lines drawn in the plane. Some of them are parallel some of them intersect each other. No three lines do intersect at one point. Is it possible to have exactly $17$ intersection points? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a_1,a_2,\ldots,a_n$ be non-negative real numbers ,$S_k= \sum\limits_{i=1}^{k}a_i $ $(1\le k\le n)$.Prove that$$\sum\limits_{i=1}^{n}\left(a_iS_i\sum\limits_{j=i}^{n}a^2_j\right)\le \sum\limits_{i=1}^{n}\left(a_iS_i\right)^2$$

Let the function $f:R \to R$ satisfies the following conditions: 1) for all $x, y\in R$, $ f(x +y) = f(x) +f(y)$ 2)$ f(1)=1$ 3) for all $x \ne 0$ , $ f(1/x) =\frac{f(x)}{x^2}$ Prove that for all $x \in R$, $f(x) = x$.

Let \[ f(x_1,x_2,x_3,x_4,x_5,x_6,x_7)=x_1x_2x_4+x_2x_3x_5+x_3x_4x_6+x_4x_5x_7+x_5x_6x_1+x_6x_7x_2+x_7x_1x_3 \] be defined for non-negative real numbers $x_1,x_2,\dots,x_7$ with sum $1$. Prove that $f(x_1,x_2,\dots,x_7)$ has a maximum value and find that value.

Find the maximum value of the positive real number $k$ such that the inequality $$\frac{1}{kab+c^2} +\frac{1} {kbc+a^2} +\frac{1} {kca+b^2} \geq \frac{k+3}{a^2+b^2+c^2} $$holds for all positive real numbers $a,b,c$ such that $a^2+b^2+c^2=2(ab+bc+ca).$

Given the positive numbers $x_1, x_2,..., x_n$, such that $x_i \le 2x_j$ with $1 \le i < j \le n$. Prove that there are positive numbers $y_1\le y_2\le...\le y_n$, such that $x_k \le y_k \le 2x_k$ for all $k=1,2,..., n$

Let $a_1,\ldots,a_8$ be reals, not all equal to zero. Let \[ c_n = \sum^8_{k=1} a^n_k\] for $n=1,2,3,\ldots$. Given that among the numbers of the sequence $(c_n)$, there are infinitely many equal to zero, determine all the values of $n$ for which $c_n = 0.$

Let $a, b, c \in [0;1]$ be reals such that $ab+bc+ca=1$. Find the minimal and maximal value of $a^3+b^3+c^3$.

Sea $f: R^+ \to R$ defined as $$f (x) = \frac{1}{\sqrt[3]{x^2 + 6x + 9} + \sqrt[3]{x^2 + 4x + 3} + \sqrt[3]{x^2 + 2x + 1}}$$ Calculate $$f (1) + f (2) + f (3) + ... + f (2016).$$

Let $x_1,x_2,\cdots,x_n \in(0,1)$ , $n\geq2$. Prove that$$\frac{\sqrt{1-x_1}}{x_1}+\frac{\sqrt{1-x_2}}{x_2}+\cdots+\frac{\sqrt{1-x_n}}{x_n}<\frac{\sqrt{n-1}}{x_1 x_2 \cdots x_n}.$$

For a positive integer $k$, define the $k$-[i]pop[/i] of a positive integer $n$ as the infinite sequence of integers $a_1, a_2, ...$ such that $a_1 = n$ and $$a_{i+1}= \left\lfloor \frac{a_i}{k} \right\rfloor , i = 1, 2, ..$$ where $ \lfloor x\rfloor $ denotes the greatest integer less than or equal to $x$. Furthermore, define a positive integer $m$ to be $k$-[i]pop avoiding[/i] if $k$ does not divide any nonzero term in the $k$-pop of $m$. For example, $14$ is 3-pop avoiding because $3$ does not divide any nonzero term in the $3$-pop of $14$, which is $14, 4, 1, 0, 0, ....$ Suppose that the number of positive integers less than $13^{2018}$ which are $13$-pop avoiding is equal to N. What is the remainder when $N$ is divided by $1000$?