Let $a,b,c$ be three integers for which the sum \[ \frac{ab}{c}+ \frac{ac}{b}+ \frac{bc}{a}\] is integer. Prove that each of the three numbers \[ \frac{ab}{c}, \quad \frac{ac}{b},\quad \frac{bc}{a}\] is integer. (Proposed by Gerhard J. Woeginger)

Let $P(x)$ be a polynomial of degree $n$ with real coefficients and let $a\geq 3$. Prove that \[\max_{0\leq j \leq n+1}\left | a^j-P(j) \right |\geq 1\]

[b](a)[/b] Find the rearrangement $\{a_1, \dots , a_n\}$ of $\{1, 2, \dots, n\}$ that maximizes \[a_1a_2 + a_2a_3 + \cdots + a_na_1 = Q.\] [b](b)[/b] Find the rearrangement that minimizes $Q.$

The polynomial $P$ is a quadratic with integer coefficients. For every positive integer $n$, the integers $P(n)$ and $P(P(n))$ are relatively prime to $n$. If $P(3) = 89$, what is the value of $P(10)$?

Find the value of the following expression: $2 + 33 + 6 + 35 + 10 + 37 + \ldots + 1194 + 629 + 1198 + 631$

Find the polynomials $ f(x)$ having the following properties: (i) $ f(0) \equal{} 1$, $ f'(0) \equal{} f''(0) \equal{} \cdots \equal{} f^{(n)}(0) \equal{} 0$ (ii) $ f(1) \equal{} f'(1) \equal{} f''(1) \equal{} \cdots \equal{} f^{(m)}(1) \equal{} 0$

Let $a$ and $b$ be two positive reals such that the following inequality \[ ax^3 + by^2 \geq xy - 1 \] is satisfied for any positive reals $x, y \geq 1$. Determine the smallest possible value of $a^2 + b$. [i]Proposed by Fajar Yuliawan[/i]

Consider the sequence $p_{1},p_{2},...$ of primes such that for each $i\geq2$, either $p_{i}=2p_{i-1}-1$ or $p_{i}=2p_{i-1}+1$. Show that any such sequence has a finite number of terms.

In a geometric sequence of real numbers, the sum of the first two terms is 7, and the sum of the first 6 terms is 91. The sum of the first 4 terms is $\text{(A)}\ 28 \qquad \text{(B)}\ 32 \qquad \text{(C)}\ 35 \qquad \text{(D)}\ 49 \qquad \text{(E)}\ 84$

Let $a_1,...,a_n$ be $n$ real numbers. If for each odd positive integer $k\leqslant n$ we have $a_1^k+a_2^k+\ldots+a_n^k=0$, then for each odd positive integer $k$ we have $a_1^k+a_2^k+\ldots+a_n^k=0$. [i]Proposed by M. Didin[/i]

Determine whether there exists a polynomial $f(x_1, x_2)$ with two variables, with integer coefficients, and two points $A=(a_1, a_2)$ and $B=(b_1, b_2)$ in the plane, satisfying the following conditions: (i) $A$ is an integer point (i.e $a_1$ and $a_2$ are integers); (ii) $|a_1-b_1|+|a_2-b_2|=2010$; (iii) $f(n_1, n_2)>f(a_1, a_2)$ for all integer points $(n_1, n_2)$ in the plane other than $A$; (iv) $f(x_1, x_2)>f(b_1, b_2)$ for all integer points $(x_1, x_2)$ in the plane other than $B$. [i]Massimo Gobbino, Italy[/i]

Let $a,b$ be real numbers ($b\ne 0$) and consider the infinite arithmetic sequence $a, a+b ,a +2b , \ldots.$ Show that this sequence contains an infinite geometric subsequence if and only if $\frac{a}{b}$ is rational.

Show that if $a \ge b \ge c \ge 0$ then $$a^2b(a - b) + b^2c(b - c) + c^2a(c - a) \ge 0.$$

Let $x_1, x_2, ... , x_n$ be non-negative numbers $n\ge3$ such that $x_1 + x_2 + ... + x_n = 1$. Determine the maximum possible value of the expression $x_1x_2 + x_2x_3 + ... + x_{n-1}x_n$.

[b]p1.[/b] Warren interrogates the $25$ members of his cabinet, each of whom always lies or always tells the truth. He asks them all, “How many of you always lie?” He receives every integer answer from $1$ to $25$ exactly once. Find the actual number of liars in his cabinet. [b]p2.[/b] Abraham thinks of distinct nonzero digits $E$, $M$, and $C$ such that $E +M = \overline{CC}$. Help him evaluate the sum of the two digit numbers $\overline{EC}$ and $\overline{MC}$. (Note that $\overline{CC}$, $\overline{EC}$, and $\overline{MC}$ are read as two-digit numbers.) [b]p3.[/b] Let $\omega$, $\Omega$, $\Gamma$ be concentric circles such that $\Gamma$ is inside $\Omega$ and $\Omega$ is inside $\omega$. Points $A,B,C$ on $\omega$ and $D,E$ on $\Omega$ are chosen such that line $AB$ is tangent to $\Omega$, line $AC$ is tangent to $\Gamma$, and line $DE$ is tangent to $\Gamma$. If $AB = 21$ and $AC = 29$, find $DE$. [b]p4.[/b] Let $a$, $b$, and $c$ be three prime numbers such that $a + b = c$. If the average of two of the three primes is four less than four times the fourth power of the last, find the second-largest of the three primes. [b]p5.[/b] At Stillwells Ice Cream, customers must choose one type of scoop and two different types of toppings. There are currently $630$ different combinations a customer could order. If another topping is added to the menu, there would be $840$ different combinations. If, instead, another type of scoop were added to the menu, compute the number of different combinations there would be. [b]p6.[/b] Eleanor the ant takes a path from $(0, 0)$ to $(20, 24)$, traveling either one unit right or one unit up each second. She records every lattice point she passes through, including the starting and ending point. If the sum of all the $x$-coordinates she records is $271$, compute the sum of all the $y$-coordinates. (A lattice point is a point with integer coordinates.) [b]p7.[/b] Teddy owns a square patch of desert. He builds a dam in a straight line across the square, splitting the square into two trapezoids. The perimeters of the trapezoids are$ 64$ miles and $76$ miles, and their areas differ by $135$ square miles. Find, in miles, the length of the segment that divides them. [b]p8.[/b] Michelle is playing Spot-It with a magical deck of $10$ cards. Each card has $10$ distinct symbols on it, and every pair of cards shares exactly $1$ symbol. Find the minimum number of distinct symbols on all of the cards in total. [b]p9.[/b] Define the function $f(n) = \frac{1}{2^n} + \frac{1}{3^n} + \frac{1}{4^n} + ...$ for integers $n \ge 2$. Find $$f(2) + f(4) + f(6) + ... .$$ [b]p10.[/b] There are $9$ indistinguishable ants standing on a $3\times 3$ square grid. Each ant is standing on exactly one square. Compute the number of different ways the ants can stand so that no column or row contains more than $3$ ants. [b]p11.[/b] Let $s(N)$ denote the sum of the digits of $N$. Compute the sum of all two-digit positive integers $N$ for which $s(N^2) = s(N)^2$. [b]p12.[/b] Martha has two square sheets of paper, $A$ and $B$. With each sheet, she repeats the following process four times: fold bottom side to top side, fold right side to left side. With sheet $A$, she then makes a cut from the top left corner to the bottom right. With sheet $B$, she makes a cut from the bottom left corner to the top right. Find the total number of pieces of paper yielded from sheets $A$ and sheets $B$. [img]https://cdn.artofproblemsolving.com/attachments/f/6/ff3a459a135562002aa2c95067f3f01441d626.png[/img] [b]p13.[/b] Let $x$ and $y$ be positive integers such that gcd $(x^y, y^x) = 2^{28}$. Find the sum of all possible values of min $(x, y)$. [b]p14.[/b] Convex hexagon $TRUMAN$ has opposite sides parallel. If each side has length $3$ and the area of this hexagon is $5$, compute $$TU \cdot RM \cdot UA \cdot MN \cdot AT \cdot NR.$$ [b]p15.[/b] Let $x$, $y$, and $z$ be positive real numbers satisfying the system $$\begin{cases} x^2 + xy + y^2 = 25\\ y^2 + yz + z^2 = 36 \\ z^2 + zx + x^2 = 49 \end{cases}$$ Compute $x^2 + y^2 + z^2$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.

Given complex numbers $a,b,c$, we have that $|az^2 + bz +c| \leq 1$ holds true for any complex number $z, |z| \leq 1$. Find the maximum value of $|bc|$.

Find all pair of constants $(a,b)$ such that there exists real-coefficient polynomial $p(x)$ and $q(x)$ that satisfies the condition below. [b]Condition[/b]: $\forall x\in \mathbb R,$ $ $ $p(x^2)q(x+1)-p(x+1)q(x^2)=x^2+ax+b$

$ c$ is a positive integer. Consider the following recursive sequence: $ a_1\equal{}c, a_{n\plus{}1}\equal{}ca_{n}\plus{}\sqrt{(c^2\minus{}1)(a_n^2\minus{}1)}$, for all $ n \in N$. Prove that all the terms of the sequence are positive integers.

Let $ \mathcal{H}_n$ be the set of all numbers of the form $ 2 \pm\sqrt{2 \pm\sqrt{2 \pm\ldots\pm\sqrt 2}}$ where "root signs" appear $ n$ times. (a) Prove that all the elements of $ \mathcal{H}_n$ are real. (b) Computer the product of the elements of $ \mathcal{H}_n$. (c) The elements of $ \mathcal{H}_{11}$ are arranged in a row, and are sorted by size in an ascending order. Find the position in that row, of the elements of $ \mathcal{H}_{11}$ that corresponds to the following combination of $ \pm$ signs: \[ \plus{}\plus{}\plus{}\plus{}\plus{}\minus{}\plus{}\plus{}\minus{}\plus{}\minus{}\]

How many real roots do $x^5 +3x^4 -4x^3 -8x^2 +6x-1$ and $x^5-3x^4 -2x^3 -3x^2 -6x+1$ share?

Let $p \geq 2$ be a natural number. Prove that there exist an integer $n_0$ such that \[ \sum^{n_0}_{i=1} \frac{1}{i \cdot \sqrt[p]{i + 1}} > p. \]

Find all functions $f:\mathbb{R}\backslash\{0\}\rightarrow\mathbb{R}$ for which $xf(xy) + f(-y) = xf(x)$ for all non-zero real numbers $x, y$.

Jack writes whole numbers starting from $ 1$ and skips all numbers that contain either a $2$ or $9$. What is the $100$th number that Jack writes down?

Let $a$ be a real number, such that $a\ne 0, a\ne 1, a\ne -1$ and $m,n,p,q$ be natural numbers . Prove that if $a^m+a^n=a^p+a^q$ and $a^{3m}+a^{3n}=a^{3p}+a^{3q}$ , then $m \cdot n = p \cdot q$.