The sum of $n > 2$ nonzero real numbers (not necessarily distinct) equals zero. For each of the $2^n - 1$ ways to choose one or more of these numbers, their sums are written in non-increasing order in a row. The first number in the row is $S$. Find the smallest possible value of the second number.

Given the equations (1) $ax^2 + bx + c = 0$ (2)$ -ax^2 + bx + c = 0$ prove that if $x_1$ and $x_2$ are some roots of equations (1) and (2), respectively, then there is a root $x_3$ of the equation $$\frac{a}{2}x^2 + bx + c = 0$$ such that either $x_1 \le x_3 \le x_2$ or $x_1 \ge x_3 \ge x_2$.

Let $S$ be a set of positive integers satisfying the following two conditions: • For each positive integer $n$, at least one of $n, 2n, \dots, 100n$ is in $S$. • If $a_1, a_2, b_1, b_2$ are positive integers such that $\gcd(a_1a_2, b_1b_2) = 1$ and $a_1b_1, a_2b_2 \in S,$ then $a_2b_1, a_1b_2 \in S.$ Suppose that $S$ has natural density $r$. Compute the minimum possible value of $\lfloor 10^5r\rfloor$. Note: $S$ has natural density $r$ if $\tfrac{1}{n}|S \cap {1, \dots, n}|$ approaches $r$ as $n$ approaches $\infty$.

Consider a matrix of size $n\times n$ whose entries are real numbers of absolute value not exceeding $1$. The sum of all entries of the matrix is $0$. Let $n$ be an even positive integer. Determine the least number $C$ such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding $C$ in absolute value. [i]Proposed by Marcin Kuczma, Poland[/i]

Let $\alpha$ and $\beta$ be real numbers with $\beta \ne 0$. Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that \[f(\alpha f(x)+f(y))=\beta x+f(y)\] holds for all real $x$ and $y$. [i](Walther Janous)[/i]

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]

Find $[ \sqrt{19992000}]$ where $[a]$ is the greatest integer less than or equal to $x$.

Let $m,n,a_1,a_2,\dots,a_n$ be positive integers and $r$ be a real number. Prove that the equation \[\lfloor a_1x\rfloor+\lfloor a_2x\rfloor+\cdots+\lfloor a_nx\rfloor=sx+r\] has exactly $ms$ solutions in $x$, where $s=a_1+a_2+\cdots+a_n+\frac1m$. [i]Linus Tang[/i]

If $n,p,q \in \mathbb{N}, p<q $ then $${{(p+q)n}\choose{n}} \sum \limits_{k=0}^n (-1)^k {{n}\choose{k}} {{(p+q-1)n}\choose{pn-k}}= {{(p+q)n}\choose{pn}} \sum \limits_{k=0}^{\left [\frac{n}{2} \right ]} (-1)^k {{pn}\choose{k}} {{(q-p)n}\choose{n-2k}} $$

[u]Round 9[/u] [b]p25.[/b] A positive integer is called spicy if it is divisible by the sum if its digits. Find the number of spicy integers between $100$ and $200$ inclusive. [b]p26.[/b] Rectangle $ABCD$ has points $E$ and $F$ on sides $AB$ and $BC$, respectively. Given that $\frac{AE}{BE} = \frac{BF}{FC} =\frac12$, $\angle ADE = 30^o$, and $[DEF] = 25$, find the area of rectangle $ABCD$. [b]p27.[/b] Find the largest value of $n$ for which $3^n$ divides ${100 \choose 33}$. [u]Round 10[/u] [b]p28.[/b] Isosceles trapezoid $ABCD$ is inscribed in a circle such that $AB \parallel CD$, $AB = 2$, $CD = 4$, and $AC = 9$. What is the radius of the circle? [b]p29.[/b] Find the product of all possible positive integers $n$ less than $11$ such that in a group of $n$ people, it is possible for every person to be friends with exactly $3$ other people within the group. Assume that friendship is amutual relationship. [b]p30.[/b] Compute the infinite product $$\left( 1+ \frac{1}{2^1} \right) \left( 1+ \frac{1}{2^2} \right) \left( 1+ \frac{1}{2^4} \right) \left( 1+ \frac{1}{2^8} \right) \left( 1+ \frac{1}{2^{16}} \right) ...$$ [u]Round 11[/u] [b]p31.[/b] Find the sum of all possible values of $x y$ if $x +\frac{1}{y}= 12$ and $\frac{1}{x}+ y = 8$. [b]p32.[/b] Find the number of ordered pairs $(a,b)$, where $0 < a,b < 1999$, that satisfy $a^2 +b^2 \equiv ab$ (mod $1999$) [b]p33.[/b] Let $f :N\to Q$ be a function such that $f(1) =0$, $f (2) = 1$ and $f (n) = \frac{f(n-1)+f (n-2)}{2}$ . Evaluate $$\lim_{n\to \infty} f (n).$$ [u]Round 12[/u] [b]p34.[/b] Estimate the sumof the digits of $2018^{2018}$. The number of points you will receive is calculated using the formula $\max \,(0,15-\log_{10}(A-E))$, where $A$ is the true value and $E$ is your estimate. [b]p35.[/b] Let $C(m,n)$ denote the number of ways to tile an $m$ by $n$ rectangle with $1\times 2$ tiles. Estimate $\log_{10}(C(100, 2))$. The number of points you will recieve is calculated using the formula $\max \,(0,15- \log_{10}(A-E))$, where $A$ is the true value and $E$ is your estimate. [b]p36.[/b] Estimate $\log_2 {1000 \choose 500}$. The number of points you earn is equal to $\max \,(0,15-|A-E|)$, where $A$ is the true value and $E$ is your estimate. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3165983p28809209]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3165992p28809294]here[/url].. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $x, y, z, t$ be real numbers such that $(x^2 + y^2 -1)(z^2 + t^2 - 1) > (xz + yt -1)^2$. Prove that $x^2 + y^2 > 1$.

If $x$ and $y$ are positive real numbers such that $(x + \sqrt{x^2 + 1})(y +\sqrt{y^2 + 1}) = 2018$: The minimum possible value of $x + y$ is A. $\frac{2017}{\sqrt{2018}}$ B. $\frac{2018}{\sqrt{2019}}$ C. $\frac{2017}{2\sqrt{2018}}$ D. $\frac{2019}{\sqrt{2018}}$ E. $\sqrt{3}$

Determine the sum of all distinct real values of $x$ such that $||| \cdots ||x|+x| \cdots |+x|+x|=1$ where there are $2017$ $x$s in the equation.

Consider a sequence of numbers between $0$ and $1$ in which the next number after $x$ is $1 - |1 - 2x|$. ($|x| = x$ if$ x \ge 0$, $|x| = -x$ if $x < 0$.) Prove that (a) if the first number of the sequence is rational, then the sequence will be periodic (i.e. the terms repeat with a certain cycle length after a certain term in the sequence); (b) if the sequence is periodic, then the first number is rational. (G Shabat)

Given a function $g:[0,1] \to \mathbb{R}$ satisfying the property that for every non empty dissection of the trivial $[0,1]$ to subsets $A,B$ we have either $\exists x \in A; g(x) \in B$ or $\exists x \in B; g(x) \in A$ and we have furthermore $g(x)>x$ for $x \in [0,1]$. Prove that there exist infinite $x \in [0,1]$ with $g(x)=1$. [i]Proposed by Ali Zamani [/i]

let ${a_{n}}$ be a sequence of integers,$a_{1}$ is odd,and for any positive integer $n$,we have $n(a_{n+1}-a_{n}+3)=a_{n+1}+a_{n}+3$,in addition,we have $2010$ divides $a_{2009}$ find the smallest $n\ge\ 2$,so that $2010$ divides $a_{n}$

Marisela is putting on a juggling show! She starts with $1$ ball, tossing it once per second. Lawrence tosses her another ball every five seconds, and she always tosses each ball that she has once per second. Compute the total number of tosses Marisela has made one minute after she starts juggling.

Let $F:\mathbb R^{\mathbb R}\to\mathbb R^{\mathbb R}$ be a function (from the set of real-valued functions to itself) such that $$F(F(f)\circ g+g)=f\circ F(g)+F(F(F(g)))$$ for all $f,g:\mathbb R\to\mathbb R$. Prove that there exists a function $\sigma:\mathbb R\to\mathbb R$ such that $$F(f)=\sigma\circ f\circ\sigma$$ for all $f:\mathbb R\to\mathbb R$.

A sequence $ (x_n)$ is given by $ x_1\equal{}2$ and $ nx_n\equal{}2(2n\minus{}1)x_{n\minus{}1}$ for $ n>1$. Prove that $ x_n$ is an integer for every $ n \in \mathbb{N}$.

Define the sequences $a_n$ and $b_n$ as follows: $a_1 = 2017$ and $b_1 = 1$. For $n > 1$, if there is a greatest integer $k > 1$ such that $a_n$ is a perfect $k$th power, then $a_{n+1} =\sqrt[k]{a_n}$, otherwise $a_{n+1} = a_n + b_n$. If $a_{n+1} \ge a_n$ then $b_{n+1} = b_n$, otherwise $b_{n+1} = b_n + 1$. Find $a_{2017}$.

Find a sequence of positive integer $f(n)$, $n \in \mathbb{N}$ such that $(1)$ $f(n) \leq n^8$ for any $n \geq 2$, $(2)$ for any pairwisely distinct natural numbers $a_1,a_2,\cdots, a_k$ and $n$, we have that $$f(n) \neq f(a_1)+f(a_2)+ \cdots + f(a_k)$$

The arithmetic mean of two distinct positive integers $x$ and $y$ is a two-digit integer. The geometric mean of $x$ and $y$ is obtained by reversing the digits of the arithmetic mean. What is $|x-y|$? $ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 54 \qquad \textbf{(D)}\ 66 \qquad \textbf{(E)}\ 70 $

If $x^{x^4}=4 $ what is the value of $x^{x^2}+x^{x^8} $ ?

We say that $H$ permeates $G$ if $G$ and $H$ are finite groups and for all subgroup $F$ of $G$ there is $H'\cong H$ with $H'\le F$ or $F\le H'\le G$. Suppose that a non-abelian group $H$ permeates $G$ and let $S=\langle H'\le G | H'\cong H\rangle$. Show that $$|\bigcap_{H'\in S} H'|>1$$

Let $a_n = 6^n + 8^n$. Determine the remainder on dividing $a_{83}$ by 49.