$n$ parallel segments of lengths $a_1 \le a_2 \le a_3 \le ... \le a_n$ were painted to mark an airport atrium. However, the architect decided that the $n$ segments should have equal length. If the cost per meter of extending the lines is equal to the cost of reducing them, how long should the lines be in order to minimize costs?

Find all quadruplets (x, y, z, w) of positive real numbers that satisfy the following system: $\begin{cases} \frac{xyz+1}{x+1}= \frac{yzw+1}{y+1}= \frac{zwx+1}{z+1}= \frac{wxy+1}{w+1}\\ x+y+z+w= 48 \end{cases}$

The given parabola $y=ax^2+bx+c$ doesn't intersect the $X$-axis and passes from the points $A(-2,1)$ and $B(2,9)$. Find all the possible values of the $x$ coordinates of the vertex of this parabola.

The real numbers $a, b, c, d$ are such that $a\geq b\geq c\geq d>0$ and $a+b+c+d=1$. Prove that \[(a+2b+3c+4d)a^ab^bc^cd^d<1\] [i]Proposed by Stijn Cambie, Belgium[/i]

[u]Round 1[/u] [b]p1.[/b] A $\$100$ TV has its price increased by $10\%$. The new price is then decreased by $10\%$. What is the current price of the TV? [b]p2.[/b] If $9w + 8x + 7y = 42$ and $w + 2x + 3y = 8$, then what is the value of $100w + 101x + 102y$? [b]p3.[/b] Find the number of positive factors of $37^3 \cdot 41^3$. [u]Round 2[/u] [b]p4.[/b] Three hoses work together to fill up a pool, and each hose expels water at a constant rate. If it takes the first, second, and third hoses 4, 6, and 12 hours, respectively, to fill up the pool alone, then how long will it take to fill up the pool if all three hoses work together? [b]p5.[/b] A semicircle has radius $1$. A smaller semicircle is inscribed in the larger one such that the two bases are parallel and the arc of the smaller is tangent to the base of the larger. An even smaller semicircle is inscribed in the same manner inside the smaller of the two semicircles, and this procedure continues indefinitely. What is the sum of all of the areas of the semicircles? [b]p6.[/b] Given that $P(x)$ is a quadratic polynomial with $P(1) = 0$, $P(2) = 0$, and $P(0) = 2012$, find $P(-1)$. [u]Round 3[/u] [b]p7.[/b] Darwin has a paper circle. He labels one point on the circumference as $A$. He folds $A$ to every point on the circumference on the circle and undoes it. When he folds $A$ to any point $P$, he makes a blue mark on the point where $\overline{AP}$ and the made crease intersect. If the area of Darwin paper circle is 80, then what is the area of the region surrounded by blue? [b]p8.[/b] Α rectangular wheel of dimensions $6$ feet by $8$ feet rolls for $28$ feet without sliding. What is the total distance traveled by any corner on the rectangle during this roll? [b]p9[/b]. How many times in a $24$-hour period do the minute hand and hour hand of a $12$-hour clock form a right angle? [u]Round 4[/u] The answers in this section all depend on each other. Find smallest possible solution set. [b]p10.[/b] Let B be the answer to problem $11$. Right triangle $ACD$ has a right angle at $C$. Squares $ACEF$ and $ADGH$ are drawn such that points $D$ and $E$ do not coincide and points $E$ and $H$ do not coincide. The midpoints of the sides of $ADGH$ are connected to form a smaller square with area $B.$ If the area of $ACEF$ is also $B$, then find the length $CD$ rounded up to the nearest integer. [b]p11.[/b] Let $C$ be the answer to problem $12$. Find the sum of the digits of $C$. [b]p12.[/b] Let $A$ be the answer to problem $10$. Given that $a_0 = 1$, $a_1 = 2$, and that $a_n = 3a_{n-1 }-a_{n-2}$ for $n \ge 2$, find $a_A$. PS. You should use hide for answers.Rounds 5-8 are [url=https://artofproblemsolving.com/community/c3h3134466p28406321]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3134489p28406583]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[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 $Q.$

Real numbers $a$ and $b$ are chosen so that each of two quadratic trinomials $x^2+ax+b$ and $x^2+bx+a$ has two distinct real roots,and the product of these trinomials has exactly three distinct real roots.Determine all possible values of the sum of these three roots. [i](S.Berlov)[/i]

Let $ P(n)$ and $ S(n)$ denote the product and the sum, respectively, of the digits of the integer $ n$. For example, $ P(23) \equal{} 6$ and $ S(23) \equal{} 5$. Suppose $ N$ is a two-digit number such that $ N \equal{} P(N) \plus{} S(N)$. What is the units digit of $ N$? $ \textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad\textbf{(E) }9$

Determine the lowest positive integer n such that following statement is true: If polynomial with integer coefficients gets value 2 for n different integers, then it can't take value 4 for any integer.

Let $a$ be a positive integer. Determine all $a$ such that the equation $$ \biggl( 1+\frac{1}{x} \biggr) \cdot \biggl( 1+\frac{1}{x+1} \biggr) \cdots \biggl( 1+\frac{1}{x+a} \biggr)=a-x$$ has at least one integer solution for $x$. For every such $a$ state the respective solutions. (Richard Henner)

Find all triples $(x, y, z)$ of real numbers that satisfy $x^2 + y^2 + z^2 + 1 = xy + yz + zx +|x - 2y + z|$.

The real numbers $a, b, c$ are such that $a^2 + b^2 = 2c^2$, and also such that $a \ne b, c \ne -a, c \ne -b$. Show that \[\frac{(a+b+2c)(2a^2-b^2-c^2)}{(a-b)(a+c)(b+c)}\] is an integer.

Let $f$ be a polynomial with real coefficients such that for each positive integer n the equation $f(x) = n$ has at least one rational solution. Find $f$.

Do there exist polynomials $f(x)$, $g(x)$ with real coefficients and a positive integer $k$ satisfying the following condition? (Here, the equation $x^2 = 0$ is considered to have $1$ distinct real roots. The equation $0 = 0$ has infinitely many distinct real roots.) For any real numbers $a, b$ with $(a,b) \neq (0,0)$, the number of distinct real roots of $a f(x) + b g(x) = 0$ is $k$.

Find all functions $f:\mathbb{Z}\rightarrow \mathbb{Z}$ satisfying the following: $i)$ $f(1)=1$; $ii)$ $f(m+n)(f(m)-f(n))=f(m-n)(f(m)+f(n))$ for all $m,n \in \mathbb{Z}$.

Find all functions $ f: \mathbb{R} \to \mathbb{R}$ satisfying \[ f\left(\frac {x \plus{} y}{x \minus{} y}\right) \equal{} \frac {f\left(x\right) \plus{} f\left(y\right)}{f\left(x\right) \minus{} f\left(y\right)} \] for all $ x \neq y$.

Determine all polynomial $P(x)\in \mathbb{R}[x]$ satisfying the following two conditions: (a) $P(2017)=2016$ and (b) $(P(x)+1)^2=P(x^2+1)$ for all real number $x$.

Let $x, y,$ and $z$ be positive real numbers such that $xy + yz + zx = 3$. Prove that $$\frac{x + 3}{y + z} + \frac{y + 3}{z + x} + \frac{z + 3}{x + y} + 3 \ge 27 \cdot \frac{(\sqrt{x} + \sqrt{y} + \sqrt{z})^2}{(x + y + z)^3}.$$ Proposed by [i]Petar Filipovski, Macedonia[/i]

Let $S$ be the set of all 3-tuples $(a, b, c)$ of positive integers such that $a + b + c = 2013$. Find $$\sum_{(a,b,c)\in S} abc.$$

Given $ a_0 \equal{} 1$, $ a_1 \equal{} 3$, and the general relation $ a_n^2 \minus{} a_{n \minus{} 1}a_{n \plus{} 1} \equal{} (\minus{}1)^n$ for $ n \ge 1$. Then $ a_3$ equals: $ \textbf{(A)}\ \frac{13}{27}\qquad \textbf{(B)}\ 33\qquad \textbf{(C)}\ 21\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ \minus{}17$

Which of the following series are convergent? $\textbf{(A)}~\sum_{n=1}^\infty\sqrt{\frac{2n^2+3}{5n^3+1}}$ $\textbf{(B)}~\sum_{n=1}^\infty\frac{(n+1)^n}{n^{n+3/2}}$ $\textbf{(C)}~\sum_{n=1}^\infty n^2x\left(1-x^2\right)^n$ $\textbf{(D)}~\text{None of the above}$

Some positive integers, sum of which is $23$, are written in sequential form. Neither one of the terms nor the sum of some consecutive terms in the sequence is equal to $3$. [b]a) [/b]Is it possible that the sequence contains exactly $11$ terms? [b]b)[/b]Is it possible that the sequence contains exactly $12$ terms?

Let $p: \mathbb C \to \mathbb C$ be a polynomial with degree $n$ and complex coefficients which satisfies \[x \in \mathbb R \iff p(x) \in \mathbb R.\] Show that $n=1$

Let the $f:\mathbb{N}\rightarrow\mathbb{N}$ be the function such that (i) For all positive integers $n,$ $f(n+f(n))=f(n)$ (ii) $f(n_o)=1$ for some $n_0$ Prove that $f(n)\equiv 1.$

Solve in the set of rational numbers the equation $$2\sqrt{3(x + 1)^2} -3 \sqrt{2(y - 2)^2}= 4\sqrt2 + 5|\sqrt2 - \sqrt3|$$