The symbol $|a|$ means $+a$ if $a$ is greater than or equal to zero, and $-a$ if $a$ is less than or equal to zero; the symbol $<$ means "less than"; the symbol $>$ means "greater than." The set of values $x$ satisfying the inequality $|3-x|<4$ consists of all $x$ such that: $ \textbf{(A)}\ x^2<49 \qquad\textbf{(B)}\ x^2>1 \qquad\textbf{(C)}\ 1<x^2<49\qquad\textbf{(D)}\ -1<x<7\qquad\textbf{(E)}\ -7<x<1 $

The numbers from$ 1$ to $2016$ are divided into three (disjoint) subsets $A, B$ and $C$, each one contains exactly $672$ numbers. Prove that you can find three numbers, each from a different subset, such that the sum of two of them is equal to the third. [hide=original wording]Skaitļi no 1 līdz 2016 ir sadalīti trīs (nešķeļošās) apakškopās A, B un C, katranotām satur tieši 672 skaitļus. Pierādīt, ka var atrast trīs tādus skaitļus, katru no citas apakškopas, ka divu no tiem summa ir vienāda ar trešo. [/hide]

Find all integers that can be written in the form $\frac{(x+y+z)^2}{xyz}$, where $x,y,z$ are positive integers.

Find all functions $f : (0, +\infty) \to (0, +\infty)$ which are increasing on $[1, +\infty)$ and for all positive reals $a, b, c$ they fulfill the following relation $f(ab)f(bc)f(ca)=f(a^2b^2c^2)+f(a^2)+f(b^2)+f(c^2)$.

The twenty-first century began on January $1$, $2001$ and runs through December $31$, $2100$. Note that March $1$, $2014$ fell on Saturday, so there were five Mondays in March $2014$. In how many years of the twenty-first century does March have five Mondays?

Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Across all $x \in \mathbb{R}$, find the maximum value of the expression $$\sin x + \sin 3x + \sin 5x.$$

Consider the sequence formed by the first digits of the powers of $5$:$$1,5,2,1,6,...$$ Prove any segment in this sequence, when written in reversed order, will be encountered in the sequence of the first digits of the powers of $2:$ $$1,2,4,8,1,3,6,1...$$

Given the nine-sided regular polygon $ A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8 A_9$, how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set $ \{A_1,A_2,...A_9\}$? $ \textbf{(A)} \ 30 \qquad \textbf{(B)} \ 36 \qquad \textbf{(C)} \ 63 \qquad \textbf{(D)} \ 66 \qquad \textbf{(E)} \ 72$

Let $ABC$ be a triangle with $\angle BAC = 75^o$ and $\angle ABC = 45^o$. If $BC =\sqrt3 + 1$, what is the perimeter of $\vartriangle ABC$?

How many different numbers with $6$ digits and multiples of $45$ can be written by adding one digit to the left and one to the right of $2008$?

In trapezoid $ABCD$, $AD$ is parallel to $BC$. Knowing that $AB=AD+BC$, prove that the bisector of $\angle A$ also bisects $CD$.

Let $ m$ and $ n$ be positive integers such that $ x^2 \minus{} mx \plus{}n \equal{} 0$ has real roots $ \alpha$ and $ \beta$. Prove that $ \alpha$ and $ \beta$ are integers [b]if and only if[/b] $ [m\alpha] \plus{} [m\beta]$ is the square of an integer. (Here $ [x]$ denotes the largest integer not exceeding $ x$)

[u]Round 1[/u] [b]p1.[/b] A circle is inscribed inside a square such that the cube of the radius of the circle is numerically equal to the perimeter of the square. What is the area of the circle? [b]p2.[/b] If the coefficient of $z^ky^k$ is $252$ in the expression $(z + y)^{2k}$, find $k$. [b]p3.[/b] Let $f(x) = \frac{4x^4-2x^3-x^2-3x-2}{x^4-x^3+x^2-x-1}$ be a function defined on the real numbers where the denominator is not zero. The graph of $f$ has a horizontal asymptote. Compute the sum of the x-coordinates of the points where the graph of $f$ intersects this horizontal asymptote. If the graph of f does not intersect the asymptote, write $0$. [u]Round 2 [/u] [b]p4.[/b] How many $5$-digit numbers have strictly increasing digits? For example, $23789$ has strictly increasing digits, but $23889$ and $23869$ do not. [b]p5.[/b] Let $$y =\frac{1}{1 +\frac{1}{9 +\frac{1}{5 +\frac{1}{9 +\frac{1}{5 +...}}}}}$$ If $y$ can be represented as $\frac{a\sqrt{b} + c}{d}$ , where $b$ is not divisible by any squares, and the greatest common divisor of $a$ and $d$ is $1$, find the sum $a + b + c + d$. [b]p6.[/b] “Counting” is defined as listing positive integers, each one greater than the previous, up to (and including) an integer $n$. In terms of $n$, write the number of ways to count to $n$. [u]Round 3 [/u] [b]p7.[/b] Suppose $p$, $q$, $2p^2 + q^2$, and $p^2 + q^2$ are all prime numbers. Find the sum of all possible values of $p$. [b]p8.[/b] Let $r(d)$ be a function that reverses the digits of the $2$-digit integer $d$. What is the smallest $2$-digit positive integer $N$ such that for some $2$-digit positive integer $n$ and $2$-digit positive integer $r(n)$, $N$ is divisible by $n$ and $r(n)$, but not by $11$? [b]p9.[/b] What is the period of the function $y = (\sin(3\theta) + 6)^2 - 10(sin(3\theta) + 7) + 13$? [u]Round 4 [/u] [b]p10.[/b] Three numbers $a, b, c$ are given by $a = 2^2 (\sum_{i=0}^2 2^i)$, $b = 2^4(\sum_{i=0}^4 2^i)$, and $c = 2^6(\sum_{i=0}^6 2^i)$ . $u, v, w$ are the sum of the divisors of a, b, c respectively, yet excluding the original number itself. What is the value of $a + b + c -u - v - w$? [b]p11.[/b] Compute $\sqrt{6- \sqrt{11}} - \sqrt{6+ \sqrt{11}}$. [b]p12.[/b] Let $a_0, a_1,..., a_n$ be such that $a_n\ne 0$ and $$(1 + x + x^3)^{341}(1 + 2x + x^2 + 2x^3 + 2x^4 + x^6)^{342} =\sum_{i=0}^n a_ix^i.$$ Find the number of odd numbers in the sequence $a_0, a_1,..., a_n$. PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2781343p24424617]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A mustache is created by taking the set of points $(x, y)$ in the $xy$-coordinate plane that satisfy $4 + 4 \cos(\pi x/24) \le y \le 6 + 6\cos(\pi x/24)$ and $-24 \le x \le 24$. What is the area of the mustache?

In the acute-angled triangle $ABC$ all angles are greater than $45^o$. Let $AM$ and $BN$ be the heights of this triangle and let $X$ and $Y$ be the points on $MA$ and $NB$, respecively, such that $|MX| =|MB|$ and $|NY| =|NA|$. Prove that $MN$ and $XY$ are parallel.

Prove that there are no real numbers $ a, b, c $, $ x_1, x_2, x_3 $ such that for every real number $ x $ $$ ax^2 + bx + c = a(x - x_2)(x - x_3) $$ $$bx^2 + cx + a = b(x - x_3) (x - x_1)$$ $$cx^2 + ax + b = c(x - x_1) (x - x_2)$$ and $ x_1 \neq x_2 $, $ x_2 \neq x_3 $, $ x_3 \neq x_1 $, $ abc \neq 0 $.

For each integer $n > 0$ show that there is a polynomial $p(x)$ such that $p(2 cos x) = 2 cos nx$.

A neat fruit arrangement on a large round dish is edged with strawberries. Between $100$ and $200$ berries are used for this border. A deliciously hungry child eats first one of the strawberries and then starts going round and round the dish, she eats strawberries in the following way: When she has eaten a berry, she leaves it next lie, then she eats the next, leaves the next, etc. Thus she continues until there is only one strawberry left. This berry is the one that was lying right after the very first thing she ate. How many berries were there originally?

Describe all functions $f: \{ 0,1\}^n \to \{ 0,1\}$ which satisfy the equation \begin{align*} & f(f(a_{11},a_{12},\dotsc ,a_{1n}),f(a_{21},a_{22},\dotsc ,a_{2n}),\dotsc ,f(a_{n1},a_{n2},\dotsc ,a_{nn}))\\ & = f(f(a_{11},a_{21},\dotsc ,a_{n1}),f(a_{12},a_{22},\dotsc ,a_{n2}),\dotsc ,f(a_{1n},a_{2n},\dotsc ,a_{nn}))\end{align*} for arbitrary $a_{ij}\in \{ 0,1\}$ where $i,j\in \{1,2,\dotsc ,n\}.$

Consider the following function. $\textbf{procedure }\textsc{M}(x)$ $\qquad\textbf{if }0\leq x\leq 1$ $\qquad\qquad\textbf{return }x$ $\qquad\textbf{return }\textsc{M}(x^2\bmod 2^{32})$ Let $f:\mathbb N\to\mathbb N$ be defined such that $f(x) = 0$ if $\textsc{M}(x)$ does not terminate, and otherwise $f(x)$ equals the number of calls made to $\textsc{M}$ during the running of $\textsc{M}(x)$, not including the initial call. For example, $f(1) = 0$ and $f(2^{31}) = 1$. Compute the number of ones in the binary expansion of \[ f(0) + f(1) + f(2) + \cdots + f(2^{32} - 1). \]

A triangle with angles of $30, 70$ and $80$ degrees is given. Cut it by a straight line into two triangles in such a way that an angle bisector in one of these triangles and a median in the other one drawn from two endpoints of the cutting segment are parallel to each other. (It suffices to find one such cutting.) (A. Shapovalov )

Let $A,B$ be two fixed points on the unit circle $\omega$, satisfying $\sqrt{2} < AB < 2$. Let $P$ be a point that can move on the unit circle, and it can move to anywhere on the unit circle satisfying $\triangle ABP$ is acute and $AP>AB>BP$. Let $H$ be the orthocenter of $\triangle ABP$ and $S$ be a point on the minor arc $AP$ satisfying $SH=AH$. Let $T$ be a point on the minor arc $AB$ satisfying $TB || AP$. Let $ST\cap BP = Q$. Show that (recall $P$ varies) the circle with diameter $HQ$ passes through a fixed point.

Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions: [list] [*] $(i)$ $f(n) \neq 0$ for at least one $n$; [*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$; [*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$. [/list]

A [i]triangulation[/i] of a convex polygon $\Pi$ is a partitioning of $\Pi$ into triangles by diagonals having no common points other than the vertices of the polygon. We say that a triangulation is a [i]Thaiangulation[/i] if all triangles in it have the same area. Prove that any two different Thaiangulations of a convex polygon $\Pi$ differ by exactly two triangles. (In other words, prove that it is possible to replace one pair of triangles in the first Thaiangulation with a different pair of triangles so as to obtain the second Thaiangulation.) [i]Proposed by Bulgaria[/i]