Let $n$ be a positive integer and $k\in[0,n]$ be a fixed real constant. Find the maximum value of $$\left|\sum_{i=1}^n\sin(2x_i)\right|$$ where $x_1,\ldots,x_n$ are real numbers satisfying $$\sum_{i=1}^n\sin^2(x_i)=k.$$

Whether exist set $A$ that contain 2016 real numbers (some of them may be equal) not all of which equal 0 such that next statement holds. For arbitrary 1008-element subset of $A$ there is a monic polynomial of degree 1008 such that elements of this subset are roots of the polynomial and other 1008 elements of $A$ are coefficients of this polynomial's degrees from 0 to 1007.

[b]p1.[/b] I open my $2010$-page dictionary, whose pages are numbered $ 1$ to $2010$ starting on page $ 1$ on the right side of the spine when opened, and ending with page $2010$ on the left. If I open to a random page, what is the probability that the two page numbers showing sum to a multiple of $6$? [b]p2.[/b] Let $A$ be the number of positive integer factors of $128$. Let $B$ be the sum of the distinct prime factors of $135$. Let $C$ be the units’ digit of $381$. Let $D$ be the number of zeroes at the end of $2^5\cdot 3^4 \cdot 5^3 \cdot 7^2\cdot 11^1$. Let $E$ be the largest prime factor of $999$. Compute $\sqrt[3]{\sqrt{A + B} +\sqrt[3]{D^C+E}}$. [b]p3. [/b] The root mean square of a set of real numbers is defined to be the square root of the average of the squares of the numbers in the set. Determine the root mean square of $17$ and $7$. [b]p4.[/b] A regular hexagon $ABCDEF$ has area $1$. The sides$ AB$, $CD$, and $EF$ are extended to form a larger polygon with $ABCDEF$ in the interior. Find the area of this larger polygon. [b]p5.[/b] For real numbers $x$, let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$. For example, $\lfloor 3\rfloor = 3$ and $\lfloor 5.2 \rfloor = 5$. Evaluate $\lfloor -2.5 \rfloor + \lfloor \sqrt 2 \rfloor + \lfloor -\sqrt 2 \rfloor + \lfloor 2.5 \rfloor$. [b]p6.[/b] The mean of five positive integers is $7$, the median is $8$, and the unique mode is $9$. How many possible sets of integers could this describe? [b]p7.[/b] How many three digit numbers x are there such that $x + 1$ is divisible by $11$? [b]p8.[/b] Rectangle $ABCD$ is such that $AD = 10$ and $AB > 10$. Semicircles are drawn with diameters $AD$ and $BC$ such that the semicircles lie completely inside rectangle $ABCD$. If the area of the region inside $ABCD$ but outside both semicircles is $100$, determine the shortest possible distance between a point $X$ on semicircle $AD$ and $Y$ on semicircle $BC$. [b]p9.[/b] $ 8$ distinct points are in the plane such that five of them lie on a line $\ell$, and the other three points lie off the line, in a way such that if some three of the eight points lie on a line, they lie on $\ell$. How many triangles can be formed using some three of the $ 8$ points? [b]p10.[/b] Carl has $10$ Art of Problem Solving books, all exactly the same size, but only $9$ spaces in his bookshelf. At the beginning, there are $9$ books in his bookshelf, ordered in the following way. $A - B - C - D - E - F - G - H - I$ He is holding the tenth book, $J$, in his hand. He takes the books out one-by-one, replacing each with the book currently in his hand. For example, he could take out $A$, put $J$ in its place, then take out $D$, put $A$ in its place, etc. He never takes the same book out twice, and stops once he has taken out the tenth book, which is $G$. At the end, he is holding G in his hand, and his bookshelf looks like this. $C - I - H - J - F - B - E - D - A$ Give the order (start to finish) in which Carl took out the books, expressed as a $9$-letter string (word). PS. You had better use hide for answers.

Given are three positive real numbers $ a,b,c$ satisfying $ abc \plus{} a \plus{} c \equal{} b$. Find the max value of the expression: \[ P \equal{} \frac {2}{a^2 \plus{} 1} \minus{} \frac {2}{b^2 \plus{} 1} \plus{} \frac {3}{c^2 \plus{} 1}.\]

[b]p1.[/b] Fill in each box with an integer from $1$ to $9$. Each number in the right column is the product of the numbers in its row, and each number in the bottom row is the product of the numbers in its column. Some numbers may be used more than once, and not every number from $1$ to $9$ is required to be used. [img]https://cdn.artofproblemsolving.com/attachments/c/0/0212181d87f89aac374f75f1f0bde6d0600037.png[/img] [b]p2.[/b] A set $X$ is called [b]prime-difference free [/b] (henceforth pdf) if for all $x, y \in X$, $|x - y|$ is not prime. Find the number n such that the following both hold. $\bullet$ There is a pdf set of size $n$ that is a subset of $\{1,..., 2016\}$, and $\bullet$ There is no pdf set of size $n + 1$ that is a subset of $\{1,..., 2016\}$. [b]p3.[/b] Let $X_1,...,X_{15}$ be a sequence of points in the $xy$-plane such that $X_1 = (10, 0)$ and $X_{15} = (0, 10)$. Prove that for some $i \in \{1, 2,..., 14\}$, the midpoint of $X_iX_{i+1}$ is of distance greater than $1/2$ from the origin. [b]p4.[/b] Suppose that $s_1, s_2,..., s_{84}$ is a sequence of letters from the set $\{A,B,C\}$ such that every four-letter sequence from $\{A,B,C\}$ occurs exactly once as a consecutive subsequence $s_k$, $s_{k+1}$, $s_{k+2}$, $s_{k+3}$. Suppose that $$(s_1, s_2, s_3, s_4, s_5) = (A,B,B,C,A).$$ What is $s_{84}$? Prove your answer. [b]p5.[/b] Determine (with proof) whether or not there exists a sequence of positive real numbers $a_1, a_2, a_3,...$ with both of the following properties: $\bullet$ $\sum^n_{i=1} a_i \le n^2$, for all $n \ge 1$, and $\bullet$ $\sum^n_{i=1} \frac{1}{a_i} \le 2016$, for all $n \ge 1$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The values of the polynomial $P(x) = 2x^3-30x^2+cx$ for any three consecutive integers are also three consecutive integers. Find these values.

[b]3A.[/b] Sudoku, the popular math game that caught on internationally before making its way here to the United States, is a game of logic based on a grid of $9$ rows and $9$ columns. This grid is subdivided into $9$ squares (“subgrids”) of length $3$. A successfully completed Sudoku puzzle fills this grid with the numbers $1$ through $9$ such that each number appears only once in each row, column, and individual $3 \times 3$ subgrid. Each Sudoku puzzle has one and only one correct solution. Complete the following Sudoku puzzle, and find the sum of the numbers represented by $X, Y$, and $Z$ in the grid. [i](1 point)[/i] $\begin{tabular}{|l|l|l|l|l|l|l|l|l|} \hline & & 2 & 9 & 7 & 4 & & & \\ \hline & Z & & & & & & 5 & 7 \\ \hline & & & & & & Y & & \\ \hline & & 4 & & 5 & & & & 2 \\ \hline & & 9 & X & 1 & & 6 & & \\ \hline 8 & & & & 3 & & 4 & & \\ \hline & & & & & & & & \\ \hline 1 & 3 & & & & & & & \\ \hline & & & 6 & 8 & 2 & 9 & & \\ \hline \end{tabular}$ [b]3B.[/b] Let $A$ equal the correct answer from [b]3A[/b]. In triangle $WXY$, $tan \angle YWX= (A + 8) / .5A$, and the altitude from $W$ divides $XY$ into segments of $3$ and $A + 3$. What is the sum of the digits of the square of the area of the triangle? [i](2 points)[/i] [b]3C.[/b] Let $B$ equal the correct answer from [b]3B[/b]. If a student team taking the $2005$ iTest solves $B$ problems correctly, and the probability that this student team makes over a $18$ is $x/y$ where $x$ and $y$ are relatively prime, find $x + y$. Assume that each chain reaction question – all $3$ parts it contains – counts as a single problem. Also assume that the student team does not attempt any tiebreakers. [i](4 points)[/i] [i][Note for problem 3C beacuse you might not know how points are given at that iTest: Part A (aka Short Answer), has 40 problems of 1 point each, total 40 Part B (aka Chain Reaction), has 3 problems of 7,6,7 points each, total 20 Part C (aka Long Answer), has 5 problems of 8 point each, total 40 all 3 parts add to 100 points totally ([url=https://artofproblemsolving.com/community/c3176431_itest_2005]here [/url] is that test)][/i] [hide=ANSWER KEY]3A.14 3B. 4 3C. 6563 [/hide]

Determine all functions $f : R_+ \to R$ such that:$$f(x)f(y) = f(xy) + \frac{1}{x} + \frac{1}{y}$$ for all $x, y$ positive reals.

Prove that given three positive numbers, we can choose two of them, say $x$ and $y,$ with $x >y$ such that $$\frac{x-y}{1 +xy }<1.$$ Prove also that if the number $1$ that appears in the second member of the previous inequality is replaced by a lower number, even if very close to $1$, the previous proposition is false.

Do there exist $2024$ non-zero reals $a_1, a_2, \ldots, a_{2024}$, such that $$\sum_{i=1}^{2024}(a_i^2+\frac{1}{a_i^2})+2\sum_{i=1}^{2024} \frac{a_i} {a_{i+1}}+2024=2\sum_{i=1}^{2024}(a_i+\frac{1}{a_i})?$$

Let $ f(x) \equal{} a \sin^2x \plus{} b \sin x \plus{} c,$ where $ a, b,$ and $ c$ are real numbers. Find all values of $ a, b$ and $ c$ such that the following three conditions are satisfied simultaneously: [b](i)[/b] $ f(x) \equal{} 381$ if $ \sin x \equal{} \frac{1}{2}.$ [b](ii)[/b] The absolute maximum of $ f(x)$ is $ 444.$ [b](iii)[/b] The absolute minimum of $ f(x)$ is $ 364.$

Does there exist a subset $M$ of positive integers such that for all positive rational numbers $r<1$ there exists exactly one finite subset of $M$ like $S$ such that sum of reciprocals of elements in $S$ equals $r$.

Let $P$ be a polynomial with integer coefficients such that $P(0)=0$ and \[\gcd(P(0), P(1), P(2), \ldots ) = 1.\] Show there are infinitely many $n$ such that \[\gcd(P(n)- P(0), P(n+1)-P(1), P(n+2)-P(2), \ldots) = n.\]

Let $n$ be a positive integer, $n \geq 1$ and $x_1,x_2,...,x_n$ positive real numbers such that $x_1+x_2+...+x_n=1$. Does the following inequality hold $$\sum_{i=1}^{n} {\frac{x_i}{1-x_1\cdot...\cdot x_{i-1} \cdot x_{i+1} \cdot ... x_n}} \leq \frac{1}{1-\left(\frac{1}{n}\right)^{n-1}} $$

Let $f$ be a strictly increasing function defined on the set of real numbers. For $x$ real and $t$ positive, set\[g(x,t)=\frac{f(x+t)-f(x)}{f(x) - f(x - t)}.\] Assume that the inequalities\[2^{-1} < g(x, t) < 2\] hold for all positive t if $x = 0$, and for all $t \leq |x|$ otherwise. Show that\[ 14^{-1} < g(x, t) < 14\] for all real $x$ and positive $t.$

Suppose $P(x) = x^{2016} + a_{2015}x^{2015} + ...+ a_1x + a_0$ satisfies $P(x)P(2x + 1) = P(-x)P(-2x - 1)$ for all $x \in R$. Find the sum of all possible values of $a_{2015}$.

Let $f : R \to R$, and let $P$ be a nonzero polynomial with degree no more than $2015$. For any nonnegative integer $n$, $f^{(n)}(x)$ denotes the function defined as $f$ composed with itself $n$ times. For example, $f^{(0)}(x) = x$, $f^{(1)}(x) = f(x)$, $f^{(2)}(x) = f(f(x))$, etc. Show that there always exists a real number $q$ such that $$f^{((2017^{2017})!)(q)} \ne (q + 2017)(qP(q) - 1).$$

Find all values of the real parameter $a$, for which the system $(|x| + |y| - 2)^2 = 1$ $y = ax + 5$ has exactly three solutions

If $ \{a_k\}$ is a sequence of real numbers, call the sequence $ \{a'_k\}$ defined by $ a_k' \equal{} \frac {a_k \plus{} a_{k \plus{} 1}}2$ the [i]average sequence[/i] of $ \{a_k\}$. Consider the sequences $ \{a_k\}$; $ \{a_k'\}$ - [i]average sequence[/i] of $ \{a_k\}$; $ \{a_k''\}$ - average sequence of $ \{a_k'\}$ and so on. If all these sequences consist only of integers, then $ \{a_k\}$ is called [i]Good[/i]. Prove that if $ \{x_k\}$ is a [i]good[/i] sequence, then $ \{x_k^2\}$ is also [i]good[/i].

Let $t$ be the area of a regular pentagon with each side equal to $1$. Let $P(x)=0$ be the polynomial equation with least degree, having integer coefficients, satisfied by $x=t$ and the $\gcd$ of all the coefficients equal to $1$. If $M$ is the sum of the absolute values of the coefficients of $P(x)$, What is the integer closest to $\sqrt{M}$ ? ($\sin 18^{\circ}=(\sqrt{5}-1)/2$)

Do there exist $2022$ polynomials with real coefficients, each of degree equal to $2021$, so that the $2021 \cdot 2022 + 1$ coefficients in their product are equal?

For a given parameter $m$, solve the equation \[x(x + 1)(x + 2)(x + 3) + 1 - m = 0.\]

Let $M \subset R$ be a finite set containing at least two elements. We say that the function $f$ has property $P$ if $f : M \to M$ and there are $a \in R^*$ and $b \in R$ such that $f(x) = ax + b$. (a) Show that there is at least a function having property $P$. (b) Show that there are at most two functions having property $P$. (c) If $M$ has $2003$ elements with sum $0$ and if there are two functions with property $P$, prove that $0 \in M$.

Consider the graphs of $y=2\log{x}$ and $y=\log{2x}$. We may say that: $ \textbf{(A)}\ \text{They do not intersect}$ $ \qquad\textbf{(B)}\ \text{They intersect at 1 point only}$ $\qquad\textbf{(C)}\ \text{They intersect at 2 points only}$ $\qquad\textbf{(D)}\ \text{They intersect at a finite number of points but greater than 2 }$ ${\qquad\textbf{(E)}\ \text{They coincide} } $

A quadratic trinomial with integer coefficients is called [i]admissible [/i] if its leading coefficient is $1$, its roots are integers and the absolute values of coefficients do not exceed $2013$. Basil has summed up all admissible quadratic trinomials. Prove that the resulting trinomial has no real roots.