For a positive integer $n \ge 1,$ let $a_n=\lfloor \sqrt[3]{n}+\tfrac{1}{2}\rfloor.$ Given a positive integer $N \ge 1,$ let $\mathcal{F}_N$ denote the set of positive integers $n \ge 1$ such that $a_n \le N.$ Let $S_N = \sum_{n \in \mathcal{F}_N} \tfrac{1}{a_n^2}.$ As $N$ goes to infinity, the quantity $S_N - 3N$ tends to $\tfrac{a\pi^2}{b}$ for relatifvely prime positive integers $a,b.$ Given that $\sum_{k=1}^{\infty} \tfrac{1}{k^2} = \tfrac{\pi^2}{6},$ find $a+b.$

Let $x$, $y$ and $z$ be positive real numbers such that $\sqrt{xy} + \sqrt{yz} + \sqrt{zx} = 3$. Prove that $\sqrt{x^3+x} + \sqrt{y^3+y} + \sqrt{z^3+z} \geq \sqrt{6(x+y+z)}$

[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]

Show that there is no function \(f:\mathbb{N}^* \rightarrow \mathbb{N}^*\) such that \(f^n(n)=n+1\) for all \(n\) (when \(f^n\) is the \(n\)th iteration of \(f\))

Let $x>1$ be a real number that is not an integer. Denote $\{x\}$ as its decimal part and $\lfloor x\rfloor$ the floor function. Prove that $$ \left(\frac{x+\{x\}}{\lfloor x\rfloor}-\frac{\lfloor x\rfloor}{x+\{x\}}\right)+\left(\frac{x+\lfloor x\rfloor}{\{x\}}-\frac{\{x\}}{x+\lfloor x\rfloor}\right)>\frac{16}{3}$$

Determine the positive integers $n$ such that the inequality \[n! > \sqrt{n^n}\] holds.

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that \[f(f(x)+x+f(y)f(z))=f(x+zf(x)+zf(y))-xf(z-1)\]for all $x,y,z\in\mathbb{R}$.

Let’s consider the real numbers $a, b, c$ satisfying the condition \[21 \cdot a \cdot b + 2 \cdot b \cdot c + 8 \cdot c \cdot a \leq 12.\] Find the minimal value of the expression \[P(a, b, c) = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.\]

Find all functions $f \colon \mathbb{R_{+}}\to \mathbb{R_{+}}$ satisfying : \[f ( f (x)-x) = 2x\] for all $x > 0$.

Determine all roots, real or complex, of the system of simultaneous equations \begin{align*} x+y+z &= 3, \\ x^2+y^2+z^2 &= 3, \\ x^3+y^3+z^3 &= 3.\end{align*}

Let $a_{ij}$ $i=1,2,3$; $j=1,2,3$ be real numbers such that $a_{ij}$ is positive for $i=j$ and negative for $i\neq j$. Prove the existence of positive real numbers $c_{1}$, $c_{2}$, $c_{3}$ such that the numbers \[a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3},\qquad a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3},\qquad a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}\] are either all negative, all positive, or all zero. [i]Proposed by Kiran Kedlaya, USA[/i]

Given a circle and $2006$ points lying on this circle. Albatross colors these $2006$ points in $17$ colors. After that, Frankinfueter joins some of the points by chords such that the endpoints of each chord have the same color and two different chords have no common points (not even a common endpoint). Hereby, Frankinfueter intends to draw as many chords as possible, while Albatross is trying to hinder him as much as he can. What is the maximal number of chords Frankinfueter will always be able to draw?

Let $a, b, c$ be positive real numbers such that $a^3 + b^3 + c^3 = 5abc$. Show that \[ \left( \frac{a + b}{c} \right) \left( \frac{b + c}{a} \right) \left( \frac{c + a}{b} \right) \geq 9. \]

Let $ \left\{x_n\right\}$, with $ n \equal{} 1, 2, 3, \ldots$, be a sequence defined by $ x_1 \equal{} 603$, $ x_2 \equal{} 102$ and $ x_{n \plus{} 2} \equal{} x_{n \plus{} 1} \plus{} x_n \plus{} 2\sqrt {x_{n \plus{} 1} \cdot x_n \minus{} 2}$ $ \forall n \geq 1$. Show that: [b](1)[/b] The number $ x_n$ is a positive integer for every $ n \geq 1$. [b](2)[/b] There are infinitely many positive integers $ n$ for which the decimal representation of $ x_n$ ends with 2003. [b](3)[/b] There exists no positive integer $ n$ for which the decimal representation of $ x_n$ ends with 2004.

The number of solutions of the equation $\tan x+\sec x=2\cos x$, where $0\le x\le\pi$, is $\textbf{(A)}~0$ $\textbf{(B)}~1$ $\textbf{(C)}~2$ $\textbf{(D)}~3$

Suppose $x_1,x_2,...,x_{60}\in [-1,1]$ , find the maximum of $$ \sum_{i=1}^{60}x_i^2(x_{i+1}-x_{i-1}),$$ where $x_{i+60}=x_i$.

Prove that there exists a convex 1990-gon with the following two properties : [b]a.)[/b] All angles are equal. [b]b.)[/b] The lengths of the 1990 sides are the numbers $ 1^2$, $ 2^2$, $ 3^2$, $ \cdots$, $ 1990^2$ in some order.

Let $k\ge2$ be an integer. The function $f:\mathbb N\to\mathbb N$ is defined by $$f(n)=n+\left\lfloor\sqrt[k]{n+\sqrt[k]n}\right\rfloor.$$Determine the set of values taken by the function $f$.

Find the coefficient of $x^2$ after expansion and collecting the terms of the following expression (there are $k$ pairs of parentheses): $$((... (((x - 2)^2 - 2)^2 -2)^2 -... -2)^2 - 2)^2$$

Find all the triples of integers $ (a, b,c)$ such that: \[ \begin{array}{ccc}a\plus{}b\plus{}c &\equal{}& 24\\ a^{2}\plus{}b^{2}\plus{}c^{2}&\equal{}& 210\\ abc &\equal{}& 440\end{array}\]

The positive real numbers $a, b, c, d$ satisfy the equality $$\left(\frac{1}{a}+ \frac{1}{b}\right) \left(\frac{1}{c}+ \frac{1}{d}\right) + \frac{1}{ab}+ \frac{1}{cd} = \frac{6}{\sqrt{abcd}}$$ Find the value of the $$\frac{a^2+ac+c^2}{b^2-bd+d^2}$$

For how many integers $k$ does the following system of equations has a solution other than $a=b=c=0$ in the set of real numbers? \begin{align*} \begin{cases} a^2+b^2=kc(a+b),\\ b^2+c^2 = ka(b+c),\\ c^2+a^2=kb(c+a).\end{cases}\end{align*}

Let $a, b, c, d$ be non-negative reals such that $\frac{1}{a+3}+\frac{1}{b+3}+\frac{1}{c+3}+\frac{1}{d+3}=1$. Show that there exists a permutation $(x_1, x_2, x_3, x_4)$ of $(a, b, c, d)$, such that $$x_1x_2+x_2x_3+x_3x_4+x_4x_1 \geq 4.$$

(a) Does there exist a polynomial $ P(x)$ with coefficients in integers, such that $ P(d) \equal{} \frac{2008}{d}$ holds for all positive divisors of $ 2008$? (b) For which positive integers $ n$ does a polynomial $ P(x)$ with coefficients in integers exists, such that $ P(d) \equal{} \frac{n}{d}$ holds for all positive divisors of $ n$?

Suppose that $n\ge3$ is a natural number. Find the maximum value $k$ such that there are real numbers $a_1,a_2,...,a_n \in [0,1)$ (not necessarily distinct) that for every natural number like $j \le k$ , sum of some $a_i$-s is $j$. [i]Proposed by Navid Safaei [/i]