Each one of $2001$ children chooses a positive integer and writes down his number and names of some of other $2000$ children to his notebook. Let $A_c$ be the sum of the numbers chosen by the children who appeared in the notebook of the child $c$. Let $B_c$ be the sum of the numbers chosen by the children who wrote the name of the child $c$ into their notebooks. The number $N_c = A_c - B_c$ is assigned to the child $c$. Determine whether all of the numbers assigned to the children could be positive.

Find all surjective functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $m,n\in \mathbb{N}$: \[m \vert n \Longleftrightarrow f(m) \vert f(n).\]

Find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that \[ f( xf(x) + 2y) = f(x^2)+f(y)+x+y-1 \] holds for all $ x, y \in \mathbb{R}$.

Let the continuous functions $ f_n(x), \; n\equal{}1,2,3,...,$ be defined on the interval $ [a,b]$ such that every point of $ [a,b]$ is a root of $ f_n(x)\equal{}f_m(x)$ for some $ n \not\equal{} m$. Prove that there exists a subinterval of $ [a,b]$ on which two of the functions are equal.

Let $S$ be the set of all rational numbers whose denominators are powers of 3. Let $a$, $b$ and $c$ be given non-zero real numbers. Determine all real-valued functions $f$ that are defined for $x \in S$, satisfy \[ f(x) = af(3x) + bf(3x - 1) + cf(3x - 2) \textrm{ if }0 \leq x \leq 1, \] and are zero elsewhere.

We are given the finite sets $ X$, $ A_1$, $ A_2$, $ \dots$, $ A_{n \minus{} 1}$ and the functions $ f_i: \ X\rightarrow A_i$. A vector $ (x_1,x_2,\dots,x_n)\in X^n$ is called [i]nice[/i], if $ f_i(x_i) \equal{} f_i(x_{i \plus{} 1})$, for each $ i \equal{} 1,2,\dots,n \minus{} 1$. Prove that the number of nice vectors is at least \[ \frac {|X|^n}{\prod\limits_{i \equal{} 1}^{n \minus{} 1} |A_i|}. \]

Let $f$ be a function such that $f(x)+f(x+1)=2^x$ and $f(0)=2010$. Find the last two digits of $f(2010)$.

Find the number of real solutions of the equation $a e^x = x^3$, where $a$ is a real parameter.

Determine all functions $ f$ defined in the set of rational numbers and taking their values in the same set such that the equation $ f(x + y) + f(x - y) = 2f(x) + 2f(y)$ holds for all rational numbers $x$ and $y$.

Let $T$ the set of the infinite sequences of integers. For two given elements in $T$: $(a_{1},a_{2},a_{3},...)$ and $(b_{1},b_{2},b_{3},...)$, define the sum $(a_{1},a_{2},a_{3},...)+(b_{1},b_{2},b_{3},...)=(a_{1}+b_{1},a_{2}+b_{2},a_{3}+b_{3},...)$. Let $f: T\rightarrow$ $\mathbb{Z}$ a function such that: i) If $x\in T$ has exactly one of your terms equal $1$ and all the others equal $0$, then $f(x)=0$. ii)$f(x+y)=f(x)+f(y)$, for all $x,y\in T$. Prove that $f(x)=0$ for all $x\in T$

Let $ f : [0, \plus{}\infty) \to \mathbb R$ be a differentiable function. Suppose that $ \left|f(x)\right| \le 5$ and $ f(x)f'(x) \ge \sin x$ for all $ x \ge 0$. Prove that there exists $ \lim_{x\to\plus{}\infty}f(x)$.

Let $r_1,r_2,\ldots,r_m$ be positive rational numbers with a sum of $1$. Find the maximum values of the function $f:\mathbb N\to\mathbb Z$ defined by $$f(n)=n-\lfloor r_1n\rfloor-\lfloor r_2n\rfloor-\ldots-\lfloor r_mn\rfloor$$

Let $r_1,r_2,\dots ,r_n$ be real numbers. Given $n$ reals $a_1,a_2,\dots ,a_n$ that are not all equal to $0$, suppose that inequality \[r_1(x_1-a_1)+ r_2(x_2-a_2)+\dots + r_n(x_n-a_n)\leq\sqrt{x_1^2+ x_2^2+\dots + x_n^2}-\sqrt{a_1^2+a_2^2+\dots +a_n^2}\] holds for arbitrary reals $x_1,x_2,\dots ,x_n$. Find the values of $r_1,r_2,\dots ,r_n$.

Prove that : $\left|\sum_{i=0}^n \left(1-\pi \sin \frac{i\pi}{4n}\cos \frac{i\pi}{4n}\right)\right|<1.$

$\mathbb{N}$ is the set of positive integers. Determine all functions $f:\mathbb{N}\to\mathbb{N}$ such that for every pair $(m,n)\in\mathbb{N}^2$ we have that: \[f(m)+f(n) \ | \ m+n .\]

Let a sequence $\{u_n\}$ be defined by $u_1=5$ and the relation $u_{n+1}-u_n=3+4(n-1)$, $n=1,2,3,\cdots$. If $u_n$ is expressed as a polynomial in $n$, the algebraic sum of its coefficients is: $\textbf{(A) }3\qquad \textbf{(B) }4\qquad \textbf{(C) }5\qquad \textbf{(D) }6\qquad \textbf{(E) }11$

Let $S$ be a set of positive integers $n_1, n_2, \cdots, n_6$ and let $n(f)$ denote the number $n_1n_{f(1)} +n_2n_{f(2)} +\cdots+n_6n_{f(6)}$, where $f$ is a permutation of $\{1, 2, . . . , 6\}$. Let \[\Omega=\{n(f) | f \text{ is a permutation of } \{1, 2, . . . , 6\} \} \] Give an example of positive integers $n_1, \cdots, n_6$ such that $\Omega$ contains as many elements as possible and determine the number of elements of $\Omega$.

Determine whether there exists a polynomial $f(x_1, x_2)$ with two variables, with integer coefficients, and two points $A=(a_1, a_2)$ and $B=(b_1, b_2)$ in the plane, satisfying the following conditions: (i) $A$ is an integer point (i.e $a_1$ and $a_2$ are integers); (ii) $|a_1-b_1|+|a_2-b_2|=2010$; (iii) $f(n_1, n_2)>f(a_1, a_2)$ for all integer points $(n_1, n_2)$ in the plane other than $A$; (iv) $f(x_1, x_2)>f(b_1, b_2)$ for all integer points $(x_1, x_2)$ in the plane other than $B$. [i]Massimo Gobbino, Italy[/i]

Find continuous functions $ f(x),\ g(x)$ which takes positive value for any real number $ x$, satisfying $ g(x)\equal{}\int_0^x f(t)\ dt$ and $ \{f(x)\}^2\minus{}\{g(x)\}^2\equal{}1$.

Show that \[ \frac{(b+c)(a^4-b^2c^2)}{ab+2bc+ca}+\frac{(c+a)(b^4-c^2a^2)}{bc+2ca+ab}+\frac{(a+b)(c^4-a^2b^2)}{ca+2ab+bc} \geq 0 \] for all positive real numbers $a, \: b , \: c.$

An infinitely long slab of glass is rotated. A light ray is pointed at the slab such that the ray is kept horizontal. If $\theta$ is the angle the slab makes with the vertical axis, then $\theta$ is changing as per the function \[ \theta(t) = t^2, \]where $\theta$ is in radians. Let the $\emph{glassious ray}$ be the ray that represents the path of the refracted light in the glass, as shown in the figure. Let $\alpha$ be the angle the glassious ray makes with the horizontal. When $\theta=30^\circ$, what is the rate of change of $\alpha$, with respect to time? Express your answer in radians per second (rad/s) to 3 significant figures. Assume the index of refraction of glass to be $1.50$. Note: the second figure shows the incoming ray and the glassious ray in cyan. [asy] fill((6,-2-sqrt(3))--(3, -2)--(3+4/sqrt(3),2)--(6+4/sqrt(3),2-sqrt(3))--cycle, gray(0.7)); draw((3+2/sqrt(3),0)--(10,0),linetype("4 4")+grey); draw((0,0)--(3+2/sqrt(3),0)); draw((3+2/sqrt(3), 0)--(7+2/sqrt(3), -1)); arrow((6.5+2/sqrt(3), -7/8), dir(180-14.04), 9); draw((3,-2)--(3,2), linetype("4 4")); draw((6,-2-sqrt(3))--(3,-2)--(3+2/sqrt(3),0)--(3+4/sqrt(3), 2)--(6+4/sqrt(3), 2-sqrt(3))); draw(anglemark((3+2/sqrt(3),0),(3,-2),(3,0),15)); label("$\theta$", (3.2, -1.6), N, fontsize(8)); label("$\alpha$", (6, -0.2), fontsize(8)); [/asy] [asy] fill((6,-2-sqrt(3))--(3, -2)--(3+4/sqrt(3),2)--(6+4/sqrt(3),2-sqrt(3))--cycle, gray(0.7)); draw((3+2/sqrt(3),0)--(10,0),linetype("4 4")+grey); draw((0,0)--(3+2/sqrt(3),0), cyan); draw((3+2/sqrt(3), 0)--(7+2/sqrt(3), -1), cyan); arrow((6.5+2/sqrt(3), -7/8), dir(180-14.04), 9, cyan); draw((3,-2)--(3,2), linetype("4 4")); draw((6,-2-sqrt(3))--(3,-2)--(3+2/sqrt(3),0)--(3+4/sqrt(3), 2)--(6+4/sqrt(3), 2-sqrt(3))); draw(anglemark((3+2/sqrt(3),0),(3,-2),(3,0),15)); label("$\theta$", (3.2, -1.6), N, fontsize(8)); label("$\alpha$", (6, -0.2), fontsize(8)); [/asy] [i]Problem proposed by Ahaan Rungta[/i]

Let $f:[0,1]\rightarrow\mathbb{R}$ be an integrable function such that: \[0<\left\vert \int_{0}^{1}f(x)\, \text{d}x\right\vert\le 1.\] Show that there exists $x_1\not= x_2, x_1,x_2\in [0,1]$, such that: \[\int_{x_1}^{x_2}f(x)\, \text{d}x=(x_1-x_2)^{2002}\]

Determine all functions $f:\mathbb{R}\to\mathbb{R}$, where $\mathbb{R}$ is the set of all real numbers, satisfying the following two conditions: 1) There exists a real number $M$ such that for every real number $x,f(x)<M$ is satisfied. 2) For every pair of real numbers $x$ and $y$, \[ f(xf(y))+yf(x)=xf(y)+f(xy)\] is satisfied.

Let $ \lambda $ be the Lebesgue measure in the plane, let $ u,v\in\mathbb{R}^2, $ let $ A\subset\mathbb{R}^2 $ such that $ \lambda (A)>0 $ and let be the function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ defined as $$ f(t)=\int_A \chi_{A+tu}\cdot\chi_{A+tv}\cdot d\lambda $$ [b]a)[/b] Show that $ f $ is continuous. [b]b)[/b] Prove that any Lebesgue measurable subset of the plane that has nonzero Lebesgue measure contains the vertices of an equilateral triangle.

Calculate the $100$th derivative of the function \[\frac{1}{x^2+3x+2}\] at $x=0$ with $10\%$ relative error.