Figures $ 0$, $ 1$, $ 2$, and $ 3$ consist of $ 1$, $ 5$, $ 13$, and $ 25$ nonoverlapping squares, respectively. If the pattern were continued, how many nonoverlapping squares would there be in figure $ 100$? [asy] unitsize(8); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((9,0)--(10,0)--(10,3)--(9,3)--cycle); draw((8,1)--(11,1)--(11,2)--(8,2)--cycle); draw((19,0)--(20,0)--(20,5)--(19,5)--cycle); draw((18,1)--(21,1)--(21,4)--(18,4)--cycle); draw((17,2)--(22,2)--(22,3)--(17,3)--cycle); draw((32,0)--(33,0)--(33,7)--(32,7)--cycle); draw((29,3)--(36,3)--(36,4)--(29,4)--cycle); draw((31,1)--(34,1)--(34,6)--(31,6)--cycle); draw((30,2)--(35,2)--(35,5)--(30,5)--cycle); label("Figure",(0.5,-1),S); label("$0$",(0.5,-2.5),S); label("Figure",(9.5,-1),S); label("$1$",(9.5,-2.5),S); label("Figure",(19.5,-1),S); label("$2$",(19.5,-2.5),S); label("Figure",(32.5,-1),S); label("$3$",(32.5,-2.5),S);[/asy]$ \textbf{(A)}\ 10401 \qquad \textbf{(B)}\ 19801 \qquad \textbf{(C)}\ 20201 \qquad \textbf{(D)}\ 39801 \qquad \textbf{(E)}\ 40801$

We may say concerning the solution of \[ |x|^2 \plus{} |x| \minus{} 6 \equal{} 0 \] that: $ \textbf{(A)}\ \text{there is only one root}\qquad \textbf{(B)}\ \text{the sum of the roots is }{\plus{}1}\qquad \textbf{(C)}\ \text{the sum of the roots is }{0}\qquad \\ \textbf{(D)}\ \text{the product of the roots is }{\plus{}4}\qquad \textbf{(E)}\ \text{the product of the roots is }{\minus{}6}$

Does there exists two functions $f,g :\mathbb{R}\rightarrow \mathbb{R}$ such that: $\forall x\not =y : |f(x)-f(y)|+|g(x)-g(y)|>1$ Time allowed for this problem was 75 minutes.

Let be a function satisfying [url=http://mathworld.wolfram.com/CauchyFunctionalEquation.html]Cauchy's functional equation,[/url] and having the property that it's monotonic on a real interval. Prove that this function is globally monotonic. [i]Florian Dumitrel[/i]

Provided that the roots of the polynom $ X^n+a_1X^{n-1} +a_2X^{n-2} +\cdots +a_{n-1}X +a_n:\in\mathbb{R}[X] , $ of degree $ n\ge 2, $ are all real and pairwise distinct, prove that there exists is a neighbourhood $ \mathcal{V} $ of $ \left( a_1,a_2,\ldots ,a_n \right) $ in $ \mathbb{R}^n $ and $ n $ functions $ x_1,x_2,\ldots ,x_n\in\mathcal{C}^{\infty } \left( \mathcal{V} \right) $ whose values at $ \left( a_1,a_2,\ldots ,a_n \right) $ are roots of the mentioned polynom.

Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy $$(f(a)-f(b))(f(b)-f(c))(f(c)-f(a)) = f(ab^2+bc^2+ca^2) - f(a^2b+b^2c+c^2a)$$for all real numbers $a$, $b$, $c$. [i]Proposed by Ankan Bhattacharya, USA[/i]

For every natural number $ r$, the set of $ r$-tuples of natural numbers is partitioned into finitely many classes. Show that if $ f(r)$ is a function such that $ f(r)\geq 1$ and $ \lim _{r\rightarrow \infty} f(r)\equal{}\plus{}\infty$, then there exists an infinite set of natural numbers that, for all $ r$, contains $ r$-triples from at most $ f(r)$ classes. Show that if $ f(r) \not \rightarrow \plus{}\infty$, then there is a family of partitions such that no such infinite set exists. [i]P. Erdos, A. Hajnal[/i]

Let $\mathcal{F}$ be the set of all functions $f : (0,\infty)\to (0,\infty)$ such that $f(3x) \geq f( f(2x) )+x$ for all $x$. Find the largest $A$ such that $f(x) \geq A x$ for all $f\in\mathcal{F}$ and all $x$.

Let $f(x)$ be a function which is differentiable twice and $f''(x)>0$ on $[0,\ 1]$. For a positive integer $n$, find $\lim_{n\to\infty} n\left\{\int_0^1 f(x)\ dx-\frac{1}{n}\sum_{k=0}^{n-1} f\left(\frac{k}{n}\right)\right\}.$

Let $ f$ be a function for which $ f(x/3) \equal{} x^2 \plus{} x \plus{} 1$. Find the sum of all values of $ z$ for which $ f(3z) \equal{} 7$. $ \textbf{(A)}\ \minus{} 1/3 \qquad \textbf{(B)}\ \minus{} 1/9 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 5/9 \qquad \textbf{(E)}\ 5/3$

Let \[F=\log\dfrac{1+x}{1-x}.\] Find a new function $G$ by replacing each $x$ in $F$ by \[\dfrac{3x+x^3}{1+3x^2},\] and simplify. The simplified expression $G$ is equal to: $\textbf{(A)}\ -F \qquad \textbf{(B)}\ F\qquad \textbf{(C)}\ 3F \qquad \textbf{(D)}\ F^3 \qquad \textbf{(E)}\ F^3-F$

Find all strictly increasing functions $f:\mathbb{N}\rightarrow\mathbb{N}$ that satisfy \[f(n+f(n))=2f(n)\quad\text{for all}\ n\in\mathbb{N} \]

Let $f \colon [0,1] \to \mathbb{R}$ be a continuous function which is differentiable on $(0,1)$. Prove that either $f(x) = ax + b$ for all $x \in [0,1]$ for some constants $a,b \in \mathbb{R}$ or there exists $t \in (0,1)$ such that $|f(1) - f(0)| < |f'(t)|$.

Let $ f$ be a continuous function on the unit interval $ [0,1]$. Show that \[ \lim_{n \rightarrow \infty} \int_0^1... \int_0^1f(\frac{x_1+...+x_n}{n})dx_1...dx_n=f(\frac12)\] and \[ \lim_{n \rightarrow \infty} \int_0^1... \int_0^1f (\sqrt[n]{x_1...x_n})dx_1...dx_n=f(\frac1e).\]

Let $S$ be the set of positive integers. For any $a$ and $b$ in the set we have $GCD(a, b)>1$. For any $a$, $b$ and $c$ in the set we have $GCD(a, b, c)=1$. Is it possible that $S$ has $2012$ elements? [i]Proposed by Ognjen Stipetić.[/i]

Let $f:[0,1]\to\mathbb R$ be an increasing function. Prove that if $\int_0^1f(x)dx=\int_0^1\left(\int_0^xf(t)dt\right)dx=0$ then $f(x)=0,\forall x\in(0,1)$. [i]Mihai Piticari[/i]

Suppose that $V$ is a finite dimensional vector space over the real numbers equipped with an inner product and $S:V\times V \longrightarrow \mathbb R$ is a skew symmetric function that is linear for each variable when others are kept fixed. Prove there exists a linear transformation $T:V \longrightarrow V$ such that $\forall u,v \in V: S(u,v)=<u,T(v)>$. We know that there always exists $v\in V$ such that $W=<v,T(v)>$ is invariant under $T$. (it means $T(W)\subseteq W$). Prove that if $W$ is invariant under $T$ then the following subspace is also invariant under $T$: $W^{\perp}=\{v\in V:\forall u\in W <v,u>=0\}$. Prove that if dimension of $V$ is more than $3$, then there exist a two dimensional subspace $W$ of $V$ such that the volume defined on it by function $S$ is zero!!!! (This is the way that we can define a two dimensional volume for each subspace $V$. This can be done for volumes of higher dimensions.)

Let $a$ and $b$ be positive real numbers such that $a+b=1$. Prove that $a^ab^b+a^bb^a\le 1$.

Find all non-decreasing functions from real numbers to itself such that for all real numbers $x,y$ $f(f(x^2)+y+f(y))=x^2+2f(y)$ holds.

If $x$ is a real number, then the quantity $(1-|x|)(1+x)$ is positive if and only if $\textbf{(A) }|x|<1\qquad\textbf{(B) }|x|>1\qquad\textbf{(C) }x<-1\text{ or }-1<x<1\qquad$ $\textbf{(D) }x<1\qquad \textbf{(E) }x<-1$

Determine all strictly increasing functions $ f: R\rightarrow R$ satisfying relationship $ f(x\plus{}f(y))\equal{}f(x\plus{}y)\plus{}2005$ for any real values of x and y.

Find all f:N $\longrightarrow$ N that: [list][b]a)[/b] $f(m)=1 \Longleftrightarrow m=1 $ [b]b)[/b] $d=gcd(m,n) f(m\cdot n)= \frac{f(m)\cdot f(n)}{f(d)} $ [b]c)[/b] $ f^{2000}(m)=f(m) $[/list]

Let $\overline{a_1a_2a_3\ldots a_n}$ be the representation of a $n-$digits number in base $10.$ Prove that there exists a one-to-one function like $f : \{0, 1, 2, 3, \ldots, 9\} \to \{0, 1, 2, 3, \ldots, 9\}$ such that $f(a_1) \neq 0$ and the number $\overline{ f(a_1)f(a_2)f(a_3) \ldots f(a_n) }$ is divisible by $3.$

Find the function $f(x)$ that satisfies the equation $$f'(x) + x^2f(x) = 0$$ knowing that $f(1) = e$. Graph this function and calculate the tangent of the curve at the point of abscissa $1$.

A multi-digit number is written on the blackboard. Susan puts in a number of plus signs between some pairs of adjacent digits. The addition is performed and the process is repeated with the sum. Prove that regardless of what number was initially on the blackboard, Susan can always obtain a single-digit number in at most ten steps.