For all $n \in {\mathbb Z^+}$ we define $$I_n=\{\frac{0}{n},\frac{1}{n},\frac{2}{n},\dotsm,\frac{n-1}{n},\frac{n}{n},\frac{n+1}{n},\dotsm\}$$ infinite cluster. For whichever $x$ and $y$ real number, we say $\mid{x-y}\mid$ is between distance of the $x$ and $y$. [b]a[/b]) For all $n$'s we find a number in $I_n$ such that, the between distance of the this number and $\sqrt 2$ is less than $\frac{1}{2n}$ [b]b[/b]) We find a $n$ such that, between distance of the a number in $I_n$ and $\sqrt 2$ is less than $\frac{1}{2011n}$

For a finite nonempty set $A$ of positive integers, $A =\{a_1, a_2,\dots , a_n\}$, we say the calamitous complement of A is the set of all positive integers $k$ for which there do not exist nonnegative integers $w_1, w_2, \dots , w_n$ with $k = a_1w_1 + a_2w_2 +\dots + a_nw_n.$ The calamitous complement of $A$ is denoted $cc(A)$. For example, $cc(\{5, 6, 9\}) = \{1, 2, 3, 4, 7, 8, 13\}$. Find all pairs of positive integers $a, b$ with $1 < a < b$ for which there exists a set $G$ satisfying all of the following properties: 1. $G$ is a set of at most three positive integers, 2. $cc(\{a, b\})$ and $cc(G)$ are both finite sets, and 3. $cc(G) = cc(\{a, b\})\cup \{m\}$ for some $m$ not in $cc(\{a, b\})$.

Given the polynomials $P(x)=px^4+qx^3+rx^2+sx+t,\ Q(x)=\frac{d}{dx}P(x)$, find the real numbers $p,\ q,\ r,\ s,\ t$ such that $P(\sqrt{-5})=0,\ Q(\sqrt{-2})=0$ and $\int_0^1 P(x)dx=-\frac{52}{5}.$

$ABC$ is an isosceles triangle such that $AB = AC$ and $\angle BAC = 96^o$. $D$ is the point for which $\angle ACD = 48^o$, $AD = BC$ and triangle $DAC$ is obtuse-angled. Find $\angle DAC$.

Consider a sequence of numbers between $0$ and $1$ in which the next number after $x$ is $1 - |1 - 2x|$. ($|x| = x$ if$ x \ge 0$, $|x| = -x$ if $x < 0$.) Prove that (a) if the first number of the sequence is rational, then the sequence will be periodic (i.e. the terms repeat with a certain cycle length after a certain term in the sequence); (b) if the sequence is periodic, then the first number is rational. (G Shabat)

Let $z_1 = 18 + 83i$, $z_2 = 18 + 39i, $ and $z_3 = 78 + 99i,$ where $i = \sqrt{-1}$. Let $z$ be the unique complex number with the properties that $\frac{z_3 - z_1}{z_2 - z_1} \cdot \frac{z - z_2}{z - z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part of $z$.

Let $l,m$ be two parallel lines in the plane. Let $P$ be a fixed point between them. Let $E,F$ be variable points on $l,m$ such that the angle $EPF$ is fixed to a number like $\alpha$ where $0<\alpha<\frac{\pi}2$. (By angle $EPF$ we mean the directed angle) Show that there is another point (not $P$) such that it sees the segment $EF$ with a fixed angle too.

Let $r = 1-\sqrt[5]{2}+ \sqrt[5]{4}-\sqrt[5]{8}+ \sqrt[5]{16}$. There exists a unique fifth-degree polynomial $P$ such that its leading coefficient is positive, all of its coefficients are integers whose greatest common factor (among all of them) is $1$, and $P(r) = 0$. Evaluate $P(10)$.

Find all positive integers $a,b,c$ such that $$\begin{cases} a^2 = 4(b+c) \\ a^3 -2b^3 -4c^3 =\frac12 abc \end {cases}$$

Three circle of unit radius passing through the point $P$ and one of the points of $A, B$ and $C$ each. What can be the radius of the circumcircle of the triangle $ABC$?

Let $b_1$, $\dots$ , $b_n$ be nonnegative integers with sum $2$ and $a_0$, $a_1$, $\dots$ , $a_n$ be real numbers such that $a_0=a_n=0$ and $|a_i-a_{i-1}|\leq b_i$ for each $i=1$, $\dots$ , $n$. Prove that $$\sum_{i=1}^n(a_i+a_{i-1})b_i\leq 2$$ [hide]I believe that the original problem was for nonnegative real numbers and it was a typo on the version of the exam paper we had but I'm not sure the inequality would hold[/hide]

Five different awards are to be given to three students. Each student will receive at least one award. In how many ways can the awards be distributed? $\textbf{(A)}\ 120 \qquad \textbf{(B)}\ 150 \qquad \textbf{(C)}\ 180 \qquad \textbf{(D)}\ 210 \qquad \textbf{(E)}\ 240$

Find the two last digits of $2012^{2012}$.

Define a [i]domino[/i] to be an ordered pair of [i]distinct[/i] positive integers. A [i]proper sequence[/i] of dominoes is a list of distinct dominoes in which the first coordinate of each pair after the first equals the second coordinate of the immediately preceding pair, and in which $(i, j)$ and $(j, i)$ do not [i]both[/i] appear for any $i$ and $j$. Let $D_n$ be the set of all dominoes whose coordinates are no larger than $n$. Find the length of the longest proper sequence of dominoes that can be formed using the dominoes of $D_n$.

In a room there are $144$ people. They are joined by $n$ other people who are each carrying $k$ coins. When these coins are shared among all $n + 144$ people, each person has $2$ of these coins. Find the minimum possible value of $2n + k$.

Given an acute-angled triangle $ABC$ whose altitudes from $B$ and $C$ intersect at $H$, let $P$ be any point on side $BC$ and $X, Y$ be points on $AB, AC$, respectively, such that $PB = PX$ and $PC = PY$. Prove that the points $A, H, X, Y$ lie on a common circle.