Let $D$ be the set of all pairs $(i,j)$, $1\le i,j\le n$. Prove there exists a subset $S \subset D$, with $|S|\ge\left \lfloor\frac{3n(n+1)}{5}\right \rfloor$, such that for any $(x_1,y_1), (x_2,y_2) \in S$ we have $(x_1+x_2,y_1+y_2) \not \in S$. (Peter Cameron)

Define $ f: \mathbb{R}\to\mathbb{R}$ by \[ f(x)\equal{}\begin{cases}x&\text{if }x\le e\\ xf(\ln x)&\text{if }x>e\end{cases}\] Does $ \displaystyle\sum_{n\equal{}1}^{\infty}\frac1{f(n)}$ converge?

A flea moves in the positive direction on the real Ox axis, starting from the origin. He can only jump over distances equal with $\sqrt 2$ or $\sqrt{2005}$. Prove that there exists $n_0$ such that the flea can reach any interval $[n,n+1]$ with $n\geq n_0$.

Given a positive integer $k$, find the least integer $n_k$ for which there exist five sets $S_1, S_2, S_3, S_4, S_5$ with the following properties: \[|S_j|=k \text{ for } j=1, \cdots , 5 , \quad |\bigcup_{j=1}^{5} S_j | = n_k ;\] \[|S_i \cap S_{i+1}| = 0 = |S_5 \cap S_1|, \quad \text{for } i=1,\cdots ,4 \]

Each term of an infinite sequence of natural numbers is obtained from the previous term by adding to it one of its nonzero digits. Prove that this sequence contains an even number.

Let $ X_1,X_2,\ldots, X_n$ be independent, identically distributed, nonnegative random variables with a common continuous distribution function $ F$. Suppose in addition that the inverse of $ F$, the quantile function $ Q$, is also continuous and $ Q(0)=0$. Let $ 0=X_{0: n} \leq X_{1: n} \leq \ldots \leq X_{n: n}$ be the ordered sample from the above random variables. Prove that if $ EX_1$ is finite, then the random variable \[ \Delta = \sup_{0\leq y \leq 1} \left| \frac 1n \sum_{i=1}^{\lfloor ny \rfloor +1} (n+1-i)(X_{i: n}-X_{i-1: n})- \int_0^y (1-u)dQ(u) \right|\] tends to zero with probability one as $ n \rightarrow \infty$. [i]S. Csorgp, L. Horvath[/i]

Let ${(a_n)_{n\ge1}} $ be a sequence with ${a_1 = 1} $ and ${a_{n+1} = \lfloor a_n +\sqrt{a_n}+\frac{1}{2}\rfloor }$ for all ${n \ge 1}$, where ${\lfloor x \rfloor}$ denotes the greatest integer less than or equal to ${x}$. Find all ${n \le 2013}$ such that ${a_n}$ is a perfect square

Is there a positive integer $n$ such that the fractional part of \[ \left(3+\sqrt{5}\right)^n >0.99 ? \]

Find all numbers $\alpha$ for which the equation \[x^2 - 2x[x] + x -\alpha = 0\] has two nonnegative roots. ($[x]$ denotes the largest integer less than or equal to x.)

In order to complete a large job, 1000 workers were hired, just enough to complete the job on schedule. All the workers stayed on the job while the first quarter of the work was done, so the first quarter of the work was completed on schedule. Then 100 workers were laid off, so the second quarter of the work was completed behind schedule. Then an additional 100 workers were laid off, so the third quarter of the work was completed still further behind schedule. Given that all workers work at the same rate, what is the minimum number of additional workers, beyond the 800 workers still on the job at the end of the third quarter, that must be hired after three-quarters of the work has been completed so that the entire project can be completed on schedule or before?

Define $\lfloor x \rfloor$ as the largest integer less than or equal to $x$. Define $\{x \} = x - \lfloor x \rfloor$. For example, $\{ 3 \} = 3-3 = 0$, $\{ \pi \} = \pi - 3$, and $\{ - \pi \} = 4-\pi$. If $\{n\} + \{ 3n\} = 1.4$, then find the sum of all possible values of $100\{n\}$. [i]Proposed by Isabella Grabski [/i]

Two sequences $(x_n)_{n\in N}$ and $(y_n)_{n\in N}$ are defined recursively as follows: $x_0 = 2015$ and $x_{n+1} =\left \lfloor x_n \cdot \frac{y_{n+1}}{y_{n-1}} \right \rfloor$ for all $n \ge 0$, $y_0 = 307$ and $y_{n+1} = y_n + 1$ for all $n \ge 0$. Compute $\lim_{n\to \infty} \frac{x_n}{(y_n)^2}$.

Let $ S$ be the set of the $ 2005$ smallest multiples of $ 4$, and let $ T$ be the set of the $ 2005$ smallest positive multiples of $ 6$. How many elements are common to $ S$ and $ T$? $ \textbf{(A)}\ 166\qquad \textbf{(B)}\ 333\qquad \textbf{(C)}\ 500\qquad \textbf{(D)}\ 668\qquad \textbf{(E)}\ 1001$

For any positive integers $a$ and $b$, define $a \oplus b$ to be the result when adding $a$ to $b$ in binary (base $2$), neglecting any carry-overs. For example, $20 \oplus 14 = 10100_2 \oplus 1110_2 = 11010_2 = 26$. (The operation $\oplus$ is called the [i]exclusive or.[/i]) Compute the sum $$\sum^{2^{2014} -1}_{k=0} \left( k \oplus \left\lfloor \frac{k}{2} \right \rfloor \right).$$ Here $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.

Let $T=\text{TNFTPP}$. How many positive integers are within $T$ of exactly $\lfloor \sqrt T\rfloor$ perfect squares? (Note: $0^2=0$ is considered a perfect square.)

The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and \[5^n<2^m<2^{m+2}<5^{n+1}?\] $\textbf{(A) }278\qquad \textbf{(B) }279\qquad \textbf{(C) }280\qquad \textbf{(D) }281\qquad \textbf{(E) }282\qquad$

In convex quadrilateral $ABCD$, $\angle A \cong \angle C$, $AB = CD = 180$, and $AD \neq BC$. The perimeter of $ABCD$ is 640. Find $\lfloor 1000 \cos A \rfloor$. (The notation $\lfloor x \rfloor$ means the greatest integer that is less than or equal to $x$.)

Show that for all integers $a \ge 1$,$ \lfloor \sqrt{a}+\sqrt{a+1}+\sqrt{a+2}\rfloor = \lfloor \sqrt{9a+8}\rfloor$

Let $c$ be a fixed positive integer, and $\{x_k\}^{\inf}_{k=1}$ be a sequence such that $x_1=c$ and $x_n=x_{n-1}+\lfloor \frac{2x_{n-1}-2}{n} \rfloor$ for $n \ge 2$. Determine the explicit formula of $x_n$ in terms of $n$ and $c$. (Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)

Let $x$ and $y$ be two real numbers. Prove that the equations \[\lfloor x \rfloor + \lfloor y \rfloor =\lfloor x +y \rfloor , \quad \lfloor -x \rfloor + \lfloor -y \rfloor =\lfloor -x-y \rfloor\] Holds if and only if at least one of $x$ or $y$ be integer.

Find all real solutions $x$ to the equation $\lfloor x^2 - 2x \rfloor + 2\lfloor x \rfloor = \lfloor x \rfloor^2$.

You are walking along a road of constant width with sidewalks on each side. You can only walk on the sidewalks or cross the road perpendicular to the sidewalk. Coming up on a turn, you realize that you are on the “outside” of the turn; i.e., you are taking the longer way around the turn. The turn is a circular arc. Assuming that your destination is on the same side of the road as you are currently, let $\theta$ be the smallest turn angle, in radians, that would justify crossing the road and then crossing back after the turn to take the shorter total path to your destination. What is $\lfloor 100 \cdot \theta \rfloor$ ?

Compute the sum of all real solutions $\alpha$ (in radians) to the equation \[ |\sin\alpha|=\left\lfloor \frac{\alpha}{20} \right\rfloor. \]

Determine the number of digits $1$ in the integer part of $\frac{10^{1992}}{10^{83}+7}$.

Set positive integer $m=2^k\cdot t$, where $k$ is a non-negative integer, $t$ is an odd number, and let $f(m)=t^{1-k}$. Prove that for any positive integer $n$ and for any positive odd number $a\le n$, $\prod_{m=1}^n f(m)$ is a multiple of $a$.