The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).

Which of the following series are convergent? $\textbf{(A)}~\sum_{n=1}^\infty\sqrt{\frac{2n^2+3}{5n^3+1}}$ $\textbf{(B)}~\sum_{n=1}^\infty\frac{(n+1)^n}{n^{n+3/2}}$ $\textbf{(C)}~\sum_{n=1}^\infty n^2x\left(1-x^2\right)^n$ $\textbf{(D)}~\text{None of the above}$

The value of $ \frac{3}{a\plus{}b}$ when $ a\equal{}4$ and $ b\equal{}\minus{}4$ is: $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac{3}{8} \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ \text{any finite number} \qquad \textbf{(E)}\ \text{meaningless}$

Let $A,B\in \mathcal{M}_n(\mathbb{R})$. Prove that $\text{rank}\ A+\text{rank}\ B\le n$ if and only if there exists an invertible matrix $X\in \mathcal{M}_n(\mathbb{R})$ such that $AXB=O_n$.

Draw the smaller number of line segments connecting points of the figure such that the new figure obtained to have exactly: [img]https://cdn.artofproblemsolving.com/attachments/d/1/098e03714904573a1eacd2d3dc28b4e8c42c7c.png[/img] i) one axis of symmetry ii) two axes of symmetry iii) four axes of symmetry Draw a new figure, at each case.

Which answer choice correctly fills the blank in the statement below? "The probability of flipping heads on a fair coin is the equal to the probability of rolling a $\underline{~~~~~~~~~~~}$ on a fair dice." $\textbf{(A) }\text{prime number}\qquad\textbf{(B) }\text{number divisible by 3}\qquad\textbf{(C) }\text{number with four factors}\qquad\textbf{(D) }2~\text{or}~3\qquad\textbf{(E) }4$

We call a matrix $\textsl{binary matrix}$ if all its entries equal to $0$ or $1$. A binary matrix is $\textsl{Good}$ if it simultaneously satisfies the following two conditions: (1) All the entries above the main diagonal (from left to right), not including the main diagonal, are equal. (2) All the entries below the main diagonal (from left to right), not including the main diagonal, are equal. Given positive integer $m$, prove that there exists a positive integer $M$, such that for any positive integer $n>M$ and a given $n \times n$ binary matrix $A_n$, we can select integers $1 \leq i_1 <i_2< \cdots < i_{n-m} \leq n$ and delete the $i_i$-th, $i_2$-th,$\cdots$, $i_{n-m}$-th rows and $i_i$-th, $i_2$-th,$\cdots$, $i_{n-m}$-th columns of $A_n$, then the resulting binary matrix $B_m$ is $\textsl{Good}$.

Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

Let $S = \{(x, y) \in Z^2 | 0 \le x \le 11, 0\le y \le 9\}$. Compute the number of sequences $(s_0, s_1, . . . , s_n)$ of elements in $S$ (for any positive integer $n \ge 2$) that satisfy the following conditions: $\bullet$ $s_0 = (0, 0)$ and $s_1 = (1, 0)$, $\bullet$ $s_0, s_1, . . . , s_n$ are distinct, $\bullet$ for all integers $2 \le i \le n$, $s_i$ is obtained by rotating $s_{i-2}$ about $s_{i-1}$ by either $90^o$ or $180^o$ in the clockwise direction.

Prove that if $k$ is an even positive integer then it is possible to write the integers from $1$ to $k-1$ in such an order that the sum of no set of successive numbers is divisible by $k$ .

In a word formed with the letters $a,b$ we can change some blocks: $aba$ in $b$ and back, $bba$ in $a$ and backwards. If the initial word is $aaa\ldots ab$ where $a$ appears 2003 times can we reach the word $baaa\ldots a$, where $a$ appears 2003 times.

Which of the following is equivalent to "If P is true, then Q is false."? $ \textbf{(A)}\ \text{``P is true or Q is false.''} \qquad$ $ \textbf{(B)}\ \text{``If Q is false then P is true.''} \qquad$ $ \textbf{(C)}\ \text{``If P is false then Q is true.''} \qquad$ $ \textbf{(D)}\ \text{``If Q is true then P is false.''} \qquad$ $ \textbf{(E)}\ \text{``If Q is true then P is true.''}$

Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity \[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\] [i]Proposed by Japan[/i]

A collection of 8 cubes consists of one cube with edge-length $k$ for each integer $k,\thinspace 1 \le k \le 8.$ A tower is to be built using all 8 cubes according to the rules: $\bullet$ Any cube may be the bottom cube in the tower. $\bullet$ The cube immediately on top of a cube with edge-length $k$ must have edge-length at most $k+2.$ Let $T$ be the number of different towers than can be constructed. What is the remainder when $T$ is divided by 1000?

If $a$ and $b$ are positive integers such that $a^2-b^4= 2009$, find $a+b$.