Albert and Kevin are playing a game. Kevin has a $10\%$ chance of winning any given round in the match. If Kevin wins the first game, he wins the match. If not, he requests that the match be extended to a best of $3$. If he wins the best of $3$, he wins the match. If not, then he requests the match be extended to a best of $5$, and so forth. What is the probability that Kevin eventually wins the match? (A best of $2n+ 1$ match consists of a series of rounds. The first person to reach $n + 1$ winning games wins the match)

In a small pond there are eleven lily pads in a row labeled $0$ through $10$. A frog is sitting on pad $1$. When the frog is on pad $N$, $0<N<10$, it will jump to pad $N-1$ with probability $\frac{N}{10}$ and to pad $N+1$ with probability $1-\frac{N}{10}$. Each jump is independent of the previous jumps. If the frog reaches pad $0$ it will be eaten by a patiently waiting snake. If the frog reaches pad $10$ it will exit the pond, never to return. What is the probability that the frog will escape being eaten by the snake? $ \textbf {(A) } \frac{32}{79} \qquad \textbf {(B) } \frac{161}{384} \qquad \textbf {(C) } \frac{63}{146} \qquad \textbf {(D) } \frac{7}{16} \qquad \textbf {(E) } \frac{1}{2} $

A sequence \(\{a_n\}\) is defined as follows: \(a_1 = 1\), \(a_2 = \frac{1}{3}\), and for all \(n \geq 1,\) \(\frac{(1+a_n)(1+a_{n+2})}{(1+a_n+1)^2} = \frac{a_na_{n+2}}{a_{n+1}^2}\). Prove that, for all \(n \geq 1\), \(a_1 + a_2 + ... + a_n < \frac{34}{21}\).

In boxes labeled $0$, $1$, $2$, $\dots$, we place integers according to the following rules: $\bullet$ If $p$ is a prime number, we place it in box $1$. $\bullet$ If $a$ is placed in box $m_a$ and $b$ is placed in box $m_b$, then $ab$ is placed in the box labeled $am_b+bm_a$. Find all positive integers $n$ that are placed in the box labeled $n$.

Let $S$ be the set of all $5^6$ positive integers, whose decimal representation consists of exactly 6 odd digits. Find the number of solutions $(x,y,z)$ of the equation $x+y=10z$, where $x\in S$, $y\in S$, $z\in S$.

Given $n\ge 4$ real numbers $a_{n}>...>a_{1} > 0$. For $r > 0$, let $f_{n}(r)$ be the number of triples $(i,j,k)$ with $1\leq i<j<k\leq n$ such that $\frac{a_{j}-a_{i}}{a_{k}-a_{j}}=r$. Prove that ${f_{n}(r)}<\frac{n^{2}}{4}$.

We call the sequence $d_1,....,d_n$ of natural numbers, not necessarily distinct, [b]covering[/b] if there exists arithmetic progressions like $c_1+kd_1$,....,$c_n+kd_n$ such that every natural number has come in at least one of them. We call this sequence [b]short[/b] if we can not delete any of the $d_1,....,d_n$ such that the resulting sequence be still covering. [b]a)[/b] Suppose that $d_1,....,d_n$ is a short covering sequence and suppose that we've covered all the natural numbers with arithmetic progressions $a_1+kd_1,.....,a_n+kd_n$, and suppose that $p$ is a prime number that $p$ divides $d_1,....,d_k$ but it does not divide $d_{k+1},....,d_n$. Prove that the remainders of $a_1,....,a_k$ modulo $p$ contains all the numbers $0,1,.....,p-1$. [b]b)[/b] Write anything you can about covering sequences and short covering sequences in the case that each of $d_1,....,d_n$ has only one prime divisor. [i]proposed by Ali Khezeli[/i]

How many sides of the convex polygon can equal its longest diagonal?

For a natural number $ n$, let $ a_n\equal{}\int_0^{\frac{\pi}{4}} (\tan x)^{2n}dx$. Answer the following questions. (1) Find $ a_1$. (2) Express $ a_{n\plus{}1}$ in terms of $ a_n$. (3) Find $ \lim_{n\to\infty} a_n$. (4) Find $ \lim_{n\to\infty} \sum_{k\equal{}1}^n \frac{(\minus{}1)^{k\plus{}1}}{2k\minus{}1}$.

Find all real solutions of the system of equations $$\begin{cases} (x_3 + x_4 + x_5)^5 = 3x_1 \\ (x_4 + x_5 + x_1)^5 = 3x_2\\ (x_5 + x _1 + x_2)^5 = 3x_3\\ (x_1 + x_2 + x_3)^5 = 3x_4\\ (x_2 + x_3 + x_4)^5 = 3x_5 \end{cases}$$ (L. Tumescu , Romania)