We are given an infinite sequence of $ 1$'s and $ 2$'s with the following properties: (1) The first element of the sequence is $ 1$. (2) There are no two consecutive $ 2$'s or three consecutive $ 1$'s. (3) If we replace consecutive $ 1$'s by a single $ 2$, leave the single $ 1$'s alone, and delete the original $ 2$'s, then we recover the original sequence. How many $ 2$'s are there among the first $ n$ elements of the sequence? [i]P. P. Palfy[/i]

There are $m$ villages on the left bank of the Lena, $n$ villages on the right bank and one village on an island. It is known that $(m+1,n+1)>1$. Every two villages separated by water are connected by ferry with positive integral number. The inhabitants of each village say that all the ferries operating in their village have different numbers and these numbers form a segment of the series of the integers. Prove that at least some of them are wrong. [i](K. Kokhas)[/i]

A quadruple $(p, a, b, c)$ of positive integers is a[i] karaka quadruple[/i] if $\bullet$ $p$ is an odd prime number $\bullet$ $a, b$ and $c$ are distinct, and $\bullet$ $ab + 1$, $bc + 1$ and $ca + 1$ are divisible by $p$. (a) Prove that for every karaka quadruple $(p, a, b, c)$ we have $p + 2 \le\frac{a + b + c}{3}$. (b) Determine all numbers $p$ for which a karaka quadruple $(p, a, b, c)$ exists with $p + 2 =\frac{a + b + c}{3}$

How many roots of the equation $z^6 +6z +10=0$ lie in each quadrant of the complex plane?

Given the numbers $a_1, a_2, ... , a_n$ such that $$0\le a_1\le a_2\le 2a_1 , a_2\le a_3\le 2a_2 , ... , a_{n-1}\le a_n\le 2a_{n-1}$$ Prove that in the sum $s=\pm a1\pm a2\pm ...\pm a_n$ You can choose appropriate signs to make $0\le s\le a_1$.

There are real numbers $x, y,$ and $z$ such that the value of $$x+y+z-\left(\frac{x^2}{5}+\frac{y^2}{6}+\frac{z^2}{7}\right)$$ reaches its maximum of $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n + x + y + z.$

Using only pennies, nickels, dimes, and quarters, what is the smallest number of coins Freddie would need so he could pay any amount of money less than one dollar? $ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 25\qquad\textbf{(E)}\ 99 $

Let $n > 3$ be an integer. Let $\Omega$ be the set of all triples of distinct elements of $\{1, 2, \ldots , n\}$. Let $m$ denote the minimal number of colours which suffice to colour $\Omega$ so that whenever $1\leq a<b<c<d \leq n$, the triples $\{a,b,c\}$ and $\{b,c,d\}$ have different colours. Prove that $\frac{1}{100}\log\log n \leq m \leq100\log \log n$.

Let $n \ge 2$ be a positive integer, and write in a digit form \[\frac{1}{n}=0.a_1a_2\dots.\] Suppose that $n = a_1 + a_2 + \cdots$. Determine all possible values of $n$.

Evaluate $\frac{1}{2+\frac{3}{1+\frac{2}{3+x}}}$ if $x = 1$.

Abe holds 1 green and 1 red jelly bean in his hand. Bea holds 1 green, 1 yellow, and 2 red jelly beans in her hand. Each randomly picks a jelly bean to show the other. What is the probability that the colors match? $\textbf{(A)}\ \frac14 \qquad \textbf{(B)}\ \frac13 \qquad \textbf{(C)}\ \frac38 \qquad \textbf{(D)}\ \frac12 \qquad \textbf{(E)}\ \frac23$

Prove that $\int_0^4 \left(1-\cos \frac{x}{2}\right)e^{\sqrt{x}}dx\leq -2e^2+30.$

Right triangular prism $ABCDEF$ with triangular faces $\vartriangle ABC$ and $\vartriangle DEF$ and edges $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ has $\angle ABC = 90^o$ and $\angle EAB = \angle CAB = 60^o$ . Given that $AE = 2$, the volume of $ABCDEF$ can be written in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/4/7/25fbe2ce2df50270b48cc503a8af4e0c013025.png[/img]

How many paths from $(0,0)$ to $(n,n)$ of length $2n$ are there with exactly $k$ steps. A step is an occurence of the pair $EN$ in the path

Let $(a_1,a_2,a_3,...,a_{2011})$ be a permutation of the numbers $1,2,3,...,2011$. Show that there exist two numbers $j,k$ such that $1\leq{j}<k\leq2011$ and $|a_j-j|=|a_k-k|$

Given eleven real numbers, prove that there exist two of them such that their decimal representations agree infinitely often.

Does there exist an increasing sequence of positive integers $a_1 , a_2 ,\cdots$ with the following two properties? (i) Every positive integer $n$ can be uniquely expressed in the form $n = a_j - a_i$ , (ii) $\frac{a_k}{k^3}$ is bounded.

Let $p$ be a prime number. We know that each natural number can be written in the form \[\sum_{i=0}^{t}a_ip^i (t,a_i \in \mathbb N\cup \{0\},0\le a_i\le p-1)\] Uniquely. Now let $T$ be the set of all the sums of the form \[\sum_{i=0}^{\infty}a_ip^i (0\le a_i \le p-1).\] (This means to allow numbers with an infinite base $p$ representation). So numbers that for some $N\in \mathbb N$ all the coefficients $a_i, i\ge N$ are zero are natural numbers. (In fact we can consider members of $T$ as sequences $(a_0,a_1,a_2,...)$ for which $\forall_{i\in \mathbb N}: 0\le a_i \le p-1$.) Now we generalize addition and multiplication of natural numbers to this set so that it becomes a ring (it's not necessary to prove this fact). For example: $1+(\sum_{i=0}^{\infty} (p-1)p^i)=1+(p-1)+(p-1)p+(p-1)p^2+...$ $=p+(p-1)p+(p-1)p^2+...=p^2+(p-1)p^2+(p-1)p^3+...$ $=p^3+(p-1)p^3+...=...$ So in this sum, coefficients of all the numbers $p^k, k\in \mathbb N$ are zero, so this sum is zero and thus we can conclude that $\sum_{i=0}^{\infty}(p-1)p^i$ is playing the role of $-1$ (the additive inverse of $1$) in this ring. As an example of multiplication consider \[(1+p)(1+p+p^2+p^3+...)=1+2p+2p^2+\cdots\] Suppose $p$ is $1$ modulo $4$. Prove that there exists $x\in T$ such that $x^2+1=0$. [i]Proposed by Masoud Shafaei[/i]

If $\phi$ is the Golden Ratio, we know that $\frac1\phi = \phi - 1$. Define a new positive real number, called $\phi_d$, where $\frac1{\phi_d} = \phi_d - d$ (so $\phi = \phi_1$). Given that $\phi_{2009} = \frac{a + \sqrt{b}}{c}$, $a, b, c$ positive integers, and the greatest common divisor of $a$ and $c$ is 1, find $a + b + c$.

Let $O$ be the circumcenter and $H$ the orthocenter of an acute-angled triangle $ABC$ such that $BC>CA$. Let $F$ be the foot of the altitude $CH$ of triangle $ABC$. The perpendicular to the line $OF$ at the point $F$ intersects the line $AC$ at $P$. Prove that $\measuredangle FHP=\measuredangle BAC$.

Let $r(m)$ be the number of positive integers a less than or equal to $m$ where $\gcd(a, m)$ is prime. Find the sum of all positive integers $m < 300$ such that $r(m) = \varphi(m),$ where $\varphi(m)$ denotes the number of positive integers $a$ less than $m$ where $\gcd(a, m) = 1.$

Let $a,b,c$ be positive real numbers. Prove that $\sqrt{\frac{a}{b+c}+\frac{b}{c+a}}+\sqrt{\frac{b}{c+a}+\frac{c}{a+b}}+\sqrt{\frac{c}{a+b}+\frac{a}{b+c}}\ge 3$

Find all triples $(a, b, c)$ of positive integers such that $a^3 + b^3 + c^3 = (abc)^2$.

Lamija and Faris are playing the following game. Cards, which are numerated from $1$ to $100$, are placed one next to other, starting from $1$ to $100$. Now Faris picks every $7$th card, and after that every card which contains number $7$. After that Lamija picks from remaining cards ones divisible with $5$, and after that cards which contain number $5$. Who will have more cards and how many ? How would game end, if Lamija started with "$5$ rule" and Faris continues with "$7$ rule"?

Given non-zero real numbers $u,v,w,x,y,z$, how many different possibilities are there for the signs of these numbers if $$(u+ix)(v+iy)(w+iz)=i?$$