If $a>1,b>1$ are the legths of the catheti of an right triangle and $c$ the length of its hypotenuse, prove that $a+b\leq c\sqrt 2$

A point $P$ lies at the center of square $ABCD$. A sequence of points $\{P_n\}$ is determined by $P_0 = P$, and given point $P_i$, point $P_{i+1}$ is obtained by reflecting $P_i$ over one of the four lines $AB$, $BC$, $CD$, $DA$, chosen uniformly at random and independently for each $i$. What is the probability that $P_8 = P$?

Determine all natural numbers $n$ so that numbers $1, 2,... , n$ can be placed on the circumference of a circle and for each natural number $s$ with $1\le s \le \frac12n(n+1)$ , there is a circular arc which has the sum of all numbers in that arc to be $s$.

Tom's age is $ T$ years, which is also the sum of the ages of his three children. His age $ N$ years ago was twice the sum of their ages then. What is $ \frac {T}{N}$? $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$

The incircle of the triangle $ABC$ touches the sides $AC$ and $BC$ at points $K$ and $L$, respectively. the $B$-excircle touches the side $AC$ of this triangle at point $P$. Line $KL$ intersects with the line passing through $A$ and parallel to $BC$ at point $M$. Prove that $PL = PM$.

A total of $ 28$ handshakes was exchanged at the conclusion of a party. Assuming that each participant was equally polite toward all the others, the number of people present was: $ \textbf{(A)}\ 14 \qquad\textbf{(B)}\ 28 \qquad\textbf{(C)}\ 56 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 7$

Let $A \text{ :}= \mathbb{Q}\setminus \{0,1\}$ denote the set of all rationals other than $0$ and $1$. A function $f:A\to \mathbb{R}$ has the property that for all $x\in A$, \[f(x)+f\left(1-\dfrac{1}{x}\right)=\log |x|.\] Compute the value of $f(2007)$.

There are two kinds of creatures living in the flatland of Pentagonia: the Spires ($S$) and the Bones ($B$). They all have the shape of an isosceles triangle: the Spiers have an apical angle of $36^o$ and the bones an apical angle of $108^o$. Every year on [i]Great Day of Division[/i] (September 11 - the day this Olympiad was held) they divide into pieces: each $S$ into two smaller $S$'s and a $B$; each $B$ in an $S$ and a $B$. Over the course of the year they then grow back to adult proportions. In the distant past, the population originated from one $B$-being. Deaths do not occur. Investigate whether the ratio between the number of Spires and the number of Bones will eventually approach a limit value and if so, calculate that limit value.

Let $ABC$ be an acute angled triangle with $\angle{BAC}=60^\circ$ and $AB > AC$. Let $I$ be the incenter, and $H$ the orthocenter of the triangle $ABC$ . Prove that $2\angle{AHI}= 3\angle{ABC}$.

Nine points of the plane, located at the vertices of a regular nonagon, are pairwise connected by segments, each of which is colored either red or blue. It is known that in any triangle with vertices at the vertices of the nonagon at least one side is red. Prove that there are four points, any two of which are connected by red lines.

Solve the equation: $x^2+2xcos(x-y)+1=0$

Tim and Allen are playing a match of [i]tenus[/i]. In a match of [i]tenus[/i], the two players play a series of games, each of which is won by one of the two players. The match ends when one player has won exactly two more games than the other player, at which point the player who has won more games wins the match. In odd-numbered games, Tim wins with probability $3/4$, and in the even-numbered games, Allen wins with probability $3/4$. What is the expected number of games in a match?

Let $f$ be the function such that \[f(x)=\begin{cases}2x & \text{if }x\leq \frac{1}{2}\\2-2x & \text{if }x>\frac{1}{2}\end{cases}\] What is the total length of the graph of $\underbrace{f(f(\ldots f}_{2012\text{ }f's}(x)\ldots))$ from $x=0$ to $x=1?$

Find the maximum constant $C$ such that, whenever $\{a_n \}_{n=1}^{\infty}$ is a sequence of positive real numbers satisfying $a_{n+1}-a_n=a_n(a_n+1)(a_n+2)$, we have $$\frac{a_{2023}-a_{2020}}{a_{2022}-a_{2021}}>C.$$

If $ a$, $ b$, and $ c$ are positive real numbers such that $ a(b \plus{} c) \equal{} 152$, $ b(c \plus{} a) \equal{} 162$, and $ c(a \plus{} b) \equal{} 170$, then abc is $ \textbf{(A)}\ 672 \qquad \textbf{(B)}\ 688 \qquad \textbf{(C)}\ 704 \qquad \textbf{(D)}\ 720 \qquad \textbf{(E)}\ 750$

The projections of the body on two planes are circles. Prove that they have the same radius.

Let ABCD be a square inscribed in a circle k and P be an arbitrary point of that circle. Prove that at least one of the numbers PA, PB, PC and PD is not rational.

Given a polynomial $p(x) =Ax^3+x^2-A$ with $A \neq 0$. Show that for every different real number $a,b,c$, at least one of $ap(b)$, $bp(a)$, and $cp(a)$ not equal to 1.

Triangle $\vartriangle ABC$ is inscribed in the circle with diameter $BC$. If $AB = 3$, $AC = 4$, and $O$ is the incenter of $\vartriangle ABC$, then find $BO \cdot OC$.

Evaluate $\prod_{k=1}^{254}\log_{k+1}(k + 2)^{u_k}$, where $u_k = \begin{cases}- k & \text{if} \,\, k \,\, \text{is odd}\\ \frac{1}{k-1} & \text{if} \,\, k \,\, \text{is even} \end{cases}$

Suppose that $x$ and $y$ are positive real numbers such that $x^3, y^3$ and $x + y$ are all rational numbers. Show that the numbers $xy, x^2+y^2, x$ and $y$ are also rational

The real positive numbers $a_1, a_2,a_3$ are three consecutive terms of an arithmetic progression, and similarly, $b_1, b_2, b_3$ are distinct real positive numbers and consecutive terms of an arithmetic progression. Is it possible to use three segments of lengths $a_1, a_2, a_3$ as bases, and other three segments of lengths $b_1, b_2, b_3$ as altitudes, to construct three rectangles of equal area ?

To open the magic chest, one needs to say a magic code of length $n$ consisting of digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9.$ Each time Griphook tells the chest a code it thinks up, the chest's talkative guardian responds by saying the number of digits in that code that match the magic code. (For example, if the magic code is $0423$ and Griphook says $3442,$ the chest's talkative guard will say $1$). Prove that there exists a number $k$ such that for any natural number $n \geq k,$ Griphook can find the magic code by checking at most $4n-2023$ times, regardless of what the magic code of the box is.

A series of figures is shown in the picture below, each one of them created by following a secret rule. If the leftmost figure is considered the first figure, how many squares will the 21st figure have? [img]http://www.artofproblemsolving.com/Forum/download/file.php?id=49934[/img] Note: only the little squares are to be counted (i.e., the $2 \times 2$ squares, $3 \times 3$ squares, $\dots$ should not be counted) Extra (not part of the original problem): How many squares will the 21st figure have, if we consider all $1 \times 1$ squares, all $2 \times 2$ squares, all $3 \times 3$ squares, and so on?.

A $2$ by $2$ square is divided into four $1$ by $1$ squares. Each of the small squares is to be painted either green or red. In how many different ways can the painting be accomplished so that no green square shares its top or right side with any red square? There may be as few as zero or as many as four small green squares. $\text{(A)}\ 4 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 16$