Solve the following equation in $\mathbb{R}$: $$\left(x-\frac{1}{x}\right)^\frac{1}{2}+\left(1-\frac{1}{x}\right)^\frac{1}{2}=x.$$

For a constant $a$, denote $C(a)$ the part $x\geq 1$ of the curve $y=\sqrt{x^2-1}+\frac{a}{x}$. (1) Find the maximum value $a_0$ of $a$ such that $C(a)$ is contained to lower part of $y=x$, or $y<x$. (2) For $0<\theta <\frac{\pi}{2}$, find the volume $V(\theta)$ of the solid $V$ obtained by revoloving the figure bounded by $C(a_0)$ and three lines $y=x,\ x=1,\ x=\frac{1}{\cos \theta}$ about the $x$-axis. (3) Find $\lim_{\theta \rightarrow \frac{\pi}{2}-0} V(\theta)$. 1992 Tokyo University entrance exam/Science, 2nd exam

A circle is divided into $2n$ equal sectors, $n \in \mathbb{N}$. Vitya and Masha are playing the following game. At first, Vitya writes one number in every sector from the set $\{1,2,\ldots,n\}$ and every number is used exatly twice. After that Masha chooses $n$ consecutive sectors and writes $1$ in the first sector, $2$ in the second, $n$ in the last. Vitya wins if at least in one sector two equal number will be written, otherwise Masha wins. Find all $n$ for which Vitya can guarantee his win.

Let $x_1, x_2, ..., x_n$ be positive numbers, with $n \ge 2$. Prove that $$\left(x_1+\frac{1}{x_1}\right)\left(x_2+\frac{1}{x_2}\right)...\left(x_n+\frac{1}{x_n}\right)\ge \left(x_1+\frac{1}{x_2}\right)\left(x_2+\frac{1}{x_3}\right)...\left(x_{n-1}+\frac{1}{x_n}\right)\left(x_n+\frac{1}{x_1}\right)$$

Let $ABC$ be a isosceles triangle with $\overline{AB}=\overline{AC}$. Let $D(\neq A, C)$ be a point on the side $AC$, and circle $\Omega$ is tangent to $BD$ at point $E$, and $AC$ at point $C$. Denote by $F(\neq E)$ the intersection of the line $AE$ and the circle $\Omega$, and $G(\neq a)$ the intersection of the line $AC$ and the circumcircle of the triangle $ABF$. Prove that points $D, E, F,$ and $G$ are concyclic.

In the following, a [i]word[/i] will mean a finite sequence of letters "$a$" and "$b$". The [i]length[/i] of a word will mean the number of the letters of the word. For instance, $abaab$ is a word of length $5$. There exists exactly one word of length $0$, namely the empty word. A word $w$ of length $\ell$ consisting of the letters $x_1$, $x_2$, ..., $x_{\ell}$ in this order is called a [i]palindrome[/i] if and only if $x_j=x_{\ell+1-j}$ holds for every $j$ such that $1\leq j\leq\ell$. For instance, $baaab$ is a palindrome; so is the empty word. For two words $w_1$ and $w_2$, let $w_1w_2$ denote the word formed by writing the word $w_2$ directly after the word $w_1$. For instance, if $w_1=baa$ and $w_2=bb$, then $w_1w_2=baabb$. Let $r$, $s$, $t$ be nonnegative integers satisfying $r + s = t + 2$. Prove that there exist palindromes $A$, $B$, $C$ with lengths $r$, $s$, $t$, respectively, such that $AB=Cab$, if and only if the integers $r + 2$ and $s - 2$ are coprime.

Six ducklings swim on the surface of a pond, which is in the shape of a circle with radius $5$ m. Show that at every moment, two of the ducklings swim on the distance of at most $5$ m from each other.

a) Let $m$ and $n$ be natural numbers. For some nonnegative integers $k_1, k_2, ... , k_n$ the number $$2^{k_1}+2^{k_2}+...+2^{k_n}$$ is divisible by $(2^m-1)$. Prove that $n \ge m$. b) Can you find a number, divisible by $111...1$ ($m$ times "$1$"), that has the sum of its digits less than $m$?

An ordinary domino tile can be identified as a pair $(k,m)$ where numbers $k$ and $m$ can get values $0, 1, 2, 3, 4, 5$ and $6.$ Pairs $(k,m)$ and $(m, k)$ determine the same tile. In particular, the pair $(k, k)$ determines one tile. We say that two domino tiles [i]match[/i], if they have a common component. [i]Generalized n-domino tiles[/i] $m$ and $k$ can get values $0, 1,... , n.$ What is the probability that two randomly chosen $n$-domino tiles match?

Let $ABCD$ be a quadrilateral circumscribed on the circle $\omega$ with center $I$. Assume $\angle BAD+ \angle ADC <\pi$. Let $M, \ N$ be points of tangency of $\omega $ with $AB, \ CD$ respectively. Consider a point $K \in MN$ such that $AK=AM$. Prove that $ID$ bisects the segment $KN$.

An odd integer $ n \ge 3$ is said to be nice if and only if there is at least one permutation $ a_{1}, \cdots, a_{n}$ of $ 1, \cdots, n$ such that the $ n$ sums $ a_{1} \minus{} a_{2} \plus{} a_{3} \minus{} \cdots \minus{} a_{n \minus{} 1} \plus{} a_{n}$, $ a_{2} \minus{} a_{3} \plus{} a_{3} \minus{} \cdots \minus{} a_{n} \plus{} a_{1}$, $ a_{3} \minus{} a_{4} \plus{} a_{5} \minus{} \cdots \minus{} a_{1} \plus{} a_{2}$, $ \cdots$, $ a_{n} \minus{} a_{1} \plus{} a_{2} \minus{} \cdots \minus{} a_{n \minus{} 2} \plus{} a_{n \minus{} 1}$ are all positive. Determine the set of all `nice' integers.

Let $f_1,f_2$ be nonnegative and concave functions. Then prove that $$(f_1f_2)^{\frac{2^n-1}{n\cdot2^n}}\left(\frac{\displaystyle\prod_{k=1}^n\left(\sqrt[2^k]{f_1}+\sqrt[2^k]{f_2}\right)}{f_1+f_2}\right)^{\frac1n}$$ is concave. [i]Proposed by Mihály Bencze and Marius Drăgan[/i]

A regular hexagon is inscribed in a circle. The ratio of the length of a side of the hexagon to the length of the shorter of the arcs intercepted by the side, is: $ \textbf{(A)}\ 1: 1 \qquad \textbf{(B)}\ 1: 6 \qquad \textbf{(C)}\ 1: \pi \qquad \textbf{(D)}\ 3: \pi \qquad \textbf{(E)}\ 6: \pi$

Point $P$ lies in the interior of a triangle $ABC$. Lines $AP$, $BP$, and $CP$ meet the opposite sides of triangle $ABC$ at $A$', $B'$, and $C'$ respectively. Let $P_A$ the midpoint of the segment joining the incenters of triangles $BPC'$ and $CPB'$, and define points $P_B$ and $P_C$ analogously. Show that if \[ AB'+BC'+CA'=AC'+BA'+CB' \] then points $P,P_A,P_B,$ and $P_C$ are concyclic. [i]Nikolai Beluhov[/i]

How many non-congruent triangles with perimeter $ 7$ have integer side lengths? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

Let $k$ and $l$ be two given positive integers and $a_{ij}(1 \leq i \leq k, 1 \leq j \leq l)$ be $kl$ positive integers. Show that if $q \geq p > 0$, then \[(\sum_{j=1}^{l}(\sum_{i=1}^{k}a_{ij}^{p})^{q/p})^{1/q}\leq (\sum_{i=1}^{k}(\sum_{j=1}^{l}a_{ij}^{q})^{p/q})^{1/p}.\]

Let $ABCD$ and $A'B'C'D'$ be two squares, both are oriented clockwise. In addition, it is assumed that all points are arranged as shown in the figure.Then it has to be shown that the sum of the areas of the quadrilaterals $ABB'A'$ and $CDD'C'$ equal to the sum of the areas of the quadrilaterals $BCC'B'$ and $DAA'D'$. [img]https://cdn.artofproblemsolving.com/attachments/0/2/6f7f793ded22fe05a7b0a912ef6c4e132f963e.png[/img]

For some integers $b$ and $c$, neither of the equations below have real solutions: \begin{align*} 2x^2+bx+c&=0\\ 2x^2+cx+b&=0. \end{align*} What is the largest possible value of $b+c$?

Let $PQ$ be the diameter of semicircle $H$. Circle $O$ is internally tangent to $H$ and tangent to $PQ$ at $C$. Let $A$ be a point on $H$ and $B$ a point on $PQ$ such that $AB\perp PQ$ and is tangent to $O$. Prove that $AC$ bisects $\angle PAB$

Find all solutions $(x_1, x_2, . . . , x_n)$ of the equation \[1 +\frac{1}{x_1} + \frac{x_1+1}{x{}_1x{}_2}+\frac{(x_1+1)(x_2+1)}{x{}_1{}_2x{}_3} +\cdots + \frac{(x_1+1)(x_2+1) \cdots (x_{n-1}+1)}{x{}_1x{}_2\cdots x_n} =0\]

The distances from a certain point inside a regular hexagon to three of its consecutive vertices are equal to $1, 1$ and $2$, respectively. Determine the length of this hexagon's side. (Mikhail Evdokimov)

Consider a fair coin and a fair 6-sided die. The die begins with the number 1 face up. A [i]step[/i] starts with a toss of the coin: if the coin comes out heads, we roll the die; otherwise (if the coin comes out tails), we do nothing else in this step. After 5 such steps, what is the probability that the number 1 is face up on the die?

The number $2$ is written on the board. Ana and Bruno play alternately. Ana begins. Each one, in their turn, replaces the number written by the one obtained by applying exactly one of these operations: multiply the number by $2$, multiply the number by $3$ or add $1$ to the number. The first player to get a number greater than or equal to $2011$ wins. Find which of the two players has a winning strategy and describe it.

A convex$N-$gon with equal sides is located inside a circle. Each side is extended in both directions up to the intersection with the circle so that it contains two new segments outside the polygon. Prove that one can paint some of these new $2N$ segments in red and the rest in blue so that the sum of lengths of all the red segments would be the same as for the blue ones. [i]($8$ points)[/i]

Let $p\ge 17$ be a prime. Prove that $t=3$ is the largest positive integer which satisfies the following condition: For any integers $a,b,c,d$ such that $abc$ is not divisible by $p$ and $(a+b+c)$ is divisible by $p$, there exists integers $x,y,z$ belonging to the set $\{0,1,2,\ldots , \left\lfloor \frac{p}{t} \right\rfloor - 1\}$ such that $ax+by+cz+d$ is divisible by $p$.