Prove that the function $f (x)$ satisfying the relation $|f (x) - f (y) | \le | x - y|^a$ for any real numbers $x, y$ and some number $a> 1$ is constant.

Let $n$ be a positive integer, $n \geq 1$ and $x_1,x_2,...,x_n$ positive real numbers such that $x_1+x_2+...+x_n=1$. Does the following inequality hold $$\sum_{i=1}^{n} {\frac{x_i}{1-x_1\cdot...\cdot x_{i-1} \cdot x_{i+1} \cdot ... x_n}} \leq \frac{1}{1-\left(\frac{1}{n}\right)^{n-1}} $$

Prove the equality:\\ $\tan (\frac{3\pi}{7})-4\sin (\frac{\pi}{7})= \sqrt{7}$ .

Prove that $(2+\sqrt{3})^{2n}+(2-\sqrt{3})^{2n}$ is an even integer and that $(2+\sqrt{3})^{2n}-(2-\sqrt{3})^{2n}=w\sqrt{3}$ for some positive integer $w$, for all integers $n \geq 1$.

$ABCD$ is a convex quadrilateral. $X$ lies on the segment $AB$ with $\frac{AX}{XB} = \frac{m}{n}$. $Y$ lies on the segment $CD$ with $\frac{CY}{YD} = \frac{m}{n}$. $AY$ and $DX$ intersect at $P$, and $BY$ and $CX$ intersect at $Q$. Show that $\frac{S_{XQYP}}{S_{ABCD}} < \frac{mn}{m^2 + mn + n^2}$.

Let $p$ be a prime number greater than $2$. Patricia wants $7$ not-necessarily different numbers from $\{1, 2, . . . , p\}$ to the black dots in the figure below, on such a way that the product of three numbers on a line or circle always has the same remainder when divided by $p$. [img]https://cdn.artofproblemsolving.com/attachments/3/1/ef0d63b8ff5341ffc340de0cc75b24c7229e23.png[/img] (a) Suppose Patricia uses the number $p$ at least once. How many times does she have the number $p$ then a minimum sum needed? (b) Suppose Patricia does not use the number $p$. In how many ways can she assign numbers? (Two ways are different if there is at least one black one dot different numbers are assigned. The figure is not rotated or mirrored.)

Let $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. Find (in terms of $n$) the smallest number of nonempty piles that one can obtain by performing a finite sequence of moves of this form.

(1) Evaluate the following definite integrals. (a) $\int_0^{\frac{\pi}{2}} \cos ^ 2 x\sin x\ dx$ (b) $\int_0^{\frac{\pi}{2}} (\pi - 2x)\cos x\ dx$ (c) $\int_0^{\frac{\pi}{2}} x\cos ^ 3 x\ dx$ (2) Let $a$ be a positive constant. Find the area of the cross section cut by the plane $z=\sin \theta \ \left(0\leq \theta \leq \frac{\pi}{2}\right)$ of the solid such that \[x^2+y^2+z^2\leq a^2,\ \ x^2+y^2\leq ax,\ \ z\geq 0\] , then find the volume of the solid. [i]1984 Yamanashi Medical University entrance exam[/i] Please slove the problem without multi integral or arcsine function for Japanese high school students aged 17-18 those who don't study them. Thanks in advance. kunny

Let $n \geq 3$ be an integer and $F$ be the focus of the parabola $y^2=2px$. A regular polygon $A_1 A_2 \ldots A_n$ has the center in $F$ and none of its vertices lie on $Ox$. $\left( FA_1 \right., \left( FA_2 \right., \ldots, \left( FA_n \right.$ intersect the parabola at $B_1,B_2,\ldots,B_n$. Prove that \[ FB_1 + FB_2 + \ldots + FB_n > np . \] [i]Calin Popescu[/i]

Find the number of finite sequences $ \{a_1,a_2,\ldots,a_{2n\plus{}1}\}$, formed with nonnegative integers, for which $ a_1\equal{}a_{2n\plus{}1}\equal{}0$ and $ |a_k \minus{}a_{k\plus{}1}|\equal{}1$, for all $ k\in\{1,2,\ldots,2n\}$.

Let $P$ be a point inside a triangle $ABC$ such that $$ \frac{AP + BP}{AB} = \frac{BP + CP}{BC} = \frac{CP + AP}{CA} .$$ Lines $AP$, $BP$, $CP$ intersect the circumcircle of triangle $ABC$ again in $A'$, $B'$, $C'$. Prove that the triangles $ABC$ and $A'B'C'$ have a common incircle.

Consider a circle $C$ with diameter $AB=1$. A point $P_0$ is chosen on $C$, $P_0 \ne A$, and starting in $P_0$ a sequence of points $P_1, P_2, \dots, P_n, \dots$ is constructed on $C$, in the following way: $Q_n$ is the symmetrical point of $A$ with respect of $P_n$ and the straight line that joins $B$ and $Q_n$ cuts $C$ at $B$ and $P_{n+1}$ (not necessary different). Prove that it is possible to choose $P_0$ such that: [b]i[/b] $\angle {P_0AB} < 1$. [b]ii[/b] In the sequence that starts with $P_0$ there are $2$ points, $P_k$ and $P_j$, such that $\triangle {AP_kP_j}$ is equilateral.

Lev and Alex are racing on a number line. Alex is much faster, so he goes to sleep until Lev reaches $100$. Lev runs at $5$ integers per minute and Alex runs at $7$ integers per minute (in the same direction). How many minutes from the START of the race will it take Alex to catch up to Lev (who is still running after Alex wakes up)?

Pasha placed numbers from 1 to 100 in the cells of the square $10\times 10$, each number exactly once. After that, Dima considered all sorts of squares, with the sides going along the grid lines, consisting of more than one cell, and painted in green the largest number in each such square (one number could be colored many times). Is it possible that all two-digit numbers are painted green?

Robin is playing notes on an 88-key piano. He starts by playing middle C, which is actually the 40th lowest note on the piano (i.e. there are 39 notes lower than middle C). After playing a note, Robin plays with probability $\tfrac12$ the lowest note that is higher than the note he just played, and with probability $\tfrac12$ the highest note that is lower than the note he just played. What is the probability that he plays the highest note on the piano before playing the lowest note?

Let $p$ be a prime number such that $\frac{28^p-1}{2p^2+2p+1}$ is an integer. Find all possible values of number of divisors of $2p^2+2p+1$.