Let $f (n)$ be a function satisfying the following three conditions for all positive integers $n$: (a) $f (n)$ is a positive integer, (b) $f (n + 1) > f (n)$, (c) $f ( f (n)) = 3n$. Find $f (2001)$.

$\overline{AB}$ and $\overline{CD}$ are segments of a circle that intersect at a point $P$ outside the circle. Calculate the value of $x$. [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNy9lL2RkZGQwNDViNTA1MzM5MDI0NDQ5MDEyOTZhZGUyNTEyYjgyZTNkLnBuZw==&rn=U2NyZWVuIFNob3QgMjAyMS0wNC0yOCBhdCAxMC4wMy4zMSBBTS5wbmc=[/img]

For a vertex $X$ of a quadrilateral, let $h(X)$ be the sum of the distances from $X$ to the two sides not containing $X$. Show that if a convex quadrilateral $ABCD$ satisfies $h(A) = h(B) = h(C) = h(D)$, then it must be a parallelogram.

Show that the numerator of \[ \frac{2^{p-1}}{p+1} - \left(\sum_{k = 0}^{p-1}\frac{\binom{p-1}{k}}{(1-kp)^2}\right) \] is a multiple of $p^3$ for any odd prime $p$. [i]Proposed by Yang Liu[/i]

Let $ABC$ be a triangle with the medians $AA_1, BB_1$ and $CC_1{}$. Prove that if the circumcircles of $BCB_1, CAC_1$ and $ABA_1$ are congruent then $ABC$ is equilateral.

There are $n\geq 2$ people at a party. Each person has at least one friend inside the party. Show that it is possible to choose a group of no more than $\frac{n}{2}$ people at the party, such that any other person outside the group has a friend inside it.

For any positive integer $n$ we define $n!!=\prod_{k=0}^{\lceil n/2\rceil -1}(n-2k)$. Prove that if the positive integers $a,b,c$ satisfy $a!=b!!+c!!$, then $b$ and $c$ are odd. [i]Proposed by Mihai Cipu[/i]

Ahmad and Salem play the following game. Ahmad writes two integers (not necessarily different) on a board. Salem writes their sum and product. Ahmad does the same thing: he writes the sum and product of the two numbers which Salem has just written. They continue in this manner, not stopping unless the two players write the same two numbers one after the other (for then they are stuck!). The order of the two numbers which each player writes is not important. Thus if Ahmad starts by writing $3$ and $-2$, the first five moves (or steps) are as shown: (a) Step 1 (Ahmad) $3$ and $-2$ (b) Step 2 (Salem) $1$ and $-6$ (c) Step 3 (Ahmad) $-5$ and $-6$ (d) Step 4 (Salem) $-11$ and $30$ (e) Step 5 (Ahmad) $19$ and $-330$ (i) Describe all pairs of numbers that Ahmad could write, and ensure that Salem must write the same numbers, and so the game stops at step 2. (ii) What pair of integers should Ahmad write so that the game finishes at step 4? (iii) Describe all pairs of integers which Ahmad could write at step 1, so that the game will finish after finitely many steps. (iv) Ahmad and Salem decide to change the game. The first player writes three numbers on the board, $u, v$ and $w$. The second player then writes the three numbers $u + v + w,uv + vw + wu$ and $uvw$, and they proceed as before, taking turns, and using this new rule describing how to work out the next three numbers. If Ahmad goes first, determine all collections of three numbers which he can write down, ensuring that Salem has to write the same three numbers at the next step.

If a right triangle is drawn in a semicircle of radius $1/2$ with one leg (not the hypotenuse) along the diameter, what is the triangle's maximum possible area?

Let $P_{1}(x)=x^{2}-2$ and $P_{j}(x)=P_{1}(P_{j-1}(x))$ for j$=2,\ldots$ Prove that for any positive integer n the roots of the equation $P_{n}(x)=x$ are all real and distinct.

Positive reals $a,b,c$ satisfy $ab+bc+ca=abc$. Prove that: $\frac{a^4+b^4}{ab(a^3+b^3)} + \frac{b^4+c^4}{bc(b^3+c^3)}+\frac{c^4+a^4}{ca(c^3+a^3)} \geq 1$

On blackboard there are $10$ different positive integers which sum is equal to $62$. Prove that product of those numbers is divisible with $60$

Prove that in any triangle $ABC$, \[ 0 < \cot { \left( \frac{A}{4} \right)} - \tan{ \left( \frac{B}{4} \right) } - \tan{ \left( \frac{C}{4} \right) } - 1 < 2 \cot { \left( \frac{A}{2} \right) }. \]

The positive integer $k$ and the set $A$ of distinct integers from $1$ to $3k$ inclusively are such that there are no distinct $a$, $b$, $c$ in $A$ satisfying $2b = a + c$. The numbers from $A$ in the interval $[1, k]$ will be called [i]small[/i]; those in $[k + 1, 2k]$ - [i]medium[/i] and those in $[2k + 1, 3k]$ - [i]large[/i]. It is always true that there are [b]no[/b] positive integers $x$ and $d$ such that if $x$, $x + d$, and $x + 2d$ are divided by $3k$ then the remainders belong to $A$ and those of $x$ and $x + d$ are different and are: a) small? $\hspace{1.5px}$ b) medium? $\hspace{1.5px}$ c) large? ([i]In this problem we assume that if a multiple of $3k$ is divided by $3k$ then the remainder is $3k$ rather than $0$[/i].)

Let $n \geq 5$ be an integer. A shop sells balls in $n$ different colors. Each of $n + 1 $ children bought three balls with different colors, but no two children bought exactly the same color combination. Show that there are at least two children who bought exactly one ball of the same color.

Suppose that \[\prod_{n=1}^{1996}(1+nx^{3^{n}}) = 1+a_{1}x^{k_{1}}+a_{2}x^{k_{2}}+\cdots+a_{m}x^{k_{m}}\] where $a_{1}$, $a_{2}$,..., $a_{m}$ are nonzero and $k_{1}< k_{2}< \cdots < k_{m}$. Find $a_{1996}$.

Let $ f(x)$ be a real function such that \[ \lim_{x \rightarrow \plus{}\infty} \frac{f(x)}{e^x}\equal{}1\] and $ |f''(x)|\leq c|f'(x)|$ for all sufficiently large $ x$. Prove that \[ \lim_{x \rightarrow \plus{}\infty} \frac{f'(x)}{e^x}\equal{}1.\] [i]P. Erdos[/i]

Find the minimum and maximum values of the function $$f(x,y,z,t)=\frac{ax^2+by^2}{ax+by}+\frac{az^2+bt^2}{az+bt},~(a>0,b>0),$$given that $x+z=y+t=1$, and $x,y,z,t\ge0$.

Garfield and Odie are situated at $(0,0)$ and $(25,0)$, respectively. Suddenly, Garfield and Odie dash in the direction of the point $(9, 12)$ at speeds of $7$ and $10$ units per minute, respectively. During this chase, the minimum distance between Garfield and Odie can be written as $\frac{m}{\sqrt{n}}$ for relatively prime positive integers $m$ and $n$. Find $m+n$. [i]Proposed by [b] Th3Numb3rThr33 [/b][/i]

Given positive integers $k, m, n$ such that $1 \leq k \leq m \leq n$. Evaluate \[\sum^{n}_{i=0} \frac{(-1)^i}{n+k+i} \cdot \frac{(m+n+i)!}{i!(n-i)!(m+i)!}.\]

Let $M,N\in M_2(\mathbb C)$ be two nonzero matrices such that $$M^2=N^2=0_2\text{ and }MN+NM=I_2$$where $0_2$ is the $2\times2$ zero matrix and $I_2$ the $2\times2$ unit matrix. Prove that there is an invertible matrix $A\in M_2(\mathbb C)$ such that $$M=A\begin{pmatrix}0&1\\0&0\end{pmatrix}A^{-1}\text{ and }N=A\begin{pmatrix}0&0\\1&0\end{pmatrix}A^{-1}.$$

There are $n$ holes in a circle. The holes are numbered $1,2,3$ and so on to $n$. In the beginning, there is a peg in every hole except for hole $1$. A peg can jump in either direction over one adjacent peg to an empty hole immediately on the other side. After a peg moves, the peg it jumped over is removed. The puzzle will be solved if all pegs disappear except for one. For example, if $n=4$ the puzzle can be solved in two jumps: peg $3$ jumps peg $4$ to hole $1$, then peg $2$ jumps the peg in $1$ to hole $4$. (See illustration below, in which black circles indicate pegs and white circles are holes.) [center][img]http://i.imgur.com/4ggOa8m.png[/img][/center] [list=a] [*]Can the puzzle be solved for $n=5$? [*]Can the puzzle be solved for $n=2014$? [/list] In each part (a) and (b) either describe a sequence of moves to solve the puzzle or explain why it is impossible to solve the puzzle.

On the graphic of the function $y=x^2$ were selected $1000$ pairwise distinct points, abscissas of which are integer numbers from the segment $[0; 100000]$. Prove that it is possible to choose six different selected points $A$, $B$, $C$, $A'$, $B'$, $C'$ such that areas of triangles $ABC$ and $A'B'C'$ are equals. [i]A. Tereshin[/i]

Prove inequallity : $$1+\frac{1}{2^3}+...+\frac{1}{2015^3}<\frac{5}{4}$$

In the figure, triangle $ADE$ is produced from triangle $ABC$ by a rotation by $90^o$ about the point $A$. If angle $D$ is $60^o$ and angle $E$ is $40^o$, how large is then angle $u$? [img]https://1.bp.blogspot.com/-6Fq2WUcP-IA/Xzb9G7-H8jI/AAAAAAAAMWY/hfMEAQIsfTYVTdpd1Hfx15QPxHmfDLEkgCLcBGAsYHQ/s0/2009%2BMohr%2Bp1.png[/img]