Initially, a calculator displays number $1$. An operation consists in pressing either key $\sin$ or key $\cos$, which calculates respectively the sine and cosine with the arguments in radians. After performing $2001$ operations, what is the greatest possible value that can be achieved?
In a convex quadrilateral, the *altitude* relative to a side is defined to be the line perpendicular to this side through the midpoint of the opposite side. Prove that the four altitudes have a common point if and only if the quadrilateral is cyclic, that is, if and only if, there exists a circle which contains its four vertices.
One has a long row of glasses and $n$ stones in the central glass (glass $0$). The following movements are allowed:
• Movement $A$
If there is at least one stone in glass $i$ and at least one in glass $i+1$, one may make one stone in glass $i+1$ jump to glass $i-1$, eliminating one stone in glass $i$.
• Movement *B*
If there are at least two stones in glass *i* one may make a stone in glass *i* jump to glass *i* + 2 and make another stone jump to glass *i* - 1.
Prove the following: performing movements A and B for a sufficiently long time, one shall always obtain a configuration from which neither movements are allowed anymore. Besides this final configuration does not depend on the choices of movements during the process.
For a given integer $a_0 \gt 1$ one defines a sequence $(a_n)_{n \ge 0}$ as follows: for each $k \ge 0$, define $a_{k+1}$ as the smallest integer greater than $a_k$ such that $\text{gcd}(a_{k+1}, a_0 a_1 \cdots a_k) = 1$. Determine all values $a_0$ for which all terms $a_k$ are primes or powers of primes.
From a sequence of integers $(a, b, c, d)$ each of the sequences
$$(c, d, a, b),(b, a, d, c),(a+n c, b+n d, c, d),(a+n b, b, c+n d, d),$$
for arbitrary integer $n$ can be obtained by one step. Is it possible to obtain $(3,4,5,7)$ from $(1,2,3,4)$ through a sequence of such steps?
Let $a$ be an odd integer. Prove that $a^{2^{n}} + 2^{2^{n}}$ and $a^{2^{m}} + 2^{2^{m}}$ are relatively prime for all positive integers $n$ and $m$ with $n \neq m$.
In a triangle $A B C$, the bisector of $\angle B A C$ meets the side $B C$ at the point $D$. Knowing that $|B D| \cdot |C D| = |A D|^{2}$ and $\angle A D B = 45^{\circ}$, determine the angles of triangle $A B C$.
Let $ABCD$ be a convex quadrilateral, and let $N$ be the midpoint of $BC$. Suppose further that $\angle AND = 135^\circ$. Prove that
$$
|AB| + |CD| + \frac{1}{\sqrt{2}} \cdot |BC| \geqslant |AD|
$$
A set of 8 problems was prepared for an examination. Each student was given 3 of them. No two students received more than one common problem. What is the largest possible number of students?
Let $a_{0}, a_{1}, a_{2}, \ldots$ be a sequence of positive real numbers satisfying $i \cdot a_{i}^{2} \geqslant (i+1) \cdot a_{i-1} a_{i+1}$ for $i=1,2, \ldots$ Furthermore, let $x$ and $y$ be positive reals, and let $b_{i}=x a_{i}+y a_{i-1}$ for $i=1,2, \ldots$ Prove that the inequality $i \cdot b_{i}^{2} \gt (i+1) \cdot b_{i-1} b_{i+1}$ holds for all integers $i \geqslant 2$.
The real-valued function $f$ is defined for all positive integers. For any integers $a \gt 1$, $b \gt 1$ with $d = \operatorname{gcd}(a, b)$, we have
$$
f(ab) = f(d) \cdot \left(f\left(\frac{a}{d}\right) + f\left(\frac{b}{d}\right)\right),
$$
Determine all possible values of $f(2001)$.
The points $A, B, C, D, E$ lie on the circle $c$ in this order and satisfy $AB \parallel EC$ and $AC \parallel ED$. The line tangent to the circle $c$ at $E$ meets the line $AB$ at $P$. The lines $BD$ and $EC$ meet at $Q$. Prove that $|AC| = |PQ|$.
There are $2n$ cards. On each card some real number $x$, $1 \leqslant x \leqslant 2$, is written (there can be different numbers on different cards). Prove that the cards can be divided into two heaps with sums $s_1$ and $s_2$ so that
$$
\frac{n}{n+1} \leqslant \frac{s_1}{s_2} \leqslant 1.
$$
Let $n$ be a positive integer. Prove that at least $2^{n-1} + n$ numbers can be chosen from the set $\{1, 2, 3, \ldots, 2^n\}$ such that for any two different chosen numbers $x$ and $y$, $x + y$ is not a divisor of $x \cdot y$.
Let $a_{0}, a_{1}, a_{2}, \ldots$ be a sequence of real numbers satisfying $a_{0}=1$ and $a_{n}=a_{\lfloor 7 n / 9\rfloor}+a_{\lfloor n / 9\rfloor}$ for $n=1,2, \ldots$ Prove that there exists a positive integer $k$ with $a_{k}\lt \frac{k}{2001 !}$.
(Here $\lfloor x\rfloor$ denotes the largest integer not greater than $x$.)
Let $a_{1}, a_{2}, \ldots, a_{n}$ be positive real numbers such that $\sum_{i=1}^{n} a_{i}^{3}=3$ and $\sum_{i=1}^{n} a_{i}^{5}=5$. Prove that $\sum_{i=1}^{n} a_{i}\gt \frac{3}{2}$.
Let $n \geqslant 2$ be a positive integer. Find whether there exist $n$ pairwise nonintersecting nonempty subsets of $\{1,2,3, \ldots\}$ such that each positive integer can be expressed in a unique way as a sum of at most $n$ integers, all from different subsets.
