\nopagenumbers
\input amssym.def
\input amssym.tex
\def\N{\Bbb N}
\def\q#1. {{\bf #1. }}
\centerline{\bf ``TUYMAADA-1999''}
\medskip
\centerline{\tt First day}
\medskip
\q1. 1999 different planes are given in the space. Prove that there
is a sphere intersecting exactly 100 of these planes.
\rightline{({\it V. Egorov}\/)}
\q2. Find all polynomials $P(x)$ such that
$$
P(x^3+1)=P(x^3)+P(x^2).
$$
\rightline{({\it A. Golovanov}\/)}
\q3. What maximum number of elements can be selected from the set
$\{1, 2, 3, \dots, 100\}$ so that no sum of any three selected numbers
is equal to a selected number?
\rightline{({\it A. Golovanov}\/)}
\q4. Prove the inequality
$$
{x\over y^2-z}+{y\over z^2-x}+{z\over x^2-y} > 1,
$$
where $2 < x, y, z < 4.$
\rightline{({\it A. Golovanov}\/)}
\medskip
\centerline{\tt Second day}
\medskip
\q5. In the triangle $ABC$ \quad $\angle ABC=100^\circ$,
$\angle ACB=65^\circ$, $M\in AB$, $N\in AC$, and
$\angle MCB=55^\circ$, $\angle NBC=80^\circ$. Find $\angle NMC$.
\rightline{({\it St.Petersburg folklore}\/)}
\q6. Can the graphs of a polynomial of degree 20 and the function
$\displaystyle y={1\over x^{40}}$ have exactly 30 points of intersection?
\rightline{({\it K. Kokhas}\/)}
\q7. A sequence of integers $a_0,\ a_1,\dots a_n \dots $ is defined
by the following rules: $a_0=0,\ a_1=1,\ a_{n+1} > a_n$ for each
$n\in \N$, and $a_{n+1}$ is the minimum number such that no three
numbers among $a_0,\ a_1,\dots a_{n+1}$ form an arithmetical progression.
Prove that $a_{2^n}=3^n$ for each $n\in \N$.
\rightline{({\it Ya. Peredriy}\/)}
\q8. A right parallelepiped (i.e. a parallelepiped one of whose edges
is perpendicular to a face) is given. Its vertices have integral coordinates,
and no other points with integral coordinates lie on its faces or edges.
Prove that the volume of this parallelepiped is a sum of three
perfect squares.
\rightline{({\it A. Golovanov}\/)}
\end