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1. Santa Claus has n 2 different gifts. He distributes these gifts in sacks (each sack can contain from 1 to n items) and puts the sacks with gifts around the Christmas tree (only the content of the sacks and their ordering on the circle around the tree are important). There are E (n) ways of doing this with an even number of sacks and O(n) ways with an odd number of sacks. Prove that E (n) = O(n). (For example, if there are three gifts A, B, C, then all possible variants are A-(BC), B-(AC), C-(AB); (ABC), A-B-C, A-C-B). 2. Given a natural number n, consider the function f (x, y ) = xn + x
n-1

y+x

n-2 2

y + · · · + xy

n-1

+y

n

of two real variables. Find the minimal number k for which there exist functions g1 , . . . , gk , h1 , . . . , hk of one real variable such that
k

f (x, y ) =
i=1

gi (x)hi (y ).

3. At a point O there is a star with three planets revolving around it. The planets have circular orbits centered at O in different two-dimensional planes and constant pairwise different angle velocities. Is it true that there is always a moment of time at which the angles between the rays from O to the planets are all at least 90 ? 4. A river falling into a sea forms a delta which is a system of branches consisting of channels without inner intersections. (a) Suppose that in a given delta there are exactly n different (i.e. differing by at least one channel) routes down the stream. Prove that there are at least 3 log3 n channels in this delta. (b) Prove that there is a function f (n) satisfying the condition lim f (n) =3 log3 n

n

and such that for every n a delta can exist with at most f (n) channels admitting precisely n different routes down the stream. 5. Prove that for every odd prime number p there is a natural number n such that the congruence n5 + n4 - 3 x2 (mod p) has no solutions.