Find the locus of points whose coordinates \((x, y)\) satisfy the relation \(\sin(x + y) = 0\).
The equations \[ax^2 + bx + c = 0 \tag{1}\] and \[- ax^2 + bx + c \tag{2}\] are given. Prove that if \(x_1\) and \(x_2\) are, respectively, any roots of the equations (1) and (2), then there is a root \(x_3\) of the equation \(\frac 12 ax^2 + bx + c\) such that either \(x_1 \leq x_3 \leq x_2\) or \(x_1 \geq x_3 \geq x_2\).
Prove that if \(x_0^4 + a_1x_0^3 + a_2x_0^2 + a_3x_0 + a_4\) and \(4x_0^3 + 3a_1x_0^2 + 2a_2x_0 + a_3 = 0\) then \(x^4 + a_1x^3 + a_2x^2 + a_3x + a_4\) is divisible by \((x - x_0)^2\).
The segment \(OA\) is given. From the end of the segment \(A\) there are 5 segments \(AB_1, AB_2, AB_3, AB_4, AB_5\). From each point \(B_i\) there can be five more new segments or not a single new segment, etc. Can the number of free ends of the constructed segments be 1001? By the free end of a segment we mean a point belonging to only one segment (except point \(O\)).
There is a system of equations \[\begin{aligned} * x + * y + * z &= 0,\\ * x + * y + * z &= 0,\\ * x + * y + * z &= 0. \end{aligned}\] Two people alternately enter a number instead of a star. Prove that the player that goes first can always ensure that the system has a non-zero solution.
Note that if you turn over a sheet on which numbers are written, then the digits 0, 1, 8 will not change and the digits 6 and 9 will switch places, whilst the others will lose their meaning. How many nine-digit numbers exist that do not change when a sheet is turned over?
Two people play a game with the following rules: one of them guesses a set of integers \((x_1, x_2, \dots , x_n)\) which are single-valued digits and can be either positive or negative. The second person is allowed to ask what is the sum \(a_1x_1 + \dots + a_nx_n\), where \((a_1, \dots ,a_n)\) is any set. What is the smallest number of questions for which the guesser recognizes the intended set?
All of the integers from 1 to 64 are written in an \(8 \times 8\) table. Prove that in this case there are two adjacent numbers, the difference between which is not less than 5. (Numbers that are in cells which share a common side are called adjacent).
Prove that in a group of 11 arbitrary infinitely long decimal numbers, it is possible to choose two whose difference contains either, in decimal form, an infinite number of zeroes or an infinite number of nines.
30 teams are taking part in a football championship. Prove that at any moment in the contest there will be two teams who have played the same number of matches up to that moment, assuming every team plays every other team exactly once by the end of the tournament.