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 \(O\)).
The expression \(ax^2+bx+c\) is an exact fourth power for all integers \(x\). Prove that \(a=b=0\).
The numbers \(\lfloor a\rfloor, \lfloor 2a\rfloor, \dots , \lfloor Na\rfloor\) are all different, and the numbers \(\lfloor 1/a\rfloor, \lfloor 2/a\rfloor,\dots , \lfloor M/a\rfloor\) are also all different. Find all such \(a\).
2022 points are selected from a cube, whose edge is equal to 13 units. Is it possible to place a cube with edge of 1 unit in this cube so that there is not one selected point inside it?
Solve the equation \(x^3 - \lfloor x\rfloor = 3\).
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.
There are two sets of numbers made up of 1s and \(-1\)s, and in each there are 2022 numbers. Prove that in some number of steps it is possible to turn the first set into the second one if for each step you are allowed to simultaneously change the sign of any 11 numbers of the starting set. (Two sets are considered the same if they have the same numbers in the same places.)