Author: I.I. Bogdanov
Peter wants to write down all of the possible sequences of 100 natural numbers, in each of which there is at least one 4 or 5, and any two neighbouring terms differ by no more than 2. How many sequences will he have to write out?
Solve the equation \(2 \sin \pi x / 2 - 2 \cos \pi x = x^5 + 10x - 54\).
Author: A.V. Shapovalov
We call a triangle rational if all of its angles are measured by a rational number of degrees. We call a point inside the triangle rational if, when we join it by segments with vertices, we get three rational triangles. Prove that within any acute-angled rational triangle there are at least three distinct rational points.
At a contest named “Ah well, monsters!”, 15 dragons stand in a row. Between neighbouring dragons the number of heads differs by 1. If the dragon has more heads than both of his two neighbors, he is considered cunning, if he has less than both of his neighbors – strong, the rest (including those standing at the edges) are considered ordinary. In the row there are exactly four cunning dragons – with 4, 6, 7 and 7 heads and exactly three strong ones – with 3, 3 and 6 heads. The first and last dragons have the same number of heads.
a) Give an example of how this could occur.
b) Prove that the number of heads of the first dragon in all potential examples is the same.
Is there a positive integer \(n\) such that \[\sqrt{n}{17\sqrt{5} + 38} + \sqrt{n}{17\sqrt{5} - 38} = 2\sqrt{5}\,?\]
Author: G. Zhukov
The square trinomial \(f (x) = ax^2 + bx + c\) that does not have roots is such that the coefficient \(b\) is rational, and among the numbers \(c\) and \(f (c)\) there is exactly one irrational.
Can the discriminant of the trinomial \(f (x)\) be rational?
Author: A. Khrabrov
Do there exist integers \(a\) and \(b\) such that
a) the equation \(x^2 + ax + b = 0\) does not have roots, and the equation \(\lfloor x^2\rfloor + ax + b = 0\) does have roots?
b) the equation \(x^2 + 2ax + b = 0\) does not have roots, and the equation \(\lfloor x^2\rfloor + 2ax + b = 0\) does have roots?
Note that here, square brackets represent integers and curly brackets represent non-integer values or 0.
Author: A.K. Tolpygo
12 grasshoppers sit on a circle at various points. These points divide the circle into 12 arcs. Let’s mark the 12 mid-points of the arcs. At the signal the grasshoppers jump simultaneously, each to the nearest clockwise marked point. 12 arcs are formed again, and jumps to the middle of the arcs are repeated, etc. Can at least one grasshopper return to his starting point after he has made a) 12 jumps; b) 13 jumps?
Solve the equation \(f (f (x)) = f (x)\) if \(f(x) = \sqrt[5]{3 - x^3 - x}\).
George drew an empty table of size \(50 \times 50\) and wrote on top of each column and to the left of each row, a number. It turned out that all 100 written numbers are different, and 50 of them are rational, and the remaining 50 are irrational. Then, in each cell of the table, he wrote down the sum of the numbers written at the start of the corresponding row and column (“addition table”). What is the largest number of sums in this table that could be rational numbers?