A numerical sequence is defined by the following conditions: \[a_1 = 1, \quad a_{n+1} = a_n + \lfloor \sqrt{a_n}\rfloor .\]
Prove that among the terms of this sequence there are an infinite number of complete squares.
The function \(f(x)\) on the interval \([a, b]\) is equal to the maximum of several functions of the form \(y = C \times 10^{- | x-d |}\) (where \(d\) and \(C\) are different, and all \(C\) are positive). It is given that \(f (a) = f (b)\). Prove that the sum of the lengths of the sections on which the function increases is equal to the sum of the lengths of the sections on which the function decreases.
An after school club was attended by 60 pupils. It turns out that in any group of 10 there will always be 3 classmates. Prove that within the group of 60 who attended there will always be at least 15 pupils from the same class.
4 points \(a, b, c, d\) lie on the segment \([0, 1]\) of the number line. Prove that there will be a point \(x\), lying in the segment \([0, 1]\), that satisfies \[\frac{1}{ | x-a |}+\frac{1}{ | x-b |}+\frac{1}{ | x-c |}+\frac{1}{ | x-d |} < 40.\]
Some points with integer co-ordinates are marked on a Cartesian plane. It is known that no four points lie on the same circle. Prove that there will be a circle of radius 1995 in the plane, which does not contain a single marked point.
Let \(n\) numbers are given together with their product \(p\). The difference between \(p\) and each of these numbers is an odd number.
Prove that all \(n\) numbers are irrational.
Some real numbers \(a_1, a_2, a_3,\dots ,a _{2022}\) are written in a row. Prove that it is possible to pick one or several adjacent numbers, so that their sum is less than 0.001 away from a whole number.
a) Could an additional \(6\) digits be added to any \(6\)-digit number starting with a \(5\), so that the \(12\)-digit number obtained is a complete square?
b) The same question but for a number starting with a \(1\).
c) Find for each \(n\) the smallest \(k = k (n)\) such that to each \(n\)-digit number you can assign \(k\) more digits so that the resulting \((n + k)\)-digit number is a complete square.
Are there such irrational numbers \(a\) and \(b\) so that \(a > 1\), \(b > 1\), and \(\lfloor a^m\rfloor\) is different from \(\lfloor b^n\rfloor\) for any natural numbers \(m\) and \(n\)?
Two players in turn paint the sides of an \(n\)-gon. The first one can paint the side that borders either zero or two colored sides, the second – the side that borders one painted side. The player who can not make a move loses. At what \(n\) can the second player win, no matter how the first player plays?