The sequence $a_1,a_2,a_3\ldots $, consisting of natural numbers, is defined by the rule: \[a_{n+1}=a_{n}+2t(n)\] for every natural number $n$, where $t(n)$ is the number of the different divisors of $n$ (including $1$ and $n$). Is it possible that two consecutive members of the sequence are squares of natural numbers?

Prove that the natural numbers can be divided into two groups in a way that both conditions are fulfilled: 1) For every prime number $p$ and every natural number $n$, the numbers $p^n,p^{n+1}$ and $p^{n+2}$ do not have the same colour. 2) There does not exist an infinite geometric sequence of natural numbers of the same colour.

We are given a real number $a$, not equal to $0$ or $1$. Sacho and Deni play the following game. First is Sasho and then Deni and so on (they take turns). On each turn, a player changes one of the “*” symbols in the equation: \[*x^4+*x^3+*x^2+*x^1+*=0\] with a number of the type $a^n$, where $n$ is a whole number. Sasho wins if at the end the equation has no real roots, Deni wins otherwise. Determine (in term of $a$) who has a winning strategy

Let $n$ be an even natural number and let $A$ be the set of all non-zero sequences of length $n$, consisting of numbers $0$ and $1$ (length $n$ binary sequences, except the zero sequence $(0,0,\ldots,0)$). Prove that $A$ can be partitioned into groups of three elements, so that for every triad $\{(a_1,a_2,\ldots,a_n), (b_1,b_2,\ldots,b_n), (c_1,c_2,\ldots,c_n)\}$, and for every $i = 1, 2,\ldots,n$, exactly zero or two of the numbers $a_i, b_i, c_i$ are equal to $1$.

Let $Q(x)$ be a quadratic trinomial. Given that the function $P(x)=x^{2}Q(x)$ is increasing in the interval $(0,\infty )$, prove that: \[P(x) + P(y) + P(z) > 0\] for all real numbers $x,y,z$ such that $x+y+z>0$ and $xyz>0$.

We are given an acute-angled triangle $ABC$ and a random point $X$ in its interior, different from the centre of the circumcircle $k$ of the triangle. The lines $AX,BX$ and $CX$ intersect $k$ for a second time in the points $A_1,B_1$ and $C_1$ respectively. Let $A_2,B_2$ and $C_2$ be the points that are symmetric of $A_1,B_1$ and $C_1$ in respect to $BC,AC$ and $AB$ respectively. Prove that the circumcircle of the triangle $A_2,B_2$ and $C_2$ passes through a constant point that does not depend on the choice of $X$.