Školjka

%V0 Determine all three-digit numbers $N$ having the property that $N$ is divisible by 11, and $\dfrac{N}{11}$ is equal to the sum of the squares of the digits of $N$.

%V0 Find all real roots of the equation $$\sqrt{x^2-p}+2\sqrt{x^2-1}=x$$ where $p$ is a real parameter.

%V0 In a mathematical contest, three problems, $A,B,C$ were posed. Among the participants ther were 25 students who solved at least one problem each. Of all the contestants who did not solve problem $A$, the number who solved $B$ was twice the number who solved $C$. The number of students who solved only problem $A$ was one more than the number of students who solved $A$ and at least one other problem. Of all students who solved just one problem, half did not solve problem $A$. How many students solved only problem $B$?

%V0 $(GDR 2)^{IMO1}$ Prove that there exist infinitely many natural numbers $a$ with the following property: The number $z = n^4 + a$ is not prime for any natural number $n.$

%V0 Let $m$ and $n$ be positive integers such that $1 \le m < n$. In their decimal representations, the last three digits of $1978^m$ are equal, respectively, so the last three digits of $1978^n$. Find $m$ and $n$ such that $m + n$ has its least value.

%V0 For each positive integer $\,n,\;S(n)\,$ is defined to be the greatest integer such that, for every positive integer $\,k\leq S(n),\;n^{2}\,$ can be written as the sum of $\,k\,$ positive squares. a.) Prove that $\,S(n)\leq n^{2}-14\,$ for each $\,n\geq 4$. b.) Find an integer $\,n\,$ such that $\,S(n)=n^{2}-14$. c.) Prove that there are infintely many integers $\,n\,$ such that $S(n)=n^{2}-14.$