SE-SKOLORNAS 1966

Let $\{x\}$ denote the fractional part of $x$ , $x - [x]$ . The sequences $x_1, x_2, x_3, ...$ and $y_1, y_2, y_3, ...$ are such that $\lim \{x_n\} = \lim \{y_n\} = 0$ . Is it true that $\lim \{x_n + y_n\} = 0$ ? $\lim \{x_n - y_n\} = 0$ ?

$a_1 + a_2 + ... + a_n = 0$ , for some $k$ we have $a_j \le 0$ for $j \le k$ and $a_j \ge 0$ for $j > k$ . If ai are not all $0$ , show that $a_1 + 2a_2 + 3a_3 + ... + na_n > 0$ .

Show that an integer $= 7 \mod 8$ cannot be sum of three squares.

Let $f(x) = 1 + \frac{2}{x}$ . Put $f_1(x) = f(x)$ , $f_2(x) = f(f_1(x))$ , $f_3(x) = f(f_2(x))$ , $...$ . Find the solutions to $x = f_n(x)$ for $n > 0$ .

Let $f(r)$ be the number of lattice points inside the circle radius $r$ , center the origin. Show that $\lim_{r\to \infty} \frac{f(r)}{r^2}$ exists and find it. If the limit is $k$ , put $g(r) = f(r) - kr^2$ . Is it true that $\lim_{r\to \infty} \frac{g(r)}{r^h} = 0$ for any $h < 2$ ?