For every positive integer $n{}$ denote $a_n$ as the last digit of the sum of the number from $1$ to $n{}$. For example $a_5=5, a_6=1.$ a) Find $a_{21}.$ b) Compute the sum $a_1+a_2+\ldots+a_{2015}.$

The triangle $ABC$ has $\angle BCA=90^{\circ}.$ Bisector of angle $\angle CAB$ intersects the side $BC$ in point $P$ and bisector of angle $\angle ABC$ intersects the side $AC$ in point $Q.$ If $M$ and $N$ are projections of $P$ and $Q$ on side $AB$, find the measure of the angle $\angle MCN.$

$$\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\dots+\frac{1}{2014\cdot2015}=\frac{m}{n},$$where $\frac{m}{n}$ is irreducible. a) Find $m+n.$ b) Find the remainder of division of $(m+3)^{1444}$ to $n{}$.

For every positive integer $n{}$, consider the numbers $a_1=n^2-10n+23, a_2=n^2-9n+31, a_3=n^2-12n+46.$ a) Prove that $a_1+a_2+a_3$ is even. b) Find all positive integers $n$ for which $a_1, a_2$ and $a_3$ are primes.

Let $x$ and $y$ be positive real numbers with $x + y =1 $. Prove that $$\frac{(3x-1)^2}{x}+ \frac{(3y-1)^2}{y} \ge1.$$For which $x$ and $y$ equality holds? (K. Czakler, GRG 21, Vienna)