## PROBLEM 1

Given the numbers

$x=\left\lfloor 3^{121}: 9^{60}+\left(5^{3}\right)^{2}:\left(5^{2}\right)^{2}\right\rfloor: 2^{2} \cdot 3-3$ and $y=100:\left\{23+34:\left\lfloor\left(2 \cdot 3^{2}\right)^{2}: 18-34^{0} \cdot 1^{2013}\right\}\right\}$

a) Compare the numbers $3^{x}$ and $5^{y}$

b) Prove that $x^{2013}+y^{2014}$ is not a perfect square.
Solution:
a) $x=18$

2 points

$y=4$

2 points

$3^{x}=27^{6}, 5^{y}=625$ 1 point

Therefore $3^{x}>5^{y}$

$.0,5$ points
b) $x^{2013}+y^{2014}=18^{2013}+4^{2014}=(16+2)^{2013}+4^{2014}=$ $\qquad$ $=4 p+2$ 1 point

Therefore $x^{2013}+y^{2014}$ is not a perfect square $\qquad$ $.0,5$ points