## Subject I

a) If $m, n \in N^{*}$, find the smallest number of the form $\left|5^{2 m}-3^{n}\right|$.

b) Compare the numbers: $\frac{\sqrt{2014}^{2014}+\sqrt{2015}^{2015}}{\sqrt{2014}^{2015}+\sqrt{2015}^{2014}}$ and $\frac{\sqrt{2014}}{\sqrt{2015}}$.
## Subject I

a) If $m, n \in N^{*}$, find the smallest number of the form $\left|5^{2 m}-3^{n}\right|$.

b) Compare the numbers: $\frac{\sqrt{2014}^{2014}+\sqrt{2015}^{2015}}{\sqrt{2014}^{2015}+\sqrt{2015}^{2014}}$ and $\frac{\sqrt{2014}}{\sqrt{2015}}$.

Scoring:

a) For $\mathrm{m}=1$ and $\mathrm{n}=3,\left|5^{2 m}-3^{n}\right|=2$............................................................................................ 1p

We have $u\left(5^{2 m}\right) \in\{5\}, \forall m \in N^{*}$ and $u\left(3^{n}\right) \in\{1,3,7,9\}, \forall n \in N^{*}, u\left(\left|5^{2 m}-3^{n}\right|\right) \in\{2,4,6,8\}, \forall m, n \in N^{*}$ $\qquad$
Therefore, the smallest number of the form $\left|5^{2 m}-3^{n}\right|$ is 2 ...............................................................1p

b) We show that: $\frac{\sqrt{2014}^{2014}+\sqrt{2015}^{2015}}{\sqrt{2014}^{2015}+\sqrt{2015}^{2014}}>1$....................................................................................... 1p

We calculate the difference: $\sqrt{2014}^{2014}+\sqrt{2015}^{2015}-\sqrt{2014}^{2015}-\sqrt{2015}^{2014}=$

$\sqrt{2015}^{2014}(\sqrt{2015}-1)-\sqrt{2014}^{2014}(\sqrt{2014}-1)>0$

Since $\sqrt{2015}-1>\sqrt{2014}-1$ and $\sqrt{2015}^{2014}>\sqrt{2014}^{2014}$, thus $\sqrt{2014}^{2014}+\sqrt{2015}^{2015}>$

$\sqrt{2014}^{2015}+\sqrt{2015}^{2014}$, which is equivalent to $\frac{\sqrt{2014}^{2014}+\sqrt{2015}^{2015}}{\sqrt{2014}^{2015}+\sqrt{2015}^{2014}}>1$.

Since $\frac{\sqrt{2014}}{\sqrt{2015}}<1$ and $\frac{\sqrt{2014}^{2014}+\sqrt{2015}^{2015}}{\sqrt{2014}^{2015}+\sqrt{2015}^{2014}}>1$, we have $\frac{\sqrt{2014}^{2014}+\sqrt{2015}^{2015}}{\sqrt{2014}^{2015}+\sqrt{2015}^{2014}}>\frac{\sqrt{2014}}{\sqrt{2015}}$.

$1 p$

## MATHEMATICS OLYMPIAD LOCAL STAGE  16.02.2014  SCORING AND GRADING CRITERIA  GRADE 7