## Problem 1.

a) Let the number $A=\frac{1}{\sqrt{4+2 \sqrt{3}}}+\frac{1}{\sqrt{8+2 \sqrt{15}}}+\cdots+\frac{1}{\sqrt{4 n+2 \sqrt{4 n^{2}-1}}}, n \in N^{\star}$. Determine the value of $n$ for which $A=10$.

b) Determine $x, y \in Z$ that satisfy the relation $4 x^{2}(y-1)-4 x-y-1=0$.
## Problem 1.

a) Let the number $A=\frac{1}{\sqrt{4+2 \sqrt{3}}}+\frac{1}{\sqrt{8+2 \sqrt{15}}}+\cdots+\frac{1}{\sqrt{4 n+2 \sqrt{4 n^{2}-1}}}, n \in N^{\star}$.

Determine the value of $n$ for which $A=10$.

b) Determine $x, y \in Z$ that satisfy the relation $4 x^{2}(y-1)-4 x-y-1=0$.

Prof. Marius Farcaş

## SCORING GUIDE

![](https://cdn.mathpix.com/cropped/2024_06_07_8b76c6b0acb9238144a7g-2.jpg?height=71&width=1564&top_left_y=1211&top_left_x=276)

Obtain $\frac{\sqrt{2 n+1}-1}{2}=10$..............................................................................................................................

Finalization: $n=220$................................................................................................................................

![](https://cdn.mathpix.com/cropped/2024_06_07_8b76c6b0acb9238144a7g-2.jpg?height=80&width=1575&top_left_y=1378&top_left_x=276)

Since $y \in Z$, obtain $x \in\{0 ; 1\}$.................................................................................................1p

Finalization: $(x, y) \in\{(0 ;-1) ;(1 ; 3)\}$............................................................................................ $1 \mathrm{p}$