[ equations in integers ]

A combination ( $x, y, z$ ) of three natural numbers, lying in the range from 10 to 20 inclusive, is an unlocking combination for a code lock if

$3 x^{2}-y^{2}-7 z=99$. Find all the unlocking combinations.
Note that $3 x^{2}-y^{2} \equiv 1(\bmod 7)$. There are 8 such pairs of remainders: $(1,3),(1,4),(2,2),(2,5),(5,2),(5,5),(6,3),(6,4)$. Considering the range of values for $x$ and $y$, we get 19 possible pairs: $(15,10),(15,17),(16,16),(16,12),(16,19)$, $(12,16),(19,16),(12,12),(12,19),(19,12),(19,19),(13,10),(13,17),(20,10),(20,17)$, $(13,11),(13,18),(20,11),(20,18)$. For each pair $(x, y)$, we find the corresponding value of $z$. Only three options fall within the range $[10,20]$:

![](https://cdn.mathpix.com/cropped/2024_05_06_be0edf3b4bb93ed2d3deg-03.jpg?height=235&width=2035&top_left_y=214&top_left_x=0)

For what value of the height does a rectangular trapezoid with an acute angle of $30^{\circ}$ and a perimeter of 6 have the maximum area?

## Hint

Express the area of the given trapezoid in terms of its height and apply the Cauchy inequality.

## Solution

Let $h$ be the height of the trapezoid, and $3x$ be the sum of the bases. Then the larger lateral side of the trapezoid is $2h$, and the perimeter is $3x + 3h = 6$. The area of the trapezoid is $\frac{3xh}{2} = \frac{3}{2}\left(\frac{x+h}{2}\right)^2 = \frac{3}{2}$, and equality is achieved when $h = x = 2 - h$, i.e., when $h = 1$.

## Answer

At $h = 1$.