Task 6. Find the value of the expression $\left(\sqrt[3]{x^{2}} \cdot x^{-0.5}\right):\left(\left(\sqrt[6]{x^{2}}\right)^{2} \cdot \sqrt{x}\right)$ at $x=\frac{1}{2}$
Answer: 2

Problem 7. The diagonals of trapezoid $ABCD (BC \| AD)$ are perpendicular to each other, and $CD = \sqrt{129}$. Find the length of the midline of the trapezoid, given that $BO = \sqrt{13}, CO = 2\sqrt{3}$, where $O$ is the point of intersection of the diagonals of the trapezoid.

Answer: 10

Problem 8. Four numbers are given, the first three of which form a geometric progression, and the last three form an arithmetic progression with a difference of 999. It is known that the first and last numbers are the same. Find these numbers and give their sum in the answer.

Answer: 2331

Problem 9. The function $f(x)$ is defined for all $x$ except 1 and satisfies the equation: $(x+2) f\left(\frac{x+1}{x-1}\right) = x - f(x)$. Find $f(-1)$.

Answer: 1

Problem 10. Find the number of integer solutions to the inequality $\frac{\left(6 x-5-x^{2}\right)\left(2-x-x^{2}\right)(1+x)}{\left(x^{2}-3 x-10\right)(1-x)} \geq 0$

Answer: 2.

Problem 11. A tourist left the camp and walked along a country road to a mountain, climbed to the top of the mountain, and then returned to the camp along the same route, spending a total of 5 hours on the entire journey. What distance did the tourist travel if his speed along the country road was 4 km/h, he climbed the mountain at a speed of 3 km/h, and descended from the mountain at a speed of 6 km/h?

Answer: 20 km

Problem 12. Find the sum of all solutions to the equation

$$
(x-2)^{2}-\frac{24}{4 x-x^{2}}=18
$$

Answer: 8

Problem 13. In trapezoid $ABCD$, the bases $BC$ and $AD$ are in the ratio $1:3$, $AC = 12 \sqrt[4]{5}$, $AD = 18 \sqrt[4]{5}$, $\cos \angle CAD = \frac{1}{9}$. Find the area of the trapezoid.

Answer: 320

Problem 14. Solve the equation in integers for all natural $n$:

$$
x^{4} + n = y(2-y) + 1
$$

In the answer, write the number of solutions.

Answer: 5

Problem 15. Two cars are participating in a race. The first car moves at a speed of 120 km/h, the second at 180 km/h. There are gas stations along the route. The first car spends $a$ minutes at each gas station, and the second car spends twice as long. They started and finished at the same time. The length of the route is 300 km. How many gas stations could there be on the route if $a$ is an integer greater than 5? In the answer, indicate the number of solutions.

Answer: 3