2. Simplify the expression:

$$
2\left(x y^{-1}-1\right)^{-p}\left(x^{2} y^{-2}-1\right)^{p}-\left(\frac{x+y}{y}\right)^{p}
$$

For $x=-2, y=-\frac{1}{2}$ and $p=-3$, calculate the value of the expression.
2. We simplify the first parenthesis $\left(x y^{-1}-1\right)^{-p}=\left(\frac{x}{y}-1\right)^{-p}=\left(\frac{x-y}{y}\right)^{-p}=\left(\frac{y}{x-y}\right)^{p}$ and the second parenthesis $\left(x^{2} y^{-2}-1\right)^{p}=\left(\frac{x^{2}-y^{2}}{y^{2}}\right)^{p}$. We multiply the parentheses, factor the numerator of the second parenthesis, and get the product $\left(\frac{y}{x-y}\right)^{p} \cdot\left(\frac{x^{2}-y^{2}}{y^{2}}\right)^{p}=\left(\frac{y}{x-y}\right)^{p} \cdot\left(\frac{(x-y)(x+y)}{y^{2}}\right)^{p}$. Since the exponent of both parentheses is the same, we put them in a common parenthesis and simplify $\left(\frac{y(x-y)(x+y)}{(x-y) y^{2}}\right)^{p}=\left(\frac{x+y}{y}\right)^{p}$. We get the expression $2\left(\frac{x+y}{y}\right)^{p}-\left(\frac{x+y}{y}\right)^{p}$. We subtract and get $\left(\frac{x+y}{y}\right)^{p}$. We substitute the values for $x, y, p$ and get the result $\frac{1}{125}$.
Writing the power $\mathbf{z}$ with a negative exponent $x y^{-1}=\frac{x}{y}$ ..... 1 point
Writing as a fraction $x y^{-1}-1=\frac{x-y}{y}$ ..... 1 point
Writing as a fraction $x^{2} y^{-2}-1=\frac{x^{2}-y^{2}}{y^{2}}$ ..... 1 point
Factoring $x^{2}-y^{2}=(x-y)(x+y)$ ..... 1 point
Simplifying $\frac{y(x-y)(x+y)}{(x-y) y^{2}}=\frac{x+y}{y}$ ..... 2 points
Simplified expression $2\left(\frac{x+y}{y}\right)^{p}-\left(\frac{x+y}{y}\right)^{p}=\left(\frac{x+y}{y}\right)^{p}$ 1 point
Substituting the values of variables $x=-2, y=-\frac{1}{2}$ and $p=-3$. ..... 1* point
Calculating $\frac{-2-\frac{1}{2}}{-\frac{1}{2}}=5$ ..... 1 point
Result $5^{-3}$ or $\frac{1}{125}$ ..... 1 point

# 18th Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools

Preliminary Competition, March 15, 2018

## Solutions for Third Year

A competitor who arrives at the solution by any correct method (even if the scoring guide does not provide for it) receives all possible points.

A correct method is one that:

- sensibly takes into account the wording of the problem,
- leads to the solution of the problem,
- is mathematically correct and complete.

If an intermediate or final result can be recognized, guessed, read from a diagram, or calculated mentally, the competitor generally receives all the points provided for. However, if the solution is guessed (cannot be arrived at without calculation), and there is no written verification or proof, it is scored with 0 points.

A competitor who has only partially solved the problem, with correct procedures but no visible path to the final solution of the problem, cannot receive more than half of the possible points.

The mark ${ }^{\prime * \prime}$ next to the points means that the point or points can be awarded to the competitor for a correct procedure, even if the calculation is incorrect.

