# Problem 7.

Find all values of $a$ for each of which the equation $x^{2}+2 a x=8 a$ has two distinct integer roots. In the answer, write the product of all such $a$, rounding to the hundredths if necessary.
Answer: 506.25

Solution. If $x_{1}$ and $x_{2}$ are the roots of this equation, then $x_{1}+x_{2}=-2a, x_{1}x_{2}=-8a$. Therefore, $x_{1}x_{2}-4(x_{1}+x_{2})=0$, which means $(x_{1}-4)(x_{2}-4)=16$. Let, for definiteness, $x_{1} \leqslant x_{2}$. Then the following cases are possible (see table)

\[
\begin{array}{rlcccccc}
(x_{1}-4) & = & -16 & -8 & -4 & 1 & 2 & 4 \\
(x_{2}-4) & = & -1 & -2 & -4 & 16 & 8 & 4 \\
x_{1} & = & -12 & -4 & 0 & 5 & 6 & 8 \\
x_{2} & = & 3 & 2 & 0 & 20 & 12 & 8 \\
a & = & 4.5 & 1 & 0 & -12.5 & -9 & -8
\end{array}
\]

We exclude the cases $a=0$ and $a=-8$, because in these cases the roots are the same. The remaining values are $a=4.5, a=1, a=-9, a=-12.5$.

## B-2

Find all such values of $a$ for which the equation $x^{2}-4ax=16a$ has two distinct integer roots. In the answer, write the product of all such $a$, rounding to the nearest hundredths if necessary.

Answer: 31.64

## B-3

Find all such values of $a$ for which the equation $x^{2}+2ax=-8a$ has two distinct integer roots. In the answer, write the product of all such $a$, rounding to the nearest hundredths if necessary.

Answer: 506.25

## B-4

Find all such values of $a$ for which the equation $x^{2}-4ax=-16a$ has two distinct integer roots. In the answer, write the product of all such $a$, rounding to the nearest hundredths if necessary.

Answer: 31.64

## Lomonosov High School Olympiad in Mathematics

Preliminary Stage 2020/21 Academic Year for 10-11 Grades

## B-1

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