Task A-2.5. (10 points)

How many quadratic functions with coefficients from the set

$$
S=\{1,2,4,8,16,32\}
$$

have no real roots?
## Solution.

The function $f(x)=a x^{2}+b x+c$ has no real roots if and only if $b^{2}-4 a c<0$.

For $a, b, c \in S$, we can set $a=2^{k}, b=2^{s}, c=2^{l}$, where $k, l, s \in\{0,1,2,3,4,5\}$.

Then,

$$
b^{2}-4 a c=\left(2^{s}\right)^{2}-4 \cdot 2^{k} \cdot 2^{l}=2^{2 s}-2^{2+k+l}
$$

so the condition $b^{2}-4 a c<0$ is equivalent to $2 s<2+k+l$, or $k+l+1 \geqslant 2 s$.

3 points

Determine the number of possible choices of numbers $k, l, s \in\{0,1,2,3,4,5\}$ such that $k+l+1 \geqslant 2 s$, depending on the sum $k+l$:

| $k+l$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| number of choices $(k, l)$ | 1 | 2 | 3 | 4 | 5 | 6 | 5 | 4 | 3 | 2 | 1 |
| number of choices $s$ | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 6 | 6 |

Therefore, the total number of functions with the desired property is

$$
1 \cdot 1+2 \cdot 2+3 \cdot 2+4 \cdot 3+5 \cdot 3+6 \cdot 4+5 \cdot 4+4 \cdot 5+3 \cdot 5+2 \cdot 6+1 \cdot 6=135
$$

## COUNTY COMPETITION IN MATHEMATICS

3rd grade - high school - A variant
March 4, 2012.

IF A STUDENT HAS A DIFFERENT APPROACH TO SOLVING THE PROBLEM, THE COMMITTEE IS OBLIGED TO GRADE AND EVALUATE THAT APPROACH APPROPRIATELY.