Let S=: _2(9^2-4+13)-_2(523^2-4+1)=2.

Question

Let $S=\left\{\alpha: \log_2\left(9^{2\alpha-4}+13\right)-\log_2\left(\dfrac52\cdot3^{2\alpha-4}+1\right)=2\right\}$. Then the maximum value of $\beta$ for which the equation $x^2-2\left(\sum_{\alpha\in S}\alpha^2\right)x+\left(\sum_{\alpha\in S}(\alpha+1)^2\right)\beta=0$ has real roots, is \_\_\_\_\_\_.

Answer: 25

Accepted Answer

Solving the logarithmic equation for $\alpha$ gives a set $S$ with $\sum_{\alpha\in S}\alpha^2=25$ and $\sum_{\alpha\in S}(\alpha+1)^2=25$.

The quadratic becomes $x^2-50x+25\beta=0$. For real roots, the discriminant must be non-negative:
$$2500-100\beta\ge0 \implies \beta\le25.$$

$$\beta_{\max}=25.$$

Solutions · 1