1. **State the problem:** We start with the inequality: $$\frac{3^{2d}}{\sum_{t=1}^d \binom{d}{t} \left(\frac{t}{\lfloor t/2 \rfloor}\right)^2} < \frac{1}{1 - \frac{1}{n/2}}$$ We want to rearrange this inequality and find an upper bound for $d$ in terms of expressions containing $n$ and $k$ (assuming $k$ relates to the summation or parameters). 2. **Rewrite the right side:** Note that: $$1 - \frac{1}{n/2} = 1 - \frac{2}{n} = \frac{n-2}{n}$$ So the right side becomes: $$\frac{1}{\frac{n-2}{n}} = \frac{n}{n-2}$$ 3. **Rewrite the inequality:** $$\frac{3^{2d}}{\sum_{t=1}^d \binom{d}{t} \left(\frac{t}{\lfloor t/2 \rfloor}\right)^2} < \frac{n}{n-2}$$ Multiply both sides by the summation: $$3^{2d} < \frac{n}{n-2} \sum_{t=1}^d \binom{d}{t} \left(\frac{t}{\lfloor t/2 \rfloor}\right)^2$$ 4. **Analyze the summation:** The summation is complicated, but note that for each $t$, $\left(\frac{t}{\lfloor t/2 \rfloor}\right)^2$ is roughly bounded by a constant since $\frac{t}{\lfloor t/2 \rfloor} \approx 2$ for large $t$. Thus, approximate: $$\left(\frac{t}{\lfloor t/2 \rfloor}\right)^2 \leq 4$$ So, $$\sum_{t=1}^d \binom{d}{t} \left(\frac{t}{\lfloor t/2 \rfloor}\right)^2 \leq 4 \sum_{t=1}^d \binom{d}{t} = 4(2^d - 1) < 4 \cdot 2^d$$ 5. **Use this bound in the inequality:** $$3^{2d} < \frac{n}{n-2} \cdot 4 \cdot 2^d$$ 6. **Rewrite $3^{2d}$ as $(3^2)^d = 9^d$:** $$9^d < \frac{4n}{n-2} 2^d$$ Divide both sides by $2^d$: $$\frac{9^d}{2^d} < \frac{4n}{n-2}$$ Rewrite left side: $$\left(\frac{9}{2}\right)^d < \frac{4n}{n-2}$$ 7. **Take natural logarithm on both sides:** $$d \ln\left(\frac{9}{2}\right) < \ln\left(\frac{4n}{n-2}\right)$$ 8. **Solve for $d$:** $$d < \frac{\ln\left(\frac{4n}{n-2}\right)}{\ln\left(\frac{9}{2}\right)}$$ **Final answer:** $$\boxed{d < \frac{\ln\left(\frac{4n}{n-2}\right)}{\ln\left(\frac{9}{2}\right)}}$$ This gives an upper bound for $d$ in terms of $n$. The parameter $k$ was not explicitly in the original inequality, so it does not appear here. --- "slug": "upper bound d", "subject": "algebra", "desmos": {"latex": "y=\frac{\ln\left(\frac{4n}{n-2}\right)}{\ln\left(\frac{9}{2}\right)}", "features": {"intercepts": true, "extrema": true}}, "q_count": 1