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6 problems match “difference of squares”. Clear

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1. Difference of squared sine and cosine ●○○○○ 10 ptsEvaluate $(\sin 72^\circ + \cos 18^\circ)(\sin 72^\circ - \cos 18^\circ)$.
2. Telescoping sum of odd-numerator fractions ●●○○○ 20 ptsCalculate $\frac{3}{4}+\frac{5}{36}+\frac{7}{144}+\frac{9}{400}+\frac{11}{900}+\frac{13}{1764}+\frac{15}{3136}$. Give your answer as a decimal.
3. Alternating block sum of squares ●●○○○ 20 ptsCompute $100^{2}+99^{2}-98^{2}-97^{2}+96^{2}+95^{2}-94^{2}-93^{2}+\ldots+4^{2}+3^{2}-2^{2}-1^{2}$.
4. Sum of conjugate fractions ●●○○○ 20 ptsSimplify $\frac{3+\sqrt{5}}{3-\sqrt{5}}+\frac{3-\sqrt{5}}{3+\sqrt{5}}$.
5. Value missing from range of a Möbius function ●●●○○ 50 ptsThe function $f$ is defined by $f(x)= \frac{ax+b}{cx+d}$, where $a$, $b$, $c$ and $d$ are nonzero real numbers. It has the properties $f(19)=19$, $f(97)=97$ and $f(f(x))=x$ for all $x$ except $-\frac{d}{c}$. Find the unique number that is not in the range of $f$.
6. 101st digit of a period-200 reciprocal ●●●○○ 50 ptsLet $p$ be a prime number such that the period of the decimal expansion of $\frac{1}{p}$ is 200. That is, $\frac{1}{p}=0.XXXX\ldots$ for some block $X$ of 200 digits, but $\frac{1}{p}\neq 0.YYYY\ldots$ for every block $Y$ with fewer than 200 digits. Find the 101st digit, counting from the left, of $X$.