\[ \begin{align*} \large 9^a&\large= 25 \\ \\ \large 3^a &\large = \ ? \end{align*} \]

What goes at the question mark in the equations above?

Hint: There's a much faster way than solving for \(a.\) You are not a robot, and you don't need a calculator to solve this!

\[ \large \frac { 1 }{ { 2 }^{ 3 } } +\frac { 1 }{ { 2 }^{ 6 } } +\frac { 1 }{ { 2 }^{ 9 } } + \cdots = \ ? \]

**Hint:** Don’t try this one on a calculator either - your fingers will wear out! This algebra problem can be solved with a geometric shortcut:

What total fraction of this image is red?

Which is greater, \(A\) or \(B?\)

\[A=\frac{99^{999}+1}{99^{1000}+1}\]

\[B=\frac{99^{1000}+1}{99^{1001}+1}\]

It’s a race! Who is the winner? Choose wisely...

Which of these sequences will exceed \(1,000,000^{1,000,000}\) first?

Sequence A: | \(1^2\) | \((1^2)^3\) | \(((1^2)^3)^4\) | \((((1^2)^3)^4)^5\) | ... |

Sequence B: | \(2^1\) | \((3^2)^1\) | \(((4^3)^2)^1\) | \((((5^4)^3)^2)^1\) | ... |

Sequence C: | \(2^1\) | \(3^{2^1}\) | \(4^{3^{2^1}}\) | \(5^{4^{3^{2^1}}}\) | ... |

Note: Towers of exponents are evaluated **from the top down**, so \(3^{2^1} = 3^{\left(2^1\right)}.\)

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