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2018-10-08 Basic

         

\[ \Huge \color{purple}\sqrt[3]{\color{green}3^{\color{blue}{3}^{\color{red}{3}}}} \color{orange}=\ ?\]

Hint:

  • \(\displaystyle \large 3^{3^3} = 3^{ \left(3^3 \right) } \)
  • \(\displaystyle \large 3^{3^3} \ne { \left(3^3 \right) }^3 \)

\[\large x = \underbrace{11111...11111}_{\text{Number of 1s = 100}} - \underbrace{22222...22222}_{\text{Number of 2s = 50}} \]

What is the sum of digits in \(x?\)

Hint: Try it with smaller numbers first. For example, find the sum of digits in \(111111 - 222.\)

I know that just adding numerators and denominators doesn’t work. For example,

\[\frac{1}{2}+\frac{2}{3} \ne \frac{1+2}{2+3}.\]

Curious if this kind of addition ever works, I look for positive integers \(a,b,c,d\) such that

\[ \dfrac ab + \dfrac cd = \dfrac{a+c}{b+d}.\]

After some searching, I conclude that such numbers don't exist. Am I right?

A system of two symmetrical, identical, water-filled jars is fixed on a turntable. Each jar has a cork—attached to the bottom by a thread—floating vertically.

What will happen to the corks when the turntable starts rotating?

Details and Assumptions:

  • The jars are closed so that no water spills out.

What is the length of \(BC?\)

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