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2017-07-31 Intermediate

         

\[ \large{\begin{array}{ccc} && & {\color{blue}E} & {\color{green}O} \\ && & & {\color{orange}O}\\ + && & & {\color{purple}O}\\ \hline & & & E & {\color{red}O}\\ \end{array}} \]

The \(E\) positions in the cryptogram above indicate even digits and the \(O\) positions indicate odd digits. No digit is repeated. While there is not a unique way to fill in the cryptogram, the red \(\color{red}O\) in the sum's result can only be one particular value. What is it?

True or False?

  1. A cyclic \( \color{blue}\text{pentagon} \) has equal angles if and only if it has equal sides.
  2. A cyclic \( \color{green} \text{hexagon} \) has equal angles if and only if it has equal sides.

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Note: A cyclic polygon is a polygon that can be inscribed in a circle. A cyclic pentagon and a cyclic hexagon are shown below.

A block of mass \(M\) is connected to a wall by two springs of respective spring constants \(k_1\) and \(k_2\). The block shows simple harmonic oscillation with amplitude \(A\).

Find the amplitude of oscillation of point \(P\) where the two springs are connected.

Suppose you walk into a room where the wall on the left, the wall in front, and the floor are all mirrors. (The walls and the floor are mutually perpendicular.)

If you hold up a ball, then how many images of the ball can you see in the mirrors?

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Note: The images shown in the figure are just the primary images.

\[\begin{cases} a=\sqrt{4-\sqrt{5-a}} \\ b=\sqrt{4+\sqrt{5-b}} \\ c=\sqrt{4-\sqrt{5+c}} \\ d=\sqrt{4+\sqrt{5+d}} \end{cases} \]

Find the product \(abcd.\)

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Hint: You don't need to find any of the individual variables!

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