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2017-07-31 Intermediate

         

EOO+OEO \large{\begin{array}{ccc} && & {\color{#3D99F6}E} & {\color{#20A900}O} \\ && & & {\color{#EC7300}O}\\ + && & & {\color{#69047E}O}\\ \hline & & & E & {\color{#D61F06}O}\\ \end{array}}

The EE positions in the cryptogram above indicate even digits and the OO positions indicate odd digits. No digit is repeated. While there is not a unique way to fill in the cryptogram, the red O\color{#D61F06}O in the sum's result can only be one particular value. What is it?

True or False?

  1. A cyclic pentagon \color{#3D99F6}\text{pentagon} has equal angles if and only if it has equal sides.
  2. A cyclic hexagon \color{#20A900} \text{hexagon} has equal angles if and only if it has equal sides.


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 MM is connected to a wall by two springs of respective spring constants k1k_1 and k2k_2. The block shows simple harmonic oscillation with amplitude AA.

Find the amplitude of oscillation of point PP 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?


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

{a=45ab=4+5bc=45+cd=4+5+d\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.abcd.


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

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