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2017-04-17 Intermediate

         

Does there exist a quadrilateral with the side lengths and diagonal lengths as indicated above?

Note: We are working in R2 \mathbb{R}^2 with the usual Euclidean geometry.

A rope of length 90 cm\SI{90}{\centi\meter} lies in a straight line on a frictionless table, except for a very small piece at one end which hangs down through a hole in the table.

This piece is released, and the rope slides down through the hole. What is the speed ((in m/s)\text{m/s}) of the rope (to 2 decimal places) at the instant it loses contact with the table?

Details: g=9.81 m/s2.g = \SI[per-mode=symbol]{9.81}{\meter\per\second\squared}.

ABCDABCD is a square with points EE and FF lying on sides CDCD and AD,AD, respectively. If the purple area is [BHGI]=120,[BHGI]=120, what is the sum of the pink areas [AHF]+[FGED]+[ICE]?[AHF]+[FGED]+[ICE]?

A particle moves in 1-dimension. If we plot its velocity and displacement over time, the trajectory forms a circle that's centered at the origin.

Which of the following relations is true regarding its acceleration (a),(a), velocity (v),(v), and displacement (x)(x)?

Note: In the options, kk is a positive constant.

There are nn people in a room, who each wear a hat of a specific color. They are able to see other people's hats but not their own.

One of them shouted, "If you can see at least 5 red hats and at least 5 white hats, raise your hand!"

Exactly 10 people raised their hands.

What is the minimum value of nn that fits this scenario?

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