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

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

A rope of length \(\SI{90}{\centi\meter}\) 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 \(\text{m/s})\) of the rope (to 2 decimal places) at the instant it loses contact with the table?

**Details**: \(g = \SI[per-mode=symbol]{9.81}{\meter\per\second\squared}.\)

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 \((a),\) velocity \((v),\) and displacement \((x)\)?

**Note:** In the options, \(k\) is a positive constant.

There are \(n\) 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 \(n\) that fits this scenario?

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