# Problems of the Week

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# 2017-02-27 Basic

In the figure to the right, $PQRSTU$ is a regular hexagon.

Sides $PQ, RS,$ and $TU$ are extended to form triangle $ABC$.

If the area of triangle $ABC$ is 90, then find the area of the shaded region.

When we take a blanket at room temperature and wrap it around ourselves, we become warmer than if we are not covered by the blanket.

When we take a blanket at room temperature and wrap it around an ice cube, would the ice melt faster or slower than if it was not covered by the blanket?


Details and Assumptions:

• Assume that the ice does not touch the blanket due to an air gap between the blanket and the ice.
• The initial temperatures of the blanket and the room are the same.

Three suspects appeared in court for the crime of murdering Jimmy. Exactly one of them is the murderer.

The judge asked the following question: "Did the murderer currently have black or brown hair?"

Alice (Female with black hair): "She has black hair."
Andrew (Male with black hair): "Alice is lying! She has brown hair."
Anna (Female with brown hair): "I speak the truth! He has brown hair."

Given that only one of them is telling the truth, any sentence a liar states is a lie, and no one accuses themselves, who is telling the truth?

True or False?

$\large \lim_{x\rightarrow\infty} \frac{x + \sin x}{x+1} = \lim_{x\rightarrow\infty} \frac{\tfrac{d}{dx}(x + \sin x)}{\tfrac{d}{dx}(x+1)}$

Six circles, each of area 1, are placed in the following manner:

Does there exist a straight line which divides the figure into 2 parts, such that the red, blue and green areas on each side are all equal to 1?

Assumption: The black border around the circles is of zero width, and the circles abut right up against each other.

Inspiration

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