Problems of the Week

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2017-05-15 Intermediate

A right trapezoid is partitioned into 4 triangles by its diagonals, as shown below:

Which colored region has a larger area?

The map below represents a neighborhood. Each node represents a house, and the segments represent the paths between those houses. Alice's house is shown in red, and Bob's house is shown in green.

Alice is going to Bob's house to celebrate his birthday. For each house that she visits along the way, she will invite one person from that house. Additionally, Alice will not visit the same house (including her own house) or pass through the same path more than once.

What is the maximum even number of people Alice can invite?


Details and Assumptions:

• Alice cannot turn while in the middle of a path represented by a black line segment, she can only turn while at one of the houses.

Melissa has had a crush on Nick for quite some time, and finally found the courage to talk to him. After some good conversation, Melissa asked Nick the following question, oblivious to the fact that he is already dating someone else:

"If I asked you out, would your answer be the same as your answer to this question?"

Assuming he responds truthfully in a yes or no fashion, will Nick go on a date with Melissa?

The diagram below shows stairs with eleven steps. Moving up only 1 or 2 steps at a time, how many different ways can a person reach the top?

The army is testing out a new prototype artillery cannon with an uncommonly high muzzle velocity of $$\SI[per-mode=symbol]{1000}{\meter\per\second}$$. The design bugs haven't been fully worked out yet, so the cannon has to wait at least a full minute between shots (reloading, etc.). The cannon can be aimed with an angle between 0 and 90 degrees with respect to the horizontal.

A particularly demanding exercise is being conducted, in which the cannon fires an initial shot into the air, and while the first shell is still in the air, a second shot is fired (at a different angle) so that both shells impact the target (at ground level) simultaneously.

What is the maximum horizontal distance from the cannon to the target, such that this is possible?


Details and Assumptions:

• Assume level ground, with no air resistance.
• The gravitational acceleration is $$\SI[per-mode=symbol]{10}{\meter\per\second\squared}$$.
• Give your answer in meters, to the nearest whole meter.
• For the sake of this problem, ignore the Earth's curvature.
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