Problems of the Week

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2017-12-11 Intermediate

         

In the regular decagon below, find the percentage of area that is shaded orange.

ab=a2+b2 ab = a^2+b^2 It is obvious that a=b=0a=b=0 satisfies the equation above. But is there another pair of real numbers (a,b)(a,b) satisfying this equation?

A slackline is stretched between two trees at a distance, d=10md = 10\,\text{m}, so that a tensioning force, F0=5000NF_0 = 5000 \,\text{N}, is applied on the empty rope. A man then balances in the middle of the rope and exerts a weight force, Fg=750NF_g = 750 \,\text{N}, onto the slackline. What is the vertical deflection, hh, of the rope?

Details and Assumptions: The rope is made of an elastic material and obeys Hooke's law F=Kll0l0,F = K \frac{l - l_0}{l_0}, where FF is the force along the rope, l0l_0 is the initial length, ll is the stretched length of the rope, and K=150 kNK = 150\text{ kN} is the force constant.

You have a set of 10 paints of distinct colors and a flag with 5 regions, shown to the right.

In how many ways can you color all 5 of the regions such that regions that share an edge are different colors?


Note: The flag is attached to a flagpole, which prevents the flag from being rotated or flipped.

What is the minimum number of soldiers needed to form 1212 lines of at least 33 soldiers each?

6 soldiers can form 3 lines of 3 soldiers each 6 soldiers can form 3 lines of 3 soldiers each


Note: The lines must be distinct. For example, if you have 5 soldiers in a line, this only counts as 1 line.

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