The MIT Program for Research in Math, Engineering, and Science (PRIMES) will award PRIMES internships to the two highest ranked participants in CrowdMath 2017 who are still in high school in Spring 2018. These students will have the opportunity to work on a research project on Skype with a mentor at MIT through PRIMES in 2018. One student will be chosen from each of the Graph Algorithms and Applications and Broken Stick projects.
CrowdMath is an online research program open to high school and college students all over the world. High school students can improve their CrowdMath rankings by posting questions, answers, ideas, or proofs on the CrowdMath message board. The rankings for the 2017 CrowdMath PRIMES prizes will be based on CrowdMath posts during the year 2017 and will be finalized on December 31, 2017. Rankings will be posted below and updated monthly.
Notice that if we do the opposite procedure, that is, place the next break in the unit length in the smaller segment, we end up with no possible triangle. This is because the larger segment must have length , and as this segment is not broken up any further, it remains larger than the semiperimeter. Thus, as given before, the area expected is , as noted before. The same applies to the second scenario in which we place the next cut in the rightmost segment.
Since the equation of a line is y=mx+c, all the possible values of m and c using an inequality should be plotted on a graph, where the x-axis is c and y-axis is m. Then another region should be drawn which points inside, (c,m), that would give the equation of a line that would cut the square into to unequal regions. Then the ratio of these two regions should be calculated. This would probably require one to look at individual cases of what some of the lines would look like.
I found some useful results about cyclic quadrilaterals that may generalize to Problem(ii)
EDIT: I'll think about it some more but I think this problem can be solved by breaking n-gons into a bunch of quadrilaterals and maybe 1 triangle (depending on n) and maximizing the areas by making all the quadrilaterals cyclic and the triangle equilateral
players play the following game consisting of turns. Alice's clones and Bob's clones go in some order. After all turns have passed, there are pieces. What is the maximum number of triangles that Alice's clones can guarantee forming given that Bob's clones are working against Alice's clones?
Clarification: One possible ordering might be , which would repeat, and so for the first turns in this example, Alice's clones and Bob's clones would follow the sequence .
Instead of asking for the maximum number of triangles Alice can make, I think it might also be interesting if we ask for the minimum. What do you guys think?