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2023 SMT Guts Round 6 p16-18 - Stanford Math Tournament

Source:

August 31, 2023
Stanford Math Tournamentalgebrageometry

Problem Statement

p16. When not writing power rounds, Eric likes to climb trees. The strength in his arms as a function of time is s(t)=t3−3t2s(t) = t^3 - 3t^2. His climbing velocity as a function of the strength in his arms is v(s)=s5+9s4+19s3−9s2−20sv(s) = s^5 + 9s^4 + 19s^3 - 9s^2 - 20s. At how many (possibly negative) points in time is Eric stationary?
p17. Consider a triangle △ABC\vartriangle ABC with angles ∠ACB=60o\angle ACB = 60^o, ∠ABC=45o\angle ABC = 45^o. The circumcircle around △ABH\vartriangle ABH, where HH is the orthocenter of △ABC\vartriangle ABC, intersects BCBC for a second time in point PP, and the center of that circumcircle is OcO_c. The line PHPH intersects ACAC in point QQ, and NN is center of the circumcircle around △AQP\vartriangle AQP. Find ∠NOcP\angle NO_cP.
p18. If x,yx, y are positive real numbers and xy3=169xy^3 = \frac{16}{9} , what is the minimum possible value of 3x+y3x + y?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
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