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Part of 2010 Harvard-MIT Mathematics Tournament

Problems(3)

2010 Algebra #1: System of Positive Reals

Source:

x

y

are positive reals such that

x-y^2=3, \qquad x^2+y^4=13.

x

2010 Calculus #1: Subtract the Derivative

Source:

p(x)

is a polynomial and that

p(x)-p^\prime (x)=x^2+2x+1

p(5)

calculusderivativealgebrapolynomial

2010 Geometry #1

Source:

Below is pictured a regular seven-pointed star. Find the measure of angle

a

in radians. [asy] size(150); draw(unitcircle, white); pair A = dir(180/7); pair B = dir(540/7); pair C = dir(900/7); pair D = dir(180); pair E = dir(-900/7); pair F = dir(-540/7); pair G = dir (-180/7); draw(A--D); draw(B--E); draw(C--F); draw(D--G); draw(E--A); draw(F--B); draw(G--C); label((-0.1,0.5), "

a