MathDB

8

Part of 2013 Harvard-MIT Mathematics Tournament

Problems(4)

2013 HMMT Algebra #8: System of Rational Equations

Source:

2/17/2013
Let x,yx,y be complex numbers such that x2+y2x+y=4\dfrac{x^2+y^2}{x+y}=4 and x4+y4x3+y3=2\dfrac{x^4+y^4}{x^3+y^3}=2. Find all possible values of x6+y6x5+y5\dfrac{x^6+y^6}{x^5+y^5}.
HMMTcomplex numbers
2013 Team #8: Circles and Points

Source:

3/1/2013
Let points AA and BB be on circle ω\omega centered at OO. Suppose that ωA\omega_A and ωB\omega_B are circles not containing OO which are internally tangent to ω\omega at AA and BB, respectively. Let ωA\omega_A and ωB\omega_B intersect at CC and DD such that DD is inside triangle ABCABC. Suppose that line BCBC meets ω\omega again at EE and let line EAEA intersect ωA\omega_A at FF. If FC⊥CD FC \perp CD , prove that OO, CC, and DD are collinear.
geometrypower of a pointradical axis
2013 HMMT Guts #8: Raj and the Balls

Source:

3/26/2013
In a game, there are three indistinguishable boxes; one box contains two red balls, one contains two blue balls, and the last contains one ball of each color. To play, Raj first predicts whether he will draw two balls of the same color or two of different colors. Then, he picks a box, draws a ball at random, looks at the color, and replaces the ball in the same box. Finally, he repeats this; however, the boxes are not shuffled between draws, so he can determine whether he wants to draw again from the same box. Raj wins if he predicts correctly; if he plays optimally, what is the probability that he will win?
HMMTcountingdistinguishabilityprobability
2013 HMMT Geometry # 8

Source:

3/3/2024
Let ABCDABCD be a convex quadrilateral. Extend line CDCD past DD to meet line ABAB at PP and extend line CBCB past BB to meet line ADAD at QQ. Suppose that line ACAC bisects ∠BAD\angle BAD. If AD=74AD = \frac{7}{4}, AP=212AP = \frac{21}{2}, and AB=1411AB = \frac{14}{11} , compute AQAQ.
geometry