MathDB

3

Part of 2019 Harvard-MIT Mathematics Tournament

Problems(4)

HMMT Team 2019/3: Trig ineq with one degree of freedom

Source:

2/17/2019
For any angle 0<θ<π/20 < \theta < \pi/2, show that 0<sin⁡θ+cos⁡θ+tan⁡θ+cot⁡θ−sec⁡θ−csc⁡θ<1.0 < \sin \theta + \cos \theta + \tan \theta + \cot \theta - \sec \theta - \csc \theta < 1.
HMMTtrigonometryalgebra
HMMT Algebra/NT 2019/3: Twist on standard t^3 + 1/t^3 problem

Source:

2/17/2019
Let xx and yy be positive real numbers. Define a=1+xya = 1 + \tfrac{x}{y} and b=1+yxb = 1 + \tfrac{y}{x}. If a2+b2=15a^2 + b^2 = 15, compute a3+b3a^3 + b^3.
HMMTalgebra
HMMT Combinatorics 2019/3: Reimu and Sanae Play a Game

Source:

2/17/2019
Reimu and Sanae play a game using 44 fair coins. Initially both sides of each coin are white. Starting with Reimu, they take turns to color one of the white sides either red or green. After all sides are colored, the four coins are tossed. If there are more red sides showing up, then Reimu wins, and if there are more green sides showing up, then Sanae wins. However, if there is an equal number of red sides and green sides, then neither of them wins. Given that both of them play optimally to maximize the probability of winning, what is the probability that Reimu wins?
HMMTprobability
HMMT Geometry 2019/3: Find X on AB with AX = 2PX

Source:

2/17/2019
Let ABAB be a line segment with length 2, and SS be the set of points PP on the plane such that there exists point XX on segment ABAB with AX=2PXAX = 2PX. Find the area of SS.
HMMTgeometry