MathDB

5

Part of 2014 Harvard-MIT Mathematics Tournament

Problems(5)

2014 HMMT #5: sum of real roots

Source:

2/23/2014
Find the sum of all real numbers xx such that 5x4−10x3+10x2−5x−11=05x^4-10x^3+10x^2-5x-11=0.
HMMTquadraticsfunction
2014 Combinatorics #5: Splitting Shares

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2/23/2014
Eli, Joy, Paul, and Sam want to form a company; the company will have 16 shares to split among the 44 people. The following constraints are imposed:
∙\bullet Every person must get a positive integer number of shares, and all 1616 shares must be given out. ∙\bullet No one person can have more shares than the other three people combined.
Assuming that shares are indistinguishable, but people are distinguishable, in how many ways can the shares be given out?
countingdistinguishability
2014 Geometry #5: 3-D Geometry

Source:

2/25/2014
Let C\mathcal{C} be a circle in the xyxy plane with radius 11 and center (0,0,0)(0, 0, 0), and let PP be a point in space with coordinates (3,4,8)(3, 4, 8). Find the largest possible radius of a sphere that is contained entirely in the slanted cone with base C\mathcal{C} and vertex PP.
geometryanalytic geometry3D geometrysphereinradius
2014 Guts #5: Lowest Number on Die

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2/25/2014
[5] If four fair six-sided dice are rolled, what is the probability that the lowest number appearing on any die is exactly 33?
probability
2014 Team #5: Magnitude of Difference of Magnitudes Constant

Source:

3/2/2014
Prove that there exists a nonzero complex number cc and a real number dd such that ∣∣11+z+z2∣−∣11+z+z2−c∣∣=d\left|\left|\dfrac1{1+z+z^2}\right|-\left|\dfrac1{1+z+z^2}-c\right|\right|=d for all zz with ∣z∣=1|z|=1 and 1+z+z2≠01+z+z^2\neq 0. (Here, ∣z∣|z| denotes the absolute value of the complex number zz, so that ∣a+bi∣=a2+b2|a+bi|=\sqrt{a^2+b^2} for real numbers a,ba,b.)
absolute value