MathDB

4

Part of 2014 Harvard-MIT Mathematics Tournament

Problems(5)

2014 Guts #4: Choosing Divisors

Source:

2/25/2014
[4] Let DD be the set of divisors of 100100. Let ZZ be the set of integers between 11 and 100100, inclusive. Mark chooses an element dd of DD and an element zz of ZZ uniformly at random. What is the probability that dd divides zz?
probabilityexpected value
2014 HMMT #4: Polynomial

Source:

2/23/2014
Let bb and cc be real numbers and define the polynomial P(x)=x2+bx+cP(x)=x^2+bx+c. Suppose that P(P(1))=P(P(2))=0P(P(1))=P(P(2))=0, and that P(1)≠P(2)P(1) \neq P(2). Find P(0)P(0).
HMMTalgebrapolynomialVietasum of roots
2014 Combinatorics #4: Triplets of Sets

Source:

2/23/2014
Find the number of triples of sets (A,B,C)(A, B, C) such that:
(a) A,B,C⊆{1,2,3,…,8}A, B, C \subseteq \{1, 2, 3, \dots , 8 \}. (b) ∣A∩B∣=∣B∩C∣=∣C∩A∣=2|A \cap B| = |B \cap C| = |C \cap A| = 2. (c) ∣A∣=∣B∣=∣C∣=4|A| = |B| = |C| = 4.
Here, ∣S∣|S| denotes the number of elements in the set SS.
HMMTcountingdistinguishability
2014 Geometry #4: Unsuccessful Angle Chase

Source:

2/25/2014
In quadrilateral ABCDABCD, ∠DAC=98∘\angle DAC = 98^{\circ}, ∠DBC=82∘\angle DBC = 82^\circ, ∠BCD=70∘\angle BCD = 70^\circ, and BC=ADBC = AD. Find ∠ACD.\angle ACD.
geometrygeometric transformationreflectiontrigonometrytrig identitiesLaw of Sinescongruent triangles
2014 Team #4: Sum Involving the Floor Function

Source:

3/2/2014
Compute ∑k=0100⌊2100250+2k⌋.\sum_{k=0}^{100}\left\lfloor\dfrac{2^{100}}{2^{50}+2^k}\right\rfloor. (Here, if xx is a real number, then ⌊x⌋\lfloor x\rfloor denotes the largest integer less than or equal to xx.)
functionfloor function