MathDB

4

Part of 2022 Harvard-MIT Mathematics Tournament

Problems(4)

2022 Algebra/NT #4

Source:

3/11/2022
Compute the sum of all 2-digit prime numbers pp such that there exists a prime number qq for which 100q+p100q + p is a perfect square.
number theory
2022 Team 4

Source:

3/14/2022
Suppose n≥3n \ge 3 is a positive integer. Let a1<a2<...<ana_1 < a_2 < ... < a_n be an increasing sequence of positive real numbers, and let an+1=a1a_{n+1} = a_1. Prove that ∑k=1nakak+1>∑k=1nak+1ak\sum_{k=1}^{n}\frac{a_k}{a_{k+1}}>\sum_{k=1}^{n}\frac{a_{k+1}}{a_k}
algebrainequalities
2022 Geometry 4

Source:

3/14/2022
Parallel lines ℓ1\ell_1, ℓ2\ell_2, ℓ3\ell_3, ℓ4\ell_4 are evenly spaced in the plane, in that order. Square ABCDABCD has the property that AA lies on ℓ1\ell_1 and CC lies on ℓ4\ell_4. Let PP be a uniformly random point in the interior of ABCDABCD and let QQ be a uniformly random point on the perimeter of ABCDABCD. Given that the probability that PP lies between ℓ2\ell_2 and ℓ3\ell_3 is 53100\frac{53}{100} , the probability that QQ lies between ℓ2\ell_2 and ℓ3\ell_3 can be expressed as ab\frac{a}{b}, where aa and bb are relatively prime positive integers. Compute 100a+b100a + b.
geometry
2022 Combinatorics 4

Source:

3/18/2022
Compute the number of nonempty subsets S⊆{−10,−9,−8,...,8,9,10}S \subseteq\{-10,-9,-8, . . . , 8, 9, 10\} that satisfy ∣S∣+ min(S)⋅max⁡(S)=0.|S| +\ min(S) \cdot \max (S) = 0.
combinatorics