MathDB

6

Part of 2022 Harvard-MIT Mathematics Tournament

Problems(4)

2022 Algebra/NT #6

Source:

3/11/2022
Let f be a function from {1,2,...,22}\{1, 2, . . . , 22\} to the positive integers such that mn∣f(m)+f(n)mn | f(m) + f(n) for all m,n∈{1,2,...,22}m, n \in \{1, 2, . . . , 22\}. If dd is the number of positive divisors of f(20)f(20), compute the minimum possible value of dd.
number theory
2022 Team 6

Source:

3/14/2022
Let P(x)=x4+ax3+bx2+xP(x) = x^4 + ax^3 + bx^2 + x be a polynomial with four distinct roots that lie on a circle in the complex plane. Prove that ab≠9ab\ne 9.
algebracomplex numberspolynomial
2022 Geometry 6

Source:

3/14/2022
Let ABCDABCD be a rectangle inscribed in circle Γ\Gamma, and let PP be a point on minor arc ABAB of Γ\Gamma. Suppose that PA⋅PB=2P A \cdot P B = 2, PC⋅PD=18P C \cdot P D = 18, and PB⋅PC=9P B \cdot P C = 9. The area of rectangle ABCDABCD can be expressed as abc\frac{a\sqrt{b}}{c} , where aa and cc are relatively prime positive integers and bb is a squarefree positive integer. Compute 100a+10b+c100a + 10b + c.
geometry
2022 Combinatorics 6

Source:

3/18/2022
The numbers 1,2,...,101, 2, . . . , 10 are randomly arranged in a circle. Let pp be the probability that for every positive integer k<10k < 10, there exists an integer k′>kk' > k such that there is at most one number between kk and k′k' in the circle. If pp can be expressed as ab\frac{a}{b} for relatively prime positive integers aa and bb, compute 100a+b100a + b.
combinatorics