MathDB

10

Part of 2010 Harvard-MIT Mathematics Tournament

Problems(3)

2010 Algebra #10: Functional Compositional Equation

Source:

7/15/2012
Let p(x)p(x) and q(x)q(x) be two cubic polynomials such that p(0)=−24p(0)=-24, q(0)=30q(0)=30, and p(q(x))=q(p(x))p(q(x))=q(p(x)) for all real numbers xx. Find the ordered pair (p(3),q(6))(p(3),q(6)).
algebrapolynomial
2010 Calculus #10: Summation with Constants

Source:

7/15/2012
Let f(n)=∑k=1n1kf(n)=\displaystyle\sum_{k=1}^n \dfrac{1}{k}. Then there exists constants γ\gamma, cc, and dd such that f(n)=ln⁡(x)+γ+cn+dn2+O(1n3),f(n)=\ln(x)+\gamma+\dfrac{c}{n}+\dfrac{d}{n^2}+O\left(\dfrac{1}{n^3}\right), where the O(1n3)O\left(\dfrac{1}{n^3}\right) means terms of order 1n3\dfrac{1}{n^3} or lower. Compute the ordered pair (c,d)(c,d).
calculus
2010 Geometry #10

Source:

1/2/2024
Circles ω1\omega_1 and ω2\omega_2 intersect at points AA and BB. Segment PQPQ is tangent to ω1\omega_1 at PP and to ω2\omega_2 at QQ, and AA is closer to PQPQ than BB. Point XX is on ω1\omega_1 such that PX∥QBPX\parallel QB, and point YY is on ω2\omega_2 such that QY∥PBQY\parallel PB. Given that ∠APQ=30∘\angle APQ=30^\circ and ∠PQA=15∘\angle PQA=15^\circ, find the ratio AX/AYAX/AY.
geometry