MathDB

3

Part of 2010 Harvard-MIT Mathematics Tournament

Problems(3)

2010 Algebra #3: Recursive Sequence

Source:

7/15/2012
Let S0=0S_0=0 and let SkS_k equal a1+2a2+…+kaka_1+2a_2+\ldots+ka_k for k≥1k\geq 1. Define aia_i to be 11 if Si−1<iS_{i-1}<i and −1-1 if Si−1≥iS_{i-1}\geq i. What is the largest k≤2010k\leq 2010 such that Sk=0S_k=0?
2010 Calculus #3: Common Zeroes

Source:

7/15/2012
Let pp be a monic cubic polynomial such that p(0)=1p(0)=1 and such that all the zeroes of p′(x)p^\prime (x) are also zeroes of p(x)p(x). Find pp. Note: monic means that the leading coefficient is 11.
calculusalgebrapolynomialVieta
2010 Geometry #3

Source:

1/2/2024
For 0≤y≤20\leq y\leq 2, let DyD_y be the half-disk of diameter 2 with one vertex at (0,y)(0,y), the other vertex on the positive xx-axis, and the curved boundary further from the origin than the straight boundary. Find the area of the union of DyD_y for all 0≤y≤20\leq y\leq 2.
geometry