MathDB

2

Part of 2010 Harvard-MIT Mathematics Tournament

Problems(3)

2010 Algebra #2: Number Ranks

Source:

7/15/2012
The rank of a rational number qq is the unique kk for which q=1a1+⋯+1akq=\frac{1}{a_1}+\cdots+\frac{1}{a_k}, where each aia_i is the smallest positive integer qq such that q≥1a1+⋯+1aiq\geq \frac{1}{a_1}+\cdots+\frac{1}{a_i}. Let qq be the largest rational number less than 14\frac{1}{4} with rank 33, and suppose the expression for qq is 1a1+1a2+1a3\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}. Find the ordered triple (a1,a2,a3)(a_1,a_2,a_3).
2010 Calculus #2: Differential Equation

Source:

7/15/2012
Let ff be a function such that f(0)=1f(0)=1, f′(0)=2f^\prime (0)=2, and f′′(t)=4f′(t)−3f(t)+1f^{\prime\prime}(t)=4f^\prime(t)-3f(t)+1 for all tt. Compute the 44th derivative of ff, evaluated at 00.
calculusfunctionderivative
2010 Geometry #2

Source:

1/2/2024
A rectangular piece of paper is folded along its diagonal (as depicted below) to form a non-convex pentagon that has an area of 710\tfrac{7}{10} of the area of the original rectangle. Find the ratio of the longer side of the rectangle to the shorter side of the rectangle. [asy] size(150); pair A = (-5,0); pair B = (5,0); pair C = (-3,4); pair D = (3,4); pair E = intersectionpoint(B--C,A--D); draw(A--B--D--cycle); draw(A--C); draw(C--E); draw(E--B,dashed); markscalefactor=0.06; draw(rightanglemark(A,C,B)); [/asy]
geometry