MathDB

6

Part of 2008 AIME Problems

Problems(2)

Triangular Array

Source: AIME 2008I Problem 6

3/23/2008
A triangular array of numbers has a first row consisting of the odd integers 1,3,5,…,99 1,3,5,\ldots,99 in increasing order. Each row below the first has one fewer entry than the row above it, and the bottom row has a single entry. Each entry in any row after the top row equals the sum of the two entries diagonally above it in the row immediately above it. How many entries in the array are multiples of 67 67? [asy]size(200); defaultpen(fontsize(10)); label("1", origin); label("3", (2,0)); label("5", (4,0)); label("⋯\cdots", (6,0)); label("97", (8,0)); label("99", (10,0));
label("4", (1,-1)); label("8", (3,-1)); label("12", (5,-1)); label("196", (9,-1)); label(rotate(90)*"⋯\cdots", (6,-2));[/asy]
inductionnumber theoryrelatively primeAMC
Two Sequences

Source: AIME 2008II Problem 6

4/3/2008
The sequence {an} \{a_n\} is defined by a_0 \equal{} 1,a_1 \equal{} 1, \text{ and } a_n \equal{} a_{n \minus{} 1} \plus{} \frac {a_{n \minus{} 1}^2}{a_{n \minus{} 2}}\text{ for }n\ge2. The sequence {bn} \{b_n\} is defined by b_0 \equal{} 1,b_1 \equal{} 3, \text{ and } b_n \equal{} b_{n \minus{} 1} \plus{} \frac {b_{n \minus{} 1}^2}{b_{n \minus{} 2}}\text{ for }n\ge2. Find b32a32 \frac {b_{32}}{a_{32}}.
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