MathDB

A triangular array of numbers has a first row consisting of the odd integers

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

? [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]

Problem Statement