2023/24 AHSMC Part 1 Problem 11

Problem 11 of 16

Jamal writes down an increasing sequence of positive integers, starting with $1$ , according to the following procedure:

Step $1$ : write down the first two numbers, skip one,

Step $2$ : write down the next two numbers, skip two,

Step $3$ : write down the next two numbers, skip three,

$\ldots$

Step $k$ : write down the next two numbers, skip $k$ ,

$\ldots$

For example, after step $5$ , Jamal will so far have written down the numbers $1$ , $2$ , $4$ , $5$ , $8$ , $9$ , $13$ , $14$ , $19$ , $20$ . On what step will Jamal first write down a four digit number?

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Solution

After step $k$ , Jamal has written $2k$ numbers and skipped $1+2+\cdots+(k-1) = k(k-1)/2$ numbers. The largest number written is $2k + k(k-1)/2 = (k^2+3k)/2$ . We want the smallest $k$ such that $(k^2+3k)/2 \ge 1000$ . Since $44 \cdot 47 \ge 2000$ while $43 \cdot 46 < 2000$ , one obtains $k = 44$ . The answer is (B).

University of Alberta

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