CEMC Hypatia, 2006

Number theory Difficulty 3.0 AMC 10/12 Prove it Canada

The odd positive integers are arranged in rows in the triangular pattern, as shown.

Row 1: $1$
Row 2: $3$ $5$
Row 3: $7$ $9$ $11$
Row 4: $13$ $15$ $17$ $19$
Row 5: $21$ $23$ $\cdots$

(a) What is the 25th odd positive integer? In which row of the pattern will this integer appear?

(b) What is the 19th integer that appears in the 21st row? Explain how you got your answer.

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