Grades 9-10 Video Solutions 2024-2024_9-10

Video Transcription

Suppose m and n are integers with 0 less than m less than n. Let p equals mn, q equals nm, and o equals 00. For how many pairs of m and n will the area of triangle OPQ be 2024?  First we'll find the area of A and we'll use the shoelace method to find it. If you haven't  learned the shoelace method you should definitely look it up and find an online guide on how to use  it. We find the area of OPQ is 1 half n squared minus m squared and therefore the area the value  of n plus n times n minus m must be 4048. We know that m and n are both integers and thus n plus m  and n minus m will both be even or both be odd. There are six ways that 4048 equals 2 to the 4  times 11 times 23 can be represented as a product of two even numbers. 2 times 2024, 2 squared times  1012, 2 to the cubed times 506, 2 times 11 times 2 cubed times 23, 2 squared times 11 times 2  times 23, and 2 times 23 times 2 to the cubed times 11. Thus there are six possible pairs of n and m as we just listed previously and thus our answer is 6.

Video Summary

The solution calculates the number of integer pairs \((m, n)\) where the area of triangle OPQ equals 2024, given \(m\) and \(n\) are integers such that \(0 < m < n\). Employing the shoelace method, the area is determined to be \(\frac{1}{2}(n^2 - m^2)\). For the area to be 2024, the expression \(n^2 - m^2\) must be 4048. Since \(n + m\) and \(n - m\) must both either be even or both be odd, this expression is broken down into six possible pairs of integer factors. Thus, there are six valid pairs of \((m, n)\).

Keywords

integer pairs

triangle area

shoelace method

factor pairs

solution calculation