Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 2.1 Junior Find the answer

How many points $(x, y)$ , with $x$ and $y$ both integers, are on the line with equation $y=4x+3$ and inside the region bounded by $x=25, x=75, y=120$ , and $y=250$ ?

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

We determine the number of integers $x$ with $25 \leq x \leq 75$ for which $120 \leq 4x+3 \leq 250$ . When $x=30$ , $4x+3=123$ and when $x=61$ , $4x+3=247$ . Therefore, $4x+3$ is between 120 and 250 exactly when $30 \leq x \leq 61$ . There are $61-30+1=32$ such values of $x$ .

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