Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.1 AIME, harder Find the answer

4. In the same Cartesian coordinate system, it is known that the line $y=k x$ intersects the graph of the function
$y=\left\{\begin{array}{ll} 2 x+4, & x<3 \\ x+7, & x \geq 3 \end{array}\right.$

at exactly three different points. Determine the range of values for $k$ .

Your answer

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

Solution

4. $\left(\frac{2}{3}, 2\right)$ .

Draw a rough sketch of the function in the coordinate system, i.e., the thick black broken line in Fig. 4.

The line $l_{1}$ : $y=2 x$ intersects the broken line at only one point;

The line $l_{2}$ : $y=\frac{2}{3} x$ intersects the broken line at only two points.

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