AMC 8 Daily Practice Round 11

Complete problem set with solutions and individual problem pages

Problem 20 Easy

As shown in the figure, a grid is formed by squares of side length 1, and 9 grid points (the intersections of the grid lines) are selected. A circle is drawn with center A and radius r. If exactly 3 of the selected points (excluding A) lie inside the circle, what is the range of r?

  • A.

    2\sqrt{2}<{}r<{}\sqrt{17}

  • B.

    \sqrt{17}<{}r\leqslant {}3\sqrt{2}

  • C.

    \sqrt{17}<{}r<{}5

  • D.

    5<{}r<{}\sqrt{29}

  • E.

    5<{}r\leqslant {}3\sqrt{2}

Answer:B

Connect A{{A}_{1}}\sim A{{A}_{8}}，

​

Then, A{{A}_{1}}=\sqrt{{{3}^{2}}+{{3}^{2}}}=3\sqrt{2},

A{{A}_{2}}=\sqrt{{{1}^{2}}+{{4}^{2}}}=\sqrt{17},

A{{A}_{3}}=\sqrt{{{3}^{2}}+{{4}^{2}}}=5,

A{{A}_{4}}=\sqrt{{{5}^{2}}+{{2}^{2}}}=\sqrt{29},

A{{A}_{5}}=\sqrt{{{1}^{2}}+{{4}^{2}}}=\sqrt{17},

A{{A}_{6}}=\sqrt{{{2}^{2}}+{{2}^{2}}}=2\sqrt{2},

A{{A}_{7}}=\sqrt{{{3}^{2}}+{{4}^{2}}}=5,

A{{A}_{8}}=\sqrt{{{3}^{2}}+{{4}^{2}}}=5,

Since 2\sqrt{2}<{}\sqrt{17}<{}3\sqrt{2}<{}5<{}\sqrt{29}.

When \sqrt{17}<{}r\leqslant 3\sqrt{2}, there are exactly 3 points \left( {{A}_{6}},{{A}_{2}},{{A}_{5}} \right) lie inside the circle.

Related Problems
Problem 18 AMC 8 Daily Practice Round 11
Problem 19 AMC 8 Daily Practice Round 11
Problem 21 AMC 8 Daily Practice Round 11
Problem 22 AMC 8 Daily Practice Round 11
Think Academy Powered by ChatGPT