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2018 MPFG Problem 10: Difference between revisions

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Created page with "== Let <imath>T_1</imath> be an isosceles triangle with sides of length <imath>8</imath>, <imath>11</imath>, and <imath>11</imath>. Let <imath>T_2</imath> be an isosceles triangle with sides of length <imath>b</imath>, <imath>1</imath>, and <imath>1</imath>. Suppose that the radius of the incircle of <imath>T_1</imath> divided by the radius of the circumcircle of <imath>T_1</imath> is equal to the radius of the incircle of <imath>T_2</imath> divided by the radius of the..."
 
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== Let <imath>T_1</imath> be an isosceles triangle with sides of length <imath>8</imath>, <imath>11</imath>, and <imath>11</imath>. Let <imath>T_2</imath> be an isosceles triangle with sides of length <imath>b</imath>, <imath>1</imath>, and <imath>1</imath>. Suppose that the radius of the incircle of <imath>T_1</imath> divided by the radius of the circumcircle of <imath>T_1</imath> is equal to the radius of the incircle of <imath>T_2</imath> divided by the radius of the circumcircle of <imath>T_2</imath>. Determine the largest possible value of <imath>b</imath>. Express your answer as a fraction in simplest form.
==Problem ==
Let <imath>T_1</imath> be an isosceles triangle with sides of length <imath>8</imath>, <imath>11</imath>, and <imath>11</imath>. Let <imath>T_2</imath> be an isosceles triangle with sides of length <imath>b</imath>, <imath>1</imath>, and <imath>1</imath>. Suppose that the radius of the incircle of <imath>T_1</imath> divided by the radius of the circumcircle of <imath>T_1</imath> is equal to the radius of the incircle of <imath>T_2</imath> divided by the radius of the circumcircle of <imath>T_2</imath>. Determine the largest possible value of <imath>b</imath>. Express your answer as a fraction in simplest form.

Revision as of 05:37, 8 November 2025

Problem

Let $T_1$ be an isosceles triangle with sides of length $8$, $11$, and $11$. Let $T_2$ be an isosceles triangle with sides of length $b$, $1$, and $1$. Suppose that the radius of the incircle of $T_1$ divided by the radius of the circumcircle of $T_1$ is equal to the radius of the incircle of $T_2$ divided by the radius of the circumcircle of $T_2$. Determine the largest possible value of $b$. Express your answer as a fraction in simplest form.

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