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2021 AIME II Problems/Problem 5

Revision as of 14:08, 23 March 2021 by MRENTHUSIASM (talk | contribs) (→‎Solution 2 (Casework: Detailed Explanation of Solution 1))

Problem

For positive real numbers $s$, let $\tau(s)$ denote the set of all obtuse triangles that have area $s$ and two sides with lengths $4$ and $10$. The set of all $s$ for which $\tau(s)$ is nonempty, but all triangles in $\tau(s)$ are congruent, is an interval $[a,b)$. Find $a^2+b^2$.

Solution 1

We start by defining a triangle. The two small sides MUST add to a larger sum than the long side. We are given 4 and 10 as the sides, so we know that the 3rd side is between 6 and 14, exclusive. We also have to consider the word OBTUSE triangles. That means that the two small sides squared is less than the 3rd side. So the triangles sides are between 6 and $\sqrt{84}$ exclusive, and the larger bound is between $\sqrt{116}$ and 14, exclusive. The area of these triangles are from 0 (straight line) to $2\sqrt{84}$ on the first "small bound" and the larger bound is between 0 and 20. $0 < s < 2\sqrt{84}$ is our first equation, and $0 < s < 20$ is our 2nd equation. Therefore, the area is between $\sqrt{336}$ and $\sqrt{400}$, so our final answer is $\boxed{736}$.

~ARCTICTURN

Solution 2 (Casework: Detailed Explanation of Solution 1)

Every obtuse triangle must satisfy both of the following:

  • Triangle Inequality Theorem: If $a,b,$ and $c$ are the side-lengths of a triangle with $a\leq b\leq c,$ then $a+b>c.$
  • Pythagorean Inequality Theorem: If $a,b,$ and $c$ are the side-lengths of an obtuse triangle with $a\leq b<c,$ then $a^2+b^2<c^2.$

For one such obtuse triangle, let $4,10,$ and $x$ be the side-lengths and $k$ be the area. We will use casework on the longest side:

Case (1): The longest side has length $\boldsymbol{10.}$

By the Triangle Inequality Theorem, ...

By the Pythagorean Inequality Theorem, ...

Case (2): The longest side has length \boldsymbol{x.}

By the Triangle Inequality Theorem, ...

By the Pythagorean Inequality Theorem, ...

Solution in progress. No edit please

~MRENTHUSIASM

See also

2021 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 4 		Followed by
Problem 6
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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