2020 AIME I Problems/Problem 1

Note: Please do not post problems here until after the AIME.

Problem

In $\triangle ABC$ with $AB=BC,$ point $D$ lies strictly between $A$ and $C$ on side $\overline{AC},$ and point $E$ lies strictly between $A$ and $B$ on side $\overline{AB}$ such that $AE=ED=DB=BC.$ The degree measure of $\angle ABC$ is $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Solution

Solution 1 (Angle Chasing) If we set $\angle{BAC}$ to $x$ , we can find all other angles through these two properties: 1. Angles in a triangle sum to $180^{\circ}$ . 2. The base angles of an isoceles triangle are congruent.

Now we angle chase. $\angle{ADE}=\angle{EAD}=x$ , $\angle{AED} = 180-2x$ , $\angle{BED}=\angle{EBD}=2x$ , $\angle{EDB} = 180-4x$ , $\angle{BDC} = \angle{BCD} = 3x$ , $\angle{CBD} = 180-6x$ . Since $AB = AC$ as given by the problem, $\angle{ABC} = \angle{ACB}$ , so $180-4x=3x$ . Therefore, $x = 180/7^{\circ}$ , and our desired angle is $180-4(\frac{180}{7}) = \frac{540}{7}$ for an answer of $\boxed{547}$ .

2020 AIME I (Problems • Answer Key • Resources)
Preceded by First Problem	Followed by Problem 2
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

Page

Toolbox

Search

2020 AIME I Problems/Problem 1

Problem

Solution

See Also