Difference between revisions of "2020 AIME I Problems/Problem 1"

Revision as of 12:51, 14 December 2020

Problem

Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule!

Solution 1

$[asy] size(10cm); pair A, B, C, D, F; A = (0, tan(3 * pi / 7)); B = (1, 0); C = (-1, 0); F = rotate(90/7, A) * (A - (0, 2)); D = rotate(900/7, F) * A; draw(A -- B -- C -- cycle); draw(F -- D); draw(D -- B); label("$A$", A, N); label("$B$", B, E); label("$C$", C, W); label("$D$", D, W); label("$E$", F, E); [/asy]$

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 isosceles 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 $180-4\left(\frac{180}{7}\right) = \frac{540}{7}$ for an answer of $\boxed{547}$ .

See here for a video solution: https://youtu.be/4e8Hk04Ax_E

Solution 2

Let $\angle{BAC}$ be $x$ in degrees. $\angle{ADE}=x$ . By Exterior Angle Theorem on triangle $AED$ , $\angle{BED}=2x$ . By Exterior Angle Theorem on triangle $ADB$ , $\angle{BDC}=3x$ . This tells us $\angle{BCA}=\angle{ABC}=3x$ and $3x+3x+x=180$ . Thus $x=\frac{180}{7}$ and we want $\angle{ABC}=3x=\frac{540}{7}$ to get an answer of $\boxed{547}$ .

Solution 3 (Official MAA)

Let $x = \angle ABC = \angle ACB$ . Because $\triangle BCD$ is isosceles, $\angle CBD = 180^\circ - 2x$ . Then $\angle DBE = x - \angle CBD = x - (180^\circ - 2x) = 3x - 180^\circ\!.$ Because $\triangle EDA$ and $\triangle DBE$ are also isosceles, $\angle BAC =\frac12(\angle EAD + \angle ADE) = \frac12(\angle BED)= \frac12(\angle DBE)$ $= \frac12 (3x - 180^\circ) = \frac32x-90^\circ\!.$ Because $\triangle ABC$ is isosceles, $\angle BAC$ is also $180^\circ-2x$ , so $\frac32x - 90^\circ = 180^\circ - 2x$ , and it follows that $\angle ABC = x = \left(\frac{540}7\right)^\circ$ . The requested sum is $540+7 = 547$ .

$[asy] unitsize(4 cm); pair A, B, C, D, E; real a = 180/7; A = (0,0); B = dir(180 - a/2); C = dir(180 + a/2); D = extension(B, B + dir(270 + a), A, C); E = extension(D, D + dir(90 - 2*a), A, B); draw(A--B--C--cycle); draw(B--D--E); label("$A$", A, dir(0)); label("$B$", B, NW); label("$C$", C, SW); label("$D$", D, S); label("$E$", E, N); [/asy]$

https://artofproblemsolving.com/wiki/index.php/1961_AHSME_Problems/Problem_25 (Almost Mirrored)

See here for a video solution:

https://youtu.be/4XkA0DwuqYk

Video solution

https://youtu.be/IH7yM3L5xjA

https://youtu.be/mgRNqSDCvgM ~yofro

@@ Line 1: / Line 1: @@
 == Problem ==
-In <math>\triangle ABC</math> with <math>AB=AC,</math> point <math>D</math> lies strictly between <math>A</math> and <math>C</math> on side <math>\overline{AC},</math> and point <math>E</math> lies strictly between <math>A</math> and <math>B</math> on side <math>\overline{AB}</math> such that <math>AE=ED=DB=BC.</math> The degree measure of <math>\angle ABC</math> is <math>\tfrac{m}{n},</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n.</math>
+Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule!
 == Solution 1==

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