Difference between revisions of "2020 AIME I Problems/Problem 1"
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== Solution == | == Solution == | ||
| + | <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, W); | ||
| + | label("$B$", B, E); | ||
| + | label("$C$", C, W); | ||
| + | label("$D$", D, W); | ||
| + | label("$E$", F, E); | ||
| + | </asy> | ||
| + | |||
If we set <math>\angle{BAC}</math> to <math>x</math>, we can find all other angles through these two properties: | If we set <math>\angle{BAC}</math> to <math>x</math>, we can find all other angles through these two properties: | ||
1. Angles in a triangle sum to <math>180^{\circ}</math>. | 1. Angles in a triangle sum to <math>180^{\circ}</math>. | ||
Revision as of 23:03, 12 March 2020
Problem
In 
with 
point 
lies strictly between 
and 
on side 
and point 
lies strictly between and 
on side 
such that 
The degree measure of 
is 
where 
and 
are relatively prime positive integers. Find 
Solution
![[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, W); label("$B$", B, E); label("$C$", C, W); label("$D$", D, W); label("$E$", F, E); [/asy]](http://latex.artofproblemsolving.com/0/0/0/000780d13c03d6c9656937da4bf687892a343c6c.png)
If we set 
to 
, we can find all other angles through these two properties:
1. Angles in a triangle sum to 
.
2. The base angles of an isoceles triangle are congruent.
Now we angle chase. 
, 
, 
, 
, 
, 
. Since 
as given by the problem, 
, so 
. Therefore, 
, and our desired angle is ![\[180-4\left(\frac{180}{7}\right) = \frac{540}{7}\]](http://latex.artofproblemsolving.com/1/c/8/1c8994eb36574f1458d2ee3824b34658f345eabc.png)
for an answer of 
.
-molocyxu
See Also
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
