Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1963 AHSME Problems/Problem 34"

(Solution to Problem 34)
 
m (→‎Solution)
 
Line 19: Line 19:
 
<cmath>\cos{x^\circ} < -\frac{1}{2}</cmath>
 
<cmath>\cos{x^\circ} < -\frac{1}{2}</cmath>
  
−
Note that <math>\cos{120^\circ} = -\frac{1}{2}</math>.  As <math>x</math> gets closer to <math>180^{\circ}</math>, <math>\cos{x}</math> decreases towards <math>-1</math>.  Thus, <math>x > 120</math>, so the answer is <math>\boxed{\textbf{(B)}}</math>
+
Note that <math>\cos{120^\circ} = -\frac{1}{2}</math>.  As <math>x</math> gets closer to <math>180^{\circ}</math>, <math>\cos{x}</math> decreases towards <math>-1</math>.  Thus, <math>x \leq 120</math>, and in order to maximize <math>x</math>, the answer has to be <math>\boxed{\textbf{(B)}}</math>
−
 
 
  
 
==See Also==
 
==See Also==

Latest revision as of 10:04, 15 July 2025

Problem

In $\triangle ABC$, side $a = \sqrt{3}$, side $b = \sqrt{3}$, and side $c > 3$. Let $x$ be the largest number such that the magnitude, in degrees, of the angle opposite side $c$ exceeds $x$. Then $x$ equals:

$\textbf{(A)}\ 150^{\circ} \qquad \textbf{(B)}\ 120^{\circ}\qquad \textbf{(C)}\ 105^{\circ} \qquad \textbf{(D)}\ 90^{\circ} \qquad \textbf{(E)}\ 60^{\circ}$

Solution

Using the Law of Cosines, \[\sqrt{3 + 3 - 2\cdot 3 \cdot \cos{x^\circ}}>3\]

Both sides are positive, so squaring both sides will not affect the inequality.

\[6 - 6 \cos{x^\circ}>9\] \[\cos{x^\circ} < -\frac{1}{2}\]

Note that $\cos{120^\circ} = -\frac{1}{2}$. As $x$ gets closer to $180^{\circ}$, $\cos{x}$ decreases towards $-1$. Thus, $x \leq 120$, and in order to maximize $x$, the answer has to be $\boxed{\textbf{(B)}}$

See Also

1963 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 33 		Followed by
Problem 35
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40
All AHSME Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. AMC Logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1963_AHSME_Problems/Problem_34&oldid=254332"