Difference between revisions of "1987 OIM Problems/Problem 5"

Revision as of 18:04, 22 March 2025

If $r$ , $s$ , and $t$ are all the roots of the equation: $x(x-2)(3x-7)=2$

(a) Prove that $r$ , $s$ , and $t$ are all positive.

(b) Calculate $\arctan r + \arctan s + \arctan t$ .

Note: The range of $\arctan x$ falls between $0$ and $\pi$ , inclusive.

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

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 == Problem ==
 If <math>r</math>, <math>s</math>, and <math>t</math> are all the roots of the equation:
-<cmath>x(x-2)3x-7)=2</cmath>
+<cmath>x(x-2)(3x-7)=2</cmath>
-(a) Prove that <math>r</math>, <math>s</math>, and <math>t</math> are all postive
+(a) Prove that <math>r</math>, <math>s</math>, and <math>t</math> are all positive.
-(b) Calculate: arctan <math>r</math> + arctan <math>s</math> + arctan <math>t</math>.
+(b) Calculate <math>\arctan r + \arctan s + \arctan t</math>.
-Note: We define arctan <math>x</math>, as the arc between <math>0</math> and <math>\pi</math>  which tangent is <math>x</math>.
+Note: The range of <math>\arctan x</math> falls between <math>0</math> and <math>\pi</math>, inclusive.
 ~translated into English by Tomas Diaz. ~orders@tomasdiaz.com