Analysis for Quiz 12: Brain Power Enrichment Quiz (I)
Question 1: If $a\gt b$ and $c\gt d$ are true, then we can say $a+c\gt b+d$ is always true.
A. Correct.
B. Incorrect.
Answer:
Yes, that is always correct, regardless if $a,\,b,\,c,\,d$ are negative or positive real number.
Take for example, we have:
Showing posts with label A. Show all posts
Showing posts with label A. Show all posts
Creative Solution for IMO Trigonometry Problem
Let $A,\,B$ be acute angles such that $\tan B=2015\sin A \cos A-2015\sin^2 A \tan B$.
Find the greatest possible value of $\tan B$.
This blog post is to highlight the fact that if we're creative enough, we can avoid the tedious calculus method to look for the maximal of $\tan B$.
You have to be aware of a few things as well:
1.
When $B$ is an acute angle and if $\sin B\le \dfrac{m}{n}$, then $\tan B\le \dfrac{m}{\sqrt{n^2-m^2}}$ must be true.
In other words, we obtain the maximal of $\tan B$ if we have obtained the maximal of $\sin B$.
Find the greatest possible value of $\tan B$.
This blog post is to highlight the fact that if we're creative enough, we can avoid the tedious calculus method to look for the maximal of $\tan B$.
You have to be aware of a few things as well:
1.
When $B$ is an acute angle and if $\sin B\le \dfrac{m}{n}$, then $\tan B\le \dfrac{m}{\sqrt{n^2-m^2}}$ must be true.
In other words, we obtain the maximal of $\tan B$ if we have obtained the maximal of $\sin B$.
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