2015 IMO contest problem: If$x,\,y,\,z$ are real numbers such that $x+2y+3z=6$ and $x^2+4y^2+9z^2=12$, evaluate $xyz$.

If$x,\,y,\,z$ are real numbers such that $x+2y+3z=6$ and $x^2+4y^2+9z^2=12$, evaluate $xyz$.

There is something that this Olympiad problem might trick us because it's obvious that $x^2,\,4y^2,\,9z^2$ are squares of $x,\,2y,\,3z$ and hence, one has reason to believe that the proper first step in solving this problem is to square the first given equation:

$x+2y+3z=6$

Math Olympiad Problem: Solve for real solutions

Solve for real solutions for the equation $(2x+1)(3x+1)(5x+1)(30x+1)=10$.

Okay, I heard you, why on earth this problem is supposed to be a delicious question that can promote higher mathematics thinking skills?

One of the most common issues math educators are struggling with is the students who underestimate the so-called trivial math problem and they think by the long and typical tedious solving method, the trivial math problem could be safely and successfully solved without a hitch.

Olympiad Trigonometric Problem $\cos^{48}\dfrac{P}{2}\sin^{23}\dfrac{Q}{2}=\cos^{48}\dfrac{Q}{2}\sin^{23}\dfrac{P}{2}$

Great and professional teaching methodology could definitely help to mold one's thinking and stimulate the inquisitive mind of students. In order to achieve this noble aim, we need good resources, such as the state of the art technologies, professional educators, competitive peers, supportive family members and above all, the challenging and intriguing math problems!

I will present one really intriguing problem here, and let's see how we are going to milk it for all its worth...

In a triangle $PQR$ the following equality holds:

$\cos^{48}\dfrac{P}{2}\sin^{23}\dfrac{Q}{2}=\cos^{48}\dfrac{Q}{2}\sin^{23}\dfrac{P}{2}$

Find the value of $\dfrac{PR}{QR}$.