Second Solution: IMO Solving Equation Problem: Solve the equation $x+a^3=\sqrt[3]{a-x}$ where a is real.

Solve the equation $x+a^3=\sqrt[3]{a-x}$ where a is real parameter.

My solution:

By observation, note that $x=a-a^3$ is a real solution for the equation $x+a^3=\sqrt[3]{a-x}$.

IMO Solving Equation Problem: Solve the equation $x+a^3=\sqrt[3]{a-x}$ where a is real (First Solution)

Solve the equation $x+a^3=\sqrt[3]{a-x}$ where $a$ is real.

If one wants to do things in haste and quickly solve the above equation for $x$ without thinking much, one would definitely raise both sides of the equation to the third power to get rid of the cube root:

$x+a^3=\sqrt[3]{a-x}$

$(x+a^3)^3=(\sqrt[3]{a-x})^3$

Optimization Contest Problem: Prove $x^4+x^3-x^2-x+1>0$ for all real $x$.

In one of my previous blog posts(optimization-contest-problem), we want to prove that [MATH]\color{yellow}\bbox[5px,blue]{x^4+x^3-x^2-x+1}[/MATH] is always greater than zero for all real $x$, or more specifically, for $x\gt 1$.

Classic Trigonometric Olympiad Problem: Evaluate $(1+\tan 1^{\circ})(1+\tan 2^{\circ})\cdots(1+\tan 43^{\circ})(1+\tan 44^{\circ})(1+\tan 45^{\circ})$

In this blog post, we will continue to manipulate the number one to looking for the most efficient and effective solution.

According to Wikipedia (Number One):

One, sometimes referred to as unity, is the integer before two and after zero. One is the first non-zero number in the natural numbers as well as the first odd number in the natural numbers.