threepointonefourone

(x^4 + x^3 + x^2 + x + 1) + (x^4 - x^3 + x^2 - x + 1)

$2x^4 + 2x^2 + 1$

So far I get $a_1 = 5$, $a_2 = 10$, and $a_3 = 20$, getting $35$. Not sure.

$1.11111 + 0.11111 + 0.01111 + 0.00111 + 0.00011 + 0.00001 = 1.23456$

it comes from the MathCounts 2022 National Sprint. Question #23.

$\frac{1}{\sqrt{36}} - \sqrt{27} - \frac{1}{\sqrt{27}} - \sqrt{18} + \frac{1}{\sqrt{18}} - \sqrt{9}$

is equal to

$-\frac{17}{6} + \frac{1}{3\sqrt2} - 3\sqrt2 - \frac{1}{3\sqrt{3}} - 3\sqrt{3}$

rationalizing the denominators and simplifying gives
$-\frac{17}{6} + \frac{\sqrt2}{6} - 3\sqrt2 - \frac{\sqrt{3}}{9} - 3\sqrt{3}$

$\text{WolframAlpha also works.}$

$\text{In the end, I was able to find the expansion of}$

${(x + y)}^{31}$ correctly.

Unfortunately, it doesn't show any steps (or minimal)

Using brute force , which is something you should NOT do (even though this time, I'm just lazy):

$11250x^6 + 12750(x^5)(y) - 1800(x^4)(y^2) - 4740(xy)^3$$- 1662(x^2)(y^4)-234(x)(y^5) - 12y^6$

$\text{The coefficient of} (x^2)(y^4) \text{​ is -1,662.}$

Hope this helped .