Problem Statement (a) Expand and simplify fully the expression $(a − 1)(6a^2 − a − 1)$. (b) Determine all values of $\theta$ with $6 \cos^3\theta −7\cos^2\theta + 1 = 0$ and $−180^{\circ} < ...

Problem Statement (a) First, determine two positive integers $x$ and $y$ with $\dfrac{2x + 11y}{3x + 4y} = 1$. Now, let $u$ and $v$ be two positive rational numbers with $u<v$. If we write $...

Problem Statement Let $x$ and $y$ be two-digit integers such that $y$ is obtained by reversing the digits of $x$. The integers $x$ and $y$ satisfy $x^2 - y^2 = m^2$ for some positive integer $m$...

Problem Statement The increasing geometric sequence $x_{0},x_{1},x_{2},\ldots$ consists entirely of integral powers of $3.$ Given that $\sum_{n=0}^{7}\log_{3}(x_{n}) = 308$ and $56 \leq \log_{3...

Problem Statement An infinite geometric series has sum 2005. A new series, obtained by squaring each term of the original series, has 10 times the sum of the original series. The common ratio of...

Problem Statement For the infinite series $1-\frac12-\frac14+\frac18-\frac{1}{16}-\frac{1}{32}+\frac{1}{64}-\frac{1}{128}-\cdots$ let $S$ be the (limiting) sum. Then $S$ equals: Problem Link ...

Problem Statement The limiting sum of the infinite series, $\frac{1}{10} + \frac{2}{10^2} + \frac{3}{10^3} + \dots$ whose $n$th term is $\frac{n}{10^n}$ is: Problem Link Solution $$\text{L...

Problem Statement Find $(\log_2 x)^2$ if $\log_2 (\log_8 x) = \log_8 (\log_2 x)$. Problem Link Solution $$\log_2 (\log_8 x) = \log_8 (\log_2 x)$$ $$\log_2 (\dfrac{\log_2(x)}{\log_2(8)}) =...

Problem Statement Find $3x^2 y^2$ if $x$ and $y$ are integers such that $y^2 + 3x^2 y^2 = 30x^2 + 517$. Problem Link Solution $$y^2 + 3x^2y^2 - 30x^2 = 517$$ $$y^2 + 3x^2(y^2 - 10) = 517$...

Problem Statement If $\tan x + \tan y=25$ and $\cot x + \cot y=30$, what is $\tan(x+y)$? Problem Link Solution $$\tan x + \tan y=25$$ $$\cot x + \cot y=30$$ $$\dfrac{1}{\tan x} + \dfrac...