December 2021

Problem
A circle has its center at (3, -27). It is tangent to the line $3x + 4y - 26 = 0$. Locate the point of tangency.
 

Problem
A battalion, 20 miles long, advances 20 miles. During this time, a messenger on a horse travels from the rear of the battalion to the front and immediately turns around, ending up precisely at the rear of the battalion upon the completion of the 20-mile journey. How far has the messenger travelled?
 

Quiz 4 - Part 1

1. The product of two consecutive negative even integers is 24. Find the sum of the numbers.
2. The sum of the digits of a two-digit number is 11. When the digits are reversed, the new number exceeds the original number by 9. From the original number, calculate the product of tens digit to the square of the units digit.
3. Find two consecutive positive even integers such that the square of the second, decreased by twice the first is 124. Find the smaller number.
4. One number is 25 greater than a second number. If the lesser number is subtracted from 3 times the greater number, the difference is 195. Find the product of the two numbers.

Quiz 3 - Part 1

1. Calculate $\sqrt{2 + \sqrt{2 + \sqrt{2 + \ldots}}}$
2. Find the product of roots of $\dfrac{x^{2002} + 4x^{2001}}{4x^{2000}} = 8$.
3. Aedan opened his science book and noticed that the product of the two pages in front of him was equal to 1122. What were the sum of the numbers of those pages?
4. If $x + y = 5$ and $xy = 6$, find the value of $x^3 + y^3$.

Quiz 2 - Part 1

1. The sum of three numbers is 7 and the sum their reciprocals is 7/5. If these three numbers form into a GP, find their product.
2. Find the sum of $1 + 2\left( \dfrac{1}{3} \right) + 3\left( \dfrac{1}{3} \right)^2 + 4\left( \dfrac{1}{3} \right)^3 + \ldots + n\left( \dfrac{1}{3} \right)^{n − 1} + \ldots$
3. A sequence of numbers is defined by the relation $\dfrac{a_{n+1}}{a_n} = 3^n$ and it is known that $a_1 = 1$. Find the value of $\log_3 a_{100}$.

Quiz 1 - Part 1

1. Find $\ln A$ if $\log_5 A = 4$.
2. $\log_3 11 = x$. What is the value of $x$?
3. Given $\log_a 2 = x$, $\log_a 5 = y$, and $\log_a 6 = z$, express $\log_a 0.15$ in terms of $x$, $y$, and $z$.
4. Solve for $x$ from $7^{2x - 1} = 3^{x + 1}$
5. Find $x$ in the expression $\left( \dfrac{x}{5} \right)^{2/3} = 5$.