Number Theory
How many even numbers are there between 1 and 50 exclusive?
Answer: 24
If cos(40) = a, what is sin(50) in terms of a?
The probability that Ali wins a round of Jeopardy is 0.4. If he plays 90 rounds, how many does he win?
Answer: 36 rounds
\[ \frac{d}{dx} \left( e^{2x} + \sin(2x) \right) \]
Answer: \( 2e^{2x} + 2\cos(2x) \)
Factorize completely:
\[ 3x^3 - 12x^2 + 9x \]
Answer: \( 3x(x-3)(x-1) \)
What is the smallest positive integer \( n \) such that \( n^2 \) is divisible by 12?
Answer: \( 6 \)
Solve the equation \( 3\cos(2\theta) = 3\cos\theta + 2 \) for \( 0 \le \theta \le 360^\circ \).
Answer: \( \theta = 134.1^\circ,\; 225.9^\circ \)
How many diagonals are there in a 12-sided polygon?
54
Evaluate the following integral:
\[ \int \sin^2 x \, dx \]
Answer: \( \frac{x}{2} - \frac{\sin(2x)}{4} + C \)
How many points of intersection are there between: \[ y = x^2 - 6x + 8 \quad \text{and} \quad y = |x - 3| \, ? \]
Answer: \( 2 \)
Bell A rings every 14 minutes and Bell B rings every 12 minutes. If both bells ring together at 10:10am, at what time will they ring together again?
Answer: 11:34am
In the equation below, the angle measures are in degrees. If \( 0 < m < 90 \), what is the value of \( m \)?
\[ \cos(32^\circ) = \sin(5m - 12) \]
Answer: \( m = 14 \)
Three distinct numbers are chosen at random between 1 and 10 inclusive. What is the probability that their sum is 15?
Answer: \(\frac{1}{12} \)
Find
\[ \int \frac{1}{4 - x^2} \, dx \]
Answer: \( \frac{1}{4} \ln\left| \frac{2+x}{2-x} \right| + C \quad \text{or} \quad \frac{1}{2} \tanh^{-1}\left(\frac{x}{2}\right) + C \)
Let the roots of the cubic polynomial \[ x^3 - 6x^2 + 11x - 6 = 0 \] be \( \alpha, \beta, \gamma \). Find: \[ \alpha^3 + \beta^3 + \gamma^3 \]
Answer: \( 36 \)
Find the smallest integer n > 1 such that n and 35 are not relatively prime.
Answer: 5
The line \( x - y + 2 = 0 \) intersects the curve \( 2x^2 - y^2 + 2x + 1 = 0 \) at the points \( A \) and \( B \). The perpendicular bisector of the line \( AB \) intersects the curve at the points \( C \) and \( D \). Find the length of the line \( CD \) in the form \( a\sqrt{5} \), where \( a \) is an integer.
Answer:\( 8\sqrt{5} \)
There are 12 lockers in a row. 4 students choose lockers such that no two chosen lockers are adjacent. In how many ways can this be done?
Answer: 126
Evaluate the following limit:
\[ \lim_{x \to 0} \frac{\sin(3x)}{x} \]
Answer: \( 3 \)
Suppose \( a, b, c, d > 0 \) satisfy \[ (a + c)(b + d) = ac + bd. \] Find the minimum value of \[ \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a}. \]
Answer: \( 4 \)
Find the remainder when \( 3^{2024} + 5^{2024} \) is divided by 6.
Answer: \( 4 \)
Find, in degrees to the nearest tenth of a degree, the values of \( x \) for which \( \sin x \tan x = 4 \), where \( 0 \le x < 360^\circ \).
\[ \sin x \tan x = 4 \]
Answer: \( x = 76.4^\circ,\; 283.6^\circ \)
Find the number of ways to place a digit in each cell of a 2 × 3 grid such that the sum of the two numbers formed by reading left to right is 999, and the sum of the three numbers formed by reading top to bottom is 99.
Answer: 45
Evaluate the following limit:
\[ \lim_{x \to 0} \frac{|x|}{x} \]
Answer: DNE (Does Not Exist)
If \( x + \frac{1}{x} = 3 \), find:
\[ x^5 + \frac{1}{x^5} \]
Answer: \( 123 \)