Practice (Basic)
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How many ways can all six numbers in the set $\{4, 3, 2, 12, 1, 6\}$ be ordered so that $a$ comes before $b$ whenever $a$ is a divisor of $b$?
What is the units digit of the product $7^{23} \times 8^{105} \times 3^{18}$?
Place 9 points in a unit square. Prove it is possible to select 3 points from them to create a triangle whose area is no more than $\frac{1}{8}$.
When $(37 \times 45) - 15$ is simplified, what is the units digit?
What is the probability that a randomly selected integer from $1$ to $81$, inclusive, is equal to the product of two one-digit numbers?
How many diagonals does a convex octagon have?
What is the largest five-digit integer such that the product of the digits is $2520$?
In $\triangle{ABC}$, segments AB and AC have each been divided into four congruent segments. We must find the fraction of the triangle that is shaded.
CDs sell for $3$ different amounts. Three customers bought $3$ CDs each but none bought three of the same price. The first customer spent $\$4$, the second spent $\$9$ and the third spent $\$12$. We must find the price of the most expensive CD.
Carol, Jane, Kim, Nancy and Vicky competed in a 400-meter race. Nancy beat Jane by 6 seconds. Carol finished 11 seconds behind Vicky. Nancy finished 2 seconds ahead of Kim, but 3 seconds behind Vicky. We are asked to find by how many seconds Kim finished ahead of Carol.
In a stack of six cards, each card is labeled with a different integer from $0$ to $5$. Two cards are selected at random without replacement. So what is the probability that their sum will be $3$?
Okta stays in the sun for $16$ minutes before getting sunburned. Using a sunscreen, he can stay in the sun $20$ times as long before getting sunburned (or $320$ minutes). If he stays in the sun for $9$ minutes and then applies the sunscreen, how much longer can he remain in the sun?
How many positive integers not exceeding $2000$ have an odd number of factors?
Four boys and four girls line up in a random order. What is the probability that both the first and last person in line is a girl?
Meena writes the numbers $1$, $2$, $3$, and $4$ in some order on a blackboard, such that she cannot swap two numbers and obtain the sequence $1$, $2$, $3$, $4$. How many sequences could she have written?
For how many ordered pairs $(x, y)$ of integers satisfying $0 \le x$, $y \le 10$, and $(x + y)^2 + (xy - 1)^2$ is a prime number?
Let $S$ be the string $0101010101010$. Determine the number of substrings containing an odd number of $1$'s. (A substring is defined by a pair of (not necessarily distinct) characters of the string and represents the characters between, inclusively, the two elements of the string.)
There are $10$ monsters, each with 6 units of health. On turn $n$, you can attack one monster, reducing its health by $n$ units. If a monster's health drops to $0$ or below, the monster dies. What is the minimum number of turns necessary to kill all of the monsters?
There are $7$ boys each of which has at least $3$ brothers among the other $6$ boys. Are these $7$ boys necessarily all brothers? Explain.
Prove: any convex pentagon must have three vertices $A$, $B$, and $C$ satisfying $\angle{ABC} \le 36^\circ$.
Let unit vectors $a$, $b$, and $c$ satisfy $a+b+c=0$, prove the angles between these vectors are all $120^\circ$.
Let complex number $z_1=2-i\cos\theta$, $z_2=2-i\sin\theta$. Find the maximum value of $|z_1z_2|$.
Show that the difference of two squares of odd numbers must be a multiple of $8$.
Show that
$$\sin^2\alpha - \sin^2\beta = \sin(\alpha + \beta)\sin(\alpha-\beta)$$
$$\cos^2\alpha - \cos^2\beta = - \sin(\alpha + \beta)\sin(\alpha-\beta)$$
Simplify $$\sin^2\alpha + \sin^2\Big(\alpha + \frac{\pi}{3}\Big)+\sin^2\Big(\alpha - \frac{\pi}{3}\Big)$$