Practice (90/1000)
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A $45^\circ$ arc of circle A is equal in length to a $30^\circ$ arc of circle B. What is the ratio of circle A's area and circle B's area?
Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let $B$ be the total area of the blue triangles, $W$ the total area of the white squares, and $R$ the area of the red square. Which of the following is correct?
There are 3 numbers A, B, and C, such that $1001C - 2002A = 4004$, and $1001B + 3003A = 5005$. What is the average of A, B, and C?
What is the sum of all of the roots of $(2x + 3) (x - 4) + (2x + 3) (x - 6) = 0$?
Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB each, 12 of the files take up 0.7 MB each, and the rest take up 0.4 MB each. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files?
Given a triangle with side lengths 15, 20, and 25, find the triangle's smallest height.
Using the digits 1, 2, 3, 4, 5, 6, 7, and 9, form 4 two-digit prime numbers, using each digit only once. What is the sum of the 4 prime numbers?
Let $a + 1 = b + 2 = c + 3 = d + 4 = a + b + c + d + 5$. What is $a + b + c + d$?
A $3\times 3\times 3$ cube is made of 27 normal dice. Each die's opposite sides sum to 7. What is the smallest possible sum of all of the values visible on the 6 faces of the large cube?
Spot's doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-yard rope. What is the area, in square yards, of the region outside of the doghouse that Spot can reach?
Points $A,B,C,D,E$ and $F$ lie, in that order, on $\overline{AF}$, dividing it into five segments, each of length 1. Point $G$ is not on line $AF$. Point $H$ lies on $\overline{GD}$, and point $J$ lies on $\overline{GF}$. The line segments $\overline{HC}, \overline{JE},$ and $\overline{AG}$ are parallel. Find $HC/JE$.
A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one?
Points $A,B,C$ and $D$ lie on a line, in that order, with $AB = CD$ and $BC = 12$. Point $E$ is not on the line, and $BE = CE = 10$. The perimeter of $\triangle AED$ is twice the perimeter of $\triangle BEC$. Find $AB$.
Tina randomly selects two distinct numbers from the set {1, 2, 3, 4, 5}, and Sergio randomly selects a number from the set {1, 2, ..., 10}. What is the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina?
In trapezoid $ABCD$ with bases $AB$ and $CD$, we have $AB = 52$, $BC = 12$, $CD = 39$, and $DA = 5$ (diagram not to scale). The area of $ABCD$ is
Compute $(1+\tan 1^\circ)(1+\tan 2^\circ)\cdots(1+\tan 44^\circ)(1+\tan 45^\circ)$
Compute $$\cos\frac{2\pi}{7} \cdot \cos \frac{4\pi}{7}\cdot \cos \frac{8\pi}{7} $$
Compute $$\cos\frac{\pi}{2n+1}\cdot\cos\frac{2\pi}{2n+1}\cdots\cos\frac{n\pi}{2n+1}$$
Compute $$\Big(1+\cos\frac{\pi}{5}\Big)\Big(1+\cos\frac{3\pi}{5}\Big)$$
Compute $$\sin^2 10^\circ + \cos^2 40^\circ + \sin 10^\circ \cos 40^\circ$$
Compute $$\sin^2 80^\circ -\sin^2 40^\circ +\sqrt{3}\sin 40^\circ \cos 80^\circ $$
Compute $$\sin^2 20^\circ -\sin 5^\circ (\sin 5^\circ +\frac{\sqrt{6}-\sqrt{2}}{2}\cos 20^\circ)$$
Let $\alpha, \beta \in (0, \frac{\pi}{2})$. Show that $\alpha + \beta = \frac{\pi}{2}$ if and only if $$\frac{\sin^4 \alpha}{\cos^2 \beta} + \frac{\cos^4\alpha}{\sin^2\beta} = 1$$
If $\sin\alpha + \sin\beta = \frac{3}{5}$ and $\cos\alpha+\cos\beta=\frac{4}{5}$, compute $\cos(\alpha -\beta)$ and $\sin(\alpha+\beta)$.
For each positive integer $n$, let $s(n)$ denote the number of ordered positive integer pair $(x, y)$ for which $\frac{1}{x} + \frac{1}{y} = \frac{1}{n}$ holds. Find all positive integers $n$ for which $s(n) = 5$.