problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
The graph of the rational function $\frac{2x^6+3x^5 - x^2 - 1}{q(x)}$ has a horizontal asymptote. What is the smallest possible degree of $q(x)$? | 6 |
John drove continuously from 8:15 a.m. until 2:45 p.m. of the same day and covered a distance of 210 miles. What was his average speed in miles per hour? | 32.31 |
$P$ is a point inside triangle $A B C$, and lines $A P, B P, C P$ intersect the opposite sides $B C, C A, A B$ in points $D, E, F$, respectively. It is given that $\angle A P B=90^{\circ}$, and that $A C=B C$ and $A B=B D$. We also know that $B F=1$, and that $B C=999$. Find $A F$. | 499 / 500 |
Mr. Lee V. Soon starts his morning commute at 7:00 AM to arrive at work by 8:00 AM. If he drives at an average speed of 30 miles per hour, he is late by 5 minutes, and if he drives at an average speed of 70 miles per hour, he is early by 4 minutes. Find the speed he needs to maintain to arrive exactly at 8:00 AM. | 32.5 |
The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$ ? | 16 |
I have 6 shirts, 4 pairs of pants, and 6 hats. The pants come in tan, black, blue, and gray. The shirts and hats come in those colors, and also white and yellow. I refuse to wear an outfit in which all 3 items are the same color. How many choices for outfits, consisting of one shirt, one hat, and one pair of pants, do ... | 140 |
Uncle Ben has 440 chickens on his farm. 39 are roosters and the rest are hens. 15 of his hens do not lay eggs, and the rest do. If each egg-laying hen lays 3 eggs, how many eggs will Uncle Ben have? | Uncle Ben has 440 chickens - 39 roosters = <<440-39=401>>401 hens.
Uncle Ben has 401 hens - 15 that don't lay eggs = <<401-15=386>>386 hens that lay eggs.
The egg-laying hens each lay 3 eggs * 386 hens = <<3*386=1158>>1158 eggs.
#### 1158 |
Jared wants to watch a series with four episodes. The first three episodes are 58 minutes, 62 minutes, and 65 minutes long respectively. If the four episodes last 4 hours, how long is the fourth episode? | The first three episodes are 58 + 62 + 65 = <<58+62+65=185>>185 minutes long.
In minutes, the four episodes last 4 x 60 = <<4*60=240>>240 minutes.
Then, the fourth episode is 240 - 185 = <<240-185=55>>55 minutes long.
#### 55 |
Let $x$ and $y$ be positive real numbers such that
\[\frac{1}{x + 2} + \frac{1}{y + 2} = \frac{1}{3}.\]Find the minimum value of $x + 2y.$ | 3 + 6 \sqrt{2} |
For every four points $P_{1},P_{2},P_{3},P_{4}$ on the plane, find the minimum value of $\frac{\sum_{1\le\ i<j\le\ 4}P_{i}P_{j}}{\min_{1\le\ i<j\le\ 4}(P_{i}P_{j})}$ . | 4 + 2\sqrt{2} |
Let $z_{1}, z_{2}, z_{3}, z_{4}$ be the solutions to the equation $x^{4}+3 x^{3}+3 x^{2}+3 x+1=0$. Then $\left|z_{1}\right|+\left|z_{2}\right|+\left|z_{3}\right|+\left|z_{4}\right|$ can be written as $\frac{a+b \sqrt{c}}{d}$, where $c$ is a square-free positive integer, and $a, b, d$ are positive integers with $\operat... | 7152 |
George is about to celebrate his 25th birthday. Since his 15th birthday, he's been given a special $1 bill from his parents. They told him that on his 25th birthday, for every bill he still has, they will give him $1.5 in exchange. He spent 20% of his special bills. How much will he receive from his parents when he exc... | He received 10 special bills because 25 - 15 = <<25-15=10>>10
He has 80% of them left because 100 - 20 = <<100-20=80>>80
He has 8 left because 10 x .8 = <<10*.8=8>>8
His parents will give her $12 for them because 8 x 1.5 = <<8*1.5=12>>12
#### 12 |
Define $\displaystyle{f(x) = x + \sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x + \ldots}}}}}$ . Find the smallest integer $x$ such that $f(x)\ge50\sqrt{x}$ .
(Edit: The official question asked for the "smallest integer"; the intended question was the "smallest positive integer".) | 2500 |
Let \(\mathbf{v}\) be a vector such that
\[
\left\| \mathbf{v} + \begin{pmatrix} 4 \\ 2 \end{pmatrix} \right\| = 10.
\]
Find the smallest possible value of \(\|\mathbf{v}\|\). | 10 - 2\sqrt{5} |
How many distinct solutions are there to the equation $|x-7| = |x+1|$? | 1 |
Given $\cos \left( \frac {π}{4}-α \right) = \frac {3}{5}$, and $\sin \left( \frac {5π}{4}+β \right) = - \frac {12}{13}$, with $α \in \left( \frac {π}{4}, \frac {3π}{4} \right)$ and $β \in (0, \frac {π}{4})$, find the value of $\sin (α+β)$. | \frac {56}{65} |
An aluminum cube with an edge length of \( l = 10 \) cm is heated to a temperature of \( t_{1} = 100^{\circ} \mathrm{C} \). After this, it is placed on ice, which has a temperature of \( t_{2} = 0^{\circ} \mathrm{C} \). Determine the maximum depth to which the cube can sink. The specific heat capacity of aluminum is \... | 0.0818 |
In equilateral triangle $ABC$ with side length 2, let the parabola with focus $A$ and directrix $BC$ intersect sides $AB$ and $AC$ at $A_{1}$ and $A_{2}$, respectively. Similarly, let the parabola with focus $B$ and directrix $CA$ intersect sides $BC$ and $BA$ at $B_{1}$ and $B_{2}$, respectively. Finally, let the para... | 66-36\sqrt{3} |
When Jeffrey walks, for every three steps forward, he takes two steps backwards. Therefore, if the distance between the house and the mailbox is 66 steps, what is the total number of steps Jeffrey takes when he goes from the house to the mailbox? | For every 3 steps forward, he adds 2 more backwards so each 'step' gained costs 5 actual steps
66 steps distance * 5 steps moved = <<66*5=330>>330 steps total
#### 330 |
Select 3 people from 5, including A and B, to form a line, and determine the number of arrangements where A is not at the head. | 48 |
Inez has $150. She spends one-half on hockey skates and a certain amount on hockey pads. If Inez has $25 remaining, how much did the hockey pads cost, together, in dollars? | She spent 150/2=$<<150/2=75>>75 on hockey skates.
Let X be the amount Inez spent on hockey pads.
150-75-X=25
75-X=25
X=<<50=50>>50
The hockey pads cost $<<50=50>>50.
#### 50 |
If a triangle has two sides of lengths 5 and 7 units, then how many different integer lengths can the third side be? | 9 |
Right triangle $PQR$ has one leg of length 9 cm, one leg of length 12 cm and a right angle at $P$. A square has one side on the hypotenuse of triangle $PQR$ and a vertex on each of the two legs of triangle $PQR$. What is the length of one side of the square, in cm? Express your answer as a common fraction. | \frac{45}{8} |
Convert the point $(-2,-2)$ in rectangular coordinates to polar coordinates. Enter your answer in the form $(r,\theta),$ where $r > 0$ and $0 \le \theta < 2 \pi.$ | \left( 2 \sqrt{2}, \frac{5 \pi}{4} \right) |
Apples used to cost $1.6 per pound. The price got raised 25%. How much does it cost to buy 2 pounds of apples for each person in a 4 member family? | The price increase raised the price by 1.6*.25=$<<1.6*.25=.4>>.4
So each pound of apples cost 1.6+.4=$2
2*4=<<2*4=8>>8 pounds of apples are bought
So the apples cost 8*2=$<<8*2=16>>16
#### 16 |
Given that $\sin\alpha = \frac{1}{2} + \cos\alpha$, and $\alpha \in (0, \frac{\pi}{2})$, find the value of $\frac{\cos 2\alpha}{\sin(\alpha - \frac{\pi}{4})}$. | -\frac{\sqrt{14}}{2} |
There is a four-digit number \( A \). By rearranging the digits (none of which are 0), the largest possible number is 7668 greater than \( A \), and the smallest possible number is 594 less than \( A \). What is \( A \)? | 1963 |
Let $N$ be the number of functions $f$ from $\{1,2, \ldots, 101\} \rightarrow\{1,2, \ldots, 101\}$ such that $f^{101}(1)=2$. Find the remainder when $N$ is divided by 103. | 43 |
Let $m$ be the smallest positive, three-digit integer congruent to 5 (mod 11). Let $n$ be the smallest positive, four-digit integer congruent to 5 (mod 11). What is $n-m$? | 902 |
A frustum of a right circular cone is formed by cutting a small cone off of the top of a larger cone. If this frustum has a lower base radius of 8 inches, an upper base radius of 5 inches, and a height of 6 inches, what is its lateral surface area? Additionally, there is a cylindrical section of height 2 inches and rad... | 39\pi\sqrt{5} + 20\pi |
Suppose
$$a(2+i)^4 + b(2+i)^3 + c(2+i)^2 + b(2+i) + a = 0,$$where $a,b,c$ are integers whose greatest common divisor is $1$. Determine $|c|$. | 42 |
Find the product of all constants $t$ such that the quadratic $x^2 + tx + 6$ can be factored in the form $(x+a)(x+b)$, where $a$ and $b$ are integers. | 1225 |
Let $n = 2^4 \cdot 3^5 \cdot 4^6\cdot 6^7$. How many natural-number factors does $n$ have? | 312 |
For how many integer values of $n$ between 1 and 180 inclusive does the decimal representation of $\frac{n}{180}$ terminate? | 20 |
Consider a circle centered at $O$ . Parallel chords $AB$ of length $8$ and $CD$ of length $10$ are of distance $2$ apart such that $AC < AD$ . We can write $\tan \angle BOD =\frac{a}{b}$ , where $a, b$ are positive integers such that gcd $(a, b) = 1$ . Compute $a + b$ . | 113 |
Compute the unique positive integer $n$ such that
\[2 \cdot 2^2 + 3 \cdot 2^3 + 4 \cdot 2^4 + \dots + n \cdot 2^n = 2^{n + 10}.\] | 513 |
Let $A := \mathbb{Q} \setminus \{0,1\}$ denote the set of all rationals other than 0 and 1. A function $f : A \rightarrow \mathbb{R}$ has the property that for all $x \in A$,
\[
f\left( x\right) + f\left( 1 - \frac{1}{x}\right) = \log\lvert x\rvert.
\]Compute the value of $f(2007)$. Enter your answer in the form "$\lo... | \log\left(\frac{2007}{2006}\right) |
Elizabetta wants to write the integers 1 to 9 in the regions of the shape shown, with one integer in each region. She wants the product of the integers in any two regions that have a common edge to be not more than 15. In how many ways can she do this? | 16 |
For each nonnegative integer $n$ we define $A_n = 2^{3n}+3^{6n+2}+5^{6n+2}$. Find the greatest common divisor of the numbers $A_0,A_1,\ldots, A_{1999}$.
[i]Romania[/i] | 7 |
Find the distance between the points $(2,1,-4)$ and $(5,8,-3).$ | \sqrt{59} |
What is $88 \div 4 \div 2$? | 11 |
If \( 9210 - 9124 = 210 - \square \), the value represented by the \( \square \) is: | 124 |
Given a fixed integer \( n \) where \( n \geq 2 \):
a) Determine the smallest constant \( c \) such that the inequality \(\sum_{1 \leq i < j \leq n} x_i x_j (x_i^2 + x_j^2) \leq c \left( \sum_{i=1}^n x_i \right)^4\) holds for all nonnegative real numbers \( x_1, x_2, \ldots, x_n \geq 0 \).
b) For this constant \( c \... | \frac{1}{8} |
In a right triangle with integer length sides, the hypotenuse has length 39 units. How many units is the length of the shorter leg? | 15 |
What is the maximum number of rooks that can be placed in an $8 \times 8 \times 8$ cube so that no two rooks attack each other? | 64 |
Find all positive integers $A$ which can be represented in the form: \[ A = \left ( m - \dfrac 1n \right) \left( n - \dfrac 1p \right) \left( p - \dfrac 1m \right) \]
where $m\geq n\geq p \geq 1$ are integer numbers.
*Ioan Bogdan* | 21 |
Only nine out of the original thirteen colonies had to ratify the U.S. Constitution in order for it to take effect. What is this ratio, nine to thirteen, rounded to the nearest tenth? | 0.7 |
Throw a fair die, and let event $A$ be that the number facing up is even, and event $B$ be that the number facing up is greater than $2$ and less than or equal to $5$. Then, the probability of the complement of event $B$ is ____, and the probability of event $A \cup B$ is $P(A \cup B) = $ ____. | \dfrac{5}{6} |
The numbers $1, 2, 3, 4, 5, 6, 7,$ and $8$ are randomly written on the faces of a regular octahedron so that each face contains a different number. The probability that no two consecutive numbers, where $8$ and $1$ are considered to be consecutive, are written on faces that share an edge is $m/n,$ where $m$ and $n$ are... | 85 |
Given that \(\theta\) is an angle in the third quadrant, and \(\sin^{4}\theta + \cos^{4}\theta = \frac{5}{9}\), determine the value of \(\sin 2\theta\). | -\frac{2\sqrt{2}}{3} |
Compute
$$\sum_{k=1}^{2000} k(\lceil \log_{2}{k}\rceil- \lfloor\log_{2}{k} \rfloor).$$ | 1998953 |
Find the remainder when $2 \times 12 \times 22 \times 32 \times \ldots \times 72 \times 82 \times 92$ is divided by $5$. | 4 |
How many positive $3$-digit numbers are multiples of $20$, but not of $55$? | 41 |
Samson derives utility according the relation $$\text{Utility} = \text{hours of math done} \times \text{hours of frisbee played}.$$On Monday he plays $t$ hours of frisbee and spends $8 - t$ hours doing math. On Tuesday, he ends up getting the same amount of utility as Monday while spending $2-t$ hours playing frisbee... | \frac{2}{3} |
Let $n$ be a positive integer. Find all $n \times n$ real matrices $A$ with only real eigenvalues satisfying $$A+A^{k}=A^{T}$$ for some integer $k \geq n$. | A = 0 |
A pea patch is twice as big as a radish patch. If one sixth of the pea patch is 5 square feet. How much is a whole radish patch in square feet? | A whole pea patch is 5*6 = <<5*6=30>>30 square feet.
A radish patch is 30/2 = <<30/2=15>>15 square feet.
#### 15 |
What is the smallest $r$ such that three disks of radius $r$ can completely cover up a unit disk? | \frac{\sqrt{3}}{2} |
A tripod has three legs each of length $5$ feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is $4$ feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let $h$ be the height in feet of the top o... | 183 |
What is the radius of the smallest sphere in which 4 spheres of radius 1 will fit? | \frac{2+\sqrt{6}}{2} |
A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is $2:1$. The ratio of the rectangle's length to its width is $2:1$. What percent of the rectangle's area is inside the square? | 12.5 |
Given an arithmetic sequence {a_n} with the sum of its first n terms denoted as S_n, and given that a_1008 > 0 and a_1007 + a_1008 < 0, find the positive integer value(s) of n that satisfy S_nS_{n+1} < 0. | 2014 |
Steve bought $25 worth of groceries. He bought a gallon of milk for $3, two boxes of cereal for $3.5 each, 4 bananas for $.25 each, four apples that cost $.5 each and a number of boxes of cookies. The cookies cost twice as much per box as the gallon of milk. How many boxes of cookies did he get? | He spent $7 on cereal because 2 x 3.5 = <<2*3.5=7>>7
He spent $1 on bananas because 4 x .25 = <<4*.25=1>>1
He spent $2 on apples because 4 x .5 = <<4*.5=2>>2
He spent 13 on everything but the cookies because 3 + 7 + 1 + 2 = 13
He spent $12 on cookies because 25 - 13 = <<25-13=12>>12
Each box of cookies is $6 because 3 ... |
On an algebra quiz, $10\%$ of the students scored $70$ points, $35\%$ scored $80$ points, $30\%$ scored $90$ points, and the rest scored $100$ points. What is the difference between the mean and median score of the students' scores on this quiz? | 3 |
Bill is sent to a donut shop to purchase exactly six donuts. If the shop has four kinds of donuts and Bill is to get at least one of each kind, how many combinations will satisfy Bill's order requirements? | 10 |
The figure shows a square in the interior of a regular hexagon. The square and regular hexagon share a common side. What is the degree measure of $\angle ABC$? [asy]
size(150);
pair A, B, C, D, E, F, G, H;
A=(0,.866);
B=(.5,1.732);
C=(1.5,1.732);
D=(2,.866);
E=(1.5,0);
F=(.5,0);
G=(.5,1);
H=(1.5,1);
draw(A--B);
draw(B... | 45 |
The novel that everyone is reading for English class has half as many pages as their history book. Their science book has 4 times the amount of pages as their novel. If the history book has 300 pages, how many pages does their science book have? | Their novel is half as long as their 300-page history book so it is 300/2 = <<300/2=150>>150 pages
Their science book has 4 times the amount of pages as their novel, which is 150 pages, so their science book has 4*150 = <<4*150=600>>600 pages
#### 600 |
Is $f(x) = 3^{x^2-3} - |x|$ an even function, odd function, or neither?
Enter "odd", "even", or "neither". | \text{even} |
Paul lives in a 5th story apartment. He makes 3 trips out from and back to his apartment throughout the day each day of a week. How many feet does he travel vertically in total over the week if each story is 10 feet tall? | Since Paul makes 3 trips per day, and each trip involves going both down and up, this means he travels the full vertical distance of his apartment complex 3*2=<<3*2=6>>6 times a day.
Since there are 7 days in a week, this means he makes this trip 6*7=42 times a week.
Since each story is 10 feet tall, that means with 5 ... |
A pyramid with a triangular base has edges of unit length, and the angles between its edges are \(60^{\circ}, 90^{\circ},\) and \(120^{\circ}\). What is the volume of the pyramid? | \frac{\sqrt{2}}{12} |
Nicole has 4 fish tanks. The first two tanks need 8 gallons of water each and the other two need 2 fewer gallons of water each than the first two tanks. If Nicole needs to change the water of the aquarium every week, how many gallons of water will she need in four weeks? | The first two tanks need 8 x 2 = <<8*2=16>>16 gallons of water.
The other two tanks need 8 - 2 = <<8-2=6>>6 gallons of water each.
So, the two other tanks need 6 x 2 = <<6*2=12>>12 gallons of water.
Nicole needs 16 + 12 = <<16+12=28>>28 gallons of water every week for her four tanks.
Therefore, she needs a total of 28 ... |
The $25$ integers from $-10$ to $14,$ inclusive, can be arranged to form a $5$-by-$5$ square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum? | 10 |
What is the equation of the oblique asymptote of the graph of $\frac{2x^2+7x+10}{2x+3}$?
Enter your answer in the form $y = mx + b.$ | y = x+2 |
Which number from the set $\{1,2,3,4,5,6,7,8,9,10,11\}$ must be removed so that the mean (average) of the numbers remaining in the set is 6.1? | 5 |
James dumps his whole collection of 500 Legos on the floor and starts building a castle out of them. He uses half the pieces before finishing and is told to put the rest away. He puts all of the leftover pieces back in the box they came from, except for 5 missing pieces that he can't find. How many Legos are in the ... | James starts with 500 Legos and uses half of them, leaving 500/2=<<500/2=250>>250 Legos unused.
He puts those unused Legos away but since he's missing 5 he only puts 250-5=<<250-5=245>>245 Legos away.
#### 245 |
Given that the area of a cross-section of sphere O is $\pi$, and the distance from the center O to this cross-section is 1, then the radius of this sphere is __________, and the volume of this sphere is __________. | \frac{8\sqrt{2}}{3}\pi |
Compute
\[\frac{\lfloor \sqrt{1} \rfloor \cdot \lfloor \sqrt{2} \rfloor \cdot \lfloor \sqrt{3} \rfloor \cdot \lfloor \sqrt{5} \rfloor \dotsm \lfloor \sqrt{15} \rfloor}{\lfloor \sqrt{2} \rfloor \cdot \lfloor \sqrt{4} \rfloor \cdot \lfloor \sqrt{6} \rfloor \dotsm \lfloor \sqrt{16} \rfloor}.\] | \frac{3}{8} |
A shop advertises everything is "half price in today's sale." In addition, a coupon gives a 20% discount on sale prices. Using the coupon, the price today represents what percentage off the original price? | 60 |
In $ xy$ plane, find the minimum volume of the solid by rotating the region boubded by the parabola $ y \equal{} x^2 \plus{} ax \plus{} b$ passing through the point $ (1,\ \minus{} 1)$ and the $ x$ axis about the $ x$ axis | \frac{16\pi}{15} |
Let $z$ be a nonreal complex number such that $|z| = 1.$ Find the real part of $\frac{1}{1 - z}.$ | \frac{1}{2} |
A circle has its center at $(2,0)$ with a radius of 2, and another circle has its center at $(5,0)$ with a radius of 1. A line is tangent to both circles in the first quadrant. The $y$-intercept of this line is closest to: | $2 \sqrt{2}$ |
Let $S$ be the set of integers of the form $2^{x}+2^{y}+2^{z}$, where $x, y, z$ are pairwise distinct non-negative integers. Determine the 100th smallest element of $S$. | 577 |
John has 6 green marbles and 4 purple marbles. He chooses a marble at random, writes down its color, and then puts the marble back. He performs this process 5 times. What is the probability that he chooses exactly two green marbles? | \frac{144}{625} |
In a certain book, there were 100 statements written as follows:
1) "In this book, there is exactly one false statement."
2) "In this book, there are exactly two false statements."
...
3) "In this book, there are exactly one hundred false statements."
Which of these statements is true? | 99 |
The common ratio of the geometric sequence \( a+\log _{2} 3, a+\log _{1} 3, a+\log _{8} 3 \) is ______. | \frac{1}{3} |
A four-digit positive integer is called [i]virtual[/i] if it has the form $\overline{abab}$, where $a$ and $b$ are digits and $a \neq 0$. For example 2020, 2121 and 2222 are virtual numbers, while 2002 and 0202 are not. Find all virtual numbers of the form $n^2+1$, for some positive integer $n$. | 8282 |
Zebadiah has 3 red shirts, 3 blue shirts, and 3 green shirts in a drawer. Without looking, he randomly pulls shirts from his drawer one at a time. What is the minimum number of shirts that Zebadiah has to pull out to guarantee that he has a set of shirts that includes either 3 of the same colour or 3 of different colou... | 5 |
Alan, Jason, and Shervin are playing a game with MafsCounts questions. They each start with $2$ tokens. In each round, they are given the same MafsCounts question. The first person to solve the MafsCounts question wins the round and steals one token from each of the other players in the game. They all have the same p... | \frac{1}{2} |
There are lily pads in a row numbered $0$ to $11$, in that order. There are predators on lily pads $3$ and $6$, and a morsel of food on lily pad $10$. Fiona the frog starts on pad $0$, and from any given lily pad, has a $\frac{1}{2}$ chance to hop to the next pad, and an equal chance to jump $2$ pads. What is the proba... | \frac{15}{256} |
Find the greatest value of $t$ such that \[\frac{t^2 - t -56}{t-8} = \frac{3}{t+5}.\] | -4 |
There are three spheres and a cube. The first sphere is tangent to each face of the cube, the second sphere is tangent to each edge of the cube, and the third sphere passes through each vertex of the cube. What is the ratio of the surface areas of these three spheres? | 1:2:3 |
John's shirt cost 60% more than his pants. His pants cost $50. How much was John's outfit? | John's pants cost 50*.6=$<<50*.6=30>>30 more than his pants
So his shirt cost 50+30=$<<50+30=80>>80
So in total he spent 80+50=$<<80+50=130>>130
#### 130 |
There are 3 math teams in the area, with 5, 7, and 8 students respectively. Each team has two co-captains. If I randomly select a team, and then randomly select two members of that team to give a copy of $\emph{Introduction to Geometry}$, what is the probability that both of the people who receive books are co-captains... | \dfrac{11}{180} |
What is half of the absolute value of the difference of the squares of 18 and 16? | 34 |
If $x$ is such that $\frac{1}{x}<2$ and $\frac{1}{x}>-3$, then: | x>\frac{1}{2} \text{ or } x<-\frac{1}{3} |
Given that \(x\) satisfies \(\log _{5x} (2x) = \log _{625x} (8x)\), find the value of \(\log _{2} x\). | \frac{\ln 5}{2 \ln 2 - 3 \ln 5} |
Let set $A=\{-1, 2, 3\}$, and set $B=\{a+2, a^2+2\}$. If $A \cap B = \{3\}$, then the real number $a=$ ___. | -1 |
For every real number $x$, what is the value of the expression $(x+1)^{2} - x^{2}$? | 2x + 1 |
In a grade, Class 1, Class 2, and Class 3 each select two students (one male and one female) to form a group of high school students. Two students are randomly selected from this group to serve as the chairperson and vice-chairperson. Calculate the probability of the following events:
- The two selected students are no... | \dfrac{2}{5} |
Nina wants to buy a new video game with her allowance money. The game cost 50 dollars. Nina also has learned that there is a 10 percent sales tax. She receives 10 dollars a week as an allowance, and thinks she can save half of that. How many weeks will it take for Nina to be able to buy the new video game with her savi... | The 10 percent tax on 50 dollars will be 10 / 100 x 50 = <<10/100*50=5>>5 dollars.
The cost of the game and the tax will be 50 + 5 = <<50+5=55>>55 dollars.
Nina will save half of her allowance every week, or 10 / 2 = <<10/2=5>>5 dollars per week.
The total cost of the game and tax, divided by Nina's weekly savings is 5... |
Let $S$ be the set \{1,2, \ldots, 2012\}. A perfectutation is a bijective function $h$ from $S$ to itself such that there exists an $a \in S$ such that $h(a) \neq a$, and that for any pair of integers $a \in S$ and $b \in S$ such that $h(a) \neq a, h(b) \neq b$, there exists a positive integer $k$ such that $h^{k}(a)=b... | 2 |
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