problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
What is the smallest positive integer $n$ such that $\frac{n}{n+150}$ is equal to a terminating decimal? | 50 |
Tom hasn't been sleeping well lately. He figures he has been getting about 5 hours of sleep each weeknight and 6 hours each night on the weekend. If Tom would ideally like to get 8 hours of sleep each night on both weeknights and weekends, how many hours of sleep is Tom behind on from the last week? | Tom ideally wants to sleep 8 hours a night for 7 days, 8 x 7 = <<8*7=56>>56 hours sleep total for the week.
Tom has actually only slept 5 hours each weeknight, 5 x 5 = <<5*5=25>>25 hours of sleep.
Tom has only slept 6 hours each night on the weekend, 6 x 2 = <<6*2=12>>12 hours of sleep.
Total, Tom has gotten 25 + 12 hours = <<25+12=37>>37 hours sleep.
The difference between how many hours Tom would like to sleep, 56, and how many hours he's actually slept, 37, is 56 - 37 = <<56-37=19>>19 hours of sleep that Tom is behind on.
#### 19 |
Lisa, a child with strange requirements for her projects, is making a rectangular cardboard box with square bases. She wants the height of the box to be 3 units greater than the side of the square bases. What should the height be if she wants the surface area of the box to be at least 90 square units while using the least amount of cardboard? | 6 |
A woman invests in a property for $12,000 with the aim of receiving a $6\%$ return on her investment after covering all expenses including taxes and insurance. She pays $360 annually in taxes and $240 annually for insurance. She also keeps aside $10\%$ of each month's rent for maintenance. Calculate the monthly rent. | 122.22 |
Janet is getting paid to moderate social media posts. She gets paid 25 cents per post she checks. If it takes her 10 seconds to check a post, how much does she earn per hour? | First find the number of seconds in an hour: 1 hour * 60 minutes/hour * 60 seconds/minute = <<1*60*60=3600>>3600 seconds/hour.
Then divide the total number of seconds per hour by the number of seconds it takes Janet to read one post: 3600 seconds/hour / 10 seconds/post = <<3600/10=360>>360 posts/hour.
Then multiply the number of posts she checks per hour by her pay per post to find her hourly pay: $0.25/post * 360 posts/hour = $<<0.25*360=90>>90/hour.
#### 90 |
A circular spinner for a game has a radius of 5 cm. The probability of winning on one spin of this spinner is $\frac{2}{5}$. What is the area, in sq cm, of the WIN sector? Express your answer in terms of $\pi$.
[asy]import graph;
draw(Circle((0,0),25),black);
draw((0,0)--(7,18),Arrow);
draw((0,0)--(0,25));
draw((0,0)--(15,-20));
label("WIN",(10,10),S);
label("LOSE",(-8,-8),N);
dot((0,0));
[/asy] | 10\pi |
Given Professor Lewis has twelve different language books lined up on a bookshelf: three Arabic, four German, three Spanish, and two French, calculate the number of ways to arrange the twelve books on the shelf keeping the Arabic books together, the Spanish books together, and the French books together. | 362,880 |
Let the function \( f(x) = 3 \sin x + 2 \cos x + 1 \). If real numbers \( a, b, c \) satisfy \( a f(x) + b f(x-c) = 1 \) for all real numbers \( x \), find the value of \( \frac{b \cos c}{a} \). | -1 |
Jay went to watch a singer in a one hour 20 minutes concert. If there was a 10-minute intermission, and all the songs were 5 minutes except for one song that lasted 10 minutes, how many songs did she sing? | The time the singer had for singing was 80 - 10 = <<80-10=70>>70 minutes.
The time the singer had for singing 5-minute songs was 70 - 10 = 60 minutes.
The number of 5-minute songs was 60 / 5 = <<60/5=12>>12 songs.
The total number of songs sung is 12 + 1 = <<12+1=13>>13 songs.
#### 13 |
Given that
\begin{eqnarray*}&(1)& x\text{ and }y\text{ are both integers between 100 and 999, inclusive;}\qquad \qquad \qquad \qquad \qquad \\ &(2)& y\text{ is the number formed by reversing the digits of }x\text{; and}\\ &(3)& z=|x-y|. \end{eqnarray*}
How many distinct values of $z$ are possible?
| 9 |
Let \[f(x) =
\begin{cases}
9x+4 &\text{if }x\text{ is an integer}, \\
\lfloor{x}\rfloor+5 &\text{if }x\text{ is not an integer}.
\end{cases}
\]Find $f(\sqrt{29})$. | 10 |
In a right triangle \( ABC \) with \( AC = 16 \) and \( BC = 12 \), a circle with center at \( B \) and radius \( BC \) is drawn. A tangent to this circle is constructed parallel to the hypotenuse \( AB \) (the tangent and the triangle lie on opposite sides of the hypotenuse). The leg \( BC \) is extended to intersect this tangent. Determine by how much the leg is extended. | 15 |
Let $n>0$ be an integer. Each face of a regular tetrahedron is painted in one of $n$ colors (the faces are not necessarily painted different colors.) Suppose there are $n^{3}$ possible colorings, where rotations, but not reflections, of the same coloring are considered the same. Find all possible values of $n$. | 1,11 |
In the cube ABCD-A<sub>1</sub>B<sub>1</sub>C<sub>1</sub>D<sub>1</sub>, the angle formed by the skew lines A<sub>1</sub>B and AC is \_\_\_\_\_\_°; the angle formed by the line A<sub>1</sub>B and the plane A<sub>1</sub>B<sub>1</sub>CD is \_\_\_\_\_\_\_\_\_°. | 30 |
Let \( z \) be a complex number such that \( |z| = 1 \). If the equation \( z x^{2} + 2 \bar{z} x + 2 = 0 \) in terms of \( x \) has a real root, find the sum of all such complex numbers \( z \). | -\frac{3}{2} |
Compute $(5+7)^3+(5^3+7^3)$. | 2196 |
Given a sequence \( a_1, a_2, a_3, \ldots, a_n \) of non-zero integers such that the sum of any 7 consecutive terms is positive and the sum of any 11 consecutive terms is negative, what is the largest possible value for \( n \)? | 16 |
Find the largest three-digit integer that is divisible by each of its digits and the sum of the digits is divisible by 6. | 936 |
Louie obtained 80% on a math quiz. He had 5 mistakes. How many items were there on the math quiz? | Louie got 100% - 80% = 20% of the total items incorrect, which is equal to 5 items.
So 1% is equal to 5/20 = 1/4 of an item.
Hence, the total number of items, represented by 100%, on the math quiz was 1/4 of an item x 100 = <<1/4*100=25>>25.
#### 25 |
Given that \( p \) is a prime number and \( r \) is the remainder when \( p \) is divided by 210, if \( r \) is a composite number that can be expressed as the sum of two perfect squares, find \( r \). | 169 |
Pete has a bag with 10 marbles. 40% are blue and the rest are red. His friend will trade him two blue marbles for every red one. If Pete keeps 1 red marble, how many total marbles does he have after trading with his friend? | He starts with 4 blue marbles because 10 x .4 = <<10*.4=4>>4
60% of the marbles are red because 100 - 40 = <<100-40=60>>60
He has 6 red marbles because 10 x .6 = <<10*.6=6>>6
He trades 5 red marbles because 6 - 1 = <<6-1=5>>5
He gets 10 blue marbles because 5 x 2 = <<5*2=10>>10
He has 15 marbles now because 4 + 1 + 10 = <<4+1+10=15>>15
#### 15 |
A certain function $f$ has the properties that $f(3x) = 3f(x)$ for all positive real values of $x$, and that $f(x) = 1-|x-2|$ for $1\le x \le 3$. Find the smallest $x$ for which $f(x) = f(2001)$. | 429 |
A room is 24 feet long and 14 feet wide. Find the ratio of the length to its perimeter and the ratio of the width to its perimeter. Express each ratio in the form $a:b$. | 7:38 |
\[\frac{\tan 96^{\circ} - \tan 12^{\circ} \left( 1 + \frac{1}{\sin 6^{\circ}} \right)}{1 + \tan 96^{\circ} \tan 12^{\circ} \left( 1 + \frac{1}{\sin 6^{\circ}} \right)} =\] | \frac{\sqrt{3}}{3} |
Determine the area of the circle described by the graph of the equation
\[r = 4 \cos \theta - 3 \sin \theta.\] | \frac{25\pi}{4} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively, and it is given that $(a+b)(\sin A-\sin B)=c(\sin C-\sin B)$.
$(1)$ Find $A$.
$(2)$ If $a=4$, find the maximum value of the area $S$ of $\triangle ABC$. | 4 \sqrt {3} |
Find the sum of the roots of $\tan^2x-9\tan x+1=0$ that are between $x=0$ and $x=2\pi$ radians. | 3 \pi |
A man has part of $4500 invested at 4% and the rest at 6%. If his annual return on each investment is the same, the average rate of interest which he realizes of the $4500 is: | 4.8\% |
Let $k$ and $n$ be positive integers and let $$ S=\left\{\left(a_{1}, \ldots, a_{k}\right) \in \mathbb{Z}^{k} \mid 0 \leq a_{k} \leq \cdots \leq a_{1} \leq n, a_{1}+\cdots+a_{k}=k\right\} $$ Determine, with proof, the value of $$ \sum_{\left(a_{1}, \ldots, a_{k}\right) \in S}\binom{n}{a_{1}}\binom{a_{1}}{a_{2}} \cdots\binom{a_{k-1}}{a_{k}} $$ in terms of $k$ and $n$, where the sum is over all $k$-tuples $\left(a_{1}, \ldots, a_{k}\right)$ in $S$. | \[
\binom{k+n-1}{k} = \binom{k+n-1}{n-1}
\] |
James works for 240 minutes. He takes a water break every 20 minutes and a sitting break every 120 minutes. How many more water breaks does he take than sitting breaks? | First find the number of water breaks James takes: 240 minutes / 20 minutes/water break = <<240/20=12>>12 water breaks
Then find the number of sitting breaks James takes: 240 minutes / 120 minutes/sitting break = <<240/120=2>>2 sitting breaks
Then subtract the number of sitting breaks from the number of water breaks to find the difference: 12 breaks - 2 breaks = <<12-2=10>>10 breaks
#### 10 |
James builds a 20 story building. The first 10 stories are each 12 feet tall each. The remaining floors are each 3 feet taller. How tall is the building? | The first 10 floors are 10*12=<<10*12=120>>120 feet
There are a remaining 20-10=<<20-10=10>>10 stories
They are each 12+3=<<12+3=15>>15 feet
So those stories are 15*10=<<15*10=150>>150 feet
So the total height is 120+150=<<120+150=270>>270 feet
#### 270 |
For some constants $a$ and $b,$ let \[f(x) = \left\{
\begin{array}{cl}
ax + b & \text{if } x < 2, \\
8 - 3x & \text{if } x \ge 2.
\end{array}
\right.\]The function $f$ has the property that $f(f(x)) = x$ for all $x.$ What is $a + b?$ | \frac{7}{3} |
At the Bertolli Farm, they grow 2073 tomatoes, 4112 cobs of corn, and 985 onions. How many fewer onions are grown than tomatoes and corn together? | 2073 + 4112 = <<2073+4112=6185>>6185
6185 - 985 = <<6185-985=5200>>5200
The Bertolli Farm grows 5200 fewer onions than tomatoes and corn.
#### 5200 |
Two sides of an isosceles triangle are 10 inches and 20 inches. If the shortest side of a similar triangle is 50 inches, what is the perimeter of the larger triangle? | 250\text{ inches} |
The 5 a.m. temperatures for seven consecutive days were $-7^{\circ}$, $-4^{\circ}$, $-4^{\circ}$, $-5^{\circ}$, $1^{\circ}$, $3^{\circ}$ and $2^{\circ}$ Celsius. What is the mean 5 a.m. temperature for the week in degrees Celsius? | -2 |
Elena has 8 lilies and 5 tulips in her garden. Each lily has 6 petals. Each tulip has 3 petals. How many flower petals are in Elena’s garden? | Elena has 8 * 6 = <<8*6=48>>48 lily petals
Elena has 5 * 3 = <<5*3=15>>15 tulip petals
Elena has a total of 48 + 15 = <<48+15=63>>63 petals
#### 63 |
John buys cans of soup for buy 1 get one free. He gets 30 cans with a normal price of $0.60. How much does he pay? | He pays for 30/2=<<30/2=15>>15 cans
So he spends 15*.6=$<<15*.6=9>>9
#### 9 |
In rectangle $ABCD$, $AB=100$. Let $E$ be the midpoint of $\overline{AD}$. Given that line $AC$ and line $BE$ are perpendicular, find the greatest integer less than $AD$.
| 141 |
The set of vectors $\mathbf{v}$ such that
\[\mathbf{v} \cdot \mathbf{v} = \mathbf{v} \cdot \begin{pmatrix} 10 \\ -40 \\ 8 \end{pmatrix}\]forms a solid in space. Find the volume of this solid. | 12348 \pi |
How many different $4\times 4$ arrays whose entries are all 1's and -1's have the property that the sum of the entries in each row is 0 and the sum of the entries in each column is 0?
| 90 |
A square sheet of paper with sides of length $10$ cm is initially folded in half horizontally. The folded paper is then folded diagonally corner to corner, forming a triangular shape. If this shape is then cut along the diagonal fold, what is the ratio of the perimeter of one of the resulting triangles to the perimeter of the original square?
A) $\frac{15 + \sqrt{125}}{40}$
B) $\frac{15 + \sqrt{75}}{40}$
C) $\frac{10 + \sqrt{50}}{40}$
D) $\frac{20 + \sqrt{100}}{40}$ | \frac{15 + \sqrt{125}}{40} |
A bag contains three balls labeled 1, 2, and 3. A ball is drawn from the bag, its number is recorded, and then it is returned to the bag. This process is repeated three times. If each ball has an equal chance of being drawn, calculate the probability of the number 2 being drawn three times given that the sum of the numbers drawn is 6. | \frac{1}{7} |
Find all $x$ such that $x^2+5x<6$. Express your answer in interval notation. | (-6, 1) |
What is the sum of all two-digit positive integers whose squares end with the digits 25? | 495 |
Cameron drives at twice the speed of his brother, Chase. But Danielle drives at three times the speed of Cameron. If it takes Danielle 30 minutes to travel from Granville to Salisbury, how long, in minutes, will it take Chase to travel from Granville to Salisbury? | If Danielle drives at three times the speed of Cameron, it will take Cameron 30*3=<<30*3=90>>90 minutes for Cameron to travel the same distance.
And since Cameron drives at twice the speed of his brother, Chase, it will take Chase 90*2=<<90*2=180>>180 minutes to travel from Granville to Salisbury.
#### 180 |
Find the number of real solutions to
\[(x^{2006} + 1)(x^{2004} + x^{2002} + x^{2000} + \dots + x^2 + 1) = 2006x^{2005}.\] | 1 |
Let the solution set of the inequality about $x$, $|x-2| < a$ ($a \in \mathbb{R}$), be $A$, and $\frac{3}{2} \in A$, $-\frac{1}{2} \notin A$.
(1) For any $x \in \mathbb{R}$, the inequality $|x-1| + |x-3| \geq a^2 + a$ always holds true, and $a \in \mathbb{N}$. Find the value of $a$.
(2) If $a + b = 1$, and $a, b \in \mathbb{R}^+$, find the minimum value of $\frac{1}{3b} + \frac{b}{a}$, and indicate the value of $a$ when the minimum is attained. | \frac{1 + 2\sqrt{3}}{3} |
A bag of fruit contains 10 fruits, including an even number of apples, at most two oranges, a multiple of three bananas, and at most one pear. How many different combinations of these fruits can there be? | 11 |
Megan is making food for a party. She has to spend 20 minutes preparing one dish in the oven. Each dish can feed 5 people. She spends 2 hours preparing as many of these dishes as she can for the party. How many people can she feed with these dishes? | First, we need to determine the total amount of minutes available for cooking. We do this by performing 2*60=<<2*60=120>>120, as there are 60 minutes in an hour.
Now that we know we have 120 minutes, we then divide the preparation time into the total amount of time by performing 120/20= <<120/20=6>>6 dishes capable of being made.
Since each dish can feed 5 people, we find the total number of people fed by performing 5*6=<<5*6=30>>30 people.
#### 30 |
Adam has an orchard. Every day for 30 days he picks 4 apples from his orchard. After a month, Adam has collected all the remaining apples, which were 230. How many apples in total has Adam collected from his orchard? | During 30 days Adam picked 4 * 30 = <<4*30=120>>120 apples.
So in total with all the remaining apples, he picked 120 + 230 = <<120+230=350>>350 apples from his orchard.
#### 350 |
A group of students during a sports meeting lined up for a team photo. When they lined up in rows of 5, there were two students left over. When they formed rows of 6 students, there were three extra students, and when they lined up in rows of 8, there were four students left over. What is the fewest number of students possible in this group? | 59 |
A right triangle has a side length of 21 inches and a hypotenuse of 29 inches. A second triangle is similar to the first and has a hypotenuse of 87 inches. What is the length of the shortest side of the second triangle? | 60\text{ inches} |
Cole wants to fence his backyard on three sides. His backyard is 9 feet along the sides and 18 feet along the back. The neighbor behind him agreed to pay for half of their shared side of his fence, and the neighbor on his left agreed to pay for a third of their shared side. Fencing costs $3 per foot. How much does Cole have to pay for the fence? | Cole has to pay 3 * 9 = $<<3*9=27>>27 for one side of the side fence.
For the shared left fence, he has to pay two-thirds of 27, so he has to pay 27 * 2 / 3 = $<<27*2/3=18>>18.
The shared back fence will cost 3 * 18 = $<<3*18=54>>54.
He has to pay for half the back fence, so it will cost 54 / 2 = $<<54/2=27>>27.
Thus, Cole has to pay 18 + 27 + 27 = $<<18+27+27=72>>72 for the fence.
#### 72 |
The bank plans to invest 40% of a certain fund in project M for one year, and the remaining 60% in project N. It is estimated that project M can achieve an annual profit of 19% to 24%, while project N can achieve an annual profit of 29% to 34%. By the end of the year, the bank must recover the funds and pay a certain rebate rate to depositors. To ensure that the bank's annual profit is no less than 10% and no more than 15% of the total investment in M and N, what is the minimum rebate rate that should be given to the depositors? | 10 |
A circle is inscribed in a right triangle. The point of tangency divides the hypotenuse into two segments of lengths $6 \mathrm{~cm}$ and $7 \mathrm{~cm}$. Calculate the area of the triangle. | 42 |
In a vertical vessel with straight walls closed by a piston, there is water. Its height is $h=2$ mm. There is no air in the vessel. To what height must the piston be raised for all the water to evaporate? The density of water is $\rho=1000$ kg / $\mathrm{m}^{3}$, the molar mass of water vapor is $M=0.018$ kg/mol, the pressure of saturated water vapor at a temperature of $T=50{ }^{\circ} \mathrm{C}$ is $p=12300$ Pa. The temperature of water and vapor is maintained constant. | 24.258 |
If $200\%$ of $x$ is equal to $50\%$ of $y$, and $x = 16$, what is the value of $y$? | 64 |
How many positive perfect squares less than $10^6$ are multiples of $24$? | 83 |
Let $T$ be the set of all positive integer divisors of $144,000$. Calculate the number of numbers that are the product of two distinct elements of $T$. | 451 |
Find the largest integer less than 2012 all of whose divisors have at most two 1's in their binary representations. | 1536 |
Calculate the value of the expression \( \sqrt{\frac{16^{12} + 8^{15}}{16^5 + 8^{16}}} \). | \frac{3\sqrt{2}}{4} |
What is $ 6 \div 3 - 2 - 8 + 2 \cdot 8$? | 8 |
Given that \( O \) is the circumcenter of \(\triangle ABC\), and \( 3 \overrightarrow{OA} + 4 \overrightarrow{OB} + 5 \overrightarrow{OC} = \overrightarrow{0} \), find the value of \( \cos \angle BAC \). | \frac{\sqrt{10}}{10} |
A projectile is launched with an initial velocity of $u$ at an angle of $\alpha$ from the ground. The trajectory can be modeled by the parametric equations:
\[
x = ut \cos \alpha, \quad y = ut \sin \alpha - \frac{1}{2} kt^2,
\]
where $t$ denotes time and $k$ denotes a constant acceleration, forming a parabolic arch.
Suppose $u$ is constant, but $\alpha$ varies over $0^\circ \le \alpha \le 90^\circ$. The highest points of each parabolic arch are plotted. Determine the area enclosed by the curve traced by these highest points, and express it in the form:
\[
d \cdot \frac{u^4}{k^2}.
\] | \frac{\pi}{8} |
A finite set $\{a_1, a_2, ... a_k\}$ of positive integers with $a_1 < a_2 < a_3 < ... < a_k$ is named *alternating* if $i+a$ for $i = 1, 2, 3, ..., k$ is even. The empty set is also considered to be alternating. The number of alternating subsets of $\{1, 2, 3,..., n\}$ is denoted by $A(n)$ .
Develop a method to determine $A(n)$ for every $n \in N$ and calculate hence $A(33)$ . | 5702887 |
Eight distinct integers are picked at random from $\{1,2,3,\ldots,15\}$. What is the probability that, among those selected, the third smallest is $5$? | \frac{72}{307} |
The distance from point P(1, -1) to the line $ax+3y+2a-6=0$ is maximized when the line passing through P is perpendicular to the given line. | 3\sqrt{2} |
Let $WXYZ$ be a rhombus with diagonals $WY = 20$ and $XZ = 24$. Let $M$ be a point on $\overline{WX}$, such that $WM = MX$. Let $R$ and $S$ be the feet of the perpendiculars from $M$ to $\overline{WY}$ and $\overline{XZ}$, respectively. Find the minimum possible value of $RS$. | \sqrt{244} |
If
\[x + \sqrt{x^2 - 1} + \frac{1}{x - \sqrt{x^2 - 1}} = 20,\]then find
\[x^2 + \sqrt{x^4 - 1} + \frac{1}{x^2 + \sqrt{x^4 - 1}}.\] | \frac{10201}{200} |
A woman is trying to decide whether it will be quicker to take an airplane or drive herself to a job interview. If she drives herself, the trip will take her 3 hours and 15 minutes. If she takes an airplane, she will first need to drive 10 minutes to the airport, and then wait 20 minutes to board the plane. After that, she will be on the airplane for one-third of the time it would have taken her to drive herself before landing in the destination city. Finally, it will take her an additional 10 minutes to get off the airplane and arrive at her interview site after the plane lands. Given this information, how many minutes faster is it for her to take the airplane? | First, we must convert the driving time of 3 hours and 15 minutes to minutes. Since there are 60 minutes in an hour, driving takes a total of 3*60 + 15 = <<3*60+15=195>>195 minutes.
Next, the woman will be on the airplane for one-third of 195 minutes, or 195/3 = <<195/3=65>>65 minutes.
Therefore, in total, the airplane trip will take the woman 10 + 20 + 65 + 10 = <<10+20+65+10=105>>105 minutes.
Thus, the airplane trip is 195 - 105 = <<195-105=90>>90 minutes faster than driving herself to the interview.
#### 90 |
When studying the operation of a new type of cyclic thermal engine, it was found that during part of the period it receives heat, and the absolute power of heat supply is expressed by the law:
\[ P_{1}(t)=P_{0} \frac{\sin (\omega t)}{100+\sin (t^{2})}, \quad 0<t<\frac{\pi}{\omega}. \]
The gas performs work, developing mechanical power
\[ P_{2}(t)=3 P_{0} \frac{\sin (2 \omega t)}{100+\sin (2 t)^{2}}, \quad 0<t<\frac{\pi}{2 \omega}. \]
The work on the gas performed by external bodies is \( \frac{2}{3} \) of the work performed by the gas. Determine the efficiency of the engine. | 1/3 |
Given an obtuse triangle \( \triangle ABC \) with the following conditions:
1. The lengths of \( AB \), \( BC \), and \( CA \) are positive integers.
2. The lengths of \( AB \), \( BC \), and \( CA \) do not exceed 50.
3. The lengths of \( AB \), \( BC \), and \( CA \) form an arithmetic sequence with a positive common difference.
Determine the number of obtuse triangles that satisfy the above conditions, and identify the side lengths of the obtuse triangle with the largest perimeter. | 157 |
Amber buys 7 guppies for her pond. Several days later, she sees 3 dozen baby guppies swimming around. Two days after that, she sees 9 more baby guppies. How many guppies does she have now? | 3 dozen baby guppies make 3 x 12 = <<3*12=36>>36 baby guppies.
There are a total of 36 + 9 = <<36+9=45>>45 baby guppies.
Adding the adult guppies, there are 7 + 45 = <<7+45=52>>52 guppies.
#### 52 |
We are given that $$54+(98\div14)+(23\cdot 17)-200-(312\div 6)=200.$$Now, let's remove the parentheses: $$54+98\div14+23\cdot 17-200-312\div 6.$$What does this expression equal? | 200 |
There are 5 girls sitting in a row on five chairs, and opposite them, on five chairs, there are 5 boys sitting. It was decided that the boys would switch places with the girls. In how many ways can this be done? | 14400 |
If we write $\sqrt{5}+\frac{1}{\sqrt{5}} + \sqrt{7} + \frac{1}{\sqrt{7}}$ in the form $\dfrac{a\sqrt{5} + b\sqrt{7}}{c}$ such that $a$, $b$, and $c$ are positive integers and $c$ is as small as possible, then what is $a+b+c$? | 117 |
Daliah picked up 17.5 pounds of garbage. Dewei picked up 2 pounds less than Daliah. Zane picked up 4 times as many pounds of garbage as Dewei. How many pounds of garbage did Zane pick up? | Daliah = <<17.5=17.5>>17.5 pounds
Dewei + 17.5 - 2 = <<+17.5-2=15.5>>15.5 pounds
Zane = 4 * 15.5 = <<4*15.5=62>>62 pounds
Zane picked up 62 pounds of garbage.
#### 62 |
Let $f(x)=|x-2|+|x-4|-|2x-6|$ for $2 \leq x \leq 8$. The sum of the largest and smallest values of $f(x)$ is | 2 |
Kylie has 34 stamps in her collection. Her friend, Nelly, has 44 more stamps than Kylie. How many stamps do Kylie and Nelly have together? | Nelly has 34 + 44 = <<34+44=78>>78 stamps.
Thus, Kylie and Nelly have 34 + 78 = <<34+78=112>>112 stamps.
#### 112 |
If the graph of the power function $y=f(x)$ passes through the point $(9, \frac{1}{3})$, find the value of $f(25)$. | \frac{1}{5} |
Given an equilateral triangle $PQR$ with a side length of 8 units, a process similar to the previous one is applied, but here each time, the triangle is divided into three smaller equilateral triangles by joining the midpoints of its sides, and the middle triangle is shaded each time. If this procedure is repeated 100 times, what is the total area of the shaded triangles?
A) $6\sqrt{3}$
B) $8\sqrt{3}$
C) $10\sqrt{3}$
D) $12\sqrt{3}$
E) $14\sqrt{3}$ | 8\sqrt{3} |
You start with a single piece of chalk of length 1. Every second, you choose a piece of chalk that you have uniformly at random and break it in half. You continue this until you have 8 pieces of chalk. What is the probability that they all have length $\frac{1}{8}$ ? | \frac{1}{63} |
In the subtraction shown, \( K, L, M \), and \( N \) are digits. What is the value of \( K+L+M+N \)?
\[
\begin{array}{llll}
5 & K & 3 & L \\
\end{array}
\]
\[
\begin{array}{r}
M & 4 & N & 1 \\
\hline
4 & 4 & 5 & 1 \\
\end{array}
\] | 20 |
If four people, A, B, C, and D, line up in a row, calculate the number of arrangements in which B and C are on the same side of A. | 16 |
Let $v$ and $w$ be distinct, randomly chosen roots of the equation $z^{1997}-1=0$. Let $\frac{m}{n}$ be the probability that $\sqrt{2+\sqrt{3}}\le\left|v+w\right|$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 582 |
Let \( p, q, r, s, t, u, v, \) and \( w \) be real numbers such that \( pqrs = 16 \) and \( tuvw = 25 \). Find the minimum value of
\[
(pt)^2 + (qu)^2 + (rv)^2 + (sw)^2.
\] | 400 |
Let $f(x) = |g(x^3)|$. If $g$ is an odd function, is $f$ odd, even, or neither?
Enter "odd", "even", or "neither". | \text{even} |
When $3x^2$ was added to the quadratic polynomial $f(x)$, its minimum value increased by 9. When $x^2$ was subtracted from it, its minimum value decreased by 9. How will the minimum value of $f(x)$ change if $x^2$ is added to it? | \frac{9}{2} |
Consider a 6x3 grid where you can move only to the right or down. How many valid paths are there from top-left corner $A$ to bottom-right corner $B$, if paths passing through segment from $(4,3)$ to $(4,2)$ and from $(2,1)$ to $(2,0)$ are forbidden? [The coordinates are given in usual (x, y) notation, where the leftmost, topmost corner is (0, 3) and the rightmost, bottommost corner is (6, 0).] | 48 |
A movie that's 1.5 hours long is being replayed 6 times in one movie theater each day. There is a 20-minute advertisement before the start of the movie. How long, in hours, does the movie theater operate each day? | The movie that is being replayed 6 times takes 1.5 x 6 = <<6*1.5=9>>9 hours.
The advertisements before the start of the movie take 20 x 6 = <<20*6=120>>120 minutes in all.
One hundred twenty minutes is equal to 120/60 = <<120/60=2>>2 hours.
Thus, the movie theater operates for 9 + 2 = <<9+2=11>>11 hours each day.
#### 11 |
Let
\[f(x) = \frac{x^2 - 6x + 6}{2x - 4}\]and
\[g(x) = \frac{ax^2 + bx + c}{x - d}.\]You are given the following properties:
$\bullet$ The graphs of $f(x)$ and $g(x)$ have the same vertical asymptote.
$\bullet$ The oblique asymptotes of $f(x)$ and $g(x)$ are perpendicular, and they intersect on the $y$-axis.
$\bullet$ The graphs of $f(x)$ and $g(x)$ have two intersection points, one of which is on the line $x = -2.$
Find the point of intersection of the graphs of $f(x)$ and $g(x)$ that does not lie on the line $x = -2.$ | \left( 4, -\frac{1}{2} \right) |
Geoff and Trevor each roll a fair eight-sided die (the sides are labeled 1 through 8). What is the probability that the product of the numbers they roll is even or a prime number? | \frac{7}{8} |
A said: "I am 10 years old, 2 years younger than B, and 1 year older than C."
B said: "I am not the youngest, C and I have a 3-year difference, and C is 13 years old."
C said: "I am younger than A, A is 11 years old, and B is 3 years older than A."
Among the three statements made by each person, one of them is incorrect. Please determine A's age. | 11 |
Given that the regular price for one backpack is $60, and Maria receives a 20% discount on the second backpack and a 30% discount on the third backpack, calculate the percentage of the $180 regular price she saved. | 16.67\% |
Jack and Jill run a 12 km circuit. First, they run 7 km to a certain point and then the remaining 5 km back to the starting point by different, uneven routes. Jack has a 12-minute head start and runs at the rate of 12 km/hr uphill and 15 km/hr downhill. Jill runs 14 km/hr uphill and 18 km/hr downhill. How far from the turning point are they when they pass each other, assuming their downhill paths are the same but differ in uphill routes (in km)?
A) $\frac{226}{145}$
B) $\frac{371}{145}$
C) $\frac{772}{145}$
D) $\frac{249}{145}$
E) $\frac{524}{145}$ | \frac{772}{145} |
Shem makes 2.5 times more money per hour than Kem. If Kem earns $4 per hour, how much does Shem earn for an 8-hour workday? | Shem earns $4 x 2.5 = $<<4*2.5=10>>10 in an hour.
So, Shem earns $10 x 8 = $<<10*8=80>>80 for an 8-hour workday.
#### 80 |
Consider a circular cone with vertex $V$, and let $A B C$ be a triangle inscribed in the base of the cone, such that $A B$ is a diameter and $A C=B C$. Let $L$ be a point on $B V$ such that the volume of the cone is 4 times the volume of the tetrahedron $A B C L$. Find the value of $B L / L V$. | \frac{\pi}{4-\pi} |
Jane bought 2 skirts for $13 each. She also bought 3 blouses for $6 each. She paid the cashier $100. How much change did she receive? | The 2 skirts cost $13 x 2 = $<<13*2=26>>26.
The 3 blouses cost $6 x 3 = $<<6*3=18>>18.
The total cost for 2 skirts and 3 blouses is $26 + $18 = $<<26+18=44>>44.
Jane received $100 - $44 = $<<100-44=56>>56 change.
#### 56 |
What is the largest integer value of $x$ for which $5-4x>17$? | -4 |
Gage skated 1 hr 15 min each day for 5 days and 1 hr 30 min each day for 3 days. How many minutes would he have to skate the ninth day in order to average 85 minutes of skating each day for the entire time? | 120 |
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