problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
For all positive numbers $a,b \in \mathbb{R}$ such that $a+b=1$, find the supremum of the expression $-\frac{1}{2a}-\frac{2}{b}$. | -\frac{9}{2} |
What is the sum of the positive solutions to \( 2x^2 - x \lfloor x \rfloor = 5 \), where \( \lfloor x \rfloor \) is the largest integer less than or equal to \( x \)? | \frac{3 + \sqrt{41} + 2\sqrt{11}}{4} |
Given that $BDEF$ is a square and $AB = BC = 1$, find the number of square units in the area of the regular octagon.
[asy]
real x = sqrt(2);
pair A,B,C,D,E,F,G,H;
F=(0,0); E=(2,0); D=(2+x,x); C=(2+x,2+x);
B=(2,2+2x); A=(0,2+2x); H=(-x,2+x); G=(-x,x);
draw(A--B--C--D--E--F--G--H--cycle);
draw((-x,0)--(2+x,0)--(2+x,2+2x... | 4+4\sqrt{2} |
In how many ways can four people line up in a straight line if the youngest person cannot be first in line? | 18 |
Find the minimum value of the expression \((\sqrt{2(1+\cos 2x)} - \sqrt{36 - 4\sqrt{5}} \sin x + 2) \cdot (3 + 2\sqrt{10 - \sqrt{5}} \cos y - \cos 2y)\). If the answer is not an integer, round it to the nearest whole number. | -27 |
Given $\tan (\alpha-\beta)= \frac {1}{2}$, $\tan \beta=- \frac {1}{7}$, and $\alpha$, $\beta\in(0,\pi)$, find the value of $2\alpha-\beta$. | - \frac {3\pi}{4} |
Borgnine wants to see 1100 legs at the zoo. He has already seen 12 chimps, 8 lions, and 5 lizards. He is next headed to see the tarantulas. How many tarantulas does he need to see to meet his goal? | He has seen 48 chimp legs because 12 x 4 = <<12*4=48>>48
He has seen 32 lion legs because 8 x 4 = <<8*4=32>>32
He has seen 20 lizard legs because 5 x 4 = <<5*4=20>>20
He has seen 100 total legs because 48 + 32 + 20 = <<48+32+20=100>>100
He has to see 1000 tarantulas legs because 1100- 100 = <<1100-100=1000>>1000
He has... |
In the central cell of a $21 \times 21$ board, there is a piece. In one move, the piece can be moved to an adjacent cell sharing a side. Alina made 10 moves. How many different cells can the piece end up in? | 121 |
I have five apples and ten oranges. If a fruit basket must contain at least one piece of fruit, how many kinds of fruit baskets can I make? (The apples are identical and the oranges are identical. A fruit basket consists of some number of pieces of fruit, and it doesn't matter how the fruit are arranged in the basket... | 65 |
Calculate the coefficient of the term containing $x^4$ in the expansion of $(x-1)(x-2)(x-3)(x-4)(x-5)$. | -15 |
8 coins are simultaneously flipped. What is the probability that heads are showing on at most 2 of them? | \dfrac{37}{256} |
Let $T$ be the set of all rational numbers $r$, $0<r<1$, that have a repeating decimal expansion in the form $0.efghefgh\ldots=0.\overline{efgh}$, where the digits $e$, $f$, $g$, and $h$ are not necessarily distinct. To write the elements of $T$ as fractions in lowest terms, how many different numerators are required? | 6000 |
Simplify $\dfrac{123}{999} \cdot 27.$ | \dfrac{123}{37} |
There are 4 boys and 3 girls lining up for a photo. The number of ways they can line up such that no two boys are adjacent is \_\_\_\_ (answer with a number), and the number of ways they can line up such that no two girls are adjacent is \_\_\_\_ (answer with a number). | 1440 |
A pyramid has a square base $ABCD$ and a vertex $E$. The area of square $ABCD$ is $256$, and the areas of $\triangle ABE$ and $\triangle CDE$ are $120$ and $136$, respectively. The distance from vertex $E$ to the midpoint of side $AB$ is $17$. What is the volume of the pyramid?
- **A)** $1024$
- **B)** $1200$
- **C)** ... | 1280 |
Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy $|\overrightarrow {a}|=5$, $|\overrightarrow {a}- \overrightarrow {b}|=6$, and $|\overrightarrow {a}+ \overrightarrow {b}|=4$, find the projection of vector $\overrightarrow {b}$ on vector $\overrightarrow {a}$. | -1 |
Let $S$ be the set of integers which are both a multiple of $70$ and a factor of $630{,}000$ . A random element $c$ of $S$ is selected. If the probability that there exists an integer $d$ with $\gcd (c,d) = 70$ and $\operatorname{lcm} (c,d) = 630{,}000$ is $\frac mn$ for some relatively prime integers... | 106 |
The entire graph of the function $f(x)$ is shown below ($f$ is only defined when $x$ is between $-4$ and $4$ inclusive). How many values of $x$ satisfy $f(f(x)) = 2$?
[asy]
import graph; size(9cm);
real lsf=0.5;
pen dps=linewidth(0.7)+fontsize(10);
defaultpen(dps); pen ds=black;
real xmin=-4.5,xmax=4.5,ymin=-0.5,y... | 3 |
Determine the number of ways to arrange the letters of the word "BALLOONIST". | 907200 |
Quinton brought 40 cupcakes to school on his birthday. He gave a cupcake to each of the 18 students in Ms. Delmont's class. He also gave a cupcake to each of the 16 students in Mrs. Donnelly's class. He also gave a cupcake to Ms. Delmont, Mrs. Donnelly, the school nurse, and the school principal. How many cupcakes did ... | Quinton had 40 cupcakes - 18 = <<40-18=22>>22 cupcakes.
He then had 22 cupcakes - 16 students = <<22-16=6>>6 cupcakes.
Quinton then gave away 6 - 1 -1 -1 -1 = <<6-1-1-1-1=2>>2 cupcakes left.
#### 2 |
The school assigns three people, A, B, and C, to participate in social practice activities in 7 different communities, with at most 2 people assigned to each community. There are $\boxed{\text{336}}$ different distribution schemes (answer in numbers). | 336 |
Given the function $f(x)=\frac{1}{1+{2}^{x}}$, if the inequality $f(ae^{x})\leqslant 1-f\left(\ln a-\ln x\right)$ always holds, then the minimum value of $a$ is ______. | \frac{1}{e} |
If a number is a multiple of 4 or contains the digit 4, we say this number is a "4-inclusive number", such as 20, 34. Arrange all "4-inclusive numbers" in the range \[0, 100\] in ascending order to form a sequence. What is the sum of all items in this sequence? | 1883 |
Convert the point $(6,2 \sqrt{3})$ in rectangular coordinates to polar coordinates. Enter your answer in the form $(r,\theta),$ where $r > 0$ and $0 \le \theta < 2 \pi.$ | \left( 4 \sqrt{3}, \frac{\pi}{6} \right) |
Trevor needs to go downtown for a restaurant date. An Uber ride downtown costs $3 more than a Lyft ride. A Lyft ride costs $4 more than a taxi ride. The Uber ride costs $22. If Trevor takes a taxi downtown and tips the taxi driver 20% of the original cost of the ride, what is the total cost of the ride downtown? | The cost of the Lyft ride is $22 - $3 = $<<22-3=19>>19
The original cost of the taxi ride is $19 - $4 = $<<19-4=15>>15
Trevor tips the taxi driver $15 * 0.20 = $<<15*0.20=3>>3
The total cost of the ride downtown is $15 + $3 = $<<15+3=18>>18
#### 18 |
Let $f(x)$ be an odd function. Is $f(f(x))$ even, odd, or neither?
Enter "odd", "even", or "neither". | \text{odd} |
A bag contains 3 tan, 2 pink and 4 violet chips. If the 9 chips are randomly drawn from the bag, one at a time and without replacement, what is the probability that the chips are drawn in such a way that the 3 tan chips are drawn consecutively, the 2 pink chips are drawn consecutively, and the 4 violet chips are drawn ... | \frac{1}{210} |
One hundred number cards are laid out in a row in ascending order: \(00, 01, 02, 03, \ldots, 99\). Then, the cards are rearranged so that each subsequent card is obtained from the previous one by increasing or decreasing exactly one of the digits by 1 (for example, after 29 there can be 19, 39 or 28, but not 30 or 20).... | 50 |
A positive five-digit integer is in the form $AB,CBA$; where $A$, $B$ and $C$ are each distinct digits. What is the greatest possible value of $AB,CBA$ that is divisible by eleven? | 96,\!569 |
Let $f(x) = x^2-2x$. How many distinct real numbers $c$ satisfy $f(f(f(f(c)))) = 3$? | 9 |
If \( \sqrt{\frac{3}{x} + 3} = \frac{5}{3} \), solve for \( x \). | -\frac{27}{2} |
A boy runs 1.5 km in 45 minutes. What is his speed in kilometers per hour? | Since 1 hour is equal to 60 minutes, then 45 minutes is equal to 45/60 = <<45/60=0.75>>0.75 hours.
Then, the speed in kilometers per hour is 1.5 km/0.75 hour = 2 kilometers per hour.
#### 2 |
For some real numbers $a$ and $b$, the equation $9x^3 + 5ax^2 + 4bx + a = 0$ has three distinct positive roots. If the sum of the base-2 logarithms of the roots is 4, what is the value of $a$? | -144 |
Paul uses 1 can of frosting to frost a layer cake. He uses a half can of frosting for a single cake, or a single pan of brownies, or a dozen cupcakes. For Saturday, he needs to have 3 layer cakes, 6 dozen cupcakes, 12 single cakes and 18 pans of brownies ready and frosted for customer pick up. How many cans of frost... | He needs 1 can of frosting per layer cake and he needs to make 3 layer cakes so that's 1*3 = <<1*3=3>>3 cans of frosting
There are 6 dozen cupcakes,12 single cakes and 18 pans of brownies for a total of 6+12+18 = <<6+12+18=36>>36 orders
Each of the 36 orders needs 1/2 can of frosting so they need 36*.5 = <<36*.5=18>>18... |
Find a necessary and sufficient condition on the natural number $ n$ for the equation
\[ x^n \plus{} (2 \plus{} x)^n \plus{} (2 \minus{} x)^n \equal{} 0
\]
to have a integral root. | n=1 |
What is the next term in the geometric sequence $$2, 6x, 18x^2, 54x^3, \ldots ?$$ Express your answer in terms of $x$. | 162x^4 |
Walter wakes up at 6:30 a.m., catches the school bus at 7:30 a.m., has 7 classes that last 45 minutes each, enjoys a 30-minute lunch break, and spends an additional 3 hours at school for various activities. He takes the bus home and arrives back at 5:00 p.m. Calculate the total duration of his bus ride. | 45 |
Expand the product $$(x^2-2x+2)(x^2+2x+2).$$ | x^4+4 |
Three volleyballs with a radius of 18 lie on a horizontal floor, each pair touching one another. A tennis ball with a radius of 6 is placed on top of them, touching all three volleyballs. Find the distance from the top of the tennis ball to the floor. (All balls are spherical in shape.) | 36 |
Suppose that \( f(x) \) is a function defined for every real number \( x \) with \( 0 \leq x \leq 1 \) with the properties that
- \( f(1-x)=1-f(x) \) for all real numbers \( x \) with \( 0 \leq x \leq 1 \),
- \( f\left(\frac{1}{3} x\right)=\frac{1}{2} f(x) \) for all real numbers \( x \) with \( 0 \leq x \leq 1 \), an... | \frac{3}{4} |
Every day, 12 tons of potatoes are delivered to the city using one type of transport from three collective farms. The price per ton is 4 rubles from the first farm, 3 rubles from the second farm, and 1 ruble from the third farm. To ensure timely delivery, the loading process for the required 12 tons should take no more... | 24.6667 |
A local farmer is paying 4 kids to help plant rows of corn. Every row of corn contains 70 ears. A bag of corn seeds contains 48 seeds and you need 2 seeds per ear of corn. He pays the kids $1.5 per row. Afterward, the kids are so hungry that they end up spending half their money on dinner. The dinner cost $36 per kid. ... | The dinner cost $144 dollars because 4 x 36 = <<4*36=144>>144
The kids earned $288 in total because 144 x 2 = <<144*2=288>>288
They planted 192 rows of corn because 288 / 1.5 = <<288/1.5=192>>192
They planted 13,440 ears of corn in total because 192 x 70 = <<192*70=13440>>13,440
They each planted 3,360 ears of corn bec... |
The gummy bear factory manufactures 300 gummy bears a minute. Each packet of gummy bears has 50 gummy bears inside. How long would it take for the factory to manufacture enough gummy bears to fill 240 packets, in minutes? | The factory creates 300 gummy bears a minute, which means it produces 300 / 50 = <<300/50=6>>6 packets of gummy bears per minute.
It would take 240 / 6 = <<240/6=40>>40 minutes to manufacture enough gummy bears to fill 240 packets.
#### 40 |
A garden store sells packages of pumpkin seeds for $2.50, tomato seeds for $1.50, and chili pepper seeds for $0.90. Harry is planning to plant three different types of vegetables on his farm. How much will Harry have to spend if he wants to buy three packets of pumpkin seeds, four packets of tomato seeds, and five pack... | The cost of three packets of pumpkin seeds is 3 x $2.50 = $<<3*2.5=7.50>>7.50.
The cost of four packets of tomato seeds is 4 x $1.50 = $<<4*1.5=6>>6.
And, the cost of five packets of chili pepper seeds is 5 x $0.90 = $4.50. Therefore, Harry will spend $7.... |
What positive integer can $n$ represent if it is known that by erasing the last three digits of the number $n^{3}$, we obtain the number $n$? | 32 |
For how many positive integers $n$ less than or equal to 500 is $$(\cos t - i\sin t)^n = \cos nt - i\sin nt$$ true for all real $t$? | 500 |
Jeanette is practicing her juggling. Each week she can juggle 2 more objects than the week before. If she starts out juggling 3 objects and practices for 5 weeks, how many objects can she juggle? | First find the total number of additional objects she learns to juggle: 2 objects/week * 5 weeks = <<2*5=10>>10 objects
Then add the initial number of objects she could juggle to find the total: 10 objects + 3 objects = <<10+3=13>>13 objects
#### 13 |
Pastor Paul prays 20 times per day, except on Sunday, when he prays twice as much. Pastor Bruce prays half as much as Pastor Paul, except on Sundays, when he prays twice as much as Pastor Paul. How many more times does Pastor Paul pray than Pastor Bruce prays in a week? | Pastor Paul prays 20*6=<<20*6=120>>120 days per week, except Sunday.
On Sunday, Pastor Paul prays 2*20=<<2*20=40>>40 times.
Per week, Pastor Paul prays 120+40=<<120+40=160>>160 times.
Pastor Bruce prays half as much as Pastor Paul does on all days except Sunday, for a total of 120/2=<<120/2=60>>60 times.
Pastor Bruce p... |
Given the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with its right focus at $(\sqrt{3}, 0)$, and passing through the point $(-1, \frac{\sqrt{3}}{2})$. Point $M$ is on the $x$-axis, and the line $l$ passing through $M$ intersects the ellipse $C$ at points $A$ and $B$ (with point $A$ above the $x$-ax... | \frac{4\sqrt{21}}{21} |
Let $\triangle ABC$ have sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$, respectively, and satisfy the equation $a\sin B = \sqrt{3}b\cos A$.
$(1)$ Find the measure of angle $A$.
$(2)$ Choose one set of conditions from the following three sets to ensure the existence and uniqueness of $\triangle ABC$, ... | 4\sqrt{3} + 3\sqrt{2} |
When three positive integers are divided by $12$, the remainders are $7,$ $9,$ and $10,$ respectively.
When the sum of the three integers is divided by $12$, what is the remainder? | 2 |
Given a sequence $a_1,$ $a_2,$ $a_3,$ $\dots,$ let $S_n$ denote the sum of the first $n$ terms of the sequence.
If $a_1 = 1$ and
\[a_n = \frac{2S_n^2}{2S_n - 1}\]for all $n \ge 2,$ then find $a_{100}.$ | -\frac{2}{39203} |
Find the greatest prime that divides $$ 1^2 - 2^2 + 3^2 - 4^2 +...- 98^2 + 99^2. $$ | 11 |
Let $n$ be a positive integer. Determine the size of the largest subset of $\{ -n, -n+1, \dots, n-1, n\}$ which does not contain three elements $a$, $b$, $c$ (not necessarily distinct) satisfying $a+b+c=0$. | 2 \left\lceil \frac{n}{2} \right\rceil |
Write $0.\overline{43}$ as a simplified fraction. | \frac{43}{99} |
Let the set \( S = \{1, 2, \cdots, 15\} \). Define \( A = \{a_{1}, a_{2}, a_{3}\} \) as a subset of \( S \), such that \( (a_{1}, a_{2}, a_{3}) \) satisfies \( 1 \leq a_{1} < a_{2} < a_{3} \leq 15 \) and \( a_{3} - a_{2} \leq 6 \). Find the number of such subsets that satisfy these conditions. | 371 |
Let the sequence $\{a_n\}$ satisfy that the sum of the first $n$ terms $S_n$ fulfills $S_n + a_1 = 2a_n$, and $a_1$, $a_2 + 1$, $a_3$ form an arithmetic sequence. Find the value of $a_1 + a_5$. | 34 |
Find the smallest natural number with the following properties:
a) It ends in digit 6 in decimal notation;
b) If you remove the last digit 6 and place this digit 6 at the beginning of the remaining number, the result is four times the original number. | 153846 |
Odell and Kershaw run for $30$ minutes on a circular track. Odell runs clockwise at $250 m/min$ and uses the inner lane with a radius of $50$ meters. Kershaw runs counterclockwise at $300 m/min$ and uses the outer lane with a radius of $60$ meters, starting on the same radial line as Odell. How many times after the sta... | 47 |
A cat is going up a stairwell with ten stairs. The cat can jump either two or three stairs at each step, or walk the last step if necessary. How many different ways can the cat go from the bottom to the top? | 12 |
Let $g$ be a function defined on the positive integers, such that
\[g(xy) = g(x) + g(y)\]
for all positive integers $x$ and $y$. Given $g(8) = 21$ and $g(18) = 26$, find $g(432)$. | 47 |
In the sequence \(\left\{a_{n}\right\}\), \(a_{1} = -1\), \(a_{2} = 1\), \(a_{3} = -2\). Given that for all \(n \in \mathbf{N}_{+}\), \(a_{n} a_{n+1} a_{n+2} a_{n+3} = a_{n} + a_{n+1} + a_{n+2} + a_{n+3}\), and \(a_{n+1} a_{n+2} a_{n+3} \neq 1\), find the sum of the first 4321 terms of the sequence \(S_{4321}\). | -4321 |
What is the value of $x$ if the three numbers $2, x$, and 10 have an average of $x$? | 6 |
At a UFO convention, there are 120 conference attendees. If there are 4 more male attendees than female attendees, how many male attendees are there? | Let's assume the number of female attendees is x.
The number of male attendees is x+4.
The sum of male attendees and female attendees is x+x+4 = 120
2x+4 = 120
2x = 116
The number of female attendees is x=116/2 = <<116/2=58>>58
The number of male attendees is 58+4 = <<58+4=62>>62
#### 62 |
Given the equation $x^2 + y^2 = |x| + 2|y|$, calculate the area enclosed by the graph of this equation. | \frac{5\pi}{4} |
Given a randomly selected number $x$ in the interval $[0,\pi]$, determine the probability of the event "$-1 \leqslant \tan x \leqslant \sqrt {3}$". | \dfrac{7}{12} |
How many integers 1-9 are divisors of the five-digit number 24,516? | 6 |
An entire floor is tiled with blue and white tiles. The floor has a repeated tiling pattern that forms every $8 \times 8$ square. Each of the four corners of this square features an asymmetrical arrangement of tiles where the bottom left $4 \times 4$ segment within each $8 \times 8$ square consists of blue tiles except... | \frac{3}{4} |
Let \( p, q, r, \) and \( s \) be positive real numbers such that
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
p^2+q^2 &=& r^2+s^2 &=& 2500, \\
pr &=& qs &=& 1200.
\end{array}
\]
If \( T = p + q + r + s \), compute the value of \( \lfloor T \rfloor \). | 120 |
Before the district play, the Unicorns had won $45$% of their basketball games. During district play, they won six more games and lost two, to finish the season having won half their games. How many games did the Unicorns play in all? | 48 |
When $k$ candies were distributed among seven people so that each person received the same number of candies and each person received as many candies as possible, there were 3 candies left over. If instead, $3 k$ candies were distributed among seven people in this way, then how many candies would have been left over? | 2 |
Our school's girls volleyball team has 14 players, including a set of 3 triplets: Missy, Lauren, and Liz. In how many ways can we choose 6 starters if the only restriction is that not all 3 triplets can be in the starting lineup? | 2838 |
Triangles $\triangle ABC$ and $\triangle A'B'C'$ lie in the coordinate plane with vertices $A(0,0)$, $B(0,12)$, $C(16,0)$, $A'(24,18)$, $B'(36,18)$, $C'(24,2)$. A rotation of $m$ degrees clockwise around the point $(x,y)$ where $0<m<180$, will transform $\triangle ABC$ to $\triangle A'B'C'$. Find $m+x+y$.
| 108 |
Stan drove 300 miles in 5 hours, 20 minutes. Next, he drove 360 miles in 6 hours, 40 minutes. What was Stan's average speed in miles per hour for the total trip? | 55 |
A large cube is made up of 27 small cubes. A plane is perpendicular to one of the diagonals of this large cube and bisects the diagonal. How many small cubes are intersected by this plane? | 19 |
At Clark's Food Store, apples cost 40 dollars for a dozen, and pears cost 50 dollars for a dozen. If Hank bought 14 dozen of each kind of fruit, how many dollars did he spend? | If Hank bought 14 dozen apples, he spent 14*$40 = $<<14*40=560>>560
Hank also spent $50*14 = $<<50*14=700>>700 for 14 dozen pears.
The total cost of the fruits that hank bought is $700+$560 = $<<700+560=1260>>1260
#### 1260 |
Henry took 9 pills a day for 14 days. Of these 9 pills, 4 pills cost $1.50 each, and the other pills each cost $5.50 more. How much did he spend in total on the pills? | There were 9-4 = <<9-4=5>>5 other pills
Each of the other pills cost 1.50+5.50 = <<1.50+5.50=7>>7 dollars each.
The 5 pills cost a total of 7*5 = <<7*5=35>>35 dollars.
The first 4 pills cost 1.50*4 = <<1.50*4=6>>6 dollars in total.
Henry spent a total of 35+6 = <<35+6=41>>41 dollars.
#### 41 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $b\sin^{2}\frac{A}{2}+a\sin^{2}\frac{B}{2}=\frac{C}{2}$.
1. If $c=2$, find the perimeter of $\triangle ABC$.
2. If $C=\frac{\pi}{3}$ and the area of $\triangle ABC$ is $2\sqrt{3}$, find $c$. | 2\sqrt{2} |
Find the smallest number which when successively divided by \( 45,454,4545 \) and 45454 leaves remainders of \( 4, 45,454 \) and 4545 respectively. | 35641667749 |
On the side of a triangle, a point is taken such that an angle equals another angle. What is the smallest possible distance between the centers of the circles circumscribed around triangles, if \( BC = 1 \)? | 1/2 |
One digit of the decimal representation of $\frac{5}{7}$ is randomly selected. What is the probability that the digit is a 4? Express your answer as a common fraction. | \frac{1}{6} |
Find $\frac{12}{15} + \frac{7}{9} + 1\frac{1}{6}$ and simplify the result to its lowest terms. | \frac{247}{90} |
Xiaoming and Xiaojun start simultaneously from locations A and B, heading towards each other. If both proceed at their original speeds, they meet after 5 hours. If both increase their speeds by 2 km/h, they meet after 3 hours. The distance between locations A and B is 30 km. | 30 |
Every 10 seconds, there is a car collision, and every 20 seconds there is a big crash. How many accidents overall will happen in 4 minutes? | 1 minute is 60 seconds, so 4 minutes is 4 * 60 = <<4*60=240>>240 seconds.
During this time there would be 240 / 20 = <<240/20=12>>12 big crashes.
And also 240 / 10 = <<240/10=24>>24 car collisions.
So in total during 4 minutes there would be 12 + 24 = <<12+24=36>>36 accidents.
#### 36 |
Celine collected twice as many erasers as Gabriel did. Julian collected twice as many erasers as Celine did. If they collected 35 erasers in total, how many erasers did Celine collect? | Let x represent the number of erasers Gabriel collected.
Celine collected 2*x erasers.
Julian collected 2*x*2=4*x erasers.
In total, Gabriel, Celine and Julian collected x+2*x+4*x=7*x erasers.
7*x=35
x=<<5=5>>5
Celine collected 5*2=<<5*2=10>>10 erasers.
#### 10 |
Given a finite arithmetic sequence \( a_{1}, a_{2}, \cdots a_{k} \), and the conditions:
$$
\begin{array}{c}
a_{4}+a_{7}+a_{10}=17, \\
a_{4}+a_{5}+a_{6}+a_{7}+a_{8}+a_{9}+a_{10}+a_{11}+a_{12}+a_{13}+a_{14}=77 .
\end{array}
$$
If \( a_{k}=13 \), then \( k \) is | 18 |
In an isosceles trapezoid \(ABCD\) (\(BC \parallel AD\)), the angles \(ABD\) and \(DBC\) are \(135^{\circ}\) and \(15^{\circ}\) respectively, and \(BD = \sqrt{6}\). Find the perimeter of the trapezoid. | 9 - \sqrt{3} |
Given that $x = \frac{5}{7}$ is a solution to the equation $56 x^2 + 27 = 89x - 8,$ what is the other value of $x$ that will solve the equation? Express your answer as a common fraction. | \frac{7}{8} |
The line $ax+(a+1)y=a+2$ passes through the point $(4,-8)$. Find $a$. | -2 |
A captain steers his ship 100 miles north on the first day of their journey. On the second day, he sails to the east three times as far as the distance as he covered on the first day. On the third day, the ship travels further east for 110 more miles than the distance it covered on the second day. What is the total dis... | If the ship traveled 100 miles to the north, it sailed for a distance of 3 * 100 miles = 300 miles to the east on the second day.
On the third day, the sheep sailed for 300 miles + 110 miles = <<300+110=410>>410 miles.
The distance covered by the ship in the three days is 410 miles + 300 miles + 100 miles = <<410+300+1... |
Carol is an aviation engineer deciding how much fuel to put in a jet. The empty plane needs 20 gallons of fuel per mile. Each person on the plane increases this amount by 3 gallons per mile, and each bag increases it by 2 gallons per mile. If there are 30 passengers and 5 flight crew, and each person brought two bags, ... | First find the total number of people by adding the number of passengers and flight crew: 30 people + 5 people = <<30+5=35>>35 people
Then find the total number of bags by doubling the total number of people: 35 people * 2 bags/person = <<35*2=70>>70 bags
Then find the fuel increase caused by the people by multiplying ... |
The sum of two numbers is $19$ and their difference is $5$. What is their product? | 84 |
The integer numbers from $1$ to $2002$ are written in a blackboard in increasing order $1,2,\ldots, 2001,2002$. After that, somebody erases the numbers in the $ (3k+1)-th$ places i.e. $(1,4,7,\dots)$. After that, the same person erases the numbers in the $(3k+1)-th$ positions of the new list (in this case, $2,5,9,\ldot... | 2,6,10 |
Express the given value of $22$ nanometers in scientific notation. | 2.2\times 10^{-8} |
Given the function $f(x)=\sin x\cos x-\sqrt{3}\cos^{2}x$.
(1) Find the smallest positive period of $f(x)$;
(2) Find the maximum and minimum values of $f(x)$ when $x\in[0,\frac{\pi }{2}]$. | -\sqrt{3} |
In a right triangle $ABC$ with equal legs $AC$ and $BC$, a circle is constructed with $AC$ as its diameter, intersecting side $AB$ at point $M$. Find the distance from vertex $B$ to the center of this circle if $BM = \sqrt{2}$. | \sqrt{5} |
Given point $M(2,0)$, draw two tangent lines $MA$ and $MB$ from $M$ to the circle $x^{2}+y^{2}=1$, where $A$ and $B$ are the tangent points. Calculate the dot product of vectors $\overrightarrow{MA}$ and $\overrightarrow{MB}$. | \dfrac{3}{2} |
Let $ABCD$ be an isosceles trapezoid with $\overline{AD}||\overline{BC}$ whose angle at the longer base $\overline{AD}$ is $\dfrac{\pi}{3}$. The diagonals have length $10\sqrt {21}$, and point $E$ is at distances $10\sqrt {7}$ and $30\sqrt {7}$ from vertices $A$ and $D$, respectively. Let $F$ be the foot of the altitud... | 32 |
A toy store manager received a large order of Mr. Slinkums just in time for the holidays. The manager places $20\%$ of them on the shelves, leaving the other 120 Mr. Slinkums in storage. How many Mr. Slinkums were in this order? | 150 |
Dorothea has a $3 \times 4$ grid of dots. She colors each dot red, blue, or dark gray. Compute the number of ways Dorothea can color the grid such that there is no rectangle whose sides are parallel to the grid lines and whose vertices all have the same color. | 284688 |
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