problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Delores has some money. She buys a computer for $400 and a printer for $40. If she has $10 left, how much money, in dollars, did Delores have at first? | The computer and printer cost 400+40=<<400+40=440>>440 dollars.
Delores had 440+10=<<440+10=450>>450 dollars at first.
#### 450 |
The graph of \(y^2 + 2xy + 60|x| = 900\) partitions the plane into several regions. What is the area of the bounded region? | 1800 |
If $\mathbf{a} \times \mathbf{b} = \begin{pmatrix} 5 \\ 4 \\ -7 \end{pmatrix},$ then compute $\mathbf{a} \times (3 \mathbf{b}).$ | \begin{pmatrix} 15 \\ 12 \\ -21 \end{pmatrix} |
Lloyd, Mark, and Michael have their Pokemon cards collection. Currently, Mark has thrice as many cards as Lloyd but has 10 fewer cards than Michael. If Michael has 100 cards now, how many more cards should they collect so that all three of them will have a total of 300 cards? | Mark has 10 fewer cards than Michael so Mark has 100 cards - 10 cards = 90 cards.
So, Lloyd has 90 cards / 3 = <<90/3=30>>30 cards.
All three of them have 90 cards + 30 cards + 100 cards = <<90+30+100=220>>220 cards.
Thus, they need to collect 300 cards - 220 cards = <<300-220=80>>80 more cards.
#### 80 |
Determine the number of minutes before Jack arrives at the park that Jill arrives at the park, given that they are 2 miles apart, Jill cycles at a constant speed of 12 miles per hour, and Jack jogs at a constant speed of 5 miles per hour. | 14 |
Let $f(x)=x^2-2x$. What is the value of $f(f(f(f(f(f(-1))))))$? | 3 |
Jerry files a lawsuit against the convenience store where he works and slipped and fell. He's asking for damages for loss of a $50,000 annual salary for 30 years, $200,000 in medical bills, and punitive damages equal to triple the medical and salary damages. If he gets 80% of what he's asking for, for much money does h... | First find the total cost of Jerry's lost salary: $50,000/year * 30 years = $<<50000*30=1500000>>1,500,000
Then add the cost of the medical bills: $1,500,000 + $200,000 = $<<1500000+200000=1700000>>1,700,000
Then triple that number to find the cost of the punitive damages: $1,700,000 * 3 = $<<1700000*3=5100000>>5,100,0... |
Given that $f(x) = x^k$ where $k < 0$, what is the range of $f(x)$ on the interval $[1, \infty)$? | (0,1] |
There are 13 3-year-olds, 20 4-year-olds, 15 5-year-olds, and 22 six-year-olds at a particular Sunday school. If the 3 and 4-year-olds are in one class and the 5 and 6-year-olds are in another class, what is the average class size? | First find the total number of 3- and 4-year-olds: 13 kids + 20 kids = <<13+20=33>>33 kids
Then find the total number of 5- and 6-year-olds: 15 kids + 22 kids = <<15+22=37>>37 kids
Then add the two class sizes: 33 kids + 37 kids = <<33+37=70>>70 kids
Then divide the number of kids by the number of classes to find the a... |
Define the sequence $a_1, a_2, a_3, \ldots$ by $a_n = \sum\limits_{k=1}^n \sin{k}$, where $k$ represents radian measure. Find the index of the 100th term for which $a_n < 0$. | 628 |
Paula has 20 candies to be given out to her six friends. She needs to buy four additional candies so she can give an equal number of candies to her friends. How many candies will each of her friends get? | Paula will have a total of 20 + 4 = <<20+4=24>>24 candies.
Thus, each of her friends will receive 24/6 = <<24/6=4>>4 candies.
#### 4 |
How many two-digit numbers have digits whose sum is a perfect square? | 17 |
The sale ad read: "Buy three tires at the regular price and get the fourth tire for 3 dollars." Sam paid 240 dollars for a set of four tires at the sale. What was the regular price of one tire? | 79 |
If the sum of the first $3n$ positive integers is $150$ more than the sum of the first $n$ positive integers, then the sum of the first $4n$ positive integers is | 300 |
Four identical regular tetrahedrons are thrown simultaneously on a table. Calculate the probability that the product of the four numbers on the faces touching the table is divisible by 4. | \frac{13}{16} |
Let $a$ , $b$ , $c$ be positive integers such that $abc + bc + c = 2014$ . Find the minimum possible value of $a + b + c$ . | 40 |
In the decimal representation of $ rac{1}{7}$, the 100th digit to the right of the decimal is? | 8 |
Let $p,$ $q,$ $r,$ $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s,$ and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q.$ Find the value of $p + q + r + s.$ | 1716 |
An equilateral hexagon with side length 1 has interior angles $90^{\circ}, 120^{\circ}, 150^{\circ}, 90^{\circ}, 120^{\circ}, 150^{\circ}$ in that order. Find its area. | \frac{3+\sqrt{3}}{2} |
Antal and Béla start from home on their motorcycles heading towards Cegléd. After traveling one-fifth of the way, Antal for some reason turns back. As a result, he accelerates and manages to increase his speed by one quarter. He immediately sets off again from home. Béla, continuing alone, decreases his speed by one qu... | 40 |
12 Smurfs are seated around a round table. Each Smurf dislikes the 2 Smurfs next to them, but does not dislike the other 9 Smurfs. Papa Smurf wants to form a team of 5 Smurfs to rescue Smurfette, who was captured by Gargamel. The team must not include any Smurfs who dislike each other. How many ways are there to form s... | 36 |
If $\sqrt{x+2}=2$, then $(x+2)^2$ equals: | 16 |
On a $12$-hour clock, an elapsed time of four hours looks the same as an elapsed time of $16$ hours. Because of this, we can say that four hours is "clock equivalent'' to its square number of hours. What is the least whole number of hours that is greater than $4$ hours and is "clock equivalent'' to its square number of... | 9 |
A ship left a port and headed due west, having 400 pounds of food for the journey's supply. After one day of sailing, 2/5 of the supplies had been used by the sailors in the ship. After another two days of sailing, the sailors used 3/5 of the remaining supplies. Calculate the number of supplies remaining in the ship to... | After one day of sailing, 2/5*400 = <<2/5*400=160>>160 pounds of food supplies had been used.
The total number of supplies remaining in the ship is 400-160 = <<400-160=240>>240 pounds.
After another 2 days of sailing, 3/5*240 = <<3/5*240=144>>144 pounds of food had been used.
The total number of supplies remaining in t... |
Find the arithmetic mean of the reciprocals of the first three prime numbers. | \frac{31}{90} |
In an isosceles triangle, one of the angles opposite an equal side is $40^{\circ}$. How many degrees are in the measure of the triangle's largest angle? [asy] draw((0,0)--(6,0)--(3,2)--(0,0)); label("$\backslash$",(1.5,1)); label("{/}",(4.5,1));
label("$40^{\circ}$",(.5,0),dir(45));
[/asy] | 100 |
In triangle $ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given that $b= \sqrt {2}$, $c=3$, $B+C=3A$.
(1) Find the length of side $a$;
(2) Find the value of $\sin (B+ \frac {3π}{4})$. | \frac{\sqrt{10}}{10} |
Find the minimum value of the function \( f(x) = \tan^2 x - 4 \tan x - 8 \cot x + 4 \cot^2 x + 5 \) on the interval \( \left( \frac{\pi}{2}, \pi \right) \). | 9 - 8\sqrt{2} |
Let point $O$ be inside $\triangle ABC$ and satisfy $4\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}=\overrightarrow{0}$. Determine the probability that a randomly thrown bean into $\triangle ABC$ lands in $\triangle OBC$. | \dfrac{2}{3} |
We label the sides of a dodecagon as \(C_{1}, C_{2}, \ldots, C_{12}\). In how many ways can we color the sides of a dodecagon with four colors such that two adjacent sides are always colored differently? (Two colorings are considered different if any one side \(C_{i}\) is colored differently in the two colorings). | 531444 |
The vertices of a cube have coordinates $(0,0,0),$ $(0,0,4),$ $(0,4,0),$ $(0,4,4),$ $(4,0,0),$ $(4,0,4),$ $(4,4,0),$ and $(4,4,4).$ A plane cuts the edges of this cube at the points $P = (0,2,0),$ $Q = (1,0,0),$ $R = (1,4,4),$ and two other points. Find the distance between these two points. | \sqrt{29} |
Let $Y$ be as in problem 14. Find the maximum $Z$ such that three circles of radius $\sqrt{Z}$ can simultaneously fit inside an equilateral triangle of area $Y$ without overlapping each other. | 10 \sqrt{3}-15 |
For a modified toothpick pattern, the first stage is constructed using 5 toothpicks. If each subsequent stage is formed by adding three more toothpicks than the previous stage, what is the total number of toothpicks needed for the $15^{th}$ stage? | 47 |
Find the value of $a_0 + a_1 + a_2 + \cdots + a_6$ given that $(2-x)^7 = a_0 + a_1(1+x)^2 + \cdots + a_7(1+x)^7$. | 129 |
Given that $\frac{{\cos 2\alpha}}{{\sin(\alpha+\frac{\pi}{4})}}=\frac{4}{7}$, find the value of $\sin 2\alpha$. | \frac{41}{49} |
If the Cesaro sum of the 50-term sequence \((b_1,\dots,b_{50})\) is 500, what is the Cesaro sum of the 51-term sequence \((2,b_1,\dots,b_{50})\)? | 492 |
The probability it will rain on Saturday is $60\%$, and the probability it will rain on Sunday is $25\%$. If the probability of rain on a given day is independent of the weather on any other day, what is the probability it will rain on both days, expressed as a percent? | 15 |
Given the letters in the word $SUCCESS$, determine the number of distinguishable rearrangements where all the vowels are at the end. | 20 |
Triangle $ABC$ is inscribed in circle $\omega$ with $AB=5$, $BC=7$, and $AC=3$. The bisector of angle $A$ meets side $\overline{BC}$ at $D$ and circle $\omega$ at a second point $E$. Let $\gamma$ be the circle with diameter $\overline{DE}$. Circles $\omega$ and $\gamma$ meet at $E$ and a second point $F$. Then $AF^2 = ... | 919 |
Egor wrote a number on the board and encoded it according to the rules of letter puzzles (different letters correspond to different digits, and identical letters correspond to identical digits). The word "GUATEMALA" was the result. How many different numbers could Egor have originally written, if his number was divisib... | 20160 |
Eight celebrities meet at a party. It so happens that each celebrity shakes hands with exactly two others. A fan makes a list of all unordered pairs of celebrities who shook hands with each other. If order does not matter, how many different lists are possible? | 3507 |
Vitya collects toy cars from the "Retro" series. The problem is that the total number of different models in the series is unknown — it's a big commercial secret. However, it is known that different cars are produced in equal quantities, so it can be assumed that all models are evenly and randomly distributed across di... | 58 |
Robin bought a four-scoop ice cream cone having a scoop each of vanilla, chocolate, strawberry and cherry. In how many orders can the four scoops be stacked on the cone if they are stacked one on top of the other? | 24 |
Let $\mathbf{a}$ and $\mathbf{b}$ be nonzero vectors such that
\[\|\mathbf{a}\| = \|\mathbf{b}\| = \|\mathbf{a} + \mathbf{b}\|.\]Find the angle between $\mathbf{a}$ and $\mathbf{b},$ in degrees. | 120^\circ |
What is $88 \div 4 \div 2$? | 11 |
In a relay race from Moscow to Petushki, two teams of 20 people each participated. Each team divided the distance into 20 segments (not necessarily equal) and assigned them among the participants so that each person ran exactly one segment (each participant's speed is constant, but the speeds of different participants ... | 38 |
Given eight distinguishable rings, let $n$ be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero digits of $n$. | 376 |
Fiona completed 36 math questions in an hour. Shirley was able to complete twice as many math questions within that same time, and Kiana completed half of the sum of Fiona and Shirley's math questions. If they each did the same number of questions the following hour, how many math questions did all three girls complete... | Shirley completed 36 x 2 = <<36*2=72>>72 questions
The sum of Fiona and Shirley's questions is 36 + 72 = <<36+72=108>>108
Kiana completed 108/2 = <<108/2=54>>54 questions
In one hour, they completed 108 +54 = <<108+54=162>>162 questions.
In two hours, they completed 162 x 2 = <<162*2=324>>324 questions.
#### 324 |
When $\sqrt[4]{5^9 \cdot 7^2}$ is fully simplified, the result is $a\sqrt[4]{b}$, where $a$ and $b$ are positive integers. What is $a+b$? | 270 |
Adam, Andrew and Ahmed all raise goats. Adam has 7 goats. Andrew has 5 more than twice as many goats as Adam. Ahmed has 6 fewer goats than Andrew. How many goats does Ahmed have? | Andrew: 5+2(7)=19 goats
Ahmed:19-6=<<19-6=13>>13 goats
#### 13 |
In $\triangle{ABC}$ with side lengths $AB = 15$, $AC = 8$, and $BC = 17$, let $O$ and $I$ denote the circumcenter and incenter, respectively. A circle with center $M$ is tangent to the legs $AC$ and $BC$ and to the circumcircle of $\triangle{ABC}$. What is the area of $\triangle{MOI}$? | 3.4 |
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 |
Given: $\because 4 \lt 7 \lt 9$, $\therefore 2 \lt \sqrt{7} \lt 3$, $\therefore$ the integer part of $\sqrt{7}$ is $2$, and the decimal part is $\sqrt{7}-2$. The integer part of $\sqrt{51}$ is ______, and the decimal part of $9-\sqrt{51}$ is ______. | 8-\sqrt{51} |
For how many integer values of $n$ between 1 and 180 inclusive does the decimal representation of $\frac{n}{180}$ terminate? | 20 |
In the fourth grade, there are 20 boys and 26 girls. The percentage of the number of boys to the number of girls is %. | 76.9 |
The tower function of twos is defined recursively as follows: $T(1) = 2$ and $T(n + 1) = 2^{T(n)}$ for $n\ge1$. Let $A = (T(2009))^{T(2009)}$ and $B = (T(2009))^A$. What is the largest integer $k$ for which $\underbrace{\log_2\log_2\log_2\ldots\log_2B}_{k\text{ times}}$ is defined? | 2010 |
Find the area of a triangle with side lengths 8, 9, and 9. | 4\sqrt{65} |
Ahmed and Emily are having a contest to see who can get the best grade in the class. There have been 9 assignments and Ahmed has a 91 in the class. Emily has a 92. The final assignment is worth the same amount as all the other assignments. Emily got a 90 on the final assignment. What is the minimum grade Ahmed needs to... | Ahmed has scored 819 total points in the class thus far because 9 x 91 = <<9*91=819>>819
Emily had scored 828 total points before the final assignments because 9 x 92 = <<9*92=828>>828
She scored 918 total points after the final assignment because 828 + 90 = <<828+90=918>>918
Ahmed needs to score a 99 to tie Emily for ... |
Given that the prime factorization of a positive integer \( A \) can be written as \( A = 2^{\alpha} \times 3^{\beta} \times 5^{\gamma} \), where \( \alpha, \beta, \gamma \) are natural numbers. If half of \( A \) is a perfect square, one-third of \( A \) is a perfect cube, and one-fifth of \( A \) is a fifth power of ... | 31 |
The complex number $(3 \operatorname{cis} 18^\circ)(-2\operatorname{cis} 37^\circ)$ is expressed in polar form as $r \operatorname{cis} \theta,$ where $r > 0$ and $0^\circ \le \theta < 360^\circ.$ Enter the ordered pair $(r, \theta).$ | (6,235^\circ) |
$(1)$ Given the function $f(x) = |x+1| + |2x-4|$, find the solution to $f(x) \geq 6$;<br/>$(2)$ Given positive real numbers $a$, $b$, $c$ satisfying $a+2b+4c=8$, find the minimum value of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$. | \frac{11+6\sqrt{2}}{8} |
In $\triangle ABC$, $E$ is the midpoint of side $BC$ and $D$ is on side $AC$.
If the length of $AC$ is $1$ and $\measuredangle BAC = 60^\circ, \measuredangle ABC = 100^\circ, \measuredangle ACB = 20^\circ$ and
$\measuredangle DEC = 80^\circ$, then the area of $\triangle ABC$ plus twice the area of $\triangle CDE$ equ... | \frac{\sqrt{3}}{8} |
We call a natural number \( b \) lucky if for any natural \( a \) such that \( a^{5} \) is divisible by \( b^{2} \), the number \( a^{2} \) is divisible by \( b \).
Find the number of lucky natural numbers less than 2010. | 1961 |
In a jumbo bag of bows, $\frac{1}{5}$ are red, $\frac{1}{2}$ are blue, $\frac{1}{10}$ are green and the remaining 30 are white. How many of the bows are green? | 15 |
Darla needs to pay $4/watt of electricity for 300 watts of electricity, plus a $150 late fee. How much does she pay in total? | First find the total cost of the electricity: $4/watt * 300 watts = $<<4*300=1200>>1200
Then add the late fee: $1200 + $150 = $<<1200+150=1350>>1350
#### 1350 |
As shown in the diagram, circles \( \odot O_{1} \) and \( \odot O_{2} \) are externally tangent. The line segment \( O_{1}O_{2} \) intersects \( \odot O_{1} \) at points \( A \) and \( B \), and intersects \( \odot O_{2} \) at points \( C \) and \( D \). Circle \( \odot O_{3} \) is internally tangent to \( \odot O_{1} ... | 1.2 |
The polynomial $P$ is a quadratic with integer coefficients. For every positive integer $n$ , the integers $P(n)$ and $P(P(n))$ are relatively prime to $n$ . If $P(3) = 89$ , what is the value of $P(10)$ ? | 859 |
A jug is filled with 5 liters of water and a bucket is filled with 4 jugs. How many liters of water are contained in 2 buckets? | 4 jugs contain 4*5 = <<4*5=20>>20 liters of water.
2 bucket contains 20*2 = <<20*2=40>>40 liters of water.
#### 40 |
A square is divided into 2016 triangles, with no vertex of any triangle lying on the sides or inside any other triangle. The sides of the square are sides of some of the triangles in the division. How many total points, which are the vertices of the triangles, are located inside the square? | 1007 |
Luna, the poodle, is supposed to eat 2 cups of kibble every day. But Luna's master, Mary, and her husband, Frank, sometimes feed Luna too much kibble. One day, starting with a new, 12-cup bag of kibble, Mary gave Luna 1 cup of kibble in the morning and 1 cup of kibble in the evening, But on the same day, Frank also g... | Mary fed Luna 1+1=2 cups of kibble.
Frank fed Luna 1-cup plus twice 1-cup, or 1+2=3 cups of kibble.
In total, they fed Luna 2+3=<<2+3=5>>5 cups of kibble.
Thus, if the new bag held 12 cups of kibble, the next morning, 12-5=<<12-5=7>>7 cups of kibble remained in the bag.
#### 7 |
A circle with a radius of 3 units has its center at $(0, 0)$. Another circle with a radius of 5 units has its center at $(12, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. Determine the value of $x$. Express your answer as a common fraction. | \frac{9}{2} |
Given a cube \( A B C D A_{1} B_{1} C_{1} D_{1} \), \( M \) is the center of the face \( A B B_{1} A_{1} \), \( N \) is a point on the edge \( B_{1} C_{1} \), \( L \) is the midpoint of \( A_{1} B_{1} \); \( K \) is the foot of the perpendicular dropped from \( N \) to \( BC_{1} \). In what ratio does point \( N \) div... | \sqrt{2} + 1 |
On the refrigerator, MATHEMATICS is spelled out with $11$ magnets, one letter per magnet. Two vowels and four consonants fall off and are put away in a bag. If the T's, M's, and A's are indistinguishable, how many distinct possible collections of letters could be put in the bag? | 72 |
Use the bisection method to find an approximate zero of the function $f(x) = \log x + x - 3$, given that approximate solutions (accurate to 0.1) are $\log 2.5 \approx 0.398$, $\log 2.75 \approx 0.439$, and $\log 2.5625 \approx 0.409$. | 2.6 |
Find all odd positive integers $n>1$ such that there is a permutation $a_{1}, a_{2}, \ldots, a_{n}$ of the numbers $1,2, \ldots, n$, where $n$ divides one of the numbers $a_{k}^{2}-a_{k+1}-1$ and $a_{k}^{2}-a_{k+1}+1$ for each $k, 1 \leq k \leq n$ (we assume $a_{n+1}=a_{1}$ ). | n=3 |
Linda is building a new hotel with two wings. The first wing has 9 floors and each floor has 6 halls each with 32 rooms. The second wing has 7 floors each with 9 halls with 40 rooms each. How many rooms are in the hotel total? | First find the number of rooms per floor in the first wing: 32 rooms/hall * 6 halls/floor = <<32*6=192>>192 rooms/floor
Then find the total number of rooms in the first wing by multiplying that number by the number of floors: 192 rooms/floor * 9 floors = <<192*9=1728>>1728 rooms
Then find the number of rooms per floor ... |
Calculate $3 \cdot 7^{-1} + 9 \cdot 13^{-1} \pmod{60}$.
Express your answer as an integer from $0$ to $59$, inclusive. | 42 |
When $\sqrt[3]{2700}$ is simplified, the result is $a\sqrt[3]{b}$, where $a$ and $b$ are positive integers and $b$ is as small as possible. What is $a+b$? | 103 |
In a Cartesian coordinate system, the "rectangular distance" between points $P\left(x_{1}, y_{1}\right)$ and $Q\left(x_{2}, y_{2}\right)$ is defined as $d(P, Q) = \left|x_{1}-x_{2}\right| + \left|y_{1}-y_{2}\right|$. If the "rectangular distance" from point $C(x, y)$ to points $A(1,3)$ and $B(6,9)$ is equal, where real... | 5(\sqrt{2} + 1) |
The geometric series $a+ar+ar^2+\cdots$ has a sum of $12$, and the terms involving odd powers of $r$ have a sum of $5.$ What is $r$? | \frac{5}{7} |
Xiao Wang loves mathematics and chose the six numbers $6$, $1$, $8$, $3$, $3$, $9$ to set as his phone's startup password. If the two $3$s are not adjacent, calculate the number of different passwords Xiao Wang can set. | 240 |
Three people, A, B, and C, stand on a staircase with 7 steps. If each step can accommodate at most 2 people, and the positions of people on the same step are not distinguished, then the number of different ways they can stand is. | 336 |
Find the greatest common divisor of $5616$ and $11609$. | 13 |
The equation
$$
(x-1) \times \ldots \times(x-2016) = (x-1) \times \ldots \times(x-2016)
$$
is written on the board. We want to erase certain linear factors so that the remaining equation has no real solutions. Determine the smallest number of linear factors that need to be erased to achieve this objective. | 2016 |
If $10$ divides the number $1\cdot2^1+2\cdot2^2+3\cdot2^3+\dots+n\cdot2^n$ , what is the least integer $n\geq 2012$ ? | 2014 |
Define $E(a,b,c) = a \cdot b^2 + c$. What value of $a$ is the solution to the equation $E(a,4,5) = E(a,6,7)$? | -\frac{1}{10} |
Find the smallest positive integer \( n \) such that every \( n \)-element subset of \( S = \{1, 2, \ldots, 150\} \) contains 4 numbers that are pairwise coprime (it is known that there are 35 prime numbers in \( S \)). | 111 |
Consider the function $g(x)=3x-4$. For what value of $a$ is $g(a)=0$? | \frac{4}{3} |
The average of the seven numbers in a list is 62. The average of the first four numbers is 58. What is the average of the last three numbers? | 67.\overline{3} |
In triangle $ABC$, angle $C$ is a right angle and the altitude from $C$ meets $\overline{AB}$ at $D$. The lengths of the sides of $\triangle ABC$ are integers, $BD=29^2$, and $\sin B = p/q$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | 17 |
Fern is checking IDs to get into an R-rated movie. She denied 20% of the 120 kids from Riverside High, 70% of the 90 kids from West Side High, and half the 50 kids from Mountaintop High. How many kids got into the movie? | First find how many kids from Riverside High are rejected: 20% * 120 kids = <<20*.01*120=24>>24 kids
Then find how many kids from West Side High are rejected: 70% * 90 kids = <<70*.01*90=63>>63 kids
Then find how many kids from Mountaintop High are rejected: 50 kids / 2 = <<50/2=25>>25 kids
Then add the number of kids ... |
Evaluate the expression:
\[4(1+4(1+4(1+4(1+4(1+4(1+4(1+4(1))))))))\] | 87380 |
Given the function $f(x)=\cos(2x+\varphi), |\varphi| \leqslant \frac{\pi}{2}$, if $f\left( \frac{8\pi}{3}-x \right)=-f(x)$, determine the horizontal shift required to obtain the graph of $y=\sin 2x$ from the graph of $y=f(x)$. | \frac{\pi}{6} |
There are 20 cards, each with a number from 1 to 20. These cards are placed in a box, and 4 people each draw one card without replacement. The two people who draw the smaller numbers form one group, and the two people who draw the larger numbers form another group. If two people draw the numbers 5 and 14, what is the p... | 7/51 |
A fair coin is flipped $8$ times. What is the probability that at least $6$ consecutive flips come up heads? | \frac{17}{256} |
Cooper makes 7 apple pies a day. He does this for 12 days. Ashley then eats 50 of his pies. How many apple pies remain with Cooper? | The number of pies Cooper makes in 12 days is 7 pies/day * 12 days = <<7*12=84>>84 apple pies.
After Ashley eats 50 of them, there are 84 pies – 50 pies = <<84-50=34>>34 apple pies.
#### 34 |
Three friends have a total of 6 identical pencils, and each one has at least one pencil. In how many ways can this happen? | 10 |
There are three machines in a factory. Machine A can put caps on 12 bottles in 1 minute. Machine B can put caps to 2 fewer bottles than Machine A. Machine C can put caps to 5 more bottles than Machine B. How many bottles can those three machines put caps on in 10 minutes? | Machine A can put caps on 12 x 10 = <<12*10=120>>120 bottles in 10 minutes.
Machine B can put caps on 12 - 2 = <<12-2=10>>10 bottles in 1 minute.
So in 10 minutes, machine B can put caps on 10 x 10 = <<10*10=100>>100 bottles.
Machine c can put caps on 10 + 5 = <<10+5=15>>15 bottles in 1 minute.
So in 10 minutes, machin... |
In a certain city, the rules for selecting license plate numbers online are as follows: The last five characters of the plate must include two English letters (with the letters "I" and "O" not allowed), and the last character must be a number. How many possible combinations meet these requirements? | 3456000 |
In a different lemonade recipe, the ratio of water to lemon juice is $5$ parts water to $2$ parts lemon juice. If I decide to make $2$ gallons of lemonade, how many quarts of water are needed? Note: there are four quarts in a gallon. | \frac{40}{7} |
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