problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
A function \( f: \{a, b, c, d\} \rightarrow \{1, 2, 3\} \) is given. If \( 10 < f(a) \cdot f(b) \) and \( f(c) \cdot f(d) < 20 \), how many such mappings exist? | 25 |
At Central Middle School the $108$ students who take the AMC 8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of $15$ cookies, lists these items:
$\bullet$ $1\frac{1}{2}$ cups of flou... | 11 |
What is the sum of all integer values $n$ for which $\binom{20}{n}+\binom{20}{10}=\binom{21}{11}$? | 20 |
The fourth term of a geometric sequence is 512, and the 9th term is 8. Determine the positive, real value for the 6th term. | 128 |
For any positive integer $n$, we define the integer $P(n)$ by :
$P(n)=n(n+1)(2n+1)(3n+1)...(16n+1)$.
Find the greatest common divisor of the integers $P(1)$, $P(2)$, $P(3),...,P(2016)$. | 510510 |
Given the function $f(x)=(ax^{2}+bx+c)e^{x}$ $(a > 0)$, the derivative $y=f′(x)$ has two zeros at $-3$ and $0$.
(Ⅰ) Determine the intervals of monotonicity for $f(x)$.
(Ⅱ) If the minimum value of $f(x)$ is $-1$, find the maximum value of $f(x)$. | \dfrac {5}{e^{3}} |
Marcus wants to buy a new pair of shoes. He decided to pay not more than $130 for them. He found a pair for $120, on which he got a discount of 30%. How much money will he manage to save by buying these shoes and not spending the assumed maximum amount? | The discount on the shoes Marcus found is 120 * 30/100 = $<<120*30/100=36>>36.
That means he only had to pay 120 - 36 = $<<120-36=84>>84.
So Marcus was able to save 130 - 84 = $<<130-84=46>>46.
#### 46 |
A rectangular garden 60 feet long and 15 feet wide is enclosed by a fence. To utilize the same fence but change the shape, the garden is altered to an equilateral triangle. By how many square feet does this change the area of the garden? | 182.53 |
In a far-off land three fish can be traded for two loaves of bread and a loaf of bread can be traded for four bags of rice. How many bags of rice is one fish worth? | 2\frac{2}{3} |
Let $m>1$ be an integer. Find the smallest positive integer $n$, such that for any integers $a_1,a_2,\ldots ,a_n; b_1,b_2,\ldots ,b_n$ there exists integers $x_1,x_2,\ldots ,x_n$ satisfying the following two conditions:
i) There exists $i\in \{1,2,\ldots ,n\}$ such that $x_i$ and $m$ are coprime
ii) $\sum^n_{i=1} a_... | 2\omega(m) + 1 |
If $x^{2y}=16$ and $x = 16$, what is the value of $y$? Express your answer as a common fraction. | \frac{1}{4} |
The older brother and the younger brother each bought several apples. The older brother said to the younger brother, "If I give you one apple, we will have the same number of apples." The younger brother thought for a moment and said to the older brother, "If I give you one apple, the number of apples you have will be ... | 12 |
1. Given that ${(3x-2)^{6}}={a_{0}}+{a_{1}}(2x-1)+{a_{2}}{(2x-1)^{2}}+ \cdots +{a_{6}}{(2x-1)^{6}}$, find the value of $\dfrac{{a_{1}}+{a_{3}}+{a_{5}}}{{a_{0}}+{a_{2}}+{a_{4}}+{a_{6}}}$.
2. A group of 6 volunteers is to be divided into 4 teams, with 2 teams of 2 people and the other 2 teams of 1 person each, to be sent... | \dfrac{ \sqrt{2}}{2} |
Caleb and his dad went fishing at the lake. Caleb caught 2 trouts and his dad caught three times as much as Caleb. How many more trouts did his dad catch compared to Caleb? | Caleb’s dad caught 2 x 3 = <<2*3=6>>6 trouts.
His dad caught 6 - 2 = <<6-2=4>>4 more trouts than Caleb.
#### 4 |
If $x=3$, what is the value of $-(5x - 6x)$? | 3 |
If the integer $a$ makes the inequality system about $x$ $\left\{\begin{array}{l}{\frac{x+1}{3}≤\frac{2x+5}{9}}\\{\frac{x-a}{2}＞\frac{x-a+1}{3}}\end{array}\right.$ have at least one integer solution, and makes the solution of the system of equations about $x$ and $y$ $\left\{\begin{array}{l}ax+2y=-4\\ x+y=4\end{array}\... | -16 |
Cameron writes down the smallest positive multiple of 30 that is a perfect square, the smallest positive multiple of 30 that is a perfect cube, and all the multiples of 30 between them. How many integers are in Cameron's list? | 871 |
Given an arithmetic sequence $\{a_n\}$, if $d < 0$, $T_n$ is the sum of the first $n$ terms of $\{a_n\}$, and $T_3=15$, also $a_1+1, a_2+3, a_3+9$ form a geometric sequence with common ratio $q$, calculate the value of $q$. | \frac{1}{2} |
If $\displaystyle\frac{q}{r} = 9$, $\displaystyle\frac{s}{r} = 6$, and $\displaystyle \frac{s}{t} = \frac{1}{2}$, then what is $\displaystyle\frac{t}{q}$? | \frac{4}{3} |
For $\mathbf{v} = \begin{pmatrix} 1 \\ y \end{pmatrix}$ and $\mathbf{w} = \begin{pmatrix} 9 \\ 3 \end{pmatrix}$,
\[\text{proj}_{\mathbf{w}} \mathbf{v} = \begin{pmatrix} -6 \\ -2 \end{pmatrix}.\]Find $y$. | -23 |
A plane flies between 4 cities; A, B, C and D. Passengers board and alight at each airport in every city when it departs and lands, respectively. The distance between city A and city B is 100 miles. The distance between city B and city C is 50 miles more than the distance between city A and city B. The distance betw... | Since the distance between city A and B is 100 miles, and city B and C are 50 more miles apart than city A and B, then city B and C are 100+50 = <<100+50=150>>150 miles apart.
The plane's total flying distance between cities A and C is 100+150= <<100+150=250>>250 miles.
The distance between city C and city D is twice t... |
$A B C$ is a triangle with $A B=15, B C=14$, and $C A=13$. The altitude from $A$ to $B C$ is extended to meet the circumcircle of $A B C$ at $D$. Find $A D$. | \frac{63}{4} |
Evaluate $3x^y + 4y^x$ when $x=2$ and $y=3$. | 60 |
In the polygon shown, each side is perpendicular to its adjacent sides, and all 28 of the sides are congruent. The perimeter of the polygon is 56. Find the area of the polygon.
[asy]
unitsize(0.5 cm);
draw((3,0)--(4,0)--(4,1)--(5,1)--(5,2)--(6,2)--(6,3)--(7,3)--(7,4)--(6,4)--(6,5)--(5,5)--(5,6)--(4,6)--(4,7)--(3,7)... | 100 |
In a game of 27 cards, each card has three characteristics: shape (square, circle, or triangle), color (blue, yellow, or red), and pattern (solid, dotted, or hatched). All cards are different. A combination of three cards is called complementary if, for each of the three characteristics, the three cards are either all ... | 117 |
Among the non-empty subsets of the set \( A = \{1, 2, \cdots, 10\} \), how many subsets have the sum of their elements being a multiple of 10? | 103 |
Given $f(x)= \frac{\ln x+2^{x}}{x^{2}}$, find $f'(1)=$ ___. | 2\ln 2 - 3 |
Xiao Ming and Xiao Hua are counting picture cards in a box together. Xiao Ming is faster, being able to count 6 cards in the same time it takes Xiao Hua to count 4 cards. When Xiao Hua reached 48 cards, he forgot how many cards he had counted and had to start over. When he counted to 112 cards, there was only 1 card le... | 169 |
Let \( a_1, a_2, \dots \) be a sequence of positive real numbers such that
\[ a_n = 7a_{n-1} - n \]for all \( n > 1 \). Find the smallest possible value of \( a_1 \). | \frac{13}{36} |
What is the area enclosed by the graph of $|x| + |3y| + |x - y| = 20$? | \frac{200}{3} |
Paige bought some new stickers and wanted to share them with 3 of her friends. She decided to share a sheet of 100 space stickers and a sheet of 50 cat stickers equally among her 3 friends. How many stickers will she have left? | Paige has 100 space stickers that she is dividing among 3 friends, so 100 stickers / 3 friends = 33 stickers for each friend, with 1 remaining.
She also has 50 cat stickers that she is diving among 3 friends, so 50 stickers / 3 friends = 16 stickers for each friend, with 2 remaining.
So Paige will have 1 remaining spac... |
What is the greatest possible sum of the digits in the base-seven representation of a positive integer less than $2019$? | 22 |
If
\[\mathbf{A} = \begin{pmatrix} 1 & 3 \\ 2 & 1 \end{pmatrix},\]then compute $\det (\mathbf{A}^2 - 2 \mathbf{A}).$ | 25 |
Let $x,$ $y,$ $z$ be real numbers, all greater than 3, so that
\[\frac{(x + 2)^2}{y + z - 2} + \frac{(y + 4)^2}{z + x - 4} + \frac{(z + 6)^2}{x + y - 6} = 36.\]Enter the ordered triple $(x,y,z).$ | (10,8,6) |
Compute
\[
\left( 1 - \sin \frac {\pi}{8} \right) \left( 1 - \sin \frac {3\pi}{8} \right) \left( 1 - \sin \frac {5\pi}{8} \right) \left( 1 - \sin \frac {7\pi}{8} \right).
\] | \frac{1}{4} |
Mimi picked up 2 dozen seashells on the beach. Kyle found twice as many shells as Mimi and put them in his pocket. Leigh grabbed one-third of the shells that Kyle found. How many seashells did Leigh have? | Mimi has 2 x 12 = <<2*12=24>>24 sea shells.
Kyle has 24 x 2 = <<24*2=48>>48 sea shells.
Leigh has 48 / 3 = <<48/3=16>>16 sea shells.
#### 16 |
In the diagram, \(PQRS\) is a square with side length 8. Points \(T\) and \(U\) are on \(PS\) and \(QR\) respectively with \(QU = TS = 1\). The length of \(TU\) is closest to | 10 |
In the quadrilateral pyramid \(P-ABCD\), given that \(AB\) is parallel to \(CD\), \(AB\) is perpendicular to \(AD\), \(AB=4\), \(AD=2\sqrt{2}\), \(CD=2\), and \(PA\) is perpendicular to the plane \(ABCD\), with \(PA=4\). Let \(Q\) be a point on line segment \(PB\) such that the sine of the angle between line \(QC\) and... | 7/12 |
For a positive integer $n,$ let
\[G_n = 1^2 + \frac{1}{2^2} + \frac{1}{3^2} + \dots + \frac{1}{n^2}.\]Compute
\[\sum_{n = 1}^\infty \frac{1}{(n + 1) G_n G_{n + 1}}.\] | 1 - \frac{6}{\pi^2} |
If $x+y = 6$ and $x^2-y^2 = 12$, then what is $x-y$? | 2 |
Each block on the grid shown in the Figure is 1 unit by 1 unit. Suppose we wish to walk from $A$ to $B$ via a 7 unit path, but we have to stay on the grid -- no cutting across blocks. How many different paths can we take?[asy]size(3cm,3cm);int w=5;int h=4;int i;for (i=0; i<h; ++i){draw((0,i) -- (w-1,i));}for (i=0; i<... | 35 |
Let $a,$ $b,$ $c$ be the roots of the cubic polynomial $x^3 - x - 1 = 0.$ Find
\[a(b - c)^2 + b(c - a)^2 + c(a - b)^2.\] | -9 |
The ellipse whose equation is
\[\frac{x^2}{25} + \frac{y^2}{9} = 1\]is graphed below. The chord $\overline{AB}$ passes through a focus $F$ of the ellipse. If $AF = \frac{3}{2},$ then find $BF.$
[asy]
unitsize (0.6 cm);
pair A, B, F;
F = (4,0);
A = (35/8,3*sqrt(15)/8);
B = (55/16,-9*sqrt(15)/16);
draw(xscale(5)*ys... | \frac{9}{4} |
Find all integers $n$ satisfying $n \geq 2$ and \(\frac{\sigma(n)}{p(n)-1}=n\), in which \(\sigma(n)\) denotes the sum of all positive divisors of \(n\), and \(p(n)\) denotes the largest prime divisor of \(n\). | n=6 |
Brett has 24 more blue marbles than red marbles. He has 5 times as many blue marbles as red marbles. How many red marbles does he have? | Let x be the number of red marbles.
5*x=x+24
4*x=24
x=<<6=6>>6
#### 6 |
Thirty teams play in a league where each team plays every other team exactly once, and every game results in a win or loss with no ties. Each game is independent with a $50\%$ chance of either team winning. Determine the probability that no two teams end up with the same number of total victories, expressed as $\frac{p... | 409 |
On Tony's map, the distance from Saint John, NB to St. John's, NL is $21 \mathrm{~cm}$. The actual distance between these two cities is $1050 \mathrm{~km}$. What is the scale of Tony's map? | 1:5 000 000 |
Given that six students are to be seated in three rows of two seats each, with one seat reserved for a student council member who is Abby, calculate the probability that Abby and Bridget are seated next to each other in any row. | \frac{1}{5} |
Given that $a$, $b$, $c \in R^{+}$ and $a + b + c = 1$, find the maximum value of $\sqrt{4a + 1} + \sqrt{4b + 1} + \sqrt{4c + 1}$. | \sqrt{21} |
A rising number, such as $34689$, is a positive integer each digit of which is larger than each of the digits to its left. There are $\binom{9}{5} = 126$ five-digit rising numbers. When these numbers are arranged from smallest to largest, the $97^{\text{th}}$ number in the list does not contain the digit | 5 |
We want to set up an electric bell. The location of the bell is at a distance of $30 \mathrm{~m}$ from the ringing spot. The internal resistance of the bell is 2 ohms. We plan to use 2 Leclanché cells connected in series, each with an electromotive force of 1.5 Volts and an internal resistance of 1 ohm. What diameter ... | 0.63 |
A pirate finds three chests on the wrecked ship S.S. Triumph, recorded in base 7. The chests contain $3214_7$ dollars worth of silver, $1652_7$ dollars worth of precious stones, $2431_7$ dollars worth of pearls, and $654_7$ dollars worth of ancient coins. Calculate the total value of these treasures in base 10. | 3049 |
Find the coefficient of \(x^8\) in the polynomial expansion of \((1-x+2x^2)^5\). | 80 |
Josh has 100 feet of rope. He cuts the rope in half, and then takes one of the halves and cuts it in half again. He grabs one of the remaining pieces and cuts it into fifths. He's stuck holding one length of the rope he's most recently cut. How long is it? | John starts with 100ft of rope that he cuts in half, meaning he has 100/2= <<100/2=50>>50-foot sections of rope.
He then takes one of these sections and cuts it in half again, leaving him with two 50/2= <<50/2=25>>25-foot sections of rope.
He then takes one of these sections and cuts it into fifths, leaving him with 25... |
Let $x$ and $y$ be positive real numbers. Find the minimum value of
\[\left( x + \frac{1}{y} \right) \left( x + \frac{1}{y} - 2018 \right) + \left( y + \frac{1}{x} \right) \left( y + \frac{1}{x} - 2018 \right).\] | -2036162 |
How many positive factors of 72 are perfect cubes? | 2 |
A square is inscribed in a circle of radius 1. Find the perimeter of the square. | 4 \sqrt{2} |
Find the sum of all real solutions to the equation \[\sqrt{x} + \sqrt{\frac{4}{x}} + \sqrt{x + \frac{4}{x}} = 6.\] | \frac{64}{9} |
In a certain country, there are 200 cities. The Ministry of Aviation requires that each pair of cities be connected by a bidirectional flight operated by exactly one airline, and that it should be possible to travel from any city to any other city using the flights of each airline (possibly with layovers). What is the ... | 100 |
If $\Diamond4_7=\Diamond1_{8}$ and $\Diamond$ represents a digit, solve for $\Diamond$. | 3 |
Given that the plane vectors $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ satisfy $|\boldsymbol{\alpha} + 2\boldsymbol{\beta}| = 3$ and $|2\boldsymbol{\alpha} + 3\boldsymbol{\beta}| = 4$, find the minimum value of $\boldsymbol{\alpha} \cdot \boldsymbol{\beta}$. | -170 |
Let set $\mathcal{A}$ be a 90-element subset of $\{1,2,3,\ldots,100\},$ and let $S$ be the sum of the elements of $\mathcal{A}.$ Find the number of possible values of $S.$
| 901 |
To make lemonade, I use a ratio of $7$ parts water to $1$ part lemon juice. If I want to make a gallon of lemonade, and there are four quarts in a gallon, how many quarts of water do I need? Write your answer as a proper or improper fraction. | \frac{7}{2} |
Simplify $2w+4w+6w+8w+10w+12$. | 30w+12 |
A racer departs from point \( A \) along the highway, maintaining a constant speed of \( a \) km/h. After 30 minutes, a second racer starts from the same point with a constant speed of \( 1.25a \) km/h. How many minutes after the start of the first racer was a third racer sent from the same point, given that the third ... | 50 |
Let $A$, $M$, and $C$ be nonnegative integers such that $A+M+C=12$. What is the maximum value of \[A\cdot M\cdot C+A\cdot M+M\cdot
C+C\cdot A?\] | 112 |
A circle with center $A$ and radius three inches is tangent at $C$ to a circle with center $B$, as shown. If point $B$ is on the small circle, what is the area of the shaded region? Express your answer in terms of $\pi$.
[asy]
filldraw(circle((0,0),6),gray,linewidth(2));
filldraw(circle(3dir(-30),3),white,linewidth(2)... | 27\pi |
Find the smallest integer $k > 1$ for which $n^k-n$ is a multiple of $2010$ for every integer positive $n$ . | 133 |
In the diagram, $\triangle ABE$, $\triangle BCE$ and $\triangle CDE$ are right-angled, with $\angle AEB=\angle BEC = \angle CED = 60^\circ$, and $AE=24$. [asy]
pair A, B, C, D, E;
A=(0,20.785);
B=(0,0);
C=(9,-5.196);
D=(13.5,-2.598);
E=(12,0);
draw(A--B--C--D--E--A);
draw(B--E);
draw(C--E);
label("A", A, N);
label("B",... | 27+21\sqrt{3} |
An integer is called parity-monotonic if its decimal representation $a_{1}a_{2}a_{3}\cdots a_{k}$ satisfies $a_{i}<a_{i+1}$ if $a_{i}$ is odd, and $a_{i}>a_{i+1}$ if $a_{i}$ is even. How many four-digit parity-monotonic integers are there? | 640 |
What is the arithmetic mean of $\frac{2}{5}$ and $\frac{4}{7}$ ? Express your answer as a common fraction. | \frac{17}{35} |
We flip a fair coin 10 times. What is the probability that we get heads in at least 8 of the 10 flips? | \dfrac{7}{128} |
Let $\otimes$ be a binary operation that takes two positive real numbers and returns a positive real number. Suppose further that $\otimes$ is continuous, commutative $(a \otimes b=b \otimes a)$, distributive across multiplication $(a \otimes(b c)=(a \otimes b)(a \otimes c))$, and that $2 \otimes 2=4$. Solve the equati... | \sqrt{2} |
Amy, Jeremy, and Chris have a combined age of 132. Amy is 1/3 the age of Jeremy, and Chris is twice as old as Amy. How old is Jeremy? | Let x represent the age of Jeremy
Amy:(1/3)x
Chris:2(1/3)x=(2/3)x
Total:x+(1/3)x+(2/3)x=132
2x=132
x=66 years old
#### 66 |
In a sequence of coin tosses, one can keep a record of instances in which a tail is immediately followed by a head, a head is immediately followed by a head, and etc. We denote these by TH, HH, and etc. For example, in the sequence TTTHHTHTTTHHTTH of 15 coin tosses we observe that there are two HH, three HT, four TH, a... | 560 |
Compute the number of ordered triples of integers $(a,b,c)$ between $1$ and $12$ , inclusive, such that, if $$ q=a+\frac{1}{b}-\frac{1}{b+\frac{1}{c}}, $$ then $q$ is a positive rational number and, when $q$ is written in lowest terms, the numerator is divisible by $13$ .
*Proposed by Ankit Bisain* | 132 |
Dean scored a total of 252 points in 28 basketball games. Ruth played 10 fewer games than Dean. Her scoring average was 0.5 points per game higher than Dean's scoring average. How many points, in total, did Ruth score? | 171 |
Jasmine bought 4 pounds of coffee beans and 2 gallons of milk. A pound of coffee beans costs $2.50 and a gallon of milk costs $3.50. How much will Jasmine pay in all? | Four pounds of coffee cost $2.50 x 4 = $<<4*2.5=10>>10.
Two gallons of milk cost $3.50 x 2 = $<<3.5*2=7>>7.
So, Jasmine will pay $10 + $7 = $<<10+7=17>>17.
#### 17 |
An auctioneer raises the price of an item he is auctioning by $5 every time someone new bids on it. Two people enter a bidding war on a desk and the price goes from $15 to $65 before the desk is sold. How many times did each person bid on the desk? | The desk price was raised 65 - 15 = $<<65-15=50>>50.
At $5 per bid, there were 50 / 5 = <<50/5=10>>10 bids on the desk.
There were two bidders, so each bid 10 / 2 = <<10/2=5>>5 times on the desk.
#### 5 |
Let $P(x)$ be a polynomial with degree 2008 and leading coefficient 1 such that $P(0)=2007, P(1)=2006, P(2)=2005, \ldots, P(2007)=0$. Determine the value of $P(2008)$. You may use factorials in your answer. | 2008!-1 |
An equilateral triangle is originally painted black. Each time the triangle is changed, the middle fourth of each black triangle turns white. After five changes, what fractional part of the original area of the black triangle remains black? | \frac{243}{1024} |
Given \( |z|=2 \) and \( u=\left|z^{2}-z+1\right| \), find the minimum value of \( u \) where \( z \in \mathbf{C} \). | \frac{3}{2} \sqrt{3} |
Lily has 5 lottery tickets to sell. She sells the first ticket for $1. She then sells each successive ticket for a dollar more than the previous ticket. She plans to keep a $4 profit and give the remaining money as the prize. How much money will the winner of the lottery receive? | The second ticket sold will cost $1 + $1 = $<<1+1=2>>2.
The third ticket sold will cost $2 + $1 = $<<2+1=3>>3.
The fourth ticket sold will cost $3 + $1 = $<<3+1=4>>4.
The fifth ticket sold will cost $4 + $1 = $<<4+1=5>>5.
The total money collected is $1 + $2 + $3 + $4 + $5 = $<<1+2+3+4+5=15>>15.
After taking profit, th... |
Given quadrilateral $ABCD,$ side $\overline{AB}$ is extended past $B$ to $A'$ so that $A'B = AB.$ Points $B',$ $C',$ and $D'$ are similarly constructed.
[asy]
unitsize(1 cm);
pair[] A, B, C, D;
A[0] = (0,0);
B[0] = (2,0);
C[0] = (1.5,2);
D[0] = (0.2,1.5);
A[1] = 2*B[0] - A[0];
B[1] = 2*C[0] - B[0];
C[1] = 2*D[0] - ... | \left( \frac{1}{15}, \frac{2}{15}, \frac{4}{15}, \frac{8}{15} \right) |
A block of iron solidifies from molten iron, and its volume reduces by $\frac{1}{34}$. Then, if this block of iron melts back into molten iron (with no loss in volume), by how much does its volume increase? | \frac{1}{33} |
A sports conference has 14 teams in two divisions of 7. How many games are in a complete season for the conference if each team must play every other team in its own division twice and every team in the other division once? | 133 |
Calculate the definite integral:
$$
\int_{0}^{2 \pi} \sin ^{2}\left(\frac{x}{4}\right) \cos ^{6}\left(\frac{x}{4}\right) d x
$$ | \frac{5\pi}{64} |
Points $A$, $B$, $Q$, $D$, and $C$ lie on the circle shown and the measures of arcs $BQ$ and $QD$ are $42^\circ$ and $38^\circ$, respectively. Find the sum of the measures of angles $P$ and $Q$, in degrees.
[asy]
import graph;
unitsize(2 cm);
pair A, B, C, D, P, Q;
A = dir(160);
B = dir(45);
C = dir(190);
D = dir(... | 40^\circ |
Find the largest value of $c$ such that $1$ is in the range of $f(x)=x^2-5x+c$. | \frac{29}{4} |
A family went out to see a movie. The regular ticket costs $9 and the ticket for children is $2 less. They gave the cashier two $20 bills and they received a $1 change. How many children are there if there are 2 adults in the family? | They gave the cashier $20 x 2 = $<<20*2=40>>40.
Since a $1 change was returned, this means the tickets cost $40 - $1 = $<<40-1=39>>39.
Two adult tickets cost $9 x 2 = $<<9*2=18>>18.
So, $39 - $18 = $<<39-18=21>>21 was spent for the children's tickets.
Each child's ticket costs $9 - $2 = $<<9-2=7>>7.
Therefore, there ar... |
What is the sum of all the four-digit positive integers? | 49495500 |
A box contains $2$ pennies, $4$ nickels, and $6$ dimes. Six coins are drawn without replacement, with each coin having an equal probability of being chosen. What is the probability that the value of coins drawn is at least $50$ cents? | \frac{127}{924} |
Let $r$ and $s$ be positive integers such that\[\frac{5}{11} < \frac{r}{s} < \frac{4}{9}\]and $s$ is as small as possible. What is $s - r$? | 11 |
Determine the time the copy machine will finish all the paperwork if it starts at 9:00 AM and completes half the paperwork by 12:30 PM. | 4:00 |
How many distinct arrangements of the letters in the word "balloon" are there, considering the repeated 'l' and 'o'? | 1260 |
A palindromic number is a number that reads the same when the order of its digits is reversed. What is the difference between the largest and smallest five-digit palindromic numbers that are both multiples of 45? | 9090 |
Given that the equations of the two asymptotes of a hyperbola are $y = \pm \sqrt{2}x$ and it passes through the point $(3, -2\sqrt{3})$.
(1) Find the equation of the hyperbola;
(2) Let $F$ be the right focus of the hyperbola. A line with a slope angle of $60^{\circ}$ intersects the hyperbola at points $A$ and $B$. Find... | 16 \sqrt{3} |
If $x+\sqrt{81}=25$, what is the value of $x$? | 16 |
Joan wants to visit her family who live 480 miles away. If she drives at a rate of 60 mph and takes a lunch break taking 30 minutes, and 2 bathroom breaks taking 15 minutes each, how many hours did it take her to get there? | The driving time is 480/60= <<480/60=8>>8 hours
The time for breaks was 30+15+15=<<30+15+15=60>>60 minutes
So she spent 60/60=<<60/60=1>>1 hour for rest stops.
So it took her 8+1=<<8+1=9>>9 hours
#### 9 |
Simplify $5(3-i)+3i(5-i)$. | 18+10i |
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