problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Find the volume of the region in space defined by
\[|x + y + 2z| + |x + y - 2z| \le 12\]
and $x, y, z \ge 0$. | 54 |
A regular triangle $EFG$ with a side length of $a$ covers a square $ABCD$ with a side length of 1. Find the minimum value of $a$. | 1 + \frac{2}{\sqrt{3}} |
How many times do the graphs $r = 4 \cos \theta$ and $r = 8 \sin \theta$ intersect? | 2 |
Given positive integers \( n \) and \( m \), let \( A = \{1, 2, \cdots, n\} \) and define \( B_{n}^{m} = \left\{\left(a_{1}, a_{2}, \cdots, a_{m}\right) \mid a_{i} \in A, i=1,2, \cdots, m\} \right. \) satisfying:
1. \( \left|a_{i} - a_{i+1}\right| \neq n-1 \), for \( i = 1, 2, \cdots, m-1 \);
2. Among \( a_{1}, a_{2}, ... | 104 |
Three-quarters of the oil from a 4000-liter tank (that was initially full) was poured into a 20000-liter capacity tanker that already had 3000 liters of oil. How many more liters of oil would be needed to make the large tanker half-full? | Three-quarters of oil from the initially full 4000-liter tank would be 4000*(3/4)= <<4000*3/4=3000>>3000 liters
In addition to the 3000 liters that were already in the large tanker, there are 3000+3000 = <<3000+3000=6000>>6000 liters in it now.
The tanker would be filled to half capacity by (1/2)*20000 = <<(1/2)*20000=... |
The quadratic equation $x^2+mx+n=0$ has roots that are twice those of $x^2+px+m=0,$ and none of $m,$ $n,$ and $p$ is zero. What is the value of $n/p?$ | 8 |
Eight teams participated in a football tournament, and each team played exactly once against each other team. If a match was drawn then both teams received 1 point; if not then the winner of the match was awarded 3 points and the loser received no points. At the end of the tournament the total number of points gained b... | 17 |
Let $a=x^3-3x^2$, then the coefficient of the $x^2$ term in the expansion of $(a-x)^6$ is $\boxed{-192}$. | -192 |
Determine the number of angles between 0 and $2 \pi,$ other than integer multiples of $\frac{\pi}{2},$ such that $\sin \theta,$ $\cos \theta$, and $\tan \theta$ form a geometric sequence in some order. | 4 |
Find the sum $\sum_{d=1}^{2012}\left\lfloor\frac{2012}{d}\right\rfloor$. | 15612 |
Let \[f(x) = \left\{
\begin{array}{cl}
2x + 7 & \text{if } x < -2, \\
-x^2 - x + 1 & \text{if } x \ge -2.
\end{array}
\right.\]Find the sum of all values of $x$ such that $f(x) = -5.$ | -4 |
Let $a, b, c$ be not necessarily distinct integers between 1 and 2011, inclusive. Find the smallest possible value of $\frac{a b+c}{a+b+c}$. | $\frac{2}{3}$ |
The number 3003 is the only number known to appear eight times in Pascal's triangle, at positions $\binom{3003}{1},\binom{3003}{3002},\binom{a}{2},\binom{a}{a-2},\binom{15}{b},\binom{15}{15-b},\binom{14}{6},\binom{14}{8}$. Compute $a+b(15-b)$. | 128 |
Apollo pulls the sun across the sky every night. Once a month, his fiery chariot’s wheels need to be replaced. He trades golden apples to Hephaestus the blacksmith to get Hephaestus to make him new wheels. Hephaestus raised his rates halfway through the year and now demands twice as many golden apples as before. He cha... | Apollo had to pay 3 * 6 = <<3*6=18>>18 golden apples for the first six months.
For the second six months, he had to pay twice that amount, or 18 * 2 = <<18*2=36>>36 golden apples.
Thus, Apollo had to pay 18 + 36 = <<18+36=54>>54 golden apples for a year of chariot wheels.
#### 54 |
Given \(5^p + 5^3 = 140\), \(3^r + 21 = 48\), and \(4^s + 4^3 = 280\), find the product of \(p\), \(r\), and \(s\). | 18 |
Let $ABCDE$ be a convex pentagon with $AB \parallel CE, BC \parallel AD, AC \parallel DE, \angle ABC=120^\circ, AB=3, BC=5,$ and $DE = 15.$ Given that the ratio between the area of triangle $ABC$ and the area of triangle $EBD$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$ | 484 |
How many two-digit prime numbers have the property that both digits are also primes? | 4 |
In the Cartesian coordinate plane $(xOy)$, the focus of the parabola $y^{2}=2x$ is $F$. Let $M$ be a moving point on the parabola, then the maximum value of $\frac{MO}{MF}$ is _______. | \frac{2\sqrt{3}}{3} |
In this diagram, both polygons are regular. What is the value, in degrees, of the sum of the measures of angles $ABC$ and $ABD$?
[asy]
draw(10dir(0)--10dir(60)--10dir(120)--10dir(180)--10dir(240)--10dir(300)--10dir(360)--cycle,linewidth(2));
draw(10dir(240)--10dir(300)--10dir(300)+(0,-10)--10dir(240)+(0,-10)--10dir(24... | 210 |
Five cards labeled $1,3,5,7,9$ are laid in a row in that order, forming the five-digit number 13579 when read from left to right. A swap consists of picking two distinct cards, and then swapping them. After three swaps, the cards form a new five-digit number $n$ when read from left to right. Compute the expected value ... | 50308 |
Find the value of cos $$\frac {π}{11}$$cos $$\frac {2π}{11}$$cos $$\frac {3π}{11}$$cos $$\frac {4π}{11}$$cos $$\frac {5π}{11}$$\=\_\_\_\_\_\_． | \frac {1}{32} |
Kristy baked cookies because her friends are coming over to her house. She ate 2 of them and gave her brother 1 cookie. Her first friend arrived and took 3 cookies. The second and third friends to arrive took 5 cookies each. If there are 6 cookies left, how many cookies did Kristy bake? | There were 6 + 5 = <<6+5=11>>11 cookies left before the third friend took 5.
There were 11 + 5 = <<11+5=16>>16 cookies left before the second friend took 5.
There were 16 + 3 = <<16+3=19>>19 cookies left before the first friend took 3.
There were 19 + 1 = <<19+1=20>>20 cookies left before Kristy gave 1 cookie to her br... |
There exist vectors $\mathbf{a}$ and $\mathbf{b}$ such that
\[\mathbf{a} + \mathbf{b} = \begin{pmatrix} 6 \\ -3 \\ -6 \end{pmatrix},\]where $\mathbf{a}$ is parallel to $\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix},$ and $\mathbf{b}$ is orthogonal to $\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}.$ Find $\mathbf{b}.$ | \begin{pmatrix} 7 \\ -2 \\ -5 \end{pmatrix} |
Tatuya, Ivanna, and Dorothy took a quiz together. Tatuya scored twice as much as Ivanna, and Ivanna scored 3/5 times as many marks as Dorothy. If Dorothy scored 90 marks, calculate the average marks scored by the three. | Since Dorothy scored 90 marks, and Ivanna scored 3/5 times as many marks, Ivanna scored 3/5*90 = <<90*3/5=54>>54 marks.
Together, Dorothy and Ivanna scored a total of 54+90 = <<54+90=144>>144 marks.
If Tayuta scored twice as many marks as Ivanna, she scored 2*54 = <<2*54=108>>108 marks.
The total marks the three scored... |
The solution to the inequality
\[y = -x^2 + ax + b \le 0\]is $(-\infty,-3] \cup [5,\infty).$ Find the vertex of the parabola $y = -x^2 + ax + b.$ | (1,16) |
Each of 8 balls is randomly and independently painted either black or white with equal probability. Calculate the probability that every ball is different in color from at least half of the other 7 balls. | \frac{35}{128} |
How many positive real solutions are there to $x^{10}+7x^9+14x^8+1729x^7-1379x^6=0$? | 1 |
The Fibonacci numbers are defined by $F_{1}=F_{2}=1$, and $F_{n}=F_{n-1}+F_{n-2}$ for $n \geq 3$. If the number $$ \frac{F_{2003}}{F_{2002}}-\frac{F_{2004}}{F_{2003}} $$ is written as a fraction in lowest terms, what is the numerator? | 1 |
Joe played catch with Derek and Tammy. He caught the ball 23 times. Derek made four less than double the catches Joe did. Tammy caught the ball sixteen more than a third of the times Derek did. How many times did Tammy catch the ball? | Derek caught the ball 2 * 23 - 4 = 46 - 4 = <<2*23-4=42>>42 times.
Tammy caught the ball 42 / 3 + 16 = 14 + 16 = <<42/3+16=30>>30 times.
#### 30 |
How many natural-number factors does $N$ have if $N = 2^4 \cdot 3^3 \cdot 5^2 \cdot 7^2$? | 180 |
Let $S$ be the set of integers $n > 1$ for which $\tfrac1n = 0.d_1d_2d_3d_4\ldots$, an infinite decimal that has the property that $d_i = d_{i+12}$ for all positive integers $i$. Given that $9901$ is prime, how many positive integers are in $S$? (The $d_i$ are digits.)
| 255 |
A tourist attraction estimates that the number of tourists $p(x)$ (in ten thousand people) from January 2013 onwards in the $x$-th month is approximately related to $x$ as follows: $p(x)=-3x^{2}+40x (x \in \mathbb{N}^{*}, 1 \leqslant x \leqslant 12)$. The per capita consumption $q(x)$ (in yuan) in the $x$-th month is a... | 3125 |
Let the complex number \( z = \cos \frac{2\pi}{13} + i \sin \frac{2\pi}{13} \). Find the value of \( \left(z^{-12} + z^{-11} + z^{-10}\right)\left(z^{3} + 1\right)\left(z^{6} + 1\right) \). | -1 |
$K$ takes $30$ minutes less time than $M$ to travel a distance of $30$ miles. $K$ travels $\frac {1}{3}$ mile per hour faster than $M$. If $x$ is $K$'s rate of speed in miles per hours, then $K$'s time for the distance is: | \frac{30}{x} |
The number of elements in a finite set $P$ is denoted as $\text{card}(P)$. It is known that $\text{card}(M) = 10$, $A \subseteq M$, $B \subseteq M$, $A \cap B = \emptyset$, and $\text{card}(A) = 2$, $\text{card}(B) = 3$. If the set $X$ satisfies $A \subseteq X \subseteq M$, then the number of such sets $X$ is ____. (An... | 256 |
In the Cartesian coordinate system, the center of circle $C$ is at $(2,0)$, and its radius is $\sqrt{2}$. Establish a polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis. The parametric equation of line $l$ is:
$$
\begin{cases}
x=-t \\
y=1+t
\end{cases} \quad (t \tex... | 3\sqrt{2} |
Keaton climbed a 30 feet ladder twenty times while working at the construction site. Reece, also working at the same site, climbed a ladder 4 feet shorter than Keaton's ladder 15 times. What's the total length of the ladders that both workers climbed in inches? | If Keaton climbed his ladder twenty times, he covered a length of 30*20 = <<30*20=600>>600 feet.
Reece's ladder, 4 feet shorter than Keaton's ladder, was 30-4 = <<30-4=26>>26 feet long.
If Reece climbed the ladder 15 times, he covered a length of 15*26 = <<15*26=390>>390 feet.
Keaton and Reece's total length covered wh... |
Barbara has 9 stuffed animals. Trish has two times as many stuffed animals as Barbara. They planned to sell their stuffed animals and donate all the money to their class funds. Barbara will sell her stuffed animals for $2 each while Trish will sell them for $1.50 each. How much money will they donate to their class fun... | Barbara will be able to sell all her stuffed animals for 9 x $2 = $<<9*2=18>>18.
Trish has 9 x 2 = <<9*2=18>>18 stuffed animals.
Trish will be able to sell all her stuffed animals for 18 x $1.50 = $<<18*1.5=27>>27.
Therefore, they will donate a total of $18 + $27 = $<<18+27=45>>45 to their class funds.
#### 45 |
Vince owns a hair salon and he earns $18 per head. His monthly expenses are $280 for rent and electricity and 20% of his earnings are allocated for recreation and relaxation. He will save the rest. How much does he save if he serves 80 customers a month? | This month, Vince earns 80 x $18 = $<<80*18=1440>>1440.
The amount he allocates for recreation and relaxation is 20/100 x $1440 = $<<20/100*1440=288>>288.
Thus, his monthly expenses is $280 + $288 =$<<280+288=568>>568.
Therefore, he saves $1440 - $568 = $<<1440-568=872>>872.
#### 872 |
Luke is planning a trip to London and wants to see how long it will take him to travel there. He decides that he will take a bus to the town center, walk for 15 minutes to the train center, wait twice this long for the train to arrive, and then take the 6-hour train ride to London. If Luke’s entire trip takes him 8 hou... | Removing the time of the train ride from the journey shows that his trip to get on the train took him 8 hours – 6 hours = <<8-6=2>>2 hours.
Converting this time into minutes shows that getting to the train took 2 hours * 60 minutes/hour = <<2*60=120>>120 minutes.
Waiting for the train to arrive took twice as long as hi... |
If $x+y=9$ and $xy=10$, what is the value of $x^3+y^3$? | 459 |
Either increasing the radius or the height of a cylinder by six inches will result in the same volume. The original height of the cylinder is two inches. What is the original radius in inches? | 6 |
A number $m$ is randomly selected from the set $\{12, 14, 16, 18, 20\}$, and a number $n$ is randomly selected from $\{2005, 2006, 2007, \ldots, 2024\}$. What is the probability that $m^n$ has a units digit of $6$?
A) $\frac{1}{5}$
B) $\frac{1}{4}$
C) $\frac{2}{5}$
D) $\frac{1}{2}$
E) $\frac{3}{5}$ | \frac{2}{5} |
Let $M$ be a point on the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$, with $F_1$ and $F_2$ as its foci. If $\angle F_1MF_2 = \frac{\pi}{6}$, calculate the area of $\triangle MF_1F_2$. | 16(2 - \sqrt{3}) |
Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order)... | 80 |
The side length of square $A$ is 36 cm. The side length of square $B$ is 42 cm. What is the ratio of the area of square $A$ to the area of square $B$? Express your answer as a common fraction. | \frac{36}{49} |
In the diagram, $O$ is the center of a circle with radii $OP=OQ=5$. What is the perimeter of the shaded region?
[asy]
size(100);
import graph;
label("$P$",(-1,0),W); label("$O$",(0,0),NE); label("$Q$",(0,-1),S);
fill(Arc((0,0),1,-90,180)--cycle,mediumgray);
draw(Arc((0,0),1,-90,180));
fill((0,0)--(-1,0)--(0,-1)--cyc... | 10 + \frac{15}{2}\pi |
The graph below shows a portion of the curve defined by the quartic polynomial $P(x)=x^4+ax^3+bx^2+cx+d$.
[asy]
unitsize(0.8 cm);
int i;
real func (real x) {
return(0.5*(x^4/4 - 2*x^3/3 - 3/2*x^2) + 2.7);
}
draw(graph(func,-4.5,4.5));
draw((-4.5,0)--(4.5,0));
draw((0,-5.5)--(0,5.5));
for (i = -4; i <= 4; ++i) {
... | \text{C} |
Given that $\sin\alpha + \sin\beta = \frac{1}{3}$, find the maximum and minimum values of $y = \sin\beta - \cos^2\alpha$. | -\frac{11}{12} |
Use each of the five digits $2, 4, 6, 7$ and $9$ only once to form a three-digit integer and a two-digit integer which will be multiplied together. What is the three-digit integer that results in the greatest product? | 762 |
The Evil League of Evil plans to set out from their headquarters at (5,1) to poison two pipes: one along the line \( y = x \) and the other along the line \( x = 7 \). They wish to determine the shortest distance they can travel to visit both pipes and then return to their headquarters. | 4\sqrt{5} |
Compute the least possible value of $ABCD - AB \times CD$ , where $ABCD$ is a 4-digit positive integer, and $AB$ and $CD$ are 2-digit positive integers. (Here $A$ , $B$ , $C$ , and $D$ are digits, possibly equal. Neither $A$ nor $C$ can be zero.) | 109 |
Chad bought 6 packages of cat food and 2 packages of dog food. Each package of cat food contained 9 cans, and each package of dog food contained 3 cans. How many more cans of cat food than dog food did Chad buy? | Chad's cat food total was 6 packages * 9 cans = <<6*9=54>>54 cans of cat food.
Chad's dog food total was 2 packages * 3 cans = <<2*3=6>>6 cans of dog food.
Chad had 54 cans of cat food - 6 cans of dog food = <<54-6=48>>48 more cans of cat food.
#### 48 |
Given the function $f\left(x\right)=x^{2}-2$, find $\lim_{{Δx→0}}\frac{{f(3)-f({3-2Δx})}}{{Δx}}$. | 12 |
In the country Betia, there are 125 cities, some of which are connected by express trains that do not stop at intermediate stations. It is known that any four cities can be visited in a circular order. What is the minimum number of city pairs connected by express trains? | 7688 |
Marcus, Humphrey, and Darrel are bird watching. Marcus sees 7 birds, Humphrey sees 11 birds, and Darrel sees 9 birds. How many birds does each of them see on average? | They see 27 total birds because 9 +11 + 7 = <<9+11+7=27>>27
They see an average of 9 birds each because 27 / 3 = <<27/3=9>>9
#### 9 |
Find the remainder when $x^{100}$ is divided by $(x + 1)^3.$ | 4950x^2 + 9800x + 4851 |
Last year, Australian Suzy Walsham won the annual women's race up the 1576 steps of the Empire State Building in New York for a record fifth time. Her winning time was 11 minutes 57 seconds. Approximately how many steps did she climb per minute? | 130 |
James buys twice as many toy soldiers as toy cars. He buys 20 toy cars. How many total toys does he buy? | He buys 2*20=<<2*20=40>>40 toy cars
So the total number of toys is 20+40=<<20+40=60>>60
#### 60 |
Rudy runs 5 miles at a rate of 10 minutes per mile. Later he runs 4 miles at a rate of 9.5 minutes per mile. What is the total length of time that Rudy runs? | Rudy first runs for 5 * 10 = <<5*10=50>>50 minutes.
Then Rudy runs for 4 * 9.5 = <<4*9.5=38>>38 minutes.
Rudy runs for a total length of 50 + 38 = <<50+38=88>>88 minutes
#### 88 |
Amanda, Ben, and Carlos share a sum of money. Their portions are in the ratio of 1:2:7, respectively. If Amanda's portion is $\$$20, what is the total amount of money shared? | 200 |
Three numbers, $a_1\,$, $a_2\,$, $a_3\,$, are drawn randomly and without replacement from the set $\{1, 2, 3, \dots, 1000\}\,$. Three other numbers, $b_1\,$, $b_2\,$, $b_3\,$, are then drawn randomly and without replacement from the remaining set of 997 numbers. Let $p\,$ be the probability that, after a suitable rotat... | 5 |
There are four positive integers that are divisors of each number in the list $$36, 72, -12, 114, 96.$$Find the sum of these four positive integers. | 12 |
What is the sum of the values of $x$ that satisfy the equation $x^2-5x+5=9$? | 5 |
The time it takes for person A to make 90 parts is the same as the time it takes for person B to make 120 parts. It is also known that A and B together make 35 parts per hour. Determine how many parts per hour A and B each make. | 20 |
Find the number of ordered pairs of integers $(a, b)$ such that $a, b$ are divisors of 720 but $a b$ is not. | 2520 |
A nine-digit integer is formed by repeating a positive three-digit integer three times. For example, 123,123,123 or 456,456,456 are integers of this form. What is the greatest common divisor of all nine-digit integers of this form? | 1001001 |
Given $f(x) = kx + \frac {2}{x^{3}} - 3$ $(k \in \mathbb{R})$, and it is known that $f(\ln 6) = 1$. Find $f\left(\ln \frac {1}{6}\right)$. | -7 |
How many ways, without taking order into consideration, can 2002 be expressed as the sum of 3 positive integers (for instance, $1000+1000+2$ and $1000+2+1000$ are considered to be the same way)? | 334000 |
Given the expression $(xy - \frac{1}{2})^2 + (x - y)^2$ for real numbers $x$ and $y$, find the least possible value. | \frac{1}{4} |
For all positive integers $n$ , let $f(n)$ return the smallest positive integer $k$ for which $\tfrac{n}{k}$ is not an integer. For example, $f(6) = 4$ because $1$ , $2$ , and $3$ all divide $6$ but $4$ does not. Determine the largest possible value of $f(n)$ as $n$ ranges over the set $\{1,2,\l... | 11 |
Yolanda scored 345 points over the entire season. There were 15 games over the season. She averaged 4 free throws (worth one point), and 5 two-point baskets per game. How many three-point baskets did she average a game? | She averaged 23 points per game because 345 / 15 = <<345/15=23>>23
She averaged 10 points a game off two-point baskets because 5 x 2 = 10
She averaged 4 points a game from free throws because 4 x 1 = <<4*1=4>>4
She averaged 14 points a game off non-three point baskets because 10 + 4 = <<10+4=14>>14
She averaged 9 point... |
A wooden model of a square pyramid has a base edge of 12 cm and an altitude of 8 cm. A cut is made parallel to the base of the pyramid that separates it into two pieces: a smaller pyramid and a frustum. Each base edge of the smaller pyramid is 6 cm and its altitude is 4 cm. How many cubic centimeters are in the volume ... | 336 |
The numbers \(2, 3, 12, 14, 15, 20, 21\) may be divided into two sets so that the product of the numbers in each set is the same.
What is this product? | 2520 |
Determine the minimum possible value of the sum
\[
\frac{a}{3b} + \frac{b}{6c} + \frac{c}{9a},
\]
where \(a\), \(b\), and \(c\) are positive real numbers. | \frac{3}{\sqrt[3]{162}} |
What is the value of $23^2 + 2(23)(2) + 2^2$? | 625 |
The endpoints of a line segment \\(AB\\) with a fixed length of \\(3\\) move on the parabola \\(y^{2}=x\\). Let \\(M\\) be the midpoint of the line segment \\(AB\\). The minimum distance from \\(M\\) to the \\(y\\)-axis is \_\_\_\_\_\_. | \dfrac{5}{4} |
Using the digits 0, 1, 2, 3, 4, 5 to form numbers without repeating any digit. Calculate:
(1) How many six-digit numbers can be formed?
(2) How many three-digit numbers can be formed that contain at least one even number?
(3) How many three-digit numbers can be formed that are divisible by 3? | 40 |
Joe sells cookies for 20% more than it costs to make them. If he sold 50 cookies and earned $60, how much do the cookies cost to make? | He sold the cookies for $1.20 each because 60 / 50 = <<60/50=1.2>>1.2
They cost $1 to make because 1.2 / 1.2 = <<1.2/1.2=1>>1
#### 1 |
Beatriz loves odd numbers. How many numbers between 0 and 1000 can she write using only odd digits? | 155 |
Given that four integers \( a, b, c, d \) are all even numbers, and \( 0 < a < b < c < d \), with \( d - a = 90 \). If \( a, b, c \) form an arithmetic sequence and \( b, c, d \) form a geometric sequence, then find the value of \( a + b + c + d \). | 194 |
In the series of activities with the theme "Labor Creates a Better Life" carried out at school, Class 1 of Grade 8 is responsible for the design, planting, and maintenance of a green corner on campus. The students agreed that each person would take care of one pot of green plants. They planned to purchase a total of 46... | $369 |
Let $x, y, z$ be real numbers such that:
\begin{align*}
y+z & = 16, \\
z+x & = 18, \\
x+y & = 20.
\end{align*}
Find $\sqrt{xyz(x+y+z)}$. | 9\sqrt{77} |
Given the sequence $\{a_n\}$ satisfies $a_1=1$, $a_2=4$, $a_3=9$, $a_n=a_{n-1}+a_{n-2}-a_{n-3}$, for $n=4,5,...$, calculate $a_{2017}$. | 8065 |
Let $a$, $b$, and $c$ be the $3$ roots of $x^3-x+1=0$. Find $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}$. | -2 |
Five students from a certain class participated in a speech competition and the order of appearance was determined by drawing lots, under the premise that student A must appear before student B. Calculate the probability of students A and B appearing adjacent to each other. | \frac{2}{5} |
A positive integer will be called "sparkly" if its smallest (positive) divisor, other than 1, equals the total number of divisors (including 1). How many of the numbers $2,3, \ldots, 2003$ are sparkly? | 3 |
In $\triangle ABC$, $D$ and $E$ are points on sides $BC$ and $AC$, respectively, such that $\frac{BD}{DC} = \frac{2}{3}$ and $\frac{AE}{EC} = \frac{3}{4}$. Find the value of $\frac{AF}{FD} \cdot \frac{BF}{FE}$. | $\frac{35}{12}$ |
Charles is wondering how much chocolate milk he can make with all the supplies he finds in the fridge. He is going to keep drinking 8-ounce glasses until he uses up all the ingredients. Each glass must contain 6.5 ounces of milk and 1.5 ounces of chocolate syrup. If he has 130 ounces of milk and 60 ounces of chocolate ... | He has enough milk for 20 glasses of chocolate milk because 130 / 6.5 = <<130/6.5=20>>20
He has enough syrup for 40 glasses of chocolate milk because 60 / 1.5 = <<60/1.5=40>>40
He can make 20 glasses of chocolate milk because 20 < 40
He will drink 160 ounces of chocolate milk because 20 x 8 = <<20*8=160>>160
#### 160 |
Francesca uses $100$ grams of lemon juice, $100$ grams of sugar, and $400$ grams of water to make lemonade. There are $25$ calories in $100$ grams of lemon juice and $386$ calories in $100$ grams of sugar. Water contains no calories. How many calories are in $200$ grams of her lemonade? | 137 |
Let $x_1,$ $x_2,$ $x_3,$ $\dots,$ $x_{100}$ be positive real numbers such that $x_1^2 + x_2^2 + x_3^2 + \dots + x_{100}^2 = 1.$ Find the minimum value of
\[\frac{x_1}{1 - x_1^2} + \frac{x_2}{1 - x_2^2} + \frac{x_3}{1 - x_3^2} + \dots + \frac{x_{100}}{1 - x_{100}^2}.\] | \frac{3 \sqrt{3}}{2} |
Find in explicit form all ordered pairs of positive integers $(m, n)$ such that $mn-1$ divides $m^2 + n^2$. | (2, 1), (3, 1), (1, 2), (1, 3) |
Out of the 150 students, 60% are girls and the rest are boys. Only 1/3 of the boys joined varsity clubs. How many of the boys did not join varsity clubs? | Out of the 150 students, 100% - 60% = 40% are boys.
So, there are 150 x 40/100 = <<150*40/100=60>>60 boys.
Out of the 60 boys, 60 x 1/3 = <<60*1/3=20>>20 joined varsity clubs.
Therefore, 60 - 20 = <<60-20=40>>40 boys did not join varsity clubs.
#### 40 |
People enter the subway uniformly from the street. After passing through the turnstiles, they end up in a small hall before the escalators. The entrance doors have just opened, and initially, the hall before the escalators was empty, with only one escalator running to go down. One escalator couldn't handle the crowd, s... | 60 |
Marlon had 42 lollipops in the beginning. On his way home, he saw Emily and gave her 2/3 of his lollipops. Then, Marlon kept 4 lollipops and gave the rest to Lou. How many lollipops did Lou receive? | Marlon gave Emily 42 x 2/3 = <<42*2/3=28>>28 lollipops.
So, he only had 42 - 28 = <<42-28=14>>14 lollipops left.
After keeping 4, Marlon gave 14 - 4 = <<14-4=10>>10 lollipops to Lou.
#### 10 |
The expression $x^2 + 13x + 30$ can be written as $(x + a)(x + b),$ and the expression $x^2 + 5x - 50$ written as $(x + b)(x - c)$, where $a$, $b$, and $c$ are integers. What is the value of $a + b + c$? | 18 |
Find the coefficient of the $x^2$ term in the expansion of the product $(ax^3 + 3x^2 - 2x)(bx^2 - 7x - 4)$. | 2 |
A national team needs to select 4 out of 6 sprinters to participate in the 4×100 m relay at the Asian Games. If sprinter A cannot run the first leg and sprinter B cannot run the fourth leg, there are a total of ______ ways to participate. | 252 |
Given that the point \( P(x, y) \) satisfies the equation \( (x-4 \cos \theta)^{2}+(y-4 \sin \theta)^{2}=4(\theta \in \mathbf{R}) \), find the area of the region where the point \( P(x, y) \) can be located. | 32\pi |
The asymptotes of a hyperbola are $y = x + 1$ and $y = 3 - x.$ Also, the hyperbola passes through $(3,3).$ Find the distance between the foci of the hyperbola. | 2 \sqrt{6} |
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