problem
stringlengths 10
5.15k
| answer
stringlengths 0
1.23k
|
|---|---|
In the Cartesian coordinate system $xOy$, the sum of distances from point $P$ to the centers of two circles $C_1$ and $C_2$ is equal to 4, where $C_1: x^2+y^2-2\sqrt{3}y+2=0$, $C_2: x^2+y^2+2\sqrt{3}y-3=0$. Let the trajectory of point $P$ be $C$.
(1) Find the equation of $C$;
(2) Suppose the line $y=kx+1$ intersects $C$ at points $A$ and $B$. What is the value of $k$ when $\overrightarrow{OA} \perp \overrightarrow{OB}$? What is the value of $|\overrightarrow{AB}|$ at this time?
|
\frac{4\sqrt{65}}{17}
|
A three-digit positive integer \( n \) has digits \( a, b, c \). (That is, \( a \) is the hundreds digit of \( n \), \( b \) is the tens digit of \( n \), and \( c \) is the ones (units) digit of \( n \).) Determine the largest possible value of \( n \) for which:
- \( a \) is divisible by 2,
- the two-digit integer \( ab \) (where \( a \) is the tens digit and \( b \) is the ones digit) is divisible by 3 but is not divisible by 6, and
- \( n \) is divisible by 5 but is not divisible by 7.
|
870
|
Find all 4-digit numbers $n$ , such that $n=pqr$ , where $p<q<r$ are distinct primes, such that $p+q=r-q$ and $p+q+r=s^2$ , where $s$ is a prime number.
|
2015
|
Which of the following could NOT be the lengths of the external diagonals of a right regular prism [a "box"]? (An $\textit{external diagonal}$ is a diagonal of one of the rectangular faces of the box.)
$\text{(A) }\{4,5,6\} \quad \text{(B) } \{4,5,7\} \quad \text{(C) } \{4,6,7\} \quad \text{(D) } \{5,6,7\} \quad \text{(E) } \{5,7,8\}$
|
\{4,5,7\}
|
What is the minimum number of points that can be chosen on a circle with a circumference of 1956 so that for each of these points there is exactly one chosen point at a distance of 1 and exactly one at a distance of 2 (distances are measured along the circle)?
|
1304
|
In January the families visiting a national park see animals 26 times. In February the families that visit the national park see animals three times as many as were seen there in January. Then in March the animals are shyer and the families who visit the national park see animals half as many times as they were seen in February. How many times total did families see an animal in the first three months of the year?
|
Animals were seen 26 times in January and three times as much in February, 26 x 3 = <<26*3=78>>78 times animals were seen in February.
In March the animals were seen 1/2 as many times as were seen in February, 78 / 2 = <<78/2=39>>39 times animals were seen in March.
If animals were seen 26 times in January + 78 times in February + 39 times in March = <<26+78+39=143>>143 times animals were seen in the first three months of the year.
#### 143
|
A line passes through point $Q(\frac{1}{3}, \frac{4}{3})$ and intersects the hyperbola $x^{2}- \frac{y^{2}}{4}=1$ at points $A$ and $B$. Point $Q$ is the midpoint of chord $AB$.
1. Find the equation of the line containing $AB$.
2. Find the length of $|AB|$.
|
\frac{8\sqrt{2}}{3}
|
Given vectors $\overrightarrow {a}$=（sinx，cosx）, $\overrightarrow {b}$=（1，$\sqrt {3}$）.
(1) If $\overrightarrow {a}$$∥ \overrightarrow {b}$, find the value of tanx;
(2) Let f(x) = $\overrightarrow {a}$$$\cdot \overrightarrow {b}$, stretch the horizontal coordinates of each point on the graph of f(x) to twice their original length (vertical coordinates remain unchanged), then shift all points to the left by φ units (0 < φ < π), obtaining the graph of function g(x). If the graph of g(x) is symmetric about the y-axis, find the value of φ.
|
\frac{\pi}{3}
|
How many $3$-digit squares are palindromes?
|
3
|
In the adjoining figure, two circles with radii $8$ and $6$ are drawn with their centers $12$ units apart. At $P$, one of the points of intersection, a line is drawn in such a way that the chords $QP$ and $PR$ have equal length. Find the square of the length of $QP$.
[asy]size(160); defaultpen(linewidth(.8pt)+fontsize(11pt)); dotfactor=3; pair O1=(0,0), O2=(12,0); path C1=Circle(O1,8), C2=Circle(O2,6); pair P=intersectionpoints(C1,C2)[0]; path C3=Circle(P,sqrt(130)); pair Q=intersectionpoints(C3,C1)[0]; pair R=intersectionpoints(C3,C2)[1]; draw(C1); draw(C2); draw(O2--O1); dot(O1); dot(O2); draw(Q--R); label("$Q$",Q,NW); label("$P$",P,1.5*dir(80)); label("$R$",R,NE); label("12",waypoint(O1--O2,0.4),S);[/asy]
|
130
|
Find the distance \( B_{1} H \) from point \( B_{1} \) to the line \( D_{1} B \), given \( B_{1}(5, 8, -3) \), \( D_{1}(-3, 10, -5) \), and \( B(3, 4, 1) \).
|
2\sqrt{6}
|
Jeff decides to play with a Magic 8 Ball. Each time he asks it a question, it has a 2/5 chance of giving him a positive answer. If he asks it 5 questions, what is the probability that it gives him exactly 2 positive answers?
|
\frac{216}{625}
|
Two skaters, Allie and Billie, are at points $A$ and $B$, respectively, on a flat, frozen lake. The distance between $A$ and $B$ is $100$ meters. Allie leaves $A$ and skates at a speed of $8$ meters per second on a straight line that makes a $60^\circ$ angle with $AB$. At the same time Allie leaves $A$, Billie leaves $B$ at a speed of $7$ meters per second and follows the straight path that produces the earliest possible meeting of the two skaters, given their speeds. How many meters does Allie skate before meeting Billie?
[asy] pointpen=black; pathpen=black+linewidth(0.7); pair A=(0,0),B=(10,0),C=6*expi(pi/3); D(B--A); D(A--C,EndArrow); MP("A",A,SW);MP("B",B,SE);MP("60^{\circ}",A+(0.3,0),NE);MP("100",(A+B)/2); [/asy]
|
160
|
Let $S = \{1, 2, \ldots, 2005\}$. If any set of $n$ pairwise coprime numbers from $S$ always contains at least one prime number, find the smallest value of $n$.
|
16
|
In triangle \(ABC\), \(AB = 13\) and \(BC = 15\). On side \(AC\), point \(D\) is chosen such that \(AD = 5\) and \(CD = 9\). The angle bisector of the angle supplementary to \(\angle A\) intersects line \(BD\) at point \(E\). Find \(DE\).
|
7.5
|
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 $g(12) = 18$ and $g(48) = 26,$ find $g(600).$
|
36
|
When $1 + 3 + 3^2 + \cdots + 3^{1004}$ is divided by $500$, what is the remainder?
|
121
|
We define a function $g(x)$ such that $g(12)=37$, and if there exists an integer $a$ such that $g(a)=b$, then $g(b)$ is defined and follows these rules:
1. $g(b)=3b+1$ if $b$ is odd
2. $g(b)=\frac{b}{2}$ if $b$ is even.
What is the smallest possible number of integers in the domain of $g$?
|
23
|
Find $k$ where $2^k$ is the largest power of $2$ that divides the product \[2008\cdot 2009\cdot 2010\cdots 4014.\]
|
2007
|
Let the sequence $a_{i}$ be defined as $a_{i+1}=2^{a_{i}}$. Find the number of integers $1 \leq n \leq 1000$ such that if $a_{0}=n$, then 100 divides $a_{1000}-a_{1}$.
|
50
|
Suppose functions $g$ and $f$ have the properties that $g(x)=3f^{-1}(x)$ and $f(x)=\frac{24}{x+3}$. For what value of $x$ does $g(x)=15$?
|
3
|
Given a geometric sequence $\left\{a_{n}\right\}$ with real terms, and the sum of the first $n$ terms is $S_{n}$. If $S_{10} = 10$ and $S_{30} = 70$, then $S_{40}$ is equal to:
|
150
|
Given a circle of radius 3, find the area of the region consisting of all line segments of length 6 that are tangent to the circle at their midpoints.
A) $3\pi$
B) $6\pi$
C) $9\pi$
D) $12\pi$
E) $15\pi$
|
9\pi
|
A large candle is $119$ centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes $10$ seconds to burn down the first centimeter from the top, $20$ seconds to burn down the second centimeter, and $10k$ seconds to burn down the $k$-th centimeter. (The candle burns down each individual centimeter at a fixed rate.)
Suppose it takes $T$ seconds for the candle to burn down completely. Compute the candle's height in centimeters $\tfrac{T}{2}$ seconds after it is lit.
|
35
|
Each morning of her five-day workweek, Jane bought either a $50$-cent muffin or a $75$-cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy?
|
2
|
In Mr. Bolton's class of 25 students, 2/5 of the students like maths, 1/3 of the remaining students like science, and the rest of the students like history. Calculate the combined total number of students who like history and those who like maths.
|
If there are 25 students in the class, then 2/5*25 = <<2/5*25=10>>10 students like maths.
The number of students who don't like math is 25-10 = <<25-10=15>>15 students.
If 1/3 of the students that don't like math like science, then 1/3*15 = <<1/3*15=5>>5 students like science.
The remaining number of students who enjoy history is 15-5 =<<15-5=10>>10
The combined total for the students who want history and maths is 10+10 = <<10+10=20>>20
#### 20
|
Let $f(r)=\sum_{j=2}^{2008} \frac{1}{j^{r}}=\frac{1}{2^{r}}+\frac{1}{3^{r}}+\cdots+\frac{1}{2008^{r}}$. Find $\sum_{k=2}^{\infty} f(k)$.
|
\frac{2007}{2008}
|
In the geometric sequence $\{a_n\}$, the common ratio $q = -2$, and $a_3a_7 = 4a_4$, find the arithmetic mean of $a_8$ and $a_{11}$.
|
-56
|
A rectangular garden 50 feet long and 10 feet wide is enclosed by a fence. To make the garden larger, while using the same fence, its shape is changed to a square. By how many square feet does this enlarge the garden?
|
400
|
What is the base $2$ representation of $125_{10}$?
|
1111101_2
|
If \(\frac{1}{4} + 4\left(\frac{1}{2013} + \frac{1}{x}\right) = \frac{7}{4}\), find the value of \(1872 + 48 \times \left(\frac{2013 x}{x + 2013}\right)\).
|
2000
|
There are 196 students numbered from 1 to 196 arranged in a line. Students at odd-numbered positions (1, 3, 5, ...) leave the line. The remaining students are renumbered starting from 1 in order. Then, again, students at odd-numbered positions leave the line. This process repeats until only one student remains. What was the initial number of this last remaining student?
|
128
|
Find $\frac{7}{17} - \frac{4}{51}$. Reduce your answer to simplest form.
|
\frac{1}{3}
|
Let $ABCD$ be a unit square. $E$ and $F$ trisect $AB$ such that $AE<AF. G$ and $H$ trisect $BC$ such that $BG<BH. I$ and $J$ bisect $CD$ and $DA,$ respectively. Let $HJ$ and $EI$ meet at $K,$ and let $GJ$ and $FI$ meet at $L.$ Compute the length $KL.$
|
\frac{6\sqrt{2}}{35}
|
In a circle with radius 7, two intersecting chords $PQ$ and $RS$ are given such that $PQ=10$ and $RS$ bisects $PQ$ at point $T$. It is also given that $PQ$ is the only chord starting at $P$ which is bisected by $RS$. Determine the sine of the central angle subtended by the minor arc $PR$ and express it as a fraction $\frac{a}{b}$ in lowest terms. Find the product $ab$.
|
15
|
Given \(\alpha, \beta \in \left(0, \frac{\pi}{2}\right)\), \(\cos \alpha = \frac{4}{5}\), \(\tan (\alpha - \beta) = -\frac{1}{3}\), find \(\cos \beta\).
|
\frac{9 \sqrt{10}}{50}
|
Minimize \(\boldsymbol{F}=\boldsymbol{x}_{2}-\boldsymbol{x}_{1}\) for non-negative \(x_{1}\) and \(x_{2}\), subject to the system of constraints:
$$
\left\{\begin{aligned}
-2 x_{1}+x_{2}+x_{3} &=2 \\
x_{1}-2 x_{2}+x_{4} &=2 \\
x_{1}+x_{2}+x_{5} &=5
\end{aligned}\right.
$$
|
-3
|
Daisy and Rose were enjoying their backyard pool with their dogs. If there are 24 legs/paws in the pool, how many dogs do Daisy and Rose have?
|
Daisy and Rose each have 2 legs so between them, they have 2*2 = <<2*2=4>>4 legs
There are 24 legs in the pool and 4 belong to Daisy and Rose so there are 24-4 = <<24-4=20>>20 legs in the pool
Dogs have 4 legs and there are 20 legs in the pool so there are 20/4 = 5 dogs in the pool
#### 5
|
Find $k$ if
\[(\sin \alpha + \csc \alpha)^2 + (\cos \alpha + \sec \alpha)^2 = k + \tan^2 \alpha + \cot^2 \alpha.\]
|
7
|
The endpoints of a line segment are (2, 3) and (8, 15). What is the sum of the coordinates of the midpoint of the segment?
|
14
|
Voldemort bought a book for $\$5$. It was one-tenth of its original price. What was the original price in dollars?
|
\$50
|
Find the number of positive integers $n,$ $1 \le n \le 1000,$ for which the polynomial $x^2 + x - n$ can be factored as the product of two linear factors with integer coefficients.
|
31
|
Suppose that the roots of $x^3+3x^2+4x-11=0$ are $a$, $b$, and $c$, and that the roots of $x^3+rx^2+sx+t=0$ are $a+b$, $b+c$, and $c+a$. Find $t$.
|
23
|
If $3 imes n=6 imes 2$, what is the value of $n$?
|
4
|
In the Cartesian coordinate system $xOy$, with the origin as the pole and the positive half-axis of the $x$-axis as the polar axis, the polar coordinate equation of the curve $C_{1}$ is $\rho \cos \theta = 4$.
$(1)$ Let $M$ be a moving point on the curve $C_{1}$, point $P$ lies on the line segment $OM$, and satisfies $|OP| \cdot |OM| = 16$. Find the rectangular coordinate equation of the locus $C_{2}$ of point $P$.
$(2)$ Suppose the polar coordinates of point $A$ are $({2, \frac{π}{3}})$, point $B$ lies on the curve $C_{2}$. Find the maximum value of the area of $\triangle OAB$.
|
2 + \sqrt{3}
|
The perimeter of triangle \( \mathrm{ABC} \) is 1. A circle touches side \( \mathrm{AB} \) at point \( P \) and the extension of side \( \mathrm{AC} \) at point \( Q \). A line passing through the midpoints of sides \( \mathrm{AB} \) and \( \mathrm{AC} \) intersects the circumcircle of triangle \( \mathrm{APQ} \) at points \( X \) and \( Y \). Find the length of segment \( X Y \).
|
\frac{1}{2}
|
Jack and Jill run 10 km. They start at the same point, run 5 km up a hill, and return to the starting point by the same route. Jack has a 10 minute head start and runs at the rate of 15 km/hr uphill and 20 km/hr downhill. Jill runs 16 km/hr uphill and 22 km/hr downhill. How far from the top of the hill are they when they pass each other going in opposite directions (in km)?
|
\frac{35}{27}
|
A chocolate bar weighs 125 g. A shopkeeper has just received a 2 kilogram box of chocolate. How many bars does this represent?
|
We have 2 kg = 2 * 1000 = <<2*1000=2000>>2000 g
So, the number of bars of chocolate is 2000 / 125 = <<2000/125=16>>16
#### 16
|
In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, with $b=1$, and $2\cos C - 2a - c = 0$.
(Ⅰ) Find the magnitude of angle $B$;
(Ⅱ) Find the distance from the circumcenter of $\triangle ABC$ to side $AC$.
|
\frac{\sqrt{3}}{6}
|
How many ways are there to place four points in the plane such that the set of pairwise distances between the points consists of exactly 2 elements? (Two configurations are the same if one can be obtained from the other via rotation and scaling.)
|
6
|
Two runners are competing in a 10-mile race. The first runs at an average pace of 8 minutes per mile, while the second runs at an average pace of 7 minutes per mile. After 56 minutes, the second runner stops for a drink of water. For how many minutes could the second runner remain stopped before the first runner catches up with him?
|
After 56 minutes, the second runner has traveled 56/7 = <<56/7=8>>8 miles.
Similarly, the first runner has traveled 56/8 = <<56/8=7>>7 miles.
Therefore, the two runners are 8 - 7 = <<8-7=1>>1 mile apart when the second runner stops.
Thus, the second runner can stop for a maximum of 8 minutes, which is the amount of time it takes the first runner to travel 1 mile.
#### 8
|
Find all integers $\,a,b,c\,$ with $\,1<a<b<c\,$ such that \[ (a-1)(b-1)(c-1) \] is a divisor of $abc-1.$
|
(2, 4, 8) \text{ and } (3, 5, 15)
|
A cube with 3-inch edges is made using 27 cubes with 1-inch edges. Nineteen of the smaller cubes are white and eight are black. If the eight black cubes are placed at the corners of the larger cube, what fraction of the surface area of the larger cube is white?
|
\frac{5}{9}
|
How many distinct digits can appear as the second to last digit (penultimate digit) of an integral perfect square number?
|
10
|
In some cases, it is not necessary to calculate the result to remove the absolute value. For example: $|6+7|=6+7$, $|6-7|=7-6$, $|7-6|=7-6$, $|-6-7|=6+7.\left(1\right)$ According to the above rule, express the following expressions in the form without absolute value symbols: <br/>①$|\frac{7}{17}-\frac{7}{18}|=$______；<br/>②$|\pi -3.15|=$______；<br/>$(2)$ Calculate using a simple method: $|\frac{1}{3}-\frac{1}{2}|+|\frac{1}{4}-\frac{1}{3}|+|\frac{1}{5}-\frac{1}{4}|+…+|\frac{1}{2022}-\frac{1}{2021}|$.
|
\frac{505}{1011}
|
Every Halloween one house in the neighborhood gives out toothbrushes instead of candy, so it always gets egged and covered in toilet paper. If the owner spends 15 seconds cleaning up each egg and 30 minutes cleaning up each roll of toilet paper, how long (in minutes) will they have to spend cleaning up 60 eggs and 7 rolls of toilet paper?
|
First find how many eggs per minute the owner can clean up: 60 seconds/minute / 15 seconds/egg = <<60/15=4>>4 eggs/minute
Then divide the total number of eggs by the number of eggs cleaned per minute to find how long the owner spends cleaning them up: 60 eggs / 4 eggs/minute = <<60/4=15>>15 minutes
Then find the total time the owner spends cleaning up toilet paper: 7 rolls * 30 minutes/roll = <<7*30=210>>210 minutes
Finally, add that amount to the egg cleaning time to find the total cleaning time: 210 minutes + 15 minutes = <<210+15=225>>225 minutes
#### 225
|
Square corners, 5 units on a side, are removed from a $20$ unit by $30$ unit rectangular sheet of cardboard. The sides are then folded to form an open box. The surface area, in square units, of the interior of the box is
|
500
|
A transgalactic ship encountered an astonishing meteor stream. Some meteors fly along a straight line at the same speed, equally spaced from each other. Another group of meteors flies in the exact same manner along another straight line, parallel to the first, but in the opposite direction, also equally spaced. The ship flies parallel to these lines. Astronaut Gavril observed that every 7 seconds the ship meets meteors flying towards it, and every 13 seconds it meets meteors flying in the same direction as the ship. Gavril wondered how often the meteors would pass by if the ship were stationary. He thought that he should take the arithmetic mean of the two given times. Is Gavril correct? If yes, write down this arithmetic mean as the answer. If not, specify the correct time in seconds, rounded to the nearest tenth.
|
4.6
|
Eight points are chosen on a circle, and chords are drawn connecting every pair of points. No three chords intersect in a single point inside the circle. How many triangles with all three vertices in the interior of the circle are created?
|
28
|
Natural numbers \( A \) and \( B \) are divisible by all natural numbers from 1 to 65. What is the smallest natural number that \( A + B \) might not be divisible by?
|
67
|
In a 5 by 5 grid, each of the 25 small squares measures 2 cm by 2 cm and is shaded. Five unshaded circles are then placed on top of the grid as shown. The area of the visible shaded region can be written in the form $A-B\pi$ square cm. What is the value $A+B$?
[asy]
for(int i = 0; i < 5; ++i)
{
for(int j = 0; j < 5; ++j)
{
filldraw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--(i,j)--cycle,gray,linewidth(2));
}
}
filldraw(circle((2.5,.5),.5),white,linewidth(2));
filldraw(circle((4.5,2.5),.5),white,linewidth(2));
filldraw(circle((2.5,4.5),.5),white,linewidth(2));
filldraw(circle((.5,2.5),.5),white,linewidth(2));
filldraw(circle((2.5,2.5),1.5),white,linewidth(2));
[/asy]
|
113
|
Masha wrote the numbers $4, 5, 6, \ldots, 16$ on the board and then erased one or more of them. It turned out that the remaining numbers on the board cannot be divided into several groups such that the sums of the numbers in the groups are equal. What is the greatest possible value that the sum of the remaining numbers on the board can have?
|
121
|
In triangle \( ABC \), let \( E \) be the point where the side \( AC \) is divided into quarters closest to \( C \), and let \( F \) be the midpoint of side \( BC \). The line passing through points \( E \) and \( F \) intersects line \( AB \) at point \( D \). What percentage of the area of triangle \( ABC \) is the area of triangle \( ADE \)?
|
112.5
|
Karen is planning writing assignments for her fifth grade class. She knows each short-answer question takes 3 minutes to answer, each paragraph takes 15 minutes to write, and each essay takes an hour to write. If Karen assigns 2 essays and 5 paragraphs, how many short-answer questions should she assign if she wants to assign 4 hours of homework total?
|
First figure out how many minutes 4 hours is: 4 hours * 60 minutes/hour = <<4*60=240>>240 minutes
Then figure out how long the two essays will take by multiplying the time per essay by the number of essays: 2 essays * 1 hour/essay * 60 minutes/hour = <<2*1*60=120>>120 minutes
Use the same method to find the total time spent writing paragraphs: 15 minutes/paragraph * 5 paragraphs = <<15*5=75>>75 minutes
Now subtract the time spent on essays and paragraphs from the total time to find the time spent on all the short answer questions: 240 minutes - 120 minutes - 75 minutes = <<240-120-75=45>>45 minutes
Finally, divide the total time spent on short-answer questions by the time per question to find the number of questions: 45 minutes / 3 minutes/question = <<45/3=15>>15 questions
#### 15
|
Compute $18\left(\frac{200}{3} + \frac{50}{6} + \frac{16}{18} + 2\right)$.
|
1402
|
Let $m$ be the smallest positive three-digit integer congruent to 7 (mod 13). Let $n$ be the smallest positive four-digit integer congruent to 7 (mod 13). What is the value of $n - m$?
|
897
|
In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. If $c\cos B + b\cos C = 2a\cos A$, $M$ is the midpoint of $BC$, and $AM=1$, find the maximum value of $b+c$.
|
\frac{4\sqrt{3}}{3}
|
Find the number of pairs $(a, b)$ of positive integers with the property that the greatest common divisor of $a$ and $ b$ is equal to $1\cdot 2 \cdot 3\cdot ... \cdot50$, and the least common multiple of $a$ and $ b$ is $1^2 \cdot 2^2 \cdot 3^2\cdot ... \cdot 50^2$.
|
32768
|
An inscribed dodecagon. A convex dodecagon is inscribed in a circle. The lengths of some six sides of the dodecagon are $\sqrt{2}$, and the lengths of the remaining six sides are $\sqrt{24}$. What is the radius of the circle?
|
\sqrt{38}
|
A line passing through the vertex \( A \) of triangle \( ABC \) perpendicular to its median \( BD \) bisects this median.
Find the ratio of the sides \( AB \) and \( AC \).
|
\frac{1}{2}
|
Philip has a farm with animals. He has 20 cows, 50% more ducks. Philip also has as many pigs as one-fifth of ducks and cows in total. How many animals does Philip have on his farm?
|
Philip has 50/100 * 20 = <<50/100*20=10>>10 more ducks than cows.
Which means he has 20 + 10 = <<20+10=30>>30 ducks.
So Philip's cows and ducks sum up to 20 + 30 = <<20+30=50>>50 animals.
If there are 50 cows and ducks, that means Philip has 1/5 * 50 = <<50*1/5=10>>10 pigs.
In total Philip has 50 + 10 = <<50+10=60>>60 animals on his farm.
#### 60
|
How many integers between 0 and 8 inclusive have an inverse modulo 9?
|
6
|
The sequence 12, 15, 18, 21, 51, 81, $\ldots$ consists of all positive multiples of 3 that contain at least one digit that is a 1. What is the $50^{\mathrm{th}}$ term of the sequence?
|
318
|
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 flour
$\bullet$ $2$ eggs
$\bullet$ $3$ tablespoons butter
$\bullet$ $\frac{3}{4}$ cups sugar
$\bullet$ $1$ package of chocolate drops
They will make only full recipes, no partial recipes.
They learn that a big concert is scheduled for the same night and attendance will be down $25\%.$ How many recipes of cookies should they make for their smaller party?
|
11
|
In triangle \(ABC\), a circle is constructed with diameter \(AC\), which intersects side \(AB\) at point \(M\) and side \(BC\) at point \(N\). Given that \(AC = 2\), \(AB = 3\), and \(\frac{AM}{MB} = \frac{2}{3}\), find \(AN\).
|
\frac{24}{\sqrt{145}}
|
Given that $a$ is an odd multiple of $1183$, find the greatest common divisor of $2a^2+29a+65$ and $a+13$.
|
26
|
How many integers, $x$, satisfy $|5x - 3| \le 7$?
|
3
|
Determine the largest square number that is not divisible by 100 and, when its last two digits are removed, is also a square number.
|
1681
|
There are 9 digits: 0, 1, 2, …, 8. Using five cards, with the two sides respectively marked as 0/8, 1/7, 2/5, 3/4, 6/9; and 6 can be used as 9. How many different four-digit numbers can be formed with these five cards?
|
1728
|
A frog located at $(x,y)$, with both $x$ and $y$ integers, makes successive jumps of length $5$ and always lands on points with integer coordinates. Suppose that the frog starts at $(0,0)$ and ends at $(1,0)$. What is the smallest possible number of jumps the frog makes?
|
3
|
Given \((n+1)^{\alpha+1}-n^{\alpha+1} < n^{\alpha}(\alpha+1) < n^{\alpha+1}-(n-1)^{\alpha+1}, -1 < \alpha < 0\). Let \(x = \sum_{k=4}^{10^{6}} \frac{1}{\sqrt[3]{k}}\), find the integer part of \(x\).
|
146
|
Jan enters a double dutch competition. After training she doubles her speed which used to be 70 skips per minute. How many skips does she do in 5 minutes?
|
She does 70*2=<<70*2=140>>140 skips a minute
So he does 140*5=<<140*5=700>>700 skips in the 5 minutes
#### 700
|
The Bulls are playing the Heat in the NBA finals. To win the championship, a team needs to secure 4 victories before the opponent does. If the Heat win each game with a probability of $\frac{3}{4}$ and there are no ties, what is the probability that the Bulls will win the NBA finals, and the series will extend to all seven games? Express your answer as a fraction.
|
\frac{540}{16384}
|
What is the hundreds digit of $(20! - 15!)?$
|
0
|
Given a hyperbola with eccentricity $e$ and an ellipse with eccentricity $\frac{\sqrt{2}}{2}$ share the same foci $F_{1}$ and $F_{2}$. If $P$ is a common point of the two curves and $\angle F_{1}PF_{2}=60^{\circ}$, then $e=$ ______.
|
\frac{\sqrt{6}}{2}
|
Trip wanted to watch the new action movie in theaters. An evening ticket cost $10 and a large popcorn & drink combo would cost him an additional $10. He noticed on their website, they had a special offer. From 12 noon to 3 pm, save 20% off tickets and 50% off any food combos. How much money could Trip save by going to the earlier movie?
|
An evening movie is $10 and the food combo would be $10 so 10 + 10 = $<<10+10=20>>20
If he goes to an earlier movie, it's 20% off the normal ticket price of $10 so 10 * .20 = $<<10*.20=2>>2
If he goes to an earlier movie, it's 50% off the normal food price of $10 so 10 * .50 = $<<10*.50=5>>5
If he saves $2 for the ticket and $5 on food, then 2+5 = $<<2+5=7>>7 saved
#### 7
|
PQR Entertainment wishes to divide their popular idol group PRIME, which consists of seven members, into three sub-units - PRIME-P, PRIME-Q, and PRIME-R - with each of these sub-units consisting of either two or three members. In how many different ways can they do this, if each member must belong to exactly one sub-unit?
|
630
|
Five marbles are distributed at a random among seven urns. What is the expected number of urns with exactly one marble?
|
6480/2401
|
The lengths of the edges of a regular tetrahedron \(ABCD\) are 1. \(G\) is the center of the base \(ABC\). Point \(M\) is on line segment \(DG\) such that \(\angle AMB = 90^\circ\). Find the length of \(DM\).
|
\frac{\sqrt{6}}{6}
|
Let $f(x)=e^{x}$, and $f(x)=g(x)-h(x)$, where $g(x)$ is an even function, and $h(x)$ is an odd function. If there exists a real number $m$ such that the inequality $mg(x)+h(x)\geqslant 0$ holds for $x\in [-1,1]$, determine the minimum value of $m$.
|
\dfrac{e^{2}-1}{e^{2}+1}
|
John is an eccentric millionaire. He decides to fill his swimming pool with bottled water. A cubic foot of water is 25 liters. His pool is 10 feet deep and 6 feet by 20 feet. A liter of water costs $3. How much does it cost to fill the pool?
|
The pool is 10*6*20=<<10*6*20=1200>>1200 cubic feet
That means it is 25*1200=<<25*1200=30000>>30000 liters
So it cost 3*30,000=$<<3*30000=90000>>90,000
#### 90000
|
Suppose in a right triangle where angle \( Q \) is at the origin and \( \cos Q = 0.5 \). If the length of \( PQ \) is \( 10 \), what is \( QR \)?
|
20
|
Let \(\triangle ABC\) be an isosceles right triangle with \(AB=AC=10\). Let \(M\) be the midpoint of \(BC\) and \(N\) the midpoint of \(BM\). Let \(AN\) hit the circumcircle of \(\triangle ABC\) again at \(T\). Compute the area of \(\triangle TBC\).
|
30
|
In a factory, there are 300 employees. 200 of them earn $12 per hour. Of the rest, 40 of them earn $14 per hour. All others earn $17 per hour. What is the cost to employ all these people for one 8-hour long shift?
|
The cost of the 200 employees is 200 employees * $12/employee/hour = $<<200*12=2400>>2400 per hour.
The cost of the 40 employees is 40 employees * $14/employee/hour = $<<40*14=560>>560 per hour.
The rest means 300 employees - 200 employees - 40 employees = <<300-200-40=60>>60 employees.
These 60 employees' employment costs are at 60 employees * $17/employee/hour = $<<60*17=1020>>1020 per hour.
So in total all employees earn $2400/hour + $560/hour + $1020/hour = $3980/hour.
During an 8-hour shift, this cost would be at 8 hours * $3980/hour = $<<8*3980=31840>>31840.
#### 31840
|
Simplify $\displaystyle\frac{2+2i}{-3+4i}$. Express your answer as a complex number in the form $a+bi$, where $a$ and $b$ are real numbers.
|
\frac{2}{25} - \frac{14}{25}i
|
For certain real values of $a, b, c,$ and $d_{},$ the equation $x^4+ax^3+bx^2+cx+d=0$ has four non-real roots. The product of two of these roots is $13+i$ and the sum of the other two roots is $3+4i,$ where $i^2 = -1.$ Find $b.$
|
51
|
In a regular tetrahedron embedded in 3-dimensional space, the centers of the four faces are the vertices of a smaller tetrahedron. If the vertices of the larger tetrahedron are located on the surface of a sphere of radius \(r\), find the ratio of the volume of the smaller tetrahedron to that of the larger tetrahedron. Express your answer as a simplified fraction.
|
\frac{1}{27}
|
Given the parabola $C: y^{2}=2px\left(p \lt 0\right)$ passing through the point $A\left(-2,-4\right)$.
$(1)$ Find the equation of the parabola $C$ and its directrix equation.
$(2)$ A line passing through the focus of the parabola, making an angle of $60^{\circ}$ with the $x$-axis, intersects the parabola at points $A$ and $B$. Find the length of segment $AB$.
|
\frac{32}{3}
|
John is 24 years younger than his dad. The sum of their ages is 68 years. How many years old is John?
|
22
|
The product of two positive three-digit palindromes is 436,995. What is their sum?
|
1332
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.