problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
A positive integer \( m \) has the property that when multiplied by 12, the result is a four-digit number \( n \) of the form \( 20A2 \) for some digit \( A \). What is the four-digit number \( n \)? | 2052 |
Tina makes $18.00 an hour. If she works more than 8 hours per shift, she is eligible for overtime, which is paid by your hourly wage + 1/2 your hourly wage. If she works 10 hours every day for 5 days, how much money does she make? | She works 8 hours a day for $18 per hour so she makes 8*18 = $<<8*18=144.00>>144.00 per 8-hour shift
She works 10 hours a day and anything over 8 hours is eligible for overtime, so she gets 10-8 = <<10-8=2>>2 hours of overtime
Overtime is calculated as time and a half so and she makes $18/hour so her overtime pay is 18... |
Given $tan({θ+\frac{π}{4}})=2tanθ-7$, determine the value of $\sin 2\theta$. | \frac{4}{5} |
Suppose that $p$ and $q$ are positive numbers for which $\log_{9}(p) = \log_{12}(q) = \log_{16}(p+q)$. What is the value of $\frac{q}{p}$? | \frac{1+\sqrt{5}}{2} |
Simplify $$\frac{11!}{9! + 2\cdot 8!}$$ | 90 |
Of the 60 students in the drama club, 36 take mathematics, 27 take physics and 20 students take both mathematics and physics. How many drama club students take neither mathematics nor physics? | 17 |
A classroom has 10 chairs arranged in a row. Tom and Jerry choose their seats at random, but they are not allowed to sit on the first and last chairs. What is the probability that they don't sit next to each other? | \frac{3}{4} |
Find the number of natural numbers \( k \) that do not exceed 291000 and such that \( k^{2} - 1 \) is divisible by 291. | 4000 |
Let $d$ be a positive number such that when $109$ is divided by $d$, the remainder is $4.$ Compute the sum of all possible two-digit values of $d$. | 71 |
Regular hexagon $ABCDEF$ has an area of $n$. Let $m$ be the area of triangle $ACE$. What is $\tfrac{m}{n}?$
A) $\frac{1}{2}$
B) $\frac{2}{3}$
C) $\frac{3}{4}$
D) $\frac{1}{3}$
E) $\frac{3}{2}$ | \frac{2}{3} |
At a roller derby, 4 teams are competing. Each team is made up of 10 members, and each member needs a pair of roller skates to compete in and another pair of skates as a backup. None of the skates have laces yet, so each member is offered 3 sets of laces per pair of skates. How many sets of laces have been handed out? | There are a total of 4 teams * 10 members per team = <<4*10=40>>40 team members.
Each person receives 1 pair of competing roller skates + 1 pair of backup roller skates = <<1+1=2>>2 pairs of roller skates.
So there are a total of 40 team members * 2 pairs of roller skates per team member = <<40*2=80>>80 roller skates.
... |
Given the function $f(x)=\sin (\omega x+\varphi)$ with $\omega > 0$ and $|\varphi| < \frac {\pi}{2}$, the function has a minimum period of $4\pi$ and, after being shifted to the right by $\frac {2\pi}{3}$ units, becomes symmetric about the $y$-axis. Determine the value of $\varphi$. | -\frac{\pi}{6} |
Let $ABC$ be an isosceles triangle with $AB=AC$ and incentre $I$ . If $AI=3$ and the distance from $I$ to $BC$ is $2$ , what is the square of length on $BC$ ? | 80 |
What is the average of all the integer values of $N$ such that $\frac{N}{84}$ is strictly between $\frac{4}{9}$ and $\frac{2}{7}$? | 31 |
Given the function $f(x)= \frac{x}{4} + \frac{a}{x} - \ln x - \frac{3}{2}$, where $a \in \mathbb{R}$, and the curve $y=f(x)$ has a tangent at the point $(1,f(1))$ which is perpendicular to the line $y=\frac{1}{2}x$.
(i) Find the value of $a$;
(ii) Determine the intervals of monotonicity and the extreme values for t... | -\ln 5 |
There are integers $x$ that satisfy the inequality $|x-2000|+|x| \leq 9999$. Find the number of such integers $x$. | 9999 |
Given the function $f\left( x \right)=\sin \left( 2x+\phi \right)\left(\left| \phi \right| < \dfrac{\pi }{2} \right)$ whose graph is symmetric about the point $\left( \dfrac{\pi }{3},0 \right)$, and $f\left( {{x}_{1}} \right)+f\left( {{x}_{2}} \right)=0$ when ${{x}_{1}},{{x}_{2}}\in \left( \dfrac{\pi }{12},\dfrac{7\pi ... | -\dfrac{\sqrt{3}}{2} |
Ms. Warren ran at 6 mph for 20 minutes. After the run, she walked at 2 mph for 30 minutes. How many miles did she run and walk in total? | 20 minutes is 20/60=1/3 of an hour.
Ms. Warren ran 6/3=<<6/3=2>>2 miles.
30 minutes is 30/60=1/2 of an hour.
Ms. Warren walked 2/2=<<2/2=1>>1 mile.
Ms Warren ran and walked 2+1=<<2+1=3>>3 miles in total.
#### 3 |
What is the least positive integer value of $x$ such that $(3x)^2 + 3 \cdot 29 \cdot 3x + 29^2$ is a multiple of 43? | 19 |
Find the limit, when $n$ tends to the infinity, of $$ \frac{\sum_{k=0}^{n} {{2n} \choose {2k}} 3^k} {\sum_{k=0}^{n-1} {{2n} \choose {2k+1}} 3^k} $$ | \sqrt{3} |
Calculate the sum of $0.\overline{6}$ and $0.\overline{7}$ as a common fraction. | \frac{13}{9} |
The altitude of an equilateral triangle is $\sqrt6$ units. What is the area of the triangle, in square units? Express your answer in simplest radical form. | 2\sqrt{3} |
If altitude $CD$ is $\sqrt3$ centimeters, what is the number of square centimeters in the area of $\Delta ABC$?
[asy] import olympiad; pair A,B,C,D; A = (0,sqrt(3)); B = (1,0);
C = foot(A,B,-B); D = foot(C,A,B); draw(A--B--C--A); draw(C--D,dashed);
label("$30^{\circ}$",A-(0.05,0.4),E);
label("$A$",A,N);label("$B$",B,E... | 2\sqrt{3} |
Find the total number of different integer values the function $$f(x)=[x]+[2 x]+\left[\frac{5 x}{3}\right]+[3 x]+[4 x]$$ takes for real numbers $x$ with $0 \leq x \leq 100$. Note: $[t]$ is the largest integer that does not exceed $t$. | 734 |
There are three pairs of real numbers $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$ that satisfy both $x^3-3xy^2=2005$ and $y^3-3x^2y=2004$. Compute $\left(1-\frac{x_1}{y_1}\right)\left(1-\frac{x_2}{y_2}\right)\left(1-\frac{x_3}{y_3}\right)$. | \frac{1}{1002} |
Ben "One Hunna Dolla" Franklin is flying a kite KITE such that $I E$ is the perpendicular bisector of $K T$. Let $I E$ meet $K T$ at $R$. The midpoints of $K I, I T, T E, E K$ are $A, N, M, D$, respectively. Given that $[M A K E]=18, I T=10,[R A I N]=4$, find $[D I M E]$. | 16 |
Given that the volume of a regular triangular pyramid $P-ABC$ is $\frac{1}{12}$, the center of its circumscribed sphere is $O$, and it satisfies $\vec{OA} + \vec{OB} + \vec{OC} = \vec{0}$, find the radius of the circumscribed sphere of the regular triangular pyramid $P-ABC$. | \frac{\sqrt{3}}{3} |
Given that the function $f(x)$ is an odd function defined on $\mathbb{R}$, and when $x > 0$, $f(x) = -x^{2} + 4x$.
- (Ⅰ) Find the analytical expression of the function $f(x)$.
- (Ⅱ) Find the minimum value of the function $f(x)$ on the interval $\left[-2,a\right]$ where $\left(a > -2\right)$. | -4 |
Find all integer $n$ such that the following property holds: for any positive real numbers $a,b,c,x,y,z$, with $max(a,b,c,x,y,z)=a$ , $a+b+c=x+y+z$ and $abc=xyz$, the inequality $$a^n+b^n+c^n \ge x^n+y^n+z^n$$ holds. | n \ge 0 |
Emilia needs 42 cartons of berries to make a berry cobbler. She already has 2 cartons of strawberries and 7 cartons of blueberries in her cupboard. She decides to go to the supermarket to get more cartons. How many more cartons of berries should Emilia buy? | She previously had 2 cartons + 7 cartons = <<2+7=9>>9 cartons of berries.
Emilia needs to buy 42 cartons - 9 cartons = <<42-9=33>>33 cartons of berries.
#### 33 |
Car A and Car B are traveling from point A to point B. Car A departs 6 hours later than Car B. The speed ratio of Car A to Car B is 4:3. 6 hours after Car A departs, its speed doubles, and both cars arrive at point B simultaneously. How many hours in total did Car A take to travel from A to B? | 8.4 |
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(... | 130 |
A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$. Three of the sides of this quadrilateral have length $200$. What is the length of the fourth side? | 500 |
Let $A B C D$ be a quadrilateral inscribed in a circle with diameter $\overline{A D}$. If $A B=5, A C=6$, and $B D=7$, find $C D$. | \sqrt{38} |
Right triangles \(ABC\) and \(ABD\) share a common hypotenuse \(AB = 5\). Points \(C\) and \(D\) are located on opposite sides of the line passing through points \(A\) and \(B\), with \(BC = BD = 3\). Point \(E\) lies on \(AC\), and \(EC = 1\). Point \(F\) lies on \(AD\), and \(FD = 2\). Find the area of the pentagon \... | 9.12 |
Given $\angle1+\angle2=180^\circ$ and $\angle3=\angle4,$ find $\angle4.$ Express your answer in degrees. [asy]
/* AMC8 1997 #12 Problem */
pair A=(0,0), B=(24,0), C=(48,0), D=(18,24), E=(12,48);
pen p=1mm+black;
draw(A--C);
draw(A--E);
draw(B--E);
draw(D--C);
label("70", A, NE);
label("40", shift(0,-7)*E, S);
label("1"... | 35^\circ |
Given a tetrahedron $P-ABC$, in the base $\triangle ABC$, $\angle BAC=60^{\circ}$, $BC=\sqrt{3}$, $PA\perp$ plane $ABC$, $PA=2$, then the surface area of the circumscribed sphere of this tetrahedron is ______. | 8\pi |
For how many integers $n$ between 1 and 11 (inclusive) is $\frac{n}{12}$ a repeating decimal? | 8 |
From 3 male students and 2 female students, calculate the number of different election results in which at least one female student is elected. | 14 |
The positive integers $A, B, C$, and $D$ form an arithmetic and geometric sequence as follows: $A, B, C$ form an arithmetic sequence, while $B, C, D$ form a geometric sequence. If $\frac{C}{B} = \frac{7}{3}$, what is the smallest possible value of $A + B + C + D$? | 76 |
Given an arithmetic sequence ${\_{a\_n}}$ with a non-zero common difference $d$, and $a\_7$, $a\_3$, $a\_1$ are three consecutive terms of a geometric sequence ${\_{b\_n}}$.
(1) If $a\_1=4$, find the sum of the first 10 terms of the sequence ${\_{a\_n}}$, denoted as $S_{10}$;
(2) If the sum of the first 100 terms of t... | 50 |
Let $a_n$ be the integer obtained by writing all the integers from $1$ to $n$ from left to right. For example, $a_3 = 123$ and $a_{11} = 1234567891011$. Compute the remainder when $a_{44}$ is divided by $45$. | 9 |
In a right triangular pyramid P-ABC, where PA, PB, and PC are mutually perpendicular and PA=1, the center of the circumscribed sphere is O. Find the distance from O to plane ABC. | \frac{\sqrt{3}}{6} |
There are $100$ piles of $400$ stones each. At every move, Pete chooses two piles, removes one stone from each of them, and is awarded the number of points, equal to the non- negative difference between the numbers of stones in two new piles. Pete has to remove all stones. What is the greatest total score Pete can get,... | 3920000 |
From point $A$ to point $B$ at 13:00, a bus and a cyclist left simultaneously. After arriving at point $B$, the bus, without stopping, returned and met the cyclist at point $C$ at 13:10. Returning to point $A$, the bus again without stopping headed towards point $B$ and caught up with the cyclist at point $D$, which is... | 40 |
For $k\ge 1$ , define $a_k=2^k$ . Let $$ S=\sum_{k=1}^{\infty}\cos^{-1}\left(\frac{2a_k^2-6a_k+5}{\sqrt{(a_k^2-4a_k+5)(4a_k^2-8a_k+5)}}\right). $$ Compute $\lfloor 100S\rfloor$ . | 157 |
If the consecutive integers from $50$ to $1$ were written as $$5049484746...,$$ what would be the $67^{\text{th}}$ digit to be written? | 1 |
What is the number of square meters in the area of a circle with diameter 4 meters? Express your answer in terms of $\pi$. | 4\pi |
To make a cherry pie, Veronica needs 3 pounds of pitted cherries. There are 80 single cherries in one pound of cherries. It takes 10 minutes to pit 20 cherries. How many hours will it take Veronica to pit all the cherries? | There are 80 cherries in a pound and she needs 3 pounds to make a pie so she needs 80*3 = <<80*3=240>>240 cherries
It takes her 10 minutes to pit a unit of 20 cherries. She has 240/20 = <<240/20=12>>12 units of cherries to pit
It takes 10 minutes to pit a unit of cherries and she has 12 units so it will take her 10*12 ... |
It's Yvette's turn to treat herself and her three best friends to a round of ice cream sundaes. Alicia orders the peanut butter sundae for $7.50. Brant orders the Royal banana split sundae for $10.00. Josh orders the death by chocolate sundae for $8.50 and Yvette orders the cherry jubilee sundae for $9.00. She leav... | Yvette spent 7.50+10+8.5+9 = $<<7.50+10+8.5+9=35.00>>35.00 on ice cream sundaes
Her bill comes to $35.00 and she leaves a 20% tip for the waiter so that's 35*.20 = $7.00
The bill was $35.00 and she added a $7.00 tip for a total of 35+7 = $<<35+7=42.00>>42.00
#### 42 |
Triangle $ABC$ has vertices $A(0, 8)$, $B(2, 0)$, $C(8, 0)$. A line through $B$ cuts the area of $\triangle ABC$ in half; find the sum of the slope and $y$-intercept of this line. | -2 |
Given the function $f(x)=2\sqrt{2}\cos x\sin\left(x+\frac{\pi}{4}\right)-1$.
(I) Find the value of $f\left(\frac{\pi}{4}\right)$;
(II) Find the maximum and minimum values of $f(x)$ in the interval $\left[0, \frac{\pi}{2}\right]$. | -1 |
Let points \( A_{1}, A_{2}, A_{3}, A_{4}, A_{5} \) be located on the unit sphere. Find the maximum value of \( \min \left\{A_{i} A_{j} \mid 1 \leq i < j \leq 5 \right\} \) and determine all cases where this maximum value is achieved. | \sqrt{2} |
Fifteen freshmen are sitting in a circle around a table, but the course assistant (who remains standing) has made only six copies of today's handout. No freshman should get more than one handout, and any freshman who does not get one should be able to read a neighbor's. If the freshmen are distinguishable but the hando... | 125 |
A river flows at a constant speed. Piers A and B are located upstream and downstream respectively, with a distance of 200 kilometers between them. Two boats, A and B, depart simultaneously from piers A and B, traveling towards each other. After meeting, they continue to their respective destinations, immediately return... | 14 |
Simplify first, then evaluate: $(\frac{{x-3}}{{{x^2}-1}}-\frac{2}{{x+1}})\div \frac{x}{{{x^2}-2x+1}}$, where $x=(\frac{1}{2})^{-1}+\left(\pi -1\right)^{0}$. | -\frac{2}{3} |
Let $f$ be a function taking the positive integers to the positive integers, such that
(i) $f$ is increasing (i.e. $f(n + 1) > f(n)$ for all positive integers $n$)
(ii) $f(mn) = f(m) f(n)$ for all positive integers $m$ and $n,$ and
(iii) if $m \neq n$ and $m^n = n^m,$ then $f(m) = n$ or $f(n) = m.$
Find the sum of al... | 900 |
Person A and Person B started working on the same day. The company policy states that Person A works for 3 days and then rests for 1 day, while Person B works for 7 days and then rests for 3 consecutive days. How many days do Person A and Person B rest on the same day within the first 1000 days? | 100 |
In a geometric sequence $\\{a\_n\\}$, $a\_n > 0 (n \in \mathbb{N}^*)$, the common ratio $q \in (0, 1)$, and $a\_1a\_5 + 2aa\_5 + a\_2a\_8 = 25$, and the geometric mean of $a\_3$ and $a\_5$ is $2$.
(1) Find the general term formula of the sequence $\\{a\_n\\}$;
(2) If $b\_n = \log_2 a\_n$, find the sum of the first $n... | 19 |
A circle with radius 1 is tangent to a circle with radius 3 at point \( C \). A line passing through point \( C \) intersects the smaller circle at point \( A \) and the larger circle at point \( B \). Find \( AC \), given that \( AB = 2\sqrt{5} \). | \frac{\sqrt{5}}{2} |
Find $(-2)^{3}+(-2)^{2}+(-2)^{1}+2^{1}+2^{2}+2^{3}$. | 8 |
Determine the value of
\[3003 + \frac{1}{3} \left( 3002 + \frac{1}{3} \left( 3001 + \dots + \frac{1}{3} \left( 4 + \frac{1}{3} \cdot 3 \right) \right) \dotsb \right).\] | 9006 |
Two athletes decided to compete to see who had the best jumping ability. They were each going to do the long jump, triple jump, and high jump to see who had the highest average jump. The first athlete jumped 26 feet in the long jump, 30 feet in the triple jump, and 7 feet in the high jump. The second athlete jumped 24 ... | The first athlete jumped a total of 63 feet because 26 + 30 + 7 = <<26+30+7=63>>63
The second athlete jumped a total of 66 feet because 24 + 34 + 8 = <<24+34+8=66>>66
The first athlete jumped 21 feet on average because 63 / 3 = <<63/3=21>>21
The second athlete jumped 22 feet on average because 66 / 3 = <<66/3=22>>22
Th... |
In the sequence $\{a_n\}$, $a_n+a_{n+1}+a_{n+2}=(\sqrt{2})^{n}$. Find the sum of the first $9$ terms of the sequence $\{a_n\}$ (express the answer as a numerical value). | 4+9\sqrt{2} |
An element is randomly chosen from among the first $15$ rows of Pascal's Triangle. What is the probability that the value of the element chosen is $1$?
Note: The 1 at the top is often labelled the "zeroth" row of Pascal's Triangle, by convention. So to count a total of 15 rows, use rows 0 through 14. | \frac{29}{120} |
How many positive integers less than $201$ are multiples of either $6$ or $8$, but not both at once? | 42 |
$X, Y$ and $Z$ are pairwise disjoint sets of people. The average ages of people in the sets
$X, Y, Z, X \cup Y, X \cup Z$ and $Y \cup Z$ are $37, 23, 41, 29, 39.5$ and $33$ respectively.
Find the average age of the people in set $X \cup Y \cup Z$. | 34 |
Three of the edges of a cube are $\overline{AB}, \overline{BC},$ and $\overline{CD},$ and $\overline{AD}$ is an interior diagonal. Points $P, Q,$ and $R$ are on $\overline{AB}, \overline{BC},$ and $\overline{CD},$ respectively, so that $AP = 5, PB = 15, BQ = 15,$ and $CR = 10.$ What is the area of the polygon that is t... | 525 |
Two identical cylindrical containers are connected at the bottom by a small tube with a tap. While the tap was closed, water was poured into the first container, and oil was poured into the second one, so that the liquid levels were the same and equal to $h = 40$ cm. At what level will the water in the first container ... | 16.47 |
If the system of equations
\[
\begin{align*}
4x + y &= a, \\
3x + 4y^2 &= 3a,
\end{align*}
\]
has a solution $(x,y)$ when $x=3$, compute $a$. | 9.75 |
A particle moves so that its speed for the second and subsequent miles varies inversely as the integral number of miles already traveled. For each subsequent mile the speed is constant. If the second mile is traversed in $2$ hours, then the time, in hours, needed to traverse the $n$th mile is: | 2(n-1) |
In the diagram, the circle has center \( O \) and square \( OPQR \) has vertex \( Q \) on the circle. If the area of the circle is \( 72 \pi \), the area of the square is: | 36 |
Tom bought 10 packages of miniature racing cars. Each package contains five cars. He gave each of his two nephews 1/5 of the cars. How many miniature racing cars are left with Tom? | Tom had 10 x 5 = <<10*5=50>>50 miniature racing cars.
Tom gave 50 x 1/5 = <<50*1/5=10>>10 miniature racing cars to each of his two nephews.
So, he gave a total of 10 x 2 = <<10*2=20>>20 cars to his two nephews.
Therefore, Tom is left with 50 - 20 = <<50-20=30>>30 miniature racing cars.
#### 30 |
A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits ($\spadesuit$, $\heartsuit$, $\diamondsuit$, and $\clubsuit$), such that there is exactly one card for any given rank and suit. Two of the suits ($\spadesuit$ and $\clubsuit$) are black and the other two suits ($\... | \dfrac{1}{52} |
What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50? | 3127 |
How many integers from 1 through 9999, inclusive, do not contain any of the digits 2, 3, 4 or 5? | 1295 |
A teacher tells the class,
"Think of a number, add 1 to it, and double the result. Give the answer to your partner. Partner, subtract 1 from the number you are given and double the result to get your answer."
Ben thinks of $6$, and gives his answer to Sue. What should Sue's answer be? | 26 |
Two distinct primes, each greater than 20, are multiplied. What is the least possible product of these two primes? | 667 |
Define: \( a \oplus b = a \times b \), \( c \bigcirc d = d \times d \times d \times \cdots \times d \) (d multiplied c times). Find \( (5 \oplus 8) \oplus (3 \bigcirc 7) \). | 13720 |
When three standard dice are tossed, the numbers $a, b, c$ are obtained. Find the probability that $abc = 72$. | \frac{1}{72} |
We roll five dice, each a different color. In how many ways can the sum of the rolls be 11? | 205 |
How many $6$ -tuples $(a, b, c, d, e, f)$ of natural numbers are there for which $a>b>c>d>e>f$ and $a+f=b+e=c+d=30$ are simultaneously true? | 364 |
If the graph of the function $f(x) = \sin \omega x \cos \omega x + \sqrt{3} \sin^2 \omega x - \frac{\sqrt{3}}{2}$ ($\omega > 0$) is tangent to the line $y = m$ ($m$ is a constant), and the abscissas of the tangent points form an arithmetic sequence with a common difference of $\pi$.
(Ⅰ) Find the values of $\omega$ and... | \frac{11\pi}{3} |
Narsa buys a package of 45 cookies on Monday morning. How many cookies are left in the package after Friday? | 15 |
Given two functions $f(x) = e^{2x-3}$ and $g(x) = \frac{1}{4} + \ln \frac{x}{2}$, if $f(m) = g(n)$ holds, calculate the minimum value of $n-m$. | \frac{1}{2} + \ln 2 |
Find the minimum value of the expression \(\frac{5 x^{2}-8 x y+5 y^{2}-10 x+14 y+55}{\left(9-25 x^{2}+10 x y-y^{2}\right)^{5 / 2}}\). If necessary, round the answer to hundredths. | 0.19 |
Solve the application problem by setting up equations:<br/>A gift manufacturing factory receives an order for a batch of teddy bears and plans to produce them in a certain number of days. If they produce $20$ teddy bears per day, they will be $100$ short of the order. If they produce $23$ teddy bears per day, they will... | 40 |
Mike decides to develop a plot of land. He bought 200 acres for $70 per acre. After development, he sold half of the acreage for $200 per acre. How much profit did he make? | He bought the land for 200*70=$<<200*70=14000>>14000
He sold 200/2=<<200/2=100>>100 acres
He got 100*200=$<<100*200=20000>>20,000 from selling this
So he made a profit of 20,000-14,000=$<<20000-14000=6000>>6,000
#### 6000 |
The foci of the ellipse \(\frac{x^2}{25} + \frac{y^2}{b^2} = 1\) and the foci of the hyperbola
\[\frac{x^2}{196} - \frac{y^2}{121} = \frac{1}{49}\] coincide. Find \(b^2\). | \frac{908}{49} |
Ms. Johnson awards bonus points to students in her class whose test scores are above the median. The class consists of 81 students. What is the maximum number of students who could receive bonus points? | 40 |
What is the least whole number that is divisible by 7, but leaves a remainder of 1 when divided by any integer 2 through 6? | 301 |
There are 20 boys and 11 girls in the second grade and twice that number in the third grade. How many students are in grades 2 and 3? | There are 20 + 11 = <<20+11=31>>31 students in second grade.
There are 31 x 2 = <<31*2=62>>62 students in third grade.
In total, there are 31 + 62 = <<31+62=93>>93 students.
#### 93 |
A $4 \times 4$ window is made out of 16 square windowpanes. How many ways are there to stain each of the windowpanes, red, pink, or magenta, such that each windowpane is the same color as exactly two of its neighbors? | 24 |
In triangle $PQR$, $QR = 24$. An incircle of the triangle trisects the median $PS$ from $P$ to side $QR$. Given that the area of the triangle is $k \sqrt{p}$, where $k$ and $p$ are integers and $p$ is square-free, find $k+p$. | 106 |
How many pairs $(x, y)$ of non-negative integers with $0 \leq x \leq y$ satisfy the equation $5x^{2}-4xy+2x+y^{2}=624$? | 7 |
Brandon has a collection of 20 baseball cards. Malcom has 8 more cards than Brandon. However, then Malcom gives half of his cards to his friend Mark. How many cards does Malcom have left? | Malcom has 20 cards + 8 cards = <<20+8=28>>28 cards.
Malcom gives away 1/2 * 28 cards = <<1/2*28=14>>14 cards to Mark.
Malcom has 28-14 cards = <<28-14=14>>14 cards remaining.
#### 14 |
Sets $A, B$ , and $C$ satisfy $|A| = 92$ , $|B| = 35$ , $|C| = 63$ , $|A\cap B| = 16$ , $|A\cap C| = 51$ , $|B\cap C| = 19$ . Compute the number of possible values of $ |A \cap B \cap C|$ . | 10 |
Compute
\[\left( 1 - \frac{1}{\cos 23^\circ} \right) \left( 1 + \frac{1}{\sin 67^\circ} \right) \left( 1 - \frac{1}{\sin 23^\circ} \right) \left( 1 + \frac{1}{\cos 67^\circ} \right).\] | 1 |
Two vertices of an obtuse triangle are $(6,4)$ and $(0,0)$. The third vertex is located on the negative branch of the $x$-axis. What are the coordinates of the third vertex if the area of the triangle is 30 square units? | (-15, 0) |
What is the area of the region defined by the equation $x^2+y^2 - 7 = 4y-14x+3$? | 63\pi |
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