problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Given that 2 students exercised for 0 days, 4 students exercised for 1 day, 2 students exercised for 2 days, 5 students exercised for 3 days, 4 students exercised for 4 days, 7 students exercised for 5 days, 3 students exercised for 6 days, and 2 students exercised for 7 days, find the mean number of days of exercise, rounded to the nearest hundredth. | 3.66 |
Find the distance between the foci of the ellipse
\[\frac{x^2}{20} + \frac{y^2}{4} = 7.\] | 8 \sqrt{7} |
In rectangle \( ABCD \), a circle \(\omega\) is constructed using side \( AB \) as its diameter. Let \( P \) be the second intersection point of segment \( AC \) with circle \(\omega\). The tangent to \(\omega\) at point \( P \) intersects segment \( BC \) at point \( K \) and passes through point \( D \). Find \( AD \), given that \( KD = 36 \). | 24 |
In the land of Draconia, there are red, green, and blue dragons. Each dragon has three heads, and every head either always tells the truth or always lies. Each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each of them said:
- 1st head: "On my left is a green dragon."
- 2nd head: "On my right is a blue dragon."
- 3rd head: "There is no red dragon next to me."
What is the maximum number of red dragons that could have been seated at the table? | 176 |
Yannick picks a number $N$ randomly from the set of positive integers such that the probability that $n$ is selected is $2^{-n}$ for each positive integer $n$. He then puts $N$ identical slips of paper numbered 1 through $N$ into a hat and gives the hat to Annie. Annie does not know the value of $N$, but she draws one of the slips uniformly at random and discovers that it is the number 2. What is the expected value of $N$ given Annie's information? | \frac{1}{2 \ln 2-1} |
Susie and Britney each keep chickens, of two different breeds. Susie has 11 Rhode Island Reds and 6 Golden Comets. Britney has twice as many Rhode Island Reds as Susie, but only half as many Golden Comets. How many more chickens are in Britney's flock than in Susie's? | Britney has 11*2=<<11*2=22>>22 Rhode Island Reds.
She has 6/2=<<6/2=3>>3 Golden Comets.
She has 22+3=<<22+3=25>>25 total.
Susie has 11+6=<<11+6=17>>17 total.
Britney has 25-17=<<25-17=8>>8 more.
#### 8 |
Emily's broken clock runs backwards at five times the speed of a regular clock. How many times will it display the correct time in the next 24 hours? Note that it is an analog clock that only displays the numerical time, not AM or PM. The clock updates continuously. | 12 |
How many distinct four-digit even numbers can be formed using the digits 0, 1, 2, 3? | 10 |
Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy $| \overrightarrow {a}|=2$, $| \overrightarrow {b}|=1$, and $\overrightarrow {b} \perp ( \overrightarrow {a}+ \overrightarrow {b})$, determine the projection of vector $\overrightarrow {a}$ onto vector $\overrightarrow {b}$. | -1 |
Let $X=\{1,2,3,...,10\}$ . Find the number of pairs of $\{A,B\}$ such that $A\subseteq X, B\subseteq X, A\ne B$ and $A\cap B=\{5,7,8\}$ . | 2186 |
For how many integers $n$ with $1 \le n \le 2012$ is the product
\[
\prod_{k=0}^{n-1} \left( \left( 1 + e^{2 \pi i k / n} \right)^n + 1 \right)
\]equal to zero? | 335 |
Let $K$ be the measure of the area bounded by the $x$-axis, the line $x=8$, and the curve defined by
\[f=\{(x,y)\quad |\quad y=x \text{ when } 0 \le x \le 5, y=2x-5 \text{ when } 5 \le x \le 8\}.\]
Then $K$ is: | 36.5 |
Given three points in space \\(A(0,2,3)\\), \\(B(-2,1,6)\\), and \\(C(1,-1,5)\\):
\\((1)\\) Find \\(\cos < \overrightarrow{AB}, \overrightarrow{AC} >\\).
\\((2)\\) Find the area of the parallelogram with sides \\(AB\\) and \\(AC\\). | 7\sqrt{3} |
Two of the roots of the equation \[ax^3+bx^2+cx+d=0\]are $3$ and $-2.$ Given that $a \neq 0,$ compute $\frac{b+c}{a}.$ | -7 |
Find the product of $218_9 \cdot 5_9$. Express your answer in base 9. | 1204_9 |
The harmonic mean of two positive integers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x,y)$ with $x<y$ is the harmonic mean of $x$ and $y$ equal to $6^{20}$?
| 799 |
In the number $2 * 0 * 1 * 6 * 0 *$, each of the 5 asterisks must be replaced with any of the digits $0,1,2,3,4,5,6,7,8$ (digits can be repeated) so that the resulting 10-digit number is divisible by 18. How many ways can this be done? | 3645 |
If 24 out of every 60 individuals like football and out of those that like it, 50% play it, how many people would you expect play football out of a group of 250? | If 50% of individuals that like football play it, and there are 24 out of 60 individuals that like it, then 24*50% = 12 out of 60 individuals play it
If 12 individuals play it out of 60 individuals then that means only 12/60 = 1/5 or 20% play it
If 20% of individuals play football out of a group of 250 people, one would expect 250*20% = <<250*20*.01=50>>50 individuals to play it
#### 50 |
How many multiples of 4 are between 100 and 350? | 62 |
Let $S$ be the set $\{1,2,3,...,19\}$. For $a,b \in S$, define $a \succ b$ to mean that either $0 < a - b \le 9$ or $b - a > 9$. How many ordered triples $(x,y,z)$ of elements of $S$ have the property that $x \succ y$, $y \succ z$, and $z \succ x$? | 855 |
Solve
\[-1 < \frac{x^2 - 14x + 11}{x^2 - 2x + 3} < 1.\] | \left( \frac{2}{3}, 1 \right) \cup (7,\infty) |
Expand $(x^{22}-3x^{5} + x^{-2} - 7)\cdot(5x^4)$. | 5x^{26}-15x^9-35x^4+5x^2 |
In triangle $∆ABC$, the angles $A$, $B$, $C$ correspond to the sides $a$, $b$, $c$ respectively. The vector $\overrightarrow{m}\left(a, \sqrt{3b}\right)$ is parallel to $\overrightarrow{n}=\left(\cos A,\sin B\right)$.
(1) Find $A$;
(2) If $a= \sqrt{7},b=2$, find the area of $∆ABC$. | \dfrac{3 \sqrt{3}}{2} |
Toss a fair coin, with the probability of landing heads or tails being $\frac {1}{2}$ each. Construct the sequence $\{a_n\}$ such that
$$
a_n=
\begin{cases}
1 & \text{if the } n\text{-th toss is heads,} \\
-1 & \text{if the } n\text{-th toss is tails.}
\end{cases}
$$
Define $S_n =a_1+a_2+…+a_n$. Find the probability that $S_2 \neq 0$ and $S_8 = 2$. | \frac {13}{128} |
Given that $a, b, c$ are positive integers satisfying $$a+b+c=\operatorname{gcd}(a, b)+\operatorname{gcd}(b, c)+\operatorname{gcd}(c, a)+120$$ determine the maximum possible value of $a$. | 240 |
Anh read a book. On the first day she read $n$ pages in $t$ minutes, where $n$ and $t$ are positive integers. On the second day Anh read $n + 1$ pages in $t + 1$ minutes. Each day thereafter Anh read one more page than she read on the previous day, and it took her one more minute than on the previous day until she completely read the $374$ page book. It took her a total of $319$ minutes to read the book. Find $n + t$. | 53 |
Let $\triangle PQR$ have side lengths $PQ=13$, $PR=15$, and $QR=14$. Inside $\angle QPR$ are two circles: one is tangent to rays $\overline{PQ}$, $\overline{PR}$, and segment $\overline{QR}$; the other is tangent to the extensions of $\overline{PQ}$ and $\overline{PR}$ beyond $Q$ and $R$, and also tangent to $\overline{QR}$. Compute the distance between the centers of these two circles. | 5\sqrt{13} |
Given a function $f(x) = 2\sin x\cos x + 2\cos\left(x + \frac{\pi}{4}\right)\cos\left(x - \frac{\pi}{4}\right)$,
1. Find the interval(s) where $f(x)$ is monotonically decreasing.
2. If $\alpha \in (0, \pi)$ and $f\left(\frac{\alpha}{2}\right) = \frac{\sqrt{2}}{2}$, find the value of $\sin \alpha$. | \frac{\sqrt{6} + \sqrt{2}}{4} |
There is a prime number which is a factor of every sum of three consecutive integers. What is the number? | 3 |
If the graph of the function $y=3\sin(2x+\phi)$ $(0 < \phi < \pi)$ is symmetric about the point $\left(\frac{\pi}{3},0\right)$, then $\phi=$ ______. | \frac{\pi}{3} |
At the end of a professional bowling tournament, the top 6 bowlers have a playoff. First #6 bowls #5. The loser receives $6^{th}$ prize and the winner bowls #4 in another game. The loser of this game receives $5^{th}$ prize and the winner bowls #3. The loser of this game receives $4^{th}$ prize and the winner bowls #2. The loser of this game receives $3^{rd}$ prize, and the winner bowls #1. The winner of this game gets 1st prize and the loser gets 2nd prize. In how many orders can bowlers #1 through #6 receive the prizes? | 32 |
James prints out 2 books. They are each 600 pages long. He prints out double-sided and 4 pages per side. How many sheets of paper does he use? | He printed 600*2=<<600*2=1200>>1200 pages
Each sheet of paper was 4*2=<<4*2=8>>8 pages
So he used 1200/8=<<1200/8=150>>150 sheets of paper
#### 150 |
Given two lines $l_{1}$: $(3+m)x+4y=5-3m$ and $l_{2}$: $2x+(5+m)y=8$ are parallel, the value of the real number $m$ is ______. | -7 |
In a circle of radius $5$ units, $CD$ and $AB$ are perpendicular diameters. A chord $CH$ cutting $AB$ at $K$ is $8$ units long. The diameter $AB$ is divided into two segments whose dimensions are: | 2,8 |
Evaluate the expression \[(5^{1001} + 6^{1002})^2 - (5^{1001} - 6^{1002})^2\] and express it in the form \(k \cdot 30^{1001}\) for some integer \(k\). | 24 |
Given a parameterized curve $ C: x\equal{}e^t\minus{}e^{\minus{}t},\ y\equal{}e^{3t}\plus{}e^{\minus{}3t}$ .
Find the area bounded by the curve $ C$ , the $ x$ axis and two lines $ x\equal{}\pm 1$ . | \frac{5\sqrt{5}}{2} |
A printer prints text pages at a rate of 17 pages per minute and graphic pages at a rate of 10 pages per minute. If a document consists of 250 text pages and 90 graphic pages, how many minutes will it take to print the entire document? Express your answer to the nearest whole number. | 24 |
Sarah uses 1 ounce of shampoo, and one half as much conditioner as shampoo daily. In two weeks, what is the total volume of shampoo and conditioner, in ounces, that Sarah will use? | Sarah uses 1+0.5=<<1+0.5=1.5>>1.5 ounces of hair care products daily.
Two weeks is 2*7=<<2*7=14>>14 days.
In 14 days, Sarah will use 14*1.5=<<14*1.5=21>>21 ounces of hair care products.
#### 21 |
Suppose $\sqrt{1 + \sqrt{2y-3}} = \sqrt{6}$; find $y$. | 14 |
The quadratic equation $ax^2+20x+c=0$ has exactly one solution. If $a+c=29$, and $a<c$ find the ordered pair $(a,c)$. | (4,25) |
Given that the numbers - 2, 5, 8, 11, and 14 are arranged in a specific cross-like structure, find the maximum possible sum for the numbers in either the row or the column. | 36 |
Find the number of ordered pairs $(m, n)$ such that $m$ and $n$ are positive integers in the set $\{1, 2, ..., 30\}$ and the greatest common divisor of $2^m + 1$ and $2^n - 1$ is not $1$. | 295 |
$\frac{2}{25}=$ | .08 |
Given polynomials $A=ax^{2}+3xy+2|a|x$ and $B=2x^{2}+6xy+4x+y+1$.
$(1)$ If $2A-B$ is a quadratic trinomial in terms of $x$ and $y$, find the value of $a$.
$(2)$ Under the condition in $(1)$, simplify and evaluate the expression $3(-3a^{2}-2a)-[a^{2}-2(5a-4a^{2}+1)-2a]$. | -22 |
Arrange for 7 staff members to be on duty from May 1st to May 7th. Each person is on duty for one day, with both members A and B not being scheduled on May 1st and 2nd. The total number of different scheduling methods is $\_\_\_\_\_\_\_$. | 2400 |
Find all real numbers $x$ satisfying the equation $x^{3}-8=16 \sqrt[3]{x+1}$. | -2,1 \pm \sqrt{5} |
One dimension of a cube is tripled, another is decreased by `a/2`, and the third dimension remains unchanged. The volume gap between the new solid and the original cube is equal to `2a^2`. Calculate the volume of the original cube. | 64 |
Divide the numbers 1 through 10 into three groups such that the difference between any two numbers within the same group does not appear in that group. If one of the groups is {2, 5, 9}, then the sum of all the numbers in the group that contains 10 is $\qquad$. | 22 |
For all real numbers $r$ and $s$, define the mathematical operation $\#$ such that the following conditions apply: $r\ \#\ 0 = r, r\ \#\ s = s\ \#\ r$, and $(r + 1)\ \#\ s = (r\ \#\ s) + s + 1$. What is the value of $11\ \#\ 5$? | 71 |
Let the complex numbers \(z\) and \(w\) satisfy \(|z| = 3\) and \((z + \bar{w})(\bar{z} - w) = 7 + 4i\), where \(i\) is the imaginary unit and \(\bar{z}\), \(\bar{w}\) denote the conjugates of \(z\) and \(w\) respectively. Find the modulus of \((z + 2\bar{w})(\bar{z} - 2w)\). | \sqrt{65} |
How many combinations of pennies (1 cent), nickels (5 cents) and/or dimes (10 cents) are there with a total value of 25 cents? | 12 |
Find the number of ordered quadruples $(a,b,c,d)$ of real numbers such that
\[\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \renewcommand{\arraystretch}{1.5} \begin{pmatrix} \frac{1}{a} & \frac{1}{b} \\ \frac{1}{c} & \frac{1}{d} \end{pmatrix} \renewcommand{\arraystretch}{1}.\] | 0 |
Given that when 81849, 106392, and 124374 are divided by an integer \( n \), the remainders are equal. If \( a \) is the maximum value of \( n \), find \( a \). | 243 |
Rhombus $ABCD$ has side length $2$ and $\angle B = 120^\circ$. Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. What is the area of $R$? | \frac{2\sqrt{3}}{3} |
Given that a three-digit positive integer (a_1 a_2 a_3) is said to be a convex number if it satisfies (a_1 < a_2 > a_3), determine the number of three-digit convex numbers. | 240 |
Given the following system of equations: $$ \begin{cases} R I +G +SP = 50 R I +T + M = 63 G +T +SP = 25 SP + M = 13 M +R I = 48 N = 1 \end{cases} $$
Find the value of L that makes $LMT +SPR I NG = 2023$ true.
| \frac{341}{40} |
Let \[f(x) = \left\{
\begin{array}{cl}
-x - 3 & \text{if } x \le 1, \\
\frac{x}{2} + 1 & \text{if } x > 1.
\end{array}
\right.\]Find the sum of all values of $x$ such that $f(x) = 0$. | -3 |
Given an isosceles triangle $ABC$ satisfying $AB=AC$, $\sqrt{3}BC=2AB$, and point $D$ is on side $BC$ with $AD=BD$, then the value of $\sin \angle ADB$ is ______. | \frac{2 \sqrt{2}}{3} |
What is the smallest positive multiple of $23$ that is $4$ more than a multiple of $89$? | 805 |
Simplify first, then evaluate: $(1- \frac {2}{x+1})÷ \frac {x^{2}-x}{x^{2}-1}$, where $x=-2$. | \frac {3}{2} |
Given the function $f(x)=2\sin ωx\cos ωx-2\sqrt{3} \cos ^{2}ωx+\sqrt{3} (ω > 0)$, and the distance between two adjacent symmetry axes of the graph of $y=f(x)$ is $\frac{π}{2}$.
(I) Find the period of the function $f(x)$;
(II) In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Angle $C$ is acute, and $f(C)=\sqrt{3}$, $c=3\sqrt{2}$, $\sin B=2\sin A$. Find the area of $\triangle ABC$. | 3\sqrt{3} |
A window is made up of 8 glass panes. Each pane has a length of 12 inches and a width of 8 inches. What is the area of the window? | The area of a glass pane is 8 x 12 = <<8*12=96>>96 sq in
The area of the window is 96 x 8 = <<96*8=768>>768 sq in
#### 768 |
Given the hyperbola $$\frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1$$ ($a>0$, $b>0$) with its right focus at $F(c, 0)$.
(1) If one of the asymptotes of the hyperbola is $y=x$ and $c=2$, find the equation of the hyperbola;
(2) With the origin $O$ as the center and $c$ as the radius, draw a circle. Let the intersection of the circle and the hyperbola in the first quadrant be $A$. Draw the tangent line to the circle at $A$, with a slope of $-\sqrt{3}$. Find the eccentricity of the hyperbola. | \sqrt{2} |
A month with $31$ days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month? | 2 |
Matt has somewhere between 1000 and 2000 pieces of paper he's trying to divide into piles of the same size (but not all in one pile or piles of one sheet each). He tries $2,3,4,5,6,7$, and 8 piles but ends up with one sheet left over each time. How many piles does he need? | 41 |
In ancient China, when determining the pentatonic scale of Gong, Shang, Jiao, Zhi, and Yu, a method of calculation involving a three-part loss and gain was initially used. The second note was obtained by subtracting one-third from the first note, and the third note was obtained by adding one-third to the second note, and so on until all five notes were obtained. For example, if the length of an instrument capable of producing the first fundamental note is 81, then the length of an instrument capable of producing the second fundamental note would be $81 \times (1-\frac{1}{3}) = 54$, and the length of an instrument capable of producing the third fundamental note would be $54 \times (1+\frac{1}{3}) = 72$, that is, decreasing by one-third first and then increasing by one-third. If the length of an instrument capable of producing the first fundamental note is $a$, and the length of an instrument capable of producing the fourth fundamental note is 32, then the value of $a$ is ____. | 54 |
A Tim number is a five-digit positive integer with the following properties:
1. It is a multiple of 15.
2. Its hundreds digit is 3.
3. Its tens digit is equal to the sum of its first three (leftmost) digits.
How many Tim numbers are there? | 16 |
On the board, two sums are written:
$$
\begin{array}{r}
1+22+333+4444+55555+666666+7777777+ \\
+88888888+999999999
\end{array}
$$
and
$9+98+987+9876+98765+987654+9876543+$
$+98765432+987654321$
Determine which of them is greater (or if they are equal). | 1097393685 |
The region consisting of all points in three-dimensional space within 4 units of line segment $\overline{CD}$, plus a cone with the same height as $\overline{CD}$ and a base radius of 4 units, has a total volume of $448\pi$. Find the length of $\textit{CD}$. | 17 |
2 diagonals of a regular heptagon (a 7-sided polygon) are chosen. What is the probability that they intersect inside the heptagon? | \dfrac{5}{13} |
Emily has 6 marbles. Megan gives Emily double the number she has. Emily then gives Megan back half of her new total plus 1. How many marbles does Emily have now? | Megan gives Emily 6*2=<<6*2=12>>12 marbles.
Emily then has 6+12=<<6+12=18>>18 marbles.
Emily gives Megan 18/2+1=<<18/2+1=10>>10 marbles.
Emily has 18-10=<<18-10=8>>8 marbles now.
#### 8 |
After Sally takes 20 shots, she has made $55\%$ of her shots. After she takes 5 more shots, she raises her percentage to $56\%$. How many of the last 5 shots did she make? | 3 |
The coordinates of the vertices of isosceles trapezoid $ABCD$ are all integers, with $A=(20,100)$ and $D=(21,107)$. The trapezoid has no horizontal or vertical sides, and $\overline{AB}$ and $\overline{CD}$ are the only parallel sides. The sum of the absolute values of all possible slopes for $\overline{AB}$ is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
| 131 |
Alice made 52 friendship bracelets over spring break to sell at school. It only cost her $3.00 in materials to make these bracelets. During the break, she gave 8 of her bracelets away. Back at school, she sells all of the remaining bracelets at $0.25 each. How much profit did she make (money earned after paying initial costs) on the sale of her bracelets? | She made 52 bracelets but gave 8 away so she had 50-8 = <<52-8=44>>44 bracelets left.
She sold 44 bracelets at $0.25 each for a total of 44 * .25 = $<<44*.25=11.00>>11.00
She made $11.00 but she spent $3.00 on the supplies. So her profit would be 11-3 = $<<11-3=8.00>>8.00
#### 8 |
Sean and Sierra invited 200 guests to their wedding. If 83% of the guests RSVP with a Yes response and 9% of the guests RSVP with a No response, how many guests did not respond at all? | Yes RSVP responses are 200 x 83% = <<200*83*.01=166>>166 guests.
No RSVP responses are 200 x 9% = <<200*9*.01=18>>18 guests.
Guests who responded are 166 + 18 = <<166+18=184>>184.
Guests who did not respond at all are 200 - 184 = <<200-184=16>>16.
#### 16 |
Find the quadratic function $f(x) = x^2 + ax + b$ such that
\[\frac{f(f(x) + x)}{f(x)} = x^2 + 1776x + 2010.\] | x^2 + 1774x + 235 |
There are several pairs of integers $ (a, b) $ satisfying $ a^2 - 4a + b^2 - 8b = 30 $ . Find the sum of the sum of the coordinates of all such points. | 60 |
The graphs of $y=|x|$ and $y=-x^2-3x-2$ are drawn. For every $x$, a vertical segment connecting these two graphs can be drawn as well. Find the smallest possible length of one of these vertical segments. | 1 |
In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a\cos B - b\cos A = c$, and $C = \frac{π}{5}$, then find the measure of angle $B$. | \frac{3\pi}{10} |
Calculate the arc lengths of the curves given by the equations in the rectangular coordinate system.
$$
y=2-e^{x}, \ln \sqrt{3} \leq x \leq \ln \sqrt{8}
$$ | 1 + \frac{1}{2} \ln \frac{3}{2} |
The energy stored by any pair of positive charges is inversely proportional to the distance between them, and directly proportional to their charges. Four identical point charges start at the vertices of a square, and this configuration stores 20 Joules of energy. How much more energy, in Joules, would be stored if one of these charges was moved to the center of the square? | 5(3\sqrt{2} - 3) |
Find the point in the plane $3x - 4y + 5z = 30$ that is closest to the point $(1,2,3).$ | \left( \frac{11}{5}, \frac{2}{5}, 5 \right) |
The graph of the equation \[ x^2 + 4y^2 - 10x + 56y = k\]is a non-degenerate ellipse if and only if $k > a.$ What is $a?$ | -221 |
Given that the curves $y=x^2-1$ and $y=1+x^3$ have perpendicular tangents at $x=x_0$, find the value of $x_0$. | -\frac{1}{\sqrt[3]{6}} |
The midpoints of the sides of a parallelogram with area $P$ are joined to form a smaller parallelogram inside it. What is the ratio of the area of the smaller parallelogram to the area of the original parallelogram? Express your answer as a common fraction. | \frac{1}{4} |
Given that $5^{2018}$ has $1411$ digits and starts with $3$ (the leftmost non-zero digit is $3$ ), for how many integers $1\leq n\leq2017$ does $5^n$ start with $1$ ?
*2018 CCA Math Bonanza Tiebreaker Round #3* | 607 |
An amusement park sells tickets for $3. This week it welcomed 100 people per day but on Saturday it was visited by 200 people and on Sunday by 300. How much money did it make overall in a week? | During the weekdays it was visited by 100 * 5= <<100*5=500>>500 people
On Sunday and Saturday, it was visited by 200 + 300 = <<200+300=500>>500 people
In total 500 + 500 = <<500+500=1000>>1000 people visited the park in a week
So it earned 1000 * 3 = <<1000*3=3000>>3000 dollars in a week
#### 3000 |
What is the smallest whole number larger than the perimeter of any triangle with a side of length $5$ and a side of length $19$? | 48 |
Three balls are randomly and independantly tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin $i$ is $2^{-i}$ for $i=1,2,3,....$ More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is $\frac pq,$ where $p$ and $q$ are relatively prime positive integers. (For example, the balls are evenly spaced if they are tossed into bins $3,17,$ and $10.$) What is $p+q?$ | 55 |
The sum of two numbers is 50 and their difference is 6. What is their product? | 616 |
In $\triangle ABC$, if $\angle B=30^{\circ}$, $AB=2 \sqrt {3}$, $AC=2$, find the area of $\triangle ABC$. | 2 \sqrt {3} |
Given the origin $O$ of a Cartesian coordinate system as the pole and the non-negative half-axis of the $x$-axis as the initial line, a polar coordinate system is established. The polar equation of curve $C$ is $\rho\sin^2\theta=4\cos\theta$.
$(1)$ Find the Cartesian equation of curve $C$;
$(2)$ The parametric equation of line $l$ is $\begin{cases} x=1+ \frac{2\sqrt{5}}{5}t \\ y=1+ \frac{\sqrt{5}}{5}t \end{cases}$ ($t$ is the parameter), let point $P(1,1)$, and line $l$ intersects with curve $C$ at points $A$, $B$. Calculate the value of $|PA|+|PB|$. | 4\sqrt{15} |
A corporation plans to expand its sustainability team to include specialists in three areas: energy efficiency, waste management, and water conservation. The company needs 95 employees to specialize in energy efficiency, 80 in waste management, and 110 in water conservation. It is known that 30 employees will specialize in both energy efficiency and waste management, 35 in both waste management and water conservation, and 25 in both energy efficiency and water conservation. Additionally, 15 employees will specialize in all three areas. How many specialists does the company need to hire at minimum? | 210 |
What is the least possible value of
\[(x+1)(x+2)(x+3)(x+4)+2019\]where $x$ is a real number? | 2018 |
What is the value of $\frac{2023^3 - 2 \cdot 2023^2 \cdot 2024 + 3 \cdot 2023 \cdot 2024^2 - 2024^3 + 1}{2023 \cdot 2024}$? | 2023 |
Find a positive integer whose last digit is 2, and if this digit is moved from the end to the beginning, the new number is twice the original number. | 105263157894736842 |
The last three digits of \( 1978^n \) and \( 1978^m \) are the same. Find the positive integers \( m \) and \( n \) such that \( m+n \) is minimized (here \( n > m \geq 1 \)). | 106 |
Lauren solved the equation $|x-5| = 2$. Meanwhile Jane solved an equation of the form $x^2+ bx + c = 0$ that had the same two solutions for $x$ as Lauren's equation. What is the ordered pair $(b, c)$? | (-10,21) |
Jared likes to draw monsters. He drew a monster family portrait. The mom had 1 eye and the dad had 3. They had 3 kids, each with 4 eyes. How many eyes did the whole family have? | The mom and dad monsters had 1 + 3 = <<1+3=4>>4 eyes.
The 3 kids had 4 eyes each, so 3 x 4 = <<3*4=12>>12 eyes.
Altogether, the monster family had 4 parent eyes + 12 child eyes = <<4+12=16>>16 eyes.
#### 16 |
Find the quadratic polynomial $p(x)$ such that $p(-7) = 0,$ $p(4) = 0,$ and $p(5) = -36.$ | -3x^2 - 9x + 84 |
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