problem stringlengths 10 5.15k | answer dict |
|---|---|
Given a regular triangular prism $ABC-A_1B_1C_1$ with side edges equal to the edges of the base, the sine of the angle formed by $AB_1$ and the lateral face $ACCA_1$ is equal to _______. | {
"answer": "\\frac{\\sqrt{6}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parametric equations of curve $C$ are
$$
\begin{cases}
x=3+ \sqrt {5}\cos \alpha \\
y=1+ \sqrt {5}\sin \alpha
\end{cases}
(\alpha \text{ is the parameter}),
$$
with the origin of the Cartesian coordinate system as the pole and the positive half-axis of $x$ as the polar axis, establish a polar coordinate system.
$(1)$ Find the polar equation of curve $C$;
$(2)$ If the polar equation of a line is $\sin \theta-\cos \theta= \frac {1}{\rho }$, find the length of the chord cut from curve $C$ by the line. | {
"answer": "\\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the base five product of the numbers $132_{5}$ and $12_{5}$? | {
"answer": "2114_5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle with radius 6 cm is tangent to three sides of a rectangle. The area of the rectangle is three times the area of the circle. Determine the length of the longer side of the rectangle, expressed in centimeters and in terms of $\pi$. | {
"answer": "9\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, $AC=2$, $D$ is the midpoint of $AB$, $CD=\frac{1}{2}BC=\sqrt{7}$, $P$ is a point on $CD$, and $\overrightarrow{AP}=m\overrightarrow{AC}+\frac{1}{3}\overrightarrow{AB}$. Find $|\overrightarrow{AP}|$. | {
"answer": "\\frac{2\\sqrt{13}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $F$ be the focus of the parabola $C: y^2=4x$, point $A$ lies on $C$, and point $B(3,0)$. If $|AF|=|BF|$, then calculate the distance of point $A$ from point $B$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that all terms are positive in the geometric sequence $\{a_n\}$, and the sum of the first $n$ terms is $S_n$, if $S_1 + 2S_5 = 3S_3$, then the common ratio of $\{a_n\}$ equals \_\_\_\_\_\_. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is known that $x^5 = a_0 + a_1 (1+x) + a_2 (1+x)^2 + a_3 (1+x)^3 + a_4 (1+x)^4 + a_5 (1+x)^5$, find the value of $a_0 + a_2 + a_4$. | {
"answer": "-16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it is given that $a\cos B - b\cos A = \frac{1}{2}c$. When $\tan(A-B)$ takes its maximum value, the value of angle $B$ is \_\_\_\_\_\_. | {
"answer": "\\frac{\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The volume of the parallelepiped generated by the vectors $\begin{pmatrix} 3 \\ 4 \\ 5 \end{pmatrix}$, $\begin{pmatrix} 2 \\ m \\ 3 \end{pmatrix}$, and $\begin{pmatrix} 2 \\ 3 \\ m \end{pmatrix}$ is 20. Find $m$, where $m > 0$. | {
"answer": "3 + \\frac{2\\sqrt{15}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define a "digitized number" as a ten-digit number $a_0a_1\ldots a_9$ such that for $k=0,1,\ldots, 9$ , $a_k$ is equal to the number of times the digit $k$ occurs in the number. Find the sum of all digitized numbers. | {
"answer": "6210001000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest composite number that has no prime factors less than 15. | {
"answer": "323",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two players, A and B, take turns shooting baskets. The probability of A making a basket on each shot is $\frac{1}{2}$, while the probability of B making a basket is $\frac{1}{3}$. The rules are as follows: A goes first, and if A makes a basket, A continues to shoot; otherwise, B shoots. If B makes a basket, B continues to shoot; otherwise, A shoots. They continue to shoot according to these rules. What is the probability that the fifth shot is taken by player A? | {
"answer": "\\frac{247}{432}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that all vertices of the tetrahedron S-ABC are on the surface of sphere O, SC is the diameter of sphere O, and if plane SCA is perpendicular to plane SCB, with SA = AC and SB = BC, and the volume of tetrahedron S-ABC is 9, find the surface area of sphere O. | {
"answer": "36\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The Fahrenheit temperature ( $F$ ) is related to the Celsius temperature ( $C$ ) by $F = \tfrac{9}{5} \cdot C + 32$ . What is the temperature in Fahrenheit degrees that is one-fifth as large if measured in Celsius degrees? | {
"answer": "-4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $f(1) = 3$, $f(2)= 12$, and $f(x) = ax^2 + bx + c$, what is the value of $f(3)$? | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Karl the old shoemaker made a pair of boots and sent his son Hans to the market to sell them for 25 talers. At the market, two people, one missing his left leg and the other missing his right leg, approached Hans and asked to buy one boot each. Hans agreed and sold each boot for 12.5 talers.
When Hans came home and told his father everything, Karl decided that he should have sold the boots cheaper to the disabled men, for 10 talers each. He gave Hans 5 talers and instructed him to return 2.5 talers to each person.
While Hans was looking for the individuals in the market, he saw sweets for sale, couldn't resist, and spent 3 talers on candies. He then found the men and gave them the remaining money – 1 taler each. On his way back home, Hans realized how bad his actions were. He confessed everything to his father and asked for forgiveness. The shoemaker was very angry and punished his son by locking him in a dark closet.
While sitting in the closet, Hans thought deeply. Since he returned 1 taler to each man, they effectively paid 11.5 talers for each boot: $12.5 - 1 = 11.5$. Therefore, the boots cost 23 talers: $2 \cdot 11.5 = 23$. And Hans had spent 3 talers on candies, resulting in a total of 26 talers: $23 + 3 = 26$. But there were initially only 25 talers! Where did the extra taler come from? | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Xiaoming's family raises chickens and pigs in a ratio of 26:5, and sheep to horses in a ratio of 25:9, while the ratio of pigs to horses is 10:3. Find the ratio of chickens, pigs, horses, and sheep. | {
"answer": "156:30:9:25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The work team was working at a rate fast enough to process $1250$ items in ten hours. But after working for six hours, the team was given an additional $150$ items to process. By what percent does the team need to increase its rate so that it can still complete its work within the ten hours? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two numbers need to be inserted between $4$ and $16$ such that the first three numbers are in arithmetic progression and the last three numbers are in geometric progression. What is the sum of those two numbers?
A) $6\sqrt{3} + 8$
B) $10\sqrt{3} + 6$
C) $6 + 10\sqrt{3}$
D) $16\sqrt{3}$ | {
"answer": "6\\sqrt{3} + 8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diameter of the semicircle $AB=4$, with $O$ as the center, and $C$ is any point on the semicircle different from $A$ and $B$. Find the minimum value of $(\vec{PA}+ \vec{PB})\cdot \vec{PC}$. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a Queen and the second card is a $\diamondsuit$? | {
"answer": "\\dfrac{1}{52}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine all three-digit numbers $N$ having the property that $N$ is divisible by $11,$ and $\frac{N}{11}$ is equal to the sum of the squares of the digits of $N.$ | {
"answer": "550",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the distance from point $P(x,y)$ to $A(0,4)$ and $B(-2,0)$ is equal, the minimum value of ${2}^{x}+{4}^{y}$ is. | {
"answer": "4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number 2017 has 7 ones and 4 zeros in binary representation. When will the nearest year come where the binary representation of the year has no more ones than zeros? (Enter the year.) | {
"answer": "2048",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a household, when someone is at home, the probability of the phone being answered at the 1st ring is 0.1, at the 2nd ring is 0.3, at the 3rd ring is 0.4, and at the 4th ring is 0.1. What is the probability that the phone is not answered within the first 4 rings? | {
"answer": "0.1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The Fibonacci sequence starts with two 1s, and each term afterwards is the sum of its two predecessors. Determine the last digit to appear in the units position of a number in the Fibonacci sequence when considered modulo 12. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A parallelogram $ABCD$ is inscribed in an ellipse $\frac{x^2}{4}+\frac{y^2}{2}=1$. The slope of line $AB$ is $k_1=1$. Calculate the slope of line $AD$. | {
"answer": "-\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Xiao Li was doing a subtraction problem and mistook the tens digit 7 for a 9 and the ones digit 3 for an 8, resulting in a difference of 76. The correct difference is ______. | {
"answer": "51",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate:<br/>$(1)-1^{2023}-\sqrt{2\frac{1}{4}}+\sqrt[3]{-1}+\frac{1}{2}$;<br/>$(2)2\sqrt{3}+|1-\sqrt{3}|-\left(-1\right)^{2022}+2$. | {
"answer": "3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What integer $n$ satisfies $0 \leq n < 201$ and $$200n \equiv 144 \pmod {101}~?$$ | {
"answer": "29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the chord common to circle C: x²+(y-4)²=18 and circle D: (x-1)²+(y-1)²=R² has a length of $6\sqrt {2}$, find the radius of circle D. | {
"answer": "2\\sqrt {7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four fair coins are tossed once. For every head that appears, two six-sided dice are rolled. What is the probability that the sum of all dice rolled is exactly ten?
A) $\frac{1} {48}$
B) $\frac{1} {20}$
C) $\frac{1} {16}$
D) $\frac{1} {30}$ | {
"answer": "\\frac{1} {20}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of an infinite geometric series is $64$ times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio? | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Some mice live in three neighboring houses. Last night, every mouse left its house and moved to one of the other two houses, always taking the shortest route. The numbers in the diagram show the number of mice per house, yesterday and today. How many mice used the path at the bottom of the diagram?
A 9
B 11
C 12
D 16
E 19 | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If there exists a real number $x$ such that the inequality $\left(e^{x}-a\right)^{2}+x^{2}-2ax+a^{2}\leqslant \dfrac{1}{2}$ holds with respect to $x$, determine the range of real number $a$. | {
"answer": "\\left\\{\\dfrac{1}{2}\\right\\}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A store sells a type of notebook. The retail price for each notebook is 0.30 yuan, a dozen (12 notebooks) is priced at 3.00 yuan, and for purchases of more than 10 dozen, each dozen can be paid for at 2.70 yuan.
(1) There are 57 students in the ninth grade class 1, and each student needs one notebook of this type. What is the minimum amount the class has to pay if they buy these notebooks collectively?
(2) There are 227 students in the ninth grade, and each student needs one notebook of this type. What is the minimum amount the grade has to pay if they buy these notebooks collectively? | {
"answer": "51.30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\underbrace{9999\cdots 99}_{80\text{ nines}}$ is multiplied by $\underbrace{7777\cdots 77}_{80\text{ sevens}}$, calculate the sum of the digits in the resulting product. | {
"answer": "720",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the regression equation $\hat{y}$=4.4x+838.19, estimate the ratio of the growth rate between x and y, denoted as \_\_\_\_\_\_. | {
"answer": "\\frac{5}{22}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alexis imagines a $2008\times 2008$ grid of integers arranged sequentially in the following way:
\[\begin{array}{r@{\hspace{20pt}}r@{\hspace{20pt}}r@{\hspace{20pt}}r@{\hspace{20pt}}r}1,&2,&3,&\ldots,&20082009,&2010,&2011,&\ldots,&40264017,&4018,&4019,&\ldots,&6024\vdots&&&&\vdots2008^2-2008+1,&2008^2-2008+2,&2008^2-2008+3,&\ldots,&2008^2\end{array}\]
She picks one number from each row so that no two numbers she picks are in the same column. She them proceeds to add them together and finds that $S$ is the sum. Next, she picks $2008$ of the numbers that are distinct from the $2008$ she picked the first time. Again she picks exactly one number from each row and column, and again the sum of all $2008$ numbers is $S$ . Find the remainder when $S$ is divided by $2008$ . | {
"answer": "1004",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given $9\sin ^{2}B=4\sin ^{2}A$ and $\cos C=\frac{1}{4}$, calculate $\frac{c}{a}$. | {
"answer": "\\frac{\\sqrt{10}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\cos (α+ \dfrac {π}{6})= \dfrac {1}{3}$, where $α∈(0\,,\,\, \dfrac {π}{2})$, find $\sin α$ and $\sin (2α+ \dfrac {5π}{6})$. | {
"answer": "-\\dfrac {7}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A car's clock is running at a constant speed but is inaccurate. One day, when the driver begins shopping, he notices both the car clock and his wristwatch (which is accurate) show 12:00 noon. After shopping, the wristwatch reads 12:30, and the car clock reads 12:35. Later that day, he loses his wristwatch and looks at the car clock, which shows 7:00. What is the actual time? | {
"answer": "6:00",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(P_1\) be a regular \(r\)-sided polygon and \(P_2\) be a regular \(s\)-sided polygon with \(r \geq s \geq 3\), such that each interior angle of \(P_1\) is \(\frac{61}{60}\) as large as each interior angle of \(P_2\). What is the largest possible value of \(s\)? | {
"answer": "121",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$908 \times 501 - [731 \times 1389 - (547 \times 236 + 842 \times 731 - 495 \times 361)] =$ | {
"answer": "5448",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\cos (\alpha - \frac{\pi }{3}) - \cos \alpha = \frac{1}{3}$, find the value of $\sin (\alpha - \frac{\pi }{6})$. | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=2 \sqrt {3}\sin x\cos x-\cos 2x$, where $x\in R$.
(1) Find the interval where the function $f(x)$ is monotonically increasing.
(2) In $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $f(A)=2$, $C= \frac {\pi}{4}$, and $c=2$, find the value of the area $S_{\triangle ABC}$. | {
"answer": "\\frac {3+ \\sqrt {3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a French class, I need to master a list of 600 vocabulary words for an upcoming test. The score on the test is based on the percentage of words I recall correctly. In this class, I have noticed that even when guessing the words I haven't studied, I have about a 10% chance of getting them right due to my prior knowledge. What is the minimum number of words I need to learn in order to guarantee at least a 90% score on this test? | {
"answer": "534",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that $a,b,c$ are real numbers such that $a < b < c$ and $a^3-3a+1=b^3-3b+1=c^3-3c+1=0$ . Then $\frac1{a^2+b}+\frac1{b^2+c}+\frac1{c^2+a}$ can be written as $\frac pq$ for relatively prime positive integers $p$ and $q$ . Find $100p+q$ .
*Proposed by Michael Ren* | {
"answer": "301",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
**How many different positive integers can be represented as a difference of two distinct members of the set $\{1, 3, 6, \ldots, 45\}$, where the numbers form an arithmetic sequence? Already given that the common difference between successive elements is 3.** | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A modified octahedron consists of two pyramids, each with a pentagonal base, glued together along their pentagonal bases, forming a polyhedron with twelve faces. An ant starts at the top vertex and walks randomly to one of the five adjacent vertices in the middle ring. From this vertex, the ant walks again to another randomly chosen adjacent vertex among five possibilities. What is the probability that the second vertex the ant reaches is the bottom vertex of the polyhedron? Express your answer as a common fraction. | {
"answer": "\\frac{1}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \(AD\), \(BE\), and \(CF\) are the altitudes of the acute triangle \(\triangle ABC\). If \(AB = 26\) and \(\frac{EF}{BC} = \frac{5}{13}\), what is the length of \(BE\)? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system $xOy$, curve $C_1$ passes through point $P(a, 1)$ with parametric equations $$\begin{cases} x=a+ \frac { \sqrt {2}}{2}t \\ y=1+ \frac { \sqrt {2}}{2}t\end{cases}$$ where $t$ is a parameter and $a \in \mathbb{R}$. In the polar coordinate system with the pole at $O$ and the non-negative half of the $x$-axis as the polar axis, the polar equation of curve $C_2$ is $\rho\cos^2\theta + 4\cos\theta - \rho = 0$.
(I) Find the Cartesian equation of the curve $C_1$ and the polar equation of the curve $C_2$;
(II) Given that the curves $C_1$ and $C_2$ intersect at points $A$ and $B$, with $|PA| = 2|PB|$, find the value of the real number $a$. | {
"answer": "\\frac{1}{36}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the real solution \( x, y, z \) to the equations \( x + y + z = 5 \) and \( xy + yz + zx = 3 \) such that \( z \) is the largest possible value. | {
"answer": "\\frac{13}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F\_1$ and $F\_2$ are the foci of a hyperbola, a line passing through $F\_2$ perpendicular to the real axis intersects the hyperbola at points $A$ and $B$. If $BF\_1$ intersects the $y$-axis at point $C$, and $AC$ is perpendicular to $BF\_1$, determine the eccentricity of the hyperbola. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A nine-joint bamboo tube has rice capacities of 4.5 *Sheng* in the lower three joints and 3.8 *Sheng* in the upper four joints. Find the capacity of the middle two joints. | {
"answer": "2.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A residential building has a construction cost of 250 yuan per square meter. Considering a useful life of 50 years and an annual interest rate of 5%, what monthly rent per square meter is required to recoup the entire investment? | {
"answer": "1.14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the volume of the pyramid whose net is shown, if the base is a square with a side length of $1$? | {
"answer": "\\frac{\\sqrt{3}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify first, then evaluate: $(1-\frac{a}{a+1})\div \frac{{a}^{2}-1}{{a}^{2}+2a+1}$, where $a=\sqrt{2}+1$. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the product
\[
\prod_{n = 1}^{15} \frac{n^2 + 5n + 6}{n+2}.
\] | {
"answer": "\\frac{18!}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $A_1,B_1,C_1,D_1$ be the midpoints of the sides of a convex quadrilateral $ABCD$ and let $A_2, B_2, C_2, D_2$ be the midpoints of the sides of the quadrilateral $A_1B_1C_1D_1$ . If $A_2B_2C_2D_2$ is a rectangle with sides $4$ and $6$ , then what is the product of the lengths of the diagonals of $ABCD$ ? | {
"answer": "96",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\sin \alpha = \frac{2\sqrt{2}}{3}$, $\cos(\alpha + \beta) = -\frac{1}{3}$, and both $\alpha$ and $\beta$ are within the interval $(0, \frac{\pi}{2})$, find the value of $\sin(\alpha - \beta)$. | {
"answer": "\\frac{10\\sqrt{2}}{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively. If the area of $\triangle ABC$ equals $8$, $a=5$, and $\tan B= -\frac{4}{3}$, then find the value of $\frac{a+b+c}{\sin A+\sin B+\sin C}$. | {
"answer": "\\frac{5 \\sqrt{65}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a bag, there are a total of 12 balls that are identical in shape and mass, including red, black, yellow, and green balls. When drawing a ball at random, the probability of getting a red ball is $\dfrac{1}{3}$, the probability of getting a black or yellow ball is $\dfrac{5}{12}$, and the probability of getting a yellow or green ball is also $\dfrac{5}{12}$.
(1) Please calculate the probabilities of getting a black ball, a yellow ball, and a green ball, respectively;
(2) When drawing a ball at random, calculate the probability of not getting a "red or green ball". | {
"answer": "\\dfrac{5}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A student, Theo, needs to earn a total of 30 homework points. For the first six homework points, he has to do one assignment each; for the next six points, he needs to do two assignments each; and so on, such that for every subsequent set of six points, the number of assignments he needs to complete doubles the previous set. Calculate the minimum number of homework assignments necessary for Theo to earn all 30 points. | {
"answer": "186",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Employees from department X are 30, while the employees from department Y are 20. Since employees from the same department do not interact, the number of employees from department X that will shake hands with the employees from department Y equals 30, and the number of employees from department Y that will shake hands with the employees from department X also equals 20. Find the total number of handshakes that occur between employees of different departments. | {
"answer": "600",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function \( y = y_1 + y_2 \), where \( y_1 \) is directly proportional to \( x^2 \) and \( y_2 \) is inversely proportional to \( x^2 \). When \( x = 1 \), \( y = 5 \); when \( x = \sqrt{3} \), \( y = 7 \). Find the value of \( x \) when \( y \) is minimized. | {
"answer": "\\sqrt[4]{\\frac{3}{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a pedestrian signal light that alternates between red and green, with the red light lasting for 50 seconds, calculate the probability that a student needs to wait at least 20 seconds for the green light to appear. | {
"answer": "\\dfrac{3}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many three-digit whole numbers contain at least one 6 or at least one 8? | {
"answer": "452",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In isosceles $\triangle ABC$, $|AB|=|AC|$, vertex $A$ is the intersection point of line $l: x-y+1=0$ with the y-axis, and $l$ bisects $\angle A$. If $B(1,3)$, find:
(I) The equation of line $BC$;
(II) The area of $\triangle ABC$. | {
"answer": "\\frac {3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a kite-shaped field with the following measurements and angles: sides AB = 120 m, BC = CD = 80 m, DA = 120 m. The angle between sides AB and BC is 120°. The angle between sides CD and DA is also 120°. The wheat harvested from any location in the field is brought to the nearest point on the field's perimeter. What fraction of the crop is brought to the longest side, which in this case is side BC? | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a tram with a starting station A and an ending station B. A tram departs from station A every 5 minutes towards station B, completing the journey in 15 minutes. A person starts cycling along the tram route from station B towards station A just as a tram arrives at station B. On his way, he encounters 10 trams coming towards him before reaching station A. At this moment, another tram is just departing from station A. How many minutes did it take for him to travel from station B to station A? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the random variable $X$ follows a normal distribution $N(0,\sigma^{2})$, if $P(X > 2) = 0.023$, determine the probability $P(-2 \leqslant X \leqslant 2)$. | {
"answer": "0.954",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = |2x+a| + |2x-2b| + 3$
(Ⅰ) If $a=1$, $b=1$, find the solution set of the inequality $f(x) > 8$;
(Ⅱ) When $a>0$, $b>0$, if the minimum value of $f(x)$ is $5$, find the minimum value of $\frac{1}{a} + \frac{1}{b}$. | {
"answer": "\\frac{3+2\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the sines of the internal angles of $\triangle ABC$ form an arithmetic sequence, what is the minimum value of $\cos C$? | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum value of
\[\frac{2x + 3y + 4}{\sqrt{x^2 + y^2 + 4}}\]
over all real numbers $x$ and $y$. | {
"answer": "\\sqrt{29}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If
\[
(1 + \tan 1^\circ)(1 + \tan 2^\circ)(1 + \tan 3^\circ) \dotsm (1 + \tan 89^\circ) = 2^m,
\]
then find $m.$ | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parabola $y^{2}=4x$, its focus intersects the parabola at two points $A(x_{1},y_{1})$ and $B(x_{2},y_{2})$. If $x_{1}+x_{2}=10$, find the length of the chord $AB$. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the sum of all the even integers between $200$ and $400$? | {
"answer": "30100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum number of white dominoes that can be cut from the board shown on the left. A domino is a $1 \times 2$ rectangle. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The output of a factory last year is denoted as $1$. If it is planned that the output of each of the next five years will increase by $10\%$ compared to the previous year, then the total output of this factory for the five years starting from this year will be approximately \_\_\_\_\_\_\_\_. (Keep one decimal place, take $1.1^{5} \approx 1.6$) | {
"answer": "6.6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $x = \frac{2}{3}$ and $y = \frac{5}{2}$, find the value of $\frac{1}{3}x^8y^9$. | {
"answer": "\\frac{5^9}{2 \\cdot 3^9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In quadrilateral ABCD, m∠B = m∠C = 120°, AB = 4, BC = 6, and CD = 7. Diagonal BD = 8. Calculate the area of ABCD. | {
"answer": "16.5\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among the following propositions, the true one is numbered \_\_\_\_\_\_.
(1) The negation of the proposition "For all $x>0$, $x^2-x\leq0$" is "There exists an $x>0$ such that $x^2-x>0$."
(2) If $A>B$, then $\sin A > \sin B$.
(3) Given a sequence $\{a_n\}$, "The sequence $a_n, a_{n+1}, a_{n+2}$ forms a geometric sequence" is a necessary and sufficient condition for $a_{n+1}^2=a_{n}a_{n+2}$.
(4) Given the function $f(x)=\lg x+ \frac{1}{\lg x}$, then the minimum value of $f(x)$ is 2. | {
"answer": "(1)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that points P1 and P2 are two adjacent centers of symmetry for the curve $y= \sqrt {2}\sin ωx-\cos ωx$ $(x\in\mathbb{R})$, if the tangents to the curve at points P1 and P2 are perpendicular to each other, determine the value of ω. | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a circle with center $O$, the measure of $\angle RIP$ is $45^\circ$ and $OR=15$ cm. Find the number of centimeters in the length of arc $RP$. Express your answer in terms of $\pi$. | {
"answer": "7.5\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the equation about $x$, $x^{2}-2a\ln x-2ax=0$ has a unique solution, find the value of the real number $a$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Matrices $A$ , $B$ are given as follows.
\[A=\begin{pmatrix} 2 & 1 & 0 1 & 2 & 0 0 & 0 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 4 & 2 & 0 2 & 4 & 0 0 & 0 & 12\end{pmatrix}\]
Find volume of $V=\{\mathbf{x}\in\mathbb{R}^3 : \mathbf{x}\cdot A\mathbf{x} \leq 1 < \mathbf{x}\cdot B\mathbf{x} \}$ . | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $({x-1})^4({x+2})^5=a_0+a_1x+a_2x^2+⋯+a_9x^9$, find the value of $a_{2}+a_{4}+a_{6}+a_{8}$. | {
"answer": "-24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The magnitude of the vector $\overset{→}{a} +2 \overset{→}{b}$, where $\overset{→}{a} =(2,0)$, $\overset{→}{b}$ is a unit vector with a magnitude of 1 and the angle between the two vectors is $60^{\circ}$. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $m, n \in \mathbb{R}$, if the line $(m+1)x + (n+1)y - 2 = 0$ is tangent to the circle $x^2 + y^2 = 1$, find the maximum value of $m - n$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\cos^{4}x-2\sin x\cos x-\sin^{4}x$.
(1) Find the smallest positive period of the function $f(x)$;
(2) When $x\in\left[0,\frac{\pi}{2}\right]$, find the minimum value of $f(x)$ and the set of $x$ values where the minimum value is obtained. | {
"answer": "\\left\\{\\frac{3\\pi}{8}\\right\\}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Abby, Bernardo, Carl, and Debra play a revised game where each starts with five coins and there are five rounds. In each round, five balls are placed in an urn—two green, two red, and one blue. Each player draws a ball at random without replacement. If a player draws a green ball, they give one coin to a player who draws a red ball. If anyone draws a blue ball, no transaction occurs for them. What is the probability that at the end of the fifth round, each of the players has five coins?
**A)** $\frac{1}{120}$
**B)** $\frac{64}{15625}$
**C)** $\frac{32}{3125}$
**D)** $\frac{1}{625}$
**E)** $\frac{4}{125}$ | {
"answer": "\\frac{64}{15625}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Katie and Allie are playing a game. Katie rolls two fair six-sided dice and Allie flips two fair two-sided coins. Katie’s score is equal to the sum of the numbers on the top of the dice. Allie’s score is the product of the values of two coins, where heads is worth $4$ and tails is worth $2.$ What is the probability Katie’s score is strictly greater than Allie’s? | {
"answer": "25/72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A Saxon silver penny, from the reign of Ethelbert II in the eighth century, was sold in 2014 for £78000. A design on the coin depicts a circle surrounded by four equal arcs, each a quarter of a circle. The width of the design is 2 cm. What is the radius of the small circle, in centimetres?
A) \(\frac{1}{2}\)
B) \(2 - \sqrt{2}\)
C) \(\frac{1}{2} \sqrt{2}\)
D) \(5 - 3\sqrt{2}\)
E) \(2\sqrt{2} - 2\) | {
"answer": "2 - \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the amount of personal income tax (НДФЛ) paid by Sergey over the past year if he is a resident of the Russian Federation and had a stable income of 30,000 rubles per month and a one-time vacation bonus of 20,000 rubles during this period. Last year, Sergey sold his car, which he inherited two years ago, for 250,000 rubles and bought a plot of land for building a house for 300,000 rubles. Sergey applied all applicable tax deductions. (Provide the answer without spaces and units of measurement.) | {
"answer": "10400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular prism measuring 20 cm by 14 cm by 12 cm has a small cube of 4 cm on each side removed from each corner. What percent of the original volume is removed? | {
"answer": "15.24\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $f(x)$ has a derivative and satisfies $\lim_{\Delta x \to 0} \, \frac{f(1)-f(1-2\Delta x)}{2\Delta x}=-1$, find the slope of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a hyperbola $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$ has only one common point with the parabola $y=x^{2}+1$, calculate the eccentricity of the hyperbola. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define an ordered triple $(D, E, F)$ of sets to be minimally intersecting if $|D \cap E| = |E \cap F| = |F \cap D| = 1$ and $D \cap E \cap F = \emptyset$. Let $M$ be the number of such ordered triples where each set is a subset of $\{1,2,3,4,5,6,7,8\}$. Find $M$ modulo $1000$. | {
"answer": "064",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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