problem stringlengths 10 5.15k | answer dict |
|---|---|
Given the event "Randomly select a point P on the side CD of rectangle ABCD, such that the longest side of ΔAPB is AB", with a probability of 1/3, determine the ratio of AD to AB. | {
"answer": "\\frac{\\sqrt{5}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The ratio of the land area to the ocean area on the Earth's surface is 29:71. If three-quarters of the land is in the northern hemisphere, then calculate the ratio of the ocean area in the southern hemisphere to the ocean area in the northern hemisphere. | {
"answer": "171:113",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Lucas wants to buy a book that costs $28.50. He has two $10 bills, five $1 bills, and six quarters in his wallet. What is the minimum number of nickels that must be in his wallet so he can afford the book? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The mass of the first cast iron ball is $1462.5\%$ greater than the mass of the second ball. By what percentage less paint is needed to paint the second ball compared to the first ball? The volume of a ball with radius $R$ is $\frac{4}{3} \pi R^{3}$, and the surface area of a ball is $4 \pi R^{2}$. | {
"answer": "84",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a regular tetrahedron $ABCD$. Find $\sin \angle BAC$. | {
"answer": "\\frac{2\\sqrt{2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $m$ be the product of all positive integers less than $5!$ which are invertible modulo $5!$. Find the remainder when $m$ is divided by $5!$. | {
"answer": "119",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence \\(\{a_n\}\) satisfies \\(a_{n+1}= \dfrac {2016a_n}{2014a_n+2016}(n\in N_+)\), and \\(a_1=1\), find \\(a_{2017}= \) ______. | {
"answer": "\\dfrac {1008}{1007\\times 2017+1}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $l_{1}$: $ρ \sin (θ- \frac{π}{3})= \sqrt {3}$, $l_{2}$: $ \begin{cases} x=-t \\ y= \sqrt {3}t \end{cases}(t$ is a parameter), find the polar coordinates of the intersection point $P$ of $l_{1}$ and $l_{2}$. Additionally, points $A$, $B$, and $C$ are on the ellipse $\frac{x^{2}}{4}+y^{2}=1$. $O$ is the coordinate origin, and $∠AOB=∠BOC=∠COA=120^{\circ}$, find the value of $\frac{1}{|OA|^{2}}+ \frac{1}{|OB|^{2}}+ \frac{1}{|OC|^{2}}$. | {
"answer": "\\frac{15}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the lengths of arcs of curves given by the parametric equations.
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=2(t-\sin t) \\
y=2(1-\cos t)
\end{array}\right. \\
& 0 \leq t \leq \frac{\pi}{2}
\end{aligned}
$$ | {
"answer": "8 - 4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The average density of pathogenic microbes in one cubic meter of air is 100. A sample of 2 cubic decimeters of air is taken. Find the probability that at least one microbe will be found in the sample. | {
"answer": "0.181",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among the numbers $85_{(9)}$, $210_{(6)}$, $1000_{(4)}$, and $111111_{(2)}$, the smallest number is __________. | {
"answer": "111111_{(2)}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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 \(DE\). | {
"answer": "7.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x > 0$, $y > 0$, and $\frac{1}{2x+y} + \frac{4}{x+y} = 2$, find the minimum value of $7x + 5y$. | {
"answer": "7 + 2\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \( AC = 1.5 \, \text{cm} \) and \( AD = 4 \, \text{cm} \), what is the relationship between the areas of triangles \( \triangle ABC \) and \( \triangle DBC \)? | {
"answer": "3/5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \(a, b, c, d\) are within the interval \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\), and \(\sin a + \sin b + \sin c + \sin d = 1\), and \(\cos 2a + \cos 2b + \cos 2c + \cos 2d \geq \frac{10}{3}\), what is the maximum value of \(a\)? \(\quad\). | {
"answer": "\\frac{\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In trapezoid \( KLMN \), the lengths of the bases are \( KN = 25 \), \( LM = 15 \), and the lengths of the legs are \( KL = 6 \), \( MN = 8 \). Find the length of the segment connecting the midpoints of the bases. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the definite integral:
$$
\int_{0}^{2 \pi} \sin^{6} x \cos^{2} x \, dx
$$ | {
"answer": "\\frac{5\\pi}{64}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a circle $O: x^{2}+y^{2}=4$.<br/>$(1)$ A tangent line is drawn from point $P(2,1)$ to circle $O$, find the equation of the tangent line $l$;<br/>$(2)$ Let $A$ and $B$ be the points where circle $O$ intersects the positive $x$-axis and positive $y$-axis, respectively. A moving point $Q$ satisfies $QA=\sqrt{2}QB$. Is the locus of the moving point $Q$ intersecting circle $O$ at two points? If yes, find the length of the common chord; if not, explain the reason. | {
"answer": "\\frac{8\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify first, then evaluate: $(1-\frac{m}{{m-3}})\div \frac{{{m^2}-3m}}{{{m^2}-6m+9}}$, where $m=4\sqrt{3}$. | {
"answer": "-\\frac{\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the task of selecting 10 individuals to participate in a quality education seminar from 7 different schools, with the condition that at least one person must be chosen from each school, determine the total number of possible allocation schemes. | {
"answer": "84",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At the mountain hut, the coach said, "If we continue at this comfortable pace of $4 \mathrm{~km}$ per hour, we will arrive at the station 45 minutes after the departure of our train."
Then he pointed to a group that had just passed us: "They are using poles, and therefore achieve an average speed of $6 \mathrm{~km}$ per hour. They will arrive at the station half an hour before the departure of our train."
How far was the station from the mountain hut? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $x$, $y$, $z$ are positive real numbers, find the maximum value of $\dfrac{xy+yz}{x^{2}+y^{2}+z^{2}}$. | {
"answer": "\\dfrac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F_1$ and $F_2$ are the left and right foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$, and line $l$ passes through points $(a, 0)$ and $(0, b)$. The sum of the distances from $F_1$ and $F_2$ to line $l$ is $\frac{4c}{5}$. Determine the eccentricity of the hyperbola. | {
"answer": "\\frac{5\\sqrt{21}}{21}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $P$ is any point on the circle $C$: $(x-2)^{2}+(y-2)^{2}=1$, and $Q$ is any point on the line $l$: $x+y=1$, find the minimum value of $| \overrightarrow{OP}+ \overrightarrow{OQ}|$. | {
"answer": "\\frac{5 \\sqrt{2}-2}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(r(x)\) have a domain of \(\{-2,-1,0,1\}\) and a range of \(\{-1,0,2,3\}\). Let \(t(x)\) have a domain of \(\{-1,0,1,2,3\}\) and be defined as \(t(x) = 2x + 1\). Furthermore, \(s(x)\) is defined on the domain \(\{1, 2, 3, 4, 5, 6\}\) by \(s(x) = x + 2\). What is the sum of all possible values of \(s(t(r(x)))\)? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that there are 5 advertisements (3 commercial advertisements and 2 Olympic promotional advertisements), the last advertisement is an Olympic promotional advertisement, and the two Olympic promotional advertisements cannot be broadcast consecutively, determine the number of different broadcasting methods. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a store, there are 21 white shirts and 21 purple shirts hanging in a row. Find the minimum number \( k \) such that, regardless of the initial order of the shirts, it is possible to take down \( k \) white shirts and \( k \) purple shirts so that the remaining white shirts are all hanging consecutively and the remaining purple shirts are also hanging consecutively. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function \( f(x) = 5(x+1)^{2} + \frac{a}{(x+1)^{5}} \) for \( a > 0 \), find the minimum value of \( a \) such that \( f(x) \geqslant 24 \) when \( x \geqslant 0 \). | {
"answer": "2 \\sqrt{\\left(\\frac{24}{7}\\right)^7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A six-digit number can be tripled by reducing the first digit by three and appending a three at the end. What is this number? | {
"answer": "428571",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rectangle \(ABCD\) has area 2016. Point \(Z\) is inside the rectangle and point \(H\) is on \(AB\) so that \(ZH\) is perpendicular to \(AB\). If \(ZH : CB = 4 : 7\), what is the area of pentagon \(ADCZB\)? | {
"answer": "1440",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A trainee cook took two buckets of unpeeled potatoes and peeled everything in an hour. Meanwhile, $25\%$ of the potatoes went into peels. How much time did it take for him to collect exactly one bucket of peeled potatoes? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the positive divisors of \( 2014^2 \) be \( d_{1}, d_{2}, \cdots, d_{k} \). Then
$$
\frac{1}{d_{1}+2014}+\frac{1}{d_{2}+2014}+\cdots+\frac{1}{d_{k}+2014} =
$$ | {
"answer": "\\frac{27}{4028}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $(xOy)$, the sum of the distances from point $P$ to two points $(0,-\sqrt{3})$ and $(0,\sqrt{3})$ is equal to $4$. Let the trajectory of point $P$ be $C$.
(I) Write the equation of $C$;
(II) Given that the line $y=kx+1$ intersects $C$ at points $A$ and $B$, for what value of $k$ is $\overrightarrow{OA} \perp \overrightarrow{OB}$? What is the value of $|\overrightarrow{AB}|$ at this time? | {
"answer": "\\frac{4\\sqrt{65}}{17}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many different ways are there to rearrange the letters in the word 'BRILLIANT' so that no two adjacent letters are the same after the rearrangement? | {
"answer": "55440",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For the "Skillful Hands" club, Pavel needs to cut several identical pieces of wire (the length of each piece is an integer number of centimeters). Initially, Pavel took a wire piece of 10 meters and was able to cut only 15 pieces of the required length. Then, Pavel took a piece that was 40 centimeters longer, but it was also enough for only 15 pieces. What length should the pieces be? Express the answer in centimeters. | {
"answer": "66",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Factorize \( n^{5} - 5n^{3} + 4n \). What can be concluded in terms of divisibility? | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Thirty people are divided into three groups (I, II, and III) with 10 people in each group. How many different compositions of groups can there be? | {
"answer": "\\frac{30!}{(10!)^3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given triangle $\triangle ABC$ with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively. If $\overrightarrow{BC} \cdot \overrightarrow{BA} + 2\overrightarrow{AC} \cdot \overrightarrow{AB} = \overrightarrow{CA} \cdot \overrightarrow{CB}$. <br/>$(1)$ Find the value of $\frac{{\sin A}}{{\sin C}}$; <br/>$(2)$ If $2a \cdot \cos C = 2b - c$, find the value of $\cos B$. | {
"answer": "\\frac{3\\sqrt{2} - \\sqrt{10}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the sequence \(\left\{a_{n}\right\}\) defined by \(a_{0}=\frac{1}{2}\) and \(a_{n+1}=a_{n}+\frac{a_{n}^{2}}{2023}\) for \(n = 0, 1, \ldots\). Find the integer \(k\) such that \(a_{k} < 1 < a_{k+1}\). | {
"answer": "2023",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( A \), \( B \), and \( C \) be pairwise independent events with equal probabilities, and \( A \cap B \cap C = \varnothing \). Find the maximum possible value for the probability \( \mathrm{P}(A) \). | {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\triangle ABC$ with the sides opposite to angles $A$, $B$, $C$ being $a$, $b$, $c$ respectively, and it satisfies $\frac {\sin (2A+B)}{\sin A}=2+2\cos (A+B)$.
(I) Find the value of $\frac {b}{a}$;
(II) If $a=1$ and $c= \sqrt {7}$, find the area of $\triangle ABC$. | {
"answer": "\\frac { \\sqrt {3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five years ago, there were 25 trailer homes on Maple Street with an average age of 12 years. Since then, a group of brand new trailer homes was added, and 5 old trailer homes were removed. Today, the average age of all the trailer homes on Maple Street is 11 years. How many new trailer homes were added five years ago? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A full container holds 150 watermelons and melons with a total value of 24,000 rubles. The total value of all watermelons is equal to the total value of all melons. How much does one watermelon cost in rubles, given that the container can hold 120 melons (without watermelons) or 160 watermelons (without melons)? | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$AB$ and $AC$ are tangents to a circle, $\angle BAC = 60^{\circ}$, the length of the broken line $BAC$ is 1. Find the distance between the points of tangency $B$ and $C$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point \( A \) lies on the line \( y = \frac{5}{12} x - 11 \), and point \( B \) lies on the parabola \( y = x^{2} \). What is the minimum length of segment \( AB \)? | {
"answer": "6311/624",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x) = x^5 + 2x^3 + 3x^2 + x + 1$, calculate the value of $v_3$ when evaluating the value of $f(3)$ using Horner's method. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $A$, $B$, and $C$ are the three interior angles of $\triangle ABC$, let vectors $\overrightarrow{m}=(\cos B,\sin C)$ and $\overrightarrow{n}=(\cos C,-\sin B)$, and $\overrightarrow{m}\cdot \overrightarrow{n}=\frac{1}{2}$.
$(1)$ Find the magnitude of the interior angle $A$;
$(2)$ If the radius of the circumcircle of $\triangle ABC$ is $R=2$, find the maximum value of the perimeter $L$ of $\triangle ABC$. | {
"answer": "2\\sqrt{3}+4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the distance from vertex \( A_4 \) to the face \( A_1 A_2 A_3 \).
\( A_1(2, 1, 4) \)
\( A_2(-1, 5, -2) \)
\( A_3(-7, -3, 2) \)
\( A_4(-6, -3, 6) \) | {
"answer": "5 \\sqrt{\\frac{2}{11}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of solutions to
\[\sin x = \left( \frac{1}{3} \right)^x\] on the interval $(0,100\pi).$ | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Represent the number 36 as the product of three whole number factors, the sum of which is equal to 4. What is the smallest of these factors? | {
"answer": "-4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the base of the tetrahedron \( S-ABC \) is an isosceles right triangle with \( AB \) as the hypotenuse, and \( SA = SB = SC = AB = 2 \). Assume that the points \( S, A, B, \) and \( C \) lie on a spherical surface with center \( O \). Find the distance from point \( O \) to the plane \( ABC \). | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $a\in[0,\pi]$, $\beta\in\left[-\frac{\pi}{4},\frac{\pi}{4}\right]$, $\lambda\in\mathbb{R}$, and $\left(\alpha -\frac{\pi}{2}\right)^{3}-\cos \alpha -2\lambda =0$, $4\beta^{3}+\sin \beta \cos \beta +\lambda =0$, then the value of $\cos \left(\frac{\alpha}{2}+\beta \right)$ is ______. | {
"answer": "\\frac{ \\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose $w$ is a complex number such that $w^3 = 64 - 48i$. Find $|w|$. | {
"answer": "2\\sqrt[3]{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A store sells chalk in three types of packages: regular, unusual, and excellent. Initially, the quantitative ratio of the types was 2:3:6. After some packages of regular and unusual chalk—totaling no more than 100 packages—were delivered to the store, and $40\%$ of the excellent chalk packages were sold, the ratio changed to 5:7:4. How many total chalk packages were sold in the store? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the area of the quadrilateral formed by the lines $y=8$, $y=x+3$, $y=-x+3$, and $x=5$? | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a board on $2013 \times 2013$ squares, what is the maximum number of chess knights that can be placed so that no $2$ attack each other? | {
"answer": "2026085",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A shopkeeper set up incorrect scales in his shop, where one side of the balance beam is longer than the other. During one weighing, 3 cans balanced with 8 packets, and during another, 1 packet balanced with 6 cans. Given that the true weight of one can is 1 kg, how much do 8 packets weigh? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $-765^\circ$, convert this angle into the form $2k\pi + \alpha$ ($0 \leq \alpha < 2\pi$), where $k \in \mathbb{Z}$. | {
"answer": "-6\\pi + \\frac{7\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that in triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and $\left(a+b\right)\left(\sin A-\sin B\right)=\left(c-b\right)\sin C$ with $a=2$, find the maximum area of triangle $\triangle ABC$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Lucy begins a sequence with the first term at 4. Each subsequent term is generated as follows: If a fair coin flip results in heads, she triples the previous term and then adds 3. If it results in tails, she subtracts 3 and divides the result by 3. What is the probability that the fourth term in Lucy's sequence is an integer?
A) $\frac{1}{4}$
B) $\frac{1}{2}$
C) $\frac{3}{4}$
D) $\frac{7}{8}$ | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a travel agency plans to arrange a trip for 900 passengers using two types of buses, A and B, with capacities 36 and 60 passengers respectively, rental costs 1600 yuan and 2400 yuan per bus respectively, and the total number of buses rented does not exceed 21, and the number of type B buses cannot exceed the number of type A buses by more than 7, calculate the minimum rental cost. | {
"answer": "36800",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cone is perfectly fitted inside a cube such that the cone's base is one face of the cube and its vertex touches the opposite face. A sphere is inscribed in the same cube. Given that one edge of the cube is 8 inches, calculate:
1. The volume of the inscribed sphere.
2. The volume of the inscribed cone.
Express your answer in terms of $\pi$. | {
"answer": "\\frac{128}{3}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the polar coordinate system, the polar coordinate equation of circle $C$ is $ρ^{2}-8ρ\sin (θ- \dfrac {π}{3})+13=0$. Given points $A(1, \dfrac {3π}{2})$ and $B(3, \dfrac {3π}{2})$, where $P$ is a point on circle $C$, find the minimum value of the area of $\triangle PAB$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The natural domain of the function \( y = f\left(\frac{2x}{3x^2 + 1}\right) \) is \(\left[\frac{1}{4}, a\right]\). Find the value of \( a \). | {
"answer": "\\frac{4}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it is given that $(a+b-c)(a+b+c)=ab$.
$(1)$ Determine the measure of angle $C$.
$(2)$ If $c=2a\cos B$ and $b=2$, find the area of $\triangle ABC$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For any natural values \( m \) and \( n \) (with \( n > 1 \)), a function is defined as \( f\left(\frac{m}{n}\right)=\frac{\sqrt[n]{3^{m}}}{\sqrt[n]{3^{m}}+3} \). Compute the sum
\[ f\left(\frac{1}{2020}\right)+f\left(\frac{2}{2020}\right)+f\left(\frac{3}{2020}\right)+\cdots+f\left(\frac{4039}{2020}\right)+f\left(\frac{4040}{2020}\right) \]. | {
"answer": "2020",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The radian measure of 300° is $$\frac {5π}{3}$$ | {
"answer": "\\frac{5\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We roll a fair die consecutively until the sum of the numbers obtained, \( S \), exceeds 100. What is the most probable value of \( S \)? | {
"answer": "101",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, $\angle C=90^\circ$, $AC=8$ and $BC=12$. Points $D$ and $E$ are on $\overline{AB}$ and $\overline{BC}$, respectively, and $\angle BED=90^\circ$. If $DE=6$, then what is the length of $BD$? | {
"answer": "3\\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Extend line $PD$ to intersect line $BC$ at point $F$. Construct lines $DG$ and $PT$ parallel to $AQ$.
Introduce the following notations: $AP = x$, $PB = \lambda x$, $BQ = y$, $QC = \mu y$, $PE = u$, $ED = v$.
From the similarity of triangles:
\[
\frac{z}{(1+\mu)y} = \lambda, \quad z = \lambda(1+\mu)y
\]
\[
\frac{BT}{\lambda x} = \frac{y}{(1+\mu)y}, \quad BT = \frac{\lambda x}{1+\mu} = \frac{\lambda y}{1+\lambda}, \quad TQ = y - \frac{\lambda y}{1+\lambda} = \frac{y}{1+\lambda}
\]
\[
QG = AD = (1+\mu)y
\]
By Thales' theorem:
\[
\frac{PE}{ED} = \frac{u}{v} = \frac{TQ}{QG} = \frac{y}{(1+\lambda)(1+\mu)y} = \frac{1}{(1+\lambda)(1+\mu)} = 3:20
\] | {
"answer": "3:20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The crafty rabbit and the foolish fox made an agreement: every time the fox crosses the bridge in front of the rabbit's house, the rabbit would double the fox's money. However, each time the fox crosses the bridge, he has to pay the rabbit a toll of 40 cents. Hearing that his money would double each time he crossed the bridge, the fox was very happy. However, after crossing the bridge three times, he discovered that all his money went to the rabbit. How much money did the fox initially have? | {
"answer": "35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate: $\frac{145}{273} \times 2 \frac{173}{245} \div 21 \frac{13}{15}=$ | {
"answer": "\\frac{7395}{112504}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
This was a highly dangerous car rally. It began with a small and very narrow bridge, where one out of five cars would fall into the water. Then followed a terrifying sharp turn, where three out of ten cars would go off the road. Next, there was a dark and winding tunnel where one out of ten cars would crash. The last part of the route was a sandy road where two out of five cars would get hopelessly stuck in the sand.
Find the total percentage of cars involved in accidents during the rally. | {
"answer": "69.76",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest natural number that can be represented in exactly two ways as \(3x + 4y\), where \(x\) and \(y\) are natural numbers. | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Six standard six-sided dice are rolled. We are told there is a pair and a three-of-a-kind, but no four-of-a-kind initially. The pair and the three-of-a-kind are set aside, and the remaining die is re-rolled. What is the probability that after re-rolling this die, at least four of the six dice show the same value? | {
"answer": "\\frac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, \( AB \) is the diameter of circle \( O \) with a length of 6 cm. One vertex \( E \) of square \( BCDE \) is on the circumference of the circle, and \( \angle ABE = 45^\circ \). Find the difference in area between the non-shaded region of circle \( O \) and the non-shaded region of square \( BCDE \) in square centimeters (use \( \pi = 3.14 \)). | {
"answer": "10.26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given right triangle $ABC$ with a right angle at vertex $C$ and $AB = 2BC$, calculate the value of $\cos A$. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a math class, each dwarf needs to find a three-digit number without any zero digits, divisible by 3, such that when 297 is added to the number, the result is a number with the same digits in reverse order. What is the minimum number of dwarfs that must be in the class so that there are always at least two identical numbers among those found? | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jane is considering buying a sweater priced at $50. A store is offering a 10% discount on the sweater. After applying the discount, Jane needs to pay a state sales tax of 7.5%, and a local sales tax of 7%. Calculate the difference between the state and local sales taxes that Jane has to pay. | {
"answer": "0.225",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We have a $6 \times 6$ square, partitioned into 36 unit squares. We select some of these unit squares and draw some of their diagonals, subject to the condition that no two diagonals we draw have any common points. What is the maximal number of diagonals that we can draw? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F\_1$ and $F\_2$ are the left and right foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$, a line parallel to one of the hyperbola's asymptotes passes through $F\_2$ and intersects the hyperbola at point $P$. If $|PF\_1| = 3|PF\_2|$, find the eccentricity of the hyperbola. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For every integer $k$ with $k > 0$, let $R(k)$ be the probability that
\[
\left[\frac{n}{k}\right] + \left[\frac{200 - n}{k}\right] = \left[\frac{200}{k}\right]
\]
for an integer $n$ randomly chosen from the interval $1 \leq n \leq 199$. What is the minimum possible value of $R(k)$ over the integers $k$ in the interval $1 \leq k \leq 199$?
A) $\frac{1}{4}$
B) $\frac{1}{2}$
C) $\frac{2}{3}$
D) $\frac{3}{4}$
E) $\frac{4}{5}$ | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
10 students (one of whom is the captain, and 9 are team members) formed a team to participate in a math competition and won first prize. The organizing committee decided to award each team member 200 yuan as a prize. The captain received 90 yuan more than the average prize of all 10 team members. How much prize money did the captain receive? | {
"answer": "300",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An investor has an open brokerage account with an investment company. In 2021, the investor received the following income from securities:
- Dividends from shares of the company PAO “Winning” amounted to 50,000 rubles.
- Coupon income from government bonds OFZ amounted to 40,000 rubles.
- Coupon income from corporate bonds of PAO “Reliable” amounted to 30,000 rubles.
In addition, the investor received a capital gain from selling 100 shares of PAO "Risky" at 200 rubles per share. The purchase price was 150 rubles per share. The investor held the shares for 4 months.
Calculate the amount of personal income tax (NDFL) on the income from the securities. | {
"answer": "11050",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Pizzas are sized by diameter. What percent increase in area results if Lorrie’s pizza increases from a 16-inch pizza to an 18-inch pizza? | {
"answer": "26.5625\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=2 \sqrt {3}\sin \frac {ωx}{2}\cos \frac {ωx}{2}-2\sin ^{2} \frac {ωx}{2}(ω > 0)$ with a minimum positive period of $3π$.
(I) Find the interval where the function $f(x)$ is monotonically increasing.
(II) In $\triangle ABC$, $a$, $b$, and $c$ correspond to angles $A$, $B$, and $C$ respectively, with $a < b < c$, $\sqrt {3}a=2c\sin A$, and $f(\frac {3}{2}A+ \frac {π}{2})= \frac {11}{13}$. Find the value of $\cos B$. | {
"answer": "\\frac {5 \\sqrt {3}+12}{26}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( x \) and \( y \) are real numbers satisfying the following equations:
\[
x + xy + y = 2 + 3 \sqrt{2} \quad \text{and} \quad x^2 + y^2 = 6,
\]
find the value of \( |x + y + 1| \). | {
"answer": "3 + \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain rectangle had its dimensions expressed in whole numbers of decimeters. Then, it changed its dimensions three times. First, one of its dimensions was doubled and the other was adjusted so that the area remained the same. Then, one dimension was increased by $1 \mathrm{dm}$ and the other decreased by $4 \mathrm{dm}$, keeping the area the same. Finally, its shorter dimension was reduced by $1 \mathrm{dm}$, while the longer dimension remained unchanged.
Determine the ratio of the lengths of the sides of the final rectangle.
(E. Novotná) | {
"answer": "4:1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given point $A$ is on line segment $BC$ (excluding endpoints), and $O$ is a point outside line $BC$, with $\overrightarrow{OA} - 2a \overrightarrow{OB} - b \overrightarrow{OC} = \overrightarrow{0}$, then the minimum value of $\frac{a}{a+2b} + \frac{2b}{1+b}$ is \_\_\_\_\_\_. | {
"answer": "2 \\sqrt{2} - 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$a$,$b$,$c$ are the opposite sides of angles $A$,$B$,$C$ in $\triangle ABC$. It is known that $\sqrt{5}a\sin B=b$.
$(1)$ Find $\sin A$;
$(2)$ If $A$ is an obtuse angle, and $b=\sqrt{5}$, $c=3$, find the perimeter of $\triangle ABC$. | {
"answer": "\\sqrt{26} + \\sqrt{5} + 3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\Gamma$ be the region formed by the points $(x, y)$ that satisfy
$$
\left\{\begin{array}{l}
x \geqslant 0, \\
y \geqslant 0, \\
x+y+[x]+[y] \leqslant 5
\end{array}\right.
$$
where $[x]$ represents the greatest integer less than or equal to the real number $x$. Find the area of the region $\Gamma$. | {
"answer": "9/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The factory's planned output value for this year is $a$ million yuan, which is a 10% increase from last year. If the actual output value this year can exceed the plan by 1%, calculate the increase in the actual output value compared to last year. | {
"answer": "11.1\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The Gregorian calendar defines a common year as having 365 days and a leap year as having 366 days. The $n$-th year is a leap year if and only if:
1. $n$ is not divisible by 100 and $n$ is divisible by 4, or
2. $n$ is divisible by 100 and $n$ is divisible by 400.
For example, 1996 and 2000 are leap years, whereas 1997 and 1900 are not. These rules were established by Pope Gregory XIII.
Given that the "Gregorian year" is fully aligned with the astronomical year, determine the length of an astronomical year. | {
"answer": "365.2425",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum value of the following function $f(x) $ defined at $0<x<\frac{\pi}{2}$ .
\[f(x)=\int_0^x \frac{d\theta}{\cos \theta}+\int_x^{\frac{\pi}{2}} \frac{d\theta}{\sin \theta}\] | {
"answer": "\\ln(3 + 2\\sqrt{2})",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\theta=\arctan \frac{5}{12}$, find the principal value of the argument of the complex number $z=\frac{\cos 2 \theta+i \sin 2 \theta}{239+i}$. | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $PQR$, $PQ = 8$, $QR = 15$, and $PR = 17$. Point $S$ is the angle bisector of $\angle QPR$. Find the length of $QS$ and then find the length of the altitude from $P$ to $QS$. | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We placed 6 different dominoes in a closed chain on the table. The total number of points on the dominoes is $D$. What is the smallest possible value of $D$? (The number of points on each side of the dominoes ranges from 0 to 6, and the number of points must be the same on touching sides of the dominoes.) | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(ABCD\) be a square of side length 5. A circle passing through \(A\) is tangent to segment \(CD\) at \(T\) and meets \(AB\) and \(AD\) again at \(X \neq A\) and \(Y \neq A\), respectively. Given that \(XY = 6\), compute \(AT\). | {
"answer": "\\sqrt{30}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the line $l: x+2y+1=0$, and the set $A=\{n|n<6, n\in \mathbb{N}^*\}$, if we randomly select 3 different elements from set $A$ to be $a$, $b$, and $r$ in the circle equation $(x-a)^2+(y-b)^2=r^2$, then the probability that the line connecting the center $(a, b)$ of the circle to the origin is perpendicular to line $l$ equals \_\_\_\_\_\_. | {
"answer": "\\frac {1}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=(a\sin x+b\cos x)\cdot e^{x}$ has an extremum at $x= \frac {\pi}{3}$, determine the value of $\frac {a}{b}$. | {
"answer": "2- \\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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