problem stringlengths 10 5.15k | answer dict |
|---|---|
Determine the value of $a$ such that the equation
\[\frac{x - 3}{ax - 2} = x\]
has exactly one solution. | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given point $P(1, 2, 3)$, the symmetric point of $P$ about the $y$-axis is $P_1$, and the symmetric point of $P$ about the coordinate plane $xOz$ is $P_2$. Find the distance $|P_1P_2|$. | {
"answer": "2\\sqrt{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate
\[\frac{1}{t_1} + \frac{1}{t_2} + \frac{1}{t_3} + ... + \frac{1}{t_{2003}}\]
where \( t_n = \frac{n(n+1)}{2} \) is the $n$th triangular number.
A) $\frac{2005}{1002}$
B) $\frac{4006}{2003}$
C) $\frac{2003}{1001}$
D) $\frac{2003}{1002}$
E) 2 | {
"answer": "\\frac{2003}{1002}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system, point \( O(0,0) \), \( A(0,6) \), \( B(-3,2) \), \( C(-2,9) \), and \( P \) is any point on line segment \( OA \) (including endpoints). Find the minimum value of \( PB + PC \). | {
"answer": "5 + \\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(a\), \(b\), and \(c\) be angles such that:
\[
\sin a = \cot b, \quad \sin b = \cot c, \quad \sin c = \cot a.
\]
Find the largest possible value of \(\cos a\). | {
"answer": "\\sqrt{\\frac{3 - \\sqrt{5}}{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, $sinB=\sqrt{2}sinA$, $∠C=105°$, and $c=\sqrt{3}+1$. Calculate the area of the triangle. | {
"answer": "\\frac{\\sqrt{3} + 1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that quadrilateral \(ABCD\) is an isosceles trapezoid with \(AB \parallel CD\), \(AB = 6\), and \(CD = 16\). Triangle \(ACE\) is a right-angled triangle with \(\angle AEC = 90^\circ\), and \(CE = BC = AD\). Find the length of \(AE\). | {
"answer": "4\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five friends were comparing how much scrap iron they brought to the collection. On average, it was $55 \mathrm{~kg}$, but Ivan brought only $43 \mathrm{~kg}$.
What is the average amount of iron brought without Ivan?
(Note: By how many kilograms does Ivan's contribution differ from the average?) | {
"answer": "58",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the quadrilateral \(ABCD\), \(AB = 2\), \(BC = 4\), \(CD = 5\). Find its area given that it is both circumscribed and inscribed. | {
"answer": "2\\sqrt{30}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Isabel wants to save 40 files onto disks, each with a capacity of 1.44 MB. 5 of the files take up 0.95 MB each, 15 files take up 0.65 MB each, and the remaining 20 files each take up 0.45 MB. Calculate the smallest number of disks needed to store all 40 files. | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a village where Glafira lives, there is a small pond that is filled by springs at the bottom. Curious Glafira found out that a herd of 17 cows completely drank this pond in 3 days. After some time, the springs filled the pond again, and then 2 cows drank it in 30 days. How many days will it take for one cow to drink this pond? | {
"answer": "75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A secret facility is a rectangle measuring $200 \times 300$ meters. Outside the facility, there is one guard at each of the four corners. An intruder approached the perimeter of the facility from the outside, and all the guards ran towards the intruder using the shortest paths along the external perimeter (the intruder remained stationary). The total distance run by three of the guards to reach the intruder was 850 meters. How many meters did the fourth guard run to reach the intruder? | {
"answer": "150",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Along a straight alley at equal intervals, there are 400 lampposts, numbered sequentially from 1 to 400. At the same time, Alla and Boris started walking towards each other from opposite ends of the alley with different constant speeds (Alla started from the first lamppost and Boris from the four-hundredth). When Alla reached the 55th lamppost, Boris was at the 321st lamppost. At which lamppost will they meet? If the meeting point is between two lampposts, indicate the smaller number of the two in your answer. | {
"answer": "163",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Zach has twelve identical-looking chocolate eggs. Exactly three of the eggs contain a special prize inside. Zach randomly gives three of the twelve eggs to each of Vince, Wendy, Xin, and Yolanda. What is the probability that only one child will receive an egg that contains a special prize (that is, that all three special prizes go to the same child)? | {
"answer": "1/55",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\cos(2x+\frac{\pi}{3})+1$, in triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively.
$(I)$ If angles $A$, $B$, and $C$ form an arithmetic sequence, find the value of $f(B)$;
$(II)$ If $f\left(\frac{B}{2}-\frac{\pi}{6}\right)=\frac{7}{4}$, and sides $a$, $b$, and $c$ form a geometric sequence, with the area of $\triangle ABC$ being $S=\frac{\sqrt{7}}{4}$, find the perimeter of $\triangle ABC$. | {
"answer": "3+\\sqrt{2}",
"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. If the area of $\triangle ABC$ equals $8$, $a=5$, and $\tan B= -\frac{4}{3}$, then $\frac{a+b+c}{\sin A+\sin B+\sin C}=$ \_\_\_\_\_\_. | {
"answer": "\\frac{5\\sqrt{65}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the largest integer \( n \) such that
$$
\frac{\sqrt{7}+2 \sqrt{n}}{2 \sqrt{7}-\sqrt{n}}
$$
is an integer? | {
"answer": "343",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the perimeter of a sector is 10cm, and its area is 4cm<sup>2</sup>, find the radian measure of the central angle $\alpha$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( b\left[\frac{1}{1 \times 3}+\frac{1}{3 \times 5}+\cdots+\frac{1}{1999 \times 2001}\right]=2 \times\left[\frac{1^{2}}{1 \times 3}+\frac{2^{2}}{3 \times 5}+\cdots+\frac{1000^{2}}{1999 \times 2001}\right] \), find the value of \( b \). | {
"answer": "1001",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $PQR$ is a right triangle with legs $PQ$ and $PR$. Points $U$ and $V$ are on legs $PQ$ and $PR$, respectively so that $PU:UQ = PV:VR = 1:3$. If $QU = 18$ units, and $RV = 45$ units, what is the length of hypotenuse $PQ$? Express your answer in simplest radical form. | {
"answer": "12\\sqrt{29}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four cars $A, B, C$ and $D$ start simultaneously from the same point on a circular track. $A$ and $B$ drive clockwise, while $C$ and $D$ drive counter-clockwise. All cars move at constant (but pairwise different) speeds. Exactly 7 minutes after the race begins, $A$ meets $C$ for the first time, and at that same moment, $B$ meets $D$ for the first time. After another 46 minutes, $A$ and $B$ meet for the first time. When will all the cars meet for the first time after the race starts? | {
"answer": "371",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $y=2\sin \left(x+ \frac {\pi}{6}\right)\cos \left(x+ \frac {\pi}{6}\right)$, determine the horizontal shift required to obtain its graph from the graph of the function $y=\sin 2x$. | {
"answer": "\\frac{\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=4\cos x\sin \left(x- \frac{\pi}{3}\right)+a$ has a maximum value of $2$.
$(1)$ Find the value of $a$ and the smallest positive period of the function $f(x)$;
$(2)$ In $\triangle ABC$, if $A < B$, and $f(A)=f(B)=1$, find the value of $\frac{BC}{AB}$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a geometric sequence with positive terms $\{a_n\}$, where $a_5= \frac {1}{2}$ and $a_6 + a_7 = 3$, find the maximum positive integer value of $n$ such that $a_1 + a_2 + \ldots + a_n > a_1 a_2 \ldots a_n$. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two concentric circles have radii of 24 and 36 units, respectively. A shaded region is formed between these two circles. A new circle is to be drawn such that its diameter is equal to the area of the shaded region. What must the diameter of this new circle be? Express your answer in simplest radical form. | {
"answer": "720 \\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
C is the complex numbers. \( f : \mathbb{C} \to \mathbb{R} \) is defined by \( f(z) = |z^3 - z + 2| \). What is the maximum value of \( f \) on the unit circle \( |z| = 1 \)? | {
"answer": "\\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the length of the shortest path on the surface of a unit cube between its opposite vertices. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the smallest positive integer \( a \) for which \( 47^n + a \cdot 15^n \) is divisible by 1984 for all odd \( n \). | {
"answer": "1055",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
January 1st of a certain non-leap year fell on a Saturday. How many Fridays are there in this year? | {
"answer": "52",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine into how many parts a regular tetrahedron is divided by six planes, each passing through one edge and the midpoint of the opposite edge of the tetrahedron. Find the volume of each part if the volume of the tetrahedron is 1. | {
"answer": "1/24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An error of $.05''$ is made in the measurement of a line $25''$ long, and an error of $.3''$ is made in the measurement of a line $150''$ long. In comparison with the relative error of the first measurement, find the relative error of the second measurement. | {
"answer": "0.2\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the cuboid ABCD-A<sub>1</sub>B<sub>1</sub>C<sub>1</sub>D<sub>1</sub>, where AB=3, AD=4, and AA<sub>1</sub>=5, point P is a moving point on the surface A<sub>1</sub>B<sub>1</sub>C<sub>1</sub>D<sub>1</sub>. Find the minimum value of |PA|+|PC|. | {
"answer": "5\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Selina takes a sheet of paper and cuts it into 10 pieces. She then takes one of these pieces and cuts it into 10 smaller pieces. She then takes another piece and cuts it into 10 smaller pieces and finally cuts one of the smaller pieces into 10 tiny pieces. How many pieces of paper has the original sheet been cut into? | {
"answer": "37",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How can 50 cities be connected with the smallest number of airline routes so that it is possible to travel from any city to any other city with no more than two layovers? | {
"answer": "49",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The plane angle at the vertex of a regular triangular pyramid is $90^{\circ}$. Find the ratio of the lateral surface area of the pyramid to the area of its base. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Positive real numbers \( a \) and \( b \) satisfy \( \log _{9} a = \log _{12} b = \log _{16}(3a + b) \). Find the value of \(\frac{b}{a}\). | {
"answer": "\\frac{1+\\sqrt{13}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the volumes of solids formed by the rotation of regions bounded by the graphs of the functions around the y-axis.
$$
y=\arcsin \frac{x}{5}, y=\arcsin x, y=\frac{\pi}{2}
$$ | {
"answer": "6 \\pi^2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangle with angles \( A, B, C \) satisfies the following conditions:
\[
\frac{\sin A + \sin B + \sin C}{\cos A + \cos B + \cos C} = \frac{12}{7},
\]
and
\[
\sin A \sin B \sin C = \frac{12}{25}.
\]
Given that \( \sin C \) takes on three possible values \( s_1, s_2 \), and \( s_3 \), find the value of \( 100 s_1 s_2 s_3 \). | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The denominator of a geometric progression \( b_{n} \) is \( q \), and for some natural \( n \geq 2 \),
$$
\log_{4} b_{2}+\log_{4} b_{3}+\ldots+\log_{4} b_{n}=4 \cdot \log_{4} b_{1}
$$
Find the smallest possible value of \( \log_{q} b_{1}^{2} \), given that it is an integer. For which \( n \) is this value achieved? | {
"answer": "-30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$(HUN 6)$ Find the positions of three points $A,B,C$ on the boundary of a unit cube such that $min\{AB,AC,BC\}$ is the greatest possible. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider $13$ marbles that are labeled with positive integers such that the product of all $13$ integers is $360$ . Moor randomly picks up $5$ marbles and multiplies the integers on top of them together, obtaining a single number. What is the maximum number of different products that Moor can obtain? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the maximum real value of $a$ for which there is an integer $b$ such that $\frac{ab^2}{a+2b} = 2019$ . Compute the maximum possible value of $a$ . | {
"answer": "30285",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mr. Wang, a math teacher, is preparing to visit a friend. Before leaving, Mr. Wang calls the friend's house, and the phone number is 27433619. After the call, Mr. Wang realizes that this phone number is exactly the product of 4 consecutive prime numbers. What is the sum of these 4 prime numbers? | {
"answer": "290",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x)= \frac{\ln x+2^{x}}{x^{2}}$, find $f'(1)=$ ___. | {
"answer": "2\\ln 2 - 3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the vertex of a parabola is at the origin and the center of the circle $(x-2)^2 + y^2 = 4$ is exactly the focus of the parabola.
1. Find the equation of the parabola.
2. A line with a slope of 2 passes through the focus of the parabola and intersects the parabola at points A and B. Find the area of triangle OAB. | {
"answer": "4\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points $\boldsymbol{A}$ and $\boldsymbol{B}$ are located on a straight highway running from west to east. Point B is 9 km east of A. A car departs from point A heading east at a speed of 40 km/h. Simultaneously, from point B, a motorcycle starts traveling in the same direction with a constant acceleration of 32 km/h². Determine the greatest distance that can be between the car and the motorcycle during the first two hours of their movement. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There were electronic clocks on the International Space Station, displaying time in the format HH:MM. Due to an electromagnetic storm, the device started malfunctioning, and each digit on the display either increased by 1 or decreased by 1. What was the actual time when the storm occurred if the clocks showed 00:59 immediately after it? | {
"answer": "11:48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \(\alpha\), \(\beta\), and \(\gamma\) are acute angles such that \(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\), find the minimum value of \(\tan \alpha \cdot \tan \beta \cdot \tan \gamma\). | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the hyperbola $C$: $x^{2}- \frac{y^{2}}{b^{2}}=1$ ($b > 0$) has two asymptotes that intersect with the circle $O$: $x^{2}+y^{2}=2$ at four points sequentially named $A$, $B$, $C$, and $D$. If the area of rectangle $ABCD$ is $b$, then the value of $b$ is. | {
"answer": "\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among A, B, C, and D comparing their heights, the sum of the heights of two of them is equal to the sum of the heights of the other two. The average height of A and B is 4 cm more than the average height of A and C. D is 10 cm taller than A. The sum of the heights of B and C is 288 cm. What is the height of A in cm? | {
"answer": "139",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( x, y, z \in [0, 1] \), find the maximum value of \( M = \sqrt{|x-y|} + \sqrt{|y-z|} + \sqrt{|z-x|} \). | {
"answer": "\\sqrt{2} + 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of the digits of the result of the expression $\underbrace{99 \cdots 99}_{2021 \text{ digits}} \times \underbrace{99 \cdots 99}_{2020 \text{ digits}}$ is $\qquad$ | {
"answer": "18189",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of positive integers less than $2000$ that are neither $5$-nice nor $6$-nice. | {
"answer": "1333",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sphere is inscribed in a cube, and the cube has a surface area of 54 square meters. A second cube is then inscribed within the sphere. A third, smaller sphere is then inscribed within this second cube. What is the surface area of the second cube and the volume of the third sphere? | {
"answer": "\\frac{\\sqrt{3}\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ten points are spaced around at equal intervals on the circumference of a regular pentagon, each side being further divided into two equal segments. Two of the 10 points are chosen at random. What is the probability that the two points are exactly one side of the pentagon apart?
A) $\frac{1}{5}$
B) $\frac{1}{9}$
C) $\frac{2}{9}$
D) $\frac{1}{18}$
E) $\frac{1}{45}$ | {
"answer": "\\frac{2}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ivan wanted to buy nails. In one store, where 100 grams of nails cost 180 rubles, he couldn't buy the required amount because he was short 1430 rubles. Then he went to another store where 100 grams cost 120 rubles. He bought the required amount and received 490 rubles in change. How many kilograms of nails did Ivan buy? | {
"answer": "3.2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A lieutenant is training recruits in marching drills. Upon arriving at the parade ground, he sees that all the recruits are arranged in several rows, with each row having the same number of soldiers, and that the number of soldiers in each row is 5 more than the number of rows. After finishing the drills, the lieutenant decides to arrange the recruits again but cannot remember how many rows there were initially. So, he orders them to form as many rows as his age. It turns out that each row again has the same number of soldiers, but in each row, there are 4 more soldiers than there were in the original arrangement. How old is the lieutenant? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( p \) and \( q \) be positive integers such that \( \frac{5}{8}<\frac{p}{q}<\frac{7}{8} \). What is the smallest value of \( p \) such that \( p+q=2005 \)? | {
"answer": "772",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the geometric sequence $\{a_n\}$, if $a_2a_5= -\frac{3}{4}$ and $a_2+a_3+a_4+a_5= \frac{5}{4}$, calculate $\frac{1}{a_2}+ \frac{1}{a_3}+ \frac{1}{a_4}+ \frac{1}{a_5}$. | {
"answer": "-\\frac{5}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive integers less than 900 can be written as a product of two or more consecutive prime numbers? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A natural number greater than 1 is called "good" if it is exactly equal to the product of its distinct proper divisors (excluding 1 and the number itself). Find the sum of the first ten "good" natural numbers. | {
"answer": "182",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 100 natural numbers written in a circle. It is known that among any three consecutive numbers, there is at least one even number. What is the minimum number of even numbers that can be among the written numbers? | {
"answer": "34",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A smooth sphere with a radius of 1 cm is dipped in red paint and released between two absolutely smooth concentric spheres with radii of 4 cm and 6 cm respectively (this sphere is outside the smaller sphere but inside the larger one). Upon contact with both spheres, the sphere leaves a red mark. During its movement, the sphere followed a closed path, resulting in a red-bordered area on the smaller sphere with an area of 37 square cm. Find the area of the region bordered by the red contour on the larger sphere. Give your answer in square centimeters, rounded to the nearest hundredth if necessary. | {
"answer": "83.25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the table, 8 is located in the 3rd row and 2nd column, 2017 is located in the $a$th row and $b$th column. Find $a - b = \quad$
\begin{tabular}{|c|c|c|}
\hline 1 & 4 & 5 \\
\hline 2 & 3 & 6 \\
\hline 9 & 8 & 7 \\
\hline 10 & 13 & 14 \\
\hline 11 & 12 & 15 \\
\hline 18 & 17 & 16 \\
\hline 19 & 22 & 23 \\
\hline 20 & 21 & $\cdots$ \\
\hline
\end{tabular} | {
"answer": "672",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many six-digit numbers are there in which only the middle two digits are the same? | {
"answer": "90000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A number $m$ is randomly selected from the set $\{12, 14, 16, 18, 20\}$, and a number $n$ is randomly selected from $\{2005, 2006, 2007, \ldots, 2024\}$. What is the probability that $m^n$ has a units digit of $6$?
A) $\frac{1}{5}$
B) $\frac{1}{4}$
C) $\frac{2}{5}$
D) $\frac{1}{2}$
E) $\frac{3}{5}$ | {
"answer": "\\frac{2}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $0 < a < 1$, and $\left[a+\frac{1}{2020}\right]+\left[a+\frac{2}{2020}\right]+\cdots+\left[a+\frac{2019}{2020}\right] = 2018$, calculate the value of $[1010 a]$. | {
"answer": "1009",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $α \in (0, \frac{π}{2})$ and $\sin (\frac{π}{4} - α) = \frac{\sqrt{10}}{10}$,
(1) find the value of $\tan 2α$;
(2) find the value of $\frac{\sin (α + \frac{π}{4})}{\sin 2α + \cos 2α + 1}$. | {
"answer": "\\frac{\\sqrt{10}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the point M ($4\sqrt {2}$, $\frac{\pi}{4}$) in the polar coordinate system, the polar coordinate equation of the curve C is $\rho^2 = \frac{12}{1+2\sin^{2}\theta}$. Point N moves on the curve C. Establish a rectangular coordinate system with the pole as the coordinate origin and the positive half of the polar axis as the x-axis. The parameter equation of line 1 is $\begin{cases} x=6+t \\ y=t \end{cases}$ (t is the parameter).
(I) Find the ordinary equation of line 1 and the parameter equation of curve C;
(II) Find the minimum distance of the distance from the midpoint P of line segment MN to line 1. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle XYZ$, $\angle X = 90^\circ$ and $\tan Z = \sqrt{3}$. If $YZ = 150$, what is $XY$? | {
"answer": "75\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \(\alpha, \beta \in \left[0, \frac{\pi}{4}\right]\), find the maximum value of \(\sin(\alpha - \beta) + 2 \sin(\alpha + \beta)\). | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the function f(x) = 2sinωx(cosωx + $\sqrt{3}$sinωx) - $\sqrt{3}$ (ω > 0) has a minimum positive period of π.
(1) Find the interval where the function f(x) is monotonically increasing.
(2) If the graph of function f(x) is shifted to the left by $\frac{\pi}{6}$ units and up by 2 units to obtain the graph of function g(x), find the sum of the zeros of function g(x) in the interval [0, 5π]. | {
"answer": "\\frac{55\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a certain exam with 50 questions, each correct answer earns 3 points, each incorrect answer deducts 1 point, and unanswered questions neither add nor deduct points. Xiaolong scored 120 points. How many questions did Xiaolong answer correctly at most? | {
"answer": "42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose $x$ and $y$ are in the interval $(0, +\infty)$, and $x^{2}+ \frac{y^{2}}{2}=1$, find the maximum value of $x \sqrt{1+y^{2}}$. | {
"answer": "\\frac{3\\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose in a right triangle $PQR$, $\cos Q = 0.5$. Point $Q$ is at the origin, and $PQ = 15$ units along the positive x-axis. What is the length of $QR$? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a point on the terminal side of angle \(\alpha\) has coordinates \((\sin \frac{2\pi }{3},\cos \frac{2\pi }{3})\), find the smallest positive angle for \(\alpha\). | {
"answer": "\\frac{11\\pi }{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let P be any point on the curve $y=x^2-\ln x$. Find the minimum distance from point P to the line $y=x-4$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( p \in \mathbb{R} \). In the complex plane, consider the equation
\[ x^2 - 2x + 2 = 0, \]
whose two complex roots are represented by points \( A \) and \( B \). Also consider the equation
\[ x^2 + 2px - 1 = 0, \]
whose two complex roots are represented by points \( C \) and \( D \). If the four points \( A \), \( B \), \( C \), and \( D \) are concyclic, find the value of \( p \). | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The right triangles $\triangle M D C$ and $\triangle A D K$ share a common right angle $\angle D$. Point $K$ is on $C D$ and divides it in the ratio $2: 3$ counting from point $C$. Point $M$ is the midpoint of side $A D$. Find the sum of $\angle A K D$ and $\angle M C D$, if $A D: C D=2: 5$. | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point $P$ is a moving point on the circle $x^2+y^2=18$, and $PQ \perp x$-axis at point $Q$, if the moving point $M$ satisfies $\overrightarrow{OM}=\frac{1}{3}\overrightarrow{OP}+\frac{2}{3}\overrightarrow{OQ}$.
(Ⅰ) Find the equation of the trajectory $C$ of the moving point $M$;
(Ⅱ) The line passing through point $E(-4,0)$ with equation $x=my-4$ $(m\ne 0)$ intersects the curve $C$ at points $A$ and $B$. The perpendicular bisector of segment $AB$ intersects the $x$-axis at point $D$. Find the value of $\frac{|DE|}{|AB|}$. | {
"answer": "\\frac{\\sqrt{2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Anya wants to buy ice cream that costs 19 rubles. She has two 10-ruble coins, two 5-ruble coins, and one 2-ruble coin in her pocket. Anya randomly picks three coins from her pocket without looking. Find the probability that the selected coins will be enough to pay for the ice cream. | {
"answer": "0.4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides opposite $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, and $c$ respectively. Given that $\tan A + \tan C = \sqrt{3}(\tan A \tan C - 1)$,
(Ⅰ) find angle $B$
(Ⅱ) If $b=2$, find the maximum area of $\triangle ABC$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Little Wang has three ballpoint pens of the same style but different colors. Each pen has a cap that matches its color. Normally, Wang keeps the pen and cap of the same color together, but sometimes he mixes and matches the pens and caps. If Wang randomly pairs the pens and caps, calculate the probability that he will mismatch the colors of two pairs. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a group of seven friends, the mean (average) age of three of the friends is 12 years and 3 months, and the mean age of the remaining four friends is 13 years and 5 months. In months, what is the mean age of all seven friends? | {
"answer": "155",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Of the following complex numbers $z$, which one has the property that $z^5$ has the greatest real part?
- $z = -3$
- $z = -2 + \frac{1}{2}i$
- $z = -\frac{3}{2} + \frac{3}{2}i$
- $z = -1 + 2i$
- $z = 3i$
A) $-243$
B) $-12.3125$
C) $-168.75$
D) $39$
E) $0$ | {
"answer": "39",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A man bought a number of ping-pong balls where a 16% sales tax is added. If he did not have to pay tax, he could have bought 3 more balls for the same amount of money. If \( B \) is the total number of balls that he bought, find \( B \). | {
"answer": "18.75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Emily and John each solved three-quarters of the homework problems individually and the remaining one-quarter together. Emily correctly answered 70% of the problems she solved alone, achieving an overall accuracy of 76% on her homework. John had an 85% success rate with the problems he solved alone. Calculate John's overall percentage of correct answers. | {
"answer": "87.25\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the surface area of the part of the paraboloid of revolution \( 3y = x^2 + z^2 \) that is located in the first octant and bounded by the plane \( y = 6 \). | {
"answer": "\\frac{39 \\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Square \(ABCD\) has points \(A\) and \(B\) on the \(x\)-axis, and points \(C\) and \(D\) below the \(x\)-axis on the parabola with equation \(y = x^{2} - 4\). What is the area of \(ABCD\)? | {
"answer": "24 - 8\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Shuai Shuai memorized more than one hundred words in seven days. The number of words memorized in the first three days is $20\%$ less than the number of words memorized in the last four days, and the number of words memorized in the first four days is $20\%$ more than the number of words memorized in the last three days. How many words did Shuai Shuai memorize in total over the seven days? | {
"answer": "198",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the numbers $1, 2, \cdots, 2005$, choose $n$ different numbers. If it is always possible to find three numbers among these $n$ numbers that can form the side lengths of a triangle, determine the minimum value of $n$. | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A tourist spends half of their money and an additional $100 \mathrm{Ft}$ each day. By the end of the fifth day, all their money is gone. How much money did they originally have? | {
"answer": "6200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute $\arccos (\sin 3).$ All functions are in radians. | {
"answer": "3 - \\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify the expression and then evaluate: $(a-2b)(a^2+2ab+4b^2)-a(a-5b)(a+3b)$, where $a=-1$ and $b=1$. | {
"answer": "-21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify $(- \sqrt {3})^{2}\;^{- \frac {1}{2}}$. | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Continue the problem statement "Petya and Kolya went to the forest to pick mushrooms. In the first hour, Petya collected twice as many mushrooms as Kolya..." so that its solution is determined by the expression $30: 3+4$. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow {m}=(a,-1)$, $\overrightarrow {n}=(2b-1,3)$ where $a > 0$ and $b > 0$. If $\overrightarrow {m}$ is parallel to $\overrightarrow {n}$, determine the value of $\dfrac{2}{a}+\dfrac{1}{b}$. | {
"answer": "8+4\\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the value of the definite integral $\int_{0}^{1} ( \sqrt{1-(x-1)^{2}}-{x}^{2})dx$. | {
"answer": "\\frac{\\pi}{4} - \\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A tetrahedron with each edge length equal to $\sqrt{2}$ has all its vertices on the same sphere. Calculate the surface area of this sphere. | {
"answer": "3\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the $2013\cdot Jining$ test, if the sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n = n^2 - 4n + 2$, then $|a_1| + |a_2| + \ldots + |a_{10}| = \_\_\_\_\_\_\_$. | {
"answer": "66",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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