problem stringlengths 10 5.15k | answer dict |
|---|---|
Quarter circles of radius 1' form a pattern as shown below. What is the area, in square feet, of the shaded region in a 3-foot length of this pattern? The quarter circles' flat sides face outward, alternating top and bottom along the length. | {
"answer": "\\frac{3}{4}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a survey of 500 students at a different school, it was found that 75 students own cats and 125 students own dogs. What percent of the students own cats? Also, what percent of the students own dogs? | {
"answer": "25\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a cone with vertex $S$, and generatrices $SA$, $SB$ perpendicular to each other, and the angle between $SA$ and the base of the cone is $30^{\circ}$. If the area of $\triangle SAB$ is $8$, then the volume of this cone is ______. | {
"answer": "8\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the line $15x + 6y = 90$ which forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?
A) $21$
B) $35$
C) $41$
D) $21 + 10\sqrt{\frac{1}{29}}$ | {
"answer": "21 + 10\\sqrt{\\frac{1}{29}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the inequality:
\[ 2 \sqrt{(4 x-9)^{2}}+\sqrt[4]{\sqrt{3 x^{2}+6 x+7}+\sqrt{5 x^{2}+10 x+14}+x^{2}+2 x-4} \leq 18-8 x \] | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a revised game of Deal or No Deal, participants choose a box at random from a set of $30$, each containing one of the following values:
\[
\begin{array}{|c|c|}
\hline
\$0.50 & \$50,000 \\
\hline
\$5 & \$100,000 \\
\hline
\$20 & \$150,000 \\
\hline
\$50 & \$200,000 \\
\hline
\$100 & \$250,000 \\
\hline
\$250 & \$300,000 \\
\hline
\$500 & \$400,000 \\
\hline
\$750 & \$500,000 \\
\hline
\$1,000 & \$750,000 \\
\hline
\$1,500 & \$1,000,000 \\
\hline
\end{array}
\]
After choosing a box, participants eliminate other boxes by opening them. What is the minimum number of boxes a participant needs to eliminate in order to have a two-thirds chance of holding at least $\$200,\!000$ as his or her chosen box? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$, $\overrightarrow{a}\cdot (\overrightarrow{b}-\overrightarrow{a})=0$, find the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many integers \( n \) between 1 and 15 (inclusive) is \(\frac{n}{18}\) a repeating decimal? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose $X$ is a discrete random variable, $P(X=x_{1})= \frac {2}{3},P(X=x_{2})= \frac {1}{3}$, and $x_{1} < x_{2}$, it is also known that $EX= \frac {4}{9}$, $DX=2$, calculate the sum of $x_{1}$ and $x_{2}$. | {
"answer": "\\frac{17}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a point \\(P(m,-\sqrt{3})\\) (\\(m \neq 0\\)) on the terminal side of angle \\(\alpha\\), and \\(\cos \alpha = \frac{\sqrt{2}m}{4}\\),
\\((i)\\) find the value of \\(m\\);
\\((ii)\\) calculate \\(\sin \alpha\\) and \\(\tan \alpha\\). | {
"answer": "-\\frac{\\sqrt{15}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify $\dfrac{123}{999} \cdot 27.$ | {
"answer": "\\dfrac{123}{37}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A whole number was increased by 2, and its square decreased by 2016. What was the number initially (before the increase)? | {
"answer": "-505",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
People are standing in a circle - there are liars, who always lie, and knights, who always tell the truth. Each of them said that among the people standing next to them, there is an equal number of liars and knights. How many people are there in total if there are 48 knights? | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Brachycephalus frogs have three toes on each foot and two fingers on each hand. The common frog has five toes on each foot and four fingers on each hand. Some Brachycephalus and common frogs are in a bucket. Each frog has all its fingers and toes. Between them they have 122 toes and 92 fingers. How many frogs are in the bucket?
A 15
B 17
C 19
D 21
E 23 | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Every day, the ram learns the same number of languages. By the evening of his birthday, he knew 1000 languages. On the first day of the same month, he knew 820 languages by evening, and on the last day of that month, he knew 1100 languages. When is the ram's birthday? | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square \(ABCD\) has a side-length of 2, and \(M\) is the midpoint of \(BC\). The circle \(S\) inside the quadrilateral \(AMCD\) touches the three sides \(AM\), \(CD\), and \(DA\). What is its radius? | {
"answer": "3 - \\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers from 1 to 8 are arranged at the vertices of a cube in such a way that the sum of the numbers at any three vertices on the same face is at least 10. What is the minimum possible sum of the numbers on the vertices of one face? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, \( Z \) lies on \( XY \) and the three circles have diameters \( XZ \), \( ZY \), and \( XY \). If \( XZ = 12 \) and \( ZY = 8 \), then the ratio of the area of the shaded region to the area of the unshaded region is | {
"answer": "12:13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, with $c=5$ and $b(2\sin B+\sin A)+(2a+b)\sin A=2c\sin C$.
(1) Find the value of $C$.
(2) If $\cos A= \frac {4}{5}$, find the value of $b$. | {
"answer": "4- \\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the line $y=x+1$ intersects with the ellipse $mx^2+my^2=1(m > n > 0)$ at points $A$ and $B$, where the x-coordinate of the midpoint of the chord $AB$ is equal to $-\frac{1}{3}$, find the eccentricity of the hyperbola $\frac{y^2}{m^2}-\frac{x^2}{n^2}=1$. | {
"answer": "\\frac{\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $F$ be the set of functions from $\mathbb{R}^{+}$ to $\mathbb{R}^{+}$ such that $f(3x) \geq f(f(2x)) + x$. Maximize $\alpha$ such that $\forall x \geq 0, \forall f \in F, f(x) \geq \alpha x$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a square $\mathrm{ABCD}$, point $\mathrm{E}$ is on $\mathrm{BC}$ with $\mathrm{BE} = 2$ and $\mathrm{CE} = 1$. Point $\mathrm{P}$ moves along $\mathrm{BD}$. What is the minimum value of $\mathrm{PE} + \mathrm{PC}$? | {
"answer": "\\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a redesign of his company's logo, Wei decided to use a larger square and more circles. Each circle is still tangent to two sides of the square and its adjacent circles, but now there are nine circles arranged in a 3x3 grid instead of a 2x2 grid. If each side of the new square measures 36 inches, calculate the total shaded area in square inches. | {
"answer": "1296 - 324\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From 125 sugar cubes, a $5 \times 5 \times 5$ cube was made. Ponchik picked all the cubes that have an odd number of neighbors and ate them (neighbors are those cubes that share a face). How many cubes did Ponchik eat in total? | {
"answer": "62",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given there are 2, 1, 3, and 4 paths leading to the top of the mountain from the east, west, south, and north sides, respectively, calculate the maximum number of ways to ascend from one side and descend from any other side. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the numbers: $8, a, b, 26, x$, where each of the first four numbers is the average of the two adjacent numbers, find the value of $x$. | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many numbers $n$ does $2017$ divided by $n$ have a remainder of either $1$ or $2$ ? | {
"answer": "43",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point P is the intersection of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ ($a > 0, b > 0$) and the circle $x^2+y^2=a^2+b^2$ in the first quadrant. $F_1$ and $F_2$ are the left and right foci of the hyperbola, respectively, and $|PF_1|=3|PF_2|$. Calculate the eccentricity of the hyperbola. | {
"answer": "\\frac{\\sqrt{10}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a, b, c, d$ be real numbers such that $a + b + c + d = 10$ and $ab + ac + ad + bc + bd + cd = 20$. Find the largest possible value of $d$. | {
"answer": "\\frac{5 + \\sqrt{105}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify and then evaluate the expression: $$(1- \frac {1}{a-2})÷ \frac {a^{2}-6a+9}{2a-4}$$, where $$a=2 \sqrt {3}+3$$ | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If two stagecoaches travel daily from Bratislava to Brașov, and likewise, two stagecoaches travel daily from Brașov to Bratislava, and considering that the journey takes ten days, how many stagecoaches will you encounter on your way when traveling by stagecoach from Bratislava to Brașov? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ΔABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $a=\sqrt{3}$, $b=\sqrt{2}$, and $A=\frac{\pi}{3}$, find the value of $B=$ _______; and the area of $ΔABC=S_{ΔABC}=$ _______. | {
"answer": "\\frac{3+ \\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the integer $n,$ $-180 \le n \le 180,$ such that $\cos n^\circ = \cos 430^\circ.$ | {
"answer": "-70",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
OKRA is a trapezoid with OK parallel to RA. If OK = 12 and RA is a positive integer, how many integer values can be taken on by the length of the segment in the trapezoid, parallel to OK, through the intersection of the diagonals? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are two islands, A and B, that are 20 nautical miles apart. When viewing Island C from Island A, the angle between Island B and Island C is 60°. When viewing Island C from Island B, the angle between Island A and Island C is 75°. Find the distance between Island B and Island C. | {
"answer": "10\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The shortest distance from a moving point P on circle C: $\rho = -4\sin\theta$ to the line $l: \rho\sin(\theta + \frac{\pi}{4}) = \sqrt{2}$ is ______. | {
"answer": "2\\sqrt{2} - 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangle \( ABC \) is given. It is known that \( AB=4 \), \( AC=2 \), and \( BC=3 \). The angle bisector of \( \angle BAC \) intersects side \( BC \) at point \( K \). A line passing through point \( B \) parallel to \( AC \) intersects the extension of the bisector \( AK \) at point \( M \). Find \( KM \). | {
"answer": "2 \\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the area of the parallelogram formed by the vectors \( a \) and \( b \).
Given:
\[ a = 3p - 4q \]
\[ b = p + 3q \]
\[ |p| = 2 \]
\[ |q| = 3 \]
\[ \text{Angle between } p \text{ and } q \text{ is } \frac{\pi}{4} \] | {
"answer": "39\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the value of the function $f(x)=3x^{6}-2x^{5}+x^{3}+1$ at $x=2$ using the Horner's method (also known as the Qin Jiushao algorithm) to determine the value of $v_{4}$. | {
"answer": "34",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mr. Lee V. Soon starts his morning commute at 7:00 AM to arrive at work by 8:00 AM. If he drives at an average speed of 30 miles per hour, he is late by 5 minutes, and if he drives at an average speed of 70 miles per hour, he is early by 4 minutes. Find the speed he needs to maintain to arrive exactly at 8:00 AM. | {
"answer": "32.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Inside a square, 100 points are marked. The square is divided into triangles such that the vertices of the triangles are only the marked 100 points and the vertices of the square, and for each triangle in the division, each marked point either lies outside the triangle or is a vertex of that triangle (such divisions are called triangulations). Find the number of triangles in the division. | {
"answer": "202",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse $C$: $\frac{x^{2}}{2}+y^{2}=1$, respectively. Point $N$ is the top vertex of the ellipse $C$. If a moving point $M$ satisfies $|\overrightarrow{MN}|^{2}=2\overrightarrow{MF_{1}}\cdot\overrightarrow{MF_{2}}$, then the maximum value of $|\overrightarrow{MF_{1}}+2\overrightarrow{MF_{2}}|$ is \_\_\_\_\_\_ | {
"answer": "6+\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A line segment is divided into three parts, $x$, $y$, and $z$, such that, $x < y < z$ and $x$ to $y$ is as $y$ to $z$. If all three parts combined form a segment of length $s$, and $x + y = z$, determine the value of the ratio $\frac{x}{y}$.
A) $\frac{-1 - \sqrt{5}}{2}$
B) $\frac{-1 + \sqrt{5}}{2}$
C) $-\frac{1 + \sqrt{5}}{2}$
D) $\frac{1 - \sqrt{5}}{2}$ | {
"answer": "\\frac{-1 + \\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain agency has 18 elderly, 12 middle-aged, and 6 young individuals. When drawing a sample of size X using systematic sampling and stratified sampling, there is no need to discard any individuals. However, if the sample size is increased by 1, then using systematic sampling requires the removal of 1 individual from the total population. Therefore, the sample size X = ______. | {
"answer": "X = 6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = \cos x \cdot \sin\left(x + \frac{\pi}{3}\right) - \sqrt{3}\cos^2x + \frac{\sqrt{3}}{4}$, where $x \in \mathbb{R}$.
(1) Find the interval of monotonic increase for $f(x)$.
(2) In an acute triangle $\triangle ABC$, where the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, if $f(A) = \frac{\sqrt{3}}{4}$ and $a = \sqrt{3}$, find the maximum area of $\triangle ABC$. | {
"answer": "\\frac{3\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose you have $6$ red shirts, $7$ green shirts, $9$ pairs of pants, $10$ blue hats, and $10$ red hats, all distinct. How many outfits can you make consisting of one shirt, one pair of pants, and one hat, if neither the hat nor the pants can match the shirt in color? | {
"answer": "1170",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( x \) and \( y \) be positive numbers, and let \( s \) be the smallest of the numbers \( x \), \( y + \frac{1}{x} \), and \( \frac{1}{y} \). Find the maximum possible value of \( s \). For which values of \( x \) and \( y \) is it achieved? | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the polar equation of curve $C_{1}$ is $\rho=2\sin \theta$, and the polar equation of curve $C_{2}$ is $\theta= \frac {\pi}{3}$ ($\rho\in\mathbb{R}$), curves $C_{1}$ and $C_{2}$ intersect at points $M$ and $N$, then the length of chord $MN$ is ______. | {
"answer": "\\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sphere is inscribed in a cube. The edge of the cube is 10 inches. Calculate both the volume and the surface area of the sphere. Express your answer for the volume in terms of \(\pi\). | {
"answer": "100\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that sinα + cosα = $\frac{7}{5}$, find the value of tanα. | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the ratio of the legs of a right triangle is $3: 4$, then the ratio of the corresponding segments of the hypotenuse made by a perpendicular upon it from the vertex is:
A) $\frac{16}{9}$
B) $\frac{9}{16}$
C) $\frac{3}{4}$
D) $\frac{4}{3}$ | {
"answer": "\\frac{16}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We are given a cone with height 6, whose base is a circle with radius $\sqrt{2}$ . Inside the cone, there is an inscribed cube: Its bottom face on the base of the cone, and all of its top vertices lie on the cone. What is the length of the cube's edge?
 | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles touch each other internally at point K. The chord \(A B\) of the larger circle is tangent to the smaller circle at point \(L\), and \(A L = 10\). Find \(B L\) if \(A K: B K = 2: 5\). | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A finite increasing sequence \(a_{1}, a_{2}, \ldots, a_{n}\) of natural numbers is given, where \(n \geq 3\), and for all \(k \leq n-2\) the following equality holds: \(a_{k+2} = 3a_{k+1} - 2a_{k} - 2\). The sequence must include \(a_{k} = 2022\). Determine the maximum number of three-digit numbers, divisible by 4, that this sequence can contain. | {
"answer": "225",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In how many ways can four black balls, four white balls, and four blue balls be distributed into six different boxes? | {
"answer": "2000376",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many integers \( n \) between 1 and 15 (inclusive) is \(\frac{n}{18}\) a repeating decimal? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point is randomly thrown on the segment \([6, 11]\) and let \( k \) be the resulting value. Find the probability that the roots of the equation \(\left(k^{2}-2k-24\right)x^{2}+(3k-8)x+2=0\) satisfy the condition \( x_{1} \leq 2x_{2} \). | {
"answer": "2/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the complex plane, the complex numbers $\frac {1}{1+i}$ and $\frac {1}{1-i}$ (where $i$ is the imaginary unit) correspond to points A and B, respectively. If point C is the midpoint of line segment AB, determine the complex number corresponding to point C. | {
"answer": "\\frac {1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $T$ be the set of all complex numbers $z$ where $z = w - \frac{1}{w}$ for some complex number $w$ of absolute value $2$. Determine the area inside the curve formed by $T$ in the complex plane. | {
"answer": "\\frac{9}{4} \\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a ship, it was decided to determine the depth of the ocean at their current location. The signal sent by the echo sounder was received on the ship after 5 seconds. The speed of sound in water is 1.5 km/s. Determine the depth of the ocean. | {
"answer": "3750",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a circle $x^2 + (y-1)^2 = 1$ with its tangent line $l$, which intersects the positive x-axis at point A and the positive y-axis at point B. Determine the y-intercept of the tangent line $l$ when the distance AB is minimized. | {
"answer": "\\frac{3+\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, if $A=120^{\circ}$, $AB=5$, $BC=7$, find the value of $\sin B$. | {
"answer": "\\frac{3 \\sqrt{3}}{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Place each of the digits 3, 4, 5, and 6 in exactly one square to make the smallest possible product. Arrange the digits such that two numbers are formed and multiplied. How should the digits be placed, and what is the minimum product? | {
"answer": "1610",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a tetrahedron \( PABC \), \(\angle APB = \angle BPC = \angle CPA = 90^\circ\). The dihedral angles formed between \(\triangle PBC, \triangle PCA, \triangle PAB\), and \(\triangle ABC\) are denoted as \(\alpha, \beta, \gamma\), respectively. Consider the following three propositions:
1. \(\cos \alpha \cdot \cos \beta + \cos \beta \cdot \cos \gamma + \cos \gamma \cdot \cos \alpha \geqslant \frac{3}{2}\)
2. \(\tan^2 \alpha + \tan^2 \beta + \tan^2 \gamma \geqslant 0\)
3. \(\sin \alpha \cdot \sin \beta \cdot \sin \gamma \leqslant \frac{1}{2}\)
Which of the following is correct? | {
"answer": "(2)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many numbers between $1$ and $2500$ are integer multiples of $4$ or $5$ but not $15$? | {
"answer": "959",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The café has enough chairs to seat $310_5$ people. If $3$ people are supposed to sit at one table, how many tables does the café have? | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Kolya and his sister Masha went to visit someone. After walking a quarter of the way, Kolya remembered that they had forgotten the gift at home and turned back, while Masha continued walking. Masha arrived at the visit 20 minutes after leaving home. How many minutes later did Kolya arrive, given that they walked at the same speeds all the time? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The radius of a sphere is \( r = 10 \text{ cm} \). Determine the volume of the spherical segment whose surface area is in the ratio 10:7 compared to the area of its base. | {
"answer": "288 \\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)={x^3}+\frac{{{{2023}^x}-1}}{{{{2023}^x}+1}}+5$, if real numbers $a$ and $b$ satisfy $f(2a^{2})+f(b^{2}-2)=10$, then the maximum value of $a\sqrt{1+{b^2}}$ is ______. | {
"answer": "\\frac{3\\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Vasya and Petya live in the mountains and like to visit each other. They ascend the mountain at a speed of 3 km/h and descend at a speed of 6 km/h (there are no flat sections of the road). Vasya calculated that it takes him 2 hours and 30 minutes to go to Petya, and 3 hours and 30 minutes to return. What is the distance between Vasya and Petya's homes? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the program flowchart on the right is executed, determine the value of the output S. | {
"answer": "55",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point is randomly dropped on the interval $[5 ; 7]$ and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}-3 k-4\right) x^{2}+(3 k-7) x+2=0$ satisfy the condition $x_{1} \leq 2 x_{2}$. | {
"answer": "1/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the positive value of $y$ which satisfies
\[\log_7 (y - 3) + \log_{\sqrt{7}} (y^2 - 3) + \log_{\frac{1}{7}} (y - 3) = 3.\] | {
"answer": "\\sqrt{\\sqrt{343} + 3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function \( f(x) = \sqrt{1 + x \sqrt{1 + (x+1) \sqrt{1 + (x+2) \sqrt{1 + (x+3) \sqrt{\cdots}}}}} \) for \( x > 1 \), \( x \in \mathbf{N}^{*} \), find \( f(2008) \). | {
"answer": "2009",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Diagonals $A C$ and $C E$ of a regular hexagon $A B C D E F$ are divided by points $M$ and $N$ such that $A M : A C = C N : C E = \lambda$. Find $\lambda$ if it is known that points $B, M$, and $N$ are collinear. | {
"answer": "\\frac{1}{\\sqrt{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate: $$\frac {1}{2}\log_{2}3 \cdot \frac {1}{2}\log_{9}8 = \_\_\_\_\_\_ ．$$ | {
"answer": "\\frac {3}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a certain sunflower cell has 34 chromosomes at the late stage of the second meiotic division when forming pollen grains, determine the number of tetrads that can be produced by this cell during meiosis. | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(\mathbb{Z}_{\geq 0}\) be the set of non-negative integers, and let \(f: \mathbb{Z}_{\geq 0} \times \mathbb{Z}_{\geq 0} \rightarrow \mathbb{Z}_{\geq 0}\) be a bijection such that whenever \(f(x_1, y_1) > f(x_2, y_2)\), we have \(f(x_1+1, y_1) > f(x_2+1, y_2)\) and \(f(x_1, y_1+1) > f(x_2, y_2+1)\).
Let \(N\) be the number of pairs of integers \((x, y)\), with \(0 \leq x, y < 100\), such that \(f(x, y)\) is odd. Find the smallest and largest possible values of \(N\). | {
"answer": "5000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose \( S = \{1,2, \cdots, 2005\} \). Find the minimum value of \( n \) such that every subset of \( S \) consisting of \( n \) pairwise coprime numbers contains at least one prime number. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)= \sqrt {3}|\cos \frac {π}{2}x|(x≥0)$, the highest points of the graph from left to right are consecutively labeled as P₁, P₃, P₅, …, and the intersection points of the function y=f(x) with the x-axis from left to right are consecutively labeled as P₂, P₄, P₆, …, Let Sₙ = $\overrightarrow {P_{1}P_{2}}\cdot \overrightarrow {P_{2}P_{3}}+ ( \overrightarrow {P_{2}P_{3}}\cdot \overrightarrow {P_{3}P_{4}})^{2}$+$( \overrightarrow {P_{3}P_{4}}\cdot \overrightarrow {P_{4}P_{5}})^{3}$+$( \overrightarrow {P_{4}P_{5}}\cdot \overrightarrow {P_{5}P_{6}})^{4}$+…+$( \overrightarrow {P_{n}P_{n+1}}\cdot \overrightarrow {p_{n+1}p_{n+2}})^{n}$, then $\overset{lim}{n\rightarrow \infty } \frac {S_{n}}{1+(-2)^{n}}$= \_\_\_\_\_\_. | {
"answer": "\\frac {2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given real numbers $a$ and $b$, satisfying $e^{2-a}=a$ and $b\left(\ln b-1\right)=e^{3}$, where $e$ is the base of natural logarithm, the value of $ab$ is ______. | {
"answer": "e^{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, $\triangle ABE$, $\triangle BCE$, and $\triangle CDE$ are right-angled with $\angle AEB = 30^\circ, \angle BEC = 45^\circ$, and $\angle CED = 45^\circ$, and $AE=30$. Find the length of $CE$. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the integral \(\int_{0}^{1} \ln x \ln (1-x) \, dx\). | {
"answer": "2 - \\frac{\\pi^2}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$ABCD$ is a rectangle; $P$ and $Q$ are the mid-points of $AB$ and $BC$ respectively. $AQ$ and $CP$ meet at $R$. If $AC = 6$ and $\angle ARC = 150^{\circ}$, find the area of $ABCD$. | {
"answer": "8\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\triangle ABC$ be a right triangle with $B$ as the right angle. A circle with diameter $AC$ intersects side $BC$ at point $D$. If $AB = 18$ and $AC = 30$, find the length of $BD$. | {
"answer": "14.4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $[x]$ denote the greatest integer not exceeding the real number $x$. If
\[ A = \left[\frac{7}{8}\right] + \left[\frac{7^2}{8}\right] + \cdots + \left[\frac{7^{2019}}{8}\right] + \left[\frac{7^{2020}}{8}\right], \]
what is the remainder when $A$ is divided by 50? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using the digits 0, 1, 2, 3, 4, how many even three-digit numbers can be formed if each digit can be used more than once, and the number must be greater than 200? | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x)= \sqrt{2}\sin \left( 2x+ \frac{π}{4} \right)$.
(1) Find the equation of the axis of symmetry of the graph of the function $f(x)$;
(2) Find the interval(s) where $f(x)$ is monotonically increasing;
(3) Find the maximum and minimum values of the function $f(x)$ when $x\in \left[ \frac{π}{4}, \frac{3π}{4} \right]$. | {
"answer": "- \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x$ and $y$ satisfy the constraints:
\[
\begin{cases}
x + 2y - 1 \geqslant 0, \\
x - y \geqslant 0, \\
0 \leqslant x \leqslant k,
\end{cases}
\]
If the minimum value of $z = x + ky$ is $-2$, then the maximum value of $z$ is ______. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among the integers from 1 to 100, how many integers can be divided by exactly two of the following four numbers: 2, 3, 5, 7? | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $c=2$, $2\sin A= \sqrt {3}a\cos C$.
(1) Find the measure of angle $C$;
(2) If $2\sin 2A+ \sin (2B+C)= \sin C$, find the area of $\triangle ABC$. | {
"answer": "\\dfrac {2 \\sqrt {3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For any real number \(x\), \(\lfloor x \rfloor\) denotes the largest integer less than or equal to \(x\). For example, \(\lfloor 4.2 \rfloor = 4\) and \(\lfloor 0.9 \rfloor = 0\).
If \(S\) is the sum of all integers \(k\) with \(1 \leq k \leq 999999\) and for which \(k\) is divisible by \(\lfloor \sqrt{k} \rfloor\), then \(S\) equals: | {
"answer": "999999000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\theta$ is an angle in the second quadrant, and $\tan(\theta + \frac{\pi}{4}) = \frac{1}{2}$, calculate the value of $\sin\theta + \cos\theta$. | {
"answer": "-\\frac{\\sqrt{10}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ctibor marked a square land plot on a map with a scale of 1:50000 and calculated that its side corresponds to $1 \mathrm{~km}$ in reality. He then resized the map on a copier such that the marked square had an area $1.44 \mathrm{~cm}^{2}$ smaller than the original.
What was the scale of the resized map?
Hint: What were the dimensions of the marked plot on the original map? | {
"answer": "1:62500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( x \) is a real number and \( y = \sqrt{x^2 - 2x + 2} + \sqrt{x^2 - 10x + 34} \). Find the minimum value of \( y \). | {
"answer": "4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain product has a purchase price of 50 yuan per item and a selling price of 60 yuan per item, with a daily sales volume of 190 items. If the selling price of each item increases by 1 yuan, then 10 fewer items are sold each day. Let the selling price increase by $x$ yuan ($x$ is a positive integer), and the daily sales profit be $y$ yuan.
(1) Find the relationship between $y$ and $x$.
(2) At what selling price per item will the maximum daily profit be obtained? What is the maximum profit? | {
"answer": "2100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point $M$ is the midpoint of line segment $AB$ in plane $\alpha$, and point $P$ is a point outside plane $\alpha$. If $AB = 2$ and the angles between lines $PA$, $PM$, $PB$ and plane $\alpha$ are $30^{\circ}$, $45^{\circ}$, and $60^{\circ}$ respectively, find the distance from point $P$ to plane $\alpha$. | {
"answer": "\\frac{\\sqrt{6}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Take a unit sphere \(S\), i.e., a sphere with radius 1. Circumscribe a cube \(C\) about \(S\), and inscribe a cube \(D\) in \(S\) such that every edge of cube \(C\) is parallel to some edge of cube \(D\). What is the shortest possible distance from a point on a face of \(C\) to a point on a face of \(D\)? | {
"answer": "1 - \\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The left and right foci of the hyperbola $E$: $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ ($a > 0, b > 0$) are $F_1$ and $F_2$, respectively. Point $M$ is a point on the asymptote of hyperbola $E$, and $MF_1 \perpendicular MF_2$. If $\sin \angle MF_1F_2 = \dfrac{1}{3}$, then the eccentricity of this hyperbola is ______. | {
"answer": "\\dfrac{9}{7}",
"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, which form a geometric progression. Also, $(2a-c)\cos B = b\cos C$.
(Ⅰ) Find the magnitude of angle $B$;
(Ⅱ) Calculate $\frac{1}{\tan A} + \frac{1}{\tan C}$. | {
"answer": "\\frac{2 \\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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