problem stringlengths 10 5.15k | answer dict |
|---|---|
Given that vectors $a$ and $b$ satisfy $(2a+3b) \perp b$, and $|b|=2\sqrt{2}$, find the projection of vector $a$ onto the direction of $b$. | {
"answer": "-3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
One day while Tony plays in the back yard of the Kubik's home, he wonders about the width of the back yard, which is in the shape of a rectangle. A row of trees spans the width of the back of the yard by the fence, and Tony realizes that all the trees have almost exactly the same diameter, and the trees look equally spaced. Tony fetches a tape measure from the garage and measures a distance of almost exactly $12$ feet between a consecutive pair of trees. Tony realizes the need to include the width of the trees in his measurements. Unsure as to how to do this, he measures the distance between the centers of the trees, which comes out to be around $15$ feet. He then measures $2$ feet to either side of the first and last trees in the row before the ends of the yard. Tony uses these measurements to estimate the width of the yard. If there are six trees in the row of trees, what is Tony's estimate in feet?
[asy]
size(400);
defaultpen(linewidth(0.8));
draw((0,-3)--(0,3));
int d=8;
for(int i=0;i<=5;i=i+1)
{
draw(circle(7/2+d*i,3/2));
}
draw((5*d+7,-3)--(5*d+7,3));
draw((0,0)--(2,0),Arrows(size=7));
draw((5,0)--(2+d,0),Arrows(size=7));
draw((7/2+d,0)--(7/2+2*d,0),Arrows(size=7));
label(" $2$ ",(1,0),S);
label(" $12$ ",((7+d)/2,0),S);
label(" $15$ ",((7+3*d)/2,0),S);
[/asy] | {
"answer": "82",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangle with dimensions $8 \times 2 \sqrt{2}$ and a circle with a radius of 2 have a common center. Find the area of their overlapping region. | {
"answer": "2 \\pi + 4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The probability of an event occurring in each of 900 independent trials is 0.5. Find a positive number $\varepsilon$ such that with a probability of 0.77, the absolute deviation of the event frequency from its probability of 0.5 does not exceed $\varepsilon$. | {
"answer": "0.02",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four steel balls, each with a radius of 1, are completely packed into a container in the shape of a regular tetrahedron. Find the minimum height of this regular tetrahedron. | {
"answer": "2+\\frac{2 \\sqrt{6}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a cube \(ABCD-A_1B_1C_1D_1\) with side length 1, and \(E\) as the midpoint of \(D_1C_1\), find the following:
1. The distance between skew lines \(D_1B\) and \(A_1E\).
2. The distance from \(B_1\) to plane \(A_1BE\).
3. The distance from \(D_1C\) to plane \(A_1BE\).
4. The distance between plane \(A_1DB\) and plane \(D_1CB_1\). | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the amount of cultural and tourism vouchers issued is $2.51 million yuan, express this amount in scientific notation. | {
"answer": "2.51 \\times 10^{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the right triangular prism $ABC - A_1B_1C_1$, $\angle ACB = 90^\circ$, $AC = 2BC$, and $A_1B \perp B_1C$. Find the sine of the angle between $B_1C$ and the lateral face $A_1ABB_1$. | {
"answer": "\\frac{\\sqrt{10}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a convex pentagon, all diagonals are drawn. For each pair of diagonals that intersect inside the pentagon, the smaller of the angles between them is found. What values can the sum of these five angles take? | {
"answer": "180",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = 2\sin^2x + \cos\left(\frac{\pi}{3} - 2x\right)$.
(1) Find the decreasing interval of $f(x)$ on $[0, \pi]$.
(2) Let $\triangle ABC$ have internal angles $A$, $B$, $C$ opposite sides $a$, $b$, $c$ respectively. If $f(A) = 2$, and the vector $\overrightarrow{m} = (1, 2)$ is collinear with the vector $\overrightarrow{n} = (\sin B, \sin C)$, find the value of $\frac{a}{b}$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Professor Severus Snape brewed three potions, each in a volume of 600 ml. The first potion makes the drinker intelligent, the second makes them beautiful, and the third makes them strong. To have the effect of the potion, it is sufficient to drink at least 30 ml of each potion. Severus Snape intended to drink his potions, but he was called away by the headmaster and left the signed potions in large jars on his table. Taking advantage of his absence, Harry, Hermione, and Ron approached the table with the potions and began to taste them.
The first to try the potions was Hermione: she approached the first jar of the intelligence potion and drank half of it, then poured the remaining potion into the second jar of the beauty potion, mixed the contents thoroughly, and drank half of it. Then it was Harry's turn: he drank half of the third jar of the strength potion, then poured the remainder into the second jar, thoroughly mixed everything in the jar, and drank half of it. Now all the contents were left in the second jar, which Ron ended up with. What percentage of the contents of this jar does Ron need to drink to ensure that each of the three potions will have an effect on him? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the sum of the following fractions: $\frac{1}{12} + \frac{2}{12} + \frac{3}{12} + \frac{4}{12} + \frac{5}{12} + \frac{6}{12} + \frac{7}{12} + \frac{8}{12} + \frac{9}{12} + \frac{65}{12} + \frac{3}{4}$.
A) $\frac{119}{12}$
B) $9$
C) $\frac{113}{12}$
D) $10$ | {
"answer": "\\frac{119}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We have created a convex polyhedron using pentagons and hexagons where three faces meet at each vertex. Each pentagon shares its edges with 5 hexagons, and each hexagon shares its edges with 3 pentagons. How many faces does the polyhedron have? | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
You have a \(2 \times 3\) grid filled with integers between 1 and 9. The numbers in each row and column are distinct. The first row sums to 23, and the columns sum to 14, 16, and 17 respectively.
Given the following grid:
\[
\begin{array}{c|c|c|c|}
& 14 & 16 & 17 \\
\hline
23 & a & b & c \\
\hline
& x & y & z \\
\hline
\end{array}
\]
What is \(x + 2y + 3z\)? | {
"answer": "49",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $\frac {x^{2}}{a^{2}} + \frac {y^{2}}{b^{2}} = 1 (a > b > 0)$ with vertex $B$ at the top, vertex $A$ on the right, and right focus $F$. Let $E$ be a point on the lower half of the ellipse such that the tangent at $E$ is parallel to $AB$. If the eccentricity of the ellipse is $\frac {\sqrt{2}}{2}$, then the slope of line $EF$ is __________. | {
"answer": "\\frac {\\sqrt {2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two adjacent faces of a tetrahedron, which are equilateral triangles with side length 1, form a dihedral angle of 60 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto a plane containing the given edge. (12 points) | {
"answer": "\\frac{\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a hyperbola $C$ with an eccentricity of $\sqrt {3}$, foci $F\_1$ and $F\_2$, and a point $A$ on the curve $C$. If $|F\_1A|=3|F\_2A|$, then $\cos \angle AF\_2F\_1=$ \_\_\_\_\_\_. | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)= \dfrac {2-\cos \left( \dfrac {\pi}{4}(1-x)\right)+\sin \left( \dfrac {\pi}{4}(1-x)\right)}{x^{2}+4x+5}(-4\leqslant x\leqslant 0)$, find the maximum value of $f(x)$. | {
"answer": "2+ \\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When \( n \) is a positive integer, the function \( f \) satisfies \( f(n+3)=\frac{f(n)-1}{f(n)+1} \), with \( f(1) \neq 0 \) and \( f(1) \neq \pm 1 \). Find the value of \( f(8) \cdot f(2018) \). | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In how many ways can a committee of three people be formed if the members are to be chosen from four married couples? | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $-\frac{\pi}{2}<\alpha<\frac{\pi}{2}, 2 \tan \beta=\tan 2\alpha, \tan (\beta-\alpha)=-2 \sqrt{2}$, find the value of $\cos \alpha$. | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a given positive integer \( k \), let \( f_{1}(k) \) represent the square of the sum of the digits of \( k \), and define \( f_{n+1}(k) = f_{1}\left(f_{n}(k)\right) \) for \( n \geq 1 \). Find the value of \( f_{2005}\left(2^{2006}\right) \). | {
"answer": "169",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A frustum of a right circular cone is formed by cutting a smaller cone from a larger cone. Suppose the frustum has a lower base radius of 8 inches, an upper base radius of 2 inches, and a height of 5 inches. Calculate the total surface area of the frustum. | {
"answer": "10\\pi \\sqrt{61} + 68\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of the interior numbers in the sixth row of Pascal's Triangle is 30. What is the sum of the interior numbers of the eighth row? | {
"answer": "126",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle with a radius of 15 is tangent to two adjacent sides \( AB \) and \( AD \) of square \( ABCD \). On the other two sides, the circle intercepts segments of 6 and 3 cm from the vertices, respectively. Find the length of the segment that the circle intercepts from vertex \( B \) to the point of tangency. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the volume of the solid bounded by the surfaces \(x + z = 6\), \(y = \sqrt{x}\), \(y = 2\sqrt{x}\), and \(z = 0\) using a triple integral. | {
"answer": "\\frac{48}{5} \\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle with a radius of 2 passes through the midpoints of three sides of triangle \(ABC\), where the angles at vertices \(A\) and \(B\) are \(30^{\circ}\) and \(45^{\circ}\), respectively.
Find the height drawn from vertex \(A\). | {
"answer": "2 + 2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a checkers tournament, students from 10th and 11th grades participated. Each player played against every other player exactly once. A win earned a player 2 points, a draw earned 1 point, and a loss earned 0 points. The number of 11th graders was 10 times the number of 10th graders, and together they scored 4.5 times more points than all the 10th graders combined. How many points did the most successful 10th grader score? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jindra collects dice, all of the same size. Yesterday he found a box in which he started stacking the dice. He managed to fully cover the square base with one layer of dice. He similarly stacked five more layers, but he ran out of dice halfway through the next layer. Today, Jindra received 18 more dice from his grandmother, which were exactly the amount he needed to complete this layer.
How many dice did Jindra have yesterday? | {
"answer": "234",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the circumference of a sector is $20\,cm$ and its area is $9\,cm^2$, find the radian measure of the central angle of the sector. | {
"answer": "\\frac{2}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $O$ is the origin of coordinates, and $M$ is a point on the ellipse $\frac{x^2}{2} + y^2 = 1$. Let the moving point $P$ satisfy $\overrightarrow{OP} = 2\overrightarrow{OM}$.
- (I) Find the equation of the trajectory $C$ of the moving point $P$;
- (II) If the line $l: y = x + m (m \neq 0)$ intersects the curve $C$ at two distinct points $A$ and $B$, find the maximum value of the area of $\triangle OAB$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
New definition: Given that $y$ is a function of $x$, if there exists a point $P(a, a+2)$ on the graph of the function, then point $P$ is called a "real point" on the graph of the function. For example, the "real point" on the line $y=2x+1$ is $P(1,3)$.
$(1)$ Determine whether there is a "real point" on the line $y=\frac{1}{3}x+4$. If yes, write down its coordinates directly; if not, explain the reason.
$(2)$ If there are two "real points" on the parabola $y=x^{2}+3x+2-k$, and the distance between the two "real points" is $2\sqrt{2}$, find the value of $k$.
$(3)$ If there exists a unique "real point" on the graph of the quadratic function $y=\frac{1}{8}x^{2}+\left(m-t+1\right)x+2n+2t-2$, and when $-2\leqslant m\leqslant 3$, the minimum value of $n$ is $t+4$, find the value of $t$. | {
"answer": "t=-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $\cos \frac{A}{2}= \frac{2 \sqrt{5}}{5}, \overrightarrow{AB} \cdot \overrightarrow{AC}=15$.
$(1)$ Find the area of $\triangle ABC$;
$(2)$ If $\tan B=2$, find the value of $a$. | {
"answer": "2 \\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the hyperbola C passes through the point (4, 3) and shares the same asymptotes with $$\frac {x^{2}}{4}$$ - $$\frac {y^{2}}{9}$$ = 1, determine the equation of C and its eccentricity. | {
"answer": "\\frac {\\sqrt {13}} {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the expression \((x+y+z)^{2024} + (x-y-z)^{2024}\), the parentheses are expanded and like terms are combined. How many monomials \(x^{a} y^{b} z^{c}\) have a non-zero coefficient? | {
"answer": "1026169",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given points $A(-2, 0)$ and $B(0, 2)$, $k$ is a constant, and $M$, $N$ are two distinct points on the circle ${x^2} + {y^2} + kx = 0$. $P$ is a moving point on the circle ${x^2} + {y^2} + kx = 0$. If $M$ and $N$ are symmetric about the line $x - y - 1 = 0$, find the maximum area of $\triangle PAB$. | {
"answer": "3 + \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \(1 \leq x^{2}+y^{2} \leq 4\), find the sum of the maximum and minimum values of \(x^{2}-xy+y^{2}\). | {
"answer": "6.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(ABCD\) be an isosceles trapezoid such that \(AD = BC\), \(AB = 3\), and \(CD = 8\). Let \(E\) be a point in the plane such that \(BC = EC\) and \(AE \perp EC\). Compute \(AE\). | {
"answer": "2\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $θ∈[0,π]$, then the probability of $\sin (θ+ \frac {π}{3}) > \frac {1}{2}$ being true is ______. | {
"answer": "\\frac {1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, $AB=3$, $AC=4$, and $\angle BAC=60^{\circ}$. If $P$ is a point in the plane of $\triangle ABC$ and $AP=2$, calculate the maximum value of $\vec{PB} \cdot \vec{PC}$. | {
"answer": "10 + 2 \\sqrt{37}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the equation of the circle $x^{2}+y^{2}-8x+15=0$, find the minimum value of $k$ such that there exists at least one point on the line $y=kx+2$ that can serve as the center of a circle with a radius of $1$ that intersects with circle $C$. | {
"answer": "- \\frac{4}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a_1 = \sqrt 7$ and $b_i = \lfloor a_i \rfloor$ , $a_{i+1} = \dfrac{1}{b_i - \lfloor b_i \rfloor}$ for each $i\geq i$ . What is the smallest integer $n$ greater than $2004$ such that $b_n$ is divisible by $4$ ? ( $\lfloor x \rfloor$ denotes the largest integer less than or equal to $x$ ) | {
"answer": "2005",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = e^{x} \cos x - x$.
(I) Find the equation of the tangent line to the curve $y = f(x)$ at the point $(0, f(0))$;
(II) Find the maximum and minimum values of the function $f(x)$ on the interval $[0, \frac{\pi}{2}]$. | {
"answer": "-\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of 8-digit numbers where the product of the digits equals 9261. Present the answer as an integer. | {
"answer": "1680",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the last two digits of base- $3$ representation of $2005^{2003^{2004}+3}$ ? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the sine and cosine values of angle $α$ are both negative, and $\cos(75^{\circ}+α)=\frac{1}{3}$, find the value of $\cos(105^{\circ}-α)+\sin(α-105^{\circ})$ = \_\_\_\_\_\_. | {
"answer": "\\frac{2\\sqrt{2}-1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\sin \alpha + \cos \beta = \frac{1}{3}$ and $\sin \beta - \cos \alpha = \frac{1}{2}$, find $\sin (\alpha-\beta)=$ ______. | {
"answer": "- \\frac{59}{72}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $\frac{x^2}{4} + y^2 = 1$ with points A and B symmetric about the line $4x - 2y - 3 = 0$, find the magnitude of the vector sum of $\overrightarrow{OA}$ and $\overrightarrow{OB}$. | {
"answer": "\\sqrt {5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a race on the same distance, two cars and a motorcycle participated. The second car took 1 minute longer to cover the entire distance than the first car. The first car moved 4 times faster than the motorcycle. What portion of the distance per minute did the second car cover if it covered $\frac{1}{6}$ of the distance more per minute than the motorcycle, and the motorcycle covered the distance in less than 10 minutes? | {
"answer": "2/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the limit of the function:
$$\lim _{x \rightarrow \frac{\pi}{2}} \frac{e^{\sin 2 x}-e^{\tan 2 x}}{\ln \left(\frac{2 x}{\pi}\right)}$$ | {
"answer": "-2\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Into how many parts do the planes of the faces of a tetrahedron divide the space? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a certain region are five towns: Freiburg, Göttingen, Hamburg, Ingolstadt, and Jena.
On a certain day, 40 trains each made a journey, leaving one of these towns and arriving at one of the other towns. Ten trains traveled either from or to Freiburg. Ten trains traveled either from or to Göttingen. Ten trains traveled either from or to Hamburg. Ten trains traveled either from or to Ingolstadt.
How many trains traveled from or to Jena? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the geometric sequence $\{a_n\}$, the common ratio $q = -2$, and $a_3a_7 = 4a_4$, find the arithmetic mean of $a_8$ and $a_{11}$. | {
"answer": "-56",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a certain middle school, there are 180 students in both the eighth and ninth grades. To understand the physical health of students in these two grades, a sampling survey was conducted as follows:
$(1)$ Data Collection:
Twenty students were randomly selected from each of the eighth and ninth grades for physical health tests. The test scores (in percentage) are as follows:
| Grade | Scores |
|-------|--------|
| Eighth | 78, 86, 74, 81, 75, 76, 87, 70, 75, 90, 75, 79, 81, 76, 74, 80, 86, 69, 83, 77 |
| Ninth | 93, 73, 88, 81, 72, 81, 94, 83, 77, 83, 80, 81, 70, 81, 73, 78, 82, 80, 70, 40 |
$(2)$ Data Organization and Description:
Organize and describe the two sets of sample data into the following score ranges:
| Score $x$ | $40\leqslant x\leqslant 49$ | $50\leqslant x\leqslant 59$ | $60\leqslant x\leqslant 69$ | $70\leqslant x\leqslant 79$ | $80\leqslant x\leqslant 89$ | $90\leqslant x\leqslant 100$ |
|-----------|-----------------------------|-----------------------------|-----------------------------|-----------------------------|-----------------------------|------------------------------|
| Eighth Grade Students | 0 | 0 | 1 | 11 | 7 | 1 |
| Ninth Grade Students | 1 | 0 | 0 | 7 | 10 | 2 |
(Note: A score of 80 or above indicates excellent physical health, 70-79 indicates good physical health, 60-69 indicates qualified physical health, and below 60 indicates poor physical health.)
$(3)$ Data Analysis:
The mean, median, and mode of the two sets of sample data are shown in the table below. Please complete the table:
| | Mean | Median | Mode |
|----------|------|--------|------|
| Eighth Grade | 78.3 | 77.5 | ①______ |
| Ninth Grade | 78 | ②______ | 81 |
$(4)$ Conclusion:
① Estimate the number of students in the ninth grade who have excellent physical health to be ______.
② It can be inferred that ______ students in each grade have better physical health. The reason is ______. (Provide at least two different perspectives to explain the reasonableness of the inference.) | {
"answer": "108",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagram shows three triangles which are formed by the five line segments \(A C D F, B C G, G D E, A B\), and \(E F\) so that \(A C = B C = C D = G D = D F = E F\). Also, \(\angle C A B = \angle E F D\). What is the size, in degrees, of \(\angle C A B\)? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence $\{a_{n}\}$, where $a_{n}$ are integers, and for $n \geq 3, n \in \mathbf{N}$, the relation $a_{n} = a_{n-1} - a_{n-2}$ holds. If the sum of the first 1985 terms of the sequence is 1000, and the sum of the first 1995 terms is 4000, then what is the sum of the first 2002 terms of the sequence? | {
"answer": "3000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given positive numbers \( a \) and \( b \) that satisfy \( 2 + \log_{2} a = 3 + \log_{3} b = \log_{6}(a+b) \), find the value of \( \frac{1}{a} + \frac{1}{b} \). | {
"answer": "108",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( ABC \), given \( a^{2} + b^{2} + c^{2} = 2\sqrt{3} \, ab \, \sin C \), find \( \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2} \). | {
"answer": "\\frac{3\\sqrt{3}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( T \) be the set of positive real numbers. Let \( g : T \to \mathbb{R} \) be a function such that
\[ g(x) g(y) = g(xy) + 2006 \left( \frac{1}{x} + \frac{1}{y} + 2005 \right) \] for all \( x, y > 0 \).
Let \( m \) be the number of possible values of \( g(3) \), and let \( t \) be the sum of all possible values of \( g(3) \). Find \( m \times t \). | {
"answer": "\\frac{6019}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parabola $y=-x^{2}+3$, there exist two distinct points $A$ and $B$ on it that are symmetric about the line $x+y=0$. Find the length of the segment $|AB|$. | {
"answer": "3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that positive real numbers $a$ and $b$ satisfy $a \gt b$ and $ab=\frac{1}{2}$, find the minimum value of $\frac{4{a}^{2}+{b}^{2}+3}{2a-b}$. | {
"answer": "2\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Based on the definition of the derivative, find $f^{\prime}(0)$:
$$
f(x)=\left\{\begin{array}{c}
\frac{2^{\operatorname{tg} x}-2^{\sin x}}{x^{2}}, x \neq 0 \\
0, x=0
\end{array}\right.
$$ | {
"answer": "\\ln \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $$\sin\left(\alpha+ \frac {\pi}{3}\right)=- \frac {4}{5}$$, and $$- \frac {\pi}{2}<\alpha<0$$, find the value of $\cos\alpha$. | {
"answer": "\\frac {3-4 \\sqrt {3}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a 14 team baseball league, each team played each of the other teams 10 times. At the end of the season, the number of games won by each team differed from those won by the team that immediately followed it by the same amount. Determine the greatest number of games the last place team could have won, assuming that no ties were allowed. | {
"answer": "52",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The product of three prime numbers. There is a number that is the product of three prime factors whose sum of squares is equal to 2331. There are 7560 numbers (including 1) less than this number and coprime with it. The sum of all the divisors of this number (including 1 and the number itself) is 10560. Find this number. | {
"answer": "8987",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the real numbers \( x \) and \( y \) satisfy the equation \( 2x^2 + 3xy + 2y^2 = 1 \), find the minimum value of \( x + y + xy \). | {
"answer": "-\\frac{9}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a natural number \( N \), if at least seven out of the nine natural numbers from 1 to 9 are factors of \( N \), \( N \) is called a "seven-star number." What is the smallest "seven-star number" greater than 2000? | {
"answer": "2016",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ in $\triangle ABC$, respectively, and $2\sin \frac {7\pi}{6}\sin \left( \frac {\pi}{6}+C\right)+\cos C= - \frac {1}{2}$.
$(1)$ Find $C$;
$(2)$ If $c= \sqrt {13}$ and the area of $\triangle ABC$ is $3 \sqrt {3}$, find the value of $\sin A+\sin B$. | {
"answer": "\\frac {7 \\sqrt {39}}{26}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
\(PQRS\) is a square. The points \(T\) and \(U\) are the midpoints of \(QR\) and \(RS\) respectively. The line \(QS\) cuts \(PT\) and \(PU\) at \(W\) and \(V\) respectively. What fraction of the total area of the square \(PQRS\) is the area of the pentagon \(RTWVU\)?
A) \(\frac{1}{3}\)
B) \(\frac{2}{5}\)
C) \(\frac{3}{7}\)
D) \(\frac{5}{12}\)
E) \(\frac{4}{15}\) | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\{a_n\}$ is an arithmetic sequence, $a_1 > 0$, $a_{23} + a_{24} > 0$, and $a_{23} \cdot a_{24} < 0$, determine the maximum positive integer $n$ for which the sum of the first $n$ terms $S_n > 0$. | {
"answer": "46",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)= \sqrt{3}\sin ωx−2{(\sin \dfrac{ωx}{2})}^{2}(ω > 0)$ with the smallest positive period of $3π$.
(1) Find the maximum and minimum values of the function $f(x)$ in the interval $[-{\dfrac{3π }{4}},π]$.
(2) Given $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively in an acute triangle $ABC$ with $b=2$, $f(A)=\sqrt{3}-1$, and $\sqrt{3}a=2b\sin A$, find the area of $\triangle ABC$. | {
"answer": "\\dfrac {3+ \\sqrt {3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence \(\left\{a_{n}\right\}\) is defined as follows:
\[
\begin{align*}
a_{1} &= 0, \\
a_{2} &= 1, \\
a_{n+1} &= \frac{\sqrt{6}-\sqrt{2}}{2} a_{n} - a_{n-1} \quad \text{for } n \geq 2.
\end{align*}
\]
Determine the value of \(a_{2019}\). | {
"answer": "\\frac{\\sqrt{6} - \\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the line $l_{1}:ax+2y+6=0$ is parallel to the line $l_{2}:x+\left(a-1\right)y+\left(a^{2}-1\right)=0$, determine the value of $a$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At a sumo wrestling tournament, 20 sumo wrestlers participated. After weighing, it was found that the average weight of the wrestlers is 125 kg. What is the maximum possible number of wrestlers weighing more than 131 kg, given that according to the rules, individuals weighing less than 90 kg cannot participate in sumo wrestling? | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( a_{1}, a_{2}, \cdots, a_{n} \) be pairwise distinct positive integers satisfying \( a_{1} + a_{2} + \cdots + a_{n} = 2014 \), where \( n \) is an integer greater than 1. Let \( d \) be the greatest common divisor of \( a_{1}, a_{2}, \cdots, a_{n} \). For all possible values of \( n \) and \( a_{1}, a_{2}, \cdots, a_{n} \) that satisfy the above conditions, find the maximum value of \( n d \). | {
"answer": "530",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangle can be formed having side lengths 4, 5, and 8. It is impossible, however, to construct a triangle with side lengths 4, 5, and 9. Ron has eight sticks, each having an integer length. He observes that he cannot form a triangle using any three of these sticks as side lengths. The shortest possible length of the longest of the eight sticks is | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest positive integer representable as the sum of the cubes of three positive integers in two different ways? | {
"answer": "251",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Give an example of an expression consisting of ones, parentheses, the symbols "+", and "×" such that:
- Its value is equal to 11
- If in this expression all "+" signs are replaced with "×" signs, and all "×" signs are replaced with "+" signs, it will still result in 11. | {
"answer": "1+1+1+1+1+1+1+1+1+1+1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $b\sin 2C=c\sin B$.
1. Find angle $C$.
2. If $\sin \left(B-\frac{\pi }{3}\right)=\frac{3}{5}$, find the value of $\sin A$. | {
"answer": "\\frac{4\\sqrt{3}-3}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system xOy, the parametric equation of line l is $$\begin{cases} x=1+t \\ y=-3+t \end{cases}$$ (where t is the parameter), and the polar coordinate system is established with the origin O as the pole and the positive semi-axis of the x-axis as the polar axis. The polar equation of curve C is ρ=6cosθ.
(I) Find the general equation of line l and the rectangular coordinate equation of curve C.
(II) If line l intersects curve C at points A and B, find the area of triangle ABC. | {
"answer": "\\frac { \\sqrt {17}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A primary school conducted a height survey. For students with heights not exceeding 130 cm, there are 99 students with an average height of 122 cm. For students with heights not less than 160 cm, there are 72 students with an average height of 163 cm. The average height of students with heights exceeding 130 cm is 155 cm. The average height of students with heights below 160 cm is 148 cm. How many students are there in total? | {
"answer": "621",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the circle with center O, and diameters AB and CD where AB is perpendicular to CD, and chord DF intersects AB at E with DE = 6 and EF = 2, find the area of the circle. | {
"answer": "24\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a regular tetrahedron with an edge length of \(2 \sqrt{6}\), a sphere is centered at the centroid \(O\) of the tetrahedron. The total length of the curves where the sphere intersects with the four faces of the tetrahedron is \(4 \pi\). Find the radius of the sphere centered at \(O\). | {
"answer": "\\frac{\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cone is formed from a 270-degree sector of a circle of radius 18 by aligning the two straight sides. What is the result when the volume of the cone is divided by $\pi$? | {
"answer": "60.75\\sqrt{141.75}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let vectors $\overrightarrow{a_{1}}=(1,5)$, $\overrightarrow{a_{2}}=(4,-1)$, $\overrightarrow{a_{3}}=(2,1)$, and let $\lambda_{1}, \lambda_{2}, \lambda_{3}$ be non-negative real numbers such that $\lambda_{1}+\frac{\lambda_{2}}{2}+\frac{\lambda_{3}}{3}=1$. Find the minimum value of $\left|\lambda_{1} \overrightarrow{a_{1}}+\lambda_{2} \overrightarrow{a_{2}}+\lambda_{3} \overrightarrow{a_{3}}\right|$. | {
"answer": "3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let point P be the intersection point in the first quadrant of the hyperbola $\frac{x^{2}}{a^{2}}- \frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ and the circle $x^{2}+y^{2}=a^{2}+b^{2}$. F\1 and F\2 are the left and right foci of the hyperbola, respectively, and $|PF_1|=3|PF_2|$. Find the eccentricity of the hyperbola. | {
"answer": "\\frac{\\sqrt{10}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cuboid with dimensions corresponding to length twice the cube's edge is painted with stripes running from the center of one edge to the center of the opposite edge, on each of its six faces. Each face's stripe orientation (either horizontal-center or vertical-center) is chosen at random. What is the probability that there is at least one continuous stripe that encircles the cuboid horizontally?
A) $\frac{1}{64}$
B) $\frac{1}{32}$
C) $\frac{1}{16}$
D) $\frac{1}{8}$
E) $\frac{1}{4}$ | {
"answer": "\\frac{1}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the maximum area of a triangle if none of its side lengths exceed 2? | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Cinderella and her fairy godmother released a collection of seven new models of crystal slippers. The storybook heroines held a presentation of the collection for some guests: the audience had to state which slippers they liked. The guests wrote in a survey which models they considered the best. It is known that no two guests chose the same set of favorite slippers. What is the maximum number of people (excluding Cinderella and the fairy godmother) that could have attended the presentation? | {
"answer": "128",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For natural numbers _m_ greater than or equal to 2, the _n_-th power of _m_ can be decomposed as follows:
2<sup>2</sup> = 1 + 3, 3<sup>2</sup> = 1 + 3 + 5, 4<sup>2</sup> = 1 + 3 + 5 + 7…
2<sup>3</sup> = 3 + 5, 3<sup>3</sup> = 7 + 9 + 11…
2<sup>4</sup> = 7 + 9…
According to this pattern, the third number in the decomposition of 5<sup>4</sup> is ______. | {
"answer": "125",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Martin is playing a game. His goal is to place tokens on an 8 by 8 chessboard in such a way that there is at most one token per square, and each column and each row contains at most 4 tokens.
a) How many tokens can Martin place, at most?
b) If, in addition to the previous constraints, each of the two main diagonals can contain at most 4 tokens, how many tokens can Martin place, at most?
The main diagonals of a chessboard are the two diagonals running from one corner of the chessboard to the opposite corner. | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A secret agent is trying to decipher a passcode. So far, he has obtained the following information:
- It is a four-digit number.
- It is not divisible by seven.
- The digit in the tens place is the sum of the digit in the units place and the digit in the hundreds place.
- The number formed by the first two digits of the code (in this order) is fifteen times the last digit of the code.
- The first and last digits of the code (in this order) form a prime number.
Does the agent have enough information to decipher the code? Justify your conclusion. | {
"answer": "4583",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the minimum possible value of the sum \[\frac{a}{3b} + \frac{b}{5c} + \frac{c}{6a},\] where \( a, b, \) and \( c \) are positive real numbers. | {
"answer": "\\frac{3}{\\sqrt[3]{90}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There exists \( x_{0} < 0 \) such that \( x^{2} + |x - a| - 2 < 0 \) (where \( a \in \mathbb{Z} \)) is always true. Find the sum of all values of \( a \) that satisfy this condition. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a parabola with vertex V and a focus F. There exists a point B on the parabola such that BF = 25 and BV = 24. Determine the sum of all possible values of the length FV. | {
"answer": "\\frac{50}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The price (in euros) of a diamond corresponds to its mass (in grams) squared and then multiplied by 100. The price (in euros) of a crystal corresponds to three times its mass (in grams).
Martin and Théodore unearth a treasure consisting of precious stones that are either diamonds or crystals and whose total value is €5,000,000. They cut each precious stone in half, and each takes one half of each stone. Martin’s total value of stones is €2,000,000. In euros, what was the total initial value of the diamonds contained in the treasure?
Only a numerical answer is expected here. | {
"answer": "2000000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a string of $n$ $7$s, $7777\cdots77,$ into which $+$ signs are inserted to produce an arithmetic expression. How many values of $n$ are possible if the inserted $+$ signs create a sum of $7350$ using groups of $7$s, $77$s, $777$s, and possibly $7777$s? | {
"answer": "117",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the number of ways to arrange the letters of the word MOREMOM. | {
"answer": "420",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the minimum of the following function defined in the interval $45^{\circ}<x<90^{\circ}$:
$$
y=\tan x+\frac{\tan x}{\sin \left(2 x-90^{\circ}\right)}
$$ | {
"answer": "3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In one month, three Wednesdays fell on even dates. On which day will the second Sunday fall in this month? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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