problem stringlengths 10 5.15k | answer dict |
|---|---|
Calculate the limit of the function:
$\lim _{x \rightarrow \pi} \frac{\sin \left(\frac{x^{2}}{\pi}\right)}{2^{\sqrt{\sin x+1}}-2}$ | {
"answer": "\\frac{2}{\\ln 2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles have centers that are \( d \) units apart, and each has a diameter \( \sqrt{d} \). For any \( d \), let \( A(d) \) be the area of the smallest circle that contains both of these circles. Find \( \lim _{d \rightarrow \infty} \frac{A(d)}{d^{2}} \). | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the polynomial $f(x) = x^5 + 4x^4 + x^2 + 20x + 16$, evaluate $f(-2)$ using the Qin Jiushao algorithm to find the value of $v_2$. | {
"answer": "-4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( a_{0}=1, a_{1}=2 \), and \( n(n+1) a_{n+1}=n(n-1) a_{n}-(n-2) a_{n-1}, n=1,2,3, \cdots \). Find the value of \( \frac{a_{0}}{a_{1}}+\frac{a_{1}}{a_{2}}+\frac{a_{2}}{a_{3}}+\cdots+\frac{a_{50}}{a_{51}} \). | {
"answer": "1326",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest natural number such that when multiplied by 9, the resulting number consists of the same digits but in some different order. | {
"answer": "1089",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum value of the following expression:
$$
|\cdots|\left|x_{1}-x_{2}\right|-x_{3}\left|-\cdots-x_{1990}\right|,
$$
where \( x_{1}, x_{2}, \cdots, x_{1990} \) are distinct natural numbers from 1 to 1990. | {
"answer": "1989",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a certain country, there are 100 cities. The Ministry of Aviation requires that each pair of cities be connected by a bidirectional flight operated by exactly one airline, and that for each airline, it must be possible to travel from any city to any other city (possibly with transfers). What is the maximum number of airlines for which this is possible? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Inside triangle $ABC$, there are 1997 points. Using the vertices $A, B, C$ and these 1997 points, into how many smaller triangles can the original triangle be divided? | {
"answer": "3995",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x$ and $y$ be nonzero real numbers. Find the minimum value of
\[ x^2 + y^2 + \frac{4}{x^2} + \frac{2y}{x}. \] | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the integral \(\int_{0}^{2}\left(x^{2}+x-1\right) e^{x / 2} \, d x\). | {
"answer": "2 (3e - 5)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Some expressions containing square roots can be written as the square of another expression, such as $3+2\sqrt{2}={(1+\sqrt{2})}^{2}$. Let $a+b\sqrt{2}=(m+n\sqrt{2})^{2}$ (where $a$, $b$, $m$, $n$ are all positive integers), then we have $a+b\sqrt{2}=m^{2}+2n^{2}+2mn\sqrt{2}$, so $a=m^{2}+2m^{2}$, $b=2mn$. This method can be used to convert some expressions of the form $a+b\sqrt{2}$ into square form. Please explore and solve the following problems using the method described above:
$(1)$ When $a$, $b$, $m$, $n$ are all positive integers, if $a+b\sqrt{3}={(m+n\sqrt{3})}^{2}$, express $a$ and $b$ in terms of $m$ and $n$: $a=$______, $b=$______;
$(2)$ Find a set of positive integers $a$, $b$, $m$, $n$ to fill in the blanks: ______$+$______$\sqrt{5}=( \_\_\_\_\_\_+\_\_\_\_\_\_\sqrt{5})^{2}$;
$(3)$ Simplify $\frac{1}{\sqrt{16-6\sqrt{7}}}-\frac{1}{\sqrt{11+4\sqrt{7}}}$. | {
"answer": "\\frac{13}{6}+\\frac{\\sqrt{7}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A function \( f \) satisfies the equation \((n - 2019) f(n) - f(2019 - n) = 2019\) for every integer \( n \).
What is the value of \( f(2019) \)?
A) 0
B) 1
C) \(2018 \times 2019\)
D) \(2019^2\)
E) \(2019 \times 2020\) | {
"answer": "2019 \\times 2018",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a positive integer \( n \), if there exist positive integers \( a \) and \( b \) such that \( n = a + b + a \times b \), then \( n \) is called a "good number". For example, \( 3 = 1 + 1 + 1 \times 1 \), so 3 is a "good number". Among the 100 positive integers from 1 to 100, there are \(\qquad\) "good numbers". | {
"answer": "74",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A bug moves in the coordinate plane, starting at $(0,0)$. On the first turn, the bug moves one unit up, down, left, or right, each with equal probability. On subsequent turns the bug moves one unit up, down, left, or right, choosing with equal probability among the three directions other than that of its previous move. For example, if the first move was one unit up then the second move has to be either one unit down or one unit left or one unit right.
After four moves, what is the probability that the bug is at $(2,2)$? | {
"answer": "1/54",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circular piece of metal with a radius of 10 units has a maximum square cut out from it. Then, the largest possible circle is cut from this square. Calculate the total amount of metal wasted.
A) $50\pi - 200$
B) $200 - 50\pi$
C) $100\pi$
D) $50\pi$
E) None of these | {
"answer": "50\\pi - 200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = \frac{x+3}{x^2+1}$, and $g(x) = x - \ln(x-p)$.
(I) Find the equation of the tangent line to the graph of $f(x)$ at the point $\left(\frac{1}{3}, f\left(\frac{1}{3}\right)\right)$;
(II) Determine the number of zeros of the function $g(x)$, and explain the reason;
(III) It is known that the sequence $\{a_n\}$ satisfies: $0 < a_n \leq 3$, $n \in \mathbb{N}^*$, and $3(a_1 + a_2 + \ldots + a_{2015}) = 2015$. If the inequality $f(a_1) + f(a_2) + \ldots + f(a_{2015}) \leq g(x)$ holds for $x \in (p, +\infty)$, find the minimum value of the real number $p$. | {
"answer": "6044",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\sqrt{2.1}=1.449$ and $\sqrt{21}=4.573$, find the value of $\sqrt{21000}$. | {
"answer": "144.9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the numbers $1, 2, \cdots, 20$, calculate the probability that 3 numbers randomly selected form an arithmetic sequence. | {
"answer": "\\frac{3}{38}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given real numbers \( x_1, x_2, \ldots, x_{2021} \) satisfy \( \sum_{i=1}^{2021} x_i^2 = 1 \), find the maximum value of \( \sum_{i=1}^{2020} x_i^3 x_{i+1}^3 \). | {
"answer": "\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The product of the positive differences of the sum of eight integers (28) is 6. To what power will it always be divisible? | {
"answer": "6^7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( S = \{1, 2, \cdots, 10\} \). If a subset \( T \) of \( S \) has at least 2 elements and the absolute difference between any two elements in \( T \) is greater than 1, then \( T \) is said to have property \( P \). Find the number of different subsets of \( S \) that have property \( P \). | {
"answer": "133",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Inside a square, 100 points are marked. The square is divided into triangles in such a way 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 this triangle or is a vertex of it (such divisions are called triangulations). Find the number of triangles in the division. | {
"answer": "202",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of different monic quadratic polynomials (i.e., with the leading coefficient equal to 1) with integer coefficients such that they have two different roots which are powers of 5 with natural exponents, and their coefficients do not exceed in absolute value $125^{48}$. | {
"answer": "5112",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute $\arccos (\sin 1.5).$ All functions are in radians. | {
"answer": "\\frac{\\pi}{2} - 1.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The largest ocean in the world is the Pacific Ocean, with an area of 17,996,800 square kilometers. Rewrite this number in terms of "ten thousand" as the unit, and round it to the nearest "ten thousand" square kilometers. | {
"answer": "1800",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$ABCDE$ is a regular pentagon inscribed in a circle of radius 1. What is the area of the set of points inside the circle that are farther from $A$ than they are from any other vertex? | {
"answer": "\\frac{\\pi}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagram shows two unshaded circles which touch each other and also touch a larger circle. Chord \( PQ \) of the larger circle is a tangent to both unshaded circles. The length of \( PQ \) is 6 units. What is the area, in square units, of the shaded region? | {
"answer": "\\frac{9\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A parabola is given by the equation $y^{2}=4x$. Let $F$ be its focus, and let $A$ and $B$ be the points where a line passing through $F$ intersects the parabola. Let $O$ be the origin of the coordinate system. If $|AF|=3$, find the area of $\triangle AOB$. | {
"answer": "\\frac{3\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
One standard balloon can lift a basket with contents weighing not more than 80 kg. Two standard balloons can lift the same basket with contents weighing not more than 180 kg. What is the weight, in kg, of the basket? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the vertex $B$ of an isosceles triangle $ABC$, a height $BD$ is dropped to its base $AC$. Each of the legs $AB$ and $BC$ of triangle $ABC$ is equal to 8. In triangle $BCD$, a median $DE$ is drawn. A circle is inscribed in triangle $BDE$, touching side $BE$ at point $K$ and side $DE$ at point $M$. Segment $KM$ is equal to 2. Find angle $A$. | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A math class has fewer than 50 students. When the students try to sit in rows of 8, 5 students are left in the last row. When the students try to sit in rows of 6, 3 students remain in the last row. How many students are in this class? | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a function $f(x) = \frac {1}{2}\sin x - \frac {\sqrt {3}}{2}\cos x$ defined on the interval $[a, b]$, the range of $f(x)$ is $[-\frac {1}{2}, 1]$. Find the maximum value of $b-a$. | {
"answer": "\\frac{4\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\Delta XYZ$, $XZ = YZ$, $m\angle DYZ = 50^{\circ}$, and $DY \parallel XZ$. Determine the number of degrees in $m\angle FDY$.
[asy] pair X,Y,Z,D,F; Y = dir(-50); X = dir(-130); D = (.5,0); F = .4 * dir(50);
draw(Z--Y--X--F,EndArrow); draw(Z--D,EndArrow);
label("$X$",X,W); label("$Z$",Z,NW);label("$Y$",Y,E);label("$D$",D,E);label("$F$",F,E);
[/asy] | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $C$: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ shares a common vertex with the hyperbola $\frac{x^{2}}{16} - \frac{y^{2}}{9} = 1$ and passes through the point $A(6, \sqrt{5})$.
1. Find the equation of the hyperbola $C$ and write out the equations of its asymptotes.
2. If point $P$ is a point on hyperbola $C$ and the distance from $P$ to the right focus is $6$, find the distance from $P$ to the left directrix. | {
"answer": "\\frac{28\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
180 grams of 920 purity gold was alloyed with 100 grams of 752 purity gold. What is the purity of the resulting alloy? | {
"answer": "860",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an equilateral triangle ∆ABC with side length 6, where all three vertices lie on the surface of sphere O with O as the center, and the angle between OA and plane ABC is 45°, find the surface area of sphere O. | {
"answer": "96\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), the side \(BC\) is 19 cm. The perpendicular \(DF\), drawn to side \(AB\) through its midpoint \(D\), intersects side \(BC\) at point \(F\). Find the perimeter of triangle \(AFC\) if side \(AC\) is 10 cm. | {
"answer": "29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( a_{1}, a_{2}, \ldots, a_{9} \) be nine real numbers, not necessarily distinct, with average \( m \). Let \( A \) denote the number of triples \( 1 \leq i<j<k \leq 9 \) for which \( a_{i}+a_{j}+a_{k} \geq 3m \). What is the minimum possible value of \( A \)? | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $XYZ$, $XY = 15$, $XZ = 35$, $YZ = 42$, and $XD$ is an angle bisector of $\angle XYZ$. Find the ratio of the area of triangle $XYD$ to the area of triangle $XZD$, and find the lengths of segments $XD$ and $ZD$. | {
"answer": "29.4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(a\), \(b\), and \(c\) represent the lengths of the sides of a triangle, with \(\alpha\), \(\beta\), and \(\gamma\) being the angles opposite these sides respectively. Given that \(a^2 + b^2 = 2023c^2\), determine the value of
\[
\frac{\cot \alpha}{\cot \beta + \cot \gamma}.
\] | {
"answer": "1011",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the circulation of the vector field
$$
\vec{a}=\frac{y}{3} \vec{i} 3-3 x \vec{j}+x \vec{k}
$$
along the closed contour $\Gamma$
$$
\left\{\begin{array}{l}
x=2 \cos t \\
y=2 \sin t \\
z=1-2 \cos t - 2 \sin t
\end{array} \quad t \in [0,2\pi]
\right.
$$ | {
"answer": "-\\frac{52 \\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Joe has a rectangular lawn measuring 120 feet by 180 feet. His lawn mower has a cutting swath of 30 inches, and he overlaps each cut by 6 inches to ensure no grass is missed. Joe mows at a rate of 4000 feet per hour. Calculate the time it will take Joe to mow his entire lawn. | {
"answer": "2.7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangular pyramid \( S-ABC \) whose base \( ABC \) is an isosceles right triangle with \( AB \) as the hypotenuse, and satisfying \( SA = SB = SC = 2 \) and \( AB = 2 \), assume that the four points \( S, A, B, C \) are all on the surface of a sphere centered at point \( O \). Find the distance from point \( O \) to the plane \( ABC \). | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the lengths of arcs of curves given by equations in polar coordinates.
$$
\rho = 3(1 + \sin \varphi), -\frac{\pi}{6} \leq \varphi \leq 0
$$ | {
"answer": "6(\\sqrt{3} - \\sqrt{2})",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate $[(15^{15} \div 15^{13})^3 \cdot 3^2] \div 2^3$. | {
"answer": "3^8 \\cdot 5^6 \\cdot 2^{-3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that \(a, b, c, d\) are real numbers satisfying \(a \geq b \geq c \geq d \geq 0\), \(a^2 + d^2 = 1\), \(b^2 + c^2 = 1\), and \(ac + bd = \frac{1}{3}\). Find the value of \(ab - cd\). | {
"answer": "\\frac{2\\sqrt{2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum value of the maximum of \( |x^2 - 2xy| \) over \( 0 \leq x \leq 1 \) for \( y \) in \( \mathbb{R} \). | {
"answer": "3 - 2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Santa Claus has a sack containing both chocolate and gummy candies, totaling 2023 pieces. The chocolate candies make up 75% of the gummy candies. How many chocolate candies does Santa Claus have in his sack? | {
"answer": "867",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A truck has new tires fitted on all four wheels. A tire is considered completely worn out if it has traveled $15000 \mathrm{~km}$ on the rear wheel or $25000 \mathrm{~km}$ on the front wheel. How far can the truck travel before all four tires are completely worn out if the front and rear pairs of tires are swapped at suitable intervals? | {
"answer": "18750",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Harry Potter is creating an enhanced magical potion called "Elixir of Life" (this is a very potent sleeping potion composed of powdered daffodil root and wormwood infusion. The concentration of the "Elixir of Life" is the percentage of daffodil root powder in the entire potion). He first adds a certain amount of wormwood infusion to the regular "Elixir of Life," making its concentration $9 \%$. If he then adds the same amount of daffodil root powder, the concentration of the "Elixir of Life" becomes $23 \%$. What is the concentration of the regular "Elixir of Life"? $\qquad$ $\%$. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ be unit vectors, and $|\overrightarrow{a}+\overrightarrow{b}|=1$. Determine the angle $\theta$ between $\overrightarrow{a}$ and $\overrightarrow{b}$. | {
"answer": "\\frac{2\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=x(x-a)(x-b)$, its derivative is $f′(x)$. If $f′(0)=4$, find the minimum value of $a^{2}+2b^{2}$. | {
"answer": "8 \\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $a, b, c$ are real numbers such that $a+b+c=6$ and $ab+bc+ca = 9$ , find the sum of all possible values of the expression $\lfloor a \rfloor + \lfloor b \rfloor + \lfloor c \rfloor$ . | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In what ratio does the line $TH$ divide the side $BC$? | {
"answer": "1:1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The curvature of a polyhedron is $2 \pi$ minus the sum of the face angles (the internal angles of the faces of the polyhedron). For example, a cube's face angle is $2 \pi - 3 \times \frac{\pi}{2} = \frac{\pi}{2}$, and its total curvature is $\frac{\pi}{2} \times 8 = 4 \pi$. What is the total curvature of a polyhedron with four triangular faces and one square face? | {
"answer": "4\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system, suppose a curve $C_1$ is given by $x^2+y^2=1$. All points on curve $C_1$ have their $x$ and $y$ coordinates stretched by a factor of $\sqrt{2}$ and $\sqrt{3}$, respectively, resulting in a new curve $C_2$.
$(1)$ Write down the parametric equations for curve $C_2$.
$(2)$ Find the maximum distance from a point on curve $C_2$ to the line $l$: $x+y-4\sqrt{5}=0$. | {
"answer": "\\frac{5\\sqrt{10}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a circle, there are 2018 points. Each of these points is labeled with an integer. Each number is greater than the sum of the two numbers immediately preceding it in a clockwise direction.
Determine the maximum possible number of positive numbers among these 2018 numbers.
(Walther Janous) | {
"answer": "1009",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $0 < x < \frac{\pi}{2}$ and $\sin(2x - \frac{\pi}{4}) = -\frac{\sqrt{2}}{10}$, find the value of $\sin x + \cos x$. | {
"answer": "\\frac{2\\sqrt{10}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let's call a number \( \mathrm{X} \) "50-supportive" if for any 50 real numbers \( a_{1}, \ldots, a_{50} \) whose sum is an integer, there is at least one number for which \( \left|a_{i} - \frac{1}{2}\right| \geq X \).
Indicate the greatest 50-supportive \( X \), rounded to the nearest hundredth based on standard mathematical rules. | {
"answer": "0.01",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the right triangle \( \triangle ABC \),
\[
\angle A = 90^\circ, \, AB = AC
\]
\( M \) and \( N \) are the midpoints of \( AB \) and \( AC \) respectively. \( D \) is an arbitrary point on the segment \( MN \) (excluding points \( M \) and \( N \)). The extensions of \( BD \) and \( CD \) intersect \( AC \) and \( AB \) at points \( F \) and \( E \) respectively. If
\[
\frac{1}{BE} + \frac{1}{CF} = \frac{3}{4}
\]
then find the length of \( BC \). | {
"answer": "4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers from 1 to 8 are placed at the vertices of a cube such that the sum of the numbers at any three vertices lying on the same face is at least 10. What is the minimum possible sum of the numbers at the vertices of one face? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Medians $\overline{DP}$ and $\overline{EQ}$ of $\triangle DEF$ intersect at an angle of $60^\circ$. If $DP = 21$ and $EQ = 27$, determine the length of side $DE$. | {
"answer": "2\\sqrt{67}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
20 points were marked inside a square and connected with non-intersecting segments to each other and to the vertices of the square, such that the square was divided into triangles. How many triangles were formed? | {
"answer": "42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the complex numbers \( z, -z, z^2 - z + 1, z^2 + z + 1 \) correspond to the points \( A, B, C, D \) respectively in the complex plane. Given that \( |z| = 2 \) and the quadrilateral \( ABCD \) is a rhombus, and let such \( z = a + bi \) ( \( \mathrm{i} \) is the imaginary unit, \( a, b \in \mathbb{R} \) ), then \( |a| + |b| = \quad \) (provided by Fu Lexin) | {
"answer": "\\frac{\\sqrt{7}+3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function \( f(x)=\sin \omega x+\cos \omega x \) where \( \omega > 0 \) and \( x \in \mathbb{R} \), if the function \( f(x) \) is monotonically increasing on the interval \( (-\omega, \omega) \) and the graph of the function \( y=f(x) \) is symmetric with respect to the line \( x=\omega \), determine the value of \( \omega \). | {
"answer": "\\frac{\\sqrt{\\pi}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a novel that consists of 530 pages, each page number is printed once, starting from page 1 up to page 530. How many more 3's are printed than 7's throughout the book? | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two moving points \( A\left(x_{1}, y_{1}\right) \) and \( B\left(x_{2}, y_{2}\right) \) on the parabola \( x^{2}=4 y \) (where \( y_{1} + y_{2} = 2 \) and \( y_{1} \neq y_{2} \))), if the perpendicular bisector of line segment \( AB \) intersects the \( y \)-axis at point \( C \), then the maximum value of the area of triangle \( \triangle ABC \) is ________ | {
"answer": "\\frac{16 \\sqrt{6}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many zeros are at the end of the product $s(1) \cdot s(2) \cdot \ldots \cdot s(100)$, where $s(n)$ denotes the sum of the digits of the natural number $n$? | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 13 weights arranged in a row on a table, ordered by mass (the lightest on the left, the heaviest on the right). It is known that the mass of each weight is an integer number of grams, the masses of any two adjacent weights differ by no more than 5 grams, and the total mass of the weights does not exceed 2019 grams. Find the maximum possible mass of the heaviest weight under these conditions. | {
"answer": "185",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The quartic equation \( x^{4} + a x^{3} + b x^{2} + a x + 1 = 0 \) has a real root. Find the minimum value of \( a^{2} + b^{2} \). | {
"answer": "4/5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( P \) is any point on the arc \( \overparen{AD} \) of the circumscribed circle of square \( ABCD \), find the value of \( \frac{PA + PC}{PB} \). | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the shortest distance from the origin to the circle defined by \( x^2 - 30x + y^2 - 8y + 325 = 0 \)? | {
"answer": "\\sqrt{241} - 2\\sqrt{21}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the positive integer \( a \) has 15 factors and the positive integer \( b \) has 20 factors, and \( a + b \) is a perfect square, find the smallest possible value of \( a + b \) that meets these conditions. | {
"answer": "576",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five million times eight million equals | {
"answer": "40,000,000,000,000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $A$ be a set of ten distinct positive numbers (not necessarily integers). Determine the maximum possible number of arithmetic progressions consisting of three distinct numbers from the set $A$. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A series of numbers were written: \(100^{100}, 101^{101}, 102^{102}, \ldots, 234^{234}\) (i.e., the numbers of the form \(n^{n}\) for natural \(n\) from 100 to 234). How many of the numbers listed are perfect squares? (A perfect square is defined as the square of an integer.) | {
"answer": "71",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A student correctly added the two two-digit numbers on the left of the board and got the answer 137. What answer will she obtain if she adds the two four-digit numbers on the right of the board? | {
"answer": "13837",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( f(x) = 2^x \) and \( g(x) = \log_{\sqrt{2}} (8x) \), find the value of \( x \) that satisfies \( f[g(x)] = g[f(x)] \). | {
"answer": "\\frac{1 + \\sqrt{385}}{64}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow {m}=(\cos\alpha- \frac { \sqrt {2}}{3}, -1)$, $\overrightarrow {n}=(\sin\alpha, 1)$, and $\overrightarrow {m}$ is collinear with $\overrightarrow {n}$, and $\alpha\in[-\pi,0]$.
(Ⅰ) Find the value of $\sin\alpha+\cos\alpha$;
(Ⅱ) Find the value of $\frac {\sin2\alpha}{\sin\alpha-\cos\alpha}$. | {
"answer": "\\frac {7}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At 9:00, a pedestrian set off on a journey. An hour later, a cyclist set off from the same starting point. At 10:30, the cyclist caught up with the pedestrian and continued ahead, but after some time, the bicycle broke down. After repairing the bike, the cyclist resumed the journey and caught up with the pedestrian again at 13:00. How many minutes did the repair take? (The pedestrian's speed is constant, and he moved without stopping; the cyclist's speed is also constant except for the repair interval.) | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The TV station is broadcasting 5 different advertisements, among which there are 3 different commercial advertisements and 2 different Olympic promotional advertisements. The last advertisement must be an Olympic promotional advertisement, and the two Olympic promotional advertisements cannot be broadcast consecutively. Determine the number of different broadcasting methods. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Beginner millionaire Bill buys a bouquet of 7 roses for $20. Then, he can sell a bouquet of 5 roses for $20 per bouquet. How many bouquets does he need to buy to "earn" a difference of $1000? | {
"answer": "125",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle is inscribed in a convex quadrilateral \(ABCD\) with its center at point \(O\), and \(AO=OC\). Additionally, \(BC=5\), \(CD=12\), and \(\angle DAB\) is a right angle.
Find the area of the quadrilateral \(ABCD\). | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\sin\theta= \frac {m-3}{m+5}$ and $\cos\theta= \frac {4-2m}{m+5}$ ($\frac {\pi}{2} < \theta < \pi$), calculate $\tan\theta$. | {
"answer": "- \\frac {5}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse with its foci on the x-axis and its lower vertex at D(0, -1), the eccentricity of the ellipse is $e = \frac{\sqrt{6}}{3}$. A line L passes through the point P(0, 2).
(Ⅰ) Find the standard equation of the ellipse.
(Ⅱ) If line L is tangent to the ellipse, find the equation of line L.
(Ⅲ) If line L intersects the ellipse at two distinct points M and N, find the maximum area of triangle DMN. | {
"answer": "\\frac{3\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a similar tournament setup, the top 6 bowlers have a playoff. First #6 bowls #5, and the loser gets the 6th prize. The winner then bowls #4, and the loser of this match gets the 5th prize. The process continues with the previous winner bowling the next highest ranked bowler until the final match, where the winner of this match gets the 1st prize and the loser gets the 2nd prize. How many different orders can bowlers #1 through #6 receive the prizes? | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence $\{a_i\}_{i \ge 1}$ is defined by $a_1 = 1$ and \[ a_n = \lfloor a_{n-1} + \sqrt{a_{n-1}} \rfloor \] for all $n \ge 2$ . Compute the eighth perfect square in the sequence.
*Proposed by Lewis Chen* | {
"answer": "64",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the sum of distances from point $P$ to the two points $(0, -\sqrt{3})$ and $(0, \sqrt{3})$ equals $4$. Let the trajectory of point $P$ be $C$.
$(1)$ Write the equation of $C$;
$(2)$ Suppose the line $y=kx+1$ intersects $C$ at points $A$ and $B$. For what value of $k$ is $\overrightarrow{OA} \bot \overrightarrow{OB}$? What is the value of $|\overrightarrow{AB}|$ at this time? | {
"answer": "\\frac{4\\sqrt{65}}{17}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sequence of two distinct numbers is extended in two ways: one to form a geometric progression and the other to form an arithmetic progression. The third term of the geometric progression coincides with the tenth term of the arithmetic progression. With which term of the arithmetic progression does the fourth term of the geometric progression coincide? | {
"answer": "74",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In \(\triangle ABC\), \(AC = AB = 25\) and \(BC = 40\). \(D\) is a point chosen on \(BC\). From \(D\), perpendiculars are drawn to meet \(AC\) at \(E\) and \(AB\) at \(F\). \(DE + DF\) equals: | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two riders simultaneously departed from points \( A \) and \( C \) towards point \( B \). Despite the fact that \( C \) was 20 km farther from \( B \) than \( A \) was from \( B \), both riders arrived at \( B \) at the same time. Find the distance from \( C \) to \( B \), given that the rider from \( C \) traveled each kilometer 1 minute and 15 seconds faster than the rider from \( A \), and the rider from \( A \) reached \( B \) in 5 hours. | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $α \in (0, \frac{π}{2})$ and $\sin α - \cos α = \frac{1}{2}$, find the value of $\frac{\cos 2α}{\sin (α + \frac{π}{4})}$.
A) $-\frac{\sqrt{2}}{2}$
B) $\frac{\sqrt{2}}{2}$
C) $-1$
D) $1$ | {
"answer": "-\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A newly designed car travels 4.2 kilometers further per liter of gasoline than an older model. The fuel consumption for the new car is 2 liters less per 100 kilometers. How many liters of gasoline does the new car consume per 100 kilometers? If necessary, round your answer to two decimal places. | {
"answer": "5.97",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose $n \ge 0$ is an integer and all the roots of $x^3 +
\alpha x + 4 - ( 2 \times 2016^n) = 0$ are integers. Find all possible values of $\alpha$ . | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The government decided to privatize civil aviation. For each of the 127 cities in the country, the connecting airline between them is sold to one of the private airlines. Each airline must make all acquired airlines one-way but in such a way as to ensure the possibility of travel from any city to any other city (possibly with several transfers). What is the maximum number of companies that can buy the airlines? | {
"answer": "63",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A parallelogram $ABCD$ has a base $\overline{BC}$ of 6 units and a height from $A$ to line $BC$ of 3 units. Extend diagonal $AC$ beyond $C$ to point $E$ such that $\overline{CE}$ is 2 units, making $ACE$ a right triangle at $C$. Find the area of the combined shape formed by parallelogram $ABCD$ and triangle $ACE$. | {
"answer": "18 + 3\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that both $\alpha$ and $\beta$ are acute angles, and $\sin \alpha = \frac{3}{5}$, $\tan (\alpha - \beta) = -\frac{1}{3}$.
(1) Find the value of $\sin (\alpha - \beta)$;
(2) Find the value of $\cos \beta$. | {
"answer": "\\frac{9\\sqrt{10}}{50}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A gold coin is worth $x\%$ more than a silver coin. The silver coin is worth $y\%$ less than the gold coin. Both $x$ and $y$ are positive integers. How many possible values for $x$ are there? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the binomial coefficient of only the fourth term in the expansion of $(x^{2} - \frac {1}{2x})^{n}$ is the largest, then the sum of all the coefficients in the expansion is $\boxed{\text{answer}}$. | {
"answer": "\\frac {1}{64}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain senior high school has a total of 3200 students, with 1000 students each in the second and third grades. A stratified sampling method is used to draw a sample of size 160. The number of first-grade students that should be drawn is ______ . | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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