problem stringlengths 10 5.15k | answer dict |
|---|---|
What is the smallest number, \( n \), which is the product of 3 distinct primes where the mean of all its factors is not an integer? | {
"answer": "130",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let triangle $ABC$ be equilateral with each side measuring 18 inches. Let $O$ be the intersection point of medians $AP$ and $CQ$ of triangle $ABC$. If $OQ$ is 6 inches, then $AQ$, in inches, is:
A) $6$ inches
B) $12$ inches
C) $15$ inches
D) $18$ inches
E) $21$ inches | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the inequality \( \ln(ax) + ax \leq x + e^x \) holds for all \( x \) and given real numbers \( a \), what is the maximum value of \( a \)? | {
"answer": "e",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Seven people stand in a row. (Write out the necessary process, and use numbers for the answers)
(1) How many ways can person A and person B stand next to each other?
(2) How many ways can person A and person B stand not next to each other?
(3) How many ways can person A, person B, and person C stand so that no two of them are next to each other?
(4) How many ways can person A, person B, and person C stand so that at most two of them are not next to each other? | {
"answer": "4320",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the price of Product A was set at 70 yuan per piece in the first year, with an annual sales volume of 118,000 pieces, starting from the second year, the price per piece increased by $$\frac {70\cdot x\%}{1-x\%}$$ yuan due to a management fee, and the annual sales volume decreased by $10^4x$ pieces, calculate the maximum value of x such that the management fee collected in the second year is not less than 1.4 million yuan. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose two arithmetic sequences $\{a\_n\}$ and $\{b\_n\}$ have the sum of their first n terms as $S\_n$ and $T\_n$, respectively. If $a\_3 = 2b\_3$ and $\frac{S\_n}{T\_n} = \frac{3n - t}{2n + 2}$ for any $n \in \mathbb{N}^*$, find the value of $\frac{a\_9}{b\_6 + b\_8} + \frac{a\_5}{b\_3 + b\_11}$. | {
"answer": "\\frac{12}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the drawing, there is a grid consisting of 25 small equilateral triangles.
How many rhombuses can be formed from two adjacent small triangles? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the complex number $z$ that satisfies $z= \frac {1-i}{i+1}$, find the value of $|1+z|$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the expanded country of Mathlandia, all automobile license plates have five symbols. The first two must be vowels (A, E, I, O, U), the next two must be different non-vowels among the 21 non-vowels in the alphabet, and the fifth must be a digit (0 through 9). Determine the probability that the plate will read "AIE19" | {
"answer": "\\frac{1}{105,000}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = \sin^2x + 2\sqrt{3}\sin x\cos x + \sin(x + \frac{\pi}{4})\sin(x - \frac{\pi}{4})$, if $x = x_0 (0 \leq x_0 \leq \frac{\pi}{2})$ is a zero of the function $f(x)$, then $\cos 2x_0 = \_\_\_\_\_\_$. | {
"answer": "\\frac{3\\sqrt{5} + 1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify and evaluate
(Ⅰ) Evaluate \\( \dfrac{ \sqrt{3}\sin (- \dfrac{20}{3}\pi)}{\tan \dfrac{11}{3}\pi}-\cos \dfrac{13}{4}\pi\cdot\tan (- \dfrac{35}{4}\pi) \).
(Ⅱ) Evaluate: \\( \dfrac{\sqrt{1-2\sin {10}^{\circ }\cos {10}^{\circ }}}{\cos {10}^{\circ }-\sqrt{1-{\cos }^{2}{170}^{\circ }}} \)
(Ⅲ) If \\( \sin \theta, \cos \theta \) are the roots of the equation \\( 2{x}^{2}-x+a=0 \) (where \\( a \) is a constant) and \\( \theta \in (0,\pi) \), find the value of \\( \cos \theta - \sin \theta \). | {
"answer": "- \\dfrac{ \\sqrt{7}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A positive integer $n$ between $1$ and $N=2007^{2007}$ inclusive is selected at random. If $a$ and $b$ are natural numbers such that $a/b$ is the probability that $N$ and $n^3-36n$ are relatively prime, find the value of $a+b$ . | {
"answer": "1109",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = A\sin(x + \varphi)$ ($A > 0$, $0 < \varphi < \pi$) has a maximum value of 1, and its graph passes through point M ($\frac{\pi}{3}$, $\frac{1}{2}$), then $f(\frac{3\pi}{4})$ = \_\_\_\_\_\_. | {
"answer": "-\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alloy $A$ of two metals has a mass of 6 kg, with the first metal being twice as abundant as the second metal. When placed in a container of water, it exerts a force of $30\ \mathrm{N}$ on the bottom. Alloy $B$ of the same metals has a mass of 3 kg, with the first metal being five times less abundant than the second metal. When placed in a container of water, it exerts a force of $10\ \mathrm{N}$ on the bottom. What force (in newtons) will the third alloy, obtained by combining the original alloys, exert on the bottom? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=a\sin x - \sqrt{3}\cos x$, one of its graphs has an axis of symmetry at $x=-\frac{\pi}{6}$, and $f(x_1) - f(x_2) = -4$, calculate the minimum value of $|x_1+x_2|$. | {
"answer": "\\frac{2\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For transportation between points located hundreds of kilometers apart on the Earth's surface, people of the future will likely dig straight tunnels through which capsules will travel frictionlessly under the influence of Earth's gravity. Let points \( A, B, \) and \( C \) lie on the same meridian, with the surface distance from \( A \) to \( B \) related to the surface distance from \( B \) to \( C \) in the ratio \( m : n \). A capsule travels through the tunnel \( AB \) in approximately 42 minutes. Estimate the travel time through tunnel \( AC \). Provide the answer in minutes. | {
"answer": "42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If three different natural numbers $a$, $b$ and $c$ each have exactly four natural-number factors, how many factors does $a^3b^4c^5$ have? | {
"answer": "2080",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ m\equal{}\left(abab\right)$ and $ n\equal{}\left(cdcd\right)$ be four-digit numbers in decimal system. If $ m\plus{}n$ is a perfect square, what is the largest value of $ a\cdot b\cdot c\cdot d$ ? | {
"answer": "600",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point \( P \) lies on the hyperbola \( \frac{x^2}{16} - \frac{y^2}{9} = 1 \), and the distance from \( P \) to the right directrix of this hyperbola is the arithmetic mean of the distances from \( P \) to the two foci of the hyperbola, find the x-coordinate of \( P \). | {
"answer": "-\\frac{64}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the equation of the directrix of the parabola \( y = \frac{x^2 - 8x + 12}{16} \). | {
"answer": "y = -\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The graph of $r = \sin \theta$ is also seen in a circular form but pertains to a vertical alignment. Find the smallest value of $t$ so that when $r = \sin \theta$ is plotted for $0 \le \theta \le t,$ the resulting graph forms a complete circle. | {
"answer": "2\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Express 0.0000006 in scientific notation. | {
"answer": "6 \\times 10^{-7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the coefficients $p$ and $q$ are integers and the roots $\alpha_{1}$ and $\alpha_{2}$ are irrational, a quadratic trinomial $x^{2} + px + q$ is called an irrational quadratic trinomial. Determine the minimum sum of the absolute values of the roots among all irrational quadratic trinomials. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the functions $f(x)=x^{2}+ax+3$, $g(x)=(6+a)\cdot 2^{x-1}$.
(I) If $f(1)=f(3)$, find the value of the real number $a$;
(II) Under the condition of (I), determine the monotonicity of the function $F(x)=\frac{2}{1+g(x)}$ and provide a proof;
(III) When $x \in [-2,2]$, $f(x) \geqslant a$, ($a \notin (-4,4)$) always holds, find the minimum value of the real number $a$. | {
"answer": "-7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an equilateral triangle $DEF$ with $DE = DF = EF = 8$ units and a circle with radius $4$ units tangent to line $DE$ at $E$ and line $DF$ at $F$, calculate the area of the circle passing through vertices $D$, $E$, and $F$. | {
"answer": "\\frac{64\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the equation \(x + 11y + 11z = n\) where \(n \in \mathbf{Z}_{+}\), there are 16,653 sets of positive integer solutions \((x, y, z)\). Find the minimum value of \(n\). | {
"answer": "2014",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Select 4 out of 7 different books to distribute to 4 students, one book per student, with the restriction that books A and B cannot be given to student C, and calculate the number of different distribution methods. | {
"answer": "600",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a linear function \( f(x) \). It is known that the distance between the points of intersection of the graphs \( y = x^2 + 1 \) and \( y = f(x) \) is \( 3\sqrt{2} \), and the distance between the points of intersection of the graphs \( y = x^2 \) and \( y = f(x) - 2 \) is \( \sqrt{10} \). Find the distance between the points of intersection of the graphs of the functions \( y = x^2 \) and \( y = f(x) \). | {
"answer": "\\sqrt{26}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $O$ is the center of the circumcircle of $\triangle ABC$, $D$ is the midpoint of side $BC$, and $BC=4$, and $\overrightarrow{AO} \cdot \overrightarrow{AD} = 6$, find the maximum value of the area of $\triangle ABC$. | {
"answer": "4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Gavrila found that the front tires of the car last for 21,000 km, and the rear tires last for 28,000 km. Therefore, he decided to swap them at some point so that the car would travel the maximum possible distance. Find this maximum distance (in km). | {
"answer": "24000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a, b, and c are the sides opposite to angles A, B, and C respectively in triangle ABC, and c = 2, sinC(cosB - $\sqrt{3}$sinB) = sinA.
(1) Find the measure of angle C;
(2) If cosA = $\frac{2\sqrt{2}}{3}$, find the length of side b. | {
"answer": "\\frac{4\\sqrt{2} - 2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the right triangle \( \triangle ABC \), \( \angle B = 90^\circ \). Point \( P \) is on the angle bisector of \( \angle A \) within \( \triangle ABC \). Point \( M \) (distinct from \( A \) and \( B \)) is a point on side \( AB \). The lines \( AP \), \( CP \), and \( MP \) intersect sides \( BC \), \( AB \), and \( AC \) at points \( D \), \( E \), and \( N \) respectively. Given that \( \angle MPB = \angle PCN \) and \( \angle NPC = \angle MBP \), find \( \frac{S_{\triangle APC}}{S_{ACDE}} \). | {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find a five-digit number that has the following property: when multiplied by 9, the result is a number represented by the same digits but in reverse order. | {
"answer": "10989",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two identical resistors $R_{0}$ are connected in series and connected to a DC voltage source. An ideal voltmeter is connected in parallel with one of the resistors. Its reading is $U=2 \text{V}$. If the voltmeter is replaced with an ideal ammeter, its reading will be $I=4 \text{A}$. Determine the value of $R_{0}$. | {
"answer": "0.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a right prism $ABC-A_{1}B_{1}C_{1}$, where $AB=3$, $AC=4$, and $AB \perp AC$, $AA_{1}=2$, find the sum of the surface areas of the inscribed sphere and the circumscribed sphere of the prism. | {
"answer": "33\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an isosceles right triangle $ABC$ with $\angle A = 90^{\circ}$ and $AB = AC = 2$, calculate the projection of the vector $\vec{AB}$ in the direction of $\vec{BC}$. | {
"answer": "-\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of the first thirteen terms of an arithmetic progression is $50\%$ of the sum of the last thirteen terms of this progression. The sum of all terms of this progression, excluding the first three terms, is to the sum of all terms excluding the last three terms in the ratio $5:4$. Find the number of terms in this progression. | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\sin(\theta + 3\pi) = -\frac{2}{3}$, find the value of $\frac{\tan(-5\pi - \theta) \cdot \cos(\theta - 2\pi) \cdot \sin(-3\pi - \theta)}{\tan(\frac{7\pi}{2} + \theta) \cdot \sin(-4\pi + \theta) \cdot \cot(-\theta - \frac{\pi}{2})} + 2 \tan(6\pi - \theta) \cdot \cos(-\pi + \theta)$. | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are $2022$ grids in a row. Two people A and B play a game with these grids. At first, they mark each odd-numbered grid from the left with A's name, and each even-numbered grid from the left with B's name. Then, starting with the player A, they take turns performing the following action: $\bullet$ One should select two grids marked with his/her own name, such that these two grids are not adjacent, and all grids between them are marked with the opponent's name. Then, change the name in all the grids between those two grids to one's own name.
Two people take turns until one can not perform anymore. Find the largest positive integer $m$ satisfying the following condition: $\bullet$ No matter how B acts, A can always make sure that there are at least $m$ grids marked with A's name. | {
"answer": "1011",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the equation \( x^{2} - a|x| + a^{2} - 3 = 0 \) has a unique real solution, then \( a = \) ______. | {
"answer": "-\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a cube $ABCD-A_{1}B_{1}C_{1}D_{1}$ with edge length 1, $E$ is the midpoint of $AB$, and $F$ is the midpoint of $CC_{1}$. Find the distance from $D$ to the plane passing through points $D_{1}$, $E$, and $F$.
| {
"answer": "\\frac{4 \\sqrt{29}}{29}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, where $b=2$.
$(1)$ If $A+C=120^{\circ}$ and $a=2c$, find the length of side $c$.
$(2)$ If $A-C=15^{\circ}$ and $a=\sqrt{2}c\sin A$, find the area of triangle $\triangle ABC$. | {
"answer": "3 - \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is an Ace and the second card is a King? | {
"answer": "\\dfrac{4}{663}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangular pyramid \( P-ABC \) with a base that is an equilateral triangle of side length \( 4 \sqrt{3} \), and with \( PA=3 \), \( PB=4 \), and \( PC=5 \). If \( O \) is the center of the triangle \( ABC \), find the length of \( PO \). | {
"answer": "\\frac{\\sqrt{6}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The total investment of the Xiangshan Port Sea-Crossing Bridge, about 7.7 billion yuan, should be expressed in scientific notation. | {
"answer": "7.7 \\times 10^9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rectangle \(PQRS\) is divided into 60 identical squares, as shown. The length of the diagonal of each of these squares is 2. The length of \(QS\) is closest to | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), the heights are given: \(h_{a} = \frac{1}{3}\), \(h_{b} = \frac{1}{4}\), \(h_{c} = \frac{1}{5}\). Find the ratio of the angle bisector \(CD\) to the circumradius. | {
"answer": "\\frac{24\\sqrt{2}}{35}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two positive integers are written on the blackboard, one being 2002 and the other being a number less than 2002. If the arithmetic mean of the two numbers is an integer $m$, then the following operation can be performed: one of the numbers is erased and replaced by $m$. What is the maximum number of times this operation can be performed? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$A$ and $B$ both live such that one lives $3 \text{ km}$ further from the city than the other. They both arrived in the city at the same time, with $A$ traveling by car and $B$ by truck. They were picked up by the vehicles halfway to their destinations. $A$ walks $1.5$ times faster than $B$, but the truck that picked up $B$ travels $1.5$ times faster than the car that picked up $A$. Additionally, the car travels twice as fast as $A$ walks. How far do $A$ and $B$ live from the city?
| {
"answer": "16.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $$\overrightarrow {m}=(\cos \frac {x}{3}, \sqrt {3}\cos \frac {x}{3})$$, $$\overrightarrow {n}=(\sin \frac {x}{3}, \cos \frac {x}{3})$$, and $$f(x)= \overrightarrow {m}\cdot \overrightarrow {n}$$.
(Ⅰ) Find the monotonic intervals of the function $f(x)$;
(Ⅱ) If the graph of $f(x)$ is first translated to the left by $\varphi$ ($\varphi>0$) units, and then, keeping the ordinate unchanged, the abscissa is scaled to $\frac {1}{3}$ of its original, resulting in the graph of the function $g(x)$. If $g(x)$ is an even function, find the minimum value of $\varphi$. | {
"answer": "\\frac {\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many integers are there from 1 to 1,000,000 that are neither perfect squares, nor perfect cubes, nor fourth powers? | {
"answer": "998910",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two runners started simultaneously in the same direction from the same point on a circular track. The first runner, moving ahead, caught up with the second runner at the moment when the second runner had only run half a lap. From that moment, the second runner doubled their speed. Will the first runner catch up with the second runner again? If so, how many laps will the second runner complete by that time? | {
"answer": "2.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) with an eccentricity of $\frac{\sqrt{5}}{5}$, and the equation of the right directrix is $x=5$.
(1) Find the equation of the ellipse;
(2) A line $l$ with a slope of 1 passes through the right focus $F$ of the ellipse $C$ and intersects the ellipse at points $A$ and $B$. Let $P$ be a moving point on the ellipse. Find the maximum area of $\triangle PAB$. | {
"answer": "\\frac{16\\sqrt{10}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given vectors $\overrightarrow{m}=(b,a-2c)$, $\overrightarrow{n}=(\cos A-2\cos C,\cos B)$, and $\overrightarrow{m} \perp \overrightarrow{n}$.
1. Find the value of $\frac{\sin C}{\sin A}$;
2. If $a=2, |m|=3 \sqrt {5}$, find the area of $\triangle ABC$, denoted as $S$. | {
"answer": "\\frac{3\\sqrt{15}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the definite integral:
$$
\int_{0}^{\frac{\pi}{2}}\left(x^{2}-5x+6\right) \sin 3x \, dx
$$ | {
"answer": "\\frac{67 - 3\\pi}{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an arithmetic sequence $\{a_{n}\}$ with the sum of the first $n$ terms as $S_{n}$, if $a_{2}+a_{4}+3a_{7}+a_{9}=24$, calculate the value of $S_{11}$. | {
"answer": "44",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a regular tetrahedron A-BCD with an edge length of 1, and $\overrightarrow{AE} = 2\overrightarrow{EB}$, $\overrightarrow{AF} = 2\overrightarrow{FD}$, calculate $\overrightarrow{EF} \cdot \overrightarrow{DC}$. | {
"answer": "-\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the cartesian coordinate system $(xOy)$, the parametric equations of line $l$ are given by: $ \begin{cases} x=1+t\cos \alpha \\ y=2+t\sin \alpha \end{cases} (t$ is the parameter, $0\leqslant \alpha < \pi)$, and the polar coordinate system is established with $O$ as the pole and the positive semi-axis of $x$ as the polar axis. The polar equation of curve $C$ is $\rho=6\sin \theta$.
(I)(i) When $\alpha=\dfrac{\pi}{4}$, write the ordinary equation of line $l$;
(ii) Write the cartesian equation of curve $C$;
(II) If point $P(1, 2)$, and curve $C$ intersects with line $l$ at points $A$ and $B$, find the minimum value of $\dfrac{1}{|PA|}+\dfrac{1}{|PB|}$. | {
"answer": "\\dfrac{2\\sqrt{7}}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area of the enclosed shape formed by the line $y=0$, $x=e$, $y=2x$, and the curve $y= \frac {2}{x}$ is $\int_{1}^{e} \frac{2}{x} - 2x \,dx$. | {
"answer": "e^{2}-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ancient astronaut theorist Nutter B. Butter claims that the Caloprians from planet Calop, 30 light years away and at rest with respect to the Earth, wiped out the dinosaurs. The iridium layer in the crust, he claims, indicates spaceships with the fuel necessary to travel at 30% of the speed of light here and back, and that their engines allowed them to instantaneously hop to this speed. He also says that Caloprians can only reproduce on their home planet. Call the minimum life span, in years, of a Caloprian, assuming some had to reach Earth to wipe out the dinosaurs, $T$ . Assume that, once a Caloprian reaches Earth, they instantaneously wipe out the dinosaurs. Then, $T$ can be expressed in the form $m\sqrt{n}$ , where $n$ is not divisible by the square of a prime. Find $m+n$ .
*(B. Dejean, 6 points)* | {
"answer": "111",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The side \( AB \) of a regular hexagon \( ABCDEF \) is equal to \( \sqrt{3} \) and serves as a chord of a certain circle, while the other sides of the hexagon lie outside this circle. The length of the tangent \( CM \), drawn to the same circle from vertex \( C \), is 3. Find the diameter of the circle. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $A=30^{\circ}$, $AB=\sqrt {3}$, $BC=1$, find the area of $\triangle ABC$. | {
"answer": "\\frac {\\sqrt {3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five standard, six-sided dice are to be rolled. If the product of their values is an even number, what is the probability that their sum is divisible by 3? | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Of the numbers $\frac{8}{12}, \frac{5}{6}$, and $\frac{9}{12}$, which number is the arithmetic mean of the other two? | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Chinese Basketball Association (CBA) Finals, a best-of-seven series is played, where the first team to win four games is declared the champion. Team A and Team B are evenly matched in the finals, with each team having a probability of $\frac{1}{2}$ of winning each game. According to historical data, the ticket revenue for the first game is 4 million yuan, and for each subsequent game, the revenue increases by 1 million yuan.
$(1)$ Find the probability that the total ticket revenue for the finals is exactly 30 million yuan.
$(2)$ Let $X$ be the total ticket revenue for the finals. Find the mathematical expectation $E(X)$ of $X$. | {
"answer": "3775",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let positive integers $a$, $b$, $c$, and $d$ be randomly and independently selected with replacement from the set $\{1, 2, 3,\dots, 2000\}$. Find the probability that the expression $ab+bc+cd+d+1$ is divisible by $4$. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $α \in (0, \frac{π}{2})$, and $\sin (\frac{π}{6} - α) = -\frac{1}{3}$, find the value of $\cos α$. | {
"answer": "\\frac{2\\sqrt{6} - 1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( n \) be a fixed integer, \( n \geq 2 \).
1. Determine the smallest constant \( c \) such that the inequality \[
\sum_{1 \leq i < j \leq n} x_i x_j (x_i^2 + x_j^2) \leq c \left( \sum_{i=1}^n x_i \right)^4
\] holds for all non-negative real numbers \( x_1, x_2, \cdots, x_n \).
2. For this constant \( c \), determine the necessary and sufficient conditions for equality to hold. | {
"answer": "\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the equation \(3 \cdot 4^{\log_{x} 2} - 46 \cdot 2^{\log_{x} 2 - 1} = 8\). | {
"answer": "\\sqrt[3]{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When I saw Eleonora, I found her very pretty. After a brief trivial conversation, I told her my age and asked how old she was. She answered:
- When you were as old as I am now, you were three times older than me. And when I will be three times older than I am now, together our ages will sum up to exactly a century.
How old is this capricious lady? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Elective 4-4: Coordinate System and Parametric Equations
In the Cartesian coordinate system $xOy$, the parametric equation of line $l_1$ is $\begin{cases} x=2+t \\ y=kt \end{cases}$ ($t$ is the parameter), and the parametric equation of line $l_2$ is $\begin{cases} x=-2+m \\ y=\frac{m}{k} \end{cases}$ ($m$ is the parameter). Let $P$ be the intersection point of $l_1$ and $l_2$. When $k$ changes, the trajectory of $P$ is curve $C$.
(1) Write the standard equation of $C$; (2) Establish a polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis,
Let $l_3: \rho (\cos \theta +\sin \theta )-\sqrt{2}=0$, $M$ be the intersection point of $l_3$ and $C$, find the polar radius of $M$. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is known that the sum of the absolute values of the pairwise differences of five nonnegative numbers is equal to one. Find the smallest possible sum of these numbers. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A laptop is originally priced at $800. The store offers a $15\%$ discount, followed by another $10\%$ discount on the discounted price. Tom also has a special membership card giving an additional $5\%$ discount on the second discounted price. What single percent discount would give the same final price as these three successive discounts? | {
"answer": "27.325\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the 2013 Zhejiang College Entrance Examination, arrange the six letters A, B, C, D, E, F in a row, with both A and B on the same side of C. How many different arrangements are there? (Answer with a number.) | {
"answer": "480",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given points $A=(4,10)$ and $B=(10,8)$ lie on circle $\omega$ in the plane, and the tangent lines to $\omega$ at $A$ and $B$ intersect at a point on the $x$-axis, find the area of $\omega$. | {
"answer": "\\frac{100\\pi}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the limit of the function:
$$
\lim _{x \rightarrow 1}\left(\frac{e^{\sin \pi x}-1}{x-1}\right)^{x^{2}+1}
$$ | {
"answer": "\\pi^2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( f(x) = x^2 + px + q \). It is known that the inequality \( |f(x)| > \frac{1}{2} \) has no solutions on the interval \([3, 5]\). Find \(\underbrace{f(f(\ldots f}_{2017}\left(\frac{7+\sqrt{15}}{2}\right)) \ldots)\). Round the answer to hundredths if necessary. | {
"answer": "1.56",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( x_1, x_2, \cdots, x_{1993} \) that satisfy
\[
\begin{array}{l}
\left|x_1 - x_2\right| + \left|x_2 - x_3\right| + \cdots + \left|x_{1992} - x_{1993}\right| = 1993, \\
y_k = \frac{x_1 + x_2 + \cdots + x_k}{k}, \quad (k=1, 2, \cdots, 1993)
\end{array}
\]
What is the maximum possible value of \( \left|y_1 - y_2\right| + \left|y_2 - y_3\right| + \cdots + \left|y_{1992} - y_{1993}\right| \)? | {
"answer": "1992",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\frac{p}{q} = \frac{5}{2}$ and $\frac{s}{u} = \frac{11}{7}$, find the value of $\frac{5ps - 3qu}{7qu - 2ps}$.
A) $-\frac{233}{14}$
B) $-\frac{233}{12}$
C) $-\frac{233}{10}$
D) $\frac{233}{12}$ | {
"answer": "-\\frac{233}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point is randomly thrown onto the interval \( [11; 18] \) and let \( k \) be the resulting value. Find the probability that the roots of the equation \((k^2 + 2k - 99)x^2 + (3k - 7)x + 2 = 0\) satisfy the condition \( x_1 \leq 2x_2 \). | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate: \(\sin 37^{\circ} \cos ^{2} 34^{\circ} + 2 \sin 34^{\circ} \cos 37^{\circ} \cos 34^{\circ} - \sin 37^{\circ} \sin ^{2} 34^{\circ}\). | {
"answer": "\\frac{\\sqrt{6} + \\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ with magnitudes $|\overrightarrow{a}| = 1$ and $|\overrightarrow{b}| = 2$, if for any unit vector $\overrightarrow{e}$, the inequality $|\overrightarrow{a} \cdot \overrightarrow{e}| + |\overrightarrow{b} \cdot \overrightarrow{e}| \leq \sqrt{6}$ holds, find the maximum value of $\overrightarrow{a} \cdot \overrightarrow{b}$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a modified design of Wei's logo, nine identical circles are placed within a square of side length 24 inches. Each circle is tangent to two sides of the square and to its adjacent circles wherever possible. How many square inches will be shaded in this new design? | {
"answer": "576 - 144\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The length of the chord cut by the line $y= \frac{1}{2}x+1$ on the ellipse $x^2+4y^2=16$ is ______. | {
"answer": "\\sqrt{35}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the definite integral:
$$
\int_{0}^{\pi}\left(x^{2}-3 x+2\right) \sin x \, dx
$$ | {
"answer": "\\pi^2 - 3\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the arc lengths of the curves given by equations in the rectangular coordinate system.
$$
y=1+\arcsin x-\sqrt{1-x^{2}}, 0 \leq x \leq \frac{3}{4}
$$ | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A positive integer has exactly 8 divisors. The sum of its smallest 3 divisors is 15. This four-digit number has a prime factor such that the prime factor minus 5 times another prime factor equals twice the third prime factor. What is this number? | {
"answer": "1221",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For the function $f(x)=\sin \left(2x+ \frac {\pi}{6}\right)$, consider the following statements:
$(1)$ The graph of the function is symmetric about the line $x=- \frac {\pi}{12}$;
$(2)$ The graph of the function is symmetric about the point $\left( \frac {5\pi}{12},0\right)$;
$(3)$ The graph of the function can be obtained by shifting the graph of $y=\sin 2x$ to the left by $\frac {\pi}{6}$ units;
$(4)$ The graph of the function can be obtained by compressing the $x$-coordinates of the graph of $y=\sin \left(x+ \frac {\pi}{6}\right)$ to half of their original values (the $y$-coordinates remain unchanged);
Among these statements, the correct ones are \_\_\_\_\_\_. | {
"answer": "(2)(4)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 8 books on a shelf, which include a trilogy that must be selected together. In how many ways can 5 books be selected from this shelf if the order in which the books are selected does not matter? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, $sin2C=\sqrt{3}sinC$.
$(1)$ Find the value of $\angle C$;
$(2)$ If $b=6$ and the perimeter of $\triangle ABC$ is $6\sqrt{3}+6$, find the area of $\triangle ABC$. | {
"answer": "6\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A factory arranges $5$ workers to three duty positions, with each worker assigned to only one position. Each position must have at least $1$ worker. Calculate the number of ways to assign workers A and B to the same position. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A school arranges five classes every morning from Monday to Friday, each lasting 40 minutes. The first class starts from 7:50 to 8:30, with a 10-minute break between classes. A student returns to school after taking leave. If he arrives at the classroom randomly between 8:50 and 9:30, calculate the probability that he listens to the second class for no less than 20 minutes. | {
"answer": "\\dfrac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the distance from the point \( M_0 \) to the plane passing through the three points \( M_1, M_2, \) and \( M_3 \).
\( M_1(2, -4, -3) \)
\( M_2(5, -6, 0) \)
\( M_3(-1, 3, -3) \)
\( M_0(2, -10, 8) \) | {
"answer": "\\frac{73}{\\sqrt{83}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum of the first 10 elements that appear both among the terms of the arithmetic progression $\{4,7,10,13, \ldots\}$ and the geometric progression $\{20,40,80,160, \ldots\}$. | {
"answer": "13981000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The polynomial $f(x)=x^3-4\sqrt{3}x^2+13x-2\sqrt{3}$ has three real roots, $a$ , $b$ , and $c$ . Find $$ \max\{a+b-c,a-b+c,-a+b+c\}. $$ | {
"answer": "2\\sqrt{3} + 2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In this version of SHORT BINGO, a $5\times5$ card is again filled by marking the middle square as WILD and placing 24 other numbers in the remaining 24 squares. Now, the card is made by placing 5 distinct numbers from the set $1-15$ in the first column, 5 distinct numbers from $11-25$ in the second column, 4 distinct numbers from $21-35$ in the third column (skipping the WILD square in the middle), 5 distinct numbers from $31-45$ in the fourth column, and 5 distinct numbers from $41-55$ in the last column. How many distinct possibilities are there for the values in the first column of this SHORT BINGO card? | {
"answer": "360360",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), the sides \(AC = 14\) and \(AB = 6\) are given. A circle with center \(O\), constructed on side \(AC\) as the diameter, intersects side \(BC\) at point \(K\). It is given that \(\angle BAK = \angle ACB\). Find the area of triangle \(BOC\). | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A supermarket has 6 checkout lanes, each with two checkout points numbered 1 and 2. Based on daily traffic, the supermarket plans to select 3 non-adjacent lanes on Monday, with at least one checkout point open in each lane. How many different arrangements are possible for the checkout lanes on Monday? | {
"answer": "108",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When \(2x^2\) is added to a quadratic polynomial \(f(x)\), its maximum value increases by 10, and when \(5x^2\) is subtracted from it, its maximum value decreases by \(\frac{15}{2}\). By how much will the maximum value of \(f(x)\) change if \(3x^2\) is added to it? | {
"answer": "\\frac{45}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the random variable $\xi$ follows a normal distribution $N(2, \sigma^2)$, and $P(\xi < 3) = 0.6$, find $P(1 < \xi < 2)$. | {
"answer": "0.1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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