problem stringlengths 10 5.15k | answer dict |
|---|---|
$A, B, C, D, E, F, G$ are seven people sitting around a circular table. If $d$ is the total number of ways that $B$ and $G$ must sit next to $C$, find the value of $d$. | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Comparing two rectangular parallelepiped bars, it was found that the length, width, and height of the second bar are each 1 cm greater than those of the first bar, and the volume and total surface area of the second bar are 18 cm³ and 30 cm² greater, respectively, than those of the first one. What is the total surface area of the first bar? | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$ (a_n)_{n \equal{} 1}^\infty$ is defined on real numbers with $ a_n \not \equal{} 0$ , $ a_na_{n \plus{} 3} \equal{} a_{n \plus{} 2}a_{n \plus{} 5}$ and $ a_1a_2 \plus{} a_3a_4 \plus{} a_5a_6 \equal{} 6$ . So $ a_1a_2 \plus{} a_3a_4 \plus{} \cdots \plus{}a_{41}a_{42} \equal{} ?$ | {
"answer": "42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(x)$ have a derivative, and satisfy $\lim_{\Delta x \to 0} \frac{f(1)-f(1-2\Delta x)}{2\Delta x}=-1$. Find the slope of the tangent line at point $(1,f(1))$ on the curve $y=f(x)$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
To meet the shopping needs of customers during the "May Day" period, a fruit supermarket purchased cherries and cantaloupes from the fruit production base for $9160$ yuan, totaling $560$ kilograms. The purchase price of cherries is $35$ yuan per kilogram, and the purchase price of cantaloupes is $6 yuan per kilogram.
$(1)$ Find out how many kilograms of cherries and cantaloupes the fruit store purchased.
$(2)$ After selling the first batch of purchased fruits, the fruit supermarket decided to reward customers by launching a promotional activity. They purchased a total of $300$ kilograms of cherries and cantaloupes for no more than $5280$ yuan. Among them, $a$ kilograms of cherries and $2a$ kilograms of cantaloupes were sold at the purchase price, while the remaining cherries were sold at $55$ yuan per kilogram and cantaloupes were sold at $10$ yuan per kilogram. If the supermarket owner plans to make a profit of at least $2120$ yuan in this transaction, find the maximum value of the positive integer $a$. | {
"answer": "35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Any six points are taken inside or on a rectangle with dimensions $1 \times 2$. Let $b$ be the smallest possible value such that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. Determine the value of $b$. | {
"answer": "\\frac{\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
**p1.** The Evergreen School booked buses for a field trip. Altogether, $138$ people went to West Lake, while $115$ people went to East Lake. The buses all had the same number of seats and every bus has more than one seat. All seats were occupied and everybody had a seat. How many seats were on each bus?**p2.** In New Scotland there are three kinds of coins: $1$ cent, $6$ cent, and $36$ cent coins. Josh has $99$ of the $36$ -cent coins (and no other coins). He is allowed to exchange a $36$ cent coin for $6$ coins of $6$ cents, and to exchange a $6$ cent coin for $6$ coins of $1$ cent. Is it possible that after several exchanges Josh will have $500$ coins?**p3.** Find all solutions $a, b, c, d, e, f, g, h$ if these letters represent distinct digits and the following multiplication is correct: $\begin{tabular}{ccccc}
& & a & b & c
+ & & & d & e
\hline
& f & a & g & c
x & b & b & h &
\hline
f & f & e & g & c
\end{tabular}$ **p4.** Is it possible to find a rectangle of perimeter $10$ m and cut it in rectangles (as many as you want) so that the sum of the perimeters is $500$ m?**p5.** The picture shows a maze with chambers (shown as circles) and passageways (shown as segments). A cat located in chamber $C$ tries to catch a mouse that was originally in the chamber $M$ . The cat makes the first move, moving from chamber $C$ to one of the neighboring chambers. Then the mouse moves, then the cat, and so forth. At each step, the cat and the mouse can move to any neighboring chamber or not move at all. The cat catches the mouse by moving into the chamber currently occupied by the mouse. Can the cat get the mouse?
PS. You should use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309). | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two lines $l_{1}$: $(a+2)x+(a+3)y-5=0$ and $l_{2}$: $6x+(2a-1)y-5=0$ are parallel, then $a=$ . | {
"answer": "-\\dfrac{5}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given Jane lists the whole numbers $1$ through $50$ once and Tom copies Jane's numbers, replacing each occurrence of the digit $3$ by the digit $2$, calculate how much larger Jane's sum is than Tom's sum. | {
"answer": "105",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In Lhota, there was an election for the mayor. Two candidates ran: Mr. Schopný and his wife, Dr. Schopná. The village had three polling stations. In the first and second stations, Dr. Schopná received more votes. The vote ratios were $7:5$ in the first station and $5:3$ in the second station. In the third polling station, the ratio was $3:7$ in favor of Mr. Schopný. The election ended in a tie, with both candidates receiving the same number of votes. In what ratio were the valid votes cast in each polling station if we know that the same number of people cast valid votes in the first and second polling stations? | {
"answer": "24 : 24 : 25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of all trees planted at a five-foot distance from each other on a rectangular plot of land, the sides of which are 120 feet and 70 feet. | {
"answer": "375",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A number is composed of 10 ones, 9 tenths (0.1), and 6 hundredths (0.01). This number is written as ____, and when rounded to one decimal place, it is approximately ____. | {
"answer": "11.0",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify first, then evaluate: $(1-\frac{m}{{m+3}})÷\frac{{{m^2}-9}}{{{m^2}+6m+9}}$, where $m=\sqrt{3}+3$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a$ and $b$ be nonnegative real numbers such that
\[\sin (ax + b) = \sin 17x\]for all integers $x.$ Find the smallest possible value of $a.$ | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\cos^2x+\cos^2\left(x-\frac{\pi}{3}\right)-1$, where $x\in \mathbb{R}$,
$(1)$ Find the smallest positive period and the intervals of monotonic decrease for $f(x)$;
$(2)$ The function $f(x)$ is translated to the right by $\frac{\pi}{3}$ units to obtain the function $g(x)$. Find the expression for $g(x)$;
$(3)$ Find the maximum and minimum values of $f(x)$ in the interval $\left[-\frac{\pi}{4},\frac{\pi}{3}\right]$; | {
"answer": "- \\frac{\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At a school cafeteria, Sam wants to buy a lunch consisting of one main dish, one beverage, and one snack. The table below lists Sam's choices in the cafeteria. How many distinct possible lunches can he buy if he avoids pairing Fish and Chips with Soda due to dietary restrictions?
\begin{tabular}{ |c | c | c | }
\hline \textbf{Main Dishes} & \textbf{Beverages}&\textbf{Snacks} \\ \hline
Burger & Soda & Apple Pie \\ \hline
Fish and Chips & Juice & Chocolate Cake \\ \hline
Pasta & & \\ \hline
Vegetable Salad & & \\ \hline
\end{tabular} | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the function $f(x) = (\sin x + \cos x)^2 - \sqrt{3}\cos 2x$.
(Ⅰ) Find the smallest positive period of $f(x)$;
(Ⅱ) Find the maximum value of $f(x)$ on the interval $\left[0, \frac{\pi}{2}\right]$ and the corresponding value of $x$ when the maximum value is attained. | {
"answer": "\\frac{5\\pi}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the sum of the square of the binomial coefficients: $C_2^2+C_3^2+C_4^2+…+C_{11}^2$. | {
"answer": "220",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the distance \( B_{1} H \) from point \( B_{1} \) to the line \( D_{1} B \), given \( B_{1}(5, 8, -3) \), \( D_{1}(-3, 10, -5) \), and \( B(3, 4, 1) \). | {
"answer": "2\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$JKLM$ is a square and $PQRS$ is a rectangle. If $JK$ is parallel to $PQ$, $JK = 8$ and $PS = 2$, then the total area of the shaded regions is: | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the function $f(x)$ be defined on $\mathbb{R}$ and satisfy $f(2-x) = f(2+x)$ and $f(7-x) = f(7+x)$. Also, in the closed interval $[0, 7]$, only $f(1) = f(3) = 0$. Determine the number of roots of the equation $f(x) = 0$ in the closed interval $[-2005, 2005]$. | {
"answer": "802",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ali Baba and the 40 thieves decided to divide a treasure of 1987 gold coins in the following manner: the first thief divides the entire treasure into two parts, then the second thief divides one of the parts into two parts, and so on. After the 40th division, the first thief picks the largest part, the second thief picks the largest of the remaining parts, and so on. The last, 41st part goes to Ali Baba. For each of the 40 thieves, determine the maximum number of coins he can secure for himself in such a division irrespective of the actions of other thieves. | {
"answer": "49",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A snowball with a temperature of $0^{\circ} \mathrm{C}$ is launched at a speed $v$ towards a wall. Upon impact, $k=0.02\%$ of the entire snowball melts. Determine what percentage of the snowball will melt if it is launched towards the wall at a speed of $\frac{v}{2}$? The specific heat of fusion of snow is $\lambda = 330$ kJ/kg. Assume that all the energy released upon impact is used for melting. | {
"answer": "0.005",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system xOy, the parametric equation of curve C is $$\begin{cases} x=3\cos\theta \\ y=3\sin\theta \end{cases}$$ (θ is the parameter). Establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of the x-axis as the polar axis. The polar coordinate equation of line l is ρ(cosθ - sinθ) = 1.
1. Find the rectangular coordinate equations of C and l.
2. Given that line l intersects the y-axis at point M and intersects curve C at points A and B, find the value of $|$$\frac {1}{|MA|}- \frac {1}{|MB|}$$|$. | {
"answer": "\\frac { \\sqrt {2}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate: $$\frac {\cos 2^\circ}{\sin 47^\circ} + \frac {\cos 88^\circ}{\sin 133^\circ}$$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the function f(x) defined on the set of real numbers ℝ satisfies f(x+1) = 1/2 + √(f(x) - f^2(x)), find the maximum value of f(0) + f(2017). | {
"answer": "1+\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sets
$$
\begin{array}{c}
M=\{x, xy, \lg (xy)\} \\
N=\{0, |x|, y\},
\end{array}
$$
and that \( M = N \), determine the value of
$$
\left(x+\frac{1}{y}\right)+\left(x^2+\frac{1}{y^2}\right)+\left(x^3+\frac{1}{y^3}\right)+\cdots+\left(x^{2001}+\frac{1}{y^{2001}}\right).
$$ | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cyclist is riding on a track at a constant speed. It is known that at 11:22, he covered a distance that is 1.4 times greater than the distance he covered at 11:08. When did he start? | {
"answer": "10:33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=( \frac {1}{3})^{x}$, the sum of the first $n$ terms of the geometric sequence $\{a\_n\}$ is $f(n)-c$, and the first term of the sequence $\{b\_n\}_{b\_n > 0}$ is $c$. The sum of the first $n$ terms, $S\_n$, satisfies $S\_n-S_{n-1}= \sqrt {S\_n}+ \sqrt {S_{n-1}}(n\geqslant 2)$.
(I) Find the general term formula for the sequences $\{a\_n\}$ and $\{b\_n\}$;
(II) If the sum of the first $n$ terms of the sequence $\{ \frac {1}{b\_nb_{n+1}}\}$ is $T\_n$, what is the smallest positive integer $n$ such that $T\_n > \frac {1005}{2014}$? | {
"answer": "252",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum number of Permutation of set { $1,2,3,...,2014$ } such that for every 2 different number $a$ and $b$ in this set at last in one of the permutation $b$ comes exactly after $a$ | {
"answer": "1007",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\tan \alpha = -\frac{1}{3}$ and $\cos \beta = \frac{\sqrt{5}}{5}$, with $\alpha, \beta \in (0, \pi)$, find:
1. The value of $\tan(\alpha + \beta)$;
2. The maximum value of the function $f(x) = \sqrt{2} \sin(x - \alpha) + \cos(x + \beta)$. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an arithmetic sequence $\{a_n\}$ with a common difference $d \neq 0$ and the first term $a_1 = d$, the sum of the first $n$ terms of the sequence $\{a_n^2\}$ is $S_n$. A geometric sequence $\{b_n\}$ has a common ratio $q$ less than 1 and consists of rational sine values, with the first term $b_1 = d^2$, and the sum of its first $n$ terms is $T_n$. If $\frac{S_3}{T_3}$ is a positive integer, then find the possible value of $q$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The height \(CH\), dropped from the vertex of the right angle of the triangle \(ABC\), bisects the angle bisector \(BL\) of this triangle. Find the angle \(BAC\). | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the sides \( AB, BC \), and \( AC \) of triangle \( ABC \), points \( M, N, \) and \( K \) are taken respectively so that \( AM:MB = 2:3 \), \( AK:KC = 2:1 \), and \( BN:NC = 1:2 \). In what ratio does the line \( MK \) divide the segment \( AN \)? | {
"answer": "6:7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The reform pilot of basic discipline enrollment, also known as the Strong Foundation Plan, is an enrollment reform project initiated by the Ministry of Education, mainly to select and cultivate students who are willing to serve the country's major strategic needs and have excellent comprehensive quality or outstanding basic subject knowledge. The school exam of the Strong Foundation Plan is independently formulated by pilot universities. In the school exam process of a certain pilot university, students can only enter the interview stage after passing the written test. The written test score $X$ of students applying to this pilot university in 2022 approximately follows a normal distribution $N(\mu, \sigma^2)$. Here, $\mu$ is approximated by the sample mean, and $\sigma^2$ is approximated by the sample variance $s^2$. It is known that the approximate value of $\mu$ is $76.5$ and the approximate value of $s$ is $5.5$. Consider the sample to estimate the population.
$(1)$ Assuming that $84.135\%$ of students scored higher than the expected average score of the university, what is the approximate expected average score of the university?
$(2)$ If the written test score is above $76.5$ points to enter the interview, and $10$ students are randomly selected from those who applied to this pilot university, with the number of students entering the interview denoted as $\xi$, find the expected value of the random variable $\xi$.
$(3)$ Four students, named A, B, C, and D, have entered the interview, and their probabilities of passing the interview are $\frac{1}{3}$, $\frac{1}{3}$, $\frac{1}{2}$, and $\frac{1}{2}$, respectively. Let $X$ be the number of students among these four who pass the interview. Find the probability distribution and the mathematical expectation of the random variable $X$.
Reference data: If $X \sim N(\mu, \sigma^2)$, then: $P(\mu - \sigma < X \leq \mu + \sigma) \approx 0.6827$, $P(\mu - 2\sigma < X \leq \mu + 2\sigma) \approx 0.9545$, $P(\mu - 3\sigma < X \leq \mu + 3\sigma) \approx 0.9973$. | {
"answer": "\\frac{5}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. Given that $b^{2}=ac$ and $a^{2}-c^{2}=ac-bc$.
1. Find the measure of $\angle A$.
2. Let $f(x)=\cos (wx-\frac{A}{2})+\sin (wx) (w > 0)$ and the smallest positive period of $f(x)$ is $\pi$. Find the maximum value of $f(x)$ on $[0, \frac{\pi}{2}]$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Olga Ivanovna, the homeroom teacher of class 5B, is staging a "Mathematical Ballet". She wants to arrange the boys and girls so that every girl has exactly 2 boys at a distance of 5 meters from her. What is the maximum number of girls that can participate in the ballet if it is known that 5 boys are participating? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A standard 52-card deck contains cards of 4 suits and 13 numbers, with exactly one card for each pairing of suit and number. If Maya draws two cards with replacement from this deck, what is the probability that the two cards have the same suit or have the same number, but not both? | {
"answer": "15/52",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We suppose that $AB=1$, and that the oblique segments form an angle of $45^{\circ}$ with respect to $(AB)$. There are $n$ vertices above $(AB)$.
What is the length of the broken line? | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using the digits 0, 1, 2, 3, and 4, how many even numbers can be formed without repeating any digits? | {
"answer": "163",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x_1$ , $x_2$ , and $x_3$ be the roots of the polynomial $x^3+3x+1$ . There are relatively prime positive integers $m$ and $n$ such that $\tfrac{m}{n}=\tfrac{x_1^2}{(5x_2+1)(5x_3+1)}+\tfrac{x_2^2}{(5x_1+1)(5x_3+1)}+\tfrac{x_3^2}{(5x_1+1)(5x_2+1)}$ . Find $m+n$ . | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( a \) is a real number, and for any \( k \in [-1,1] \), when \( x \in (0,6] \), the following inequality is always satisfied:
\[ 6 \ln x + x^2 - 8 x + a \leq k x. \]
Find the maximum value of \( a \). | {
"answer": "6 - 6 \\ln 6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A ship sails on a river. After 6 hours, it returns to its starting point, having covered a distance of 36 km according to the map (naturally, the ship had to move in one direction and then in the opposite direction).
What is the speed of the ship if we assume that it did not spend any time turning around and the speed of the river current is $3 \mathrm{~km} / \mathrm{h}$? | {
"answer": "3 + 3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two dice are differently designed. The first die has faces numbered $1$, $1$, $2$, $2$, $3$, and $5$. The second die has faces numbered $2$, $3$, $4$, $5$, $6$, and $7$. What is the probability that the sum of the numbers on the top faces of the two dice is $3$, $7$, or $8$?
A) $\frac{11}{36}$
B) $\frac{13}{36}$
C) $\frac{1}{3}$
D) $\frac{1}{2}$ | {
"answer": "\\frac{13}{36}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a circle with a diameter of $1$ unit, if a point $P$ on the circle starts from point $A$ representing $3$ on the number line and rolls one round to the left along the number line, find the real number represented by the point $B$ where $P$ arrives on the number line. | {
"answer": "3 - \\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Adding two dots above the decimal 0.142857 makes it a repeating decimal. The 2020th digit after the decimal point is 5. What is the repeating cycle? $\quad$ . | {
"answer": "142857",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the plane containing $\triangle PAD$ is perpendicular to the plane containing rectangle $ABCD$, and $PA = PD = AB = 2$, with $\angle APD = 60^\circ$. If points $P, A, B, C, D$ all lie on the same sphere, find the surface area of this sphere. | {
"answer": "\\frac{28}{3}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Petya and Vasya were collecting mushrooms. It turned out that Petya collected as many mushrooms as the percentage of mushrooms Vasya collected from the total number of mushrooms they collected together. Additionally, Vasya collected an odd number of mushrooms. How many mushrooms did Petya and Vasya collect together? | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $C$: $\frac{x^{2}}{4}-y^{2}=1$ with left and right foci $F\_1$ and $F\_2$ respectively, find the area of the quadrilateral $P\_1P\_2P\_3P\_4$ for a point $P$ on the hyperbola that satisfies $\overrightarrow{PF_{1}}\cdot \overrightarrow{PF_{2}}=0$. | {
"answer": "\\frac{8\\sqrt{6}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system, the parametric equations of the line $C_{1}$ are $\left\{\begin{array}{l}x=1+t\cos\alpha\\ y=t\sin\alpha\end{array}\right.$ (where $t$ is the parameter). Using the origin $O$ as the pole and the positive x-axis as the polar axis, the polar equation of the curve $C_{2}$ is ${\rho}^{2}=\frac{4}{3-\cos2\theta}$.
$(1)$ Find the Cartesian equation of the curve $C_{2}$.
$(2)$ If the line $C_{1}$ intersects the curve $C_{2}$ at points $A$ and $B$, and $P(1,0)$, find the value of $\frac{1}{|PA|}+\frac{1}{|PB|}$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Transform the following expression into a product: 447. \(\sin 75^{\circ} + \sin 15^{\circ}\). | {
"answer": "\\frac{\\sqrt{6}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a convex quadrilateral \( ABCD \) with an interior point \( P \) such that \( P \) divides \( ABCD \) into four triangles \( ABP, BCP, CDP, \) and \( DAP \). Let \( G_1, G_2, G_3, \) and \( G_4 \) denote the centroids of these triangles, respectively. Determine the ratio of the area of quadrilateral \( G_1G_2G_3G_4 \) to the area of \( ABCD \). | {
"answer": "\\frac{1}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a$, $b$, $c$ are three positive real numbers, and $a(a+b+c)=bc$, find the maximum value of $\frac{a}{b+c}$. | {
"answer": "\\frac{\\sqrt{2} - 1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0} \ln \left(\left(e^{x^{2}}-\cos x\right) \cos \left(\frac{1}{x}\right)+\operatorname{tg}\left(x+\frac{\pi}{3}\right)\right)
$$ | {
"answer": "\\frac{1}{2} \\ln (3)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the vertices of a regular 100-gon, 100 chips numbered $1, 2, \ldots, 100$ are placed in exactly that order in a clockwise direction. During each move, it is allowed to swap two chips placed at adjacent vertices if the numbers on these chips differ by no more than $k$. What is the smallest $k$ such that, in a series of such moves, every chip is shifted one position clockwise relative to its initial position? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two lines $l_{1}$: $(3+m)x+4y=5-3m$ and $l_{2}$: $2x+(5+m)y=8$ are parallel, the value of the real number $m$ is ______. | {
"answer": "-7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a four-digit positive integer whose thousand's place is 2. If the digit 2 is moved to the unit's place, the new number formed is 66 greater than twice the original number. Let x be the original number in the units and tens places. Express the original number as 2000 + 100x + 10y + 2, and the new number formed as 2000 + 100x + 2 + 10y. | {
"answer": "2508",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $|a|=1$, $|b|=2$, and $a+b=(1, \sqrt{2})$, the angle between vectors $a$ and $b$ is _______. | {
"answer": "\\frac{2\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ a_1, a_2,\ldots ,a_8$ be $8$ distinct points on the circumference of a circle such that no three chords, each joining a pair of the points, are concurrent. Every $4$ of the $8$ points form a quadrilateral which is called a *quad*. If two chords, each joining a pair of the $8$ points, intersect, the point of intersection is called a *bullet*. Suppose some of the bullets are coloured red. For each pair $(i j)$ , with $ 1 \le i < j \le 8$ , let $r(i,j)$ be the number of quads, each containing $ a_i, a_j$ as vertices, whose diagonals intersect at a red bullet. Determine the smallest positive integer $n$ such that it is possible to colour $n$ of the bullets red so that $r(i,j)$ is a constant for all pairs $(i,j)$ . | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Eighty percent of dissatisfied customers leave angry reviews about a certain online store. Among satisfied customers, only fifteen percent leave positive reviews. This store has earned 60 angry reviews and 20 positive reviews. Using this data, estimate the probability that the next customer will be satisfied with the service in this online store. | {
"answer": "0.64",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among all the factors of $11!$ (where $11! = 11 \times 10 \times \cdots \times 1$), the largest factor that can be expressed in the form $6k + 1$ (where $k$ is a natural number) is $\qquad$. | {
"answer": "385",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum value of the function \( f(x) = x^{2} + 2x + \frac{6}{x} + \frac{9}{x^{2}} + 4 \) for \( x > 0 \). | {
"answer": "10 + 4 \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two lines $l_1: ax+2y+6=0$, and $l_2: x+(a-1)y+a^2-1=0$.
(1) If $l_1 \perp l_2$, find the value of $a$;
(2) If $l_1 \parallel l_2$, find the value of $a$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the triangular prism $P-ABC$, the three edges $PA$, $PB$, and $PC$ are mutually perpendicular, with $PA=1$, $PB=2$, and $PC=2$. If $Q$ is any point on the circumsphere of the triangular prism $P-ABC$, what is the maximum distance from $Q$ to the plane $ABC$? | {
"answer": "\\frac{3}{2} + \\frac{\\sqrt{6}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Positive numbers \(a\), \(b\), and \(c\) satisfy the equations
\[
a^{2} + ab + b^{2} = 1, \quad b^{2} + bc + c^{2} = 3, \quad c^{2} + ca + a^{2} = 4
\]
Find the value of \(a + b + c\). | {
"answer": "\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a plane intersects all 12 edges of a cube at an angle $\alpha$, find $\sin \alpha$. | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the isosceles trapezoid \(ABCD\), the bases \(AD\) and \(BC\) are related by the equation \(AD = (1 + \sqrt{15}) BC\). A circle is constructed with its center at point \(C\) and radius \(\frac{2}{3} BC\), which cuts a chord \(EF\) on the base \(AD\) of length \(\frac{\sqrt{7}}{3} BC\). In what ratio does the circle divide side \(CD\)? | {
"answer": "2:1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given real numbers \( x \) and \( y \) satisfy
\[
\left\{
\begin{array}{l}
x - y \leq 0, \\
x + y - 5 \geq 0, \\
y - 3 \leq 0
\end{array}
\right.
\]
If the inequality \( a(x^2 + y^2) \leq (x + y)^2 \) always holds, then the maximum value of the real number \( a \) is $\qquad$. | {
"answer": "25/13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( n \geq 2 \) be a fixed integer. Find the smallest constant \( C \) such that for all non-negative reals \( x_1, x_2, \ldots, x_n \):
\[ \sum_{i < j} x_i x_j (x_i^2 + x_j^2) \leq C \left( \sum_{i=1}^n x_i \right)^4. \]
Determine when equality occurs. | {
"answer": "\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The endpoints of a line segment $MN$ with a fixed length of $4$ move on the parabola $y^{2}=x$. Let $P$ be the midpoint of the line segment $MN$. The minimum distance from point $P$ to the $y$-axis is ______. | {
"answer": "\\dfrac{7}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A and B play a game as follows. Each throws a dice. Suppose A gets \(x\) and B gets \(y\). If \(x\) and \(y\) have the same parity, then A wins. If not, they make a list of all two-digit numbers \(ab \leq xy\) with \(1 \leq a, b \leq 6\). Then they take turns (starting with A) replacing two numbers on the list by their non-negative difference. When just one number remains, it is compared to \(x\). If it has the same parity A wins, otherwise B wins. Find the probability that A wins. | {
"answer": "3/4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of ordered pairs of positive integers $(a, b)$ such that $a < b$ and the harmonic mean of $a$ and $b$ is equal to $12^4$. | {
"answer": "67",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, $AC=2AB=4$ and $\cos A=\frac{1}{8}$. Calculate the length of side $BC$. | {
"answer": "3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Robots Robert and Hubert assemble and disassemble coffee grinders. Each of them assembles a grinder four times faster than they disassemble one. When they arrived at the workshop in the morning, several grinders were already assembled.
At 7:00 AM, Hubert started assembling and Robert started disassembling. Exactly at 12:00 PM, Hubert finished assembling a grinder and Robert finished disassembling another one. In total, 70 grinders were added during this shift.
At 1:00 PM, Robert started assembling and Hubert started disassembling. Exactly at 10:00 PM, Robert finished assembling the last grinder and Hubert finished disassembling another one. In total, 36 grinders were added during this shift.
How long would it take for Robert and Hubert to assemble 360 grinders if both of them worked together assembling? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The equations of the asymptotes of the hyperbola $\frac{x^2}{2}-y^2=1$ are ________, and its eccentricity is ________. | {
"answer": "\\frac{\\sqrt{6}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
They paid 100 rubles for a book and still need to pay as much as they would need to pay if they had paid as much as they still need to pay. How much does the book cost? | {
"answer": "200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( 2^{a} \times 3^{b} \times 5^{c} \times 7^{d} = 252000 \), what is the probability that a three-digit number formed by any 3 of the natural numbers \( a, b, c, d \) is divisible by 3 and less than 250? | {
"answer": "1/4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A fair 10-sided die is rolled repeatedly until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number?
- **A** $\frac{1}{300}$
- **B** $\frac{1}{252}$
- **C** $\frac{1}{500}$
- **D** $\frac{1}{100}$ | {
"answer": "\\frac{1}{252}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A function $f(x)$ defined on $\mathbb{R}$ satisfies $f(0)=0$, $f(x)+f(1-x)=1$, $f(\frac{x}{5})=\frac{1}{2}f(x)$, and $f(x_1)\leqslant f(x_2)$ when $0\leqslant x_1 < x_2\leqslant 1$. Find the value of $f(\frac{1}{2007})$.
A) $\frac{1}{2}$
B) $\frac{1}{16}$
C) $\frac{1}{32}$
D) $\frac{1}{64}$ | {
"answer": "\\frac{1}{32}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Complete the table below, discover the patterns of square roots and cube roots, and apply the patterns to solve the problem.
| $x$ | $\ldots $ | $0.064$ | $0.64$ | $64$ | $6400$ | $64000$ | $\ldots $ |
|---------|-----------|---------|--------|-------|--------|---------|-----------|
| $\sqrt{x}$ | $\ldots $ | $0.25298$ | $0.8$ | $8$ | $m$ | $252.98$ | $\ldots $ |
| $\sqrt[3]{x}$ | $\ldots $ | $n$ | $0.8618$ | $4$ | $18.566$ | $40$ | $\ldots $ |
$(1)$ $m=$______, $n= \_\_\_\_\_\_.$
$(2)$ From the numbers in the table, it can be observed that when finding the square root, if the decimal point of the radicand moves two places to the left (or right), the decimal point of its square root moves one place to the left (or right). Please describe in words the pattern of cube root: ______.
$(3)$ If $\sqrt{a}≈14.142$, $\sqrt[3]{700}≈b$, find the value of $a+b$. (Reference data: $\sqrt{2}≈1.4142$, $\sqrt{20}≈4.4721$, $\sqrt[3]{7}≈1.9129$, $\sqrt[3]{0.7}≈0.8879$) | {
"answer": "208.879",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all \( a_{0} \in \mathbb{R} \) such that the sequence defined by
\[ a_{n+1} = 2^{n} - 3a_{n}, \quad n = 0, 1, 2, \cdots \]
is increasing. | {
"answer": "\\frac{1}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the limit of the numerical sequence:
\[
\lim _{n \rightarrow \infty}\left(\frac{n^{2}-6 n+5}{n^{2}-5 n+5}\right)^{3 n+2}
\] | {
"answer": "e^{-3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the system of inequalities $\left\{\begin{array}{l}9x - a \geqslant 0, \\ 8x - b < 0\end{array}\right.$ has integer solutions only for $1, 2, 3$, how many ordered pairs of integers $(a, b)$ satisfy the system? | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the chord length intercepted by the circle $x^{2}+y^{2}+2x-4y+1=0$ on the line $ax-by+2=0$ $(a > 0, b > 0)$ is 4, find the minimum value of $\frac{1}{a} + \frac{1}{b}$. | {
"answer": "\\frac{3}{2} + \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( n \) be a positive integer, \( [x] \) be the greatest integer less than or equal to the real number \( x \), and \( \{x\}=x-[x] \).
(1) Find all positive integers \( n \) satisfying \( \sum_{k=1}^{2013}\left[\frac{k n}{2013}\right]=2013+n \);
(2) Find all positive integers \( n \) that make \( \sum_{k=1}^{2013}\left\{\frac{k n}{2013}\right\} \) attain its maximum value, and determine this maximum value. | {
"answer": "1006",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Set \( A = \{1, 2, \cdots, n\} \). If there exist nonempty sets \( B \) and \( C \) such that \( B \cap C = \emptyset \), \( B \cup C = A \), and the sum of the squares of the elements in \( B \) is \( M \), and the sum of the squares of the elements in \( C \) is \( N \), and \( M - N = 2016 \), find the smallest value of \( n \). | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a random variable $\xi \sim N(1,4)$, and $P(\xi < 3)=0.84$, then $P(-1 < \xi < 1)=$ \_\_\_\_\_\_. | {
"answer": "0.34",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Out of 500 participants in a remote math olympiad, exactly 30 did not like the problem conditions, exactly 40 did not like the organization of the event, and exactly 50 did not like the method used to determine the winners. A participant is called "significantly dissatisfied" if they were dissatisfied with at least two out of the three aspects of the olympiad. What is the maximum number of "significantly dissatisfied" participants that could have been at this olympiad? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the geometric sequence $\{a\_n\}$, it is known that $a\_1+a\_2+a\_3=1$, $a\_4+a\_5+a\_6=-2$, find the sum of the first 15 terms of the sequence, denoted as $S\_{15}$. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In response to the call of the commander, 55 soldiers came: archers and swordsmen. All of them were dressed either in golden or black armor. It is known that swordsmen tell the truth when wearing black armor and lie when wearing golden armor, while archers do the opposite.
- To the question "Are you wearing golden armor?" 44 people responded affirmatively.
- To the question "Are you an archer?" 33 people responded affirmatively.
- To the question "Is today Monday?" 22 people responded affirmatively.
How many archers in golden armor came in response to the commander's call? | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the value of the following expressions:
$(1)(2 \frac{7}{9})^{0.5}+0.1^{-2}+(2 \frac{10}{27})\,^{- \frac{2}{3}}-3π^{0}+ \frac{37}{48}$;
$(2)(-3 \frac{3}{8})\,^{- \frac{2}{3}}+(0.002)\,^{- \frac{1}{2}}-10( \sqrt{5}-2)^{-1}+( \sqrt{2}- \sqrt{3})^{0}$. | {
"answer": "- \\frac{167}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a chess tournament, students from the 9th and 10th grades participated. There were 10 times more 10th graders than 9th graders. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The ratio of the sum of the first $n$ terms of two arithmetic sequences $\{a_n\}$ and $\{b_n\}$ is $\frac{S_n}{T_n} = \frac{7n+1}{4n+2}$. Calculate the value of $\frac{a_{11}}{b_{11}}$. | {
"answer": "\\frac{74}{43}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In order to purchase new headphones costing 275 rubles, Katya decided to save money by spending less on sports activities. Until now, she had bought a single-visit pass to the swimming pool, including a trip to the sauna, for 250 rubles to warm up. However, now that summer has arrived, there is no longer a need to visit the sauna. Visiting only the swimming pool costs 200 rubles more than visiting the sauna. How many times must Katya visit the swimming pool without the sauna to save enough to buy the headphones? (Give the answer as a whole number, without spaces and units of measurement.) | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the number of ordered pairs of positive integers \((a, b)\) satisfying the equation
\[ 100(a + b) = ab - 100. \] | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a hyperbola with the equation $\frac{y^2}{4} - \frac{x^2}{a} = 1$ and asymptote equations $y = \pm\frac{2\sqrt{3}}{3}x$, find the eccentricity of this hyperbola. | {
"answer": "\\frac{\\sqrt{7}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles with centers $A$ and $B$ and radii 2 and 1, respectively, are tangent to each other. Point $C$ lies on a line that is tangent to each of the circles and is at a distance of $\frac{3 \sqrt{3}}{2 \sqrt{2}}$ from the midpoint of segment $AB$. Find the area $S$ of triangle $ABC$, given that $S > 2$. | {
"answer": "\\frac{15 \\sqrt{2}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a power function $y = f(x)$ whose graph passes through the point $(4, 2)$, find $f\left( \frac{1}{2} \right)$. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The base of isosceles $\triangle XYZ$ is 30 units and its area is 60 square units. | {
"answer": "\\sqrt{241}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For the line $l_1: ax - y - a + 2 = 0$ (where $a \in \mathbb{R}$), the line $l_2$ passing through the origin $O$ is perpendicular to $l_1$, and the foot of the perpendicular from $O$ is $M$. Then, the maximum value of $|OM|$ is ______. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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