Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
23,300 | Consider a modified function $C' = \frac{2en}{R + 2nr}$, where $e = 4$, $R = 6$, and $r = 3$. Determine the behavior of $C'$ as $n$ increases from a positive starting point. | \frac{4}{3} | 85.15625 |
23,301 | Each of the integers \(1, 2, 3, \ldots, 9\) is assigned to a vertex of a regular 9-sided polygon (every vertex receives exactly one integer from \(\{1, 2, \ldots, 9\}\), and no two vertices receive the same integer) so that the sum of the integers assigned to any three consecutive vertices does not exceed some positive integer \(n\). What is the least possible value of \(n\) for which this assignment can be done? | 15 | 28.90625 |
23,302 | Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is an Ace, the second card is a diamond, and the third card is a Jack? | \frac{1}{650} | 3.125 |
23,303 | Given $\tan\alpha= \dfrac {1}{3}$, find the values of the following expressions:
1. $\dfrac {\sin \alpha+\cos\alpha}{5\cos\alpha-\sin\alpha}$
2. $\dfrac {1}{2\sin\alpha\cdot \cos\alpha+\cos ^{2}\alpha}$. | \dfrac {2}{3} | 96.875 |
23,304 | Let $k$ be a positive integer, and the coefficient of the fourth term in the expansion of $(1+ \frac{x}{k})^{k}$ is $\frac{1}{16}$. Consider the functions $y= \sqrt{8x-x^{2}}$ and $y= \frac{1}{4}kx$, and let $S$ be the shaded region enclosed by their graphs. Calculate the probability that the point $(x,y)$ lies within the shaded region $S$ for any $x \in [0,4]$ and $y \in [0,4]$. | \frac{\pi}{4} - \frac{1}{2} | 0 |
23,305 | Given the circle $(x+1)^2+(y-2)^2=1$ and the origin O, find the minimum value of the distance |PM| if the tangent line from point P to the circle has a point of tangency M such that |PM|=|PO|. | \frac {2 \sqrt {5}}{5} | 0 |
23,306 | $1.$ A bottle in the shape of a cone lies on its base. Water is poured into the bottle until its level reaches a distance of 8 centimeters from the vertex of the cone (measured vertically). We now turn the bottle upside down without changing the amount of water it contains; This leaves an empty space in the upper part of the cone that is 2 centimeters high.
Find the height of the bottle. | 10 | 13.28125 |
23,307 | Ivan and Peter are running in opposite directions on circular tracks with a common center, initially positioned at the minimal distance from each other. Ivan completes one full lap every 20 seconds, while Peter completes one full lap every 28 seconds. In the shortest possible time, when will they be at the maximum distance from each other? | 35/6 | 65.625 |
23,308 | A point $M$ on the parabola $y=4x^{2}$ is at a distance of $1$ from the focus. The ordinate of point $M$ is __________. | \frac{15}{16} | 35.15625 |
23,309 | Given that $O$ is the circumcenter of $\triangle ABC$, and $D$ is the midpoint of $BC$, if $\overrightarrow{AO} \cdot \overrightarrow{AD} = 4$ and $BC = 2\sqrt{6}$, find the length of $AD$. | \sqrt{2} | 8.59375 |
23,310 | Calculate $3 \cdot 7^{-1} + 9 \cdot 13^{-1} \pmod{60}$.
Express your answer as an integer from $0$ to $59$, inclusive. | 42 | 32.03125 |
23,311 | Compute the value of $p$ such that the equation
\[\frac{2x + 3}{px - 2} = x\]
has exactly one solution. | -\frac{4}{3} | 93.75 |
23,312 | A point $Q$ is chosen in the interior of $\triangle DEF$ such that when lines are drawn through $Q$ parallel to the sides of $\triangle DEF$, the resulting smaller triangles $t_1$, $t_2$, and $t_3$ have areas of $16$, $25$, and $36$, respectively. Find the area of $\triangle DEF$. | 225 | 8.59375 |
23,313 | Simplify first, then evaluate: $(\frac{{x-3}}{{{x^2}-1}}-\frac{2}{{x+1}})\div \frac{x}{{{x^2}-2x+1}}$, where $x=(\frac{1}{2})^{-1}+\left(\pi -1\right)^{0}$. | -\frac{2}{3} | 88.28125 |
23,314 | Given that the function $f(x)=\sin x+a\cos x$ has a symmetry axis on $x=\frac{5π}{3}$, determine the maximum value of the function $g(x)=a\sin x+\cos x$. | \frac {2\sqrt {3}}{3} | 0 |
23,315 | In a lottery game, the host randomly selects one of the four identical empty boxes numbered $1$, $2$, $3$, $4$, puts a prize inside, and then closes all four boxes. The host knows which box contains the prize. When a participant chooses a box, before opening the chosen box, the host randomly opens another box without the prize and asks the participant if they would like to change their selection to increase the chances of winning. Let $A_{i}$ represent the event that box $i$ contains the prize $(i=1,2,3,4)$, and let $B_{i}$ represent the event that the host opens box $i$ $(i=2,3,4)$. Now, if it is known that the participant chose box $1$, then $P(B_{3}|A_{2})=$______; $P(B_{3})=______.$ | \frac{1}{3} | 28.90625 |
23,316 | Simplify the expression:
\[
\frac{4 + 2i}{4 - 2i} + \frac{4 - 2i}{4 + 2i} + \frac{4i}{4 - 2i} - \frac{4i}{4 + 2i}.
\] | \frac{2}{5} | 57.8125 |
23,317 | Several energy-saving devices with a total weight of 120 kg were delivered to the factory. It is known that the total weight of the three lightest devices is 31 kg, and the total weight of the three heaviest devices is 41 kg. How many energy-saving devices were delivered to the factory if the weights of any two devices are different? | 10 | 10.15625 |
23,318 | Given that $\tan \alpha +\tan \beta -\tan \alpha \tan \beta +1=0$, and $\alpha ,\beta \in \left(\frac{\pi }{2},\pi \right)$, calculate $\alpha +\beta$. | \frac{7\pi}{4} | 0 |
23,319 | Given the hyperbola $\dfrac {x^{2}}{9}- \dfrac {y^{2}}{27}=1$ with its left and right foci denoted as $F_{1}$ and $F_{2}$ respectively, and $F_{2}$ being the focus of the parabola $y^{2}=2px$, find the area of $\triangle PF_{1}F_{2}$. | 36 \sqrt {6} | 0 |
23,320 | Starting from $2$, the sum of consecutive even numbers is shown in the table below:
| Number of terms $n$ | $S$ |
|----------------------|-----------|
| $1$ | $2=1\times 2$ |
| $2$ | $2+4=6=2\times 3$ |
| $3$ | $2+4+6=12=3\times 4$ |
| $4$ | $2+4+6+8=20=4\times 5$ |
| $5$ | $2+4+6+8+10=30=5\times 6$ |
$(1)$ If $n=8$, then the value of $S$ is ______.<br/>$(2)$ According to the pattern in the table, it is conjectured that the formula for expressing $S$ in terms of $n$ is: $S=2+4+6+8+\ldots +2n=\_\_\_\_\_\_.$<br/>$(3)$ Calculate the value of $102+104+106+\ldots +210+212$ based on the pattern in the previous question. | 8792 | 84.375 |
23,321 | Given that the odd function $f(x)$ is also a periodic function, and the smallest positive period of $f(x)$ is $\pi$, when $x \in \left(0, \frac{\pi}{2}\right)$, $f(x) = 2\sin x$. Find the value of $f\left(\frac{11\pi}{6}\)$. | -1 | 90.625 |
23,322 | A rubber ball is released from a height of 120 feet and rebounds to three-quarters of the height it falls each time it bounces. How far has the ball traveled when it strikes the ground for the fifth time? | 612.1875 | 10.9375 |
23,323 | Given that a four-digit integer $MMMM$, with all identical digits, is multiplied by the one-digit integer $M$, the result is the five-digit integer $NPMPP$. Assuming $M$ is the largest possible single-digit integer that maintains the units digit property of $M^2$, find the greatest possible value of $NPMPP$. | 89991 | 5.46875 |
23,324 | Straw returning to the field is a widely valued measure for increasing soil fertility and production in the world today, which eliminates the air pollution caused by straw burning and also has the effect of increasing fertility and production. A farmer spent $137,600 to buy a new type of combine harvester to achieve straw returning to the field while harvesting. The annual income from harvesting is $60,000 (fuel costs deducted); the harvester requires regular maintenance, with the first year's maintenance being free of charge provided by the manufacturer, and from the second year onwards, the farmer pays for the maintenance, with the cost $y$ (in yuan) related to the number of years $n$ used as: $y=kn+b$ ($n\geqslant 2$, and $n\in N^{\ast}$), knowing that the second year's maintenance costs $1,800 yuan, and the fifth year's costs $6,000 yuan.
(Ⅰ) Try to find the relationship between the maintenance cost $f(n)$ (in yuan) and the number of years $n$ ($n\in N^{\ast}$) used;
(Ⅱ) How many years should this harvester be used to maximize the average profit? (Profit = Income - Maintenance cost - Cost of machinery) | 14 | 37.5 |
23,325 | A sequence consists of $3000$ terms. Each term after the first is $3$ larger than the previous term. The sum of the $3000$ terms is $12000$. Calculate the sum when every third term is added up, starting with the first term and ending with the third to last term. | 1000 | 28.90625 |
23,326 | A sphere with volume $V$ is inside a closed right triangular prism $ABC-A_{1}B_{1}C_{1}$, where $AB \perp BC$, $AB=6$, $BC=8$, and $AA_{1}=3$. Find the maximum value of $V$. | \frac{9\pi}{2} | 54.6875 |
23,327 | The positive integers \( r \), \( s \), and \( t \) have the property that \( r \times s \times t = 1230 \). What is the smallest possible value of \( r + s + t \)? | 52 | 31.25 |
23,328 | Convert the binary number $10110100$ to a decimal number. | 180 | 99.21875 |
23,329 | There are five students, A, B, C, D, and E, arranged to participate in the volunteer services for the Shanghai World Expo. Each student is assigned one of four jobs: translator, guide, etiquette, or driver. Each job must be filled by at least one person. Students A and B cannot drive but can do the other three jobs, while students C, D, and E are capable of doing all four jobs. The number of different arrangements for these tasks is _________. | 108 | 14.0625 |
23,330 | Given that the arithmetic square root of $m$ is $3$, and the square roots of $n$ are $a+4$ and $2a-16$.
$(1)$ Find the values of $m$ and $n$.
$(2)$ Find $\sqrt[3]{{7m-n}}$. | -1 | 35.9375 |
23,331 | Let the number $9999\cdots 99$ be denoted by $N$ with $94$ nines. Then find the sum of the digits in the product $N\times 4444\cdots 44$. | 846 | 73.4375 |
23,332 | A function \( g(x) \) is defined for all real numbers \( x \). For all non-zero values \( x \), we have
\[ 3g(x) + g\left(\frac{1}{x}\right) = 7x + 6. \]
Let \( T \) denote the sum of all of the values of \( x \) for which \( g(x) = 2005 \). Compute the integer nearest to \( T \). | 763 | 75 |
23,333 | Aws plays a solitaire game on a fifty-two card deck: whenever two cards of the same color are adjacent, he can remove them. Aws wins the game if he removes all the cards. If Aws starts with the cards in a random order, what is the probability for him to win? | \frac{\left( \binom{26}{13} \right)^2}{\binom{52}{26}} | 0 |
23,334 | Given the function $f(x)=2\cos ^{2} \frac{x}{2}- \sqrt {3}\sin x$.
(I) Find the smallest positive period and the range of the function;
(II) If $a$ is an angle in the second quadrant and $f(a- \frac {π}{3})= \frac {1}{3}$, find the value of $\frac {\cos 2a}{1+\cos 2a-\sin 2a}$. | \frac{1-2\sqrt{2}}{2} | 90.625 |
23,335 | Five positive integers (not necessarily all different) are written on five cards. Boris calculates the sum of the numbers on every pair of cards. He obtains only three different totals: 57, 70, and 83. What is the largest integer on any card? | 48 | 18.75 |
23,336 | Given the function $f(x)=\sqrt{3}\sin\left(x+\frac{\pi}{4}\right)\sin\left(x-\frac{\pi}{4}\right)+\sin x\cos x$.
$(Ⅰ)$ Find the smallest positive period and the center of symmetry of $f(x)$;
$(Ⅱ)$ Given an acute triangle $\triangle ABC$ with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively, if $f(A)=\frac{\sqrt{3}}{2}$ and $a=4$, find the maximum perimeter of $\triangle ABC$. | 12 | 85.15625 |
23,337 | In triangle \(ABC\), the sides are known: \(AB = 6\), \(BC = 4\), and \(AC = 8\). The angle bisector of \(\angle C\) intersects side \(AB\) at point \(D\). A circle is drawn through points \(A\), \(D\), and \(C\), intersecting side \(BC\) at point \(E\). Find the area of triangle \(ADE\). | \frac{3 \sqrt{15}}{2} | 5.46875 |
23,338 | In the Cartesian coordinate system $xOy$, the equation of circle $C$ is $(x- \sqrt {3})^{2}+(y+1)^{2}=9$. Establish a polar coordinate system with $O$ as the pole and the non-negative half-axis of $x$ as the polar axis.
$(1)$ Find the polar equation of circle $C$;
$(2)$ The line $OP$: $\theta= \frac {\pi}{6}$ ($p\in R$) intersects circle $C$ at points $M$ and $N$. Find the length of segment $MN$. | 2 \sqrt {6} | 0 |
23,339 | Among all integers that alternate between 1 and 0, starting and ending with 1 (e.g., 101, 10101, 10101…), how many are prime numbers? Why? And list all the prime numbers. | 101 | 20.3125 |
23,340 | Given the function\\(f(x)= \\begin{cases} (-1)^{n}\\sin \\dfrac {πx}{2}+2n,\\;x∈\[2n,2n+1) \\\\ (-1)^{n+1}\\sin \\dfrac {πx}{2}+2n+2,\\;x∈\[2n+1,2n+2)\\end{cases}(n∈N)\\)，if the sequence\\(\\{a\_{m}\\}\\) satisfies\\(a\_{m}=f(m)\\;(m∈N^{\*})\\)，and the sum of the first\\(m\\) terms of the sequence is\\(S\_{m}\\)，then\\(S\_{105}-S\_{96}=\\) \_\_\_\_\_\_ ． | 909 | 48.4375 |
23,341 | Given that rectangle ABCD has dimensions AB = 7 and AD = 8, and right triangle DCE shares the same height as rectangle side DC = 7 and extends horizontally from D towards E, and the area of the right triangle DCE is 28, find the length of DE. | \sqrt{113} | 0 |
23,342 | Given that $F_1$ and $F_2$ are the common foci of the ellipse $C_1: \frac{x^2}{4} + y^2 = 1$ and the hyperbola $C_2$, and $A, B$ are the common points of $C_1$ and $C_2$ in the second and fourth quadrants, respectively. If the quadrilateral $AF_1BF_2$ is a rectangle, determine the eccentricity of $C_2$. | \frac{\sqrt{6}}{2} | 15.625 |
23,343 | The photographer wants to arrange three boys and three girls in a row such that a boy or a girl could be at each end, and the rest alternate in the middle, calculate the total number of possible arrangements. | 72 | 82.03125 |
23,344 | In the polar coordinate system, the polar equation of curve \\(C\\) is given by \\(\rho = 6\sin \theta\\). The polar coordinates of point \\(P\\) are \\((\sqrt{2}, \frac{\pi}{4})\\). Taking the pole as the origin and the positive half-axis of the \\(x\\)-axis as the polar axis, a Cartesian coordinate system is established.
\\((1)\\) Find the Cartesian equation of curve \\(C\\) and the Cartesian coordinates of point \\(P\\);
\\((2)\\) A line \\(l\\) passing through point \\(P\\) intersects curve \\(C\\) at points \\(A\\) and \\(B\\). If \\(|PA| = 2|PB|\\), find the value of \\(|AB|\\). | 3\sqrt{2} | 8.59375 |
23,345 | Given in the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is $\begin{cases} x=3+5\cos \alpha \\ y=4+5\sin \alpha \end{cases}$, ($\alpha$ is the parameter), points $A$ and $B$ are on curve $C$. With the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the polar coordinates of points $A$ and $B$ are respectively $A(\rho_{1}, \frac{\pi}{6})$, $B(\rho_{2}, \frac{\pi}{2})$
(Ⅰ) Find the polar equation of curve $C$;
(Ⅱ) Suppose the center of curve $C$ is $M$, find the area of $\triangle MAB$. | \frac{25\sqrt{3}}{4} | 80.46875 |
23,346 | If the line $l_{1}$: $x+my-2=0$ intersects the line $l_{2}$: $mx-y+2=0$ at point $P$, and a tangent line passing through point $P$ is drawn to the circle $C: (x+2)^{2} + (y+2)^{2} = 1$, with the point of tangency being $M$, then the maximum value of $|PM|$ is ____. | \sqrt{31} | 5.46875 |
23,347 | Evaluate the expression $\log_{10} 60 + \log_{10} 80 - \log_{10} 15$. | 2.505 | 3.90625 |
23,348 | An inverted frustum with a bottom diameter of 12 and height of 18, filled with water, is emptied into another cylindrical container with a bottom diameter of 24. Assuming the cylindrical container is sufficiently tall, what will be the height of the water level in the cylindrical container? | 1.5 | 79.6875 |
23,349 | Given that $i$ is the imaginary unit, $a\in\mathbb{R}$, if $\frac{1-i}{a+i}$ is a pure imaginary number, calculate the modulus of the complex number $z=(2a+1)+ \sqrt{2}i$. | \sqrt{11} | 92.96875 |
23,350 | Given that the terminal side of angle \\(\alpha\\) passes through the point \\(P(m,2\sqrt{2})\\), \\(\sin \alpha= \frac{2\sqrt{2}}{3}\\) and \\(\alpha\\) is in the second quadrant.
\\((1)\\) Find the value of \\(m\\);
\\((2)\\) If \\(\tan \beta= \sqrt{2}\\), find the value of \\( \frac{\sin \alpha\cos \beta+3\sin \left( \frac{\pi}{2}+\alpha\right)\sin \beta}{\cos (\pi+\alpha)\cos (-\beta)-3\sin \alpha\sin \beta}\\). | \frac{\sqrt{2}}{11} | 38.28125 |
23,351 | Mr. Rose, Mr. Stein, and Mr. Schwartz start at the same point around a circular track and run clockwise. Mr. Stein completes each lap in $6$ minutes, Mr. Rose in $10$ minutes, and Mr. Schwartz in $18$ minutes. How many minutes after the start of the race are the runners at identical points around the track (that is, they are aligned and are on the same side of the track) for the first time? | 90 | 99.21875 |
23,352 | Calculate the value of $(2345 + 3452 + 4523 + 5234) \times 2$. | 31108 | 0 |
23,353 | If $(1+x+x^2)^6 = a_0 + a_1x + a_2x^2 + \ldots + a_{12}x^{12}$, then find the value of $a_2 + a_4 + \ldots + a_{12}$. | 364 | 64.84375 |
23,354 | Given that $\sin x \cdot \cos x = -\frac{1}{4}$ and $\frac{3\pi}{4} < x < \pi$, find the value of $\sin x + \cos x$. | -\frac{\sqrt{2}}{2} | 81.25 |
23,355 | An experimenter selects 4 out of 8 different chemical substances to place in 4 distinct bottles. If substances A and B should not be placed in bottle 1, the number of different ways of arranging them is ____. | 1260 | 14.0625 |
23,356 | Given that $α$ is an acute angle and $\sin α= \frac {3}{5}$, find the value of $\cos α$ and $\cos (α+ \frac {π}{6})$. | \frac {4\sqrt {3}-3}{10} | 0 |
23,357 | Find the sum of distinct residues of the number $2012^n+m^2$ on $\mod 11$ where $m$ and $n$ are positive integers. | 39 | 66.40625 |
23,358 | There is a room with four doors. Determine the number of different ways for someone to enter and exit this room. | 16 | 95.3125 |
23,359 | A certain shopping mall purchased a batch of daily necessities. If they are sold at a price of $5$ yuan per item, they can sell $30,000$ items per month. If they are sold at a price of $6$ yuan per item, they can sell $20,000$ items per month. It is assumed that the monthly sales quantity $y$ (items) and the price $x$ (yuan per item) satisfy a linear function relationship.
1. Find the function relationship between $y$ and $x.
2. If the unit price of these daily necessities is $4$ yuan when purchased, what price should be set for sales to maximize the monthly profit? What is the maximum monthly profit? | 40000 | 75 |
23,360 | A shooter's probability of hitting the 10, 9, and 8 rings in a single shot are respectively 0.2, 0.3, and 0.1. Express the probability that the shooter scores no more than 8 in a single shot as a decimal. | 0.5 | 57.8125 |
23,361 | Given a circle that is tangent to the side \(DC\) of a regular pentagon \(ABCDE\) at point \(D\) and tangent to the side \(AB\) at point \(A\), what is the degree measure of the minor arc \(AD\)? | 144 | 8.59375 |
23,362 | There are 10 steps, and one can take 1, 2, or 3 steps at a time to complete them in 7 moves. Calculate the total number of different ways to do this. | 77 | 10.9375 |
23,363 | Given $a$, $b$, $c \in \mathbb{R}$, and $2a+2b+c=8$, find the minimum value of $(a-1)^2+(b+2)^2+(c-3)^2$. | \frac{49}{9} | 17.96875 |
23,364 | Calculate:<br/>$(1)-3+8-15-6$;<br/>$(2)-35\div \left(-7\right)\times (-\frac{1}{7})$;<br/>$(3)-2^{2}-|2-5|\div \left(-3\right)$;<br/>$(4)(\frac{1}{2}+\frac{5}{6}-\frac{7}{12})×(-24)$;<br/>$(5)(-99\frac{6}{11})×22$. | -2190 | 72.65625 |
23,365 | In the triangle \( \triangle ABC \), if \(\sin^2 A + \sin^2 B + \sin^2 C = 2\), calculate the maximum value of \(\cos A + \cos B + 2 \cos C\). | \sqrt{5} | 0.78125 |
23,366 | Given that line $l_{1}$ passes through points $A(m,1)$ and $B(-3,4)$, and line $l_{2}$ passes through points $C(1,m)$ and $D(-1,m+1)$, for what value of $m$ do the lines $l_{1}$ and $l_{2}$
$(1)$ intersect at a right angle;
$(2)$ are parallel;
$(3)$ the angle of inclination of $l_{1}$ is $45^{\circ}$ | -6 | 98.4375 |
23,367 | Given $a=\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$ and $b=\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}$, find $\frac{b}{a}+\frac{a}{b}$. | 62 | 98.4375 |
23,368 | When $n$ is a positive integer, $n! = n \times (n - 1) \times \ldots \times 2 \times 1$ is defined as the factorial of $n$ (for example, $10! = 10 \times 9 \times \ldots \times 2 \times 1 = 3,628,800$). Then, how many zeros are there at the end of $2010!$? | 501 | 100 |
23,369 | Let \( n \in \mathbf{Z}_{+} \), and
$$
\begin{array}{l}
a, b, c \in \{x \mid x \in \mathbf{Z} \text{ and } x \in [1,9]\}, \\
A_{n} = \underbrace{\overline{a a \cdots a}}_{n \text{ digits}}, B_{n} = \underbrace{b b \cdots b}_{2n \text{ digits}}, C_{n} = \underbrace{c c \cdots c}_{2n \text{ digits}}.
\end{array}
$$
The maximum value of \(a + b + c\) is ( ), given that there exist at least two \(n\) satisfying \(C_{n} - B_{n} = A_{n}^{2}\). | 18 | 39.84375 |
23,370 | The integers that can be expressed as a sum of three distinct numbers chosen from the set $\{4,7,10,13, \ldots,46\}$. | 37 | 3.125 |
23,371 | In a certain country, there are 47 cities. Each city has a bus station from which buses travel to other cities in the country and possibly abroad. A traveler studied the schedule and determined the number of internal bus routes originating from each city. It turned out that if we do not consider the city of Ozerny, then for each of the remaining 46 cities, the number of internal routes originating from it differs from the number of routes originating from other cities. Find out how many cities in the country have direct bus connections with the city of Ozerny.
The number of internal bus routes for a given city is the number of cities in the country that can be reached from that city by a direct bus without transfers. Routes are symmetric: if you can travel by bus from city $A$ to city $B$, you can also travel by bus from city $B$ to city $A$. | 23 | 23.4375 |
23,372 | Let $F_n$ be the nth Fibonacci number, defined by $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n > 2$ . Let $A_0, A_1, A_2,\cdots$ be a sequence of points on a circle of radius $1$ such that the minor arc from $A_{k-1}$ to $A_k$ runs clockwise and such that
\[\mu(A_{k-1}A_k)=\frac{4F_{2k+1}}{F_{2k+1}^2+1}\]
for $k \geq 1$ , where $\mu(XY )$ denotes the radian measure of the arc $XY$ in the clockwise direction. What is the limit of the radian measure of arc $A_0A_n$ as $n$ approaches infinity? | 2\pi | 14.84375 |
23,373 | Determine the minimum possible value of the sum
\[\frac{a}{3b} + \frac{b}{5c} + \frac{c}{7a},\]
where $a,$ $b,$ and $c$ are positive real numbers. | \frac{3}{\sqrt[3]{105}} | 67.1875 |
23,374 | Given the function $f(x) = 2\sin x\cos x - 2\sin^2 x + 1$, determine the smallest positive value of $\varphi$ such that the graph of $f(x)$ shifted to the right by $\varphi$ units is symmetric about the y-axis. | \frac{3\pi}{8} | 54.6875 |
23,375 | In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l$ are $\left\{\begin{array}{l}x=1+\frac{\sqrt{2}}{2}t,\\ y=\frac{\sqrt{2}}{2}t\end{array}\right.$ (where $t$ is the parameter). Using $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis, the polar coordinate equation of the curve $C$ is $\rho ^{2}+2\rho ^{2}\sin ^{2}\theta -3=0$. <br/>$(1)$ Find a parametric equation of the curve $C$; <br/>$(2)$ Let $A$ and $B$ be the points where $l$ intersects $C$, and $M$ be the point where $l$ intersects the $x$-axis. Find the value of $|MA|^{2}+|MB|^{2}$. | \frac{5}{2} | 87.5 |
23,376 | Alex thought of a two-digit number (from 10 to 99). Grisha tries to guess it by naming two-digit numbers. It is considered that he guessed the number if he correctly guessed one digit, and the other digit is off by no more than one (for example, if the number thought of is 65, then 65, 64, and 75 are acceptable, but 63, 76, and 56 are not). Devise a method that guarantees Grisha's success in 22 attempts (regardless of the number Alex thought of). | 22 | 42.96875 |
23,377 | Given the base of a triangle is $24$ inches, two lines are drawn parallel to the base, with one of the lines dividing the triangle exactly in half by area, and the other line dividing one of the resulting triangles further into equal areas. If the total number of areas the triangle is divided into is four, find the length of the parallel line closer to the base. | 12\sqrt{2} | 30.46875 |
23,378 | Given a sequence $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$, if $a_1=-2$, $a_2=2$, and $a_{n+2}-a_n=1+(-1)^n$, then $S_{50}=$ ______. | 600 | 84.375 |
23,379 | Given the function $f(x)=\sqrt{3}\sin \omega x+\cos \omega x (\omega > 0)$, where the x-coordinates of the points where the graph of $f(x)$ intersects the x-axis form an arithmetic sequence with a common difference of $\frac{\pi}{2}$, determine the probability of the event "$g(x) \geqslant \sqrt{3}$" occurring, where $g(x)$ is the graph of $f(x)$ shifted to the left along the x-axis by $\frac{\pi}{6}$ units, when a number $x$ is randomly selected from the interval $[0,\pi]$. | \frac{1}{6} | 88.28125 |
23,380 | Warehouse A and Warehouse B originally stored whole bags of grain. If 90 bags are transferred from Warehouse A to Warehouse B, then the grain in Warehouse B will be twice that in Warehouse A. If a certain number of bags are transferred from Warehouse B to Warehouse A, then the grain in Warehouse A will be six times that in Warehouse B. What is the minimum number of bags originally stored in Warehouse A? | 153 | 78.90625 |
23,381 | Given points \(A(4,5)\), \(B(4,0)\) and \(C(0,5)\), compute the line integral of the second kind \(\int_{L}(4 x+8 y+5) d x+(9 x+8) d y\) where \(L:\)
a) the line segment \(OA\);
b) the broken line \(OCA\);
c) the parabola \(y=k x^{2}\) passing through the points \(O\) and \(A\). | \frac{796}{3} | 46.875 |
23,382 | Given that $\{1, a, \frac{b}{a}\} = \{0, a^2, a+b\}$, find the value of $a^{2015} + b^{2014}$. | -1 | 64.84375 |
23,383 | A pirate is tallying his newly plundered wealth from the vessel G.S. Legends, where all values are counted in base 8. The treasure chest includes $5267_{8}$ dollars worth of silks, $6712_{8}$ dollars worth of precious stones, and $327_{8}$ dollars worth of spices. What is the total dollar amount the pirate has accumulated? Express your answer in base 10. | 6488 | 10.15625 |
23,384 | Given the line $mx - y + m + 2 = 0$ intersects with circle $C\_1$: $(x + 1)^2 + (y - 2)^2 = 1$ at points $A$ and $B$, and point $P$ is a moving point on circle $C\_2$: $(x - 3)^2 + y^2 = 5$. Determine the maximum area of $\triangle PAB$. | 3\sqrt{5} | 5.46875 |
23,385 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, where $a=2$, $c=3$, and it satisfies $(2a-c)\cdot\cos B=b\cdot\cos C$. Find the value of $\overrightarrow{AB}\cdot\overrightarrow{BC}$. | -3 | 68.75 |
23,386 | In obtuse triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given $a=7$, $b=3$, and $\cos C= \frac{ 11}{14}$.
1. Find the values of $c$ and angle $A$.
2. Find the value of $\sin (2C- \frac{ \pi }{6})$. | \frac{ 71}{98} | 60.15625 |
23,387 | The area of a rhombus with diagonals of 6cm and 8cm is in cm<sup>2</sup>, and its perimeter is in cm. | 20 | 36.71875 |
23,388 | Given the set \( M = \{1, 3, 5, 7, 9\} \), find the non-empty set \( A \) such that:
1. Adding 4 to each element in \( A \) results in a subset of \( M \).
2. Subtracting 4 from each element in \( A \) also results in a subset of \( M \).
Determine the set \( A \). | {5} | 0 |
23,389 | Steven subtracts the units digit from the tens digit for each two-digit number. He then finds the sum of all his answers. What is the value of Steven's sum? | 45 | 30.46875 |
23,390 | Three fair, six-sided dice are rolled. What is the probability that the sum of the three numbers showing is less than 16? | \frac{103}{108} | 13.28125 |
23,391 | Let $f(x)=2\sqrt{3}\sin(\pi-x)\sin x-(\sin x-\cos x)^{2}$.
(I) Determine the intervals on which $f(x)$ is increasing;
(II) On the graph of $y=f(x)$, if every horizontal coordinate of the points is stretched to twice its original value (vertical coordinate remains unchanged) and then the resulting graph is translated to the left by $\frac{\pi}{3}$ units, we get the graph of the function $y=g(x)$. Find the value of $g\left(\frac{\pi}{6}\right)$. | \sqrt{3} | 68.75 |
23,392 | 3 points $ O(0,\ 0),\ P(a,\ a^2), Q( \minus{} b,\ b^2)\ (a > 0,\ b > 0)$ are on the parabpla $ y \equal{} x^2$ .
Let $ S_1$ be the area bounded by the line $ PQ$ and the parabola and let $ S_2$ be the area of the triangle $ OPQ$ .
Find the minimum value of $ \frac {S_1}{S_2}$ . | 4/3 | 32.03125 |
23,393 | From the set $\{1,2,3, \cdots, 10\}$, six distinct integers are chosen at random. What is the probability that the second smallest number among the chosen integers is 3? | $\frac{1}{3}$ | 0 |
23,394 | In triangle $ABC$, it is known that $BC = 1$, $\angle B = \frac{\pi}{3}$, and the area of $\triangle ABC$ is $\sqrt{3}$. The length of $AC$ is __________. | \sqrt{13} | 92.1875 |
23,395 | Let $ABC$ be a triangle with sides $51, 52, 53$ . Let $\Omega$ denote the incircle of $\bigtriangleup ABC$ . Draw tangents to $\Omega$ which are parallel to the sides of $ABC$ . Let $r_1, r_2, r_3$ be the inradii of the three corener triangles so formed, Find the largest integer that does not exceed $r_1 + r_2 + r_3$ . | 15 | 30.46875 |
23,396 | A bag contains 7 red chips and 4 green chips. One chip is drawn from the bag, then placed back into the bag, and a second chip is drawn. What is the probability that the two selected chips are of different colors? | \frac{56}{121} | 96.875 |
23,397 | Five fair ten-sided dice are rolled. Calculate the probability that at least four of the five dice show the same value. | \frac{23}{5000} | 50.78125 |
23,398 | An integer is called "octal" if it is divisible by $8$ or if at least one of its digits is $8$ .
How many integers between $1$ and $100$ are octal? | 27 | 7.8125 |
23,399 | Given that $\frac{\cos \alpha}{1 + \sin \alpha} = \sqrt{3}$, find the value of $\frac{\cos \alpha}{\sin \alpha - 1}$. | -\frac{\sqrt{3}}{3} | 78.125 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.