task_type stringclasses 4
values | problem stringlengths 14 5.23k | solution stringlengths 1 8.29k | problem_tokens int64 9 1.02k | solution_tokens int64 1 1.98k |
|---|---|---|---|---|
math | A sequence begins with 3, and each subsequent term is triple the sum of all preceding terms. Determine the first term in the sequence that exceeds 15000. | 36864 | 36 | 5 |
math | Given a cube \( ABCD-A_{1}B_{1}C_{1}D_{1} \) with edge length 2, connect \( D_{1}A \) and \( D_{1}B \). Let \( E \) be the midpoint of \( D_{1}B \), \( F \) be the midpoint of \( BC \), and \( G \) be the midpoint of \( AD_{1} \). Find the angle between the skew lines \( DG \) and \( EF \). | 60^\circ | 109 | 4 |
math | Given the function $f(x)=a^{2}\sin 2x+(a-2)\cos 2x$, if its graph is symmetric about the line $x=-\frac{\pi}{8}$, determine the maximum value of $f(x)$. | 4\sqrt{2} | 54 | 6 |
math | When the mean, median, and mode of the list 1, 2, 5, 2, 3, 2, x are arranged in increasing order, they form a non-constant arithmetic progression. Also, including x, the list remains symmetric. Find the sum of all possible real values of $x$. | 6 | 67 | 1 |
math | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. The following conclusions are given:
1. If $A > B > C$, then $\sin A > \sin B > \sin C$;
2. There exist $A$, $B$, and $C$ such that $\tan A\tan B\tan C < \tan A+\tan B+\tan C$;
3. If $\sin ^{2}A+\sin ^{2}B > \sin ^{2}... | 1, 4 | 201 | 4 |
math | Given the complex number $z=1-2i$ (where $i$ is the imaginary unit), calculate the expression $z\cdot \bar{z}+z$. | 6-2i | 37 | 4 |
math | In the Cartesian coordinate system $xOy$, the parametric equations of curve $C$ are $\begin{cases}x=2\cos \alpha \\ y=\sin \alpha \\ \end{cases}$ (where $\alpha$ is the parameter). Establish a polar coordinate system with the origin $O$ as the pole and the positive $x$-axis as the polar axis.
1. Find the polar coordin... | \frac{4}{5} | 130 | 7 |
math | Starting at $(0,0),$ an object moves in the coordinate plane via a sequence of steps, each of length one. Each step is to the left, right, up, or down, all four equally likely. Let $q$ be the probability that the object reaches $(3,3)$ in eight or fewer steps. Write $q$ in the form $a/b$, where $a$ and $b$ are relative... | 4151 | 96 | 4 |
math | The number 12320 is written on the board. Petya appended $10n + 1$ threes to it, where $n$ is a nonnegative integer. Vasya thought this was a base-4 number representing a natural number $x$, and he factored $x$ into prime factors. It turned out that there were exactly two distinct prime factors among them. For which $n... | n = 0 | 93 | 4 |
math | A circle of radius $r$ and a right-angled isosceles triangle are drawn such that one of the shorter sides of the triangle is a diameter of the circle. What is the shaded area?
A) $\sqrt{2} r$
B) $r^{2}$
C) $2 \pi r$
D) $\frac{\pi r^{2}}{4}$
E) $(\sqrt{2}-1) \pi r^{2}$ | r^2 | 97 | 3 |
math | In the plane, 2013 red points and 2014 blue points are marked so that no three of the marked points are collinear. One needs to draw \( k \) lines not passing through the marked points and dividing the plane into several regions. The goal is to do it in such a way that no region contains points of both colors.
Find th... | 2013 | 103 | 4 |
math | Given α, β, γ ∈ [0, 2π] and $\sin(α - β) = \frac{1}{4}$, find the maximum value of $\sin(α - γ) + \cos(β - γ)$. | \frac{\sqrt{10}}{2} | 52 | 11 |
math | Given the functions $f(x)=x^{2}+2bx$ and $g(x)=|x-1|$, if for any $x_{1}$, $x_{2} \in [0,2]$, when $x_{1} < x_{2}$, we always have $f(x_{1})-f(x_{2}) < g(x_{1})-g(x_{2})$, then the minimum value of the real number $b$ is $\_\_\_\_\_\_$. | -\frac{1}{2} | 108 | 7 |
math | Given a right circular cone with a base radius of \(1 \, \text{cm}\) and a slant height of \(3 \, \text{cm}\), point \(P\) is on the circumference of the base. Determine the shortest distance from the vertex \(V\) of the cone to the shortest path from \(P\) back to \(P\). | 1.5 | 75 | 3 |
math | Given a modified Fibonacci sequence that starts with $F_1 = 2$ and $F_2 = 2$, and each subsequent term is the sum of its two predecessors, modulo 7, determine the last digit from 0 to 6 that appears in the units position of a number in this sequence. | 0 | 64 | 1 |
math | Given that the center of the ellipse $C$ is at the origin and its foci lie on the $x$-axis. If the eccentricity of the ellipse $C$ is equal to $\frac{1}{2}$, and one of its vertices coincides with the focus of the parabola $x^{2}=8 \sqrt{3}y$, find the standard equation of the ellipse $C$. | \frac{x^{2}}{16} + \frac{y^{2}}{12} = 1 | 86 | 25 |
math | If $2 - \sin^{2}(x + 2y - 1) = \frac{x^{2} + y^{2} - 2(x + 1)(y - 1)}{x - y + 1}$, then the minimum value of the product $xy$ is $\qquad$ . | 1/9 | 68 | 3 |
math | Given that in $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $b = a \tan B$, with $A$ being an obtuse angle, find $A - B$. | \frac{\pi}{2} | 62 | 7 |
math | Let point $O$ be the origin in a three-dimensional coordinate system, and let points $A$, $B$, and $C$ be located on the positive $x$, $y$, and $z$ axes respectively. If $OA = \sqrt[3]{50}$ and $\angle BAC = 45^\circ$, compute the area of triangle $ABC$. | 25 | 78 | 2 |
math | The domain of the function \\(f(x)=\dfrac{1}{x}\ln (\sqrt{x^{2}-3x+2}+\sqrt{-x^{2}-3x+4})\\) is __________. | [-4,0) \cup (0,1) | 48 | 12 |
math | Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E[X]=1,$ $E[X^2]=2,$ and $E[X^3]=5.$ (Here $E[Y]$ denotes the expectation of the random variable $Y.$ ) Determine the smallest possible value of the probability of the event $X=0.$ | \frac{1}{3} | 87 | 7 |
math | Given the universal set $U={1,3,5,7,9}$, set $A={1,|a-5|,9}$, and the complement of $A$ in $U$, $∁_{U}A={5,7}$, find the value of $a$. | 2, 8 | 64 | 4 |
math | In 2016, the fourth-grade class of Hua Sheng Education carried out extracurricular reading activities. If they read 800 characters every day, then in 7 days a week they will read ______ characters, and in 20 weeks, they will need to read ______ characters. After omitting the digits following the ten-thousands place, th... | 5600, 112000, 11 | 87 | 16 |
math | Given that each of the triangles $\triangle QPT$, $\triangle QTS$, and $\triangle QSR$ is an isosceles right-angled triangle with $\angle QPT = \angle QTS = \angle QSR = 90^{\circ}$, find the value of $k$ if the combined area of the three triangles is 56, where $QP = PT = k$. | 4 | 84 | 1 |
math | Given the function $f(x)=|x-a|+4x$, where $a > 0$.
(1) When $a=2$, find the solution set of the inequality $f(x)\geqslant 2x+1$.
(2) If $x\in(-2,+\infty)$ always satisfies $f(2x) > 7x+a^{2}-3$, find the range of values for $a$. | a\in(0,2) | 96 | 8 |
math | Let the right focus of the hyperbola $\dfrac {x^{2}}{a^{2}} - \dfrac {y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ be $F$. A line $l$, perpendicular to the $x$-axis and passing through point $F$, intersects the two asymptotes at points $A$ and $B$, and intersects the hyperbola in the first quadrant at point $P$. Let $O$ be the or... | 2 | 176 | 1 |
math | What is the smallest integer $n$ , greater than one, for which the root-mean-square of the first $n$ positive integers is an integer?
$\mathbf{Note.}$ The root-mean-square of $n$ numbers $a_1, a_2, \cdots, a_n$ is defined to be \[\left[\frac{a_1^2 + a_2^2 + \cdots + a_n^2}n\right]^{1/2}\] | \(\boxed{337}\) | 104 | 9 |
math | The ratio of girls to boys in Mr. Green's science class is 4:3. If there are a total of 56 students in the class, how many girls and boys are there in Mr. Green's science class? | 24 | 48 | 2 |
math | Let \( a \in \mathbf{R}_{+} \). The equation \( x^{2} - 2a x - 2a \ln x = 0 \) has a unique solution in the interval \( (0, +\infty) \). Find the value of \( a \). | \frac{1}{2} | 65 | 7 |
math | In $\triangle ABC$, the angles $A$, $B$, $C$ have opposite sides $a$, $b$, $c$ respectively, with $B=\frac{\pi}{3}$, $\cos A=\frac{4}{5}$, and $b=\sqrt{3}$.
(1) Find the value of $\sin C$;
(2) Find the area of $\triangle ABC$. | \frac{36 + 9\sqrt{3}}{50} | 84 | 17 |
math | The number $(\sqrt{5} + 2)^{1997}$ rounded to 100 digits after the decimal point is calculated to find the 100th digit after the decimal point. | 0 | 45 | 1 |
math | Let $x$ represent the cost of one copy of the newsletter. Fourteen copies cost 14$x$ dollars and nineteen copies cost 19$x$ dollars. It is given that 14$x$ < $16.00 and 19$x$ > $21.00. Find the value of $x$. | 1.11 | 72 | 4 |
math | Find the minimum positive integer $k$ such that there exists a function $f$ from the set $\Bbb{Z}$ of all integers to $\{1, 2, \ldots k\}$ with the property that $f(x) \neq f(y)$ whenever $|x-y| \in \{5, 7, 12\}$ . | 4 | 89 | 1 |
math | How many positive multiples of 3 that are less than 150 have a units digit of 3? | 5 | 23 | 1 |
math | Find the sum of all complex numbers $z$ that satisfy
\[z^3 + z^2 - |z|^2 + 2z = 0.\] | -2 | 35 | 2 |
math | Given $3^{7} + 1$ and $3^{15} + 1$ inclusive, how many perfect cubes lie between these two values? | 231 | 33 | 3 |
math | Find the minimum value of the distance $|AB|$ where point $A$ is the intersection of the line $y=a$ and the line $y=2x+2$, and point $B$ is the intersection of the line $y=a$ and the curve $y=x+\ln x$. | \frac{3}{2} | 62 | 7 |
math | An air force squadron is holding a stunt flying performance, during which a plane takes off and completes 4 performance maneuvers after flying $0.5km$. The changes in height of the plane are shown in the table below.
| Maneuver | Height Change | Notation |
|----------|---------------|----------|
| Maneuver 1 | Ascend ... | 49L | 225 | 3 |
math | Given the function $f(x)=e^{x}+ax+1$ $(a\in \mathbb{R})$. If $x=0$ is an extremum point of $f(x)$,
(I) find $a$, and determine the minimum value of $f(x)$ on the interval $[-2,1]$;
(II) if the inequality $kf'(x) < xe^{x}+1$ holds for any $x > 0$, where $k$ is an integer and $f'(x)$ is the derivative of $f(x)$, find... | 2 | 130 | 1 |
math | Given an arithmetic sequence $\{a\_n\}$, the sum of its first $n$ terms, $S\_n$, satisfies $S\_3=0$ and $S\_5=-5$. The sum of the first 2016 terms of the sequence $\{ \frac{1}{a_{2n-1}a_{2n+1}} \}$ is $\_\_\_\_\_\_\_\_.$ | -\frac{2016}{4031} | 91 | 13 |
math | Given points P(-2,-3) and Q(5,3) in the xy-plane; point R(x,m) is such that x=2 and PR+RQ is a minimum. Find m. | \frac{3}{7} | 43 | 7 |
math | Let \( a, b, c \) be the roots of \( x^3 - 9x^2 + 11x - 1 = 0 \), and let \( s = \sqrt{a} + \sqrt{b} + \sqrt{c} \). Find \( s^4 - 18s^2 - 8s \). | -37 | 78 | 3 |
math | Given unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ forming an acute angle, and the magnitude $|\overrightarrow{a} - t\overrightarrow{b}|$ (where $t \in \mathbb{R}$) has a minimum value of $\frac{\sqrt{3}}{2}$,
1. Find the value of $(\overrightarrow{a} + \overrightarrow{b})(\overrightarrow{a} - 2\overright... | \frac{\sqrt{3}}{2} | 163 | 10 |
math | The perimeter of a rectangle is 40 meters and its area is exactly 96 square meters. What are the possible dimensions of this rectangle? | (8, 12) | 30 | 7 |
math | A fair 6-sided die is rolled. If I roll $n$, then I win $(n^3 - 2n)$ dollars if $n < 4$, and $n^3$ dollars if $n \geq 4$. What is the expected value of my winnings? Express your answer as a dollar value rounded to the nearest cent. | \$71.50 | 73 | 6 |
math | If the function $f(x)=\frac{{a{x^3}+(a-1)x+a-2b}}{x}$ is an even function defined on $\left(-2a+2,0\right)\cup \left(0,a\right)$, find the value of $f\left(1\right)$. | 3 | 70 | 1 |
math | There are five volunteers, including A and B, who are assigned to serve at different positions in the China Pavilion, the UK Pavilion, the Australia Pavilion, and the Russia Pavilion at the Shanghai World Expo. Each position must be staffed by at least one volunteer. How many ways are there for A and B to each independ... | 72 | 76 | 2 |
math | Find the repetend in the decimal representation of $\frac{5}{17}.$ | 294117 | 18 | 6 |
math | The lengths of the legs of a certain right triangle are the roots of the equation \( a x^{2} + b x + c = 0 \). Find the radius of the circle inscribed in this triangle. | -\frac{b + \sqrt{b^2 - 2ac}}{2a} | 44 | 20 |
math | A gumball machine contains $9$ red, $7$ white, and $8$ blue gumballs. The least number of gumballs a person must buy to be sure of getting four gumballs of the same color is | 10 | 51 | 2 |
math | As shown in Figure 3, in \(\triangle ABC\), \(O\) is the midpoint of side \(BC\). A line through point \(O\) intersects lines \(AB\) and \(AC\) at different points \(M\) and \(N\) respectively. If
$$
\begin{array}{l}
\overrightarrow{AB}=m \overrightarrow{AM}, \\
\overrightarrow{AC}=n \overrightarrow{AN},
\end{array}
$$... | 2 | 108 | 1 |
math | Given a geometric sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$, and it is known that $a_4=2a_5$, and $S_6= \frac{63}{64}$;
$(1)$ Find the sum of the first $n$ terms of the sequence $\{a_n\}$, $S_n$;
$(2)$ Let $b_n= \frac{2^n a_n}{n^2+n}$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, $... | \frac{n}{n+1} | 130 | 8 |
math | Given vectors $a=\left( \sin\left( \alpha+\frac{\pi}{6} \right),3 \right)$ and $b=(1,4\cos \alpha)$, where $\alpha\in(0,\pi)$.
(1) If $a\perp b$, find the value of $\tan \alpha$;
(2) If $a\parallel b$, find the value of $\alpha$. | \frac{\pi}{6} | 90 | 7 |
math | Given the complex number $z=\frac{2i}{3+i}$, determine the conjugate complex number of $z$. | \frac{1}{5}-\frac{3}{5}i | 26 | 15 |
math | Five numbers, \( c_1, c_2, c_3, c_4, c_5 \), are drawn randomly and without replacement from the set of even numbers between 1 and 100, inclusive. Let \( p \) be the probability that, after a suitable rotation, a brick of dimensions \( c_1 \times c_2 \) can be enclosed in a box of dimensions \( c_3 \times c_4 \), with ... | 3 | 171 | 1 |
math | In a larger square grid of 10 x 10 units, a shaded region is constructed using the following points of the large square grid: lower left corner, midpoint of right edge, midpoint of top edge, and center of the square. What is the ratio of the area of the shaded region to the area of the large square? | \frac{1}{8} | 69 | 7 |
math | Find the number of permutations $x_1, x_2, x_3, x_4, x_5$ of numbers $1, 2, 3, 4, 5$ such that the sum of five products $$ x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_1 + x_5x_1x_2 $$ is divisible by $3$ . | 80 | 115 | 2 |
math | What is the average speed for the entire race where the triathlete swims 5 km at 3 kilometers per hour, bikes 30 km at 25 kilometers per hour, and runs 15 km at 8 kilometers per hour? | 10.54 | 50 | 5 |
math | Given the function $f\left( x \right)=\frac{{{x}^{2}}}{1+{{x}^{2}}}$.
(1) Find the values of $f\left( 2 \right)+f\left( \frac{1}{2} \right),f\left( 3 \right)+f\left( \frac{1}{3} \right),f\left( 4 \right)+f\left( \frac{1}{4} \right)$ and conjecture a general conclusion (proof is not required).
(2) Evaluate: $2f\left(2... | 4032 | 266 | 4 |
math | An Ultraman is fighting a group of monsters. It is known that Ultraman has one head and two legs. Initially, each monster has two heads and five legs. During the battle, some monsters split, with each splitting monster creating two new monsters, each with one head and six legs (they cannot split again). At a certain mo... | 13 | 93 | 2 |
math | There are 5 female students and 2 male students in a class. Find the number of different distribution schemes in which they can be divided into two groups, with each group having both female and male students. | 60 | 42 | 2 |
math | Solve for $y$ in terms of $b$, where $b \neq 0$, given the determinant:
\[
\begin{vmatrix}
y + b & -y & y \\
-y & y + b & -y \\
y & -y & y + b
\end{vmatrix} = 0.
\] | y = -\frac{b}{3} | 72 | 10 |
math | Professor Fernando's tree grows according to the following rule:
- In the first week, the tree starts to grow with only one branch;
- After growing for two weeks, this branch produces a new branch every week;
- Each new branch continues to grow and, after growing for two weeks, produces a new branch every week.
The f... | 233 | 152 | 3 |
math | Let $\triangle XYZ$ be a right triangle with $Y$ as the right angle. A circle with diameter $YZ$ intersects side $XZ$ at point $W$. If $XW = 3$ and $YW = 9$, find the length of segment $ZW$. | 27 | 59 | 2 |
math | Consider an unusual biased coin, with probability $p$ of landing heads, probability $q \leq p$ of landing tails, and probability \frac{1}{6}$ of landing on its side (i.e. on neither face). It is known that if this coin is flipped twice, the likelihood that both flips will have the same result is \frac{1}{2}$. Find $p$. | \frac{2}{3} | 83 | 7 |
math | A schematic diagram of a computer device is shown. \( J_1 \) and \( J_2 \) represent data inputs, and \( C \) is the output for the calculation result. The calculation process involves inputting natural numbers \( m \) and \( n \) into \( J_1 \) and \( J_2 \) respectively, after which a natural number \( k \) is output... | 2^{2001} + 16 | 297 | 11 |
math | Given the one-variable quadratic equation in $x$, $(k+2)x^{2}-2x-1=0$, determine the range of real number $k$ for which the equation has real roots. | k\geqslant -3 \text{ and } k\neq -2 | 42 | 19 |
math | Let \( D, E, \) and \( F \) be points on a circle of radius \( 25 \). If \( \angle EFD = 120^\circ \), what is the length of the circumference of the minor arc \( DE \)? Express your answer in terms of \( \pi \). | \frac{50\pi}{3} | 67 | 10 |
math | Two distinct positive integers \( a \) and \( b \) are factors of 48. If \( a \cdot b \) is not a factor of 48, what is the smallest possible value of \( a \cdot b \)? | 18 | 51 | 2 |
math | Let $\binom{n}{j}=\frac{n!}{j!(n-j)!}$. Evaluate the value of $\sum_{k=1}^{49}(-1)^{k}\binom{99}{2k}$. | -2^{49} | 51 | 6 |
math | The force required to loosen bolts varies inversely with the length of the wrench handle used. Using a wrench with a handle length of 10 inches requires 300 pounds of force to loosen Bolt A. To loosen Bolt B, which is tighter and requires 400 pounds of force with a 10-inch handle, how many pounds of force will be neede... | 200 | 85 | 3 |
math | What are the first three digits to the right of the decimal point in the decimal representation of $\left(10^{2003}+1\right)^{11/8}$? | 375 | 41 | 3 |
math | Given the sequence {a<sub>n</sub>} with the sum of its first n terms S<sub>n</sub> = n<sup>2</sup> - 2n + 2, find a<sub>1</sub> and a<sub>n</sub>. | 1 | 60 | 1 |
math | Given the ellipse $\dfrac{x^2}{m^2} + y^2 = 1$ ($m > 1$) and the hyperbola $\dfrac{x^2}{n^2} - y^2 = 1$ ($n > 0$), both sharing a common focus $F_1$. Find the area of the triangle $\triangle F_1 P F_2$. | 1 | 86 | 1 |
math | What amount can a worker, who is a tax resident, expect after paying personal income tax if they are credited with 45000 for their work? | 39150 | 33 | 5 |
math | Given a rectangular prism with edge dimensions 4, 2, and 3, calculate how many unordered pairs of edges determine a plane. | 42 | 28 | 2 |
math | A real number $x$ is randomly selected from the interval $[-π,π]$. The probability of the event "$\sin x \geq \frac{1}{2}$" occurring is $\_\_\_\_\_\_\_.$ | \frac{1}{3} | 50 | 7 |
math | Let $x$ and $y$ be positive integers such that $xy - 10x + 3y = 670$. What is the smallest possible value of $|x - y|$? | 16 | 44 | 2 |
math | Given an ellipse with foci on the $x$-axis and a minor axis length of 4, the eccentricity is $\frac{\sqrt{5}}{5}$.
$(1)$ Find the standard equation of the ellipse;
$(2)$ If line $l$ passes through the left focus of the ellipse and intersects the ellipse at points $M$ and $N$, with $\left| MN \right|=\dfrac{16}{9}\... | y = x + 1 \quad \text{or} \quad y = -x - 1 | 109 | 22 |
math | Given \( x_{i} \in \{-1,1\}, i=1,2,\cdots,2021 \), and \( x_{1}+x_{2}+\cdots+x_{k} \geq 0 \) for \( k=1,2,\cdots,2020 \), with \( x_{1}+x_{2}+\cdots+x_{2021}=-1 \). How many ordered arrays \( (x_{1}, x_{2}, \cdots, x_{2021}) \) are there? | \frac{1}{1011} \binom{2020}{1010} | 127 | 24 |
math | Compute the value of $\cos 80^{\circ}\cos 20^{\circ}+\sin 80^{\circ}\sin 20^{\circ}$. | \dfrac{1}{2} | 39 | 7 |
math | The school organized a summer camp during the summer vacation. The school contacted two travel agencies, both of which quoted a price of $200 per person. After negotiation, Travel Agency A's discount condition was: a 25% discount for all teachers and students; Travel Agency B's discount condition was: the fee for one t... | x > 16 | 146 | 5 |
math | There are 8 different books, including 3 math books, 3 foreign language books, and 2 literature books. If these books are arranged in a row on a bookshelf, in how many ways can the arrangement be made such that all math books are together and all foreign language books are also together? | 864 | 63 | 3 |
math | Given points P<sub>1</sub>(x<sub>1</sub>, y<sub>1</sub>) and P<sub>2</sub>(x<sub>2</sub>, y<sub>2</sub>) on a unit circle with center O, and ∠P<sub>1</sub>OP<sub>2</sub>=θ (θ is an obtuse angle). If sin(θ + $\frac{π}{4}$) = $\frac{3}{5}$, find the value of x<sub>1</sub>x<sub>2</sub> + y<sub>1</sub>y<sub>2</sub>. | -$\frac{\sqrt{2}}{10}$ | 144 | 12 |
math | The area of triangle \( ABC \) is 1. On the rays \( AB \), \( BC \), and \( CA \), points \( B' \), \( C' \), and \( A' \) are marked respectively, such that:
\[ BB' = AB, \quad CC' = 2BC, \quad AA' = 3CA \]
Calculate the area of triangle \( A'B'C' \). | 18 | 89 | 2 |
math | In the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is $\begin{cases}x=2\cos \alpha \\ y=2+2\sin \alpha\end{cases}$ (where $\alpha$ is the parameter), and the parametric equation of line $l$ is $\begin{cases}x= \sqrt{3}- \dfrac{ \sqrt{3}}{2}t \\ y=3+ \dfrac{1}{2}t\end{cases}$ (where $t$ is th... | 2 \sqrt{3} | 258 | 6 |
math | The towns in one country are connected with bidirectional airlines, which are paid in at least one of the two directions. In a trip from town A to town B there are exactly 22 routes that are free. Find the least possible number of towns in the country. | 7 | 56 | 1 |
math | Given $f(x)=4x^{5}+3x^{4}+2x^{3}-x^{2}-x-\frac{1}{2}$, use Horner's method to find $f(-2)$. | -\frac{197}{2} | 48 | 9 |
math | Find the value of $\frac{2468_{7}}{123_{5}} - 3214_{6} + 7891_{8}$ and express your answer in base 10. | 3464 | 49 | 4 |
math | Given that $1,3,5,7,\ldots,\beta$ and $1,6,11,16,\ldots,\beta+1$ are two finite sequences of positive integers, determine $\gamma$, the number of positive integers common to both sequences. | 6 | 57 | 1 |
math | There are $24$ different pencils, $4$ different colors, and $6$ pencils of each color. They were given to $6$ children in such a way that each got $4$ pencils. What is the least number of children that you can randomly choose so that you can guarantee that you have pencils of all colors.
P.S. for 10 grade gi... | 5 | 117 | 3 |
math | What is the modular inverse of $13$, modulo $2000$?
Express your answer as an integer from $0$ to $1999$, inclusive. | 1077 | 37 | 4 |
math | Point \( M \) lies on the edge \( AB \) of cube \( ABCD A_1 B_1 C_1 D_1 \). A rectangle \( MNLK \) is inscribed in square \( ABCD \) such that one of its vertices is point \( M \) and the other three vertices are located on different sides of the square base. Rectangle \( M_1 N_1 L_1 K_1 \) is the orthogonal projection... | 1:2 | 187 | 3 |
math | The equations of $L_1$ and $L_2$ are $y=mx$ and $y=nx$, respectively. Suppose $L_1$ makes twice as large of an angle with the horizontal (measured counterclockwise from the positive x-axis ) as does $L_2$, and that $L_1$ has 4 times the slope of $L_2$. If $L_1$ is not horizontal, then $mn$ is | 2 | 98 | 1 |
math | A bike travels $\dfrac{b}{4}$ feet every $t$ seconds. There are $3$ feet in a yard. How many yards does the bike travel in $4$ minutes?
A) $\frac{15b}{t}$
B) $\frac{20b}{t}$
C) $\frac{25b}{t}$
D) $\frac{30b}{t}$ | \frac{20b}{t} | 88 | 9 |
math | Let $\omega$ be a nonreal root of the equation $z^4 = 1$. Let $b_1, b_2, \dots, b_n$ be real numbers such that
\[
\frac{1}{b_1 + \omega} + \frac{1}{b_2 + \omega} + \dots + \frac{1}{b_n + \omega} = 3 + 4i.
\]
Compute
\[
\frac{2b_1 - 1}{b_1^2 - b_1 + 1} + \frac{2b_2 - 1}{b_2^2 - b_2 + 1} + \dots + \frac{2b_n - 1}{b_n^2 -... | 6 | 174 | 1 |
math | Positive real numbers \( x, y, z \) satisfy
\[
\left\{
\begin{array}{l}
\frac{2}{5} \leqslant z \leqslant \min \{x, y\}, \\
xz \geqslant \frac{4}{15}, \\
yz \geqslant \frac{1}{5}.
\end{array}
\right.
\]
Find the maximum value of \( \frac{1}{x} + \frac{2}{y} + \frac{3}{z} \). | 13 | 123 | 2 |
math | Determine the largest value of \(n\) such that \(6x^2 + nx + 48\) can be factored into the product of two linear factors with integer coefficients. | n = 289 | 38 | 6 |
math | Calculate:<br/>$(1)15+(-22)$;<br/>$(2)(-13)+(-8)$;<br/>$(3)(-0.9)+1.5$;<br/>$(4)\frac{1}{2}+(-\frac{2}{3})$. | -\frac{1}{6} | 64 | 7 |
math | The lateral surface of a cylinder unfolds into a rectangle with sides of length $6\pi$ and $4\pi$. Then, the surface area of the cylinder is ______. | 24\pi^2 + 8\pi | 36 | 11 |
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