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 boy presses his thumb along a vertical rod that rests on a rough horizontal surface. Then he gradually tilts the rod, keeping the component of the force along the rod constant, which is applied to its end. When the tilt angle of the rod to the horizontal is $\alpha=80^{\circ}$, the rod begins to slide on the surface.... | 0.17 | 115 | 4 |
math | A real number sequence \( a_{0}, a_{1}, a_{2}, \cdots, a_{n}, \cdots \) satisfies the following equations: \( a_{0} = a \), where \( a \) is a real number,
\[ a_{n} = \frac{a_{n-1} \sqrt{3} + 1}{\sqrt{3} - a_{n-1}}, \quad n \in \mathbb{N}. \]
Find \( a_{1994} \). | \frac{a + \sqrt{3}}{1 - a \sqrt{3}} | 114 | 19 |
math | Given that $\frac{1+\sin x}{\cos x}=-\frac{1}{2}$, find the value of $\frac{\cos x}{\sin x-1}$. | \frac{1}{2} | 40 | 7 |
math | A right-angled isosceles triangle $ABP$ with $AB = BP = 4$ inches is placed inside a square $AXYZ$ with side length $8$ inches so that point $B$ is on side $AX$ of the square. The triangle is rotated clockwise about the midpoint of each side ($AB$, then $BP$, etc.), continuing this process along the sides of the square... | 12\pi\sqrt{2} | 156 | 9 |
math | For what value of the real number $m$ is the complex number $(m^{2}-5m+6)+(m^{2}-3m)i$
$(1)$ a real number;
$(2)$ a complex number;
$(3)$ a pure imaginary number? | 2 | 59 | 1 |
math | Let $c$ be a constant. The simultaneous equations
\begin{align*}x-y = &\ 2 \\
cx+y = &\ 3 \\
\end{align*}have a solution $(x, y)$ inside Quadrant I if and only if | $-1<c<\frac{3}{2}$ | 56 | 12 |
math | A resident randomly chooses to watch the performance for consecutive 3 days out of 8 days, calculate the probability that they watch the performance for consecutive 3 days during the first to the fourth day. | \dfrac {1}{3} | 40 | 7 |
math | Vanya believes that fractions can be "reduced" by canceling out the same digits in the numerator and the denominator. Seryozha noticed that sometimes Vanya gets correct equalities, for example, \( \frac{49}{98} = \frac{4}{8} \). Find all correct fractions with... | \frac{26}{65}, \frac{16}{64}, \frac{19}{95}, \frac{49}{98} | 68 | 36 |
math | For the all $(m,n,k)$ positive integer triples such that $|m^k-n!| \le n$ find the maximum value of $\frac{n}{m}$ *Proposed by Melih Üçer* | 2 | 51 | 1 |
math | A two-digit integer $AB$ equals $\frac{1}{8}$ of the three-digit integer $AAB$, where $A$ and $B$ represent distinct digits from 1 to 9. What is the smallest possible value of the three-digit integer $AAB$? | 773 | 58 | 3 |
math | Given \( x \cdot y \cdot z + y + z = 12 \), find the maximum value of \( \log_{4} x + \log_{2} y + \log_{2} z \). | 3 | 47 | 1 |
math | In the triangular prism ABC-A₁B₁C₁, AB, AC, AA₁ form 60° angles with each other. Points E, F, G are on the line segments AB, AC, AA₁, respectively, with AE = $\frac {1}{2}$AB, AF = $\frac {1}{3}$AC, AG = $\frac {2}{3}$AA₁. The volume ratio of the triangular pyramid G-AEF to the triangular prism ABC-A₁B₁C₁ is _______. | \frac {1}{27} | 107 | 8 |
math | The functions given are $p(x) = x^2 - 4$ and $q(x) = -|x| + 1$. Evaluate $q(p(x))$ at $x = -3, -2, -1, 0, 1, 2, 3$ and find the sum of the values. | -13 | 70 | 3 |
math | Let $x_1, x_2, \ldots , x_n$ be a sequence of integers such that
(i) $-1 \le x_i \le 2$ for $i = 1,2, \ldots n$
(ii) $x_1 + \cdots + x_n = 19$; and
(iii) $x_1^2 + x_2^2 + \cdots + x_n^2 = 99$.
Let $m$ and $M$ be the minimal and maximal possible values of $x_1^3 + \cdots + x_n^3$, respectively. Then $\frac Mm =$
$\mathr... | 7 | 198 | 1 |
math | Given a sequence $\{a_n\}$ that satisfies $(a_1 + 4a_2 + 4^2a_3 + \cdots + 4^{n-1}a_n = \frac{n}{4})$ $(n\in \mathbb{N}^*)$.
(I) Find the general formula for the sequence $\{a_n\}$;
(II) Let $(b_n = 2^n\log_4 a_n)$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$. | T_n = (1-n)2^{n+1} - 2 | 127 | 16 |
math | Given the infinite series $1-\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}-\frac{1}{32}+\frac{1}{64}-\frac{1}{128}-\cdots$, find the limiting sum of the series. | \frac{2}{7} | 72 | 7 |
math | In triangle $PQR,$ point $G$ is on $\overline{PQ}$ such that $PG:GQ = 4:1,$ and point $H$ is on $\overline{QR}$ such that $QH:HR = 4:1.$ If lines $GH$ and $PR$ intersect at $J,$ find the ratio $\frac{GH}{HJ}.$ | \frac{4}{1} | 84 | 7 |
math | Given that the polar coordinate equation of curve C is $ρ - 2\sin θ = 0$, establish a rectangular coordinate system with the pole as the origin and the polar axis as the positive half of the x-axis. Line l passes through point M(1,0) with a slope angle of $\frac{2π}{3}$.
(1) Find the rectangular coordinate equation of ... | \sqrt{3} + 1 | 120 | 8 |
math | A traffic light cycles as follows: green for 45 seconds, then yellow for 5 seconds, and then red for 50 seconds. Mark chooses a random five-second interval to observe the light. What is the probability that the color changes during his observation? | \frac{3}{20} | 54 | 8 |
math | Given the function $f(x)=-\cos x + \cos\left(\frac{\pi}{2} - x\right)$,
(1) If $x \in [0, \pi]$, find the maximum and minimum values of the function $f(x)$ and the corresponding values of $x$;
(2) If $x \in \left(0, \frac{\pi}{6}\right)$ and $\sin 2x = \frac{1}{3}$, find the value of $f(x)$. | f(x) = -\frac{\sqrt{6}}{3} | 112 | 15 |
math | Determine the following sums (as functions of $n$):
$$
\begin{aligned}
& A=2^{2}+4^{2}+6^{2}+\ldots+(2n-2)^{2} \\
& B=1^{2}+3^{2}+5^{2}+\ldots+(2n-1)^{2}
\end{aligned}
$$ | B = \frac{n}{3}(4n^2-1) | 85 | 15 |
math | Select $5$ people from $7$ people including $A$, $B$, and $C$ to line up in a row (all answers should be numerical).<br/>$(1)$ If $A$ must be included, how many ways are there to line them up?<br/>$(2)$ If $A$, $B$, and $C$ are not all included, how many ways are there to line them up?<br/>$(3)$ If $A$, $B$, and $C$ ar... | 144 | 145 | 3 |
math | The equations $x^3 + Ax + 10 = 0$ and $x^3 + Bx^2 + 50 = 0$ have two roots in common. Then the product of these common roots can be expressed in the form $a \sqrt[b]{c},$ where $a,$ $b,$ and $c$ are positive integers, when simplified. Find $a + b + c.$ | 12 | 90 | 2 |
math | Calculate the number of trailing zeroes in $1500!$. Additionally, determine the smallest non-zero digit at the end of $1500!$. | 1 | 33 | 1 |
math | Graph the set of solutions for the system of equations:
\[
\begin{cases}
x - 2y = 1 \\
x^3 - 8y^3 - 6xy = 1
\end{cases}
\] | y = \frac{x-1}{2} | 51 | 10 |
math | Let $a,$ $b,$ $c,$ $d$ be real numbers such that
\begin{align*}
a + b + c + d &= 6, \\
a^2 + b^2 + c^2 + d^2 &= 12.
\end{align*}Let $m$ and $M$ denote minimum and maximum values of
\[4(a^3 + b^3 + c^3 + d^3) - (a^4 + b^4 + c^4 + d^4),\]respectively. Find $m + M.$ | 84 | 126 | 2 |
math | Given the function \( f(n) = k \), where \( k \) is the \( n \)-th digit after the decimal point of the repeating decimal \( 0.\dot{9}1827364\dot{5} \), determine the value of \( \underbrace{f\{f \cdots f[f}_{1996 \uparrow f}(1)]\} \). | 4 | 88 | 1 |
math | $XYZ$ is a right triangle with $\angle YXZ = 90^\circ$, and $XY = 3$ cm, $XZ = 4$ cm. Line segment $\overline{YZ}$ is extended beyond $Z$ by its own length to point $W$, and $M$ is the midpoint of $\overline{XZ}$. Suppose $\overline{MW}$ meets $\overline{XY}$ at point $N$. Find the area of the triangle $MYN$ in square ... | 3\, \text{cm}^2 | 111 | 10 |
math | Given vectors in the plane $\overset{→}{OA}=(-1,-3)$, $\overset{→}{OB}=(5,3)$, and $\overset{→}{OM}=(2,2)$, point $P$ is on line $OM$, and $\overset{→}{PA} \cdot \overset{→}{PB}=-16$.
$(1)$ Find the coordinates of $\overset{→}{OP}$.
$(2)$ Find the cosine of $\angle APB$.
$(3)$ Let $t \in \mathbb{R}$, find the min... | \sqrt{2} | 152 | 5 |
math | A five-digit number is formed using the digits $1$, $3$, $4$, $6$, and $x (1 \leqslant x \leqslant 9, x \in \mathbb{N}^*)$ without any repetition. The sum of all the digits of all such five-digit numbers is $2640$. Find the value of $x$. | 8 | 82 | 1 |
math | Lou's Fine Shoes undergoes another pricing strategy to boost sales. The price of a pair of shoes on Thursday is $50$. On Friday, prices are boosted by $20\%$ to set up for a later discount. An advertising campaign states: "Twenty percent off the new price starting Monday!" How much does a pair of shoes cost on Monday t... | 48 | 122 | 2 |
math | Write down the general formula of a geometric sequence $\{a_{n}\}$ that satisfies both conditions ① and ②.①$a_{n}a_{n+1} \lt 0$；②$|a_{n}| \gt |a_{n+1}|$. | \left(-\frac{1}{2}\right)^{n-1} | 64 | 17 |
math | Six years ago, Mr. Li's age was the same as Xiao Ming's age seven years from now. The sum of Mr. Li's age in four years and Xiao Ming's age five years ago is 32 years old. Mr. Li is \_\_\_\_\_\_ years old this year, and Xiao Ming is \_\_\_\_\_\_ years old this year. | 10 | 80 | 2 |
math | In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are denoted by $a$, $b$, and $c$, respectively, and it is given that $a < b < c$ and $$\frac{a}{\sin A} = \frac{2b}{\sqrt{3}}$$.
(1) Find the size of angle $B$;
(2) If $a=2$ and $c=3$, find the length of side $b$ and the area of $\triangle ABC$. | \frac{3\sqrt{3}}{2} | 117 | 12 |
math | The numbers from 1 to 200, inclusive, are placed in a bag and a number is randomly selected from the bag. What is the probability it is not a perfect power (integers that can be expressed as $x^{y}$, where $x$ is an integer and $y$ is an integer greater than 1)? Express your answer as a common fraction. | \frac{181}{200} | 79 | 11 |
math | What is the median number of moons for the listed celestial bodies, taking into account both planets and prominent dwarf planets? Include the distance from the sun for each celestial body. Use the following data:
\begin{tabular}{c|c|c}
Celestial Body & Number of Moons & Distance from Sun (AU)\\
\hline
Mercury & 0 & 0.3... | 2 | 259 | 1 |
math | In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, and $1+ \frac{\tan C}{\tan B}= \frac{2a}{b}$.
(1) Find the measure of angle $C$;
(2) If $\cos (B+ \frac{\pi}{6})= \frac{1}{3}$, find the value of $\sin A$;
(3) If $(a+b)^{2}-c^{2}=4$, find the minimum value of $3a+b$. | 4 | 127 | 1 |
math | In the Cartesian coordinate plane $(xOy)$, there is an ellipse $(E)$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with an eccentricity of $\frac{1}{2}$ and a right focus $F$. The minimum distance from a point on the ellipse $(E)$ to point $F$ is $2$.
$(1)$ Find the values of $a$ and $b$;
$(2)$ Let the left... | \frac{1}{2} \times 8 \times 3 = 12 | 238 | 19 |
math | Given \\(x > 0\\), \\(y > 0\\), and \\(\lg {{2}^{x}}+\lg {{8}^{y}}=\lg 2\\), find the minimum value of \\(\dfrac{1}{x}+\dfrac{1}{y}\\). | 4+2 \sqrt{3} | 64 | 8 |
math | Given the function $y=x^{3}-3x$ in the interval $[a,a+1]$ $(a\geqslant 0)$, the difference between its maximum and minimum values is $2$. Find all possible real values of $a$. | \sqrt{3}-1 | 54 | 6 |
math | Given two plane vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfying $|\overrightarrow{a}+2\overrightarrow{b}|=\sqrt{5}$, and $\overrightarrow{a}+2\overrightarrow{b}$ is parallel to the line $y=2x+1$. If $\overrightarrow{b}=(2,-1)$, find $\overrightarrow{a}$. | \overrightarrow{a}=(-3,4) \text{ or } (-5,0) | 89 | 21 |
math | A circular region has a radius of 4 units. Determine the probability that a point randomly chosen within this circle is closer to the center than it is to a point that is exactly 3 units away from the boundary. | \frac{1}{16} | 44 | 8 |
math | If the solution set of the inequality $ax^{2}-bx+c < 0$ is $(-2,3)$, then the solution set of the inequality $bx^{2}+ax+c < 0$ is ______. | (-3,2) | 48 | 5 |
math | Two circles with equal radii intersect as shown. The area of the shaded region equals the sum of the areas of the two unshaded regions. If the area of the shaded region is \(216 \pi\), what is the circumference of each circle? | 36 \pi | 54 | 4 |
math | A triangle has sides of lengths 8 and 15 units. Determine if it is possible to form a right triangle with these sides, and if so, calculate the length of the third side. | \sqrt{161} | 40 | 7 |
math | Schools A and B are having a sports competition with three events. In each event, the winner gets 10 points and the loser gets 0 points, with no draws. The school with the highest total score after the three events wins the championship. It is known that the probabilities of school A winning the three events are 0.5, 0... | 13 | 135 | 2 |
math | Soda is now sold in packs of 8, 15, and 30 cans. Find the minimum number of packs needed to buy exactly 130 cans of soda. | 6 | 39 | 1 |
math | Given a point $P$ in the plane $P \in \{(x, y) | (x-2\cos\alpha)^2 + (y-2\sin\alpha)^2 = 16, \alpha \in \mathbb{R}\}$, then the area of the shape formed by all such points $P$ in the plane is ____. | 32\pi | 78 | 4 |
math | Given any two positive real numbers $x$ and $y$, define $x \, \Diamond \, y$ as a positive real number according to some fixed rule. Assume this operation satisfies the equations $(xy) \, \Diamond \, y=x(y \, \Diamond \, y)$ and $(x \, \Diamond \, 2) \, \Diamond \, x = x \, \Diamond \, 2$ for all $x,y>0$. Also, suppose... | 20 | 132 | 2 |
math | The number of triangles with integer side lengths and a perimeter of 20 is unknown, determine the value. | 8 | 22 | 1 |
math | Given $f(x)$ and $g(x)$ be odd and even functions defined on $\mathbb{R}$, respectively, and $f(x) + g(x) = 2^x$. If the inequality $a \cdot f(x) + g(2x) \geq 0$ holds for all $x \in [1, 2]$, determine the range of the real number $a$. | \left[-\frac{17}{6}, +\infty\right) | 87 | 18 |
math | Choose four different numbers from the set $\{3, 5, 7, 9, 11, 13\}$, add the first two and the last two of these numbers together, and then multiply the two sums. What is the smallest result that can be obtained from this process? | 128 | 63 | 3 |
math | Janet played a song on her clarinet in 3 minutes and 20 seconds. Janet’s goal is to play the song at a 25%
faster rate. Find the number of seconds it will take Janet to play the song when she reaches her goal.
| 160 \text{ seconds} | 56 | 8 |
math | Triangle $ABC$ has side-lengths $AB = 15, BC = 25,$ and $AC = 20.$ The line through the incenter of $\triangle ABC$ parallel to $\overline{BC}$ intersects the extensions of $\overline{AB}$ at $M$ and $\overline{AC}$ at $N$. Determine the perimeter of $\triangle AMN$. | 35 | 82 | 2 |
math | In front of the elevator doors, there are people with masses of 50, 51, 55, 57, 58, 59, 60, 63, 75, and 140 kg. The elevator's load capacity is 180 kg. What is the minimum number of trips needed to transport all the people? | 4 | 82 | 1 |
math | Without using a calculator or table, determine the smallest integer that is greater than \((\sqrt{3}+\sqrt{2})^{6}\). | 970 | 31 | 3 |
math | A bag contains 6 balls of the same shape and size, among which 4 are white and 2 are red. Two balls are randomly drawn from the bag.
(1) Calculate the probability that both balls are red;
(2) Calculate the probability that at least one ball is red. | \frac{3}{5} | 60 | 7 |
math | (1) Simplify: $$\frac {(2a^{ \frac {2}{3}}b^{ \frac {1}{2}})(-6 \sqrt {a} 3b )}{3a^{ \frac {1}{6}}b^{ \frac {5}{6}}}$$;
(2) Evaluate: $\log_{5}35+2\log_{0.5} \sqrt {2}-\log_{5} \frac {1}{50}-\log_{5}14+10^{\lg3}$. | 5 | 119 | 1 |
math | If $x^3 + \frac{1}{x^3} = C$, and $x - \frac{1}{x} = D$, where $C$ and $D$ are real numbers, find the minimum possible numerical value for $\frac{C}{D}$. | 3 | 59 | 1 |
math | It is given the sequence defined by $$ \{a_{n+2}=6a_{n+1}-a_n\}_{n \in \mathbb{Z}_{>0}},a_1=1, a_2=7 \text{.} $$ Find all $n$ such that there exists an integer $m$ for which $a_n=2m^2-1$ . | n = 1 | 91 | 4 |
math | Let $a$, $b$, and $c$ be positive integers with $a \ge b \ge c$ such that
$a^2-b^2-c^2+ab=2011$ and
$a^2+3b^2+3c^2-3ab-2ac-2bc=-1997$.
What is $a$? | 253 | 81 | 3 |
math | The houses of Andrey, Borya, Vova, and Gleb are located in some order on a single straight street. The distance between Andrey's and Borya's houses, as well as the distance between Vova's and Gleb's houses, is 600 meters. What can be the distance in meters between Andrey's and Gleb's houses, knowing that it is 3 times ... | 1800 | 104 | 4 |
math | A certain large theater is hosting a cultural performance with the following pricing standards:
| Number of Tickets | Pricing |
|-------------------|---------|
| Up to 30 people | $400 per person |
| More than 30 people | For each additional person, the price per ticket decreases by $5, but not less than $280 per tick... | 40 | 143 | 2 |
math | In parallelogram $EFGH$, the measure of angle $EFG$ is 4 times the measure of angle $FGH$. Calculate the degree measure of angle $EHG$. | 144^\circ | 39 | 5 |
math | In the number triangle shown, each disc is to be filled with a positive integer. Each disc in the top or middle row contains the number which is the product of the two numbers immediately below. What is the value of \( n \)?
A) 1
B) 2
C) 3
D) 6
E) 33 | 1 | 73 | 1 |
math | There is a sequence of curves \( P_{0}, P_{1}, P_{2}, \cdots \). It is known that the shape enclosed by \( P_{0} \) is an equilateral triangle with an area of 1. \( P_{k+1} \) is obtained from \( P_{k} \) by the following operation: each side of \( P_{k} \) is divided into three equal parts, an equilateral triangle is ... | \frac{8}{5} | 193 | 7 |
math | Given $x$ is the smallest angle in a triangle, determine the range of the function $y=\sqrt{2}\sin(x+45^{\circ})$. | (1, \sqrt {2}] | 35 | 8 |
math | In a bag, there are $4$ different red balls and $6$ different white balls.
$(1)$ If $4$ balls are randomly selected from the bag, how many ways are there to select the balls such that the number of red balls is not less than the number of white balls?
$(2)$ If a red ball is worth $2$ points and a white ball is wort... | 186 \text{ ways} | 114 | 8 |
math | A group of 12 friends decides to form a committee of 5. Calculate the number of different committees that can be formed. Additionally, if there are 4 friends who refuse to work together, how many committees can be formed without any of these 4 friends? | 56 | 55 | 2 |
math | In the arithmetic sequence $\{a\_n\}$, $4a_{12}=-3a_{23} > 0$. Let $b\_n= \frac {a\_n a_{n+1}}{a_{n+2}}$, $S\_n$ is the sum of the first $n$ terms of $\{b\_n\}$, and $S\_{n_0}$ is the maximum term of the sequence $\{S\_n\}$. Find $n\_0$. | 14 | 110 | 2 |
math | On a trip from the United States to Canada, Isabella took $d$ U.S. dollars. At the border she exchanged them all, receiving $10$ Canadian dollars for every $7$ U.S. dollars. After spending $60$ Canadian dollars, she had $d$ Canadian dollars left. What is the sum of the digits of $d$? | 5 | 76 | 1 |
math | Let $\mathbf{v}$ and $\mathbf{w}$ be vectors such that
\[
\operatorname{proj}_{\mathbf{w}} \mathbf{v} = \begin{pmatrix} 4 \\ 3 \end{pmatrix}.
\]
Compute $\operatorname{proj}_{\mathbf{w}} (7 \mathbf{v} + 2\mathbf{w})$. | \begin{pmatrix} 28 \\ 21 \end{pmatrix} + 2\mathbf{w} | 90 | 28 |
math | Consider two solid spherical balls, one centered at $\left(0, 0, 10\right)$ with a radius of 5, and the other centered at $(0,0,4)$ with a radius of 6. Determine how many points $(x, y, z)$ with integer coordinates are there in the intersection of the balls. | 1 | 72 | 1 |
math | Given the probability distribution table of a discrete random variable $\xi$ as shown below, calculate the mathematical expectation $E(\xi)$ where P($\xi$ = 1) = 0.5, P($\xi$ = 3) = m, and P($\xi$ = 5) = 0.2. | 2.4 | 70 | 3 |
math | Calculate the remainder when the sum of $1! + 2! + 3! + \cdots + 10!$ is divided by $7$. | 6 | 34 | 1 |
math | In the sequence $\{a_n\}$, $a_1=1$, $a_{n+1}-(1+ \frac {1}{n})a_n- \frac {n}{2^{n}}- \frac {1}{2^{n}}=0$, then the sum of the first $n$ terms of the sequence $\{a_n\}$, $S_n=$ ______. | n(n+1)+ \frac {n+2}{2^{n-1}}-4 | 85 | 20 |
math | Given that the probability that each ball falls into bin k is 3^(-k) for k = 1,2,3,..., find the probability that the blue ball falls into a lower-numbered bin than the yellow ball. | \frac{7}{16} | 48 | 8 |
math | A box contains $6$ balls, of which $3$ are yellow, $2$ are blue, and $1$ is red. Three balls are drawn from the box.
$(1)$ If the $3$ yellow balls are numbered as $A$, $B$, $C$, the $2$ blue balls are numbered as $d$, $e$, and the $1$ red ball is numbered as $x$, use $(a,b,c)$ to represent the basic event. List all th... | \frac{4}{5} | 137 | 7 |
math | $(1)$ $f(n)$ is a function defined on the set of positive integers, satisfying:<br/>① When $n$ is a positive integer, $f(f(n))=4n+9$;<br/>② When $k$ is a non-negative integer, $f(2^{k})=2^{k+1}+3$. Find the value of $f(1789)$.<br/>$(2)$ The function $f$ is defined on the set of ordered pairs of positive integers, and s... | 364 | 173 | 3 |
math | Consider the sequence $1, -4, 9, -16, 25, -36, \ldots,$ where the $n$th term is $(-1)^{n+1} \cdot n^2$. Calculate the average of the first 100 terms of this sequence. | -50.5 | 66 | 5 |
math | Given the expansion of ${(2x^2 - \frac{1}{x})^n}$ where $n \in \mathbb{N}^*$, the sum of the binomial coefficients of all terms is $64$.
(I) Find the term with the maximum binomial coefficient in the expansion.
(II) Find the constant term in the expansion of $(2 - x^3)(2x^2 - \frac{1}{x})^n$. | 132 | 98 | 3 |
math | In $\triangle ABC$, $BC= a$, $AC= b$, $AB = c$, ${{a}^{2}} -{{c}^{2}} = {{b}^{2}} - bc$, find the angle $A$. | \frac{π}{3} | 50 | 7 |
math | We have 1000 solid cubes with edge lengths of 1 unit each. We want to use these small cubes to create a hollow cube with a wall thickness of 1 unit. The small cubes can be glued together but not cut. What is the maximum (external) volume of the cube we can thus create? | 2197 | 66 | 4 |
math | Given that $F\_1$ and $F\_2$ are the left and right foci of the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a > b > 0$), the ellipse $C$ passes through the point $(-\sqrt{3}, 1)$ and shares a focus with the parabola $y^2 = -8x$.
1. Find the equation of the ellipse $C$.
2. The line $l$ passing through ... | \sqrt{6} | 160 | 5 |
math | In a right-angled triangle, the sum of the hypotenuse and one leg is \(c + b = 14 \mathrm{~cm}\), and \(\operatorname{tg} \frac{\alpha}{2}=\frac{1}{2}\), where \(\alpha\) is the angle opposite to leg \(a\). Calculate the measurements of the sides of the triangle without using trigonometric tables. | a = 7 \ \text{cm}, \ b = 5.25 \ \text{cm}, \ c = 8.75 \ \text{cm} | 87 | 38 |
math | Determine the value of $\log^{9} + \log_{3}^{27}$. | 5 | 21 | 1 |
math | In the ancient Chinese mathematical treatise "Nine Chapters on the Mathematical Art," it is recorded: "There is a field in the shape of a sector, with a circumference of 30 steps and a diameter of 16 steps. What is the area of the field?" Note: "宛田" refers to a field in the shape of a sector, "下周" refers to the circumf... | 120 | 111 | 3 |
math | If $x > 0$, $y > 0$, then the minimum value of $$\frac {x}{x+2y}+ \frac {y}{x}$$ is \_\_\_\_\_\_. | \sqrt {2} - \frac {1}{2} | 46 | 13 |
math | Given an ellipse $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with eccentricity $\frac{\sqrt{2}}{2}$, point $A\left(2,1\right)$ lies on the ellipse $C$.
$(1)$ Find the standard equation of the ellipse $C$;
$(2)$ A line $l$ passing through point $M\left(0,1\right)$ intersects the ellipse $C$ at point... | 2 | 142 | 1 |
math | If the function $f(x)=\frac{a-\sin x}{\cos x}$ is monotonically increasing in the interval $(\frac{\pi}{6}, \frac{\pi}{3})$, then the range of values for the real number $a$ is \_\_\_\_\_\_. | [2,+\infty) | 63 | 7 |
math | In a modified staircase-shaped region, all angles that appear as right angles are right angles, and each of the twelve congruent sides marked with a tick mark have length 1 foot. The larger rectangle has dimensions such that its length is modified to 12 feet. If the region has an area of 85 square feet, what is the num... | 41 | 379 | 2 |
math | There are 4 students and 1 teacher standing in a row for a photo. If the teacher stands in the middle, and the male student A does not stand on the far left, and the male student B does not stand on the far right, then the number of different possible arrangements is ______. | 6 + 8 = 14 | 61 | 8 |
math | Choose three people from five, including A and B, to form a line. What is the probability that person A is not in the first position and person B is not in the last position? | \frac {13}{20} | 39 | 9 |
math | The 5 a.m. temperatures for seven consecutive days were $-6^{\circ}$, $-3^{\circ}$, $-3^{\circ}$, $-4^{\circ}$, $2^{\circ}$, $4^{\circ}$, and $0^{\circ}$ Celsius. Calculate the mean 5 a.m. temperature for the week in degrees Celsius. | -1.43^\circ \text{C} | 83 | 12 |
math | Triangle $PQR$ has side-lengths $PQ = 20, QR = 40,$ and $PR = 30.$ The line through the incenter of $\triangle PQR$ parallel to $\overline{QR}$ intersects $\overline{PQ}$ at $X$ and $\overline{PR}$ at $Y.$ What is the perimeter of $\triangle PXY?$ | 50 | 84 | 2 |
math | Let $IJKLMNOP$ be a rectangular prism where $IJKLMN$ is the base rectangle and $OP$ lies vertically above this base in the third dimension. Suppose each edge of the base rectangle is 2 units, and the height from $IJKLMN$ to $OP$ is also 2 units. Find the volume of pyramid $IJKOP$. | \frac{4}{3} | 74 | 7 |
math | We are given $2021$ points on a plane, no three of which are collinear. Among any $5$ of these points, at least $4$ lie on the same circle. Is it necessarily true that at least $2020$ of the points lie on the same circle? | 2020 | 72 | 4 |
math | A bookshelf has $6$ different English books and $2$ different math books. If you randomly pick $1$ book, calculate the number of different ways to pick the book. | 6 + 2 = 8 | 38 | 7 |
math | The $42$ points $P_1,P_2,\ldots,P_{42}$ lie on a straight line, in that order, so that the distance between $P_n$ and $P_{n+1}$ is $\frac{1}{n}$ for all $1\leq n\leq41$ . What is the sum of the distances between every pair of these points? (Each pair of points is counted only once.) | 861 | 106 | 3 |
math | A $\textit{palindrome}$ is a number which reads the same forward as backward, like 121 or 2442. What is the smallest natural number that can be subtracted from 56,789 to create a palindrome? | 24 | 55 | 2 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.