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 | The opposite of $-5$ is $5$. | 5 | 11 | 1 |
math | Let S<sub>n</sub> be the sum of the first n terms of the sequence {a<sub>n</sub>}, given that a<sub>1</sub>\=8, a<sub>n+1</sub>\=S<sub>n</sub>+8.
(Ⅰ) Find the general term formula for the sequence {a<sub>n</sub>}.
(Ⅱ) Find the smallest positive integer n such that the inequality a<sub>1</sub>•a<sub>2</sub>•a<sub>3</sub... | n = 3 | 136 | 4 |
math | Given a triangle $ABC$ with sides $a$, $b$, and $c$ forming an arithmetic sequence and vertices $A$ and $C$ located at $(-1, 0)$ and $(1, 0)$ respectively, find the equation for the trajectory of vertex $B$. | \frac {x^2}{4} + \frac {y^2}{3} = 1 \quad \text{(for $x \neq \pm2$)} | 60 | 38 |
math | Given the vertices of a triangle A(-5, 0), B(3, -3), C(0, 2), find:
(1) The equation of the line on which side AC lies
(2) The equation of the line on which the median to side BC lies. | x+13y+5=0 | 60 | 9 |
math | Find the smallest positive integer \( a \) such that for some integers \( b \) and \( c \), the polynomial \( ax^2 - bx + c \) has two distinct zeros in the interval \( (0, 1) \). | a = 5 | 51 | 4 |
math | In a regular tetrahedron, the cosine of the dihedral angle between a lateral face and the base is equal to | \frac{1}{3} | 26 | 7 |
math | The sequence of natural numbers $1, 5, 6, 25, 26, 30, 31,...$ is made up of powers of $5$ with natural exponents or sums of powers of $5$ with different natural exponents, written in ascending order. Determine the term of the string written in position $167$ . | 81281 | 85 | 5 |
math | It is now 3:15:30 PM, as displayed on a 12-hour digital clock. In 174 hours, 58 minutes, and 16 seconds, the time will be $A:B:C$. What is the value of $A+B+C$? | 69 | 62 | 2 |
math | Regions I, II and III are bounded by squares. The perimeter of region I is 12 units and the perimeter of region II is 24 units. What is the ratio of the area of region I to the area of region III? Express your answer as a common fraction.
[asy]
draw((0,0)--(9,0)--(9,9)--(0,9)--(0,0)--cycle,linewidth(2));
draw((9,0)--(... | \frac{1}{9} | 193 | 7 |
math | In triangle $PQR$, where $PQ$ is the hypotenuse of the right triangle, a line segment starts moving from vertex $P$ to vertex $R$. The segment starts with zero length at $P$, continuously increases in length until it reaches a specific position, and then decreases back to zero length as it reaches $R$. Where does the s... | \sqrt{91} | 131 | 6 |
math | For each positive integer $n$, let $f(n)$ denote the last digit of the sum $1+2+3+\ldots+n$.
For example: $f(1)=1$, $f(2)=3$ (the last digit of $1+2$), $f(5)=5$ (the last digit of $1+2+3+4+5$), $f(7)=8$ (the last digit of $1+2+3+4+5+6+7$)
Then, the value of $f(1)+f(2)+f(3)+\ldots+f(2005)$ is . | 7015 | 145 | 4 |
math | The solution set for the inequality $|x - 1| < 2$ is | (-1, 3) | 18 | 6 |
math | Let $a$ and $b$ be real numbers. Consider the following five statements:
1. $a < b$
2. $b < 0$
3. $a < 0$
4. $\frac{1}{a} < \frac{1}{b}$
5. $a^2 < b^2$
What is the maximum number of these statements that can be true for any values of $a$ and $b$? | 3 | 94 | 1 |
math | Given the ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$, its right focus $F$, and the line passing through $F$ with a slope of $1$ intersects the ellipse at points $M$ and $N$. The perpendicular bisector of $MN$ intersects the $x$-axis at point $P$. If $\frac{|MN|}{|PF|}=4$, find the eccentricity... | \frac{1}{2} | 116 | 7 |
math | Given the ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 \left( a > b > 0 \right)$ intersects with the negative half of the $x$-axis at point $A$, $P$ is a point on the ellipse in the first quadrant, the line $OP$ intersects the ellipse at another point $Q$, the left focus of the ellipse is $F$, if the line $PF$ bisects ... | \dfrac{1}{3} | 118 | 7 |
math | In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are respectively $a$, $b$, $c$, given that the vectors $\vec{m}=(\cos A,\cos B)$, $\vec{n}=(a,2c-b)$, and $\vec{m}\parallel \vec{n}$.
(I) Find the magnitude of angle $A$;
(II) Find the maximum value of $\sin B + \sin C$ and determine the shape of $\tria... | \sqrt{3} | 114 | 5 |
math | Eight cubes, with volumes $1$, $8$, $27$, $64$, $125$, $216$, $343$, and $512$ cubic units, are stacked vertically to form a tower with decreasing volumes from the bottom to the top. Each cube except for the bottom one is placed perfectly on top of the cube below. Compute the total surface area of the tower.
- **A)** $... | 1021 | 126 | 4 |
math | Given a sequence $\{a_n\}$ satisfying $a_1=1$ and $a_{n+1}=a_n+2$, and a sequence $\{b_n\}$ whose sum of the first $n$ terms is $S_n$, also given $S_n=2-b_n$,
(Ⅰ) Find the general formula for the terms of sequences $\{a_n\}$ and $\{b_n\}$;
(Ⅱ) Let $c_n=a_nb_n$, find the sum of the first $n$ terms $T_n$ of the sequence ... | 6 - \frac{2n+3}{2^{n-1}} | 127 | 16 |
math | Let $a_{10} = 10$, and for each positive integer $n >10$ let $a_n = 100a_{n - 1} + n$. Find the least positive $n > 10$ such that $a_n$ is a multiple of $99$. | 45 | 67 | 2 |
math | Given that a line passes through point $P(2,0)$, and the chord length intercepted by the circle $(x-3)^{2}+(y-2)^{2}=4$ is $2 \sqrt {3}$, what is the equation of this line? | x=2 \text{ and } 3x-4y-6=0 | 58 | 18 |
math | Compute the sum:
\[\sum_{n = 1}^\infty \frac{2n + 1}{n(n + 1)(n + 2)}.\] | 1 | 38 | 1 |
math | How many ways are there to paint each of the integers $2, 3, \cdots , 9$ either red, green, or blue so that each number has a different color from each of its proper divisors? | 432 | 47 | 3 |
math | Let $\max (a; c)$ denote the minimum of the numbers $a$ and $c$. Construct the graph of the function $y = \max \left(x - 11; x^2 - 8x + 7\right)$ and use it to solve the inequality $\max \left(x - 11; x^2 - 8x + 7\right) < 0$. | x \in (1, 7) | 87 | 9 |
math | Given the circumference of a circular garden border is increased from 30 inches to 40 inches, calculate the increase in the radius. | \frac{5}{\pi}\text{ inches} | 28 | 12 |
math | Given the division calculation \( 952473 \div 18 \), determine the two adjacent digits that should be swapped in order to increase the result by 100. | 2 and 4 | 40 | 4 |
math | The Rockets are playing the Bulls in the NBA finals. To win the finals, a team must win 4 games. The probability that the Bulls win any given game is $\frac{3}{5}$. Assuming no ties, what is the probability that the Rockets will win the NBA finals and that the contest will need all seven games to be determined? Provide... | \frac{8640}{78125} | 78 | 14 |
math | If $x^{2}-\left(m-3\right)x+25$ is a perfect square trinomial, then the value of $m$ is ______. | -7 \text{ or } 13 | 36 | 10 |
math | ### Part ①
In the Cartesian coordinate system, with the origin as the pole and the positive half of the $x$-axis as the polar axis, a polar coordinate system is established. The polar coordinates of point $A$ are $(4\sqrt{2},\frac{\pi }{4})$, and the polar equation of line $l$ is $\rho \cos (\theta -\frac{\pi }{4})=a$,... | [-3,0] | 327 | 5 |
math | There are exactly 120 ways to color five cells in a $5 \times 5$ grid such that exactly one cell in each row and each column is colored.
There are exactly 96 ways to color five cells in a $5 \times 5$ grid without the corner cell, such that exactly one cell in each row and each column is colored.
How many ways are th... | 78 | 114 | 2 |
math | In triangle \( ABC \), side \( AB \) is longer than side \( BC \), and angle \( B \) is \( 40^\circ \). Point \( P \) is taken on side \( AB \) such that \( BP = BC \). The angle bisector \( BM \) intersects the circumcircle of triangle \( ABC \) at point \( T \). Find the angle \( MPT \). | 20^\circ | 87 | 4 |
math | In an isosceles triangle $\triangle ABC$ with vertex angle $A = \frac{2\pi}{3}$ and base $BC = 2\sqrt{3}$, find the dot product $\vec{BA} \cdot \vec{AC}$. | -2 | 56 | 2 |
math | Find the length of side $XY$ in the triangle below.
[asy]
unitsize(1inch);
pair X,Y,Z;
X = (0,0);
Y= (2,0);
Z = (0,sqrt(3));
draw (X--Y--Z--X,linewidth(0.9));
draw(rightanglemark(Y,X,Z,3));
label("$X$",X,S);
label("$Y$",Y,S);
label("$Z$",Z,N);
label("$12$",Z/2,W);
label("$60^\circ$",(1.2,0),N);
[/asy] | 24 | 127 | 2 |
math | Let $[x]$ denote the greatest integer less than or equal to the real number $x$. Determine how many elements are in the set
$$
\{[x]+[2x]+[3x] \mid x \in \mathbb{R}\} \cap \{1, 2, \ldots, 100\}.
$$ | 67 | 76 | 2 |
math | Seven standard dice are glued together to make a solid. The pairs of faces of the dice that are glued together have the same number of dots on them. How many dots are on the surface of the solid?
A) 24
B) 90
C) 95
D) 105
E) 126 | 105 | 73 | 3 |
math | Given that $a$ and $b$ are positive real numbers, and $a+2b=1$, determine the minimum value of $\frac{{a}^{2}+1}{a}+\frac{2{b}^{2}+1}{b}$. | 4+2\sqrt{2} | 57 | 8 |
math | Point D is a point on segment AC and point P is a point on segment BD in triangle △ABC, given that ℑ overrightarrow{CD} = overrightarrow{DA} and overrightarrow{AP} = λ overrightarrow{AB} + 2/6 overrightarrow{AC}. Find the value of λ. | \frac{2}{3} | 67 | 7 |
math | Calculate the value of $C_n^1+2C_n^2+2^2C_n^3+\dots+2^{n-1}C_n^n$. | S_n = \frac{1}{2}(3^n - 1) | 36 | 16 |
math | What is the minimum test score Josanna would need on her next test to increase her test average by $10$ points given that she received scores of $75, 85, 65, 95, 70$ on her previous tests? | 138 | 56 | 3 |
math | A square with side length 12 is cut 3 units from one of its sides, creating two rectangles. Determine the dimensions of the smaller rectangle. | 3 \text{ by 12} | 31 | 9 |
math | Given a sequence $\{a\_n\}$ that satisfies $a\_1=-1$, $a\_2=1$, and $a\_{n+2}= \frac {2+(-1)^{n}}{2}a\_n$ for all $n\in\mathbb{N}^*$.
1. Find the value of $a\_5+a\_6$.
2. Let $S\_n$ denote the sum of the first $n$ terms of the sequence $\{a\_n\}$. Find an expression for $S\_n$.
3. Let $b\_n=a\_{2n-1}+a\_{2n}$. Determi... | i=1, j=2, k=3 | 215 | 11 |
math | Given that the point $P(2,2\sqrt{2})$ lies on the parabola $y^2=2px$, find the distance from point $P$ to the directrix of the parabola. | 3 | 48 | 1 |
math | A group of $n$ friends takes $r$ distinct photos (no two photos have exactly the same people) each containing at least one person. Find the largest $r$ such that for each pair of photos there is at least one person appearing in both. | 2^{n-1} | 53 | 6 |
math | Some $n>2$ lamps are cyclically connected: lamp $1$ with lamp $2$ , ..., lamp $k$ with lamp $k+1$ ,..., lamp $n-1$ with lamp $n$ , lamp $n$ with lamp $1$ . At the beginning all lamps are off. When one pushes the switch of a lamp, that lamp and the two ones connected to it change status (from off to on, or... | 2^n | 131 | 2 |
math | Petya's bank account contains 500 dollars. The bank allows only two types of transactions: withdrawing 300 dollars or adding 198 dollars. What is the maximum amount Petya can withdraw from the account if he has no other money? | 498 | 56 | 3 |
math | Let t(x) has domain \{-2, -1, 0, 1\} and range \{-4, -2, 0, 2\}. Let u(x) has domain \{0, 1, 2, 3\} and is defined by u(x) = x - 1. What is the sum of all possible values of u(t(x))? | 0 | 82 | 1 |
math | Given set $A=\{x||x-4| \lt 2x\}$ and set $B=\{x|x(x-a)\geqslant (a+6)(x-a)\}$. Choose one of the following conditions to solve, and the answer should not be empty.<br/>① If $A$⋂$B=A$, find the range of real number $a$;<br/>② If $A$⋂$B=B$, find the range of real number $a$. | a\leq-\frac{14}{3} | 108 | 12 |
math | If ${x^2}+x=5+\sqrt{5}$, then the value of $x$ is ____. | \sqrt{5} \text{ or } -\sqrt{5} - 1 | 26 | 19 |
math | If 700 were expressed as a sum of at least three distinct powers of 2, what would be the least possible sum of the exponents of these powers? | 30 | 35 | 2 |
math | A manager plans to meet three interns at a café. Each of them can arrive randomly between 3:00 and 5:00 p.m. The manager will leave if not all interns are present upon his arrival. Each intern will wait for up to 30 minutes for the others before leaving. What is the probability that the meeting successfully takes place... | 0.25 | 74 | 4 |
math | Given that the probability of drawing a red ball from bag A is $\frac{1}{3}$, and from bag B is $\frac{1}{2}$, when one ball is drawn from each bag, determine the condition under which the equality of probability $\frac{2}{3}$ occurs. | \frac{2}{3} | 61 | 7 |
math | A chord passing through the focus of the parabola \( y^2 = 12x \), which makes an angle of \(60^\circ\) with the x-axis, is drawn. Determine the length of this chord and the angle between the tangents drawn at the endpoints of this chord. | 90^\circ | 62 | 4 |
math | Given the parabola $x^{2}=8y$, find the equation of its directrix and the coordinates of its focus. | (0,2) | 27 | 5 |
math | The standard equation of a parabola with focus F(0,1) is to be determined. | x^2=4y | 21 | 6 |
math | Determine the smallest positive angle $\theta$ that fulfills $\sin 3\theta \sin 4\theta = \cos 3\theta \cos 4\theta$. | \frac{90^\circ}{7} | 38 | 10 |
math | The line $kx+y+1=2k$ passes through the fixed point $(x, y)$. | (2, -1) | 23 | 6 |
math | The Gropkas of Papua New Guinea have ten letters in their alphabet: A, E, G, I, K, O, R, U, and V. Suppose license plates of four letters use only the letters in the Gropka alphabet. How many possible license plates are there of four letters that begin with either A or E, end with V, cannot contain P, and have no lette... | 84 | 84 | 2 |
math | For how many integers $x$ is the number $x^4-51x^2+50$ negative? | 12 | 26 | 2 |
math | Given that $\cos (α-\frac{\pi }{6}) +\sin α=\frac{4\sqrt{3}}{5}$, find the value of $\sin (α+\frac{7\pi }{6})$. | -\frac{4}{5} | 50 | 7 |
math | Given: The solution to the equation $\dfrac{x+m}{3}-\dfrac{2x-1}{2}=m$ is a non-positive number. Find the range of values for $m$. | m \geqslant \dfrac{3}{4} | 43 | 14 |
math | Find the angle of inclination $\theta$ of the line $\sqrt{3}x - y + 3 = 0$. | \theta = \frac{\pi}{3} | 26 | 10 |
math | Mario is once again on a quest to save Princess Peach. Mario enters Peach's castle and finds himself in a room with 4 doors. This room is the first in a sequence of 2 indistinguishable rooms. In each room, 1 door leads to the next room in the sequence (or, for the second room, into Bowser's level), while the other 3 do... | 20 | 132 | 2 |
math | Find the ordered pair $(a,b)$ of positive integers, with $a < b,$ for which
\[
\sqrt{1 + \sqrt{45 + 20 \sqrt{5}}} = \sqrt{a} + \sqrt{b}.
\] | (1,5) | 56 | 5 |
math | If the function $f(x) = x^3 - 3bx + b$ has a local minimum in the interval $(0, 1)$, then $b$ must satisfy the condition ; | b \in (0, 1) | 44 | 9 |
math | Given a line through the focus $F$ of the parabola $y^2 = 4x$ intersects the parabola at points $A$ and $B$, and $|AF| = 2$, then $|BF| =$ __________. | 2 | 55 | 1 |
math | Given the positive odd integers less than $500$, find the product. | \frac{500!}{2^{250} \cdot 250!} | 16 | 22 |
math | In the expansion of $(\sqrt{x} + \frac{1}{2\sqrt{x}})^n$, the coefficients of the first three terms form an arithmetic sequence.
(1) Find the constant term in the expansion;
(2) Find the term with the maximum coefficient in the expansion. | T_{4} = 7x | 62 | 8 |
math | The approximate interval where the zero of the function $f(x) = \log_{2}(x+2) - \frac{3}{x}$ ($x > 0$) is located, find the approximate interval. | (1,2) | 46 | 5 |
math | Let $x, y, z$, and $w$ be positive real numbers such that
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
x^2+y^2&=&z^2+w^2&=&2500,\\
xz&=&yw&=&1200.
\end{array}
\]
If $T = x + y + z + w$, compute the value of $\lfloor T \rfloor$. | 140 | 129 | 3 |
math | Say that an integer $B$ is delicious if there exist several consecutive integers, starting from $B$, that add up to 2020. What is the smallest delicious integer, given that $B$ must be less than $-2020$? | -2021 | 55 | 5 |
math | Points \( P \) and \( Q \) lie in a plane with \( PQ = 10 \). How many locations for point \( R \) in this plane are there such that the triangle with vertices \( P \), \( Q \), and \( R \) is a right triangle with area \( 16 \) square units?
A) 2
B) 4
C) 6
D) 8
E) 12 | 8 | 96 | 1 |
math | If $x^{2}-\left(m-3\right)x+16(m$ is a constant) is a perfect square trinomial, then the value of $m$ is ____. | -5 \text{ or } 11 | 41 | 10 |
math | Zibo barbecue is famous throughout China. A specialty barbecue restaurant hopes to make good profits during the National Day holiday. After calculation, the cost of lamb skewers is $3$ yuan per skewer. Drawing on past experience, if each skewer is sold for $10$ yuan, on average, they can sell 300 skewers per day. If th... | 7 \text{ yuan} | 194 | 6 |
math | For nonnegative integers $a$ and $b$ with $a + b \leq 6$, let $T(a, b) = \binom{6}{a} \binom{6}{b} \binom{6}{a + b}$. Let $S$ denote the sum of all $T(a, b)$, where $a$ and $b$ are nonnegative integers with $a + b \leq 6$. Find the remainder when $S$ is divided by $1000$.
Major Note
Most solutions use committee formin... | 564 | 145 | 3 |
math | For how many integer values of $b$ does the equation $$x^2 + bx + 12b = 0$$ have integer solutions for $x$? | 16 | 35 | 2 |
math | Given \( a, b, c, d \in [0, \sqrt[4]{2}) \), such that \( a^{3} + b^{3} + c^{3} + d^{3} = 2 \). | 2 | 50 | 1 |
math | A park has two parallel paths 60 feet apart. A crosswalk bounded by two parallel lines crosses the paths at an angle. The length of the path between the lines is 20 feet and each line is 75 feet long. Find the distance, in feet, between the lines. | 16 | 61 | 2 |
math | Suppose $R_{n}$ is the product of the first $n$ terms of the geometric sequence $\{a_{n}\}$. If $25(a_{1}+a_{3})=1$ and $a_{5}=27a_{2}$, find the value of $n$ when $R_{n}$ is minimized. | 6 | 74 | 1 |
math | Let $Q$ be the point on the line segment $\overline{CD}$ such that $CQ:QD = 2:3.$ Then express $\overrightarrow{Q}$ in terms of $\overrightarrow{C}$ and $\overrightarrow{D}$, i.e., find constants $s$ and $v$ such that
\[\overrightarrow{Q} = s \overrightarrow{C} + v \overrightarrow{D}.\] | \left( \frac{3}{5}, \frac{2}{5} \right) | 96 | 20 |
math | Given the sequence $\{a\_n\}$ with the sum of its first $n$ terms $S\_n = 3^n - 1$,
(1) Find the sum of $a\_1 + a\_4 + a\_7 + ... + a_{3n+1}$.
(2) Let $b\_n = a\_n (\log\_3 a_{n+1} - \log\_3 2)$, find the sum of the first $n$ terms of the sequence $\{b\_n\}$, denoted as $T\_n$. | \frac{1 + (2n - 1) \cdot 3^n}{2} | 123 | 20 |
math | In a kindergarten, there are two small Christmas trees and five children. The teachers want to divide the children into two circles around each of the Christmas trees, with at least one child in each circle. The teachers can distinguish between the children but not between the trees: two such divisions into circles are... | 50 | 102 | 2 |
math | Given a connected simple graph \( G \) with \( e \) edges and pieces placed on each vertex of \( G \) (where each piece can only be placed on a single vertex of \( G \)), you are allowed to perform the following operation: if the number of pieces on a vertex \( v \) is at least the number of vertices adjacent to \( v \... | e | 170 | 1 |
math | $2019$ students are voting on the distribution of $N$ items. For each item, each student submits a vote on who should receive that item, and the person with the most votes receives the item (in case of a tie, no one gets the item). Suppose that no student votes for the same person twice. Compute the maximum possibl... | 1009 | 100 | 4 |
math | Given that each of the numbers $n$ , $n+1$ , $n+2$ , $n+3$ is divisible by its sum of digits in its decimal representation and the number in ones column of $n$ is $8$ , determine the number of different values the tens column of $n$ can have. | 1 | 79 | 1 |
math | (1) Solve the equation $4^x - 2^x - 2 = 0$.
(2) Solve the inequality $\log_2(2x+3) > \log_2(5x-6)$.
(3) Find the range of the function $y = \left(\frac{1}{3}\right)^{x^2-4x}$, where $x \in [0, 5)$. | \left(\frac{1}{243}, 81\right] | 96 | 17 |
math | Determine the solution set for the following inequalities:
(1) $2x^2 + x - 3 < 0$;
(2) $x(9 - x) > 0$. | (0, 9) | 42 | 6 |
math | In trapezoid \(ABCD\), a point \(X\) is taken on the base \(AD\) such that segments \(XB\) and \(XC\) divide the trapezoid into three similar, yet pairwise unequal, non-isosceles triangles. The side \(AB\) has a length of 6. Find \(AX \cdot DX\). | 36 | 74 | 2 |
math | Given the sequence ${a}$, where $a = \frac{4}{3}$ and $a_{n+1} - 1 = a_{n}^{2} - a_{n}$ $(n \in \mathbb{N}^*)$, determine the integer part of $m = \frac{1}{a_{1}} + \frac{1}{a_{2}} + ... + \frac{1}{a_{2017}}$. | 2 | 98 | 1 |
math | A cylindrical tank with radius $6$ feet and height $10$ feet is lying on its side. The tank is filled with water to a depth of $3$ feet. Calculate the volume of water in the tank. | 120\pi - 90\sqrt{3} | 46 | 14 |
math | Let $O$ and $H$ denote the circumcenter and orthocenter of triangle $ABC,$ respectively. If $AO = AH,$ then enter all possible values of $\angle A$ (in degrees), separated by commas. | 60^\circ, 120^\circ | 48 | 11 |
math | Regardless of the real numbers $m$ and $n$, the curve represented by the equation $x^2+y^2-2mx-2ny+4(m-n-2)=0$ always passes through a fixed point. Find the coordinates of this fixed point. | (2, -2) | 55 | 6 |
math | Find the maximum number of real roots to a polynomial of the form
\[ ax^n + x^{n-1} + \dots + x + 1 = 0, \]
where \( n \) is a positive even integer, and \( a \) is a real number not equal to 0. | 1 | 64 | 1 |
math | Calculate $(3 \cos 30^\circ + 3i \sin 30^\circ)^{8}$. Enter your answer in rectangular form. | -3280.5 - 3280.5i \sqrt{3} | 33 | 21 |
math | Given the function $f(x)=({m}^{2}-2m-2){x}^{{m}^{2}+m-1}$ is a power function, and it is increasing on $\left(0,+\infty \right)$, find the real number $m$. | 3 | 61 | 1 |
math | The Lantern Festival is a traditional festival in China. People have the custom of eating tangyuan. As the Lantern Festival approaches, a certain supermarket plans to purchase a batch of tangyuan for sale.
$(1)$ If the supermarket purchases a total of 1000 bags of two brands, brand $A$ and brand $B$, and the brand $A$ ... | 15 | 207 | 2 |
math | Find the second-order derivative \( y_{xx}^{\prime \prime} \) of the function defined parametrically by:
\[
\begin{cases}
x = \cos t \\
y = \ln (\sin t)
\end{cases}
\] | -\frac{1 + \cos^2 t}{\sin^4 t} | 55 | 17 |
math | Solve the fractional equation $\frac{3x-a}{x-3}+\frac{x+1}{3-x}=1$ for $x$ to find the value of $a$, given that the solution is a positive number. | 4 | 48 | 1 |
math | A positive number and its square sum to 245. Additionally, the sum is three times the number itself. What is the number? | 14 | 29 | 2 |
math | Determine how many integers $n$ between 1 and 15 (inclusive) result in a fraction $\frac{n}{30}$ that has a repeating decimal. | 10 | 35 | 2 |
math | Given that $min\{ a,b\}$ represents the smaller value between the real numbers $a$ and $b$, and the vectors $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ satisfy $(\vert\overrightarrow{a}\vert=1,\vert\overrightarrow{b}\vert=2,\overrightarrow{a}\cdot\overrightarrow{b}=0,\overrightarrow{c}=\lambda\overrigh... | \frac{2\sqrt{5}}{5} | 163 | 12 |
math | Given the function $f(x) = a^x$ ($a > 0$, $a \neq 1$), when $m > n$, $f(m) < f(n)$, then the range of the real number $a$ is | (0, 1) | 53 | 6 |
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