problem stringlengths 14 4.09k | solution stringlengths 1 802k | task_type stringclasses 3
values | problem_tokens int64 5 895 |
|---|---|---|---|
If a die is rolled 500 times, what is the most probable number of times that the face showing 1 dot will appear? | 83 | math | 29 |
Given that a school selects five teachers: A, B, C, D, and E to support teaching in three remote areas, with at least one person in each area, where Teacher A and B cannot go to the same area, while Teacher A and C must go to the same area, calculate the total number of different assignment plans. | 36 | math | 68 |
Determine the coefficient of $x^{3}$ in the expansion of \\((2x-1)( \frac {1}{x}+x)^{6}\\). | 30 | math | 35 |
Let point $P$ be a moving point on the ellipse $x^{2}+4y^{2}=36$, and let $F$ be the left focus of the ellipse. The maximum value of $|PF|$ is _________. | 6 + 3\sqrt{3} | math | 51 |
Find the value of $p$ for which the center of the circle $x^2+y^2-6x=0$ is exactly the focus of the parabola $y^2=2px$ ($p>0$). | 6 | math | 50 |
Selecting 3 distinct numbers from 1, 2, 3, 4, 5, what is the probability that the three numbers can form the sides of a triangle? | \frac{3}{10} | math | 38 |
A survey of participants was conducted at the Olympiad. $ 90\%$ of the participants liked the first round, $60\%$ of the participants liked the second round, $90\%$ of the participants liked the opening of the Olympiad. Each participant was known to enjoy at least two of these three events. Determine the percenta... | 40\% | math | 89 |
From a group of 10 members of a school's mathematics competition team, 3 are to be selected to participate in a provincial mathematics competition. There must be at least one of two members, A and B, in the selected group and member C is not to be selected. The number of different selection methods is __________ (answe... | 49 | math | 73 |
A school conducted a survey on the extracurricular physical exercise situation of 1,000 students. Using stratified sampling by gender, a sample of 100 students was selected. It is known that 51 of the sampled students are female. What is the total number of male students in the school? | 490 | math | 67 |
The values of $p$, $q$, $r$ and $s$ are 3, 4, 5 and 6, but not necessarily in that order. What is the largest possible value of the sum of the four products $pq$, $qr$, $rs$, and $ps$? | 80 | math | 63 |
Consider a rectangle with vertices at $(2,0)$, $(7,0)$, $(7,4)$, and $(2,4)$. A line joining the points $(2,1)$ and $(7,3)$ divides this rectangle. What fraction of the area of the rectangle is above this line? | \frac{3}{4} | math | 64 |
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} | math | 108 |
The square \(ABCD\) is to be decomposed into \(n\) nonoverlapping triangles, all of whose angles are acute. Find the smallest integer \(n\) for which there exists a solution to this problem and construct at least one decomposition for this \(n\). Additionally, answer whether it is possible to require that (at least) on... | 8 | math | 84 |
In an exam, there are 4 multiple-choice questions, each with 3 possible answers. A group of students takes the exam, and it is found that for any 3 students, there is at least 1 question for which their answers are all different. What is the maximum number of students that could have taken the exam? | 9 | math | 67 |
Given the sets $P=\{x\in\mathbb{R}|0\leqslant x\leqslant 4\}$ and $Q=\{x\in\mathbb{R}||x| < 3\}$, find the union of the sets $P$ and $Q$. | (-3,4] | math | 68 |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 3.$ Find the minimum value of \[\frac{x + y}{xyz}.\] | \frac{16}{9} | math | 43 |
Simplify and find the value: $\left(\frac{{2x-2}}{x}-1\right) \div \frac{{x^2-4x+4}}{{x^2-x}}$, where $x=4$. | \frac{3}{2} | math | 50 |
Consider the infinite arithmetic sequence $A$ with first term $5$ and common difference $-2$. Now define the infinite sequence $B$ so that the $k^{th}$ term of $B$ is $2$ raised to the $k^{th}$ term of $A$. Find the sum of all of the terms of $B$. | \frac{128}{3} | math | 71 |
Given the parabola $C$: $y^{2}=2px (p > 0)$ with a focus at $F$ and a point $M(x_{0},4)$ on the parabola $C$. A circle with center $M$ and radius $|MF|$ is intercepted by the line $x=-1$ with a chord length of $2 \sqrt {7}$. Determine the value of $|MF|$. | 4 | math | 92 |
Perform the calculations:
72×54+28×54
60×25×8
2790÷(250×12-2910)
(100-1456÷26)×78. | 3432 | math | 61 |
A large rectangular region measures 15 units by 20 units. One fourth of this rectangle is shaded. What fraction of the shaded section is one half of this quarter rectangle? What is the fraction of the entire large rectangle that is shaded?
A) $\frac{1}{12}$
B) $\frac{1}{10}$
C) $\frac{1}{8}$
D) $\frac{1}{6}$
E) $\frac{... | \frac{1}{8} | math | 101 |
Solve for the smaller root of the quadratic equation:
$$ \left(x-\frac{4}{5}\right)\left(x-\frac{4}{5}\right) + \left(x-\frac{4}{5}\right)\left(x-\frac{2}{3}\right) + \frac{1}{15} = 0 $$
A) $\frac{11}{15}$
B) $\frac{4}{5}$
C) $\frac{5}{8}$
D) $1$ | \frac{11}{15} | math | 110 |
A road is 400 meters long. On both sides of the road, a trash can is placed every 20 meters. Trash cans are not placed at the start and end points, as they are marked with signboards. How many trash cans are placed in total? | 38 | math | 57 |
Three fair eight-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value shown on the remaining die?
A) $\frac{267}{512}$
B) $\frac{89}{512}$
C) $\frac{1}{3}$
D) $\frac{89}{216}$
E) $\frac{5}{24}$ | \frac{267}{512} | math | 91 |
Given that a non-zero common difference arithmetic sequence $\{a_n\}$ has a sum of its first 4 terms equal to 10, and $a_2$, $a_3$, and $a_7$ form a geometric sequence,
(I) Find the general term $a_n$ formula.
(II) Let $b_n = 2^{a_n}$, determine the sum of the first $n$ terms of the sequence $\{b_n\}$, $S_n$. | \frac{8^n - 1}{28} | math | 103 |
An equation $x^{-2} = \left(\frac{1}{2}\right)^x$ has how many solutions. | 3 | math | 26 |
Given the parabola $y^{2}=4x$ whose directrix intersects with the hyperbola $\frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1$ ($a > 0,b > 0$) at points $A$ and $B$, and point $F$ is the focus of the parabola. If $\triangle FAB$ is a right-angled triangle, then the range of the eccentricity of the hyperbola is \_\_\_\_\_\_... | (\sqrt {5},+\infty) | math | 117 |
Given a moving point M whose distance to the point (8, 0) is twice its distance to the point (2, 0).
(1) Find the equation of the trajectory C of the moving point M;
(2) If the line $y=kx-5$ has no intersection with trajectory C, find the range of values for $k$;
(3) Given that the circle $x^2+y^2-8x-8y+16=0$ int... | 4\sqrt{2} | math | 119 |
Senya has three straight sticks, each 24 centimeters long. Senya broke one of them into two pieces so that using the two pieces of the broken stick and the two whole sticks, he could form the outline of a right triangle. How many square centimeters is the area of this triangle? | 216 | math | 62 |
In the Cartesian coordinate system $xOy$, the parametric equations of curve $C$ are $\{\begin{array}{l}x=\sqrt{3}\cos2t,\\ y=2\sin t\end{array}(t$ is the parameter). Establish a polar coordinate system with the origin as the pole and the positive x-axis as the polar axis. It is known that the polar equation of the line... | \left[-\frac{19}{12}, \frac{5}{2}\right] | math | 147 |
A number y is given. x is p percent less than y and z is q percent less than y. Express p and q in terms of y given that x is 10 units less than y and z is 20 units less than y. | p = \frac{1000}{y}, q = \frac{2000}{y} | math | 52 |
Find all rational roots of the polynomial equation:
\[3x^3 - 7x^2 - 8x + 4 = 0.\] | \frac{1}{3} | math | 32 |
Given the sequence $\{a_n\}$ with the general term formula $a_n = -n^2 + 12n - 32$, and the sum of the first $n$ terms is $S_n$, determine the maximum value of $S_n - S_m$ when $n > m$. | 10 | math | 65 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. The vectors $\overrightarrow{m}=(\sin B+\sin C,\sin A-\sin B)$ and $\overrightarrow{n}=(\sin B-\sin C,\sin A)$, and it is given that $\overrightarrow{m}\perp \overrightarrow{n}$.
$(I)$ Find the magnitude ... | \frac {4 \sqrt {3}-3}{10} | math | 133 |
What is the degree measure of angle $VXZ$ when polygon $VWXYZABCD$ is a regular octagon?
```
[asy]
draw((-2,0)--(-1.414,1.414)--(0,2)--(1.414,1.414)--(2,0)--(1.414,-1.414)--(0,-2)--(-1.414,-1.414)--cycle);
draw((-1.414,-1.414)--(1.414,1.414)--(0,-2)--cycle);
label("V",(-1.414,-1.414),SW);
label("W",(-2,0),W);
label("X"... | 135^\circ | math | 278 |
Given that $S_n$ is the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$, if $\dfrac{a_5}{a_3} = \dfrac{5}{9}$, determine the value of $\dfrac{S_9}{S_5}$. | 1 | math | 67 |
In the Cartesian coordinate system Oxyz, find the distance between point A, which is symmetrical to the point (1, 2, -3) with respect to the y-axis, and point (-1, -2, -1). | 4\sqrt{2} | math | 49 |
Given that $a$ and $b$ are positive real numbers satisfying $a + b = 2$, find the minimum value of $\frac{1}{a} + \frac{2}{b}$. | \frac{3 + 2\sqrt{2}}{2} | math | 43 |
There are thirty-five red, yellow, orange, and white marbles in a bag. If half the number of red marbles equals two less than the number of yellow marbles, equals a third the number of orange marbles, and equals a third of three more than the number of white marbles, how many red marbles are there? | 8 | math | 70 |
Given that the function $f(x)$ is an even function defined on $\mathbb{R}$, and the odd function $g(x)$ defined on $\mathbb{R}$ passes through the point $(-1, 1)$, and $g(x) = f(x-1)$, find the value of $f(7) + f(8)$. | -1 | math | 76 |
Through the altitude of an equilateral triangle with side length \(a\), a plane is drawn perpendicular to the plane of the triangle, and within this plane a line \(l\) is taken parallel to the altitude of the triangle. Find the volume of the solid obtained by rotating the triangle around the line \(l\). | \frac{\pi a^3 \sqrt{3}}{24} | math | 64 |
Given $(1-2x)^5 = a + a_1x + a_2x^2 + a_3x^3 + a_4x^4 + a_5x^5$, find the value of $a_1 + a_2 + a_3 + a_4 + a_5$. | -2 | math | 69 |
The quadratic $x^2 + 784x + 500$ can be written in the form $(x+b)^2 + c$, where $b$ and $c$ are constants. What is $\frac{c}{b}$? | -391 | math | 53 |
From the set \(\{1, 2, 3, \cdots, 99, 100\}\), a number \(a\) is randomly selected, and then a number \(b\) is randomly selected from the same set. The probability that the last digit of \(3^a + 7^b\) is 8 is \(\qquad\) . | \frac{3}{16} | math | 80 |
A group of artists gathers daily for a collaborative project. When they try to form groups of 5 each, there is one person left over. When they form groups of 6, there are two people left over. Finally, when grouped by 8, they have three people left over. What is the minimum number of artists present? | 236 | math | 68 |
Convert the binary number $10101_{(2)}$ into a quaternary number. The result is \_\_\_\_\_\_. The greatest common divisor of 918 and 714 is \_\_\_\_\_\_. | 111_{(4)}, 102 | math | 54 |
Shift the graph of the function $y=\sin 2x$ to the left by $\varphi$ ($\varphi > 0$) units so that it coincides with the graph of the function $$y=\cos\left(2x- \frac{\pi}{3}\right)$$, then the minimum value of $\varphi$ is \_\_\_\_\_\_. | \frac{\pi}{12} | math | 81 |
Calculate the simplified value of the sum: $-1^{2008} + (-1)^{2009} + 1^{2010} -1^{2011}$. | -2 | math | 45 |
A package of seeds was passed around a table. The first person took 1 seed, the second person took 2 seeds, the third took 3 seeds, and so forth, with each subsequent person taking one more seed than the previous one. It is known that during the second round a total of 100 more seeds were taken than during the first ro... | 10 | math | 84 |
Given that point P(2, 2) is on the curve $y=ax^3+bx$, if the slope of the tangent line at point P is 9, then (i) $ab=\boxed{-3}$;
(ii) the range of the function $f(x)=ax^3+bx$, where $x\in \left[-\frac{3}{2}, 3\right]$, is $\boxed{[-2, 18]}$. | [-2, 18] | math | 99 |
In quadrilateral $PQRS,$ $PQ = 6,$ $QR = 10$, and $RS = 25$ units. Both angle $Q$ and angle $R$ are right angles. Determine the length of segment $PS$. | \sqrt{461} | math | 53 |
An isosceles triangle has two sides measuring 15 cm and one side measuring 24 cm. If a similar triangle has its longest side measuring 72 cm, find the perimeter of this larger triangle. | 162 \text{ cm} | math | 45 |
Find the value of $\cos 160^{\circ}\sin 10^{\circ}-\sin 20^{\circ}\cos 10^{\circ}$. | -\dfrac{1}{2} | math | 40 |
Circles of radius 3 and 4 are externally tangent and are circumscribed by a third circle. Find the area of the shaded region between these circles. Express your answer in terms of \(\pi\). | 24\pi | math | 44 |
Given that the real numbers $x$ and $y$ satisfy $x > y > 0$ and $x + y = 2$, find the minimum value of $$\frac {4}{x+3y}+ \frac {1}{x-y}$$. | \frac {9}{4} | math | 56 |
If income is defined as positive and expenses as negative, what is the value of expenses of $400$ yuan recorded as? | -400 | math | 27 |
Forty slips are placed into a hat, each bearing a number 1 through 8, with each number entered on five slips. Four slips are drawn from the hat at random and without replacement. Let $p'$ be the probability that all four slips bear the same number. Let $q'$ be the probability that two of the slips bear a number $a$ and... | 70 | math | 98 |
When $\frac{1}{1001}$ is expressed as a decimal, what is the sum of the first 50 digits after the decimal point? | 216 | math | 33 |
Determine all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for all real numbers \( x \) and \( y \),
\[ f(f(x) + y) = f(x^2 - y) + 4y f(x) \] | f(x) = 0 \text{ or } f(x) = x^2 | math | 63 |
Determine the first term in the geometric sequence $a, b, c, d, e, 32, 64$. | 1 | math | 28 |
The Sharks initially won 2 out of 3 games, and later played $N$ more games, with a total of $3 + N$ games played. The winning percentage of the Sharks is now no less than 90%, so $\frac{2 + k}{3 + N} \ge 0.9$, where $k$ is the number of the additional games won by the Sharks. | \textbf{1} | math | 83 |
At 3:20 o'clock, calculate the angle formed by the hour and minute hands of a clock. | 20 | math | 23 |
Given curve $C\_1$: $\begin{cases}x= & -4+\cos t \
y= & 3+\sin t\end{cases}$ ($t$ is the parameter), and curve $C\_2$: $\dfrac {x^{2}}{64}+ \dfrac {y^{2}}{9}=1$.
(1) Convert $C\_1$ into a general equation, $C\_2$ into a parametric equation, and explain what kind of curves they represent.
(2) If the parameter correspond... | d_{min}= \dfrac {8\sqrt{5}}{5} | math | 187 |
We have a standard deck of 52 cards, divided equally into 4 suits of 13 cards each. Determine the probability that a randomly chosen 5-card poker hand is a flush (all cards of the same suit). | \frac{33}{16660} | math | 47 |
In a basketball game, a certain team played a total of 8 games and scored 29, 30, 38, 25, 37, 40, 42, 32 points respectively. What is the 75th percentile of this data set? | 39 | math | 64 |
What is the least possible value of $(xy-1)^2+(x+y)^2$ for real numbers $x$ and $y$? | 1 | math | 30 |
The boys and girls must sit alternately, and there are 3 boys. The number of such arrangements is the product of the number of ways to choose 3 positions out of a total of 7, and the number of ways to arrange the girls for the remaining spots. | 144 | math | 56 |
Investigate the function \( y = x^{3} - 3x^{2} - 9x + 11 \) for maximum, minimum, and inflection point. | (-1, 16), \ (3, -16), \ (1, 0) | math | 38 |
Suppose $a<c$ and $b<0$. Which of the following must be true?
$ab < ac$
$a+b<c+b$
$a-b < c-b$
$c/a > 1$
Enter your answer as a list of those options that are always true. For instance, if you think only the first and third are true, enter 'A, C'. | B, C | math | 76 |
Let $f(x)$ be a function defined on $\mathbb{R}$ such that for any real number $x$, it always satisfies $f(x+2)=f(-x)$, $f(x)=-f(4-x)$, and when $x\in[0,2]$, $f(x)=2x-x^2$.
(1) Find the analytic expression of $f(x)$ when $x\in[2,4]$;
(2) Calculate the sum $f(0)+f(1)+f(2)+\cdots+f(2019)$. | 0 | math | 126 |
Given the parametric equations of curve $C_1$: $$\begin{cases} x=\cos\theta \\ y=1+\sin\theta \end{cases}$$ (where $\theta$ is the parameter), and establishing a polar coordinate system with the origin as the pole and the positive half-axis of the x-axis as the polar axis, the polar equation of curve $C_2$ is: $\rho=4\... | 2 | math | 200 |
Given the proposition $p: \frac{1}{2} \leq x \leq 1$, and the proposition $q: (x-a)(x-a-1) \leq 0$, if $\neg p$ is a necessary but not sufficient condition for $\neg q$, then the range of the real number $a$ is \_\_\_\_\_\_. | [0, \frac{1}{2}] | math | 79 |
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 | math | 122 |
If $|a|=2$, $b^{2}=9$, and $a \lt b$, find the value of $a-b$. | -1 \text{ or } -5 | math | 29 |
Given $sin(\alpha+\frac{\pi}{6})=\frac{\sqrt{3}}{3}$, calculate the value of $sin(2\alpha-\frac{\pi}{6})$. | -\frac{1}{3} | math | 41 |
What is the maximal dimension of a linear subspace $ V$ of the vector space of real $ n \times n$ matrices such that for all $ A$ in $ B$ in $ V$ , we have $ \text{trace}\left(AB\right) \equal{} 0$ ? | \frac{n(n-1)}{2} | math | 74 |
In a right prism with triangular bases, given that the sum of the areas of three mutually adjacent faces (two lateral faces and one base) is 30, find the maximum volume of the prism. | 10\sqrt{5} | math | 41 |
There are several (more than three) balls in a box. Each is painted in some color. If you take any three balls out of the box, among them there will definitely be at least one red ball and at least one blue ball. How many balls can be in the box? | 4 | math | 58 |
Evaluate the expression $\frac{(0.5)^{3.5}}{(0.05)^3}$.
A) $158.11\sqrt{5}$
B) $316.23\sqrt{5}$
C) $632.46\sqrt{5}$
D) $948.69\sqrt{5}$
E) $500\sqrt{5}$ | 316.23\sqrt{5} | math | 95 |
Given that $\vec{a}$ and $\vec{b}$ are unit vectors, and the angle between them is $60^{\circ}$, calculate the dot product of the vectors $(2 \vec{a}- \vec{b})$ and $\vec{b}$. | 0 | math | 58 |
1. Find the equation of the hyperbola that shares a common focus and has an eccentricity of $\frac{\sqrt{5}}{2}$ with the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$.
2. A line $l$ with a slope of 1 passes through the right focus of the ellipse $\frac{x^2}{4} + y^2 = 1$ and intersects the ellipse at points A and B. Fin... | \frac{8}{5} | math | 111 |
Given that the sequence $\{a_n\}$ is a geometric sequence, where $a_5$ and $a_9$ are the two roots of the equation $x^2+2016x+9=0$, calculate the value of $a_7$. | -3 | math | 58 |
Given a triangle \( \triangle ABC \) with area \( S \) and \(\angle C = \gamma \), find the minimum value of the side \( C \) opposite to \(\angle C\). | 2 \sqrt{S \tan\left(\frac{\gamma}{2}\right)} | math | 44 |
If the graph of the function $f(x)$ passes through the point $(0, 1)$, then the graph of the inverse function of $f(x+3)$ must pass through the point ______. | (1, -3) | math | 42 |
Given that in $\triangle ABC$, $\cos A= \frac{ \sqrt {6}}{3}$, and $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively.
(1) Find $\tan 2A$;
(2) If $\sin ( \frac {π}{2}+B)= \frac {2 \sqrt {2}}{3}, c=2 \sqrt {2}$, find the area of $\triangle ABC$. | \frac {2 \sqrt {2}}{3} | math | 107 |
A painting measures 20 inches by 30 inches, with the longer side placed vertically. The frame width at the top and bottom is three times the width of the wood on the sides. If the total area of the frame is the same as the area of the painting itself, find the ratio of the shorter to the longer dimension of the framed ... | 1:2 \text{ (B)} | math | 107 |
What is \[3 + 5x - 7x^2 - 9 + 11x - 13x^2 + 15 - 17x + 19x^2\] in terms of $x$? | 9 - x - x^2 | math | 55 |
For every integer $n \ge 2$ let $B_n$ denote the set of all binary $n$ -nuples of zeroes and ones, and split $B_n$ into equivalence classes by letting two $n$ -nuples be equivalent if one is obtained from the another by a cyclic permutation.(for example 110, 011 and 101 are equivalent). Determine the integers ... | n = 2 | math | 123 |
Find the derivative.
\[
y = \sqrt[3]{\operatorname{ctg} 2} - \frac{1}{20} \cdot \frac{\cos^{2} 10x}{\sin 20x}
\] | \frac{1}{4 \sin^{2}(10x)} | math | 56 |
Given points $A(x,5-x,2x-1)$ and $B(1,x+2,2-x)$, the minimum value of $|AB|$ is ______. | \frac { \sqrt {35}}{7} | math | 38 |
If $\log_{10}2=a$ and $\log_{10}3=b$, then $\log_{5}12=?$ | \frac{2a+b}{1-a} | math | 30 |
How many of the following propositions are true: "If $|m| > |n|$, then $m^2 > n^2$" and its converse, inverse, and contrapositive? | 4 | math | 41 |
The average age of 8 people in Room A is 38, and the average age of 2 people in Room B is 30. Find the average age of all the people when the two groups are combined. | 36.4 | math | 46 |
Given constants $a$ and $b$, the function $f(x) = ax^3 + b\ln(x + \sqrt{1+x^2}) + 3$ has a maximum value of 10 on the interval $(-\infty, 0)$. Find the minimum value of $f(x)$ on the interval $(0, +\infty)$. | -4 | math | 79 |
There are $100$ white points on a circle. Asya and Borya play the following game: they alternate, starting with Asya, coloring a white point in green or blue. Asya wants to obtain as much as possible pairs of adjacent points of distinct colors, while Borya wants these pairs to be as less as possible. What is the maxi... | 50 | math | 97 |
The sum of the original set of $60$ numbers is $45 \cdot 60 = 2700$. If three numbers, $40$, $50$, and $60$, are discarded, the sum of the remaining set of numbers is $2700 - 150 = 2550$. Given that there are $57$ numbers remaining, calculate the arithmetic mean of the remaining set. | 44.74 | math | 94 |
Given an angle $\theta$ whose terminal side contains a point $(a, a)$, where $a \in \mathbb{R}$ and $a \neq 0$, find the value of $\sin \theta$. | \frac{\sqrt{2}}{2} \text{ or } -\frac{\sqrt{2}}{2} | math | 47 |
Given an arithmetic sequence $\{a_n\}$, where $a_2 + a_6 = 14$, and $S_n$ is the sum of the first $n$ terms, with $S_5 = 25$.
$(1)$ Find the general formula for $\{a_n\}$.
$(2)$ Let $b_n = \frac{2}{a_n \cdot a_{n+1}}$, find the minimum value of the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n... | \frac{2}{3} | math | 122 |
Definition: If $x_{1}$ and $x_{2}$ are two integer roots of the equation $ax^{2}+bx+c=0\left(a\neq 0\right)$, and satisfy $|x_{1}-x_{2}|=1$, then this type of equation is called a "natural equation". For example, $\left(x-2\right)\left(x-3\right)=0$ is a "natural equation".<br/>$(1)$ The following equations are "natura... | m=2 \text{ or } 0 | math | 206 |
What is the minimum number of equilateral triangles needed to cover an equilateral triangle of side length 12 units, using triangles of side lengths 1 unit and 2 units? | 36 | math | 37 |
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