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 | One hundred students, each wearing a shirt with a number, stood in a circle in the following sequence: \(1, 2, 3, \ldots, 100\). Upon a command, they started leaving in this manner: the student with the number 1 stays, the student with number 2 leaves, the student with number 3 stays, the student with number 4 leaves, ... | 73 | 113 | 2 |
math | Find the number of 10-tuples $(x_1, x_2, \dots, x_{10})$ of real numbers such that
\[(1 - x_1)^2 + (x_1 - x_2)^2 + (x_2 - x_3)^2 + \dots + (x_9 - x_{10})^2 + x_{10}^2 = \frac{5}{22}.\] | 2 | 99 | 1 |
math | If the line $ax + 2y + 6 = 0$ is parallel to but does not coincide with the line $x + (a-1)y + (a^2 - 1) = 0$, then $a$ equals | -1 | 52 | 2 |
math | Let \( p = 2^{24036583} - 1 \), the largest prime currently known. For how many positive integers \( c \) do the quadratics \( \pm x^2 \pm p x \pm c \) all have rational roots? | 0 | 60 | 1 |
math | Triangle $\triangle DEF$ has a right angle at $F$, $\angle D = 45^\circ$, and $DF=12$. Find the radius of the incircle of $\triangle DEF$. | 6 - 3\sqrt{2} | 42 | 9 |
math | A large candle is 150 centimeters tall. It burns at a variable rate, taking 15 seconds to burn the first centimeter, 30 seconds for the second centimeter, and generally, \( 15k \) seconds to burn the \( k \)-th centimeter. Compute the height of the candle in centimeters \( \frac{T}{2} \) seconds after it is lit, where ... | 44 | 103 | 2 |
math | Find the smallest natural number that starts with the digit 4 and decreases four times by moving this digit to the end of the number. | 410256 | 27 | 6 |
math | The number of liters of tea a writer drinks on any given day is inversely proportional to the number of hours she spends writing. On Sunday, she spent 8 hours writing and consumed 3 liters of tea. On Wednesday, she spent 4 hours writing. How many liters of tea did she drink on Wednesday? And if she drinks only 2 liters... | 12 | 85 | 2 |
math | Xiao Zhao, Xiao Qian, Xiao Sun, and Xiao Li, four students, signed up to visit Longhushan, Sanqingshan, Jinggangshan, and Lushan. Each person only visited one of the four scenic spots. Let event $A$ be "the four people visited different scenic spots," and event $B$ be "only Xiao Zhao visited Longhushan." Find $P\left(A... | \frac{2}{9} | 97 | 7 |
math | Observe the following series of monomials: $-2x$, $4x^{3}$, $-8x^{5}$, $16x^{7}$, $\ldots$ According to the pattern you discovered, the $n$th monomial is ____. | (-1)^n \cdot 2^n \cdot x^{2n-1} | 59 | 18 |
math | In $\triangle ABC$, $\tan A + \tan B + \sqrt{3} = \sqrt{3} \tan A \tan B$. Find the value of $A + B$. | \frac{2\pi}{3} | 40 | 9 |
math | Jun Jun is looking at an incorrect one-digit multiplication equation \( A \times B = \overline{\mathrm{CD}} \) (where the digits \( A \), \( B \), \( C \), and \( D \) are all different). Jun Jun smartly discovers that if only one digit is changed, there are 3 ways to correct the equation. Additionally, if the order of... | 17 | 123 | 2 |
math | A group of five couples needs to be divided into two groups. The first group should consist of six people, including at least two couples. In how many ways can this division be accomplished? | 130 | 38 | 3 |
math | A two-digit prime number, which remains a prime number after swapping its tens and units digits, is called a "flawless prime". The sum of all "flawless primes" is ____. | 429 | 41 | 3 |
math | Given the function $f(x)= \frac {m+\ln x}{x}$, where $m\in\mathbb{R}$, and $x > 1$.
(Ⅰ) Discuss the monotonic intervals of $f(x)$.
(Ⅱ) If $m=4$, $k\in\mathbb{N}^*$, and $\frac {k}{x+1} < f(x)$ always holds. Find the maximum value of $k$. | 6 | 102 | 1 |
math | Sally has 12 blue marbles and 8 red marbles in a bag. She removes a marble at random, records the color, puts it back, and then repeats this process until she has withdrawn 8 marbles. What is the probability that exactly five of the marbles that she removes are blue? Express your answer as a decimal rounded to the near... | 0.279 | 78 | 5 |
math | In the cube $ABCD-A_{1}B_{1}C_{1}D_{1}$, what is the measure of the dihedral angle $A-BD_{1}-A_{1}$? | 60^\circ | 45 | 4 |
math | $\sin 20^{\circ}\cos 40^{\circ}+\cos 20^{\circ}\sin 40^{\circ}=$ | \frac{\sqrt{3}}{2} | 35 | 10 |
math |
Given the equation
\[ x^{2} + p x + q = 0 \]
with roots \( x_{1} \) and \( x_{2} \), determine the coefficients \( r \) and \( s \) as functions of \( p \) and \( q \) such that the equation
\[ y^{2} + r y + s = 0 \]
has roots \( y_{1} \) and \( y_{2} \) which are related to the roots \( x_{1} \) and \( x_{2} \) b... | p + q = 0 | 197 | 6 |
math | Given the function $f$ that satisfies $f(x+5)+f(x-5) = f(x)$ for all real $x$, determine the smallest positive period $p$ for these functions. | 30 | 41 | 2 |
math | 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 | 278 | 5 |
math | If the algebraic expression $\frac{\sqrt{x+4}}{x-2}$ is meaningful within the real number range, then the range of values for $x$ is ______. | x \geqslant -4 \text{ and } x \neq 2 | 38 | 19 |
math | Given a list of $2018$ positive integers with a unique mode that occurs exactly $10$ times, calculate the least number of distinct values that can occur in the list. | 225 | 39 | 3 |
math | Given that $\frac{1}{4}$ of all the ninth graders are paired with $\frac{1}{3}$ of all the sixth graders, what fraction of the total number of sixth and ninth graders have a buddy? | \frac{1}{7} | 48 | 7 |
math | If the tangent line at point P of the function $f(x) = 2\sin x$ ($x \in [0, \pi]$) is parallel to the tangent line at point Q of the function $g(x) = 2\sqrt{x} \left( \frac{x}{3} + 1 \right)$, calculate the slope of line PQ. | \frac{8}{3} | 78 | 7 |
math | Determine the increasing interval of the function $y= \sqrt{3}\cos \left( \frac{1}{2}x+ \frac{2}{3}\pi \right)$, where $x \in [0,2\pi]$. | \left[\frac{2\pi}{3}, 2\pi\right] | 54 | 18 |
math | What is \(5^{-1} + 5^{-2} \pmod{31}\)? Express your answer as an integer from 0 to 30, inclusive. | 19 | 37 | 2 |
math | The average of a list of three consecutive odd integers is 7. When a fourth positive integer, \( m \) , different from the first three, is included in the list, the average of the list is an integer. What is the sum of the three smallest possible values of \( m \)? | 29 | 61 | 2 |
math | Suppose the roots of $x^3 + 5x^2 + 6x - 7 = 0$ are $p$, $q$, and $r$. If the roots of $x^3 + ux^2 + vx + w = 0$ are $p+q$, $q+r$, and $r+p$. Find the value of $w$. | 37 | 79 | 2 |
math | Determine the number of solutions to the equation \(\sin \left( \frac{\pi}{4} \sin x\right) = \cos \left( \frac{\pi}{4} \cos x\right)\) in the closed interval \([0, 2\pi]\). | 4 | 62 | 1 |
math | Kira has $3$ blocks with the letter $A$ , $3$ blocks with the letter $B$ , and $3$ blocks with the letter $C$ . She puts these $9$ blocks in a sequence. She wants to have as many distinct distances between blocks with the same letter as possible. For example, in the sequence $ABCAABCBC$ the blocks with the... | 7 | 226 | 1 |
math | Given that the sum of two numbers is 15 and their product is 36, find the sum of their reciprocals and the sum of their squares. | 153 | 34 | 3 |
math | A pizza with 10 equally-sized slices costs 10 dollars. Adding anchovies to one-third of the pizza costs an additional 3 dollars. Dave ate all the slices with anchovies and two plain slices. Doug ate the remainder. Each paid for what he had eaten. Find the difference, in dollars, between the amount Dave paid and the amo... | 8 - 5 = 3 \text{ dollars} | 75 | 12 |
math | Find the number of pairs (m, n) of integers such that the equation $m^3 + 3m^2 + 2m = 8n^3 + 12n^2 + 6n + 1$ holds. | 0 | 53 | 1 |
math | Find the real solution \( x, y, z \) to the equations \( x + y + z = 5 \) and \( xy + yz + zx = 3 \) such that \( z \) is the largest possible value. | \frac{13}{3} | 51 | 8 |
math | Each row of a $24 \times 8$ table contains some permutation of the numbers $1, 2, \cdots , 8.$ In each column the numbers are multiplied. What is the minimum possible sum of all the products?
*(C. Wu)* | 8 * (8!)^3 | 60 | 7 |
math | The function \( f(x) \) is such that for all \( x \), the equation \( f(x+1) = f(x) + 2x + 3 \) holds. It is known that \( f(0) = 1 \). Find \( f(2012) \). | 4052169 | 65 | 7 |
math | Let \( p, q, r, \) and \( s \) be positive real numbers such that
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
p^2+q^2&=&r^2+s^2&=&2512, \\
pr&=&qs&=&1225.
\end{array}
\]
If \( T = p+q+r+s \), compute the value of \( \lfloor T \rfloor \). | 140 | 133 | 3 |
math | There are approximately 0.9072 kilograms in 2 pounds. To the nearest whole pound, how many pounds does an object that weighs 350 kg weigh? | 386\text{ pounds} | 37 | 8 |
math | Given a random variable $X\sim N(2, \sigma ^{2})$, $P(X\leqslant 0)=0.15$, calculate $P(2\leqslant X\leqslant 4)$. | 0.35 | 55 | 4 |
math | In a Cartesian coordinate system, the parametric equation of line $l$ is $\begin{cases} x=t \\ y= \sqrt{3}t+2\end{cases}$ (where $t$ is the parameter), and the parametric equation of circle $C$ is $\begin{cases} x=a\cos \theta \\ y=a\sin \theta \end{cases}$ (where $a > 0$ and $\theta$ is the parameter). Let point $P$ b... | 1 | 136 | 1 |
math | Given two angles $\alpha$ and $\beta$ in space, and the two sides of $\alpha$ are parallel to the two sides of $\beta$. If $\alpha = 30^{\circ}$, then the size of $\beta$ is ______. | 30^{\circ} \text{ or } 150^{\circ} | 53 | 19 |
math | Given that the number $2,310,000$ can be expressed in scientific notation, express it as a product of a number between 1 and 10 and a power of 10. | 2.31\times 10^{6} | 45 | 12 |
math | The function $f(x)=ax^{2}-2020x+2021 (a>0)$, the maximum value of the function $f(x)$ on the interval $\left[t-1,t+1\right] (t\in R)$ is $M$, and the minimum value is $N$. When $t$ takes any real number, the minimum value of $M-N$ is $2$. Find the value of $a$. | a=2 | 96 | 3 |
math | In the diagram, the circle has radius $\sqrt 7$ and and centre $O.$ Points $A, B$ and $C$ are on the circle. If $\angle BOC=120^\circ$ and $AC = AB + 1,$ determine the length of $AB.$ [asy]
import graph; size(120); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen ... | 4 | 485 | 1 |
math | The parabola $y = ax^2 + bx + c$ has a vertex at $(q,q)$ and a $y$-intercept at $(0, -2q)$, where $q \neq 0$. What is the value of $b$? | \frac{6}{q} | 58 | 7 |
math | Find a necessary and sufficient condition on the positive integer $n$ that the equation \[x^n + (2 + x)^n + (2 - x)^n = 0\] have a rational root. | n = 1 | 46 | 4 |
math | A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form $a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6},$ where $a^{}_{}$, $b^{}_{}$, $c^{}_{}$, and $d^{}_{}$ are positive integers. Find $a + b + c + d^{}_{}$. | 720 | 110 | 3 |
math | Find all positive integers $n>1$ for which $\frac{n^{2}+7 n+136}{n-1}$ is the square of a positive integer. | 5, 37 | 37 | 5 |
math | Let O be the coordinate origin, $F_{1}$ and $F_{2}$ be the left and right foci of the ellipse $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1\left(a \gt b \gt 0\right)$. If there exists a point P on the ellipse such that $\angle F_{1}PF_{2}=\dfrac{\pi }{3}$, and $|OP|=\dfrac{\sqrt {3}}{2}a$, determine the eccentricity of ... | \dfrac{1}{2} | 126 | 7 |
math | You are given $n \ge 2$ distinct positive integers. Let's call a pair of these integers *elegant* if their sum is an integer power of $2$ . For every $n$ find the largest possible number of elegant pairs.
*Proposed by Oleksiy Masalitin* | n-1 | 68 | 4 |
math | A table can seat 6 people. Two tables joined together can seat 10 people. Three tables joined together can seat 14 people. Following this pattern, if 10 tables are arranged in two rows with 5 tables in each row, how many people can sit? | 44 | 58 | 2 |
math | Given a student observed during an n-day holiday, it rained 7 times, either in the morning or afternoon, when it rained in the afternoon the morning was sunny, there were a total of 5 sunny afternoons, and there were a total of 6 sunny mornings, determine the minimum value of n. | 9 | 66 | 1 |
math | Find the derivative of the function.
\[ y = \ln \left(4x - 1 + \sqrt{16x^2 - 8x + 2}\right) - \sqrt{16x^2 - 8x + 2} \cdot \arctan(4x - 1) \] | y' = \frac{4(1-4 x)}{\sqrt{16 x^{2} - 8 x + 2}} \cdot \arctan(4 x - 1) | 71 | 43 |
math | 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 | 58 | 1 |
math | Given matrices $A= \begin{pmatrix} 1 & 1 \\ 2 & 3\end{pmatrix}$, $B= \begin{pmatrix} 1 & 2 \\ 2 & 3\end{pmatrix}$.
(I) Find the inverse matrix $A^{-1}$ of matrix A;
(II) Find the equation of the curve obtained by the linear transformation corresponding to $A^{-1}B$ acting on the line $x+y-1=0$. | x-2y-1=0 | 108 | 8 |
math | In triangle \(ABC\), \(\angle BAC\) is \(120^\circ\). The length of \(AB\) is 123. The point \(M\) is the midpoint of side \(BC\). The line segments \(AB\) and \(AM\) are perpendicular. What is the length of side \(AC\)? | 246 | 70 | 3 |
math | Given the function $f(x) = \lg(-x^2 + 4x + 5)$, find the interval where the function is monotonically decreasing and the maximum value of the function within its domain. | \lg 9 | 45 | 4 |
math | Let \(\mathbf{p}, \mathbf{q},\) and \(\mathbf{r}\) be unit vectors such that
\[
\mathbf{p} + 2\mathbf{q} + \mathbf{r} = \mathbf{0}.
\]
Find the angle between \(\mathbf{p}\) and \(\mathbf{r}\), in degrees. | 0^\circ | 87 | 3 |
math | A certain school's second-year junior high students from classes A and B participated in the Hua Cup competition, totaling $a$ students. The average score of class A was 71 points per student, and the average score of class B was 69 points per student. The total score of the two classes was 3480 points. How many studen... | 50 | 85 | 2 |
math | In recent years, the vegetation in the Yangtze and Yellow River basins in our country has been damaged, leading to land desertification and frequent floods. In response to and in support of the "Protect the Mother River" initiative, a protective forest measuring 100km in length and 0.5km in width was established in a c... | 6482100 | 228 | 7 |
math | Let $p(x)$ be a monic, quartic polynomial, such that $p(1) = 3,$ $p(3) = 11,$ and $p(5) = 27.$ Find
\[p(-2) + 7p(6).\] | 1112 | 62 | 4 |
math | It is known that in the trapezoid \(ABCD\), where the diagonal \(BD\) forms an angle of \(45^\circ\) with the base, it is possible to inscribe a circle and circumscribe a circle around it. Find the ratio of the area of the trapezoid to the area of the circle inscribed in it. Find the ratio of the area of the trapezoid ... | \frac{2}{\pi} | 99 | 8 |
math | Ms. Alice Swift needs to drive to her office which is a certain distance away. If she drives at 50 miles per hour, she is 5 minutes late. If she travels at 70 miles per hour, she is 5 minutes early. Find the speed at which Ms. Swift must travel to arrive exactly on time. | \frac{350}{6} | 69 | 9 |
math | A projection takes $\begin{pmatrix} 2 \\ -1 \end{pmatrix}$ to $\begin{pmatrix} 1 \\ -\frac{1}{2} \end{pmatrix}.$ Which vector does the projection take $\begin{pmatrix} -3 \\ 2 \end{pmatrix}$ to? | \begin{pmatrix} -\frac{16}{5} \\ \frac{8}{5} \end{pmatrix} | 70 | 29 |
math | What is the 121st digit to the right of the decimal point in the decimal representation of $\frac{17}{110}$? | 5 | 32 | 1 |
math | Given the right focus F of the ellipse C: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a>b>0) and a point N(0, \sqrt{2}b), let M be a moving point on C. If the maximum perimeter of \triangle MNF is (\sqrt{6}+2)a, find the eccentricity of C. | \frac{\sqrt{2}}{2} | 92 | 10 |
math | Let $\mathbf{w}$ be a vector such that
\[\left\| \mathbf{w} + \begin{pmatrix} 4 \\ -2 \end{pmatrix} \right\| = 10.\]
Find the smallest possible value of $\|\mathbf{w}\|.$ | 10 - 2\sqrt{5} | 67 | 10 |
math | Given the areas of three squares in the diagram, find the area of the triangle formed. The triangle shares one side with each of two squares and the hypotenuse with the third square.
[asy]
/* Modified AMC8-like Problem */
draw((0,0)--(10,0)--(10,10)--cycle);
draw((10,0)--(20,0)--(20,10)--(10,10));
draw((0,0)--(0,-10)--... | 50 | 216 | 2 |
math | Write 1 as a sum of 5 distinct unit fractions. | \frac{1}{2} + \frac{1}{3} + \frac{1}{7} + \frac{1}{43} + \frac{1}{43 \cdot 42} | 13 | 46 |
math | Given a geometric sequence {$a_n$} where the sum of the first n terms is $S_n = (x^2 + 3x)2^n - x + 1$, find the value of $a_3$. | -8 | 48 | 2 |
math | In the plane rectangular coordinate system $xOy$, the parametric equations of curve $C_{1}$ are $\left\{{\begin{array}{l}{x=t+\frac{8}{t},}\\{y=t-\frac{8}{t}}\end{array}}\right.$ (where $t$ is the parameter), and point $P(4,0)$. Establish a polar coordinate system with $O$ as the pole and the positive $x$-axis as the p... | 5 | 225 | 1 |
math | Given that $O$ is the coordinate origin, $F$ is the left focus of the ellipse $(C): \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$, $A$ and $B$ are the left and right vertices of $(C)$, respectively. $P$ is a point on $(C)$, and $PF$ is perpendicular to the $x$-axis. The line $l$ passing through point $A$ inte... | \frac{1}{3} | 161 | 7 |
math | Given that the line $l_{1}$: $2x+\left(a-1\right)y+a=0$ is parallel to the line $l_{2}$: $ax+y+2=0$, determine the value of $a$. | -1 | 51 | 2 |
math | For an upcoming holiday weekend, the weather forecaster predicts that there will be a $30\%$ probability of rain on Friday, a $60\%$ probability of rain on Saturday, and a $40\%$ probability of rain on Sunday. Assuming that the probabilities are independent for each day, what is the probability that it will rain on at ... | 83.2\% | 89 | 6 |
math | A shop specializing in small refrigerators sells an average of 50 refrigerators per month. The cost of each refrigerator is $1200. Since shipping is included in the sale, the shop needs to pay a $20 shipping fee for each refrigerator. In addition, the shop has to pay a "storefront fee" of $10,000 to the JD website each... | 1824 | 122 | 4 |
math | Given the set of numbers $\{-8,-6,-4,0,3,5,7\}$, find the minimum possible product of three different numbers from this set. | -280 | 36 | 4 |
math | Let (b_1, b_2, ... b_7) be a list of the first 7 odd positive integers such that for each 2 ≤ i ≤ 7, either b_i + 2 or b_i - 2 (or both) must appear before b_i in the list. How many such lists are there? | 64 | 70 | 2 |
math | In the example where a common measure fits into one segment 15 times and into another segment 12 times, can a larger segment, which will also be a common measure, be found? | 3 | 40 | 1 |
math | The function $f(x)$ is a differentiable function defined on the interval $(0,+\infty)$, with its derivative being $f'(x)$. It satisfies $f'(x) + \frac{2}{x}f(x) > 0$. The solution set of the inequality $\frac{(x+1)f(x+1)}{4} < \frac{f(2)}{x+1}$ is ______. | (-1,1) | 91 | 5 |
math | Given $|x+1|=3$, $|y|=5$, and $-\frac{y}{x} \gt 0$, calculate the value of $y-x$. | -7 \text{ or }9 | 37 | 8 |
math | The slope angle of the tangent line to the function $y= \frac {1}{3}x^{3}-x^{2}+5$ at $x=1$ is what. | \frac{3\pi}{4} | 40 | 9 |
math | Andrew the ant starts at vertex $A$ of square $ABCD$ . Each time he moves, he chooses the clockwise vertex with probability $\frac{2}{3}$ and the counter-clockwise vertex with probability $\frac{1}{3}$ . What is the probability that he ends up on vertex $A$ after $6$ moves?
*2015 CCA Math Bonanza Lightning R... | \frac{160}{729} | 97 | 11 |
math | In trapezoid $ABCD$, the parallel sides $AB$ and $CD$ have lengths of 10 and 24 units, respectively, and the altitude is 15 units. Points $E$ and $F$ are the midpoints of sides $AD$ and $BC$, respectively. Diagonals $AC$ and $BD$ intersect at point $G$, which divides each diagonal in the ratio 1:2. Determine the area o... | 153.75 | 103 | 6 |
math | There are five students taking a graduation photo in a row, among them, student A cannot stand next to student B, and students B and C must stand together. The number of different arrangements is \_\_\_\_\_\_ (answer in digits). | 36 | 51 | 2 |
math | If the lines $ax - y + 2 = 0$ and $3x - y - b = 0$ are symmetric about the line $y = x$, determine the values of $a$ and $b$. | \frac{1}{3}, 6 | 47 | 9 |
math | Let $k \ge 2$ be an integer and let $T = \{2, 4, 8, \ldots, 2^k\}$. Find the smallest value of $k$ such that for every partition of $T$ into two subsets, at least one of the subsets contains integers $a$, $b$, and $c$ (not necessarily distinct) such that $ab = c$. | k = 5 | 88 | 4 |
math | Mary told John her score on the American High School Mathematics Examination (AHSME), which was over 80. From this, John was able to determine the number of problems Mary solved correctly. If Mary's score had been any lower, but still over 80, John could not have determined this. What was Mary's score? (Recall that... | 119 | 154 | 3 |
math | The degree of any term of the polynomial is less than or equal to 6. | 6 | 17 | 1 |
math | Given a natural number $\overline{62 \alpha \beta 427}$ which is a multiple of 99, find the values of $\alpha$ and $\beta$. | \alpha = 2, \beta = 4 | 39 | 11 |
math | The rules for a race require that all runners start at $A$, touch any part of the 1500-meter wall, and stop at $B$. What is the number of meters in the minimum distance a participant must run? Express your answer to the nearest meter. Assume the distances from A to the nearest point on the wall is 400 meters, and from ... | 1803 | 92 | 4 |
math | If the least common multiple of two 7-digit integers has 13 digits, then their greatest common divisor has at most how many digits? | 2 | 29 | 1 |
math | Given that a number $a$ is randomly selected in the interval $\left[-3,3\right]$, determine the probability that the equation $x^{2}=-a-3x$ has at least one positive root for $x$. | \frac{1}{2} | 50 | 7 |
math | Let $x,y,z \in \mathbb{R}$, and $5^x = 9^y = 225^z$. Express $\frac{1}{z}$ in terms of $\frac{1}{x}$ and $\frac{1}{y}$. | \dfrac{1}{z} = \dfrac{2}{x} + \dfrac{1}{y} | 59 | 25 |
math | Given \\(a \in \mathbb{R}\\), the equation \\(a^{2}x^{2}+(a+2)y^{2}+4x+8y+5a=0\\) represents a circle. The coordinates of the center of the circle are \_\_\_\_\_\_, and the radius is \_\_\_\_\_\_. | 5 | 77 | 1 |
math | A model of a water tower is made at a scale such that a 3-foot tall water tower is represented by a model of height 6 inches. If the real water tower is 90 feet tall, find out how many cubic feet of the water tower does one cubic inch of the model represent? | 3375 | 62 | 4 |
math | Given the function $f\left(x\right)=(2m^{2}-m)x^{2m+3}$ is a power function, and the graph of the function $f\left(x\right)$ is symmetric about the $y$-axis.
$(1)$ Find the value of the real number $m$;
$(2)$ If the inequality $\left(a-1\right)^{m} \lt \left(2a-3\right)^{m}$ holds, find the range of real numbers fo... | a \in \left(\frac{3}{2}, 2\right) | 113 | 17 |
math | Given that $\frac{1 + \sin \alpha}{\cos \alpha} = -\frac{1}{2}$, find the value of $\frac{\cos \alpha}{\sin \alpha - 1}$. | \frac{1}{2} | 47 | 7 |
math | After the implementation of the "double reduction" policy, schools have attached importance to extended services and increased the intensity of sports activities in these services. A sports equipment store seized the opportunity and planned to purchase 300 sets of table tennis rackets and badminton rackets for sale. Th... | 150 | 394 | 3 |
math | Given the ellipse $C: \frac{x^2}{16} + \frac{y^2}{12} = 1$ with foci $F_1$ and $F_2$, find the largest angle in $\triangle F_1MF_2$ for a point $M$ that lies on $C$ and satisfies $|MF_1| - |MF_2| = 2$. | 90^{\circ} | 88 | 6 |
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