problem stringlengths 14 4.09k | solution stringlengths 1 802k | task_type stringclasses 3
values | problem_tokens int64 5 895 |
|---|---|---|---|
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be $2$ : $1$ ? | 4 | math | 47 |
1. \(\lim _{x \rightarrow 1}(1-x) \operatorname{tg} \frac{\pi x}{2}\)
2. \(\lim _{x \rightarrow \frac{\pi}{4}}\left(\frac{\pi}{4}-x\right) \operatorname{cosec}\left(\frac{3}{4} \pi+x\right)\)
3. \(\lim _{x \rightarrow+\infty} x \operatorname{arcctg} x\)
4. \(\lim _{x \rightarrow-\infty} x\left(\frac{\pi}{2}+\operatorna... | -1 | math | 147 |
Given that $f(x)$ is an even function defined on $\mathbb{R}$, and when $x \geqslant 0$, $f(x) = x^{2}-4x$, solve the inequality $f(x-2) < 5$. | (-3,7) | math | 56 |
Find the quadratic equation whose roots sum up to 12 and the absolute difference of whose roots is 4. | x^2 - 12x + 32 = 0 | math | 23 |
Crystal has a running course marked out for her daily run. She starts this run by heading due north for one mile. She then runs northeast for one mile, then southeast for one mile. The last portion of her run takes her on a straight line back to where she started. How far, in miles is this last portion of her run? | \sqrt{3} | math | 69 |
In triangle $ABC$ , side $AC$ is $8$ cm. Two segments are drawn parallel to $AC$ that have their ends on $AB$ and $BC$ and that divide the triangle into three parts of equal area. What is the length of the parallel segment closest to $AC$ ? | \frac{8 \sqrt{3}}{3} | math | 75 |
Calculate the area of an isosceles trapezoid with vertices at (0, 0), (8, 0), (6, 10), and (2, 10). | 60 | math | 43 |
Find the first term in the geometric sequence $x, y, z, 81, 162$. | 10.125 | math | 24 |
Archaeologists have discovered through Archaeopteryx fossil specimens that the linear regression equation between its femur length $x$ (cm) and humerus length $y$ (cm) is $$\hat y=1.197x-3.660$$. Based on this, estimate the humerus length when the femur length is 50 cm. | 56.19 | math | 78 |
Find the sum of all positive integers $n$ such that $\sqrt{n^2+85n+2017}$ is an integer. | 195 | math | 31 |
Given the equation $a \cdot b + 125 = 30 \cdot \text{lcm}(a, b) + 24 \cdot \text{gcd}(a, b) + a \mod b$, where $\text{gcd}(a, b)$ denotes the greatest common divisor of $a$ and $b$, and $\text{lcm}(a, b)$ denotes their least common multiple, assuming $a \geq b$, calculate the number of ordered pairs $(a, b)$ of positiv... | 0 | math | 115 |
If the inequality $x^2 + |2x - 6| \geq a$ holds for all real numbers $x$, then the maximum value of the real number $a$ is | 5 | math | 40 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $b\sin 2C=c\sin B$.
1. Find angle $C$.
2. If $\sin \left(B-\frac{\pi }{3}\right)=\frac{3}{5}$, find the value of $\sin A$. | \frac{4\sqrt{3}-3}{10} | math | 89 |
A point $P$ is chosen in the interior of $\triangle ABC$ such that when lines are drawn through $P$ parallel to the sides of $\triangle ABC$ , the resulting smaller triangles $t_{1}$ , $t_{2}$ , and $t_{3}$ in the figure, have areas $4$ , $9$ , and $49$ , respectively. Find the area of $\triangle ABC$ . [asy] size(200)... | 144 | math | 347 |
Given the function $f(x)=x^{3}-3x^{2}$.
(Ⅰ) Find the intervals of monotonicity for $f(x)$;
(Ⅱ) If the domain of $f(x)$ is $[-1,m]$ and its range is $[-4,0]$, find the maximum value of $m$. | 3 | math | 72 |
Given a point M(a, b) in the Cartesian coordinate system xOy, where a is chosen from the numbers 1, 2, 3, and b is chosen from the numbers 1, 2, 3, 4. Define the event "point M(a, b) lies on the line x+y=n" as event $Q_n$ ($2 \leq n \leq 7$, n is an integer). Then, when the probability of $Q_n$ is the highest, all poss... | 4 \text{ or } 5 | math | 114 |
Given an arithmetic sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$, and $a_1 = -20$. If the minimum value of $S_n$ is only $S_6$, then the range of the common difference $d$ is \_\_\_\_\_\_. | \left(\frac{10}{3}, 4\right) | math | 69 |
Given an geometric sequence $\left\{ {{a}_{n}} \right\}$ that satisfies ${{a}_{1}}+{{a}_{3}}=10,{{a}_{2}}+{{a}_{4}}=5$, find the maximum value of ${{a}_{1}}{{a}_{2}}...{{a}_{n}}$. | 64 | math | 71 |
In the Cartesian coordinate system $(xOy)$, the parametric equation of curve $C_1$ is given by $ \begin{cases} x=2\cos φ \\ y=\sin φ \end{cases} (φ \text{ is the parameter})$, and in the polar coordinate system with $O$ as the pole and the positive semi-axis of $x$ as the polar axis, curve $C_2$ is a circle with center... | [1,5] | math | 176 |
Given the solution set of the inequality $x^{2}-mx-6n < 0$ is $\{x|-3 < x < 6\}$, then $m+n=$ ______. | 6 | math | 41 |
A train display shows letters by lighting cells in a grid, such as the letter 'o' shown. A letter is made bold by also lighting any unlit cell immediately to the right of one in the normal letter. How many cells are lit in a bold 'o'?
A) 22
B) 24
C) 26
D) 28
E) 30 | 24 | math | 85 |
Given the cylindrical coordinates of point P as ($\sqrt{2}$, $\frac{\pi}{4}$, 1), find the Cartesian coordinates of point P. | (1, 1, 1) | math | 34 |
Find the maximum value of the expression \( x + y \), where \( x \) and \( y \) are integer solutions of the equation \( 3x^{2} + 5y^{2} = 345 \). | 13 | math | 50 |
For circles $C_1: (x-1)^2+(y-3)^2=9$ and $C_2: x^2+(y-2)^2=1$, let $M$ and $N$ be points on circles $C_1$ and $C_2$, respectively. Let $P$ be a point on the line $y=-1$. Find the minimum value of the sum of the distances from $P$ to $M$ and $P$ to $N$. | 5 \sqrt {2}-4 | math | 106 |
For which pairs of numbers \( (x, y) \) is the equation
$$
3 \sin x - 4 \cos x = 4 y^2 + 4 y + 6
$$
satisfied? | (x, y) = \left(-\arccos\left(-\frac{4}{5}\right) + (2k + 1)\pi, -\frac{1}{2} \right) | math | 48 |
I wrote a two-digit natural number on a card. The sum of its digits is divisible by three. If I subtract 27 from the written number, I get another two-digit natural number formed by the same digits in reverse order. What numbers could I have written on the card? | 63 \text{ and } 96 | math | 58 |
Given a natural number $x$ satisfies $3A_{x+1}^{3}=2A_{x+2}^{2}+6A_{x+1}^{2}$, determine the value of $x$. | 4 | math | 48 |
In a space experiment, six procedures need to be implemented in sequence, where Procedure A can only appear in the first or last step, and Procedures B and C must be adjacent. Calculate the total number of possible arrangements for the experiment sequence. | 96 | math | 48 |
Given that the area of \\(\triangle ABC\\) is \\(\frac{\sqrt{3}}{2}\\), and \\(\overrightarrow{AB} \cdot \overrightarrow{AC} = -3\\), then \\(A=\\) \_\_\_\_\_\_. | \frac{5\pi}{6} | math | 59 |
Two distinct positive integers \( a \) and \( b \) are factors of 48. If \( a \cdot b \) is not a factor of 48, what is the smallest possible value of \( a \cdot b \)? | 18 | math | 51 |
A line passing through the point $(1,0)$ and parallel to the line $x-\sqrt{2}y+3=0$ intersects the circle ${(x-6)}^{2}+{(y-\sqrt{2})}^{2}=12$ to form a chord of length ______. | 6 | math | 63 |
Calculate the sum of all integer values \( n \) for which \( \binom{28}{16} + \binom{28}{n} = \binom{29}{17} \). | 28 | math | 47 |
Samantha departs from Town A at 7:45 AM heading towards Town B at a steady speed of 15 miles per hour, and Adam leaves Town B at 8:15 AM heading towards Town A on the same 75-mile route at a constant speed of 20 miles per hour. Determine the time they meet and the distance traveled by Samantha at this time. | 10:11 \text{ AM, } 36 \text{ miles} | math | 82 |
The distance between points $A$ and $B$ is 500 kilometers. Two cyclists, Alpha and Beta, start from point $A$ to point $B$ at the same time. Alpha cycles 30 kilometers per day, and Beta cycles 50 kilometers per day but rests every other day. At the end of the $\qquad$ day, the remaining distance Beta has to travel to p... | 15 | math | 98 |
Point $O$ lies on plane $\alpha$, and $A$, $B$, and $C$ are three non-collinear points on plane $\alpha$. The moving point $P$ on plane $\alpha$ satisfies $\overrightarrow{OP} = \overrightarrow{OA} + \lambda (\overrightarrow{AB} + \overrightarrow{AC})$. When the value of $\lambda$ makes $\overrightarrow{PA} \cdot (\ove... | \frac{1}{4} | math | 115 |
Given that the domain of $f(x)$ is $(0, +\infty)$, satisfying $f(x) > 0$, and $f′(x)$ is its derivative, $\frac{f′(x)}{f(x)} < -1$.
(Ⅰ) Discuss the monotonicity of the function $F(x) = e^x f(x)$;
(Ⅱ) Let $0 < x < 1$, compare the magnitude of the function $xf(x)$ and $\frac{1}{x}f\left(\frac{1}{x}\right)$. | xf(x) > \frac{1}{x}f\left(\frac{1}{x}\right) | math | 125 |
For a function $f(x)$ defined on the interval $[0,1]$ that satisfies the following two conditions, we call it a G-function:
1. For any $x \in [0,1]$, $f(x) \geq 0$;
2. For $x_1 \geq 0, x_2 \geq 0, x_1 + x_2 \leq 1$, it holds that $f(x_1+x_2) \geq f(x_1) + f(x_2)$. Given the functions $g(x)=x^2$ and $h(x) = 2^x - b$, bo... | \{1\} | math | 192 |
Find the coefficient of the $x^2$ term in the expansion of the product $$(4x^3 + 3x^2 + 2x + 1)(2x^3 + x^2 + 6x + 5).$$ | 5 | math | 54 |
The digits from 1 to 9 are written in order so that the digit \( n \) is written \( n \) times, forming the block of digits \( 1223334444 \cdots 999999999 \). This block is written 100 times. Calculate the 1953rd digit written. | 6 | math | 81 |
Calculate the areas of the regions bounded by the curves given in polar coordinates.
$$
r=\cos 2 \phi
$$ | \frac{\pi}{2} | math | 27 |
Given $2\sin^2x + \cos^2y = 1$, find the range of values for $\sin^2x + \cos^2y$. | [\frac{1}{2}, 1] | math | 36 |
The function $f(x)=-x^{3}+3x+1$ has a minimum value and a maximum value. Determine these two values. | -1, 3 | math | 31 |
Admiral Ackbar needs to send a 5-character message through hyperspace to the Rebels. Each character is a lowercase letter, and the same letter may appear more than once in a message. When the message is beamed through hyperspace, the characters come out in a random order. Ackbar chooses his message so that the Rebels h... | 26 | math | 98 |
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(f(x)+y)+xf(y)=f(xy+y)+f(x)$$ for reals $x, y$. | f(x) = x \text{ or } f(x) = 0 | math | 48 |
Solve the fractional equations and simplify the fractions to find the values.<br/>$(1)\frac{2}{x-2}-\frac{4}{x^2-4}=\frac{1}{x+2}$;<br/>$(2)(\frac{x^2}{x+1}-x+1)÷\frac{x+2}{x^2+2x+1}$, where $x$ is an integer that satisfies $-3 \lt x\leqslant 0$. | \frac{1}{2} | math | 107 |
The bases $AB$ and $CD$ of the trapezoid $ABCD$ are equal to 101 and 20, respectively, and its diagonals are mutually perpendicular. Find the dot product of the vectors $\overrightarrow{AD}$ and $\overrightarrow{BC}$. | 2020 | math | 62 |
A curvilinear triangle is formed by three equal arcs of circles, each with a radius $R$, touching each other pairwise. Find the area of this triangle. | \frac{R^2 (2 \sqrt{3} - \pi)}{2} | math | 33 |
Given circles $P, Q,$ and $R$ where $P$ has a radius of 1 unit, $Q$ a radius of 2 units, and $R$ a radius of 1 unit. Circles $Q$ and $R$ are tangent to each other externally, and circle $R$ is tangent to circle $P$ externally. Compute the area inside circle $Q$ but outside circle $P$ and circle $R$. | 2\pi | math | 93 |
Given a monotonically increasing geometric sequence $\{a_n\}$ that satisfies $a_2 + a_3 + a_4 = 28$ and $a_3 + 2$ is the arithmetic mean of $a_2$ and $a_4$,
(1) Find the general term formula for the sequence $\{a_n\}$;
(2) Let $b_n = -na_n$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted by $S_n... | S_n = (1 - n) \cdot 2^{n+1} - 2 | math | 116 |
The expression \(\frac{a^{-2}}{a^{5}} \times \frac{4 a}{\left(2^{-1} a\right)^{-3}}\) where \(a \neq 0\), is equal to:
(a) \(\frac{a^{3}}{2}\)
(b) \(\frac{2}{a^{3}}\)
(c) \(\frac{1}{2 a^{3}}\)
(d) \(\frac{a^{5}}{2}\)
(e) \(\frac{2}{a^{5}}\) | \frac{1}{2 a^{3}} | math | 122 |
Given triangle $\triangle ABC$ with internal angles $A$, $B$, $C$, and opposite sides $a$, $b$, $c$ respectively, where $A < \frac{\pi}{2}$ and $\sin\left(A - \frac{\pi}{4}\right) = \frac{\sqrt{2}}{10}$.
(1) Find the value of $\sin A$;
(2) If the area of $\triangle ABC$, denoted as $s=24$, and $b=10$, find the value... | 8 | math | 119 |
Find all quadratic trinomials \( p(x) \) that attain a minimum value of \(-\frac{49}{4}\) at \(x=\frac{1}{2}\), and the sum of the fourth powers of its roots equals 337. | p(x) = x^2 - x - 12 | math | 57 |
Given the sequence $\{a\_n\}$ satisfies $a\_1=1$, and for any $n∈N^∗$, $a_{n+1}=a\_n+n+1$, find the value of $$\frac {1}{a_{1}}+ \frac {1}{a_{2}}+…+ \frac {1}{a_{2017}}+ \frac {1}{a_{2016}}+ \frac {1}{a_{2019}}$$. | \frac{2019}{1010} | math | 109 |
Given points $P$ and $Q$ are $8$ units apart in a plane, determine the number of lines containing $P$ and $Q$ that are $4$ units from $P$ and $6$ units from $Q$. | 2 | math | 51 |
Calculate the average age of the entire population given that the ratio of the number of women to men is $5:4$, the average age of women is $30$, and the average age of men is $36$. | 32\frac{2}{3} | math | 46 |
Given vectors $\overrightarrow{AP} = (1, \sqrt{3})$ and $\overrightarrow{PB} = (-\sqrt{3}, 1)$, find the angle between vector $\overrightarrow{AP}$ and vector $\overrightarrow{AB}$. | \frac{\pi}{4} | math | 56 |
Let $ \left( a_n \right)_{n\ge 1} $ be a sequence of real numbers such that $ a_1>2 $ and $ a_{n+1} =a_1+\frac{2}{a_n} , $ for all natural numbers $ n. $ **a)** Show that $ a_{2n-1} +a_{2n} >4 , $ for all natural numbers $ n, $ and $ \lim_{n\to\infty} a_n =2. $ **b)** Find the biggest real number $ a $ fo... | 2 | math | 240 |
Given the function $f(x) = x^2 - bx + 3$, and $f(0) = f(4)$.
(1) Find the zeros of the function $y = f(x)$, and write the set of $x$ values for which $f(x) < 0$;
(2) Find the maximum and minimum values of the function $y = f(x)$ in the interval $[0, 3]$. | 3 | math | 96 |
Given the function $f(x)=\sin 2x$, shift its graph to the right by $\dfrac{\pi}{6}$ units to obtain the graph of the function $g(x)$, then determine the interval of monotonic increase for the function $g(x)$. | \left[k\pi- \dfrac{\pi}{12},k\pi+ \dfrac{5\pi}{12}\right] | math | 57 |
Emily throws six identical darts, each hitting one of five differently sized dartboards. After throwing, she notes the number of darts that hit each board, from greatest to least. How many different lists are possible, considering different list represents a genuinely different outcome due to differing dartboard sizes? | 11 | math | 59 |
Given a point $M$ on the ellipse $\frac{{{x}^{{2}}}}{{25}}+ \frac{{{y}^{{2}}}{16}}={1}$, let $F\_{{1}}$ and $F\_{{2}}$ be the foci, and $\angle {{F}\_{{1}}}M{{F}\_{{2}}}=\frac{{ }\,{ }\pi }{6}$. Determine the area of $\triangle MF\_{1}F\_{{2}}$. | 16(2 - \sqrt{3}) | math | 108 |
The equation of one of the axes of symmetry for the graph of the function $y=2\sin^2x$ can be found by solving for the value of $x$ where the function is symmetric. | x=\frac{\pi}{2} | math | 43 |
Which of the numbers $-5, \frac{3}{2}, 2, \frac{3}{5}, 8$ is larger than its square? | \frac{3}{5} | math | 34 |
Given the function $f(x)=\dfrac{b-2^{x}}{2^{x+1}+a}$ is an odd function defined on $\mathbb{R}$.
$(1)$ Find the values of real numbers $a$ and $b$;
$(2)$ Determine the monotonicity of $f(x)$ on $\mathbb{R}$;
$(3)$ If $f(k\cdot 3^{x})+f(3^{x}-9^{x}+2) > 0$ holds for any $x \geqslant 1$, find the range of the real num... | k < \dfrac{4}{3} | math | 133 |
Two congruent isosceles right triangles are placed so that they overlap partly and their hypotenuses coincide, each with a hypotenuse of 10. Find the area of the region where the triangles overlap. | 12.5 | math | 45 |
In the triangle shown below, suppose $\cos R = \frac{3}{5}$. If RS = 10, what is the length of QS?
[asy]
pair Q,R,S;
S = (0,0);
Q = (sqrt(91),0);
R = (sqrt(91),-6);
draw(S--Q--R--S);
draw(rightanglemark(S,Q,R,20));
label("$S$",S,NW);
label("$Q$",Q,NE);
label("$R$",R,SE);
label("$10$",(R+S)/2,SW);
[/asy] | 8 | math | 130 |
Given that $|\overrightarrow{a}| = 3$, $|\overrightarrow{b}| = 2$, and $|2\overrightarrow{a} + \overrightarrow{b}| = 2\sqrt{13}$, find the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | \dfrac{\pi}{3} | math | 70 |
Given that points $F_1$ and $F_2$ are the left and right foci of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$, respectively, a line passing through $F_1$ perpendicular to the $x$-axis intersects the ellipse at points $A$ and $B$. If $\triangle ABF_2$ is an acute triangle, determine the range of the... | (\sqrt{2}-1, 1) | math | 117 |
In triangle ABC, the sides opposite to angles A, B, C are a, b, c respectively. Given that $B= \frac {π}{3}, b=2 \sqrt {3}$, find the range of the perimeter of triangle ABC. | (4 \sqrt {3}, 6 \sqrt {3}] | math | 53 |
A bookstore is considering the price to set for a new novel. The store has determined that if the novel is priced at \( p \) dollars (where \( p \le 25 \)), then the number of novels they can sell per month is \( 200 - 8p \). Determine the price that should be set for the novel to maximize the store's monthly revenue. | 12.5 | math | 80 |
Find the maximum distance from a point on the circle $x^{2}+y^{2}-2x-2y+1=0$ to the line $x-y=2$. | 1 + \sqrt{2} | math | 39 |
Given the function $f(x)$ is an odd function defined on $\mathbb{R}$ and for all $x \in \mathbb{R}$, $f(x) = f(x+4)$, and $f(x) = 2^{x}$ when $x \in (-2,0)$, calculate the value of $f(2015) - f(2013)$. | 1 | math | 87 |
Find all prime numbers that can be represented both as the sum and the difference of two prime numbers. | 5 | math | 20 |
A certain school's art department has 4 students, with 2 students each from the first and second grades. If 2 students are randomly selected from these 4 students, calculate the probability that these 2 students are from different grades. | \frac{2}{3} | math | 49 |
The largest number by which the expression $n^4 - n^2$ is divisible for all possible integral values of $n$. | 12 | math | 27 |
The fictional country of Isoland uses a 6-letter license plate system using the same 12-letter alphabet as the Rotokas of Papua New Guinea (A, E, G, I, K, O, P, R, T, U, V). Design a license plate that starts with a vowel (A, E, I, O, U), ends with a consonant (G, K, P, R, T, V), contains no repeated letters and does n... | 151200 | math | 104 |
In the Cartesian coordinate system xOy, with the origin as the pole O and the positive x-axis as the polar axis, the polar equation of circle C is $\rho=4\sqrt{2}\cos(\theta+ \frac{\pi}{4})$.
(1) Convert the polar equation of circle C into a Cartesian coordinate equation;
(2) A line l with a slope of 1 passes through p... | \frac{\sqrt{6}}{2} | math | 126 |
Under normal circumstances, for people aged between 18 and 38 years old, the regression equation for weight $y$ (kg) based on height $x$ (cm) is $y=0.72x-58.5$. Zhang Honghong, who is neither fat nor thin, has a height of 1.78 meters. His weight should be around \_\_\_\_\_ kg. | 70 | math | 89 |
Let $\theta$ be an acute angle, and let
\[\cos \frac{\theta}{2} = \sqrt{\frac{x - 2}{2x}}.\]
Express $\tan \theta$ in terms of $x.$ | -\frac{1}{2} \sqrt{x^2 - 4} | math | 49 |
Given the curve $y=\frac{2-\cos x}{\sin x}$, find the value of $a$ such that the tangent line to the curve at the point $(\frac{\pi }{2}, 2)$ is perpendicular to the line $x+ay+1=0$. | a = 1 | math | 62 |
A line is parameterized by a parameter $t$. The vector on the line at $t = 1$ is $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$, at $t = 4$ is $\begin{pmatrix} 8 \\ -5 \end{pmatrix}$, and at $t = 5$ is $\begin{pmatrix} 10 \\ -9 \end{pmatrix}$. Find the vector on the line at $t = 0.$ | \begin{pmatrix} 0 \\ \frac{17}{3} \end{pmatrix} | math | 109 |
Given the observed pattern where \( 9^2 = 81 \) contains no zeros, \( 999^2 = 998001 \) contains two zeros, and \( 9999^2 = 99980001 \) contains three zeros, determine how many zeros are in the decimal expansion of \( 99999999^2 \). | 7 | math | 90 |
Find maximal positive integer $p$ such that $5^7$ is sum of $p$ consecutive positive integers | 125 | math | 29 |
Let $y=f(x)$ be an odd function defined on $\mathbb{R}$, satisfying $f(x+2)=-f(x)$. When $x \in [0,1]$, $f(x)=x+1$. Find $f(7.5)$. | -1.5 | math | 59 |
Among 9 consecutive positive odd numbers, what is the maximum number of prime numbers? Answer: . | 7 | math | 20 |
Given that point $P$ lies on the unit circle $O$ (with $O$ as the origin), and point $A(-2,0)$, the range of $\overrightarrow{AO} \cdot \overrightarrow{AP}$ is ______. | [2,6] | math | 53 |
The United States Postal Service has updated its postage rules. An extra charge of $\$0.11$ is added if the length of an envelope, in inches, divided by its height, in inches, is less than $1.4$ or greater than $2.4$ or if the area of the envelope (length times height) is less than $18$ square inches. Determine the num... | 3 | math | 157 |
Calculate \(\left(\frac{3}{4}\right)^3\). | \frac{27}{64} | math | 16 |
We are allowed to remove exactly one integer from the list $$ -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 $$ and then we choose two distinct integers at random from the remaining list. What number should we remove if we wish to maximize the probability that the sum of the two chosen numbers is 12? | 6 | math | 104 |
During the Qingming Festival holiday, it is known that the probability of student A visiting Wuyuan Ancient Town is $\frac{2}{3}$, the probability of student B visiting Wuyuan Ancient Town is $\frac{1}{4}$, and the probability of student C visiting Wuyuan Ancient Town is $\frac{2}{5}$. Moreover, the actions of students... | \frac{9}{20} | math | 150 |
The equation of the line passing through point P(-1, 2) and perpendicular to the line $x+2y-1=0$ is to be found. | 2x-y+4=0 | math | 35 |
Given the sequence $\{a_n\}$, where $a_1=t$ and $a_{n+1}= \frac{a_n}{2}+ \frac{2}{a_n}$, if $\{a_n\}$ is a monotonically decreasing sequence, determine the range of the real number $t$. | (2,+\infty) | math | 68 |
Two students, A and B, participate in a three-point basketball shooting competition. The probability of A making a shot is $\frac{1}{3}$, while the probability of B making a shot is $\frac{1}{2}$. Each of them attempts three shots.
1. Find the probability that B makes at most 2 shots.
2. Find the probability that B ma... | \frac{1}{6} | math | 86 |
Among prime numbers not exceeding $12$, determine the probability of randomly selecting two different numbers whose sum is an even number. | \frac{3}{5} | math | 25 |
In the diagram, each circle is divided into two equal areas and $O$ is the center of the larger circle. The area of the larger circle is $64\pi.$ What is the total area of the shaded regions? [asy]
size(100);
import graph;
fill(Arc((0,0),2,180,360)--cycle,mediumgray);fill(Arc((0,1),1,0,180)--cycle,mediumgray);
draw(Cir... | 40\pi | math | 165 |
The following $100$ numbers are written on the board: $$ 2^1 - 1, 2^2 - 1, 2^3 - 1, \dots, 2^{100} - 1. $$ Alice chooses two numbers $a,b,$ erases them and writes the number $\dfrac{ab - 1}{a+b+2}$ on the board. She keeps doing this until a single number remains on the board.
If the sum of all possible number... | 100 | math | 162 |
Determine the number of terms in the polynomial of $x$ obtained from the expansion of $(\sqrt{3}x+\sqrt[3]{2})^{100}$ where the coefficient is a rational number. | 17 | math | 45 |
Given a convex polygon, except for one interior angle, the sum of the remaining n-1 interior angles is $2008^\circ$. Calculate the value of n. | 14 | math | 36 |
Given the complex number $z=(2+i)(a+2i^3)$ corresponds to a point in the fourth quadrant on the complex plane, determine the range of the real number $a$. | (-1,4) | math | 40 |
Find all \( n \in\{1,2, \ldots, 999\} \) such that \( n^{2} \) is equal to the cube of the sum of the digits of \( n \). | 1 \text{ and } 27 | math | 49 |
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