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 | In a certain place, four people, $A$, $B$, $C$, and $D$, successively contracted the novel coronavirus, with only $A$ having visited an epidemic area.
1. If the probabilities of $B$, $C$, and $D$ being infected by $A$ are $\frac{1}{2}$ each, what is the probability that exactly one of $B$, $C$, and $D$ is infected with... | \frac{11}{6} | 248 | 8 |
math | Given two positive integers \(x\) and \(y\), \(xy - (x + y) = \operatorname{HCF}(x, y) + \operatorname{LCM}(x, y)\), where \(\operatorname{HCF}(x, y)\) and \(\operatorname{LCM}(x, y)\) are respectively the greatest common divisor and the least common multiple of \(x\) and \(y\). If \(c\) is the maximum possible value o... | 10 | 114 | 2 |
math | Find the sum of all positive integers $n$ such that $\sqrt{n^2+85n+2017}$ is an integer. | 195 | 31 | 3 |
math | What is the smallest positive integer \(n\) such that all the roots of the equation \(z^5 - z^3 + z = 0\) are \(n^\text{th}\) roots of unity? | 12 | 44 | 2 |
math | Given that $|3a+7|+|3a-5|=12$, determine the number of integer values of $a$ such that the equation holds true. | 4 | 36 | 1 |
math | Given $x$ is a real number and $|x-5| + |x-7| < b$ where $b > 0$, determine the range of values that $b$ must fall into for the inequality to hold for some $x$. | b > 2 | 53 | 4 |
math | Determine the quadrant in the complex plane that corresponds to the complex number $$\frac {2-i}{i}$$ (where $i$ is the imaginary unit). | 3 | 34 | 1 |
math | Given the unit price of a book is 8 yuan and a 10% shipping fee is added, determine the total cost in yuan for ordering $a$ books. | 8(1+10\%)a | 35 | 9 |
math | In a certain location, the Olympic torch relay is divided into 6 segments, carried out by 6 torchbearers. If the first torchbearer can only be selected from among three people: A, B, and C, and the last torchbearer can only be selected from among two people: A and B, then the total number of different relay schemes is ... | 96 | 83 | 2 |
math | In the arithmetic sequence $\{a_n\}$, $a_2 = 5$, $a_1 + a_3 + a_4 = 19$.
(Ⅰ) Find the general term formula for the sequence $\{a_n\}$;
(Ⅱ) Suppose for the sequence $\{b_n\}$, the sum of the first n terms is $S_n$, and $S_n + \frac{a_n - 1}{2^n} = \lambda$ (where $\lambda$ is a constant). Let $c_n = b_{n+1}$ (for $n \in... | T_n = 1 - (n+1)\left(\frac{1}{2}\right)^n | 161 | 22 |
math | The pages of a booklet are numbered from 1 to 2n. A single sheet (of 2 pages) is removed. The numbers of the remaining pages sum to 963. How many pages did the booklet have originally and which pages were removed? | 13,14 | 54 | 5 |
math | Observe the following equations and complete the following questions: $\sqrt{2+\frac{2}{3}}=2\sqrt{\frac{2}{3}}$, $\sqrt{3+\frac{3}{8}}=3\sqrt{\frac{3}{8}}$, $\sqrt{4+\frac{4}{{15}}}=4\sqrt{\frac{4}{{15}}}$, $\ldots \ldots $.<br/>$(1)$ Based on the above equations, what pattern can you find? Please express the pattern ... | \sqrt{5+\frac{5}{{24}}}=5\sqrt{\frac{5}{{24}}} | 163 | 26 |
math | Given that $\{a_n\}$ is an arithmetic sequence with a common difference of $\frac{1}{2}$, and $S_n$ represents the sum of the first $n$ terms of $\{a_n\}$, if $a_2$, $a_6$, and $a_{14}$ form a geometric sequence, calculate the value of $S_5$. | 10 | 80 | 2 |
math | Let \( P \) and \( A \) denote the perimeter and area respectively of a right triangle with relatively prime integer side-lengths. Find the largest possible integral value of \(\frac{P^{2}}{A}\). | 45 | 47 | 2 |
math | A high school has a total of 1000 students, among which there are 380 students in the first grade. There are 180 male students in the second grade. If one student is randomly selected from all the students, the probability of selecting a female student from the second grade is 0.19. Now, using stratified sampling (by g... | 25 | 108 | 2 |
math | If the power function $f(x) = x^{k}$ is a decreasing function on $(0, +\infty)$, determine the value of $k$. | -1 | 34 | 2 |
math | In the trapezoid \(ABCD\) with an area of 1, the bases \(BC\) and \(AD\) are in the ratio \(1:2\). Let \(K\) be the midpoint of the diagonal \(AC\). The line \(DK\) intersects the side \(AB\) at point \(L\). Find the area of the quadrilateral \(BCLK\). | \frac{2}{9} | 79 | 7 |
math | There are 7 safes and 7 codes for them, but it is unknown which code matches which safe. What is the minimum number of attempts needed to accurately match all the codes to the safes? | 21 | 42 | 2 |
math | In a rectangular grid where grid lines are spaced $1$ unit apart, the acronym XYZ is depicted below. The X is formed by two diagonal lines crossing, the Y is represented with a 'V' shape starting from a bottom point going up to join two endpoints with horizontal lines, the Z is drawn with a top horizontal line, a diago... | 4 + 5\sqrt{2} | 147 | 9 |
math | Given the ellipse \( C_{1}: \frac{x^{2}}{4} + y^{2} = 1 \) and the parabola \( C_{2}: x^{2} = 2py \) (where \( p > 0 \)), suppose \( C_{1} \) and \( C_{2} \) intersect at points \( A \) and \( B \), with \( O \) as the origin of coordinates.
(1) If the circumcenter of \( \triangle A B O \) lies on the ellipse, find th... | 3 | 173 | 1 |
math | In triangle $ABC$, the sides $AC = 14$ and $AB = 6$ are known. A circle with center $O$ constructed on side $AC$ as the diameter intersects side $BC$ at point $K$. It is given that $\angle BAK = \angle ACB$. Find the area of triangle $BOC$. | 21 | 73 | 2 |
math | For every $4^\circ$ rise in temperature, the volume of a certain gas expands by $5$ cubic centimeters. Initially, the gas had a volume of $35$ cubic centimeters at a temperature of $25^\circ$. Given that the pressure remains constant, what was the volume of the gas when the temperature was $11^\circ$?
A) $15$ cm³
B) ... | 17.5 | 115 | 4 |
math | An ellipse has foci at $(1, 1)$ and $(1, 7)$, and it passes through the point $(12, -4)$. Write the equation of the ellipse in standard form as \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1,\]where $a, b, h, k$ are constants, and $a$ and $b$ are positive. Find the ordered quadruple $(a, b, h, k)$. | \left(\frac{\sqrt{146} + \sqrt{242}}{2}, \sqrt{\left(\frac{\sqrt{146} + \sqrt{242}}{2}\right)^2 - 9}, 1, 4\right) | 114 | 61 |
math | Given that $a$, $b$, and $c$ represent the sides opposite to angles $A$, $B$, and $C$ respectively in $\triangle ABC$, and the altitude on side $BC$ is $\frac{a}{2}$. Determine the maximum value of $\frac{c}{b}$. | \sqrt{2} + 1 | 64 | 8 |
math | Simplify the expression $\frac{3^{n+5} - 3(3^n)}{3(3^{n+4}) - 6}$. | \frac{80}{81} | 34 | 9 |
math | Given an ellipse C with its center at the origin and foci on the x-axis and a focal distance of 2, and the length of the major axis is $$\sqrt {2}$$ times the length of the minor axis.
(1) Determine the standard equation of ellipse C.
(2) Let P(2, 0), a line l passing through the left focus F of the ellipse C intersect... | \frac{17}{2} | 145 | 8 |
math | Find the smallest positive integer \( n \) such that for any positive integer \( k \geq n \), in the set \( M = \{1, 2, \ldots, k\} \) of the first \( k \) positive integers, for any \( x \in M \), there is always another number \( y \in M \) (where \( y \neq x \)) such that \( x + y \) is a perfect square. | 7 | 99 | 1 |
math | Let the function $f(x) = a_1 + a_2x + a_3x^2 + \ldots + a_nx^{n-1}$, where $f(0) = \frac{1}{2}$, and the sequence $\{a_n\}$ satisfies $f(1) = n^2a_n$ ($n \in \mathbb{N}^*$). Find the general term $a_n$ of the sequence $\{a_n\}$. | a_n = \frac{1}{(n+1)n} | 105 | 14 |
math | Calculate the sum $$\frac{2^1}{8^1 - 1} + \frac{2^2}{8^2 - 1} + \frac{2^3}{8^3 - 1} + \frac{2^4}{8^4 - 1} + \cdots.$$ | \frac{1}{3} | 68 | 7 |
math | Find the zeroes of the function \( f(z) = e^z - 1 - z \) and determine their order. | z = 0 | 26 | 4 |
math | A given improper fraction \(\frac{6 x^{3}+5 x^{2}+3 x-4}{x^{2}+4}\) needs to be expressed as the sum of a polynomial and a proper fraction. | 6x + 5 - \frac{21x + 24}{x^2 + 4} | 48 | 24 |
math | Given that the sum of the first $n$ terms ($S_n$) of an arithmetic sequence {$a_n$} has a maximum value, and $\frac{a_{15}}{a_{14}} < -1$, determine the maximum value of $n$ that makes $S_n > 0$. | 27 | 66 | 2 |
math | The product of the lengths of the two congruent sides of an obtuse isosceles triangle is equal to the product of the base and twice the triangle's height to the base. What is the measure, in degrees, of the vertex angle of this triangle? | 150 | 54 | 3 |
math | Given the statements $p: |4x-3| \leqslant 1$ and $q: x^2 - (2a+1)x + a(a+1) \leqslant 0$, if the statement "$\neg p \Rightarrow \neg q$" is false and "$\neg q \Rightarrow \neg p$" is true, find the range of real number values for $a$. | a \in [0, \frac{1}{2}] | 88 | 13 |
math | Given the curve $C$: $\begin{cases}x= \frac{ \sqrt{3}}{2}t \\ y=a+ \frac{ \sqrt{2}}{2}t\end{cases} (t$ is a parameter $)$, and the points $A(-1,0)$, $B(1,0)$, determine the range of the real number $a$ such that there exists a point $P$ on the curve $C$ such that $\overrightarrow{AP} \cdot \overrightarrow{BP}=0$. | \left[- \sqrt {2}, \sqrt {2}\right] | 117 | 15 |
math | Simplify completely: $$\sqrt[3]{20^3 + 30^3 + 40^3 + 60^3}.$$ | 10 \cdot \sqrt[3]{315} | 34 | 13 |
math | Given that one vertex of the ellipse $\dfrac{{x}^{2}}{{a}^{2}}+ \dfrac{{y}^{2}}{{b}^{2}}=1$ ($a > b > 0$) and its two foci form an equilateral triangle, calculate the eccentricity. | \dfrac{1}{2} | 64 | 7 |
math | Pyramid $OABCD$ has square base $ABCD,$ congruent edges $\overline{OA}, \overline{OB}, \overline{OC},$ and $\overline{OD},$ and $\angle AOB=45^\circ.$ Let $\theta$ be the measure of the dihedral angle formed by faces $OAB$ and $OBC.$ Given that $\cos \theta=m+\sqrt{n},$ where $m$ and $n$ are integers, find $m+n.$
| 5 | 109 | 1 |
math | Tom and Geri have a competition. Initially, each player has one attempt at hitting a target. If one player hits the target and the other does not then the successful player wins. If both players hit the target, or if both players miss the target, then each has another attempt, with the same rules applying. If the proba... | \frac{2}{3} | 167 | 7 |
math | Consider the equation \[ 3x^2 + 2y^2 - 6x - 16y = m \]. Determine the value of $m$ for which the graph of the equation is a non-degenerate ellipse. | -35 | 50 | 3 |
math | Each bag of sugar has a standard weight of $500$ grams. The quality supervision department wants to understand the weight condition of a batch of sugar. They randomly selected $9$ bags and weighed each bag (in grams) as follows:
$490$ $495$ $493$ $498$ $499$ $500$ $503$ $507$ $506$
(I) Calculate the mea... | 1 | 152 | 1 |
math | If \( r \) is the radius of the circle that touches the sides of the triangle, then the area of the triangle is given by:
$$
t=r^{2} \operatorname{ctg} \frac{\alpha}{2} \operatorname{ctg} \frac{\beta}{2} \operatorname{ctg} \frac{\gamma}{2}
$$ | T = r^2 \cot \frac{\alpha}{2} \cot \frac{\beta}{2} \cot \frac{\gamma}{2} | 79 | 32 |
math | Given the function $f(x)= \frac {1}{3}x^{3}+ \frac {1-a}{2}x^{2}-ax-a$, where $x\in\mathbb{R}$ and $a > 0$.
$(1)$ Find the intervals of monotonicity for the function $f(x)$.
$(2)$ If the function $f(x)$ has exactly two zeros in the interval $(-2,0)$, find the range of values for $a$. | (0, \frac {1}{3}) | 105 | 10 |
math | Given the function $f(x)=2\sin x \cdot \cos x + 2\sqrt{3}\cos^2 x - \sqrt{3}$.
$(1)$ Find the smallest positive period of the function $f(x)$.
$(2)$ Find the solution set of the inequality $f(x) \geq 1$ on the interval $\left[0,\pi \right]$. | [0, \frac{\pi}{4}] \cup [\frac{11\pi}{12}, \pi] | 86 | 26 |
math | In $\triangle ABC$, if $\overrightarrow{BC} \cdot \overrightarrow{BA} + 2 \overrightarrow{AC} \cdot \overrightarrow{AB} = \overrightarrow{CA} \cdot \overrightarrow{CB}$, then the value of $\frac {\sin A}{\sin C}$ is ______. | \sqrt {2} | 70 | 5 |
math | There are three bags labeled $A$, $B$, and $C$. Bag $A$ contains $1$ red ball, bag $B$ contains $2$ different white balls, and bag $C$ contains $3$ different yellow balls. Two balls are drawn from these bags.
$(1)$ How many ways are there to draw two balls of different colors?
$(2)$ How many ways are there to draw tw... | 4 | 92 | 1 |
math | Find the equation of the line that passes through the intersection point of $l_1: 2x-3y+2=0$ and $l_2: 3x-4y+2=0$, and is parallel to the line $4x+y-4=0$. | 4x+y-10=0 | 61 | 8 |
math | Find $4 \cdot 7^{-1} + 12 \cdot 13^{-1} - 6 \cdot 17^{-1} \pmod{60}$.
Express your answer as an integer from $0$ to $59$, inclusive. | 58 | 58 | 2 |
math | Consider the function $f(x) = -2m + 2m\sin(x+ \frac{3\pi}{2}) - 2\cos^2(x - \frac{\pi}{2}) + 1$, where $x \in \left[-\frac{\pi}{2}, 0\right]$ and its minimum value is denoted as $h(m)$.
1. Find $h(m)$.
2. If $h(m) = \frac{1}{2}$, determine the value of $m$ and the maximum value of $f(x)$ under this condition. | 4 | 125 | 1 |
math | What is the ratio of the area of an equilateral triangle inscribed in a semicircle with radius \(r\) to the area of an equilateral triangle inscribed in a circle with radius \(r\)? Express your answer as a common fraction. | \frac{4}{9} | 51 | 7 |
math | Let $g$ be a function taking the nonnegative integers to the nonnegative integers, such that
\[2g(a^2 + 2b^2) = [g(a)]^2 + 3[g(b)]^2\]
for all nonnegative integers $a$ and $b.$
Let $n$ be the number of possible values of $g(50),$ and let $s$ be the sum of the possible values of $g(50).$ Find $n \times s.$ | 0 | 109 | 1 |
math | Luna, Ginny, and Hermione are having a race on their broomsticks. In this race, there could be a tie for the first place but no ties afterward. In how many different possible orders can they finish considering the tie possibility? | 12 | 49 | 2 |
math | The quadratic $-3x^2 + 27x - 153$ can be written in the form $a(x+b)^2+c$, where $a$, $b$, and $c$ are constants. What is $a+b+c$? | -99.75 | 55 | 6 |
math | A fair coin is tossed 4 times. What is the probability of getting at least two consecutive heads? | \frac{5}{8} | 21 | 7 |
math | The sum of the following seven numbers is exactly 19: $a_1 = 2.56,$ $a_2 = 2.61,$ $a_3 = 2.65,$ $a_4 = 2.71,$ $a_5 = 2.79,$ $a_6 = 2.82,$ $a_7 = 2.86.$ Each $a_i$ is approximated by some integer $A_i,$ for $1 \le i \le 7,$ such that the sum of the $A_i$'s is also $19.$ Let $M$ be the maximum of the seven "errors" $|A_i... | 0.61 | 167 | 4 |
math | Determine the smallest non-negative integer $n$ such that $0 \le n < 53$ and $$50n \equiv 47 \pmod{53}.$$ | 2 | 40 | 1 |
math | In a tetrahedron \(ABCD\), \(AD = \sqrt{2}\) and all other edge lengths are 1. Find the shortest path distance from the midpoint \(M\) of edge \(AB\) to the midpoint \(N\) of edge \(CD\) along the surface of the tetrahedron. | \frac{\sqrt{3}}{2} | 66 | 10 |
math | Find the derivative $y_{x}^{\prime}$.
$$
\left\{\begin{array}{l}
x=\arcsin (\sin t) \\
y=\arccos (\cos t)
\end{array}\right.
$$ | 1 | 52 | 1 |
math | Given that $\{a\_n\}$ is an increasing geometric sequence, if $a\_2=2$, $a\_4-a\_3=4$,
(I) Find the value of the first term $a\_1$ and the common ratio $q$;
(II) Find the value of the fifth term $a\_5$ and the sum of the first five terms $S\_5$. | 31 | 84 | 2 |
math | A school bus departs from a school with 60 students on board. At the first two stops, one-third of the students get off the bus. At the third stop, one-quarter of the remaining students get off. How many students are left on the bus after the third stop? | 20 | 59 | 2 |
math | Let $ m,n > 1$ are integers which satisfy $ n|4^m \minus{} 1$ and $ 2^m|n \minus{} 1$ . Is it a must that $ n \equal{} 2^{m} \plus{} 1$ ? | n = 2^m + 1 | 67 | 9 |
math | Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(x^2 - yf(z)) = xf(x) - zf(y)\] for all real numbers $x,$ $y,$ and $z.$
Let $n$ be the number of possible values of $f(5),$ and let $s$ be the sum of all possible values of $f(5).$ Find $n \times s.$ | 10 | 101 | 2 |
math | Let $S$ be the set of all real values of $x$ with $0 < x < \frac{\pi}{4}$ such that $\sin x$, $\cos x$, and $\cot x$ form the side lengths (in some order) of a right triangle. Compute the sum of $\cot^2 x$ over all $x$ in $S$. | 2 | 76 | 1 |
math | Given that the three interior angles $A$, $B$, $C$ of $\triangle ABC$ form an arithmetic sequence, and the side $b$ opposite to angle $B$ equals $\sqrt{3}$, and the function $f(x)=2 \sqrt{3}\sin ^{2}x+2\sin x\cos x- \sqrt{3}$ reaches its maximum value at $x=A$, then the area of $\triangle ABC$ is __________. | \frac{3+ \sqrt{3}}{4} | 98 | 13 |
math | Given two numbers, $x$ and $y$, are randomly selected from the interval $[0,π]$, determine the probability that the event "$y \leqslant \sin x$" occurs. | \dfrac{2}{\pi ^{2}} | 43 | 11 |
math | The Hoopers, coached by Coach Loud, have 15 players. George and Alex are the two players who refuse to play together in the same lineup. Additionally, if George plays, another player named Sam refuses to play. How many starting lineups of 6 players can Coach Loud create, provided the lineup does not include both George... | 3795 | 71 | 4 |
math | The extreme points of the function $f(x)= \frac {x^{4}}{4}- \frac {x^{3}}{3}$ are $0$ or $1$. | 1 | 38 | 1 |
math | Provide an example of a non-zero polynomial with integer coefficients that has the number $\cos 18^{\circ}$ as one of its roots. | 16x^4 - 20x^2 + 5 | 30 | 15 |
math | Restore the digits represented by stars in the multiplication example $* * \cdot * *=1 * 1$. Find all solutions. | 11 \times 11 = 121 | 26 | 12 |
math | A sequence of 2003 digits starts with the digit 1. Any two-digit number formed by consecutive digits within this sequence is divisible by 17 or 23. Determine the largest possible last digit in this sequence. | 8 | 48 | 1 |
math | Teams A and B each have 7 players who will compete in a Go tournament in a predetermined order. The match starts with player 1 from each team competing against each other. The loser is eliminated, and the winner next competes against the loser’s teammate. This process continues until all players of one team are elimina... | 3432 | 81 | 4 |
math | Triangle $ABC$ has vertices with coordinates $A(3,4),$ $B(8,9),$ and $C(-3,7)$. After a transformation, the image points are $A'(-2,-6),$ $B'(-7,-11),$ and $C'(2,-9)$. Determine the operation performed, and if it includes reflection, find the equation of the line of reflection. | y = -1 | 85 | 4 |
math | The roots of $64x^3-144x^2+92x-15=0$ are in arithmetic progression. The difference between the largest and smallest roots is: | 1 | 42 | 1 |
math | The students in the mathematics interest group of a 9th-grade class exchanged greeting cards before New Year's Day, with each student giving a card to every other member of the group. A total of 182 cards were exchanged within the group. Determine the number of students in this mathematics interest group. | 14 | 62 | 2 |
math | Given a natural number \( n \geq 2 \). Consider all colorings of the cells of an \( n \times n \) board with \( k \) colors, such that each cell is painted in exactly one color and all \( k \) colors are used. Determine the smallest \( k \) for which, in any such coloring, there will be four cells painted in four diffe... | 2n | 93 | 2 |
math | The function $f(x) = x^2 + 2(a - 1)x + 2$ is decreasing on the interval $(-\infty, 4]$, then the range of the real number $a$ is | (-\infty, -3] | 48 | 8 |
math | On a table, there are 10 cards numbered $1, 1, 2, 2, 3, 3, 4, 4, 5, 5$. These 10 cards are shuffled and arranged in a row from left to right. Then, the number of cards between the two 1s, the two 2s, the two 3s, the two 4s, and the two 5s are counted. What is the maximum sum of these 5 numbers? | 20 | 109 | 2 |
math | Consider a convex 20-sided polygon where the degree measures of the angles are given as an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle. | 143^\circ | 35 | 5 |
math | Given the function $f(x) = a(x-5)^2 + 6\ln x$, where $a \in \mathbb{R}$, the tangent line to the curve $y = f(x)$ at the point $(1, f(1))$ intersects the $y$-axis at the point $(0,6)$.
(1) Determine the value of $a$;
(2) Find the intervals of monotonicity and the extreme values of the function $f(x)$. | 2 + 6\ln 3 | 105 | 8 |
math | Let $S = \{1,2,3,\ldots,n\}$ . Consider a function $f\colon S\to S$ . A subset $D$ of $S$ is said to be invariant if for all $x\in D$ we have $f(x)\in D$ . The empty set and $S$ are also considered as invariant subsets. By $\deg (f)$ we define the number of invariant subsets $D$ of $S$ for the function... | 2^k | 202 | 3 |
math | If the function $f(x)=kx-\cos x$ is monotonically increasing in the interval $\left(\frac{\pi}{6},\frac{2\pi}{3}\right)$, determine the range of $k$. | \left[-\frac{1}{2},+\infty\right) | 49 | 16 |
math | It takes Clea 70 seconds to walk down an escalator when it is not moving, and 30 seconds when it is moving. Her walking speed increases by 50% when the escalator is moving. Determine the time it would take Clea to ride the escalator down when she is not walking. | 84 | 66 | 2 |
math | Given a monotonically decreasing function $f(x)$ defined on $(-\infty, 3]$, such that for all real numbers $x$, the inequality $f(1+\sin^2x) \leq f(a-2\cos x)$ holds, determine the range of the real number $a$. | (-\infty, -1] | 67 | 8 |
math | Given $n$ new students, among any 3 students, there are 2 who know each other, and among any 4 students, there are 2 who do not know each other. Determine the maximum value of $n$. | 8 | 48 | 1 |
math | (1) A factory produces three different models of products, $A$, $B$, and $C$, with the ratio of their quantities being $2:3:5$. Now, using stratified sampling, a sample of size $n$ is drawn, and the sample contains $16$ units of model $A$. Then, the sample size $n = \_\_\_\_\_\_\_\_$.
(2) If a real number $a$ is rando... | \frac{5}{16} | 317 | 8 |
math | Nine lines parallel to the base of a triangle divide the other sides each into $10$ equal segments and the area into $10$ distinct parts. If the area of the largest of these parts is $38$, calculate the area of the original triangle. | 200 | 59 | 3 |
math | Determine the largest integer $n$ such that $2^n$ divides the decimal representation given by some permutation of the digits $2$ , $0$ , $1$ , and $5$ . (For example, $2^1$ divides $2150$ . It may start with $0$ .) | 4 | 80 | 1 |
math | What is the greatest possible value of the expression \(\frac{1}{a+\frac{2010}{b+\frac{1}{c}}}\), where \(a, b, c\) are distinct non-zero digits? | 1/203 | 48 | 5 |
math | Let $n \geqslant 4$ be an integer. In the set $\{1, 2, 3, ..., n\}$, we pick two distinct elements $a$ and $b$ ($a > b$). Let $A_n$ denote the number of ways to pick $a$ and $b$ such that $a + b$ is divisible by $2$.
1. Find $A_n$ when $n = 6$.
2. Express $A_n$ in general. | 6 | 109 | 1 |
math | A hyperbola with its center shifted to $(1,1)$ passes through point $(4, 2)$. The hyperbola opens horizontally, with one of its vertices at $(3, 1)$. Determine $t^2$ if the hyperbola also passes through point $(t, 4)$. | 36 | 66 | 2 |
math | Given functions $f\left(x\right)$ and $g\left(x\right)$ are provided in the table below:
| $x$ | $1$ | $2$ | $3$ |
|-----|-----|-----|-----|
| $f\left(x\right)$ | $1$ | $3$ | $1$ |
| $g\left(x\right)$ | $3$ | $2$ | $1$ |
Find the value of $f\left[g\left(1\right)\right]$ as _______; then determine the solution set of ... | \{2\} | 155 | 5 |
math | Let $a_n$ be the number obtained by writing the integers 1 to $n$ from left to right. For instance, $a_4 = 1234$ and \[a_{12} = 123456789101112.\]For $1 \le k \le 120$, how many $a_k$ are divisible by 11? | 11 | 91 | 2 |
math | Given that the terminal side of angle $\alpha$ intersects the unit circle at point $(-\frac{\sqrt{3}}{2}, \frac{1}{2})$, determine the value of $\cos 2\alpha$. | \frac{1}{2} | 47 | 7 |
math | Given a quiz with three questions, each worth one mark, and the following distribution of student performance: 20% got 0 questions correct, 5% got 1 question correct, 40% got 2 questions correct, and 35% got all 3 questions correct, calculate the overall class mean (average) mark. | 1.9 | 71 | 3 |
math | In a small office, each worker has a probability of being late once every 40 days due to traffic. Calculate the probability that among three randomly chosen workers on a given day, exactly two are late while the third one is on time. Express your answer as a percent to the nearest tenth. | 0.2\% | 60 | 5 |
math | The number 42524 is a palindrome, because it reads the same backwards as forward. How many integer palindromes are between 10,000 and 70,000, and are even? | 300 | 50 | 3 |
math | A student needs to choose a program of five courses from a list consisting of English, Algebra, Geometry, Calculus, History, Art, Science, and Latin. This program must include English, at least one mathematics course (from Algebra, Geometry, or Calculus), and at least one humanities course (from History, Art, or Latin)... | 33 | 80 | 2 |
math | What is the greatest three-digit number "abc'' such that $4,a,b$ forms a geometric sequence and $b,c,5$ forms an arithmetic sequence? | 697 | 33 | 3 |
math | The lengths of the two diagonals of a rhombus are the two real roots of the equation $x^{2}-21x+30=0$. Find the area of the rhombus. | 15 | 43 | 2 |
math | By expanding the expression \((1+\sqrt{11})^{214}\) using the binomial theorem, we obtain terms of the form \(C_{214}^{k}(\sqrt{11})^{k}\). Find the value of \( k \) for which this term has the greatest value. | 165 | 68 | 3 |
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