Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
24,000 | Determine the difference between the sum of the first one hundred positive even integers and the sum of the first one hundred positive multiples of 3. | -5050 | 28.125 |
24,001 | The second hand on a clock is 8 cm long. How far in centimeters does the tip of the second hand travel during a period of 45 minutes? Express your answer in terms of $\pi$. | 720\pi | 93.75 |
24,002 | The difference between two numbers is 7.02. If the decimal point of the smaller number is moved one place to the right, it becomes the larger number. The larger number is \_\_\_\_\_\_, and the smaller number is \_\_\_\_\_\_. | 0.78 | 24.21875 |
24,003 | How many positive four-digit integers are divisible by both 13 and 7? | 99 | 29.6875 |
24,004 | The coefficients of the polynomial
\[ a_{12} x^{12} + a_{11} x^{11} + \dots + a_2 x^2 + a_1 x + a_0 = 0 \]
are all integers, and its roots $s_1, s_2, \dots, s_{12}$ are all integers. Furthermore, the roots of the polynomial
\[ a_0 x^{12} + a_1 x^{11} + a_2 x^{10} + \dots + a_{11} x + a_{12} = 0 \]
are also $s_1, s_2, \dots, s_{12}.$ Find the number of possible multisets $S = \{s_1, s_2, \dots, s_{12}\}.$ | 13 | 76.5625 |
24,005 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. The function $f(x)=2\cos x\sin (x-A)+\sin A (x\in R)$ reaches its maximum value at $x=\frac{5\pi}{12}$.
(1) Find the range of the function $f(x)$ when $x\in(0,\frac{\pi}{2})$;
(2) If $a=7$ and $\sin B+\sin C=\frac{13\sqrt{3}}{14}$, find the area of $\triangle ABC$. | 10\sqrt{3} | 63.28125 |
24,006 | The hyperbola $M$: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ has left and right foci $F_l$ and $F_2$. The parabola $N$: $y^{2} = 2px (p > 0)$ has a focus at $F_2$. Point $P$ is an intersection point of hyperbola $M$ and parabola $N$. If the midpoint of $PF_1$ lies on the $y$-axis, calculate the eccentricity of this hyperbola. | \sqrt{2} + 1 | 17.96875 |
24,007 | Given that $a$, $b$, and $c$ are the lengths of the sides opposite the angles $A$, $B$, and $C$ in $\triangle ABC$ respectively, with $a=2$, and $$\frac{\sin A - \sin B}{\sin C} = \frac{c - b}{2 + b}.$$ Find the maximum area of $\triangle ABC$. | \sqrt{3} | 40.625 |
24,008 | Maria needs to build a circular fence around a garden. Based on city regulations, the garden's diameter needs to be close to 30 meters, with an allowable error of up to $10\%$. After building, the fence turned out to have a diameter of 33 meters. Calculate the area she thought she was enclosing and the actual area enclosed. What is the percent difference between these two areas? | 21\% | 68.75 |
24,009 | In rectangle $ABCD,$ $P$ is a point on side $\overline{BC}$ such that $BP = 18$ and $CP = 6.$ If $\tan \angle APD = 2,$ find $AB.$ | 18 | 17.1875 |
24,010 | An odd function $f(x)$ defined on $R$ satisfies $f(x) = f(2-x)$. When $x \in [0,1]$, $f(x) = ax^{3} + 2x + a + 1$. Find $f(2023)$. | -1 | 31.25 |
24,011 | A bag contains 20 candies: 4 chocolate, 6 mint, and 10 butterscotch. Candies are removed randomly from the bag and eaten. What is the minimum number of candies that must be removed to be certain that at least two candies of each flavor have been eaten? | 18 | 54.6875 |
24,012 | Each row of a seating arrangement seats either 9 or 10 people. A total of 100 people are to be seated. How many rows seat exactly 10 people if every seat is occupied? | 10 | 70.3125 |
24,013 | Xiao Wang loves mathematics and chose the six numbers $6$, $1$, $8$, $3$, $3$, $9$ to set as his phone's startup password. If the two $3$s are not adjacent, calculate the number of different passwords Xiao Wang can set. | 240 | 55.46875 |
24,014 | Maria buys computer disks at a price of 5 for $6 and sells them at a price of 4 for $7. Find how many computer disks Maria must sell to make a profit of $120. | 219 | 50.78125 |
24,015 | Given three points in space \\(A(0,2,3)\\), \\(B(-2,1,6)\\), and \\(C(1,-1,5)\\):
\\((1)\\) Find \\(\cos < \overrightarrow{AB}, \overrightarrow{AC} >\\).
\\((2)\\) Find the area of the parallelogram with sides \\(AB\\) and \\(AC\\). | 7\sqrt{3} | 87.5 |
24,016 | The coefficient of the $x^2$ term in the expansion of $\sum\limits_{k=1}^{n}{(x+1)}^{k}$ is equal to the coefficient of the $x^{10}$ term. Determine the positive integer value of $n$. | 13 | 32.03125 |
24,017 | Determine the value of $\sin 135^{\circ}\cos 15^{\circ}-\cos 45^{\circ}\sin (-15^{\circ})$. | \frac{\sqrt{3}}{2} | 62.5 |
24,018 | Calculate: $-{1^{2023}}+|{\sqrt{3}-2}|-3\tan60°$. | 1 - 4\sqrt{3} | 92.1875 |
24,019 | In the Cartesian coordinate system $xOy$, the curve $C$ is given by $\frac{x^2}{4} + \frac{y^2}{3} = 1$. Taking the origin $O$ of the Cartesian coordinate system $xOy$ as the pole and the positive half-axis of $x$ as the polar axis, and using the same unit length, a polar coordinate system is established. It is known that the line $l$ is given by $\rho(\cos\theta - 2\sin\theta) = 6$.
(I) Write the Cartesian coordinate equation of line $l$ and the parametric equation of curve $C$;
(II) Find a point $P$ on curve $C$ such that the distance from point $P$ to line $l$ is maximized, and find this maximum value. | 2\sqrt{5} | 35.9375 |
24,020 | The value of $a$ is chosen so that the number of roots of the first equation $4^{x}-4^{-x}=2 \cos(a x)$ is 2007. How many roots does the second equation $4^{x}+4^{-x}=2 \cos(a x)+4$ have for the same value of $a$? | 4014 | 14.0625 |
24,021 | Calculate the volume in cubic centimeters of a truncated cone formed by cutting a smaller cone from a larger cone. The larger cone has a diameter of 8 cm at the base and a height of 10 cm. The smaller cone, which is cut from the top, has a diameter of 4 cm and a height of 4 cm. Express your answer in terms of \(\pi\). | 48\pi | 59.375 |
24,022 | Elective 4-4: Coordinate System and Parametric Equations
In the Cartesian coordinate system $xOy$, the curve $C$ is given by the parametric equations $\begin{cases} & x=2\sqrt{3}\cos \alpha \\ & y=2\sin \alpha \end{cases}$, where $\alpha$ is the parameter, $\alpha \in (0,\pi)$. In the polar coordinate system with the origin $O$ as the pole and the positive $x$-axis as the polar axis, the point $P$ has polar coordinates $(4\sqrt{2},\frac{\pi}{4})$, and the line $l$ has the polar equation $\rho \sin \left( \theta -\frac{\pi}{4} \right)+5\sqrt{2}=0$.
(I) Find the Cartesian equation of line $l$ and the standard equation of curve $C$;
(II) If $Q$ is a moving point on curve $C$, and $M$ is the midpoint of segment $PQ$, find the maximum distance from point $M$ to line $l$. | 6 \sqrt{2} | 79.6875 |
24,023 | Given that $a_1$, $a_2$, $a_3$, $a_4$, $a_5$ are five different integers satisfying the condition $a_1 + a_2 + a_3 + a_4 + a_5 = 9$, if $b$ is an integer root of the equation $(x - a_1)(x - a_2)(x - a_3)(x - a_4)(x - a_5) = 2009$, then the value of $b$ is. | 10 | 44.53125 |
24,024 | In the diagram, $ABCD$ is a trapezoid with an area of $18.$ $CD$ is three times the length of $AB.$ What is the area of $\triangle ABD?$
[asy]
draw((0,0)--(2,5)--(8,5)--(15,0)--cycle);
draw((8,5)--(0,0));
label("$D$",(0,0),W);
label("$A$",(2,5),NW);
label("$B$",(8,5),NE);
label("$C$",(15,0),E);
[/asy] | 4.5 | 75 |
24,025 | A club has between 300 and 400 members. The members gather every weekend and are divided into eight distinct groups. If two members are absent, the groups can all have the same number of members. What is the sum of all possible numbers of members in the club? | 4200 | 11.71875 |
24,026 | Given the circle $C:(x-1)^2+(y-2)^2=25$ and the line $l:(2m+1)x+(m+1)y-7m-4=0$, determine the length of the shortest chord intercepted by line $l$ on circle $C$. | 4\sqrt{5} | 17.1875 |
24,027 | The opposite of the arithmetic square root of $\sqrt{81}$ is ______. | -9 | 95.3125 |
24,028 | Complex numbers \( a \), \( b \), and \( c \) form an equilateral triangle with side length 24 in the complex plane. If \( |a + b + c| = 48 \), find \( |ab + ac + bc| \). | 768 | 39.84375 |
24,029 | From June to August 1861, a total of 1026 inches of rain fell in Cherrapunji, India, heavily influenced by the monsoon season, and the total duration of these summer months is 92 days and 24 hours per day. Calculate the average rainfall in inches per hour. | \frac{1026}{2208} | 0 |
24,030 | In $\triangle ABC$, $a=2$, $b=3$, $c=4$, find the cosine value of the largest internal angle. | -\frac{1}{4} | 64.0625 |
24,031 | The increasing sequence of positive integers $a_1, a_2, a_3, \dots$ is defined by the rule
\[a_{n + 2} = a_{n + 1} + a_n\]
for all $n \ge 1.$ If $a_7 = 210$, then find $a_8.$ | 340 | 16.40625 |
24,032 | The Euler family has four girls aged $6$, $6$, $9$, and $11$, and two boys aged $13$ and $16$. What is the mean (average) age of the children? | \frac{61}{6} | 11.71875 |
24,033 | In the rectangular coordinate system xOy, the parametric equations of the curve C1 are given by $$\begin{cases} x=t\cos\alpha \\ y=1+t\sin\alpha \end{cases}$$, and the polar coordinate equation of the curve C2 with the origin O as the pole and the positive semi-axis of the x-axis as the polar axis is ρ=2cosθ.
1. If the parameter of curve C1 is α, and C1 intersects C2 at exactly one point, find the Cartesian equation of C1.
2. Given point A(0, 1), if the parameter of curve C1 is t, 0<α<π, and C1 intersects C2 at two distinct points P and Q, find the maximum value of $$\frac {1}{|AP|}+\frac {1}{|AQ|}$$. | 2\sqrt{2} | 12.5 |
24,034 | Given $\tan \alpha=4$, calculate the value of $\frac{1+\cos 2\alpha+8\sin^2\alpha}{\sin 2\alpha}$. | \frac{65}{4} | 83.59375 |
24,035 | Given the function $f(x)=\sqrt{3}\sin x \cos x - \cos^2 x, (x \in \mathbb{R})$.
$(1)$ Find the intervals where $f(x)$ is monotonically increasing.
$(2)$ Find the maximum and minimum values of $f(x)$ on the interval $[-\frac{\pi}{4}, \frac{\pi}{4}]$. | -\frac{3}{2} | 52.34375 |
24,036 | The edge of a regular tetrahedron is equal to \(\sqrt{2}\). Find the radius of the sphere whose surface touches all the edges of the tetrahedron. | 1/2 | 80.46875 |
24,037 | Given that $\tan x = -\frac{12}{5}$ and $x \in (\frac{\pi}{2}, \pi)$, find the value of $\cos(-x + \frac{3\pi}{2})$. | -\frac{12}{13} | 59.375 |
24,038 | In an isosceles trapezoid \(ABCD\), \(AB\) is parallel to \(CD\), \(AB = 6\), \(CD = 14\), \(\angle AEC\) is a right angle, and \(CE = CB\). What is \(AE^2\)? | 84 | 7.03125 |
24,039 | If point P lies on the graph of the function $y=e^x$ and point Q lies on the graph of the function $y=\ln x$, then the minimum distance between points P and Q is \_\_\_\_\_\_. | \sqrt{2} | 85.9375 |
24,040 | Given that there are 6 balls of each of the four colors: red, blue, yellow, and green, each numbered from 1 to 6, calculate the number of ways to select 3 balls with distinct numbers, such that no two balls have the same color or consecutive numbers. | 96 | 64.0625 |
24,041 | Find the smallest solution to the equation \[\lfloor x^2 \rfloor - \lfloor x \rfloor^2 = 19.\] | \sqrt{109} | 0 |
24,042 | Let $G$ be the set of points $(x, y)$ such that $x$ and $y$ are positive integers less than or equal to 20. Say that a ray in the coordinate plane is *ocular* if it starts at $(0, 0)$ and passes through at least one point in $G$ . Let $A$ be the set of angle measures of acute angles formed by two distinct ocular rays. Determine
\[
\min_{a \in A} \tan a.
\] | 1/722 | 0 |
24,043 | Given a rectangular grid constructed with toothpicks of equal length, with a height of 15 toothpicks and a width of 12 toothpicks, calculate the total number of toothpicks required to build the grid. | 387 | 12.5 |
24,044 | In triangle $ABC$, the sides opposite to angles $A$, $B$, $C$ are respectively $a$, $b$, $c$. It is known that $2a\cos A=c\cos B+b\cos C$.
(Ⅰ) Find the value of $\cos A$;
(Ⅱ) If $a=1$ and $\cos^2 \frac{B}{2}+\cos^2 \frac{C}{2}=1+ \frac{\sqrt{3}}{4}$, find the value of side $c$. | \frac{\sqrt{3}}{3} | 7.8125 |
24,045 | Given two intersecting circles O: $x^2 + y^2 = 25$ and C: $x^2 + y^2 - 4x - 2y - 20 = 0$, which intersect at points A and B, find the length of the common chord AB. | \sqrt{95} | 84.375 |
24,046 | Points $M$ and $N$ are located on side $AC$ of triangle $ABC$, and points $K$ and $L$ are on side $AB$, with $AM : MN : NC = 1 : 3 : 1$ and $AK = KL = LB$. It is known that the area of triangle $ABC$ is 1. Find the area of quadrilateral $KLNM$. | 7/15 | 11.71875 |
24,047 | Solve for $\log_{3} \sqrt{27} + \lg 25 + \lg 4 + 7^{\log_{7} 2} + (-9.8)^{0} = \_\_\_\_\_\_\_\_\_\_\_$. | \frac{13}{2} | 85.9375 |
24,048 | Find the positive integer $n$ so that $n^2$ is the perfect square closest to $8 + 16 + 24 + \cdots + 8040.$ | 2011 | 56.25 |
24,049 | Find the sum $m + n$ where $m$ and $n$ are integers, such that the positive difference between the two roots of the quadratic equation $2x^2 - 5x - 12 = 0$ can be expressed as $\frac{\sqrt{m}}{n}$, and $m$ is not divisible by the square of any prime number. | 123 | 58.59375 |
24,050 | Find the smallest positive real number $x$ such that
\[\lfloor x^2 \rfloor - x \lfloor x \rfloor = 10.\] | \frac{131}{11} | 6.25 |
24,051 | Find the maximum value of
\[\frac{2x + 3y + 4}{\sqrt{x^2 + 4y^2 + 2}}\]
over all real numbers \( x \) and \( y \). | \sqrt{29} | 61.71875 |
24,052 | Given $f(x)=\cos x(\sqrt{3}\sin x-\cos x)+\frac{3}{2}$.
$(1)$ Find the interval on which $f(x)$ is monotonically decreasing on $[0,\pi]$.
$(2)$ If $f(\alpha)=\frac{2}{5}$ and $\alpha\in(\frac{\pi}{3},\frac{5\pi}{6})$, find the value of $\sin 2\alpha$. | \frac{-3\sqrt{3} - 4}{10} | 0 |
24,053 | Given that $\sin \alpha - 2\cos \alpha = \frac{\sqrt{10}}{2}$, find $\tan 2\alpha$. | \frac{3}{4} | 61.71875 |
24,054 | Six people enter two rooms, with the conditions that: ①each room receives three people; ②each room receives at least one person. How many distribution methods are there for each condition? | 62 | 87.5 |
24,055 | Given $\tan \theta =2$, find $\cos 2\theta =$____ and $\tan (\theta -\frac{π}{4})=$____. | \frac{1}{3} | 99.21875 |
24,056 | Given points $A$, $B$, and $C$ are on the curve $y=\sqrt{x}$ $(x \geqslant 0)$, with x-coordinates $1$, $m$, and $4$ $(1 < m < 4)$, when the area of $\triangle ABC$ is maximized, $m$ equals: | $\frac{9}{4}$ | 0 |
24,057 | Let $ABCD$ be a square with side length $2$ , and let a semicircle with flat side $CD$ be drawn inside the square. Of the remaining area inside the square outside the semi-circle, the largest circle is drawn. What is the radius of this circle? | 4 - 2\sqrt{3} | 3.90625 |
24,058 | We write the equation on the board:
$$
(x-1)(x-2) \ldots (x-2016) = (x-1)(x-2) \ldots (x-2016) .
$$
We want to erase some of the 4032 factors in such a way that the equation on the board has no real solutions. What is the minimal number of factors that need to be erased to achieve this? | 2016 | 71.875 |
24,059 | If $p$, $q$, $r$, $s$, $t$, and $u$ are integers such that $1728x^3 + 64 = (px^2 + qx + r)(sx^2 + tx + u)$ for all $x$, then what is $p^2+q^2+r^2+s^2+t^2+u^2$? | 23456 | 2.34375 |
24,060 | In rectangle $ABCD$, $AB=2$, $BC=4$, and points $E$, $F$, and $G$ are located as follows: $E$ is the midpoint of $\overline{BC}$, $F$ is the midpoint of $\overline{CD}$, and $G$ is one fourth of the way down $\overline{AD}$ from $A$. If point $H$ is the midpoint of $\overline{GE}$, what is the area of the shaded region defined by triangle $EHF$?
A) $\dfrac{5}{4}$
B) $\dfrac{3}{2}$
C) $\dfrac{7}{4}$
D) 2
E) $\dfrac{9}{4}$ | \dfrac{5}{4} | 64.0625 |
24,061 | Given the function $f(x)=2x^{2}-3x-\ln x+e^{x-a}+4e^{a-x}$, where $e$ is the base of the natural logarithm, if there exists a real number $x_{0}$ such that $f(x_{0})=3$ holds, then the value of the real number $a$ is \_\_\_\_\_\_. | 1-\ln 2 | 35.15625 |
24,062 | In a new diagram, triangle $A'B'C'$ has an area of 36 square units. The points $A', B', C', D'$ are aligned such that $A'C' = 12$ units and $C'D' = 30$ units. What is the area of triangle $B'C'D'$? | 90 | 77.34375 |
24,063 | Suppose
\[\frac{1}{x^3 - 3x^2 - 13x + 15} = \frac{A}{x+3} + \frac{B}{x-1} + \frac{C}{(x-1)^2}\]
where $A$, $B$, and $C$ are real constants. What is $A$? | \frac{1}{16} | 73.4375 |
24,064 | What is the smallest square number that, when divided by a cube number, results in a fraction in its simplest form where the numerator is a cube number (other than 1) and the denominator is a square number (other than 1)? | 64 | 6.25 |
24,065 | A square has vertices at \((-2a, -2a), (2a, -2a), (-2a, 2a), (2a, 2a)\). The line \( y = -\frac{x}{2} \) cuts this square into two congruent quadrilaterals. Calculate the perimeter of one of these quadrilaterals divided by \( 2a \). Express your answer in simplified radical form. | 4 + \sqrt{5} | 56.25 |
24,066 | Determine the constant $d$ such that
$$\left(3x^3 - 2x^2 + x - \frac{5}{4}\right)(ex^3 + dx^2 + cx + f) = 9x^6 - 5x^5 - x^4 + 20x^3 - \frac{25}{4}x^2 + \frac{15}{4}x - \frac{5}{2}$$ | \frac{1}{3} | 44.53125 |
24,067 | There are six unmarked envelopes on a table, each containing a letter for a different person. If the mail is randomly distributed among these six people, with each person getting one letter, what is the probability that exactly three people get the right letter? | \frac{1}{18} | 94.53125 |
24,068 | Let set $\mathcal{C}$ be a 70-element subset of $\{1,2,3,\ldots,120\},$ and let $P$ be the sum of the elements of $\mathcal{C}.$ Find the number of possible values of $P.$ | 3501 | 95.3125 |
24,069 | Find the sum of $245_8$, $174_8$, and $354_8$ in base 8. | 1015_8 | 65.625 |
24,070 | If $\sin(\frac{\pi}{6} - \alpha) = \frac{2}{3}$, find the value of $\cos(\frac{2\pi}{3} - \alpha)$. | -\frac{2}{3} | 89.0625 |
24,071 | A line passes through the point $(-2, \sqrt{3})$ at an angle of inclination of $\frac{\pi}{3}$. Find the parametric equation of the line, and if it intersects the curve $y^2 = -x - 1$ at points A and B, find the length $|AB|$. | \frac{10}{3} | 45.3125 |
24,072 | Let $ ABCD$ be a unit square (that is, the labels $ A, B, C, D$ appear in that order around the square). Let $ X$ be a point outside of the square such that the distance from $ X$ to $ AC$ is equal to the distance from $ X$ to $ BD$ , and also that $ AX \equal{} \frac {\sqrt {2}}{2}$ . Determine the value of $ CX^2$ . | 5/2 | 46.09375 |
24,073 | Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ have an angle of $45^\circ$ between them, and $|\overrightarrow{a}|=1$, $|2\overrightarrow{a}-\overrightarrow{b}|=\sqrt{10}$, find the magnitude of vector $\overrightarrow{b}$. | 3\sqrt{2} | 94.53125 |
24,074 | The average of the numbers $1, 2, 3, \dots, 44, 45, x$ is $50x$. What is $x$? | \frac{1035}{2299} | 63.28125 |
24,075 | Grandma has two balls of yarn: one large and one small. From the large ball, she can either knit a sweater and three socks, or five identical hats. From the small ball, she can either knit half a sweater or two hats. (In both cases, all the yarn will be used up.) What is the maximum number of socks Grandma can knit using both balls of yarn? Justify your answer. | 21 | 5.46875 |
24,076 | A set of 36 square blocks is arranged into a 6 × 6 square. How many different combinations of 4 blocks can be selected from that set so that no two blocks are in the same row or column? | 5400 | 94.53125 |
24,077 | Given points $A$, $B$, and $C$ are on the curve $y=\sqrt{x}$ $(x \geqslant 0)$, with x-coordinates $1$, $m$, and $4$ $(1 < m < 4)$, find the value of $m$ that maximizes the area of $\triangle ABC$. | \frac{9}{4} | 92.1875 |
24,078 | Class 2 of the second grade has 42 students, including $n$ male students. They are numbered from 1 to $n$. During the winter vacation, student number 1 called 3 students, student number 2 called 4 students, student number 3 called 5 students, ..., and student number $n$ called half of the students. Determine the number of female students in the class. | 23 | 18.75 |
24,079 | Given the set $\{2,3,5,7,11,13\}$, add one of the numbers twice to another number, and then multiply the result by the third number. What is the smallest possible result? | 22 | 0 |
24,080 | Consider the sum
\[
S_n = \sum_{k = 1}^n \frac{1}{\sqrt{2k-1}} \, .
\]
Determine $\lfloor S_{4901} \rfloor$ . Recall that if $x$ is a real number, then $\lfloor x \rfloor$ (the *floor* of $x$ ) is the greatest integer that is less than or equal to $x$ .
| 98 | 75 |
24,081 | The arithmetic mean of these six expressions is 30. What is the value of $y$? $$y + 10 \hspace{.5cm} 20 \hspace{.5cm} 3y \hspace{.5cm} 18 \hspace{.5cm} 3y + 6 \hspace{.5cm} 12$$ | \frac{114}{7} | 69.53125 |
24,082 | (Ⅰ) Find the equation of the line that passes through the intersection point of the two lines $2x-3y-3=0$ and $x+y+2=0$, and is perpendicular to the line $3x+y-1=0$.
(Ⅱ) Given the equation of line $l$ in terms of $x$ and $y$ as $mx+y-2(m+1)=0$, find the maximum distance from the origin $O$ to the line $l$. | 2 \sqrt {2} | 0 |
24,083 | Jane places six ounces of tea into a ten-ounce cup and six ounces of milk into a second cup of the same size. She then pours two ounces of tea from the first cup to the second and, after stirring thoroughly, pours two ounces of the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now milk?
A) $\frac{1}{8}$
B) $\frac{1}{6}$
C) $\frac{1}{4}$
D) $\frac{1}{3}$
E) $\frac{1}{2}$ | \frac{1}{4} | 86.71875 |
24,084 | Given that $θ∈[0,π]$, find the probability that $\sin (θ+ \frac {π}{3}) < \frac {1}{2}$. | \frac{1}{2} | 36.71875 |
24,085 | A student needs to provide his waist size in centimeters for a customized lab coat, based on the measurements in inches. If there are $10$ inches in a foot and $25$ centimeters in a foot, then what size should the student specify in centimeters if his waist size is $40$ inches? | 100 | 59.375 |
24,086 | Given that $\tan \alpha=-2$, the focus of the parabola $y^{2}=2px (p > 0)$ is $F(-\sin \alpha\cos \alpha,0)$, and line $l$ passes through point $F$ and intersects the parabola at points $A$ and $B$ with $|AB|=4$, find the distance from the midpoint of segment $AB$ to the line $x=-\frac{1}{2}$. | \frac{21}{10} | 38.28125 |
24,087 | In triangle $\triangle ABC$, $b\sin 2A = \sqrt{3}a\sin B$.
$(Ⅰ)$ Find $\angle A$;
$(Ⅱ)$ If the area of $\triangle ABC$ is $3\sqrt{3}$, choose one of the three conditions, condition ①, condition ②, or condition ③, as the given condition to ensure the existence and uniqueness of $\triangle ABC$, and find the value of $a$.
Condition ①: $\sin C = \frac{2\sqrt{7}}{7}$;
Condition ②: $\frac{b}{c} = \frac{3\sqrt{3}}{4}$;
Condition ③: $\cos C = \frac{\sqrt{21}}{7}$
Note: If the chosen condition does not meet the requirements, no points will be awarded for question $(Ⅱ)$. If multiple suitable conditions are chosen and answered separately, the first answer will be scored. | \sqrt{7} | 55.46875 |
24,088 | Let's consider the sum of five consecutive odd numbers starting from 997 up to 1005. If the sum $997 + 999 + 1001 + 1003 + 1005 = 5100 - M$, find $M$. | 95 | 69.53125 |
24,089 | Find the sum of the \(1005\) roots of the polynomial \((x-1)^{1005} + 2(x-2)^{1004} + 3(x-3)^{1003} + \cdots + 1004(x-1004)^2 + 1005(x-1005)\). | 1003 | 64.0625 |
24,090 | For a positive integer $n$ , define $n?=1^n\cdot2^{n-1}\cdot3^{n-2}\cdots\left(n-1\right)^2\cdot n^1$ . Find the positive integer $k$ for which $7?9?=5?k?$ .
*Proposed by Tristan Shin* | 10 | 25.78125 |
24,091 | The 5 on the tenths place is \_\_\_\_\_ more than the 5 on the hundredths place. | 0.45 | 85.9375 |
24,092 | Given the set S={1, 2, 3, ..., 40}, and a subset A⊆S containing three elements, find the number of such sets A that can form an arithmetic progression. | 380 | 71.09375 |
24,093 | If the binomial coefficient of only the sixth term in the expansion of $(\sqrt{x} - \frac{2}{x^{2}})^{n}$ is the largest, then the constant term in the expansion is _______. | 180 | 76.5625 |
24,094 | The graph of the function $f(x)=\frac{x}{x+a}$ is symmetric about the point $(1,1)$, and the function $g(x)=\log_{10}(10^x+1)+bx$ is even. Find the value of $a+b$. | -\frac{3}{2} | 46.09375 |
24,095 | Jessica has exactly one of each of the first 30 states' new U.S. quarters. The quarters were released in the same order that the states joined the union. The graph below shows the number of states that joined the union in each decade. What fraction of Jessica's 30 coins represents states that joined the union during the decade 1800 through 1809? Express your answer as a common fraction.
[asy]size(200);
label("1780",(6,0),S);
label("1800",(12,0),S);
label("1820",(18,0),S);
label("1840",(24,0),S);
label("1860",(30,0),S);
label("1880",(36,0),S);
label("1900",(42,0),S);
label("1950",(48,0),S);
label("to",(6,-4),S);
label("to",(12,-4),S);
label("to",(18,-4),S);
label("to",(24,-4),S);
label("to",(30,-4),S);
label("to",(36,-4),S);
label("to",(42,-4),S);
label("to",(48,-4),S);
label("1789",(6,-8),S);
label("1809",(12,-8),S);
label("1829",(18,-8),S);
label("1849",(24,-8),S);
label("1869",(30,-8),S);
label("1889",(36,-8),S);
label("1909",(42,-8),S);
label("1959",(48,-8),S);
draw((0,0)--(50,0));
draw((0,2)--(50,2));
draw((0,4)--(50,4));
draw((0,6)--(50,6));
draw((0,8)--(50,8));
draw((0,10)--(50,10));
draw((0,12)--(50,12));
draw((0,14)--(50,14));
draw((0,16)--(50,16));
draw((0,18)--(50,18));
fill((4,0)--(8,0)--(8,12)--(4,12)--cycle,gray(0.8));
fill((10,0)--(14,0)--(14,5)--(10,5)--cycle,gray(0.8));
fill((16,0)--(20,0)--(20,7)--(16,7)--cycle,gray(0.8));
fill((22,0)--(26,0)--(26,6)--(22,6)--cycle,gray(0.8));
fill((28,0)--(32,0)--(32,7)--(28,7)--cycle,gray(0.8));
fill((34,0)--(38,0)--(38,5)--(34,5)--cycle,gray(0.8));
fill((40,0)--(44,0)--(44,4)--(40,4)--cycle,gray(0.8));
[/asy] | \frac{1}{6} | 81.25 |
24,096 | Tim is organizing a week-long series of pranks. On Monday, he gets his friend Joe to help. On Tuesday, he can choose between two friends, either Ambie or John. For Wednesday, there are four new people willing to help. However, on Thursday, none of these previous individuals can participate, but Tim has convinced five different friends to help. On Friday, Tim decides to go solo again. Additionally, this time, whoever helps on Wednesday cannot help on Thursday. How many different combinations of people could Tim involve in his pranks during the week? | 40 | 13.28125 |
24,097 | A point is chosen at random within a rectangle in the coordinate plane whose vertices are $(0, 0), (3030, 0), (3030, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{3}{4}$. Find the value of $d$ to the nearest tenth. | 0.5 | 32.03125 |
24,098 | Calculate the value of $v_2$ when $x = 2$ for $f(x) = 3x^4 + x^3 + 2x^2 + x + 4$ using Horner's method. | 16 | 79.6875 |
24,099 | A certain shopping mall sells two types of products, A and B. The profit margin for each unit of product A is $40\%$, and for each unit of product B is $50\%$. When the quantity of product A sold is $150\%$ of the quantity of product B sold, the total profit margin for selling these two products in the mall is $45\%$. Determine the total profit margin when the quantity of product A sold is $50\%$ of the quantity of product B sold. | 47.5\% | 50.78125 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.