Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
23,900 | Given that the power function $y=x^{m}$ is an even function and is a decreasing function when $x \in (0,+\infty)$, determine the possible value of the real number $m$. | -2 | 94.53125 |
23,901 | Remove all perfect squares from the sequence of positive integers \(1, 2, 3, \cdots\) to get a new sequence. What is the 2003rd term of this new sequence? | 2048 | 96.09375 |
23,902 | In a box, there are 22 kg of cranberries. How, using a single 2-kilogram weight and a two-pan scale, can you measure out 17 kg of cranberries in two weighings? | 17 | 62.5 |
23,903 | Read the text below and answer the questions. Everyone knows that $\sqrt{2}$ is an irrational number, and irrational numbers are infinite non-repeating decimals. Therefore, we cannot write out all the decimal parts of $\sqrt{2}$, but since $1 \lt \sqrt{2} \lt 2$, the integer part of $\sqrt{2}$ is $1$. Subtracting the integer part $1$ from $\sqrt{2}$ gives the decimal part as $(\sqrt{2}-1)$. Answer the following questions:
$(1)$ The integer part of $\sqrt{10}$ is ______, and the decimal part is ______;
$(2)$ If the decimal part of $\sqrt{6}$ is $a$, and the integer part of $\sqrt{13}$ is $b$, find the value of $a+b-\sqrt{6}$;
$(3)$ Given $12+\sqrt{3}=x+y$, where $x$ is an integer and $0 \lt y \lt 1$, find the opposite of $x-y$. | \sqrt{3} - 14 | 8.59375 |
23,904 | Three distinct real numbers form (in some order) a 3-term arithmetic sequence, and also form (in possibly a different order) a 3-term geometric sequence. Compute the greatest possible value of the common ratio of this geometric sequence. | -2 | 11.71875 |
23,905 | If 500 were expressed as a sum of at least two distinct powers of 2, what would be the least possible sum of the exponents of these powers? | 32 | 21.875 |
23,906 | The domain of the function \( f(x) \) is \( D \). If for any \( x_{1}, x_{2} \in D \), when \( x_{1} < x_{2} \), it holds that \( f(x_{1}) \leq f(x_{2}) \), then \( f(x) \) is called a non-decreasing function on \( D \). Suppose that the function \( f(x) \) is non-decreasing on \( [0,1] \) and satisfies the following three conditions:
1. \( f(0)=0 \);
2. \( f\left(\frac{x}{3}\right)=\frac{1}{2}f(x) \);
3. \( f(1-x)=1-f(x) \).
What is \( f\left(\frac{5}{12}\right) + f\left(\frac{1}{8}\right) \)? | \frac{3}{4} | 46.875 |
23,907 | The average of seven numbers in a list is 62. The average of the first four numbers is 55. What is the average of the last three numbers? | 71.\overline{3} | 0 |
23,908 | Circle I is externally tangent to Circle II and passes through the center of Circle II. Given that the area of Circle I is increased to 16 square inches, determine the area of Circle II, in square inches. | 64 | 42.1875 |
23,909 | Given $60\%$ of students like dancing and the rest dislike it, $80\%$ of those who like dancing say they like it and the rest say they dislike it, also $90\%$ of those who dislike dancing say they dislike it and the rest say they like it. Calculate the fraction of students who say they dislike dancing but actually like it. | 25\% | 0 |
23,910 | Find the sum of the first eight prime numbers that have a units digit of 3. | 394 | 0.78125 |
23,911 | Given the sequence ${a_{n}}$ where ${a_{1}}=1$, ${a_{2}}=2$, and ${a_{n+2}}-{a_{n}}=2-2{(-1)^{n}}$, $n\in {N^{*}}$, find the value of ${S_{2017}}$. | 2017\times1010-1 | 0 |
23,912 | In right triangle $ABC$, $\sin A = \frac{8}{17}$ and $\sin B = 1$. Find $\sin C$. | \frac{15}{17} | 71.09375 |
23,913 | Cat and Claire are having a conversation about Cat's favorite number.
Cat says, "My favorite number is a two-digit positive integer with distinct nonzero digits, $\overline{AB}$ , such that $A$ and $B$ are both factors of $\overline{AB}$ ."
Claire says, "I don't know your favorite number yet, but I do know that among four of the numbers that might be your favorite number, you could start with any one of them, add a second, subtract a third, and get the fourth!"
Cat says, "That's cool, and my favorite number is among those four numbers! Also, the square of my number is the product of two of the other numbers among the four you mentioned!"
Claire says, "Now I know your favorite number!"
What is Cat's favorite number?
*Proposed by Andrew Wu* | 24 | 44.53125 |
23,914 | Point D is from AC of triangle ABC so that 2AD=DC. Let DE be perpendicular to BC and AE
intersects BD at F. It is known that triangle BEF is equilateral. Find <ADB? | 90 | 20.3125 |
23,915 | A dormitory of a certain high school senior class has 8 people. In a health check, the weights of 7 people were measured to be 60, 55, 60, 55, 65, 50, 50 (in kilograms), respectively. One person was not measured due to some reasons, and it is known that the weight of this student is between 50 and 60 kilograms. The probability that the median weight of the dormitory members in this health check is 55 is __. | \frac{1}{2} | 9.375 |
23,916 | If the function $f(x)=\sin \omega x+\sqrt{3}\cos \omega x$ $(x\in \mathbb{R})$, and $f(\alpha)=-2,f(\beta)=0$, with the minimum value of $|\alpha -\beta|$ being $\frac{3\pi}{4}$, determine the value of the positive number $\omega$. | \frac{2}{3} | 50.78125 |
23,917 | For how many integer values of $n$ between 1 and 180 inclusive does the decimal representation of $\frac{n}{180}$ terminate? | 60 | 36.71875 |
23,918 | A person's commute times (in minutes) for 5 trips were 12, 8, 10, 11, and 9, respectively. The standard deviation of this set of data is ______. | \sqrt{2} | 18.75 |
23,919 | Given the hyperbola $\frac{y^2}{4}-x^2=1$, find the value of $\cos 2\alpha$, where $\alpha$ is the acute angle between its two asymptotes. | -\frac{7}{25} | 92.1875 |
23,920 | Find the area of a triangle with angles $\frac{1}{7} \pi$ , $\frac{2}{7} \pi$ , and $\frac{4}{7} \pi $ , and radius of its circumscribed circle $R=1$ . | \frac{\sqrt{7}}{4} | 68.75 |
23,921 | A right circular cone is sliced into five pieces of equal height by planes parallel to its base. Determine the ratio of the volume of the second-largest piece to the volume of the largest piece. | \frac{37}{61} | 23.4375 |
23,922 | Bangladesh National Mathematical Olympiad 2016 Higher Secondary
<u>**Problem 2:**</u>
(a) How many positive integer factors does $6000$ have?
(b) How many positive integer factors of $6000$ are not perfect squares? | 34 | 75.78125 |
23,923 | 10 - 1.05 ÷ [5.2 × 14.6 - (9.2 × 5.2 + 5.4 × 3.7 - 4.6 × 1.5)] = ? | 9.93 | 3.125 |
23,924 | Express the quotient $1021_3 \div 11_3$ in base $3$. | 22_3 | 100 |
23,925 | Given $a\in \mathbb{R}$, $b\in \mathbb{R}$, if the set $\{a, \frac{b}{a}, 1\} = \{a^{2}, a-b, 0\}$, calculate the value of $a^{2019}+b^{2019}$. | -1 | 76.5625 |
23,926 | Let's consider two positive real numbers $a$ and $b$, where an operation $a \, \blacktriangle \, b$ is defined such that $(ab) \, \blacktriangle \, b = a(b \, \blacktriangle \, b)$ and $(a \, \blacktriangle \, 1) \, \blacktriangle \, a = a \, \blacktriangle \, 1$ for all $a,b>0$. Additionally, it is given that $1 \, \blacktriangle \, 1 = 2$. Find the value of $23 \, \blacktriangle \, 45$. | 2070 | 25 |
23,927 | Five six-sided dice are rolled. It is known that after the roll, there are two pairs of dice showing the same number, and one odd die. The odd die is rerolled. What is the probability that after rerolling the odd die, the five dice show a full house? | \frac{1}{3} | 31.25 |
23,928 | If five people are selected at random from a group of ten men and five women, what is the probability that at least one woman is selected? Express your answer as a common fraction. | \frac{917}{1001} | 3.125 |
23,929 | Evaluate the expression $\log_{y^8}{x^2}\cdot\log_{x^7}{y^3}\cdot\log_{y^5}{x^4}\cdot\log_{x^4}{y^5}\cdot\log_{y^3}{x^7}$ and express it as $b\log_y{x}$ for some constant $b$. | \frac{1}{4} | 10.15625 |
23,930 | In a "sing, read, speak, and spread" performance activity participated by six units, including units A and B, each unit's program is arranged together. If a lottery method is used to randomly determine the order of performance for each unit (numbered 1, 2, …, 6), calculate:
(Ⅰ) The probability that both units A and B have even numbers as their performance order;
(Ⅱ) The probability that the performance order numbers of units A and B are not adjacent. | \frac{2}{3} | 90.625 |
23,931 | How many four-digit positive integers $y$ satisfy $5678y + 123 \equiv 890 \pmod{29}$? | 310 | 64.84375 |
23,932 | A younger brother leaves home and walks to the park at a speed of 4 kilometers per hour. Two hours later, the older brother leaves home and rides a bicycle at a speed of 20 kilometers per hour to catch up with the younger brother. How long will it take for the older brother to catch up with the younger brother? | 0.5 | 78.125 |
23,933 | Suppose $f(x)$ and $g(x)$ are functions satisfying $f(g(x)) = x^2$ and $g(f(x)) = x^4$ for all $x \ge 1.$ If $g(81) = 81,$ compute $[g(9)]^4.$ | 81 | 50.78125 |
23,934 | A tangent line is drawn to the moving circle $C: x^2 + y^2 - 2ay + a^2 - 2 = 0$ passing through the fixed point $P(2, -1)$. If the point of tangency is $T$, then the minimum length of the line segment $PT$ is \_\_\_\_\_\_. | \sqrt {2} | 0 |
23,935 | How many times does the digit 9 appear in the list of all integers from 1 to 1000? | 300 | 100 |
23,936 | A circle with center $D$ and radius four feet is tangent at $E$ to a circle with center $F$, as shown. If point $F$ is on the small circle, what is the area of the shaded region? Express your answer in terms of $\pi$.
[asy]
filldraw(circle((0,0),8),gray,linewidth(2));
filldraw(circle(4dir(-30),4),white,linewidth(2));
dot((0,0));
dot(4dir(-30));
dot(8dir(-30));
label("$F$",(0,0),NW);
label("$D$",4dir(-30),NE);
label("$E$",8dir(-30),SE);
[/asy] | 48\pi | 93.75 |
23,937 | A positive real number $x$ is such that \[
\sqrt[3]{1-x^4} + \sqrt[3]{1+x^4} = 1.
\]Find $x^8.$ | \frac{28}{27} | 78.125 |
23,938 | A math test consists of 12 multiple-choice questions, each worth 5 points. It is known that a student is confident in correctly answering 6 of these questions. For another three questions, the student can eliminate one incorrect option. For two questions, the student can eliminate two incorrect options. For the last question, due to a lack of understanding, the student has to guess randomly. Estimate the score of this student in this test. | 41.25 | 44.53125 |
23,939 | Given the function $y=x^{2}+bx+3$ (where $b$ is a real number), the range of $y$ is $\left[0,+\infty \right)$. Find the value of the real number $c$ if the solution set of the inequality $x^{2}+bx+3 \lt c$ is $m-8 \lt x \lt m$. | 16 | 83.59375 |
23,940 | Given a group with the numbers $-3, 0, 5, 8, 11, 13$, and the following rules: the largest isn't first, and it must be within the first four places, the smallest isn't last, and it must be within the last four places, and the median isn't in the first or last position, determine the average of the first and last numbers. | 5.5 | 5.46875 |
23,941 | In the complex plane, the distance between the points corresponding to the complex numbers $-3+i$ and $1-i$ is $\boxed{\text{answer}}$. | \sqrt{20} | 0 |
23,942 | In a cube $ABCDEFGH$, the coordinates of vertices are set in a conventional cube alignment with $A(0, 0, 0)$, $B(2, 0, 0)$, $C(2, 0, 2)$, $D(0, 0, 2)$, $E(0, 2, 0)$, $F(2, 2, 0)$, $G(2, 2, 2)$, and $H(0, 2, 2)$. Let $M$ and $N$ be the midpoints of the segments $\overline{EB}$ and $\overline{HD}$, respectively. Determine the ratio $S^2$ where $S$ is the ratio of the area of triangle $MNC$ to the total surface area of the cube.
A) $\frac{1}{144}$
B) $\frac{17}{2304}$
C) $\frac{1}{48}$
D) $\frac{1}{96}$ | \frac{17}{2304} | 43.75 |
23,943 | Given that the sequence {a<sub>n</sub>} is an arithmetic sequence, if $\frac {a_{11}}{a_{10}}$ < -1, and its first n terms sum S<sub>n</sub> has a maximum value, determine the maximum value of n that makes S<sub>n</sub> > 0. | 19 | 67.96875 |
23,944 | A volunteer organizes a spring sports event and wants to form a vibrant and well-trained volunteer team. They plan to randomly select 3 people from 4 male volunteers and 3 female volunteers to serve as the team leader. The probability of having at least one female volunteer as the team leader is ____; given the condition that "at least one male volunteer is selected from the 3 people drawn," the probability of "all 3 people drawn are male volunteers" is ____. | \frac{2}{17} | 11.71875 |
23,945 | Calculate the area of the polygon with vertices at $(2,1)$, $(4,3)$, $(6,1)$, $(4,-2)$, and $(3,4)$. | \frac{11}{2} | 0.78125 |
23,946 | If $$\sin\alpha= \frac {4}{7} \sqrt {3}$$ and $$\cos(\alpha+\beta)=- \frac {11}{14}$$, and $\alpha$, $\beta$ are acute angles, then $\beta= \_\_\_\_\_\_$. | \frac {\pi}{3} | 55.46875 |
23,947 | Xiaoming has several 1-yuan, 2-yuan, and 5-yuan banknotes. He wants to use no more than 10 banknotes to buy a kite priced at 18 yuan, and he must use at least two different denominations. How many different payment methods are possible? | 11 | 7.03125 |
23,948 | Given the line $l: 4x+3y-8=0$ passes through the center of the circle $C: x^2+y^2-ax=0$ and intersects circle $C$ at points A and B, with O as the origin.
(I) Find the equation of circle $C$.
(II) Find the equation of the tangent to circle $C$ at point $P(1, \sqrt {3})$.
(III) Find the area of $\triangle OAB$. | \frac{16}{5} | 57.8125 |
23,949 | A circle with its center at point $M$ on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$) is tangent to the $x$-axis exactly at one of the foci $F$ of the hyperbola, and intersects the $y$-axis at points $P$ and $Q$. If $\triangle MPQ$ is an equilateral triangle, calculate the eccentricity of the hyperbola. | \sqrt{3} | 22.65625 |
23,950 | In triangle $ABC$ , let $P$ and $R$ be the feet of the perpendiculars from $A$ onto the external and internal bisectors of $\angle ABC$ , respectively; and let $Q$ and $S$ be the feet of the perpendiculars from $A$ onto the internal and external bisectors of $\angle ACB$ , respectively. If $PQ = 7, QR = 6$ and $RS = 8$ , what is the area of triangle $ABC$ ? | 84 | 49.21875 |
23,951 | Given there are 1001 red marbles and 1001 black marbles in a box, find the absolute value of the difference between the probability that two marbles drawn at random from the box are the same color and the probability that they are different colors. | \frac{1}{2001} | 32.8125 |
23,952 | A hairdresser moved from Vienna to Debrecen to continue his trade. Over the course of 3 years, he became impoverished despite having some money originally. In the first year, he had to spend half of his money. In the second year, he spent a third of what he initially took with him. In the third year, he spent 200 forints, leaving him with only 50 forints for returning. How many forints did he have when he moved, and how much did he spend each year? | 1500 | 63.28125 |
23,953 | Filling the gas tank of a small car cost, in updated values, $\mathrm{R} \$ 29.90$ in 1972 and $\mathrm{R} \$ 149.70$ in 1992. Which of the following values best approximates the percentage increase in the price of gasoline during this 20-year period?
(a) $20 \%$
(b) $125 \%$
(c) $300 \%$
(d) $400 \%$
(e) $500 \%$ | 400\% | 100 |
23,954 | Let \( A, B, C, \) and \( D \) be positive real numbers such that
\[
\log_{10} (AB) + \log_{10} (AC) = 3, \\
\log_{10} (CD) + \log_{10} (CB) = 4, \\
\log_{10} (DA) + \log_{10} (DB) = 5.
\]
Compute the value of the product \( ABCD \). | 10000 | 49.21875 |
23,955 | Given \\(a, b, c > 0\\), the minimum value of \\(\frac{a^{2} + b^{2} + c^{2}}{ab + 2bc}\\) is \_\_\_\_\_\_. | \frac{2 \sqrt{5}}{5} | 0.78125 |
23,956 | Read the following material before solving the problem: In mathematics, there are numbers with square roots that contain another square root, which can be simplified by using the complete square formula and the properties of quadratic surds. For example, $\sqrt{3+2\sqrt{2}}=\sqrt{3+2×1×\sqrt{2}}=\sqrt{{1^2}+2×1×\sqrt{2}+{{({\sqrt{2}})}^2}}=\sqrt{{{({1+\sqrt{2}})}^2}}=|1+\sqrt{2}|=1+\sqrt{2}$.
Solve the following problems:
$(1) \sqrt{7+4\sqrt{3}}$;
$(2) \sqrt{9-4\sqrt{5}}$. | \sqrt{5}-2 | 87.5 |
23,957 | Convert the binary number $111011001001_{(2)}$ to its corresponding decimal number. | 3785 | 17.1875 |
23,958 | Given that $\{a_n\}$ is a sequence of positive numbers, and the sum of its first $n$ terms $2S_n = a_n^2 + a_n$ ($n \in \mathbb{N}^*$), the sequence $\{b_n\}$ satisfies $b_1 = \frac{3}{2}$, $b_{n+1} = b_n + 3^{a_n}$ ($n \in \mathbb{N}^*$).
(I) Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$.
(II) If $c_n = a_n b_n$ ($n \in \mathbb{N}^*$), and the sum of the first $n$ terms of the sequence $\{c_n\}$ is $T_n$, find $\lim_{n \to \infty} \frac{T_n}{c_n}$. | \frac{3}{2} | 56.25 |
23,959 | Given $\sin(3\pi + \alpha) = -\frac{1}{2}$, find the value of $\cos\left(\frac{7\pi}{2} - \alpha\right)$. | -\frac{1}{2} | 99.21875 |
23,960 | Sector $OAB$ is a quarter of a circle with a radius of 6 cm. A circle is inscribed within this sector, tangent to both the radius lines $OA$ and $OB$, and the arc $AB$. Determine the radius of the inscribed circle in centimeters. Express your answer in simplest radical form. | 6\sqrt{2} - 6 | 3.125 |
23,961 | The sides of a triangle have lengths of $13$, $84$, and $85$. Find the length of the shortest altitude. | 12.8470588235 | 0 |
23,962 | Given the function $f(x)=ax^{3}+bx+1$, and $f(-2)=3$, find $f(2)=$ _____ . | -1 | 100 |
23,963 | Three players are playing table tennis. The player who loses a game gives up their spot to the player who did not participate in that game. In the end, it turns out that the first player played 10 games, and the second player played 21 games. How many games did the third player play? | 11 | 43.75 |
23,964 | Read the following text and answer the questions:<br/>$\because \sqrt{1}＜\sqrt{2}＜\sqrt{4}$, which means $1＜\sqrt{2}＜2$,<br/>$\therefore$ The integer part of $\sqrt{2}$ is $1$, and the decimal part is $\sqrt{2}-1$.<br/>Please answer:<br/>$(1)$ The integer part of $\sqrt{33}$ is ______, and the decimal part is ______;<br/>$(2)$ If the decimal part of $\sqrt{143}$ is $a$, and the integer part of $\sqrt{43}$ is $b$, find the value of $a+|2b-\sqrt{143}|$;<br/>$(3)$ Given: $10+\sqrt{5}=2x+y$, where $x$ is an integer, and $0 \lt y \lt 1$, find the opposite of $x-y$. | \sqrt{5} - 8 | 19.53125 |
23,965 |
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0}\left(3-\frac{2}{\cos x}\right)^{\operatorname{cosec}^{2} x}
$$ | e^{-1} | 22.65625 |
23,966 | Given \\(x \geqslant 0\\), \\(y \geqslant 0\\), \\(x\\), \\(y \in \mathbb{R}\\), and \\(x+y=2\\), find the minimum value of \\( \dfrac {(x+1)^{2}+3}{x+2}+ \dfrac {y^{2}}{y+1}\\). | \dfrac {14}{5} | 0 |
23,967 | In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively.
$(1)$ If $2a\sin B = \sqrt{3}b$, find the measure of angle $A$.
$(2)$ If the altitude on side $BC$ is equal to $\frac{a}{2}$, find the maximum value of $\frac{c}{b} + \frac{b}{c}$. | 2\sqrt{2} | 37.5 |
23,968 | How many of the fractions $ \frac{1}{2023}, \frac{2}{2023}, \frac{3}{2023}, \cdots, \frac{2022}{2023} $ simplify to a fraction whose denominator is prime? | 22 | 3.125 |
23,969 | $O$ and $I$ are the circumcentre and incentre of $\vartriangle ABC$ respectively. Suppose $O$ lies in the interior of $\vartriangle ABC$ and $I$ lies on the circle passing through $B, O$ , and $C$ . What is the magnitude of $\angle B AC$ in degrees? | 60 | 51.5625 |
23,970 | A rectangle has a perimeter of 80 inches and each side has an integer length. Additionally, one dimension must be at least twice as long as the other. How many non-congruent rectangles meet these criteria? | 13 | 39.84375 |
23,971 | Given the expression $(1296^{\log_6 4096})^{\frac{1}{4}}$, calculate its value. | 4096 | 39.0625 |
23,972 | Let \(\mathbf{v}\) be a vector such that
\[
\left\| \mathbf{v} + \begin{pmatrix} 4 \\ 2 \end{pmatrix} \right\| = 10.
\]
Find the smallest possible value of \(\|\mathbf{v}\|\). | 10 - 2\sqrt{5} | 75.78125 |
23,973 | Given the function $f(x)$ ($x \in \mathbb{R}$) that satisfies $f(x+\pi)=f(x)+\sin x$, and $f(x)=0$ when $0 \leqslant x < \pi$, determine the value of $f(\frac{23\pi}{6})$. | \frac{1}{2} | 53.125 |
23,974 | Let $\triangle PQR$ be a right triangle such that $Q$ is a right angle. A circle with diameter $QR$ meets side $PR$ at point $S$. If $PS = 3$ and $QS = 9$, then what is $RS$? | 27 | 60.15625 |
23,975 | Determine the number of ways to arrange the letters of the word SUCCESS. | 420 | 23.4375 |
23,976 | Let a, b be positive integers such that $5 \nmid a, b$ and $5^5 \mid a^5+b^5$ . What is the minimum possible value of $a + b$ ? | 25 | 12.5 |
23,977 | Let $S_n$ be the sum of the first $n$ terms of the sequence $\{a_n\}$. Given that $3S_n=a_{n+1}-2$, and $a_2=1$, find $a_6$. | 256 | 43.75 |
23,978 | Define two binary operations on real numbers where $a \otimes b = \frac{a+b}{a-b}$ and $b \oplus a = \frac{b-a}{b+a}$. Compute the value of $(8\otimes 6) \oplus 2$.
A) $\frac{5}{9}$
B) $\frac{7}{9}$
C) $\frac{12}{9}$
D) $\frac{1}{9}$
E) $\frac{14}{9}$ | \frac{5}{9} | 75 |
23,979 | Given $\sin\alpha= \frac{1}{2}+\cos\alpha$, and $\alpha\in(0, \frac{\pi}{2})$, then $\sin2\alpha= \_\_\_\_\_\_$, $\cos2\alpha= \_\_\_\_\_\_$. | -\frac{\sqrt{7}}{4} | 55.46875 |
23,980 | Divide a 7-meter-long rope into 8 equal parts, each part is meters, and each part is of the whole rope. (Fill in the fraction) | \frac{1}{8} | 30.46875 |
23,981 | Determine the area of the triangle bounded by the axes and the curve $y = (x-5)^2 (x+3)$. | 300 | 38.28125 |
23,982 | Given a function $f(x) = \cos x \sin \left( x + \frac{\pi}{3} \right) - \sqrt{3} \cos^2 x + \frac{\sqrt{3}}{4}$, where $x \in \mathbb{R}$,
(1) Find the smallest positive period of $f(x)$ and the interval where $f(x)$ is monotonically decreasing;
(2) Find the maximum and minimum values of $f(x)$ in the closed interval $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$. | -\frac{1}{2} | 67.96875 |
23,983 | The ratio of the areas of two squares is $\frac{300}{147}$. Find the simplified form of the ratio of their side lengths, expressed as $\frac{a\sqrt{b}}{c}$ where $a$, $b$, and $c$ are integers. Additionally, if the perimeter of the larger square is 60 units, determine the side length of the smaller square. | 10.5 | 71.875 |
23,984 | In the polar coordinate system, given the curve $C: \rho = 2\cos \theta$, the line $l: \left\{ \begin{array}{l} x = \sqrt{3}t \\ y = -1 + t \end{array} \right.$ (where $t$ is a parameter), and the line $l$ intersects the curve $C$ at points $A$ and $B$.
$(1)$ Find the rectangular coordinate equation of curve $C$ and the general equation of line $l$.
$(2)$ Given the polar coordinates of point $P$ as $({1, \frac{3\pi}{2}})$, find the value of $\left(|PA|+1\right)\left(|PB|+1\right)$. | 3 + \sqrt{3} | 36.71875 |
23,985 | Among three-digit numbers, if the digit in the tens place is smaller than both the digit in the units place and the digit in the hundreds place, then this three-digit number is called a "concave number," such as 402, 745, etc. How many three-digit concave numbers with no repeating digits are there? | 240 | 93.75 |
23,986 | Once in a restaurant ***Dr. Strange*** found out that there were 12 types of food items from 1 to 12 on the menu. He decided to visit the restaurant 12 days in a row and try a different food everyday. 1st day, he tries one of the items from the first two. On the 2nd day, he eats either item 3 or the item he didn’t tried on the 1st day. Similarly, on the 3rd day, he eats either item 4 or the item he didn’t tried on the 2nd day. If someday he's not able to choose items that way, he eats the item that remained uneaten from the menu. In how many ways can he eat the items for 12 days? | 2048 | 57.03125 |
23,987 | Given the function $f(x) = 4\cos x \cos\left(x - \frac{\pi}{3}\right) - 2$.
(I) Find the smallest positive period of the function $f(x)$.
(II) Find the maximum and minimum values of the function $f(x)$ in the interval $\left[-\frac{\pi}{6}, \frac{\pi}{4}\right]$. | -2 | 60.15625 |
23,988 | What is the value of $\frac{2023^3 - 2 \cdot 2023^2 \cdot 2024 + 3 \cdot 2023 \cdot 2024^2 - 2024^3 + 1}{2023 \cdot 2024}$? | 2023 | 80.46875 |
23,989 | Given $\tan \alpha = -\frac{1}{2}$, find the value of $\frac{1+2\sin \alpha \cos \alpha}{\sin^2 \alpha - \cos^2 \alpha}$. | -\frac{1}{3} | 98.4375 |
23,990 | In parallelogram $ABCD$, $AD=1$, $\angle BAD=60^{\circ}$, and $E$ is the midpoint of $CD$. If $\overrightarrow{AD} \cdot \overrightarrow{EB}=2$, then the length of $AB$ is \_\_\_\_\_. | 12 | 39.0625 |
23,991 | Given $\cos\alpha =\frac{\sqrt{5}}{5}$ and $\sin\beta =\frac{3\sqrt{10}}{10}$, with $0 < \alpha$, $\beta < \frac{\pi}{2}$, determine the value of $\alpha +\beta$. | \frac{3\pi}{4} | 90.625 |
23,992 | Given two random variables $X$ and $Y$, where $X\sim B(8, \frac{1}{2})$ and $Y\sim N(\mu, \sigma^2)$, find the probability $P(4 \leq Y \leq 8)$, given that $\mu = E(X)$ and $P(Y < 0) = 0.2$. | 0.3 | 60.9375 |
23,993 | Select two distinct diagonals at random from a regular octagon. What is the probability that the two diagonals intersect at a point strictly within the octagon? Express your answer as $a + b$ , where the probability is $\tfrac{a}{b}$ and $a$ and $b$ are relatively prime positive integers. | 7 + 19 | 0 |
23,994 | Given numbers \( x_{1}, \ldots, x_{n} \in (0,1) \), find the maximum value of the expression
$$
A = \frac{\sqrt[4]{1-x_{1}} + \ldots + \sqrt[4]{1-x_{n}}}{\frac{1}{\sqrt[4]{x_{1}}} + \ldots + \frac{1}{\sqrt[4]{x_{n}}}}
$$ | \frac{\sqrt{2}}{2} | 10.9375 |
23,995 | Given the expression \( \left(1-\frac{1}{2^{2}}\right)\left(1-\frac{1}{3^{2}}\right)\ldots\left(1-\frac{1}{12^{2}}\right) \), compute its value. | \frac{13}{24} | 91.40625 |
23,996 | The positive integers $A, B, C$, and $D$ form an arithmetic and geometric sequence as follows: $A, B, C$ form an arithmetic sequence, while $B, C, D$ form a geometric sequence. If $\frac{C}{B} = \frac{7}{3}$, what is the smallest possible value of $A + B + C + D$? | 76 | 9.375 |
23,997 | If two lines $l$ and $m$ have equations $y = -2x + 8$, and $y = -3x + 9$, what is the probability that a point randomly selected in the 1st quadrant and below $l$ will fall between $l$ and $m$? | 0.15625 | 0 |
23,998 | Simplify the expression $(-\frac{1}{343})^{-2/3}$. | 49 | 89.0625 |
23,999 | Given that \(ABCD-A_{1}B_{1}C_{1}D_{1}\) is a cube and \(P-A_{1}B_{1}C_{1}D_{1}\) is a regular tetrahedron, find the cosine of the angle between the skew lines \(A_{1}P\) and \(BC_{1}\), given that the distance from point \(P\) to plane \(ABC\) is \(\frac{3}{2}AB\). | \frac{\sqrt{6}}{3} | 42.1875 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.