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24,100 | Given a right circular cone $(P-ABC)$ with lateral edges $(PA)$, $(PB)$, $(PC)$ being pairwise perpendicular, and base edge $AB = \sqrt{2}$, find the surface area of the circumscribed sphere of the right circular cone $(P-ABC)$. | 3\pi | 64.84375 |
24,101 | Let $S$ be a set of size $11$ . A random $12$ -tuple $(s_1, s_2, . . . , s_{12})$ of elements of $S$ is chosen uniformly at random. Moreover, let $\pi : S \to S$ be a permutation of $S$ chosen uniformly at random. The probability that $s_{i+1}\ne \pi (s_i)$ for all $1 \le i \le 12$ (where $s_{13} = s_1$ ) can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Compute $a$ . | 1000000000004 | 0 |
24,102 | Given the function $f(x)=\cos (2x+ \frac {\pi}{4})$, if we shrink the x-coordinates of all points on the graph of $y=f(x)$ to half of their original values while keeping the y-coordinates unchanged; and then shift the resulting graph to the right by $|\varphi|$ units, and the resulting graph is symmetric about the origin, find the value of $\varphi$. | \frac {3\pi}{16} | 56.25 |
24,103 | If four different positive integers $m$, $n$, $p$, $q$ satisfy $(7-m)(7-n)(7-p)(7-q)=4$, find the value of $m+n+p+q$. | 28 | 92.96875 |
24,104 | Given triangle \( \triangle ABC \) with \( Q \) as the midpoint of \( BC \), \( P \) on \( AC \) such that \( CP = 3PA \), and \( R \) on \( AB \) such that \( S_{\triangle PQR} = 2 S_{\triangle RBQ} \). If \( S_{\triangle ABC} = 300 \), find \( S_{\triangle PQR} \). | 100 | 37.5 |
24,105 | (1) Given that $0 < x < \dfrac{4}{3}$, find the maximum value of $x(4-3x)$.
(2) Point $(x,y)$ moves along the line $x+2y=3$. Find the minimum value of $2^{x}+4^{y}$. | 4 \sqrt{2} | 62.5 |
24,106 | Let $T_1$ and $T_2$ be the points of tangency of the excircles of a triangle $ABC$ with its sides $BC$ and $AC$ respectively. It is known that the reflection of the incenter of $ABC$ across the midpoint of $AB$ lies on the circumcircle of triangle $CT_1T_2$ . Find $\angle BCA$ . | 90 | 97.65625 |
24,107 | Given the polar equation of curve $C$ is $\rho=1$, with the pole as the origin of the Cartesian coordinate system, and the polar axis as the positive half-axis of $x$, establish the Cartesian coordinate system. The parametric equation of line $l$ is $\begin{cases} x=-1+4t \\ y=3t \end{cases}$ (where $t$ is the parameter), find the length of the chord cut by line $l$ on curve $C$. | \dfrac {8}{5} | 97.65625 |
24,108 | Let $\{a_{n}\}$ be an arithmetic sequence with the sum of the first $n$ terms denoted as $S_{n}$. Given $a_{1} \gt 0$, $a_{8}$ and $a_{9}$ are the two roots of the equation $x^{2}+x-2023=0$. Calculate the maximum value of $n$ that satisfies $S_{n} \gt 0$. | 15 | 60.15625 |
24,109 | Given the curves $C\_1$: $\begin{cases} & x=-4+\cos t, \ & y=3+\sin t \ \end{cases}$ (with $t$ as the parameter), and $C\_2$: $\begin{cases} & x=6\cos \theta, \ & y=2\sin \theta \ \end{cases}$ (with $\theta$ as the parameter).
(1) Convert the equations of $C\_1$ and $C\_2$ into general form and explain what type of curves they represent.
(2) If the point $P$ on $C\_1$ corresponds to the parameter $t=\frac{\pi }{2}$, and $Q$ is a moving point on $C\_2$, find the minimum distance from the midpoint $M$ of $PQ$ to the line $C\_3$: $\begin{cases} & x=-3\sqrt{3}+\sqrt{3}\alpha, \ & y=-3-\alpha \ \end{cases}$ (with $\alpha$ as the parameter). | 3\sqrt{3}-1 | 25.78125 |
24,110 | Find the value of $c$ such that all the roots of the polynomial $x^3 - 5x^2 + 2bx - c$ are real and positive, given that one root is twice another and four times the third. | \frac{1000}{343} | 60.15625 |
24,111 | The line $L_{1}$: $ax+(1-a)y=3$ and $L_{2}$: $(a-1)x+(2a+3)y=2$ are perpendicular to each other, find the values of $a$. | -3 | 40.625 |
24,112 | Given the parametric equation of line $l$ as $\begin{cases} x = \frac{\sqrt{2}}{2}t \\ y = \frac{\sqrt{2}}{2}t + 4\sqrt{2} \end{cases}$ (where $t$ is the parameter) and the polar equation of circle $C$ as $\rho = 2\cos (\theta + \frac{\pi}{4})$,
(I) Find the rectangular coordinates of the center of circle $C$.
(II) Find the minimum length of a tangent line drawn from a point on line $l$ to circle $C$. | 2\sqrt {6} | 0 |
24,113 | The diagram shows a right-angled triangle \( ACD \) with a point \( B \) on the side \( AC \). The sides of triangle \( ABD \) have lengths 3, 7, and 8. What is the area of triangle \( BCD \)? | 2\sqrt{3} | 4.6875 |
24,114 | Let $P$ be a point on the ellipse $\frac{x^2}{16} + \frac{y^2}{9} =1$, and $F_1$, $F_2$ be the left and right foci of the ellipse, respectively. If $\angle F_1 PF_2 = \frac{\pi}{3}$, find the area of $\triangle F_1 PF_2$. | 3\sqrt{3} | 87.5 |
24,115 | Determine the number of different ways to schedule volleyball, basketball, and table tennis competitions in 4 different gyms, given that each sport must be held in only one gym and that no more than 2 sports can take place in the same gym. | 60 | 71.875 |
24,116 | Given the equation $5^{12} = \frac{5^{90/x}}{5^{50/x} \cdot 25^{30/x}}$, find the value of $x$ that satisfies this equation. | -\frac{5}{3} | 97.65625 |
24,117 | Let vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ satisfy $|\overrightarrow{a}|=|\overrightarrow{b}|=1$, $\overrightarrow{a}\cdot \overrightarrow{b}= \frac{1}{2}$, and $(\overrightarrow{a}- \overrightarrow{c})\cdot(\overrightarrow{b}- \overrightarrow{c})=0$. Then, calculate the maximum value of $|\overrightarrow{c}|$. | \frac{\sqrt{3}+1}{2} | 13.28125 |
24,118 | What is the product of the [real](https://artofproblemsolving.com/wiki/index.php/Real) [roots](https://artofproblemsolving.com/wiki/index.php/Root) of the [equation](https://artofproblemsolving.com/wiki/index.php/Equation) $x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}$? | 20 | 62.5 |
24,119 | How many integers \(n\) satisfy \((n+5)(n-9) \le 0\)? | 15 | 100 |
24,120 | Given a circle with a radius of 5, its center lies on the x-axis with an integer horizontal coordinate and is tangent to the line 4x + 3y - 29 = 0.
(1) Find the equation of the circle;
(2) If the line ax - y + 5 = 0 (a ≠ 0) intersects the circle at points A and B, does there exist a real number a such that the line l passing through point P(-2, 4) is perpendicularly bisecting chord AB? If such a real number a exists, find its value; otherwise, explain the reason. | \frac{3}{4} | 29.6875 |
24,121 | Simplify the expression first, then evaluate it: \\(5(3a^{2}b-ab^{2})-(ab^{2}+3a^{2}b)\\), where \\(a= \frac {1}{2}\\), \\(b= \frac {1}{3}\\). | \frac {2}{3} | 72.65625 |
24,122 | Compute the value of $0.25 \cdot 0.8 - 0.12$. | 0.08 | 100 |
24,123 | A teacher received letters on Monday to Friday with counts of $10$, $6$, $8$, $5$, $6$ respectively. Calculate the standard deviation of this data set. | \dfrac {4 \sqrt {5}}{5} | 0 |
24,124 | A lighthouse emits a yellow signal every 15 seconds and a red signal every 28 seconds. The yellow signal is first seen 2 seconds after midnight, and the red signal is first seen 8 seconds after midnight. At what time will both signals be seen together for the first time? | 92 | 0.78125 |
24,125 | A rectangular solid has three adjacent faces with areas of $1$, $2$, and $2$, respectively. All the vertices of the rectangular solid are located on the same sphere. Find the volume of this sphere. | \sqrt{6}\pi | 4.6875 |
24,126 | 1. Simplify and evaluate the expression: $\log_{\frac{1}{3}} \sqrt{27} + \lg 25 + \lg 4 + 7^{-\log_{7} 2} + (-0.98)^0$
2. Given a point $P(\sqrt{2}, -\sqrt{6})$ on the terminal side of angle $\alpha$, evaluate: $\frac{\cos \left( \frac{\pi}{2} + \alpha \right) \cos \left( 2\pi - \alpha \right) + \sin \left( -\alpha - \frac{\pi}{2} \right) \cos \left( \pi - \alpha \right)}{\sin \left( \pi + \alpha \right) \cos \left( \frac{\pi}{2} - \alpha \right)}$ | \frac{-\sqrt{3} - 1}{3} | 0 |
24,127 | Given that the terminal side of angle $θ$ lies on the ray $y=2x(x≤0)$, find the value of $\sin θ + \cos θ$. | -\frac{3\sqrt{5}}{5} | 98.4375 |
24,128 | Given that $x$ is a multiple of $3456$, what is the greatest common divisor of $f(x)=(5x+3)(11x+2)(14x+7)(3x+8)$ and $x$? | 48 | 47.65625 |
24,129 | Given a random variable $\xi \sim N(1, \sigma ^{2})$, $a \gt 0$, $b \gt 0$, if $P(\xi \leq a) = P(\xi \geq b)$, then the minimum value of $\frac{{4a+b}}{{ab}}$ is ______. | \frac{9}{2} | 71.09375 |
24,130 | Given the hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$), perpendiculars are drawn from the right focus $F(2\sqrt{2}, 0)$ to the two asymptotes, with the feet of the perpendiculars being $A$ and $B$, respectively. Let point $O$ be the origin. If the area of quadrilateral $OAFB$ is $4$, determine the eccentricity of the hyperbola. | \sqrt{2} | 57.8125 |
24,131 | Let $ABC$ be an equilateral triangle and a point M inside the triangle such that $MA^2 = MB^2 +MC^2$ . Draw an equilateral triangle $ACD$ where $D \ne B$ . Let the point $N$ inside $\vartriangle ACD$ such that $AMN$ is an equilateral triangle. Determine $\angle BMC$ .
| 150 | 59.375 |
24,132 | The Eagles and the Hawks play 5 times. The Hawks, being the stronger team, have an 80% chance of winning any given game. What is the probability that the Hawks will win at least 4 out of the 5 games? Express your answer as a common fraction. | \frac{73728}{100000} | 0 |
24,133 | In the rectangular coordinate system, a pole coordinate system is established with the origin as the pole and the positive semi-axis of the $x$-axis as the polar axis. Given the curve $C$: ${p}^{2}=\frac{12}{2+{\mathrm{cos}}^{}θ}$ and the line $l$: $2p\mathrm{cos}\left(θ-\frac{π}{6}\right)=\sqrt{3}$.
1. Write the rectangular coordinate equations for the line $l$ and the curve $C$.
2. Let points $A$ and $B$ be the two intersection points of line $l$ and curve $C$. Find the value of $|AB|$. | \frac{4\sqrt{10}}{3} | 50.78125 |
24,134 | Find the mathematical expectation of the area of the projection of a cube with edge of length $1$ onto a plane with an isotropically distributed random direction of projection. | \frac{3}{2} | 14.84375 |
24,135 | Let $\triangle ABC$ have sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively. The areas of the equilateral triangles with side lengths $a$, $b$, $c$ are $S_{1}$, $S_{2}$, $S_{3}$ respectively. Given $S_{1}-S_{2}+S_{3}=\frac{{\sqrt{3}}}{2}$ and $\sin B=\frac{1}{3}$.<br/>$(1)$ Find the area of $\triangle ABC$;<br/>$(2)$ If $\sin A\sin C=\frac{{\sqrt{2}}}{3}$, find $b$. | \frac{1}{2} | 32.8125 |
24,136 | The stem-and-leaf plot shows the number of minutes and seconds of one ride on each of the 21 top-rated water slides in the world. In the stem-and-leaf plot, $1 \ 45$ represents 1 minute, 45 seconds, which is equivalent to 105 seconds. What is the median of this data set? Express your answer in seconds.
\begin{tabular}{c|cccccc}
0&15&30&45&55&&\\
1&00&20&35&45&55&\\
2&10&15&30&45&50&55\\
3&05&10&15&&&\\
\end{tabular} | 135 | 18.75 |
24,137 | Find the number of real solutions of the equation
\[\frac{x}{50} = \cos x.\] | 31 | 21.875 |
24,138 | Given the function $f(x)= \sqrt {3}\sin (\omega x+\varphi)-\cos (\omega x+\varphi)$ $(\omega > 0,0 < \varphi < \pi)$ is an even function, and the distance between two adjacent axes of symmetry of its graph is $\dfrac {\pi}{2}$, then the value of $f(- \dfrac {\pi}{8})$ is \_\_\_\_\_\_. | \sqrt {2} | 0 |
24,139 | Find the number of integers \( n \) that satisfy
\[ 20 < n^2 < 200. \] | 20 | 42.96875 |
24,140 | Given the real numbers $a$, $b$, $c$, $d$ that satisfy $b=a-2e^{a}$ and $c+d=4$, where $e$ is the base of the natural logarithm, find the minimum value of $(a-c)^{2}+(b-d)^{2}$. | 18 | 14.0625 |
24,141 | In $\triangle ABC$, $\angle C= \frac{\pi}{2}$, $\angle B= \frac{\pi}{6}$, and $AC=2$. $M$ is the midpoint of $AB$. $\triangle ACM$ is folded along $CM$ such that the distance between $A$ and $B$ is $2\sqrt{2}$. The surface area of the circumscribed sphere of the tetrahedron $M-ABC$ is \_\_\_\_\_\_. | 16\pi | 15.625 |
24,142 | If 2006 integers $a_1, a_2, \ldots a_{2006}$ satisfy the following conditions: $a_1=0$, $|a_2|=|a_1+2|$, $|a_3|=|a_2+2|$, $\ldots$, $|a_{2006}|=|a_{2005}+2|$, then the minimum value of $a_1+a_2+\ldots+a_{2005}$ is. | -2004 | 40.625 |
24,143 | A five-digit natural number $\overline{a_1a_2a_3a_4a_5}$ is considered a "concave number" if and only if $a_1 > a_2 > a_3$ and $a_3 < a_4 < a_5$, with each $a_i \in \{0,1,2,3,4,5\}$ for $i=1,2,3,4,5$. Calculate the number of possible "concave numbers". | 146 | 0 |
24,144 | Find the smallest positive integer $n$ that has at least $7$ positive divisors $1 = d_1 < d_2 < \ldots < d_k = n$ , $k \geq 7$ , and for which the following equalities hold: $$ d_7 = 2d_5 + 1\text{ and }d_7 = 3d_4 - 1 $$ *Proposed by Mykyta Kharin* | 2024 | 0.78125 |
24,145 | A triangle has three sides that are three consecutive natural numbers, and the largest angle is twice the smallest angle. The perimeter of this triangle is __________. | 15 | 36.71875 |
24,146 | Given $\sin (\frac{\pi }{3}-\theta )=\frac{3}{4}$, find $\cos (\frac{\pi }{3}+2\theta )$. | \frac{1}{8} | 55.46875 |
24,147 | A plane has no vertex of a regular dodecahedron on it,try to find out how many edges at most may the plane intersect the regular dodecahedron? | 10 | 13.28125 |
24,148 | Four circles of radius 1 are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle? Express your answer as a common fraction in simplest radical form. | 1 + \sqrt{2} | 29.6875 |
24,149 | Consider the following infinite geometric series: $$\frac{7}{8}-\frac{14}{27}+\frac{28}{81}-\dots$$ Find the common ratio of this series. | -\frac{2}{3} | 1.5625 |
24,150 | Jo, Blair, and Parker take turns counting from 1, increasing by one more than the last number said by the previous person. What is the $100^{\text{th}}$ number said? | 100 | 15.625 |
24,151 | Define a new operation $\star$ such that for positive integers $a, b, c$, $a \star b \star c = \frac{a \times b + c}{a + b + c}$. Calculate the value of $4 \star 8 \star 2$.
**A)** $\frac{34}{14}$ **B)** $\frac{16}{7}$ **C)** $\frac{17}{7}$ **D)** $\frac{32}{14}$ **E)** $2$ | \frac{17}{7} | 73.4375 |
24,152 | There are $7$ different books to be distributed among three people, A, B, and C.<br/>$(1)$ If one person gets $1$ book, another gets $2 books, and the third gets $4 books, how many different ways can the books be distributed?<br/>$(2)$ If one person gets $3 books, and the other two each get $2 books, how many different ways can the books be distributed? | 630 | 64.0625 |
24,153 | Let \( m = 2^{40}5^{24} \). How many positive integer divisors of \( m^2 \) are less than \( m \) but do not divide \( m \)? | 959 | 53.90625 |
24,154 | What is the coefficient of $x^3y^5$ in the expansion of $\left(\frac{2}{3}x - \frac{y}{3}\right)^8$? | -\frac{448}{6561} | 89.0625 |
24,155 | Two different numbers are randomly selected from the set $\{-3, -2, 0, 0, 5, 6, 7\}$. What is the probability that the product of these two numbers is $0$?
**A)** $\frac{1}{4}$
**B)** $\frac{1}{5}$
**C)** $\frac{5}{21}$
**D)** $\frac{1}{3}$
**E)** $\frac{1}{2}$ | \frac{5}{21} | 47.65625 |
24,156 | For the function $f(x)=a- \frac {2}{2^{x}+1}(a\in\mathbb{R})$
$(1)$ Determine the monotonicity of the function $f(x)$ and provide a proof;
$(2)$ If there exists a real number $a$ such that the function $f(x)$ is an odd function, find $a$;
$(3)$ For the $a$ found in $(2)$, if $f(x)\geqslant \frac {m}{2^{x}}$ holds true for all $x\in[2,3]$, find the maximum value of $m$. | \frac {12}{5} | 52.34375 |
24,157 | Calculate:<br/>$(1)-1^{2023}+8×(-\frac{1}{2})^{3}+|-3|$;<br/>$(2)(-25)×\frac{3}{2}-(-25)×\frac{5}{8}+(-25)÷8($simplified calculation). | -25 | 93.75 |
24,158 | In acute triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\frac{\cos A}{\cos C}=\frac{a}{2b-c}$, find:<br/>
$(1)$ The measure of angle $A$;<br/>
$(2)$ If $a=\sqrt{7}$, $c=3$, and $D$ is the midpoint of $BC$, find the length of $AD$. | \frac{\sqrt{19}}{2} | 74.21875 |
24,159 | In a geometric sequence \(\{z_{n}\}\), given \(z_{1}=1\), \(z_{2}=a+b \mathrm{i}\), \(z_{3}=b+a \mathrm{i}\) where \(a, b \in \mathbf{R}\) and \(a>0\), find the smallest value of \(n\) such that \(z_{1}+z_{2}+\cdots+z_{n}=0\), and compute the value of \(z_{1} z_{2} \cdots z_{n}\). | -1 | 46.875 |
24,160 | Let $f(x) = 4x - 9$ and $g(f(x)) = x^2 + 6x - 7$. Find $g(-8)$. | \frac{-87}{16} | 0 |
24,161 | Extend a rectangular pattern of 12 black and 18 white square tiles by attaching a border of white tiles around the rectangle. The original rectangle is 5x6 tiles and the border adds one tile to each side. Calculate the ratio of black tiles to white tiles in the extended pattern. | \frac{12}{44} | 0 |
24,162 | Given an isolated ground state iron atom with atomic number Z = 26, determine the number of orbitals that contain one or more electrons. | 15 | 64.84375 |
24,163 | Each of the twelve letters in "STATISTICS" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word "TEST"? Express your answer as a common fraction. | \frac{1}{2} | 30.46875 |
24,164 | In the sequence $\{a_{n}\}$, where $a_{1}=1$, $a_{n} \gt 0$, and the sum of the first $n$ terms is $S_{n}$. If ${a_n}=\sqrt{{S_n}}+\sqrt{{S_{n-1}}}$ for $n \in \mathbb{N}^*$ and $n \geqslant 2$, then the sum of the first $15$ terms of the sequence $\{\frac{1}{{{a_n}{a_{n+1}}}}\}$ is ____. | \frac{15}{31} | 85.9375 |
24,165 | Let $f(x) = x^2 + 5x + 4$ and $g(x) = 2x - 3$. Calculate the value of $f(g(-3)) - 2 \cdot g(f(2))$. | -26 | 74.21875 |
24,166 | Determine the exact value of the series
\[
\frac{1}{3 + 1} + \frac{2}{3^2 + 1} + \frac{4}{3^4 + 1} + \frac{8}{3^8 + 1} + \frac{16}{3^{16} + 1} + \dotsb.
\] | \frac{1}{2} | 82.8125 |
24,167 | Let $r(x)$ be a monic quartic polynomial such that $r(1) = 0,$ $r(2) = 3,$ $r(3) = 8,$ and $r(4) = 15$. Find $r(5)$. | 48 | 57.8125 |
24,168 | Convert 89 into a binary number. | 1011001_{(2)} | 0 |
24,169 | What is the base-10 integer 789 when expressed in base 7? | 2205_7 | 5.46875 |
24,170 | The line $2x+3y-6=0$ intersects the $x$-axis and $y$-axis at points A and B, respectively. Point P is on the line $y=-x-1$. The minimum value of $|PA|+|PB|$ is ________. | \sqrt{37} | 7.8125 |
24,171 | Suppose that \[\operatorname{lcm}(1024,2016)=\operatorname{lcm}(1024,2016,x_1,x_2,\ldots,x_n),\] with $x_1$ , $x_2$ , $\cdots$ , $x_n$ are distinct postive integers. Find the maximum value of $n$ .
*Proposed by Le Duc Minh* | 64 | 67.1875 |
24,172 | Simplify first, then evaluate: $({\frac{{x-1}}{x}-\frac{{x-2}}{{x+1}}})÷\frac{{2{x^2}-x}}{{{x^2}+2x+1}}$, where $x$ satisfies $x^{2}-2x-2=0$. | \frac{1}{2} | 81.25 |
24,173 | A class has a group of 7 people, and now 3 of them are chosen to swap seats with each other, while the remaining 4 people's seats remain unchanged. Calculate the number of different rearrangement plans. | 70 | 8.59375 |
24,174 | Given that \( a > 0 \), if \( f(g(h(a))) = 17 \), where \( f(x) = x^2 + 5 \), \( g(x) = x^2 - 3 \), and \( h(x) = 2x + 1 \), what is the value of \( a \)? | \frac{-1 + \sqrt{3 + 2\sqrt{3}}}{2} | 10.9375 |
24,175 | A bicycle factory plans to produce a batch of bicycles of the same model, planning to produce $220$ bicycles per day. However, due to various reasons, the actual daily production will differ from the planned quantity. The table below shows the production situation of the workers in a certain week: (Exceeding $220$ bicycles is recorded as positive, falling short of $220$ bicycles is recorded as negative)
| Day of the Week | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday |
|-----------------|--------|---------|-----------|----------|--------|----------|--------|
| Production Change (bicycles) | $+5$ | $-2$ | $-4$ | $+13$ | $-10$ | $+16$ | $-9$ |
$(1)$ According to the records, the total production in the first four days was ______ bicycles;<br/>
$(2)$ How many more bicycles were produced on the day with the highest production compared to the day with the lowest production?<br/>
$(3)$ The factory implements a piece-rate wage system, where each bicycle produced earns $100. For each additional bicycle produced beyond the daily planned production, an extra $20 is awarded, and for each bicycle less produced, $20 is deducted. What is the total wage of the workers for this week? | 155080 | 9.375 |
24,176 | [help me] Let m and n denote the number of digits in $2^{2007}$ and $5^{2007}$ when expressed in base 10. What is the sum m + n? | 2008 | 97.65625 |
24,177 | Determine the smallest positive real $K$ such that the inequality
\[ K + \frac{a + b + c}{3} \ge (K + 1) \sqrt{\frac{a^2 + b^2 + c^2}{3}} \]holds for any real numbers $0 \le a,b,c \le 1$ .
*Proposed by Fajar Yuliawan, Indonesia* | \frac{\sqrt{6}}{3} | 8.59375 |
24,178 | Amy and Bob choose numbers from $0,1,2,\cdots,81$ in turn and Amy choose the number first. Every time the one who choose number chooses one number from the remaining numbers. When all $82$ numbers are chosen, let $A$ be the sum of all the numbers Amy chooses, and let $B$ be the sum of all the numbers Bob chooses. During the process, Amy tries to make $\gcd(A,B)$ as great as possible, and Bob tries to make $\gcd(A,B)$ as little as possible. Suppose Amy and Bob take the best strategy of each one, respectively, determine $\gcd(A,B)$ when all $82$ numbers are chosen. | 41 | 59.375 |
24,179 | How many unordered pairs of edges of a given square pyramid determine a plane? | 18 | 4.6875 |
24,180 | A cube with an edge length of 6 units has the same volume as a triangular-based pyramid with a base having equilateral triangle sides of 10 units and a height of $h$ units. What is the value of $h$? | \frac{216\sqrt{3}}{25} | 85.9375 |
24,181 | Given that $\tan \alpha$ and $\frac{1}{\tan \alpha}$ are the two real roots of the equation $x^2 - kx + k^2 - 3 = 0$, and $3\pi < \alpha < \frac{7}{2}\pi$, find $\cos \alpha + \sin \alpha$. | -\sqrt{2} | 30.46875 |
24,182 | Given a rectangle $ABCD$ with all vertices on a sphere centered at $O$, where $AB = \sqrt{3}$, $BC = 3$, and the volume of the pyramid $O-ABCD$ is $4\sqrt{3}$, find the surface area of the sphere $O$. | 76\pi | 62.5 |
24,183 | We write one of the numbers $0$ and $1$ into each unit square of a chessboard with $40$ rows and $7$ columns. If any two rows have different sequences, at most how many $1$ s can be written into the unit squares? | 198 | 0.78125 |
24,184 | Given a right circular cone with three mutually perpendicular side edges, each with a length of $\sqrt{3}$, determine the surface area of the circumscribed sphere. | 9\pi | 92.96875 |
24,185 | In right triangle $XYZ$, we have $\angle X = \angle Z$ and $XZ = 8\sqrt{2}$. What is the area of $\triangle XYZ$? | 32 | 87.5 |
24,186 | In triangle ABC, the sides opposite to angles A, B, and C are a, b, and c, respectively. Given that $$\frac {sin2B}{ \sqrt {3}cos(B+C)-cosCsinB}= \frac {2b}{c}$$.
(I) Find the measure of angle A.
(II) If $$a= \sqrt {3}$$, find the maximum area of triangle ABC. | \frac { \sqrt {3}}{4} | 0 |
24,187 | Given that the base edge length of a right prism is $1$ and the side edge length is $2$, and all the vertices of the prism lie on a sphere, find the radius of the sphere. | \frac{\sqrt{6}}{2} | 82.03125 |
24,188 | Four members of Barnett family, including two grandparents, one adult parent, and one child, visit a zoo. The grandparents, being senior citizens, get a 20% discount. The child receives a 60% discount due to being under the age of 12, while the adult pays the full ticket price. If the ticket for an adult costs $10.00, and one of the grandparents is paying for everyone, how much do they need to pay in total?
A) $38
B) $30
C) $42
D) $28
E) $34 | 30 | 42.96875 |
24,189 | Given an arithmetic sequence $\{a\_n\}$, where $a\_1+a\_2=3$, $a\_4+a\_5=5$.
(I) Find the general term formula of the sequence.
(II) Let $[x]$ denote the largest integer not greater than $x$ (e.g., $[0.6]=0$, $[1.2]=1$). Define $T\_n=[a\_1]+[a\_2]+…+[a\_n]$. Find the value of $T\_30$. | 175 | 31.25 |
24,190 | The distance between two parallel lines $l_1: 3x + 4y - 2 = 0$ and $l_2: ax + 6y = 5$ is _______. | \frac{4}{15} | 74.21875 |
24,191 | In a cube $ABCDEFGH$ where each side has length $2$ units. Find $\sin \angle GAC$. (Consider this by extending the calculations needed for finding $\sin \angle HAC$) | \frac{\sqrt{3}}{3} | 54.6875 |
24,192 | Given circles ${C}_{1}:{x}^{2}+{y}^{2}=1$ and ${C}_{2}:(x-4)^{2}+(y-2)^{2}=1$, a moving point $M\left(a,b\right)$ is used to draw tangents $MA$ and $MB$ to circles $C_{1}$ and $C_{2}$ respectively, where $A$ and $B$ are the points of tangency. If $|MA|=|MB|$, calculate the minimum value of $\sqrt{(a-3)^{2}+(b+2)^{2}}$. | \frac{\sqrt{5}}{5} | 67.96875 |
24,193 | Compute the sum \[\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{25} \rfloor.\] | 75 | 100 |
24,194 | Parallelogram $ABCD$ is given such that $\angle ABC$ equals $30^o$ . Let $X$ be the foot of the perpendicular from $A$ onto $BC$ , and $Y$ the foot of the perpendicular from $C$ to $AB$ . If $AX = 20$ and $CY = 22$ , find the area of the parallelogram.
| 880 | 52.34375 |
24,195 | From the numbers \\(1\\), \\(2\\), \\(3\\), \\(4\\), \\(5\\), \\(6\\), two numbers are selected to form an ordered pair of real numbers \\((x, y)\\). The probability that \\(\dfrac{x}{y+1}\\) is an integer is equal to \_\_\_\_\_\_ | \dfrac{4}{15} | 0.78125 |
24,196 | The sum of the first and the third of three consecutive odd integers is 152. What is the value of the second integer? | 76 | 42.96875 |
24,197 | Given $n \in \mathbb{N}^*$, in the expansion of $(x+2)^n$, the coefficient of the second term is $\frac{1}{5}$ of the coefficient of the third term.
(1) Find the value of $n$;
(2) Find the term with the maximum binomial coefficient in the expansion;
(3) If $(x+2)^n=a_0+a_1(x+1)+a_2(x+1)^2+\ldots+a_n(x+1)^n$, find the value of $a_0+a_1+\ldots+a_n$. | 64 | 83.59375 |
24,198 | Determine the exact value of the series
\[\frac{1}{3 + 1} + \frac{2}{3^2 + 1} + \frac{4}{3^4 + 1} + \frac{8}{3^8 + 1} + \frac{16}{3^{16} + 1} + \dotsb.\] | \frac{1}{2} | 64.0625 |
24,199 | What is the value of $2468 + 8642 + 6824 + 4286$? | 22220 | 83.59375 |
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