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40.3k
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100
|
|---|---|---|---|
12,100 |
If the two real roots of the equation (lgx)<sup>2</sup>\-lgx+lg2•lg5=0 with respect to x are m and n, then 2<sup>m+n</sup>\=\_\_\_\_\_\_.
|
128
| 74.21875 |
12,101 |
Given that the terminal side of angle $\alpha$ passes through point $P(\sqrt{3}, m)$ ($m \neq 0$), and $\cos \alpha = \frac{m}{6}$, find the value of $\sin \alpha$ ___.
|
\frac{\sqrt{3}}{2}
| 2.34375 |
12,102 |
Points were marked on the sides of triangle \(ABC\): 12 points on side \(AB\), 9 points on side \(BC\), and 10 points on side \(AC\). None of the vertices of the triangle are marked. How many triangles can be formed with vertices at the marked points?
|
4071
| 21.09375 |
12,103 |
For how many positive integers $n$ less than or equal to 500 is $$(\cos t - i\sin t)^n = \cos nt - i\sin nt$$ true for all real $t$?
|
500
| 86.71875 |
12,104 |
Let $b_n$ be the number obtained by writing the integers 1 to $n$ from left to right, where each integer is squared. For example, $b_3 = 149$ (since $1^2 = 1$, $2^2 = 4$, $3^2 = 9$), and $b_5 = 1491625$. For $1 \le k \le 100$, determine how many $b_k$ are divisible by 4.
|
50
| 7.8125 |
12,105 |
For any real numbers $x,y$ that satisfies the equation $$ x+y-xy=155 $$ and $$ x^2+y^2=325 $$ , Find $|x^3-y^3|$
|
4375
| 75 |
12,106 |
How many zeros are there at the end of $\frac{2018!}{30!\times 11!}$?
|
493
| 36.71875 |
12,107 |
On the $x O y$ coordinate plane, there is a Chinese chess "knight" at the origin $(0,0)$. The "knight" needs to be moved to the point $P(1991,1991)$ using the movement rules of the chess piece. What is the minimum number of moves required?
|
1328
| 26.5625 |
12,108 |
A rectangle has dimensions 12 by 15, and a circle centered at one of its vertices has a radius of 15. What is the area of the union of the regions enclosed by the rectangle and the circle? Express your answer in terms of \( \pi \).
|
180 + 168.75\pi
| 0.78125 |
12,109 |
Three triangles. Inside triangle $ABC$, a random point $M$ is chosen. What is the probability that the area of one of the triangles $ABM$, $BCM$, or $CAM$ will be greater than the sum of the areas of the other two?
|
0.75
| 0 |
12,110 |
Find the 150th term of the sequence that consists of all those positive integers which are either powers of 3 or sums of distinct powers of 3.
|
2280
| 21.875 |
12,111 |
The diagonals AC and CE of the regular hexagon ABCDEF are divided by inner points M and N respectively, so that AM/AC = CN/CE = r. Determine r if B, M, and N are collinear.
|
\frac{1}{\sqrt{3}}
| 0 |
12,112 |
Given $α∈(\frac{π}{2},π)$, and $sin(α+\frac{π}{3})=\frac{12}{13}$, determine the value of $sin(\frac{π}{6}-α)+sin(\frac{2π}{3}-α)$.
|
\frac{7}{13}
| 24.21875 |
12,113 |
What is the minimum length of the second longest side of a triangle with an area of one unit?
|
\sqrt{2}
| 34.375 |
12,114 |
Let the set
\[ S=\{1, 2, \cdots, 12\}, \quad A=\{a_{1}, a_{2}, a_{3}\} \]
where \( a_{1} < a_{2} < a_{3}, \quad a_{3} - a_{2} \leq 5, \quad A \subseteq S \). Find the number of sets \( A \) that satisfy these conditions.
|
185
| 86.71875 |
12,115 |
Calculate the following infinite product: $3^{\frac{1}{3}} \cdot 9^{\frac{1}{9}} \cdot 27^{\frac{1}{27}} \cdot 81^{\frac{1}{81}} \dotsm.$
|
3^{\frac{3}{4}}
| 82.8125 |
12,116 |
Given the function $f(x)=\frac{1}{3}x^{3}+ax^{2}+bx-\frac{2}{3}$, the equation of the tangent line at $x=2$ is $x+y-2=0$.
(I) Find the values of the real numbers $a$ and $b$.
(II) Find the extreme values of the function $f(x)$.
|
-\frac{2}{3}
| 9.375 |
12,117 |
Given the function \( f(x) = \lg \frac{1 + x}{1 - x} \), if \( f\left(\frac{y + z}{1 + y z}\right) = 1 \) and \( f\left(\frac{y - z}{1 - y z}\right) = 2 \), where \( -1 < y, z < 1 \), find the value of \( f(y) \cdot f(z) \).
|
-3/4
| 0 |
12,118 |
The shortest distance from a point on the curve $f(x) = \ln(2x-1)$ to the line $2x - y + 3 = 0$ is what?
|
\sqrt{5}
| 57.8125 |
12,119 |
Given $f(\sin \alpha + \cos \alpha) = \sin \alpha \cdot \cos \alpha$, determine the domain of $f(x)$ and the value of $f\left(\sin \frac{\pi}{6}\right)$.
|
-\frac{3}{8}
| 61.71875 |
12,120 |
Find the maximum value of the product \(x^{2} y^{2} z^{2} u\) given the condition that \(x, y, z, u \geq 0\) and:
\[ 2x + xy + z + yz u = 1 \]
|
1/512
| 1.5625 |
12,121 |
Point \( M \) is the midpoint of side \( BC \) of the triangle \( ABC \), where \( AB = 17 \), \( AC = 30 \), and \( BC = 19 \). A circle is constructed with diameter \( AB \). A point \( X \) is chosen arbitrarily on this circle. What is the minimum possible length of the segment \( MX \)?
|
6.5
| 12.5 |
12,122 |
What is the remainder when $5x^8 - 3x^7 + 4x^6 - 9x^4 + 3x^3 - 5x^2 + 8$ is divided by $3x - 6$?
|
1020
| 8.59375 |
12,123 |
Given the ellipse \(\frac{x^{2}}{16}+\frac{y^{2}}{9}=1\) with three points \(P\), \(Q\), and \(R\) on it, where \(P\) and \(Q\) are symmetric with respect to the origin. Find the maximum value of \(|RP| + |RQ|\).
|
10
| 61.71875 |
12,124 |
Given that \( a \) and \( b \) are positive integers, and \( b - a = 2013 \). If the equation \( x^{2} - a x + b = 0 \) has a positive integer solution, what is the smallest value of \( a \)?
|
93
| 45.3125 |
12,125 |
Let $ABCD$ be a parallelogram with $\angle ABC=135^\circ$, $AB=14$ and $BC=8$. Extend $\overline{CD}$ through $D$ to $E$ so that $DE=3$. If $\overline{BE}$ intersects $\overline{AD}$ at $F$, then find the length of segment $FD$.
A) $\frac{6}{17}$
B) $\frac{18}{17}$
C) $\frac{24}{17}$
D) $\frac{30}{17}$
|
\frac{24}{17}
| 43.75 |
12,126 |
Let \( a \star b = ab + a + b \) for all integers \( a \) and \( b \). Evaluate \( 1 \star (2 \star (3 \star (4 \star \ldots (99 \star 100) \ldots))) \).
|
101! - 1
| 0 |
12,127 |
A basketball team consists of 12 players, including two pairs of twins, Alex and Brian, and Chloe and Diana. In how many ways can we choose a team of 5 players if no pair of twins can both be in the team lineup simultaneously?
|
560
| 23.4375 |
12,128 |
How many different positive three-digit integers can be formed using only the digits in the set $\{1, 2, 2, 3, 4, 4, 4\}$ if no digit may be used more times than it appears in the given set of available digits?
|
43
| 67.1875 |
12,129 |
Given the parabola $y=ax^{2}+bx+c$ ($a\neq 0$) with its axis of symmetry to the left of the $y$-axis, where $a$, $b$, $c \in \{-3,-2,-1,0,1,2,3\}$, let the random variable $X$ be the value of "$|a-b|$". Then, the expected value $EX$ is \_\_\_\_\_\_.
|
\dfrac {8}{9}
| 2.34375 |
12,130 |
Given the function \( f(x) = \cos x + \log_2 x \) for \( x > 0 \), if the positive real number \( a \) satisfies \( f(a) = f(2a) \), then find the value of \( f(2a) - f(4a) \).
|
-1
| 79.6875 |
12,131 |
The base of the pyramid is an isosceles right triangle, where each leg measures 8. Each of the pyramid's lateral edges is 9. Find the volume of the pyramid.
|
224/3
| 70.3125 |
12,132 |
Given that the function $f(x+1)$ is an odd function, and the function $f(x-1)$ is an even function, and $f(0)=2$, then $f(4)=$ ?
|
-2
| 11.71875 |
12,133 |
For a positive number $x$, define $f(x)=\frac{2x}{x+1}$. Calculate: $f(\frac{1}{101})+f(\frac{1}{100})+f(\frac{1}{99})+\ldots +f(\frac{1}{3})+f(\frac{1}{2})+f(1)+f(2)+f(3)+\ldots +f(99)+f(100)+f(101)$.
|
201
| 72.65625 |
12,134 |
Given that $α$ is an angle in the third quadrant and $\cos(α+π)=\frac{4}{5}$, find the value of $\tan 2α$.
|
\frac{24}{7}
| 94.53125 |
12,135 |
Let \( x \neq y \), and the two sequences \( x, a_{1}, a_{2}, a_{3}, y \) and \( b_{1}, x, b_{2}, b_{3}, y, b_{4} \) are both arithmetic sequences. Then \(\frac{b_{4}-b_{3}}{a_{2}-a_{1}}\) equals $\qquad$.
|
2.6666666666666665
| 0 |
12,136 |
Given the complex number $z=(2m^{2}+3m-2)+(m^{2}+m-2)i$ where $(m\in\mathbb{R})$, find the value of $m$ under the following conditions:
$(1) z$ is a real number;
$(2) z$ is an imaginary number;
$(3) z$ is a pure imaginary number;
$(4) z=0$.
|
-2
| 27.34375 |
12,137 |
Given a sequence \( x_{n} \), satisfying \( (n+1) x_{n+1}=x_{n}+n \), and \( x_{1}=2 \), find \( x_{2009} \).
|
\frac{2009! + 1}{2009!}
| 0 |
12,138 |
Given a four-digit number \(\overline{abcd}\), when divided by 2, 3, 4, 5, 6, and 7, the remainders are all different and none of them are 0. Find the minimum value of \(\overline{abcd}\).
|
1259
| 100 |
12,139 |
Factorize the expression $27x^6 - 512y^6$ and find the sum of all integer coefficients in its factorized form.
|
92
| 53.90625 |
12,140 |
There is a peculiar four-digit number (with the first digit not being 0). It is a perfect square, and the sum of its digits is also a perfect square. Dividing this four-digit number by the sum of its digits results in yet another perfect square. Additionally, the number of divisors of this number equals the sum of its digits, which is a perfect square. If all the digits of this four-digit number are distinct, what is this four-digit number?
|
2601
| 91.40625 |
12,141 |
Given four points \( O, A, B, C \) on a plane, with \( OA=4 \), \( OB=3 \), \( OC=2 \), and \( \overrightarrow{OB} \cdot \overrightarrow{OC}=3 \), find the maximum area of triangle \( ABC \).
|
2 \sqrt{7} + \frac{3\sqrt{3}}{2}
| 2.34375 |
12,142 |
Suppose $p$ and $q$ are both real numbers, and $\sin \alpha$ and $\cos \alpha$ are the two real roots of the equation $x^{2}+px+q=0$ with respect to $x$. Find the minimum value of $p+q$.
|
-1
| 78.125 |
12,143 |
Find the area of triangle ABC, whose vertices have coordinates A(0,0), B(1424233,2848467), C(1424234,2848469). Round the answer to two decimal places.
|
0.50
| 63.28125 |
12,144 |
The value of $\sin 210^\circ$ is equal to $\frac{-\sqrt{3}}{2}$.
|
-\frac{1}{2}
| 64.0625 |
12,145 |
Find the smallest natural number ending in the digit 6, which increases fourfold when its last digit is moved to the beginning of the number.
|
153846
| 92.1875 |
12,146 |
Does there exist an integer \( n \) such that \( 21n \equiv 1 \mod 74 \)?
|
67
| 9.375 |
12,147 |
One of the angles in a triangle is $120^{\circ}$, and the lengths of the sides form an arithmetic progression. Find the ratio of the lengths of the sides of the triangle.
|
3 : 5 : 7
| 57.8125 |
12,148 |
Parallelogram $PQRS$ has vertices $P(4,4)$, $Q(-2,-2)$, $R(-8,-2)$, and $S(2,4)$. If a point is selected at random from the region determined by the parallelogram, what is the probability that the point is not above the $x$-axis?
|
\frac{1}{2}
| 32.03125 |
12,149 |
A straight one-way city street has 8 consecutive traffic lights. Every light remains green for 1.5 minutes, yellow for 3 seconds, and red for 1.5 minutes. The lights are synchronized so that each light turns red 10 seconds after the preceding one turns red. What is the longest interval of time, in seconds, during which all 8 lights are green?
|
20
| 16.40625 |
12,150 |
The sum of the maximum and minimum values of the function $y=2\sin \left( \frac{\pi x}{6}- \frac{\pi}{3}\right)$ where $(0\leqslant x\leqslant 9)$ is to be determined.
|
2-\sqrt{3}
| 5.46875 |
12,151 |
By how many zeros does the number 2012! end?
|
501
| 100 |
12,152 |
Given the complex numbers \( z_1 \) and \( z_2 \) such that \( \left| z_1 + z_2 \right| = 20 \) and \( \left| z_1^2 + z_2^2 \right| = 16 \), find the minimum value of \( \left| z_1^3 + z_2^3 \right| \).
|
3520
| 82.03125 |
12,153 |
Find the limits:
1) \(\lim_{x \to 3}\left(\frac{1}{x-3}-\frac{6}{x^2-9}\right)\)
2) \(\lim_{x \to \infty}\left(\sqrt{x^2 + 1}-x\right)\)
3) \(\lim_{n \to \infty} 2^n \sin \frac{x}{2^n}\)
4) \(\lim_{x \to 1}(1-x) \tan \frac{\pi x}{2}\).
|
\frac{2}{\pi}
| 89.0625 |
12,154 |
To encourage residents to conserve water, a city charges residents for domestic water use in a tiered pricing system. The table below shows partial information on the tiered pricing for domestic water use for residents in the city, each with their own water meter:
| Water Sales Price | Sewage Treatment Price |
|-------------------|------------------------|
| Monthly Water Usage per Household | Unit Price: yuan/ton | Unit Price: yuan/ton |
| 17 tons or less | $a$ | $0.80$ |
| More than 17 tons but not more than 30 tons | $b$ | $0.80$ |
| More than 30 tons | $6.00$ | $0.80$ |
(Notes: 1. The amount of sewage generated by each household is equal to the amount of tap water used by that household; 2. Water bill = tap water cost + sewage treatment fee)
It is known that in April 2020, the Wang family used 15 tons of water and paid 45 yuan, and in May, they used 25 tons of water and paid 91 yuan.
(1) Find the values of $a$ and $b$;
(2) If the Wang family paid 150 yuan for water in June, how many tons of water did they use that month?
|
35
| 26.5625 |
12,155 |
From an arbitrary tetrahedron, four smaller tetrahedra are separated by four planes passing through the midpoints of the edges emanating from each vertex. Calculate the ratio of the volume of the remaining body to the volume of the original tetrahedron.
|
1/2
| 96.875 |
12,156 |
The product of all integers whose absolute value is greater than 3 but not greater than 6 is ____.
|
-14400
| 24.21875 |
12,157 |
Find the sum of $453_6$, $436_6$, and $42_6$ in base 6.
|
1415_6
| 78.90625 |
12,158 |
The cheetah takes strides of 2 meters each and the fox takes strides of 1 meter each. The time it takes for the cheetah to run 2 strides is the same as the time it takes for the fox to run 3 strides. Given that the distance between the cheetah and the fox is 30 meters, find the distance the cheetah must run to catch up with the fox.
|
120
| 51.5625 |
12,159 |
Given that the odd function $f(x)$ is a monotonically increasing function defined on $\mathbb{R}$ and the sequence $\{x_n\}$ is an arithmetic sequence with a common difference of 2, satisfying $f(x_8) + f(x_9) + f(x_{10}) + f(x_{11}) = 0$, find the value of $x_{2012}$.
|
4005
| 90.625 |
12,160 |
In the diagram, $D$ and $E$ are the midpoints of $\overline{AC}$ and $\overline{BC}$ respectively, where $A(0,8)$, $B(0,0)$, and $C(10,0)$. Find the sum of the slope and $y$-intercept of the line passing through the points $C$ and $D$.
|
\frac{36}{5}
| 73.4375 |
12,161 |
The integers \( m \) and \( n \) satisfy the equation \( 3^{m} \times n = 7! + 8! + 9! \). What is the smallest possible value for \( n \)?
|
560
| 52.34375 |
12,162 |
A sequence of numbers is arranged in the following pattern: \(1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 6, \cdots\). Starting from the first number on the left, find the sum of the first 99 numbers.
|
1782
| 39.84375 |
12,163 |
In the six-digit number $1 A B C D E$, each letter represents a digit. Given that $1 A B C D E \times 3 = A B C D E 1$, calculate the value of $A+B+C+D+E$.
|
26
| 68.75 |
12,164 |
In the interior of a triangle \( ABC \) with area 1, points \( D \), \( E \), and \( F \) are chosen such that \( D \) is the midpoint of \( AE \), \( E \) is the midpoint of \( BF \), and \( F \) is the midpoint of \( CD \). Find the area of triangle \( DEF \).
|
1/7
| 57.03125 |
12,165 |
Two cars start from the same location at the same time, moving in the same direction at a constant speed. Each car can carry a maximum of 24 barrels of gasoline, and each barrel of gasoline allows a car to travel 60km. Both cars must return to the starting point, but they do not have to return at the same time. The cars can lend gasoline to each other. To maximize the distance one car can travel away from the starting point, the other car should turn back at a distance of $\boxed{360}$ km from the starting point.
|
360
| 69.53125 |
12,166 |
In the tetrahedron $P-ABC$, edges $PA$, $AB$, and $AC$ are mutually perpendicular, and $PA = AB = AC$. Points $E$ and $F$ are the midpoints of segments $AB$ and $PC$, respectively. Find the sine of the angle between line $EF$ and plane $PBC$.
|
\frac{1}{3}
| 57.8125 |
12,167 |
What is the result of $120 \div (6 \div 2 \times 3)$?
|
\frac{120}{9}
| 0 |
12,168 |
How many pairs of two-digit positive integers have a difference of 50?
|
40
| 53.125 |
12,169 |
The diagonals \(AC\) and \(BD\) of the symmetric trapezoid \(ABCD\) intersect at point \(O\). The area of triangle \(AOB\) is \(52 \, \text{m}^2\) and the area of triangle \(COD\) is \(117 \, \text{m}^2\). Calculate the area of the trapezoid.
|
325
| 50 |
12,170 |
Calculate the value of $3^{12} \cdot 3^3$ and express it as some integer raised to the third power.
|
243
| 17.1875 |
12,171 |
In the finals of a beauty contest among giraffes, there were two finalists: the Tall one and the Spotted one. There are 135 voters divided into 5 districts, each district is divided into 9 precincts, and each precinct has 3 voters. Voters in each precinct choose the winner by majority vote; in a district, the giraffe that wins in the majority of precincts is the winner; finally, the giraffe that wins in the majority of districts is declared the winner of the final. The Tall giraffe won. What is the minimum number of voters who could have voted for the Tall giraffe?
|
30
| 64.0625 |
12,172 |
Given distinct natural numbers \( k, l, m, n \), it is known that there exist three natural numbers \( a, b, c \) such that each of the numbers \( k, l, m, n \) is a root of either the equation \( a x^{2} - b x + c = 0 \) or the equation \( c x^{2} - 16 b x + 256 a = 0 \). Find \( k^{2} + l^{2} + m^{2} + n^{2} \).
|
325
| 7.8125 |
12,173 |
Alexio now has 100 cards numbered from 1 to 100. He again randomly selects one card from the box. What is the probability that the number on the chosen card is a multiple of 3, 5, or 7? Express your answer as a common fraction.
|
\frac{11}{20}
| 67.96875 |
12,174 |
In rectangle $ABCD$, side $AB$ measures $8$ units and side $BC$ measures $4$ units. Points $F$ and $G$ are on side $CD$ such that segment $DF$ measures $2$ units and segment $GC$ measures $2$ units, and lines $AF$ and $BG$ intersect at $E$. What is the area of triangle $AEB$?
|
32
| 77.34375 |
12,175 |
\[
\sum_{k=1}^{70} \frac{k}{x-k} \geq \frac{5}{4}
\]
is a union of disjoint intervals the sum of whose lengths is 1988.
|
1988
| 31.25 |
12,176 |
When two fair dice are thrown, the numbers obtained are $a$ and $b$, respectively. Express the probability that the slope $k$ of the line $bx+ay=1$ is greater than or equal to $-\dfrac{2}{5}$.
|
\dfrac{1}{6}
| 20.3125 |
12,177 |
You are in a place where 99% of the inhabitants are vampires and 1% are regular humans. On average, 90% of the vampires are correctly identified as vampires, and 90% of humans are correctly identified as humans. What is the probability that someone identified as a human is actually a human?
|
1/12
| 84.375 |
12,178 |
Given the parabola C: x² = 2py (p > 0), draw a line l: y = 6x + 8, which intersects the parabola C at points A and B. Point O is the origin, and $\overrightarrow{OA} \cdot \overrightarrow{OB} = 0$. A moving circle P has its center on the parabola C and passes through a fixed point D(0, 4). If the moving circle P intersects the x-axis at points E and F, and |DE| < |DF|, find the minimum value of $\frac{|DE|}{|DF|}$.
|
\sqrt{2} - 1
| 0 |
12,179 |
Given an $8 \times 6$ grid, consider a triangle with vertices at $D=(2,1)$, $E=(7,1)$, and $F=(5,5)$. Determine the fraction of the grid covered by this triangle.
|
\frac{5}{24}
| 77.34375 |
12,180 |
The real numbers \( x, y, z \) satisfy \( x + y + z = 2 \) and \( xy + yz + zx = 1 \). Find the maximum possible value of \( x - y \).
|
\frac{2 \sqrt{3}}{3}
| 42.96875 |
12,181 |
The quadratic \( x^2 + 1800x + 1800 \) can be written in the form \( (x+b)^2 + c \), where \( b \) and \( c \) are constants. What is \( \frac{c}{b} \)?
|
-898
| 89.84375 |
12,182 |
A deck consists of 32 cards divided into 4 suits, each containing 8 cards. In how many ways can we choose 6 cards such that all four suits are represented among them?
|
415744
| 0 |
12,183 |
A positive integer sequence has its first term as 8 and its second term as 1. From the third term onwards, each term is the sum of the two preceding terms. What is the remainder when the 2013th term in this sequence is divided by 105?
|
16
| 27.34375 |
12,184 |
Observing the equations:<br/>$1\times 3+1=4=2^{2}$;<br/>$2\times 4+1=9=3^{2}$;<br/>$3\times 5+1=16=4^{2}$;<br/>$4\times 6+1=25=5^{2}$;<br/>$\ldots $<br/>$(1)7\times 9+1=( $______)$^{2}$;<br/>$(2)$ Using the pattern you discovered, calculate: $(1+\frac{1}{1×3})×(1+\frac{1}{2×4})×(1+\frac{1}{3×5})×⋅⋅⋅×(1+\frac{1}{198×200})$.
|
\frac{199}{100}
| 89.84375 |
12,185 |
If $\alpha, \beta, \gamma$ are the roots of the equation $x^3 - x - 1 = 0$, find the value of $\frac{1 + \alpha}{1 - \alpha} + \frac{1 + \beta}{1 - \beta} + \frac{1 + \gamma}{1 - \gamma}$.
|
-7
| 64.0625 |
12,186 |
Xiaopang arranges the 50 integers from 1 to 50 in ascending order without any spaces in between. Then, he inserts a "+" sign between each pair of adjacent digits, resulting in an addition expression: \(1+2+3+4+5+6+7+8+9+1+0+1+1+\cdots+4+9+5+0\). Please calculate the sum of this addition expression. The result is ________.
|
330
| 60.15625 |
12,187 |
Lucy surveyed a group of people about their knowledge of mosquitoes. To the nearest tenth of a percent, she found that $75.3\%$ of the people surveyed thought mosquitoes transmitted malaria. Of the people who thought mosquitoes transmitted malaria, $52.8\%$ believed that mosquitoes also frequently transmitted the common cold. Since mosquitoes do not transmit the common cold, these 28 people were mistaken. How many total people did Lucy survey?
|
70
| 40.625 |
12,188 |
Digital clocks display hours and minutes (for example, 16:15). While practicing arithmetic, Buratino finds the sum of the digits on the clock $(1+6+1+5=13)$. Write down such a time of day when the sum of the digits on the clock will be the greatest.
|
19:59
| 33.59375 |
12,189 |
Xiao Gang goes to buy milk and finds that it's on special offer that day: each bag costs 2.5 yuan, and there is a "buy two, get one free" promotion. Xiao Gang has 30 yuan. What is the maximum number of bags of milk he can buy?
|
18
| 74.21875 |
12,190 |
The interior angles of a convex polygon form an arithmetic sequence, with the smallest angle being $120^\circ$ and the common difference being $5^\circ$. Determine the number of sides $n$ for the polygon.
|
n = 9
| 86.71875 |
12,191 |
Given three points $A$, $B$, $C$ on a straight line in the Cartesian coordinate system, satisfying $\overrightarrow{OA}=(-3,m+1)$, $\overrightarrow{OB}=(n,3)$, $\overrightarrow{OC}=(7,4)$, and $\overrightarrow{OA} \perp \overrightarrow{OB}$, where $O$ is the origin.
$(1)$ Find the values of the real numbers $m$, $n$;
$(2)$ Let $G$ be the centroid of $\triangle AOC$, and $\overrightarrow{OG}= \frac{2}{3} \overrightarrow{OB}$, find the value of $\cos \angle AOC$.
|
-\frac{\sqrt{5}}{5}
| 64.84375 |
12,192 |
Given that \( x \) is a multiple of \( 7200 \), what is the greatest common divisor of \( f(x)=(5x+6)(8x+3)(11x+9)(4x+12) \) and \( x \)?
|
72
| 33.59375 |
12,193 |
There are four distinct codes $A, B, C, D$ used by an intelligence station, with one code being used each week. Each week, a code is chosen randomly with equal probability from the three codes that were not used the previous week. Given that code $A$ is used in the first week, what is the probability that code $A$ is also used in the seventh week? (Express your answer as a simplified fraction.)
|
61/243
| 65.625 |
12,194 |
If $x^2 + 3x - 1 = 0$, then $x^3 + 5x^2 + 5x + 18 =$ ?
|
20
| 94.53125 |
12,195 |
In the sequence \(\left\{a_{n}\right\}\), it is known that \(a_{1}=1\) and \(a_{n+1}>a_{n}\), and that \(a_{n+1}^{2}+a_{n}^{2}+1=2\left(a_{n+1}+a_{n}+2 a_{n+1} a_{n}\right)\). Find \(\lim \limits_{n \rightarrow \infty} \frac{S_{n}}{n a_{n}}\).
|
1/3
| 35.9375 |
12,196 |
Simplify $\sqrt{18} \times \sqrt{32} \times \sqrt{2}$.
|
24\sqrt{2}
| 91.40625 |
12,197 |
There are 19 weights with values $1, 2, 3, \ldots, 19$ grams: nine iron, nine bronze, and one gold. It is known that the total weight of all the iron weights is 90 grams more than the total weight of the bronze weights. Find the weight of the gold weight.
|
10
| 79.6875 |
12,198 |
Given that $A, B, C, D, E, F$ are the vertices of a regular hexagon with a side length of 2, and a parabola passes through the points $A, B, C, D$, find the distance from the focus of the parabola to its directrix.
|
\frac{\sqrt{3}}{2}
| 15.625 |
12,199 |
How many positive integers less than $201$ are multiples of either $6$ or $8$, but not both at once?
|
42
| 26.5625 |
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