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40.3k
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100
|
|---|---|---|---|
19,100 |
Given \(x \geqslant 1\), the minimum value of the function \(y=f(x)= \frac {4x^{2}-2x+16}{2x-1}\) is \_\_\_\_\_\_, and the corresponding value of \(x\) is \_\_\_\_\_\_.
|
\frac {5}{2}
| 33.59375 |
19,101 |
A nine-digit number is formed by repeating a three-digit number three times. For example, 123,123,123 or 456,456,456 are numbers of this form. What is the greatest common divisor of all nine-digit numbers of this form?
|
1001001
| 49.21875 |
19,102 |
Given the array: $(1,1,1)$, $(2,2,4)$, $(3,4,12)$, $(4,8,32)$, $\ldots$, $(a_{n}, b_{n}, c_{n})$, find the value of $c_{7}$.
|
448
| 90.625 |
19,103 |
The perimeter of triangle \( ABC \) is 1. Circle \( \omega \) is tangent to side \( BC \) and the extensions of side \( AB \) at point \( P \) and side \( AC \) at point \( Q \). The line passing through the midpoints of \( AB \) and \( AC \) intersects the circumcircle of triangle \( APQ \) at points \( X \) and \( Y \). Find the length of segment \( XY \).
|
\frac{1}{2}
| 53.90625 |
19,104 |
The graph of the function $f(x)=\sin(\omega x+\varphi)$, where $(\omega>0, |\varphi|<\frac{\pi}{2})$, passes through the point $(0,-\frac{1}{2})$. Find the minimum value of $\omega$ if the graph of this function is shifted to the right by $\frac{\pi}{3}$ units and becomes symmetric about the origin.
|
\frac{5}{2}
| 56.25 |
19,105 |
On a board, the 2014 positive integers from 1 to 2014 are written. The allowed operation is to choose two numbers \( a \) and \( b \), erase them, and write in their place the numbers \( \text{gcd}(a, b) \) (greatest common divisor) and \( \text{lcm}(a, b) \) (least common multiple). This operation can be performed with any two numbers on the board, including numbers that resulted from previous operations. Determine the largest number of 1's that we can leave on the board.
|
1007
| 3.90625 |
19,106 |
Let $ABC$ be an isosceles triangle with $AB=AC$ and incentre $I$ . If $AI=3$ and the distance from $I$ to $BC$ is $2$ , what is the square of length on $BC$ ?
|
80
| 19.53125 |
19,107 |
A $37$-gon $Q_1$ is drawn in the Cartesian plane, and the sum of the $x$-coordinates of its $37$ vertices equals $185$. The midpoints of the sides of $Q_1$ form a second $37$-gon, $Q_2$. Then, the midpoints of the sides of $Q_2$ form a third $37$-gon, $Q_3$. Find the sum of the $x$-coordinates of the vertices of $Q_3$.
|
185
| 92.96875 |
19,108 |
For the odd function $f(x)$ defined on domain $\mathbb{R}$ that satisfies $f(4 - x) + f(x) = 0$, given that $f(x) = 2^x$ for $-2 < x < 0$, calculate $f(\log_2 20)$.
|
-\frac{4}{5}
| 60.9375 |
19,109 |
In $\triangle ABC$, $P$ is a point on the side $BC$ such that $\overrightarrow{BP} = \frac{1}{2}\overrightarrow{PC}$. Points $M$ and $N$ lie on the line passing through $P$ such that $\overrightarrow{AM} = \lambda \overrightarrow{AB}$ and $\overrightarrow{AN} = \mu \overrightarrow{AC}$ where $\lambda, \mu > 0$. Find the minimum value of $\lambda + 2\mu$.
|
\frac{8}{3}
| 21.09375 |
19,110 |
Given the line y=b intersects with the function f(x)=2x+3 and the function g(x)=ax+ln x (where a is a real constant in the interval [0, 3/2]), find the minimum value of |AB|.
|
2 - \frac{\ln 2}{2}
| 0 |
19,111 |
In a right triangle $DEF$ with $\angle D = 90^\circ$, we have $DE = 8$ and $DF = 15$. Find $\cos F$.
|
\frac{15}{17}
| 88.28125 |
19,112 |
A number has 6 on both its tens and hundredths places, and 0 on both its ones and tenths places. This number is written as \_\_\_\_\_\_.
|
60.06
| 90.625 |
19,113 |
To arrange 5 volunteers and 2 elderly people in a row, where the 2 elderly people are adjacent but not at the ends, calculate the total number of different arrangements.
|
960
| 28.90625 |
19,114 |
Using only once each of the digits $1, 2, 3, 4, 5, 6, 7$ and $ 8$ , write the square and the cube of a positive integer. Determine what that number can be.
|
24
| 77.34375 |
19,115 |
Dr. Math's four-digit house number $ABCD$ contains no zeroes and can be split into two different two-digit primes ``$AB$'' and ``$CD$''. Moreover, both these two-digit primes are greater than 50 but less than 100. Find the total number of possible house numbers for Dr. Math.
|
90
| 3.90625 |
19,116 |
Given the inequality $x^{2}+ax+1\geqslant 0$, if this inequality holds for all $x\in(0, \frac {1}{2}]$, find the minimum value of the real number $a$.
|
-\frac {5}{2}
| 83.59375 |
19,117 |
Compute $\dbinom{60}{3}$.
|
57020
| 0 |
19,118 |
A four-digit natural number $M$, where the digits in each place are not $0$, we take its hundreds digit as the tens digit and the tens digit as the units digit to form a new two-digit number. If this two-digit number is greater than the sum of the thousands digit and units digit of $M$, then we call this number $M$ a "heart's desire number"; if this two-digit number can also be divided by the sum of the thousands digit and units digit of $M$, then we call this number $M$ not only a "heart's desire" but also a "desire fulfilled". ["Heart's desire, desire fulfilled" comes from "Analects of Confucius. On Governance", meaning that what is desired in the heart becomes wishes, and all wishes can be fulfilled.] For example, $M=3456$, since $45 \gt 3+6$, and $45\div \left(3+6\right)=5$, $3456$ is not only a "heart's desire" but also a "desire fulfilled". Now there is a four-digit natural number $M=1000a+100b+10c+d$, where $1\leqslant a\leqslant 9$, $1\leqslant b\leqslant 9$, $1\leqslant c\leqslant 9$, $1\leqslant d\leqslant 9$, $a$, $b$, $c$, $d$ are all integers, and $c \gt d$. If $M$ is not only a "heart's desire" but also a "desire fulfilled", where $\frac{{10b+c}}{{a+d}}=11$, let $F\left(M\right)=10\left(a+b\right)+3c$. If $F\left(M\right)$ can be divided by $7$, then the maximum value of the natural number $M$ that meets the conditions is ____.
|
5883
| 89.0625 |
19,119 |
A particle starts from the origin on the number line, and at each step, it can move either 1 unit in the positive direction or 1 unit in the negative direction. After 10 steps, if the distance between the particle and the origin is 4, then the total number of distinct ways the particle can move is (answer in digits).
|
240
| 48.4375 |
19,120 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. The function $f(x) = 2\cos x \sin (x - A) (x \in \mathbb{R})$ reaches its minimum value at $x = \frac{11\pi}{12}$.
1. Find the measure of angle $A$.
2. If $a = 7$ and $\sin B + \sin C = \frac{13\sqrt{3}}{14}$, find the area of $\triangle ABC$.
|
10\sqrt{3}
| 59.375 |
19,121 |
What is the least possible value of
\[(x+2)(x+3)(x+4)(x+5) + 2024\] where \( x \) is a real number?
A) 2022
B) 2023
C) 2024
D) 2025
E) 2026
|
2023
| 94.53125 |
19,122 |
What is the largest four-digit negative integer congruent to $2 \pmod{17}$?
|
-1001
| 35.15625 |
19,123 |
Given that $\cos \alpha =-\dfrac{3}{4}, \sin \beta =\dfrac{2}{3}$, with $\alpha$ in the third quadrant and $\beta \in (\dfrac{\pi }{2}, \pi )$.
(I) Find the value of $\sin 2\alpha$;
(II) Find the value of $\cos (2\alpha + \beta )$.
|
-\dfrac{\sqrt{5} + 6\sqrt{7}}{24}
| 63.28125 |
19,124 |
The number of integer solutions for the inequality \( |x| < 3 \pi \) is ( ).
|
19
| 96.875 |
19,125 |
When $\frac{1}{2222}$ is expressed as a decimal, what is the sum of the first 60 digits after the decimal point?
|
108
| 32.8125 |
19,126 |
What is \(1\tfrac{1}{2}\) divided by \(\tfrac{5}{6}\)?
|
\tfrac{9}{5}
| 88.28125 |
19,127 |
Compute the sum:
\[ 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2)))))))) \]
|
1022
| 42.96875 |
19,128 |
In acute \\(\triangle ABC\\), the sides opposite to angles \\(A\\), \\(B\\), and \\(C\\) are \\(a\\), \\(b\\), and \\(c\\) respectively, with \\(a=4\\), \\(b=5\\), and the area of \\(\triangle ABC\\) is \\(5\sqrt{3}\\). Find the value of side \\(c=\\) ______.
|
\sqrt{21}
| 99.21875 |
19,129 |
If the graph of the power function $f(x)=x^{\alpha}$ ($\alpha$ is a constant) always passes through point $A$, and the line ${kx}{-}y{+}2k{+}1{+}\sqrt{3}{=}0$ always passes through point $B$, then the angle of inclination of line $AB$ is _____.
|
\frac{5\pi}{6}
| 60.15625 |
19,130 |
Calculate $45 \cdot 68 \cdot 99 \equiv m \pmod{25}$, where $0 \leq m < 25$.
|
15
| 41.40625 |
19,131 |
In the sequence $00$ , $01$ , $02$ , $03$ , $\cdots$ , $99$ the terms are rearranged so that each term is obtained from the previous one by increasing or decreasing one of its digits by $1$ (for example, $29$ can be followed by $19$ , $39$ , or $28$ , but not by $30$ or $20$ ). What is the maximal number of terms that could remain on their places?
|
50
| 19.53125 |
19,132 |
Given $$\frac{1}{C_5^m} - \frac{1}{C_6^m} = \frac{7}{10C_7^m}$$, find $C_{21}^m$.
|
210
| 55.46875 |
19,133 |
Given the ellipse $C_{1}: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ and the hyperbola $C_{2}: \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ with asymptote equations $x \pm \sqrt{3}y = 0$, determine the product of the eccentricities of $C_{1}$ and $C_{2}$.
|
\frac{2\sqrt{2}}{3}
| 85.15625 |
19,134 |
In the expansion of \((x + y + z)^8\), determine the sum of the coefficients of all terms of the form \(x^2 y^a z^b\) (\(a, b \in \mathbf{N}\)).
|
1792
| 83.59375 |
19,135 |
Let the sequence \( A = (a_1, a_2, \dots, a_{110}) \) be such that its Cesaro sum is 1200. Calculate the Cesaro sum of the 111-term sequence \( (0, a_1, a_2, \dots, a_{110}) \).
|
\frac{132000}{111}
| 1.5625 |
19,136 |
Lisa drew graphs of all functions of the form \( y = ax + b \), where \( a \) and \( b \) take all natural values from 1 to 100. How many of these graphs pass through the point \((3, 333)\)?
|
33
| 29.6875 |
19,137 |
In a lathe workshop, parts are turned from steel blanks, one part from one blank. The shavings left after processing three blanks can be remelted to get exactly one blank. How many parts can be made from nine blanks? What about from fourteen blanks? How many blanks are needed to get 40 parts?
|
27
| 37.5 |
19,138 |
For a given list of three numbers, the operation "changesum" replaces each number in the list with the sum of the other two. For example, applying "changesum" to \(3,11,7\) gives \(18,10,14\). Arav starts with the list \(20,2,3\) and applies the operation "changesum" 2023 times. What is the largest difference between two of the three numbers in his final list?
|
18
| 16.40625 |
19,139 |
In a bag, there are 7 blue chips, 5 yellow chips, and 4 red chips. One chip is drawn from the bag and then replaced. A second chip is then drawn. What is the probability that the two selected chips are of different colors?
|
\frac{83}{128}
| 64.84375 |
19,140 |
In $\triangle XYZ$, angle XZY is a right angle. There are three squares constructed such that each side adjacent to angle XZY has a square on it. The sum of the areas of these three squares is 512 square centimeters. Also, XZ is 20% longer than ZY. What's the area of the largest square?
|
256
| 31.25 |
19,141 |
What is the smallest positive integer $n$ such that all the roots of $z^5 - z^3 + z = 0$ are $n^{\text{th}}$ roots of unity?
|
12
| 83.59375 |
19,142 |
What was Tony's average speed, in miles per hour, during the 3-hour period when his odometer increased from 12321 to the next higher palindrome?
|
33.33
| 39.0625 |
19,143 |
Miki extracts 12 ounces of juice from 4 pears and 6 ounces of juice from 3 oranges. Determine the percentage of pear juice in a blend using 8 pears and 6 oranges.
|
66.67\%
| 37.5 |
19,144 |
In rectangle $PQRS$, $PQ = 150$. Let $T$ be the midpoint of $\overline{PS}$. Given that line $PT$ and line $QT$ are perpendicular, find the greatest integer less than $PS$.
|
212
| 21.09375 |
19,145 |
Let \\(\{a_n\}\\) be a geometric sequence where each term is positive, and let \\(S_n\\) be the sum of the first \\(n\\) terms with \\(S_{10}=10\\) and \\(S_{30}=70\\). Find \\(S_{40}=\\)_______.
|
150
| 90.625 |
19,146 |
Given the points M(2,0) and N(a,b) in the Cartesian coordinate system, with the Manhattan distance between M and N defined as d(M,N) = |x₁ - x₂| + |y₁ - y₂|, and d(M,N) = 2, find the sum of the minimum and maximum values of a² + b² - 4a.
|
-2
| 15.625 |
19,147 |
$(1)$ State the operation laws or rules used in each step of the following calculation:<br/>$(-0.4)\times \left(-0.8\right)\times \left(-1.25\right)\times 2.5$<br/>$=-\left(0.4\times 0.8\times 1.25\times 2.5\right) \text{(Step 1)}$<br/>$=-\left(0.4\times 2.5\times 0.8\times 1.25\right) \text{(Step 2)}$<br/>$=-\left[\left(0.4\times 2.5\right)\times \left(0.8\times 1.25\right)\right] \text{(Step 3)}$<br/>$=-\left(1\times 1\right)=-1$.<br/>$(2)$ Calculate the following expression using a simpler method: $(-\frac{5}{8})×\frac{3}{14}×(-\frac{16}{5})×(-\frac{7}{6})$.
|
-\frac{1}{2}
| 58.59375 |
19,148 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $a > b$, $a=5$, $c=6$, and $\sin B= \frac{3}{5}$.
(Ⅰ) Find the values of $b$ and $\sin A$;
(Ⅱ) Find the value of $\sin \left(2A+ \frac{\pi}{4}\right)$.
|
\frac{7\sqrt{2}}{26}
| 11.71875 |
19,149 |
On the grid shown, Jane starts at dot $A$. She tosses a fair coin to determine which way to move. If she tosses a head, she moves up one dot. If she tosses a tail, she moves right one dot. After four tosses of the coin, Jane will be at one of the dots $P, Q, R, S$, or $T$. What is the probability that Jane will be at dot $R$?
|
$\frac{3}{8}$
| 0 |
19,150 |
A group of 12 friends decides to form a committee of 5. Calculate the number of different committees that can be formed. Additionally, if there are 4 friends who refuse to work together, how many committees can be formed without any of these 4 friends?
|
56
| 71.875 |
19,151 |
Find the value of $s$ for which the vector
\[\bold{u} = \begin{pmatrix} 1 \\ -2 \\ -4 \end{pmatrix} + s \begin{pmatrix} 5 \\ 3 \\ -2 \end{pmatrix}\] is closest to
\[\bold{b} = \begin{pmatrix} 3 \\ 3 \\ 4 \end{pmatrix}.\]
|
\frac{9}{38}
| 63.28125 |
19,152 |
During a math competition organized in a certain city, the scores of all participating students approximately follow a normal distribution $N(60, 100)$. It is known that there are 13 students who scored 90 or above.
(1) Calculate the total number of students who participated in the competition.
(2) If it is planned to reward the top 228 students in the competition, what is the score cutoff for the awarded students?
|
80
| 50.78125 |
19,153 |
The hare and the tortoise had a race over 100 meters, in which both maintained constant speeds. When the hare reached the finish line, it was 75 meters in front of the tortoise. The hare immediately turned around and ran back towards the start line. How far from the finish line did the hare and the tortoise meet?
|
60
| 13.28125 |
19,154 |
Let $A$ , $M$ , and $C$ be nonnegative integers such that $A+M+C=10$ . Find the maximum value of $A\cdot M\cdot C+A\cdot M+M\cdot C+C\cdot A$.
|
69
| 100 |
19,155 |
The famous skater Tony Hawk rides a skateboard (segment \( AB \)) on a ramp, which is a semicircle with diameter \( PQ \). Point \( M \) is the midpoint of the skateboard, and \( C \) is the foot of the perpendicular dropped from point \( A \) to the diameter \( PQ \). What values can the angle \( \angle ACM \) take, given that the angular measure of arc \( AB \) is \( 24^\circ \)?
|
12
| 79.6875 |
19,156 |
Determine the maximum integer value of the expression
\[\frac{3x^2 + 9x + 28}{3x^2 + 9x + 7}.\]
|
85
| 66.40625 |
19,157 |
Find the number of solutions to the equation
\[\sin x = \left( \frac{1}{3} \right)^x\]
on the interval \( (0, 150 \pi) \).
|
75
| 16.40625 |
19,158 |
Agricultural researchers randomly selected $50$ plants of a certain crop in an experimental field for research. The frequency distribution of the mass (in grams) of a single plant falling into each group is shown in the table below:
| Mass of a Single Plant (in grams) | Frequency |
|-----------------------------------|-----------|
| $[12.5,15.5)$ | $2$ |
| $[15.5,18.5)$ | $5$ |
| $[18.5,21.5)$ | $11$ |
| $[21.5,24.5)$ | $14$ |
| $[24.5,27.5)$ | $11$ |
| $[27.5,30.5)$ | $4$ |
| $[30.5,33.5]$ | $3$ |
$(1)$ Based on the frequency distribution table, calculate the probability that the mass of a single plant of this crop falls in the interval $[27.5,33.5]$ (using frequency to estimate probability).
$(2)$ Calculate the sample mean of the mass of these $50$ plants, denoted by $\overline{x}$. (Represent each group of data by the midpoint of the interval)
$(3)$ If the mass $X$ of a single plant of this crop follows a normal distribution $N(\mu, \sigma^2)$, where $\mu$ is approximately the sample mean $\overline{x}$ and $\sigma^2$ is approximately the sample variance $S^2$, and it is calculated that $S^2 = 18.5364$, find $P(X < 27.37)$.
Note:
1. If $X$ follows a normal distribution $N(\mu, \sigma^2)$, then $P(\mu - \sigma < X < \mu + \sigma) = 0.6826$, $P(\mu - 2\sigma < X < \mu + 2\sigma) = 0.9544$;
2. $\sqrt{18.5364} \approx 4.31$.
|
0.8413
| 24.21875 |
19,159 |
In a math test, the scores of 6 students are as follows: 98, 88, 90, 92, 90, 94. The mode of this set of data is ______; the median is ______; the average is ______.
|
92
| 9.375 |
19,160 |
In 500 kg of ore, there is a certain amount of iron. After removing 200 kg of impurities, which contain on average 12.5% iron, the iron content in the remaining ore increased by 20%. What amount of iron remains in the ore?
|
187.5
| 0 |
19,161 |
A smaller regular tetrahedron is formed by joining the midpoints of the edges of a larger regular tetrahedron. Determine the ratio of the volume of the smaller tetrahedron to the volume of the larger tetrahedron.
|
\frac{1}{8}
| 70.3125 |
19,162 |
Given the product sequence $\frac{5}{3} \cdot \frac{6}{5} \cdot \frac{7}{6} \cdot \ldots \cdot \frac{a}{b} = 12$, determine the sum of $a$ and $b$.
|
71
| 95.3125 |
19,163 |
Jenny wants to distribute 450 cookies among $p$ boxes such that each box contains an equal number of cookies. Each box must contain more than two cookies, and there must be more than one box. For how many values of $p$ can this distribution be made?
|
14
| 12.5 |
19,164 |
Given that $-\frac{\pi}{2} < x < 0$ and $\sin x + \cos x = \frac{1}{5}$.
(I) Find the value of $\sin x - \cos x$.
(II) Find the value of $\frac{3\sin^2{\frac{x}{2}} - 2\sin{\frac{x}{2}}\cos{\frac{x}{2}} + \cos^2{\frac{x}{2}}}{\tan x + \cot x}$.
|
-\frac{108}{125}
| 39.0625 |
19,165 |
Solve for $c$:
$$\sqrt{9+\sqrt{27+9c}} + \sqrt{3+\sqrt{3+c}} = 3+3\sqrt{3}$$
|
33
| 8.59375 |
19,166 |
Find the volume of the region in space defined by
\[ |z + x + y| + |z + x - y| \leq 10 \]
and \(x, y, z \geq 0\).
|
62.5
| 1.5625 |
19,167 |
If \(\frac{\left(\frac{a}{c}+\frac{a}{b}+1\right)}{\left(\frac{b}{a}+\frac{b}{c}+1\right)}=11\), where \(a, b\), and \(c\) are positive integers, the number of different ordered triples \((a, b, c)\) such that \(a+2b+c \leq 40\) is:
|
42
| 20.3125 |
19,168 |
How many distinct divisors do the following numbers have:
1) \(2^{7}\);
2) \(5^{4}\);
3) \(2^{7} \cdot 5^{4}\);
4) \(2^{m} \cdot 5^{n} \cdot 3^{k}\);
5) 3600;
6) \(42^{5}\)?
|
216
| 94.53125 |
19,169 |
The numbers 1 to 2031 are written on a blackboard.
1. Select any two numbers on the blackboard, find the absolute value of their difference, and erase these two numbers.
2. Then select another number on the blackboard, find the absolute value of its difference from the previous absolute value obtained, and erase this number.
3. Repeat step (2) until all numbers on the blackboard are erased.
What is the maximum final result?
|
2030
| 55.46875 |
19,170 |
Suppose $F$ is a polygon with lattice vertices and sides parralell to x-axis and y-axis.Suppose $S(F),P(F)$ are area and perimeter of $F$ .
Find the smallest k that:
$S(F) \leq k.P(F)^2$
|
1/16
| 95.3125 |
19,171 |
Find
\[
\cos \left( 8 \arccos \frac{1}{5} \right).
\]
|
\frac{-15647}{390625}
| 10.15625 |
19,172 |
Compute the sum:
\[\sin^2 6^\circ + \sin^2 12^\circ + \sin^2 18^\circ + \dots + \sin^2 174^\circ.\]
|
15
| 21.875 |
19,173 |
A dart board is shaped like a regular dodecagon divided into regions, with a square at its center. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?
**A)** $\frac{2+\sqrt{3}}{6}$
**B)** $\frac{\sqrt{3}}{6}$
**C)** $\frac{2 - \sqrt{3}}{6}$
**D)** $\frac{3 - 2\sqrt{3}}{6}$
**E)** $\frac{2\sqrt{3} - 3}{6}$
|
\frac{2 - \sqrt{3}}{6}
| 35.15625 |
19,174 |
Mia sells four burritos and five empanadas for $\$$4.00 and she sells six burritos and three empanadas for $\$$4.50. Assuming a fixed price per item, what is the cost, in dollars, of five burritos and seven empanadas? Express your answer as a decimal to the nearest hundredth.
|
5.25
| 18.75 |
19,175 |
Given polynomials $A=ax^{2}+3xy+2|a|x$ and $B=2x^{2}+6xy+4x+y+1$.
$(1)$ If $2A-B$ is a quadratic trinomial in terms of $x$ and $y$, find the value of $a$.
$(2)$ Under the condition in $(1)$, simplify and evaluate the expression $3(-3a^{2}-2a)-[a^{2}-2(5a-4a^{2}+1)-2a]$.
|
-22
| 31.25 |
19,176 |
**Sarah has misplaced her friend Layla's phone number. Sarah recalls that the first four digits are either 3086, 3089, or 3098. The remaining digits are 0, 1, 2, and 5, but she is not sure about their order. If Sarah randomly calls a seven-digit number based on these criteria, what is the probability that she correctly dials Layla's phone number? Express the answer as a common fraction.**
|
\frac{1}{72}
| 69.53125 |
19,177 |
Given the quadratic function $f(x)=ax^{2}+bx+c$, where $a$, $b$, and $c$ are constants, if the solution set of the inequality $f(x) \geqslant 2ax+b$ is $\mathbb{R}$, find the maximum value of $\frac{b^{2}}{a^{2}+c^{2}}$.
|
2\sqrt{2}-2
| 21.875 |
19,178 |
Let $f$ be a polynomial such that, for all real number $x$ , $f(-x^2-x-1) = x^4 + 2x^3 + 2022x^2 + 2021x + 2019$ .
Compute $f(2018)$ .
|
-2019
| 23.4375 |
19,179 |
Given that \\(F_1\\) and \\(F_2\\) are the left and right foci of the hyperbola \\( \frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1(a > 0,b > 0)\\), if there exists a point \\(P\\) on the left branch of the hyperbola that is symmetric to point \\(F_2\\) with respect to the line \\(y= \frac {bx}{a}\\), then the eccentricity of this hyperbola is \_\_\_\_\_\_.
|
\sqrt {5}
| 0 |
19,180 |
Let $x$ be the largest root of $x^4 - 2009x + 1$ . Find the nearest integer to $\frac{1}{x^3-2009}$ .
|
-13
| 27.34375 |
19,181 |
A quadrilateral \(A B C D\) is inscribed in a circle with radius 6 and center at point \(O\). Its diagonals \(A C\) and \(B D\) are mutually perpendicular and intersect at point \(K\). Points \(E\) and \(F\) are the midpoints of \(A C\) and \(B D\), respectively. The segment \(O K\) is equal to 5, and the area of the quadrilateral \(O E K F\) is 12. Find the area of the quadrilateral \(A B C D\).
|
48
| 33.59375 |
19,182 |
In a right triangle, medians are drawn from point $A$ to segment $\overline{BC}$, which is the hypotenuse, and from point $B$ to segment $\overline{AC}$. The lengths of these medians are 5 and $3\sqrt{5}$ units, respectively. Calculate the length of segment $\overline{AB}$.
|
2\sqrt{14}
| 18.75 |
19,183 |
Triangle $PQR$ has vertices at $P(7, 5)$, $Q(1, -3)$, and $R(4, 4)$. The point $S$ with coordinates $(x, y)$ is chosen inside the triangle so that the three smaller triangles $PQS$, $PRS$, and $QRS$ all have equal areas. What is the value of $12x + 3y$?
|
54
| 95.3125 |
19,184 |
In parallelogram $ABCD$, if $\overrightarrow{AE}=2\overrightarrow{ED}$, $\overrightarrow{BF}=\overrightarrow{FC}$, and $\overrightarrow{AC}=λ\overrightarrow{AE}+\overrightarrow{AF}$, then $\lambda =$____.
|
\frac{3}{4}
| 60.15625 |
19,185 |
Given the planar vectors $\overrightarrow {a}$ and $\overrightarrow {b}$, where $\overrightarrow {a}$ = (2cosα, 2sinα) and $\overrightarrow {b}$ = (cosβ, sinβ), if the minimum value of $|\overrightarrow {a} - λ\overrightarrow {b}|$ for any positive real number λ is $\sqrt{3}$, calculate $|\overrightarrow {a} - \overrightarrow {b}|$.
|
\sqrt{3}
| 62.5 |
19,186 |
Rounded to 3 decimal places, what is $\frac{8}{11}$?
|
0.727
| 60.15625 |
19,187 |
Given a parabola with vertex \( V \) and a focus \( F \), and points \( B \) and \( C \) on the parabola such that \( BF=25 \), \( BV=24 \), and \( CV=20 \), determine the sum of all possible values of the length \( FV \).
|
\frac{50}{3}
| 1.5625 |
19,188 |
For any $x \in \mathbb{R}$, the function $f(x)$ represents the minimum value among the three function values $y_{1}=4x+1$, $y_{2}=x+2$, $y_{3}=-2x+4$. The maximum value of $f(x)$ is \_\_\_\_\_\_.
|
\frac{8}{3}
| 49.21875 |
19,189 |
Given that the sequence $\{a_{n}\}$ is an arithmetic sequence with a non-zero common difference, and $a_{1}+a_{10}=a_{9}$, find $\frac{{a}_{1}+{a}_{2}+…+{a}_{9}}{{a}_{10}}$.
|
\frac{27}{8}
| 92.96875 |
19,190 |
Calculate $8 \cdot 5\frac{2}{5} - 3$.
|
40.2
| 5.46875 |
19,191 |
Determine the coefficient of the $x^{3}$ term in the expansion of $(2x+1)(x-1)^{5}$.
|
-10
| 60.9375 |
19,192 |
Given that the domain of $f(x)$ is $\mathbb{R}$, $f(1)=\frac{1}{4}$, and it satisfies the equation $4f(x)f(y) = f(x+y) + f(x-y)$, find the value of $f(2016)$.
|
\frac{1}{2}
| 60.9375 |
19,193 |
Simplify first, then evaluate: $1-\frac{a-b}{a+2b}÷\frac{a^2-b^2}{a^2+4ab+4b^2}$. Given that $a=2\sin 60^{\circ}-3\tan 45^{\circ}$ and $b=3$.
|
-\sqrt{3}
| 75 |
19,194 |
A wire is cut into two pieces: one of length $a$ bent to form a square, and another of length $b$ bent to form a regular octagon. The square and the octagon have equal areas. What is the ratio $\frac{a}{b}$?
|
\frac{\sqrt{2(1+\sqrt{2})}}{2}
| 0 |
19,195 |
The coefficient of $x^5$ in the expansion of $(1+x)^2(1-x)^5$ is \_\_\_\_\_\_ (Answer with a number).
|
-1
| 92.96875 |
19,196 |
If $f(x) = (x-1)^3 + 1$, calculate the value of $f(-5) + f(-4) + \ldots + f(0) + \ldots + f(7)$.
|
13
| 3.90625 |
19,197 |
(1) Simplify: $f(α)= \dfrac {\sin (α+ \dfrac {3}{2}π)\sin (-α+π)\cos (α+ \dfrac {π}{2})}{\cos (-α -π )\cos (α - \dfrac {π}{2})\tan (α +π )}$
(2) Evaluate: $\tan 675^{\circ}+\sin (-330^{\circ})+\cos 960^{\circ}$
|
-1
| 68.75 |
19,198 |
Given the regression equation $y=0.75x-68.2$ and a student's height of $x=170$ cm, calculate the student's weight in kg.
|
59.3
| 0 |
19,199 |
Given a die is rolled consecutively three times, determine the probability that the sequence of numbers facing upwards forms an arithmetic progression.
|
\frac{1}{12}
| 45.3125 |
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