problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Simplify first, then evaluate: $x^{2}(x-1)-x(x^{2}+x-1)$, where $x=\frac{1}{2}$. | 0 |
Example 1 Express the symmetric polynomial in three variables $x_{1}^{3}+x_{2}^{3}+x_{3}^{3}$ as a polynomial in $\sigma_{1}, \sigma_{2}, \sigma_{3}$. | x_{1}^{3}+x_{2}^{3}+x_{3}^{3}=\sigma_{1}^{3}-3\sigma_{1}\sigma_{2}+3\sigma_{3} |
1. Let three complex numbers $1, \mathrm{i}, z$ correspond to three collinear points in the complex plane, and $|z|=5$. Then $z=$ $\qquad$ . | -3+4\mathrm{i} |
Given the function $f(x) = \frac{1}{2}x^2 - 5x + 4\ln x$.
(1) Determine the domain of the function $f(x)$ and its intervals of monotonicity.
(2) Find the extremal values of the function $f(x)$. | -12 + 4\ln 4 |
The number of real values of $x$ satisfying the equation $2^{2x^2 - 7x + 5} = 1$ is:
$\textbf{(A)}\ 0 \qquad \textbf{(B) }\ 1 \qquad \textbf{(C) }\ 2 \qquad \textbf{(D) }\ 3 \qquad \textbf{(E) }\ \text{more than 4}$ | \textbf{(C)}2 |
## Task 4
Note the largest four-digit and the smallest three-digit number that can be written with different digits. Calculate their sum. | 9978 |
Example 2. Find the solution of the equation
$$
y^{\prime \prime}+y=0
$$
satisfying the initial conditions
$$
\left.y\right|_{x=0}=1,\left.\quad y^{\prime}\right|_{x=0}=0
$$ | y(x)=\cosx |
Given the function $f\left(x\right)=\sin \omega x\left(\sin \omega x+\cos \omega x\right)-\frac{1}{2}(\omega\ \ \gt 0)$ has exactly $1$ maximum point and $1$ minimum point on the interval $\left(0,\pi \right)$, then the range of $\omega$ is ( )
A: $(\frac{7}{8},\frac{11}{8})$
B: $(\frac{7}{8},\frac{11}{8}]$
C: $(\fr... | B |
Higher Secondary P4
If the fraction $\dfrac{a}{b}$ is greater than $\dfrac{31}{17}$ in the least amount while $b<17$, find $\dfrac{a}{b}$. | \frac{11}{6} |
Determine all pairs $(m, n)$ of non-negative integers that satisfy the equation
$$
3^{m}-7^{n}=2
$$ | (1,0) \text{ and } (2,1) |
11. On a circle, there is 1 red point and 2009 blue points. Xiao Dan calculates the number of convex polygons where all vertices are blue points, while Xiao Dong calculates the number of convex polygons where one vertex is the red point. The difference between the two numbers they get is $\qquad$ | 2017036 |
In a frequency distribution histogram, there are a total of 11 rectangles. If the area of the middle rectangle equals the sum of the areas of the other 10 rectangles and the sample size is 160, then the frequency of the middle group is ______. | 80 |
Given three real numbers $a$, $b$, $c$ satisfying $a-2b+c=0$ and $a+2b+c \lt 0$, then which of the following is true?
A: $b \gt 0$, $b^{2}-ac\leqslant 0$
B: $b \lt 0$, $b^{2}-ac\leqslant 0$
C: $b \gt 0$, $b^{2}-ac\geqslant 0$
D: $b \lt 0$, $b^{2}-ac\geqslant 0$ | D |
Given $\overrightarrow{a}=(1,0), \overrightarrow{b}=(1,-\sqrt{3})$, if the angle between the non-zero vector $\overrightarrow{c}$ and $\overrightarrow{a}$ is equal to the angle between $\overrightarrow{c}$ and $\overrightarrow{b}$, then the coordinates of $\overrightarrow{c}$ can be _______. (Write down the coordinates... | (\sqrt{3},-1) |
2. (5 points) Two different natural numbers end with 5 zeros and have exactly 42 divisors. Find their sum. | 700000 |
$[\underline{\text { Properties of Sections }}]$
The edge of the cube $A B C D A 1 B 1 C 1 D 1$ is 12. Point $K$ lies on the extension of edge $B C$ at a distance of 9 from vertex $C$. Point $L$ on edge $A B$ is 5 units away from $A$. Point $M$ divides the segment $A 1 C 1$ in the ratio $1: 3$, counting from $A 1$. Fi... | 156 |
2.222. $\left(\frac{2-n}{n-1}+4 \cdot \frac{m-1}{m-2}\right):\left(n^{2} \cdot \frac{m-1}{n-1}+m^{2} \cdot \frac{2-n}{m-2}\right) ; m=\sqrt[4]{400}, n=\sqrt{5}$. | \frac{\sqrt{5}}{5} |
A type of merchandise is sold at a price that is 1.1 times its cost price. After a price reduction of 10 yuan per item, the current selling price is 210 yuan per item. What is the profit rate of this merchandise? | 5\% |
Let the function $f(x) = x^2 - 5x + 6$, and $h(x) = 1 + \log_2{x}$. Define the set $P = \{x | h(x) > 2\}$, and the set $Q = \{x | f(h(x)) \geq 0, \text{ and } x \in \mathbb{R}\}$. Then, the set $M = \{x | x \in P \text{ and } x \notin Q\}$ is
A: $(2, 4)$
B: $(4, 8)$
C: $(4, +\infty)$
D: $(8, +\infty)$ | \text{A} |
Given the functions $f(x) = kx$ and $g(x) = \frac{\ln x}{x}$, if the equation $f(x) = g(x)$ has two real solutions in the interval $\left[\frac{1}{e}, e\right]$, then the range of the real number $k$ is ( )
A: $\left[\frac{1}{e^2}, \frac{1}{2e}\right)$
B: $\left(\frac{1}{2e}, \frac{1}{e}\right]$
C: $\left(0, \frac{... | A |
4. Factorize: $(x+1)(x+2)(x+3)(x+4)+x(x+5)=$ | (x^{2}+5x+3)(x^{2}+5x+8) |
Let $p \ge 3$ be a prime. For $j = 1,2 ,... ,p - 1$, let $r_j$ be the remainder when the integer $\frac{j^{p-1}-1}{p}$ is divided by $p$.
Prove that $$r_1 + 2r_2 + ... + (p - 1)r_{p-1} \equiv \frac{p+1}{2} (\mod p)$$ | \frac{p+1}{2} |
Given that the function $f(x)$ is a monotonic function on $\mathbb{R}$ and satisfies $f[f(x) - 3^x] = 4$ for any $x \in \mathbb{R}$, the value of $f(2)$ is ( )
A: 4
B: 8
C: 10
D: 12 | C |
10. (20 points) Let the line $y=2 \sqrt{2} x+2 \sqrt{3}$ intersect the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ at points $A$ and $B$, with the origin of coordinates being $O$. If $\angle A O B=90^{\circ}$ and $A B=\frac{12 \sqrt{11}}{17}$, find the equation of the ellipse. | \frac{x^{2}}{4}+\frac{y^{2}}{2}=1 |
By how much should ${ }^{4} \log 8$ be multiplied to obtain ${ }^{32} \log 8$? | \frac{2}{5} |
2. Given sets:
$$
\begin{array}{l}
A=\left\{(x, y) \mid x+a^{2} y+6=0\right\}, \\
B=\{(x, y) \mid(a-2) x+3 a y+2 a=0\} .
\end{array}
$$
If $A \cap B=\varnothing$, then the value of $a$ is ( ).
(A) 3 or -1
(B) 0
(C) -1
(D) 0 or -1 | D |
7.1. A team of athletes, one third of whom are snowboarders, descended from the mountain. Some of them got into a cable car carriage, which can accommodate no more than 10 people, while all the others descended on their own, and their number turned out to be more than $45 \%$, but less than $50 \%$ of the total number.... | 5 |
21.2.1 * Find the number of natural numbers whose digits sum to $n$ and whose digits are 1 or 2. | \frac{1}{\sqrt{5}}[(\frac{1+\sqrt{5}}{2})^{n+1}-(\frac{1-\sqrt{5}}{2})^{n+1}] |
\section*{Problem 6 - 311046}
Let \(q\) be the larger of the two solutions to the equation \(x^{2}-4 x+1=0\).
Determine the last digit (units place) in the decimal digit representation of the number \(\left[q^{1992}\right]\).
Hint: If \(z\) is a real number, the integer \(g\) such that \(g \leq z < g+1\) is denoted ... | 1 |
5. Now arrange for 7 students to participate in 5 sports events, requiring that students A and B cannot participate in the same event, each event must have participants, and each person can only participate in one event. The number of different arrangements that meet the above requirements is $\qquad$ . (Answer in numb... | 15000 |
For the propositions p: $\forall x \in \mathbb{R}$, $\log_2{x} > 0$, and q: $\exists x_0 \in \mathbb{R}$, $2^{x_0} < 0$, determine which of the following propositions is true:
A: $p \vee q$
B: $p \wedge q$
C: $\neg p \wedge q$
D: $p \vee \neg q$ | D |
One. (20 points) Find all integers $a$ such that the quadratic equation in $x$
$$
x^{2}-\sqrt{5 a^{2}-26 a-8} x-\left(a^{2}-4 a+9\right)=0
$$
has both roots as integers. | a=6 |
12. Given that $a$, $b$, and $x$ are real numbers, the function $f(x) = x^{2} - 2 a x + 1$ does not intersect with the function $g(x) = 2 b(a - x)$. Let the set of points $(a, b)$ be denoted as $A$. Then the area of the plane figure represented by the set $A$ is $\qquad$. | \pi |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively, and they satisfy $2\cos ^{2} \frac {A}{2}= \frac { \sqrt {3}}{3}\sin A$, $\sin (B-C)=4\cos B\sin C$, then $\frac {b}{c}=$ \_\_\_\_\_\_. | \frac {b}{c}=1+ \sqrt {6} |
4. Given the functions
$$
f_{n}(x)=\frac{1-\cos x \cos 2 x \ldots \cos n x}{x^{2}}, n \in \mathbb{N}
$$
a) Prove that for every $n \in \mathbb{N}$, $\lim _{x \rightarrow 0} f_{n}(x)=f_{n}$ exists.
b) Determine the relationship between $f_{n}$ and $f_{n-1}$.
c) Calculate $f_{n}$. | f_{n}=\frac{n(n+1)(2n+1)}{12} |
The equation $x^{2}+y^{2}+2x-4y-6=0$ represents the graph of ( )
A: A circle with center at $(1,-2)$ and radius $11$
B: A circle with center at $(-1,2)$ and radius $11$
C: A circle with center at $(-1,2)$ and radius $\sqrt {11}$
D: A circle with center at $(1,2)$ and radius $\sqrt {11}$ | C |
If there is no tax, which of the following costs more than $\$ 18$ to purchase?
(A) Five $\$ 1$ items and five $\$ 2$ items
(B) Nine $\$ 1$ items and four $\$ 2$ items
(C) Nine $\$ 1$ items and five $\$ 2$ items
(D) Two $\$ 1$ items and six $\$ 2$ items
(E) Sixteen $\$ 1$ items and no $\$ 2$ items | C |
‥ For the equation greater than $x$, $2 x^{2}+3 a x+a^{2}-a=0$ to have at least one root with a modulus equal to 1, the value of the real number $a$ ().
(A) does not exist
(B) has one
(C) has two
(D) has four | C |
Given points $A(1,1,1)$ and $P(1,1,0)$ in the spatial coordinate system. Rotate the point $P$ by $60^{\circ}$ in the positive direction around the ray $O A$. Determine the coordinates of the rotated point. | P'(\frac{1}{3};\frac{4}{3};\frac{1}{3}) |
8. Let $A$ be a 20-element subset of the set
$$
M=\{1,2, \cdots, 2012\}
$$
such that the difference between any two elements in $A$ is a multiple of 12. Then the number of such subsets $A$ is $\qquad$ | 8 C_{168}^{20}+4 C_{167}^{20} |
Let $\mathbb{Z}_{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that
$$ m^{2}+f(n) \mid m f(m)+n $$
for all positive integers $m$ and $n$. (Malaysia) | f(n)=n |
[ Numerical inequalities. Comparing numbers.]
What is greater, $2^{700}$ or $5^{300}$? | 2^{700} |
6. (2006 National Winter Camp) Positive integers $a_{1}, a_{2}, \cdots$, $a_{2006}$ (which can be the same) such that $\frac{a_{1}}{a_{2}}, \frac{a_{2}}{a_{3}}, \cdots, \frac{a_{2005}}{a_{2006}}$ are all distinct. How many different numbers are there at least among $a_{1}, a_{2}, \cdots, a_{2006}$? | 46 |
Given that $f(x)=ax^{2}+bx$ is an even function defined on the interval $[a-1,2a]$, the value of $a+b$ is ( ).
A: $- \dfrac {1}{3}$
B: $ \dfrac {1}{3}$
C: $- \dfrac {1}{2}$
D: $ \dfrac {1}{2}$ | B |
If a sequence $A_n$: $a_1$, $a_2$, ..., $a_n$ ($n \geq 2$) satisfies $|a_{k+1} - a_k| = 1$ ($k=1, 2, ..., n-1$), then $A_n$ is called an E sequence, and let $S(A_n) = a_1 + a_2 + ... + a_n$.
- (I) Write an E sequence $A_5$ that satisfies $a_1 = a_3 = 0$;
- (II) If $a_1 = 12$, $n = 2000$, prove that: The necessary and s... | 9 |
Find the focus of the curve $y^2 = -16x + 64$. | (0, 0) |
7. Xiao Dong is 8 years younger than his sister. In 3 years, his sister's age will be twice Xiao Dong's age. His sister is $\qquad$ years old this year. | 13 |
1.1. The Eiffel Tower has a height of 324 m and weighs 10000 tons. How many kilograms will a copy with a height of 1.62 m weigh? | 1.25 |
The function is given as $y=\sin(x- \frac {π}{3})$. If all the points on the function's graph have their x-coordinates stretched to twice their original length (y-coordinates remain unchanged), and the resulting graph is then shifted to the left by $\frac {π}{3}$ units, what is the equation of the axis of symmetry for ... | \text{C} |
66. In a Bashkir village, every resident speaks either Bashkir, or Russian, or both languages. 912 residents of the village speak Bashkir, 653 - Russian, and 435 people speak both languages. How many residents are there in this village? | 1130 |
Let $f(x)=x^3+ax^2+bx+c$ and $g(x)=x^3+bx^2+cx+a$, where $a,b,c$ are integers with $c\not=0$. Suppose that the following conditions hold:
[list=a][*]$f(1)=0$,
[*]the roots of $g(x)=0$ are the squares of the roots of $f(x)=0$.[/list]
Find the value of $a^{2013}+b^{2013}+c^{2013}$. | -1 |
Given an ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ with its left and right foci labeled as $F_1$ and $F_2$ respectively, and a point $P$ on the ellipse. If $P$, $F_1$, and $F_2$ are the three vertices of a right-angled triangle, then the distance from point $P$ to the x-axis is
A: $\frac{9}{5}$
B: 3
C: $\frac{... | D |
If the equation $(m-5)x=0$ holds true for any value of $x$, then the value of $m$ is ____. | 5 |
Give a triangle $ABC$ with heights $h_a = 3$ cm, $h_b = 7$ cm and $h_c = d$ cm, where $d$ is an integer. Determine $d$. | d = 3, 4, \text{ or } 5 |
## 30. New Mathematics
My son has learned to count in a numeral system different from the decimal system, and instead of 136, he writes 253 in this system. In what numeral system is my son counting? | 7 |
The slope angle of the line $x-\sqrt{3}y-1=0$ is ( ).
A: $60^{\circ}$
B: $30^{\circ}$
C: $120^{\circ}$
D: $150^{\circ}$ | B |
A tetrahedron \(P-ABC\) is machined from a spherical blank. In the base \(\triangle ABC\), \(\angle A=150^\circ\), \(BC=3\), and \(PA=2\), with \(PA\) perpendicular to plane \(ABC\). Find the minimum surface area of the spherical blank. | 40\pi |
Given the hyperbola \\(x^{2}- \dfrac{y^{2}}{b^{2}}=1(b > 0)\\) with a focal length of \\(4\\), then \\(b=\\) \_\_\_\_\_\_. | \sqrt{3} |
1. (1) Use the Factor Theorem to prove that $a-b, b-c, c-a$ are all factors of $a^{2}(b-c)+b^{2}(c-a)+c^{2}(a-b)$.
(2) Using the conclusion from (1), factorize $a^{2}(b-c)+b^{2}(c-a)+c^{2}(a-b)$.
(3) Factorize: $(x+y+z)^{3}-x^{3}-y^{3}-z^{3}$. | 3(x+y)(y+z)(z+x) |
Calculate: $2\cos 45^{\circ}-\left(\pi -2023\right)^{0}+|3-\sqrt{2}|$ | 2 |
5. In the number $2 * 0 * 1 * 6 * 0 * 2 *$, each of the 6 asterisks needs to be replaced with any of the digits $1,2,3,4,5,6,7,8,9$ (digits can repeat) so that the resulting 12-digit number is divisible by 18. In how many ways can this be done | 26244 |
Given the function $f(x)=\sin (\omega x+\varphi)+1$ ($\omega > 0$, $0 \leqslant \varphi \leqslant \frac {\pi}{2}$), the distance between two adjacent axes of symmetry of its graph is $\pi$, and it reaches its maximum value of $2$ at $x= \frac {\pi}{6}$. If $f(\alpha)= \frac {9}{5}$, and $ \frac {\pi}{6} < \alpha < \fra... | D |
4. Given in $\triangle A B C$, side lengths $A C=6, B C=9, \angle C=120^{\circ}$, find the length of the angle bisector $C D$ of $\angle C$. | \frac{18}{5} |
Given the function $f(x) = \frac{1 + \ln(x + 1)}{x}, (x > 0)$.
(I) Determine whether the function $f(x)$ is increasing or decreasing on the interval $(0, +\infty)$. Prove your conclusion;
(II) When $x > 0$, the inequality $f(x) > \frac{k}{x + 1}$ always holds. Find the maximum integer value of $k$; | k_\text{max} = 3 |
Given that the odd function $f(x)$ defined on $R$ satisfies $f(x) + f(x+2) = 0$, and $f(1) = -2$, find the value of $f(2019) + f(2018)$. | 2 |
The ratio $\frac{2^{2001}\cdot3^{2003}}{6^{2002}}$ is:
$\mathrm{(A) \ } 1/6\qquad \mathrm{(B) \ } 1/3\qquad \mathrm{(C) \ } 1/2\qquad \mathrm{(D) \ } 2/3\qquad \mathrm{(E) \ } 3/2\qquad$ | \frac{3}{2} |
8. A batch of disaster relief supplies is loaded onto 26 trucks, which travel at a constant speed of $v$ kilometers per hour from a city to the disaster area. If the highway between the two places is 400 kilometers long, and the distance between every two trucks must not be less than $\left(\frac{v}{20}\right)^{2}$ kil... | 10 |
A store sells shirts, water bottles, and chocolate bars. Shirts cost $\$ 10$ each, water bottles cost $\$ 5$ each, and chocolate bars cost $\$ 1$ each. On one particular day, the store sold $x$ shirts, $y$ water bottles, and $z$ chocolate bars. The total revenue from these sales was $\$ 120$. If $x, y$ and $z$ are inte... | 121 |
2. If the equation $\sin ^{2} x+\cos x+m=0$ has real solutions, then the range of values for $m$ is $\qquad$ | -\frac{5}{4} \leqslant m \leqslant 1 |
2. Let $A_{1}$ be the midpoint of the segment $A B$. We denote by $A_{2}$ the midpoint of $A A_{1}$, by $A_{3}$ the midpoint of $A_{1} A_{2}$, by $A_{4}$ the midpoint of $A_{2} A_{3}$, etc. Find the limit to which the sequence of lengths of the segments $A A_{1}, A A_{2}, A A_{3}, \ldots$ tends. | \frac{}{3} |
87 Given $\sin \theta+\cos \theta=\frac{1}{5}$, and $\theta \in(0, \pi)$. Then $\tan \theta=$ $\qquad$ | -\frac{4}{3} |
10. If real numbers $a, b, c, d, e$ satisfy $a+b+c+d+e=8, a^{2}+b^{2}+c^{2}+d^{2}+e^{2}=16$, then the sum of the maximum and minimum values of $e$ is $\qquad$ | \frac{16}{5} |
20) The sum of the digits of the square of 999999999999999995 is:
(A) 150
(B) 160
(C) 170
(D) 180
(E) 190. | 160 |
Roy is starting a baking company and decides that he will sell cupcakes. He sells $n$ cupcakes for $(n + 20)(n + 15)$ cents. A man walks in and buys $\$10.50$ worth of cupcakes. Roy bakes cupcakes at a rate of $10$ cupcakes an hour. How many minutes will it take Roy to complete the order? | 90 \text{ minutes} |
Among the following relationships, the one that holds is ( )
A: ${{\log }_{3}}4 > {{(\frac{1}{5})}^{0}} > {{\log }_{\frac{1}{3}}}10$
B: ${{\log }_{\frac{1}{3}}}10 > {{(\frac{1}{5})}^{0}} > {{\log }_{3}}4$
C: ${{\log }_{3}}4 > {{\log }_{\frac{1}{3}}}10 > {{(\frac{1}{5})}^{0}}$
D: ${{\log }_{\frac{1}{3}}}10 > {{\log }... | A |
7・10 A fishing boat is fishing in the territorial sea of a foreign country without permission. Each time it casts a net, it causes the same value of loss to the country's fishing yield. The probability of the boat being detained by the foreign coast patrol during each net casting is equal to $1 / k$, where $k$ is a nat... | n=k-1 |
In the set of real numbers $\mathbb{R}$, the differentiable function $f(x)=\frac{1}{3}x^3 + \frac{1}{2}ax^2 + 2bx + c$ attains a maximum value when $x \in (0, 1)$ and a minimum value when $x \in (1, 2)$. Determine the range of $\frac{b-2}{a-1}$. | (\frac{1}{4}, 1) |
12. If $x \in \mathbf{N}^{*}$, and $x^{5}+x+1$ is a prime number, then $x=$ | 1 |
Let $ABC$ be a triangle with $AB=17$ and $AC=23$. Let $G$ be the centroid of $ABC$, and let $B_1$ and $C_1$ be on the circumcircle of $ABC$ with $BB_1\parallel AC$ and $CC_1\parallel AB$. Given that $G$ lies on $B_1C_1$, the value of $BC^2$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively pr... | 100m + n = 100 \times 818 + 3 = 81803 |
3. The cunning rabbit and the foolish fox agreed: if the fox passes the bridge in front of the rabbit's house every time, the rabbit will give the fox money, doubling the fox's money, but the fox has to pay the rabbit $2^{2019}$ cents as a toll each time it crosses the bridge. The fox was very happy to hear that its mo... | 2^{2019}-1 |
The slope of the line passing through points M(-2, a) and N(a, 4) is 1. The value of the real number $a$ is ( )
A: 1
B: 2
C: 1 or 4
D: 1 or 2 | A |
17. Five numbers $A, B, C, D, E$. Each time, one number is removed, and the average of the remaining 4 numbers is calculated. This process is repeated 5 times, resulting in the following five numbers: $23, 26, 30, 33, 38$. Find the average of $A, B, C, D, E$. | 30 |
Example 6 For all $a, b, c \in \mathbf{R}_{+}$, and $abc=1$, find the minimum value of $S=\frac{1}{2a+1}+\frac{1}{2b+1}+\frac{1}{2c+1}$. | 1 |
2.48. In an oblique triangular prism, the distances between the lateral edges are equal to $a, b$ and $c$. The lateral edge is equal to $l$, and the height of the prism is $h$. Determine the total surface area of the prism. | \frac{2}{}(\sqrt{p(p-)(p-b)(p-)}+p) |
How many even whole numbers between 1 and 99 are multiples of 5 ?
(A) 5
(B) 7
(C) 9
(D) 11
(E) 13 | 9 |
8. It is known that two engineering teams, Team A and Team B, have several people each. If 90 people are transferred from Team A to Team B, then the total number of people in Team B will be twice that of Team A; If some people are transferred from Team B to Team A, then the total number of people in Team A will be 6 ti... | 153 |
Among the following expressions, the one that is a simplest radical form is ( )
A: $\sqrt{4}$
B: $\sqrt{6}$
C: $\sqrt{8}$
D: $\sqrt{12}$ | \text{B} |
10. (20 points) The rules of a card game are as follows: Arrange nine cards labeled $1,2, \cdots, 9$ in a random row. If the number on the first card (from the left) is $k$, then reverse the order of the first $k$ cards, which is considered one operation. The game stops when no operation can be performed (i.e., the num... | \frac{103}{2520} |
Given that the function $f(x)$ is an odd function on $\mathbb{R}$, and when $x > 0$, $f(x) = 1 + \log_2{x}$, find the value of $f(-4)$. | -3 |
Given that in the expansion of $(1-2x)^n$ ($n \in \mathbb{N}^*$), the binomial coefficients of the third and the eighth terms are equal, then the sum of the coefficients of all terms in the expansion is \_\_\_\_\_\_. | -1 |
A sufficient but not necessary condition for "$\lg m < 1$" to be true is ( )
A: $m\in(0,+\infty)$
B: $m\in\{1,2\}$
C: $0 < m < 10$
D: $m < 1$ | B |
7. Given $a>0, b>0$ and
$$
\sqrt{a}(\sqrt{a}+4 \sqrt{b})=3 \sqrt{b}(\sqrt{a}+2 \sqrt{b}) \text {. }
$$
Then the value of $\frac{a+6 \sqrt{a b}-8 b}{2 a-3 \sqrt{a b}+2 b}$ is $(\quad)$.
(A) 1
(B) 2
(C) $\frac{19}{11}$.
(D) $\sqrt{2}$ | B |
【Example 2】As shown in Figure $1, \triangle A B C$ has $n_{1}, n_{2}, n_{3}$ points on its sides (excluding points $A, B, C$). By selecting one point from each of the three sides as the vertices of a triangle, how many different triangles can be formed? | n_{1}n_{2}n_{3} |
Let the function $f(x)=|2x-1|-|x+2|$.
- (I) Solve the inequality $f(x) > 0$;
- (II) If there exists $x_{0} \in \mathbb{R}$, such that $f(x_{0}) + 2m^2 < 4m$, find the range of the real number $m$. | -\frac{1}{2} < m < \frac{5}{2} |
## Task Condition
Approximately calculate using the differential.
$y=\arcsin x, x=0.08$ | 0.08 |
3. Let $n$ be a positive integer. If a sequence of $n$ positive integers (which can be the same) is called "full", then this sequence should satisfy the condition: for each positive integer $k (k \geqslant 2)$, if $k$ is in this sequence, then $k-1$ is also in this sequence, and the first occurrence of $k-1$ is before ... | n! |
9. (10 points) The natural number 2015 can be expressed as the sum of $\qquad$ consecutive odd numbers.
Expressing the above text in English, please retain the original text's line breaks and format, and output the translation result directly. | 31 |
The number $(\sqrt{6}+\sqrt{2})(\sqrt{3}-2) \sqrt{\sqrt{3}+2}$ is equal to:
(a) $-\sqrt{3}$;
(b) $-\sqrt{2}$;
(c) -2 ;
(d) 1 ;
(e) 2. | -2 |
1. For what values of $a, b$, and $c$ do the lines $y=a x+b, y=b x+c, y=c x+a$ pass through the point $(1 ; 3)$? | =b==1.5 |
If the graph of the power function $f(x)$ passes through the point $(4, 2)$, then $f(9) = \_\_\_\_\_\_$. | 3 |
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