problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
【Example 3】A party needs to arrange 10 singing programs and 4 dance programs, hoping that at least two singing programs are inserted between every two dances. How many different performance orders can be arranged? | \frac{8!\cdot10!}{4!} |
If the range of the function $f(x) = (a+2b)x^{2} - 2\sqrt{3}x + a + 2c \ (a, b, c \in \mathbb{R})$ is $[0, +\infty)$, determine the minimum value of $a + b + c$. | \sqrt{3} |
Given $a=\frac{1}{2}$, $b=2^{\frac{1}{2}}$, $c=\log_{3}2$, then ( )
A: $b > a > c$
B: $c > b > a$
C: $b > c > a$
D: $a > b > c$ | C |
7. A digital clock displays hours, minutes, and seconds using two digits each (e.g., 10:09:18). Between 05:00:00 and 22:59:59 on the same day, the probability that all six digits on the clock are different is | \frac{16}{135} |
When evaluated, the sum of the digits of the integer equal to $10^{2021}-2021$ is
(A) 18194
(B) 18176
(C) 18167
(D) 18153
(E) 18185 | 18185 |
2. Solve the inequality $x^{\log _{3} x}-2 \leqslant(\sqrt[3]{3})^{\log _{\sqrt{3}}^{2} x}-2 \cdot x^{\log _{3}} \sqrt[3]{x}$. | x\in(0;3^{-\sqrt{\log_{3}2}}]\cup{1}\cup[3^{\sqrt{\log_{3}2}};+\infty) |
Example 7. Compute the integral
$$
I=\int_{|z|=3} \frac{z^{17}}{\left(z^{2}+2\right)^{3}\left(z^{3}+3\right)^{4}} d z
$$ | 2\pii |
Find the coordinates of the focus of the parabola $x^{2}=4y$. | (0, 1) |
4. (5 points) In the third grade, there are two classes, Class A and Class B. If 4 students are transferred from Class A to Class B, the number of students in both classes will be equal. Class A has $\qquad$ more students than Class B. | 8 |
6.219. $\left\{\begin{array}{l}x+y+z=3, \\ x+2 y-z=2, \\ x+y z+z x=3 .\end{array}\right.$ | (1;1;1),(7;-3;-1) |
Which of the following situations is suitable for a comprehensive survey?
A: Investigating whether five villages around a chemical plant are polluted.
B: Examining the quality of a batch of drugs produced by a pharmaceutical manufacturer.
C: Conducting a public opinion poll.
D: Surveying the water quality of the Y... | A |
1. After pulling out the turnip, a lot of earth was left. The Grandpa removed half of the pile, Grandma removed a third of the remaining earth, the Granddaughter removed a quarter of the remainder, the Dog removed a fifth of the remainder, the Cat removed a sixth, and the Mouse removed a seventh. What part of the pile ... | \frac{1}{7} |
3. According to the pattern, the time on the fourth clock is ( ).
A. 6:20
B. 7:20
C. 8:20
D. 8:30
E. 9:30 | C |
Given the set $M=\{a,b,c,d\}$, and $N=\{p|p⊆M\}$, the number of elements in set $N$ is ($\quad$).
A: $4$
B: $8$
C: $16$
D: $32$ | C |
2. A natural number, its double is exactly equal to the sum of all its divisors, such a number is called a "perfect number". There is a "perfect number" between $25 \sim 30$, it is $\qquad$ . | 28 |
## Task A-3.2.
Determine all natural numbers $a$ and $b$ such that
$$
a^{2}=4 b+3 \cdot \text{LCM}(a, b)
$$
where $\text{LCM}(m, n)$ denotes the least common multiple of the numbers $m$ and $n$. | (7q,7q^{2})foranynaturalq,(4q,q^{2})foranyoddnaturalq |
2. Find $\delta_{41}(10), \delta_{43}(7), \delta_{55}(2), \delta_{65}(8), \delta_{91}(11), \delta_{69}(4), \delta_{231}(5)$. | 5,6,20,4,12,11,30 |
Given that $a$, $b$, and $c$ are the lengths of the sides opposite the angles $A$, $B$, and $C$ in $\triangle ABC$ respectively, with $a=2$, and $$\frac{\sin A - \sin B}{\sin C} = \frac{c - b}{2 + b}.$$ Find the maximum area of $\triangle ABC$. | \sqrt{3} |
4. There is an unlimited supply of square glasses in 10 colors. In how many ways can 4 glasses be inserted into a $2 \times 2$ window frame so that some color appears in both the upper and lower halves of the window. | 3430 |
Let $S_n$ be the sum of the first $n$ terms of the sequence $\{a_n\}$, where $a_n = 2n - 49$. When $S_n$ reaches its minimum value, the value of $n$ is ( )
A: 12
B: 13
C: 24
D: 25 | \text{C} |
Whether it is the international situation or the domestic consumption situation, the year $2023$ is a challenging year. In order to cope with the complex economic situation, various regions have introduced various policies to promote economic development, actively responding to the current economic situation and achiev... | \frac{4}{5} |
Problem 4.6. Fourth-grader Vasya goes to the cafeteria every school day and buys either 9 marshmallows, or 2 meat pies, or 4 marshmallows and 1 meat pie. Sometimes Vasya is so busy socializing with classmates that he doesn't buy anything at all. Over 15 school days, Vasya bought 30 marshmallows and 9 meat pies. How man... | 7 |
20. [11] There exist unique nonnegative integers $A, B$ between 0 and 9 , inclusive, such that
$$
(1001 \cdot A+110 \cdot B)^{2}=57,108,249 \text {. }
$$
Find $10 \cdot A+B$. | 75 |
1.1. For what least natural value of \( b \) does the equation
$$
x^{2}+b x+25=0
$$
have at least one root? | 10 |
Given $a$ is a real number, the number of subsets of the set $M=\{x| x^2-3x-a^2+2=0, x\in \mathbb{R}\}$ is
(A) 1 (B) 2 (C) 4 (D) Uncertain | \text{C} |
1. The sequence is given by $a_{n}=\frac{c^{2}+n-2}{2}, n \in \mathbb{N}$.
a) Determine the sum $S_{n}$ of the first $n$ terms.
b) Prove that the numerator of the sum $S_{c}$ is divisible by 25 if $c=10 k+1$, where $k$ is a natural number. | 25k(10k+1)(8k+2) |
Given that $| \overrightarrow{a}|=1$, $| \overrightarrow{b}|=2$, and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $120^{\circ}$. Find:
$(1) \overrightarrow{a} \cdot \overrightarrow{b}$;
$(2)( \overrightarrow{a}-3 \overrightarrow{b}) \cdot (2 \overrightarrow{a}+ \overrightarrow{b})$;
$(3)|2 \overri... | 2 \sqrt{3} |
75. In the six-digit number $M=\overline{a b c 321}$, $a, b, c$ are three different numbers, all greater than 3. If $M$ is a multiple of 7, then the smallest value of $M$ is $\qquad$ | 468321 |
Roll three dice once each, let event A = "All three outcomes show different numbers", and event B = "At least one 6 is rolled". Calculate the probability P(A). | \frac{5}{9} |
4. The integer values of $n$ that satisfy the equation $\left(n^{2}-5 n+5\right)^{n+1}=1$ are $\qquad$ .
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | 1,4,3,-1 |
10. Let the monotonic increasing sequence $\left\{a_{n}\right\}$ consist of positive integers, and $a_{7}=120, a_{n+2}=a_{n}+a_{n+1}\left(n \in \mathbf{Z}_{+}\right)$. Then $a_{8}=$ . $\qquad$ | 194 |
11. Given $a, b$ satisfy $a^{3}-3 a^{2}+5 a=1, b^{3}-3 b^{2}+$ $5 b=5$. Then $a+b=$ $\qquad$ . | 2 |
Let $x$, $y$, $z$ be positive integers satisfying $x<y<z$ and $x+xy+xyz=37$. Find the greatest possible value of $x+y+z$. | 20 |
## Task 4 - 300524
In a sports competition, 10 teams are to participate. They should be equipped with monochrome T-shirts and monochrome shorts so that they can be distinguished from each other by the resulting color combinations.
a) What is the smallest number of colors with which this can be achieved?
b) What is t... | 4 |
In \\(\triangle ABC\\), the sides opposite to angles \\(A\\), \\(B\\), and \\(C\\) are \\(a\\), \\(b\\), and \\(c\\) respectively. Given that \\(2c\cos B=2a+b\\) and the area of \\(\triangle ABC\\) is \\(S= \dfrac{\sqrt{3}}{12}c\\), find the minimum value of \\(ab\\). | \dfrac{1}{3} |
32. Let $1 \leqslant r \leqslant n$, in the ordered tuple $\left(x_{1}, x_{2}, \cdots, x_{r}\right)$ each $x_{j} \in\{1,2, \cdots, n\}$, and for each $i=1$, $2, \cdots, n, x_{1}, x_{2}, \cdots, x_{r}$ contains at most $i-1$ elements $\leqslant i$. Find the number of such $r$-tuples. | (n-r)\cdotn^{r-1} |
97 If $\cos A+\cos B+\cos C=0$, then the value of $\frac{\cos 3 A+\cos 3 B+\cos 3 C}{\cos A \cos B \cos C}$ is $\qquad$ | 12 |
The $4$ code words
$$\square * \otimes \,\,\,\, \oplus \rhd \bullet \,\,\,\, * \square \bullet \,\,\,\, \otimes \oslash \oplus$$
they are in some order
$$AMO \,\,\,\, SUR \,\,\,\, REO \,\,\,\, MAS$$
Decrypt $$\otimes \oslash \square * \oplus \rhd \square \bullet \otimes $$ | \text{SUMAREMOS} |
5. Tim has a working analog 12 -hour clock with two hands that run continuously (instead of, say, jumping on the minute). He also has a clock that runs really slow-at half the correct rate, to be exact. At noon one day, both clocks happen to show the exact time. At any given instant, the hands on each clock form an ang... | 33 |
526 ** The 10 complex roots of the equation $x^{10}+(13 x-1)^{10}=0$ are $r_{1}, \overline{r_{1}}, r_{2}, \overline{r_{2}}$, $r_{3}, \overline{r_{3}}, r_{4}, \overline{r_{4}}, r_{5}, \overline{r_{5}}$. Find the value of the expression
$$
\frac{1}{r_{1} \overline{r_{1}}}+\frac{1}{r_{2} \overline{r_{2}}}+\cdots+\frac{1}{... | 850 |
Class 9 (1) students are practicing for the midterm exam on the playground. One-third of the students are practicing long-distance running, one-fourth are practicing jump rope, one-eighth are practicing sit-and-reach, and less than 15 students are practicing standing long jump. It is known that the number of students i... | 48 |
B4. On each of the 10,000 fields of a $100 \times 100$ chessboard, there is a number. On the top row, from left to right, the numbers are 0 to 99. In the left column, from top to bottom, the numbers are 0 to 99. The sum of four numbers in a $2 \times 2$ block is always 20. What number is in the bottom right corner of t... | -178 |
2. In $\triangle A B C$, points $A_{0}, B_{0}$, and $C_{0}$ are the feet of the altitudes from vertices $A, B$, and $C$, respectively, and it is given that $\triangle A B C \sim \triangle A_{0} B_{0} C_{0}$.
a) If it is known that $\triangle A B C$ is acute, calculate its angles.
b) If it is known that $\triangle A B... | \alpha=\frac{\pi}{3},\beta=\frac{\pi}{3},\gamma=\frac{\pi}{3} |
5. The smallest four-digit number that has exactly 14 divisors (including 1 and itself), and one of its prime factors has a units digit of 3 is $\qquad$ . | 1458 |
Let the complex number $z$ satisfy $(1+i)z=2$, where $i$ is the imaginary unit. Find the complex number $z$. | 1-i |
8. Four cars $A, B, C$, and $D$ start simultaneously from the same point on a circular track. $A$ and $B$ drive clockwise, while $C$ and $D$ drive counterclockwise. All cars move at constant (but pairwise different) speeds. Exactly 7 minutes after the start of the race, $A$ meets $C$ for the first time, and at the same... | 53 |
Example 24 (Problem from the 48th Slovenian Mathematical Olympiad) Find all five-digit numbers $\overline{a b c d e}$, such that the number is divisible by 9, and $\overline{a c e}-\overline{b d a}=760$. | 81828,91269 |
Given $$\sin\theta =- \frac {3}{5}$$, and $$(3\pi <\theta < \frac {7}{2}\pi )$$, then $$\tan \frac {\theta }{2}$$ = \_\_\_\_\_\_\_\_\_\_\_\_. | -3 |
Given the complex number $z=1+i$, then $\dfrac{z^2}{1-z} =$ ()
A: $2$
B: $-2$
C: $2i$
D: $-2i$ | \text{B} |
8.3 The number $\sqrt{1+2019^{2}+\frac{2019^{2}}{2020^{2}}}+\frac{2019}{2020}$ is an integer. Find it. | 2020 |
Calculate $\lg 5 + \lg 0.2 = \_\_\_\_\_\_$. | 0 |
China was the first country in the world to use negative numbers. Li Heng, in the book "Fa Jing" written during the Warring States period, already used negative numbers. If the year 500 BC is written as $-500$ years, then the year 2024 AD should be written as ______ years. | +2024 |
Suppose the proposition $p: \exists n \in \mathbb{N}$, $n^{2} \gt 2n-1$. The negation of proposition $p$ is ( ).
A: $\forall n \in \mathbb{N}$, $n^{2} \gt 2n-1$
B: $\forall n \in \mathbb{N}$, $n^{2} \leqslant 2n-1$
C: $\exists n \in \mathbb{N}$, $n^{2} \leqslant 2n-1$
D: $\exists n \in \mathbb{N}$, $n^{2}=2n-1$ | B |
6. C1 (UKR) On a $5 \times 5$ board, two players alternately mark numbers on empty cells. The first player always marks 1 's, the second 0 's. One number is marked per turn, until the board is filled. For each of the nine $3 \times 3$ squares the sum of the nine numbers on its cells is computed. Denote by A the maximum... | 6 |
4. (6 points) If the product of the sum and difference of two natural numbers is 23, then the quotient of the sum of these two numbers divided by their difference is . $\qquad$ | 23 |
13.039. For the first quarter, the car factory completed $25 \%$ of the annual production plan for cars. The number of cars produced in the second, third, and fourth quarters turned out to be proportional to the numbers $11,25,12$ and 13.5. Determine the overfulfillment of the annual plan in percentages, if in the seco... | 13.2 |
What is the least possible value of
\[(x+1)(x+2)(x+3)(x+4)+2019\]where $x$ is a real number?
$\textbf{(A) } 2017 \qquad\textbf{(B) } 2018 \qquad\textbf{(C) } 2019 \qquad\textbf{(D) } 2020 \qquad\textbf{(E) } 2021$ | 2018 |
The solution set of the inequality system $\left\{\begin{array}{l}x+5＜4\\ \frac{{3x+1}}{2}≥2x-1\end{array}\right.$ is ______. | x < -1 |
Use the Euclidean algorithm to find the greatest common divisor (GCD) of 15 and 2015. | 5 |
1. Calculate the following expressions
(1) $1+2-3+4+5-6+\ldots+220+221-222=$ $\qquad$ .
(2) $(22+23+24+25+26) \times(25+26+27)-(22+23+24+25+26+27)$
$\times(25+26)=$ $\qquad$
(3) $1^{2}+3^{2}+5^{2}+\ldots .+19^{2}=$ $\qquad$
(4) $\frac{1}{2}+\frac{1}{3}+\frac{2}{3}+\frac{1}{4}+\frac{2}{4}+\frac{3}{4}+\ldots+\frac{2019}{... | 2 |
The ratio between the number of men and women in the city of Campo Verde is $\frac{2}{3}$. The average age of men is 37 years and that of women is 42 years. What is the average age of the inhabitants of Campo Verde? | 40 |
Given that $a$, $b$, and $c$ are positive real numbers, and $a^{2}+b^{2}+c^{2}=2$.
$(1)$ Find the maximum value of $a+b+c$;
$(2)$ Find the minimum value of $\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}$. | \frac{3\sqrt{6}}{4} |
Given vectors $\overrightarrow{a}=(\cos x,\sin x)$ and $\overrightarrow{b}=(\sqrt{2}, \sqrt{2})$, if $\overrightarrow{a}\cdot \overrightarrow{b}= \frac{8}{5}$ and $\frac{\pi}{4} < x < \frac{\pi}{2}$,
$(1)$ try to find the values of $\cos (x- \frac{\pi}{4})$ and $\tan (x- \frac{\pi}{4})$;
$(2)$ find the value of $\fra... | - \frac{21}{100} |
4. In the game "clock", at the beginning, the arrow points to one of the numbers from 1 to 7 (drawing on the right). In each step, the arrow moves in the direction of the clock hands by as many fields as the number written in the field before the start of the step. For example, in the drawing, the arrow points to the n... | 5 |
## 39. Memories of the Trip
During the trip, I saw Romanesque cathedrals, triumphal arches, waterfalls, and medieval castles. I photographed half of these tourist attractions. I saw three times as many cathedrals as triumphal arches, and as many medieval castles as waterfalls. One quarter of the photographs I took dep... | \frac{1}{4} |
The graph of the function $f(x)=2\sin (\omega x+ \frac {\pi}{3})$ ($\omega > 0$) has exactly two maximum points in the interval $[0,1]$. The range of $\omega$ is __( )__.
A: $[2\pi,4\pi]$
B: $[2\pi, \frac {9\pi}{2})$
C: $[\frac {13\pi}{6}, \frac {25\pi}{6})$
D: $[2\pi, \frac {25\pi}{6})$ | C |
Given point P ($\tan\alpha$, $\cos\alpha$) is in the third quadrant, then the terminal side of angle $\alpha$ is in ( )
A: Fourth quadrant
B: Third quadrant
C: Second quadrant
D: First quadrant | C |
A company has 120 employees, of which 72 are female. To conduct a certain survey, it is planned to use stratified sampling to draw a sample of 15 people. The number of male employees to be selected is ( )
A: 5
B: 6
C: 7
D: 8 | B |
$\frac{{A}_{7}^{2}}{{C}_{10}^{2}}=\_\_\_\_\_\_.$ | \frac{14}{15} |
Given $\cos\left(\frac{\pi}{4} - \theta\right) = \frac{\sqrt{3}}{3}$, find the value of $\cos\left(\frac{3\pi}{4} + \theta\right) - \sin^2\left(\theta - \frac{\pi}{4}\right)$. | -\frac{2 + \sqrt{3}}{3} |
Which of the following conclusions is valid?
$(\quad)$
A: $\tan \alpha=2$ and $\dfrac {\cos \alpha}{\sin \alpha}=- \dfrac {1}{2}$
B: $\tan \alpha=1$ and $\cos \alpha=- \dfrac { \sqrt {2}}{2}$
C: $\sin \alpha=1$ and $\tan \alpha\cdot\cos \alpha= \dfrac {1}{2}$
D: $\sin \alpha= \dfrac {1}{2}$ and $\cos \alpha= \dfrac {1}... | B |
1. Let $\lfloor x\rfloor$ denote the greatest integer not exceeding the real number $x$. Find all positive integers $M$ such that the sequence defined by
$$
\begin{array}{l}
a_{0}=\frac{2 M+1}{2}, \\
a_{k+1}=a_{k}\left\lfloor a_{k}\right\rfloor(k=0,1, \cdots)
\end{array}
$$
contains at least one integer term. | M\geqslant2 |
81. Sheep Buyers. A farmer died, leaving a herd of cows to his two sons. The sons sold the herd, receiving as many dollars for each head as there were heads in the herd. With the money, the brothers bought sheep at 10 dollars each and one lamb, which cost less than 10 dollars. Then they divided the sheep and the lamb b... | 2 |
Let $\{a_n\}$ be a sequence, and $S_n$ be the sum of its first $n$ terms. Given that $a_1 = -2$, and $a_{n+1} = S_n$ for $n \geq 2$, find $a_n$ and $S_n$. | -2^n |
The angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $120^{\circ}$, and $|\overrightarrow{a}|=4$, $|\overrightarrow{b}|=2$. Find:① $\overrightarrow{a} \cdot \overrightarrow{b}$;② $(\overrightarrow{a}+\overrightarrow{b}) \cdot (\overrightarrow{a}-2\overrightarrow{b})$. | 12 |
Let $x \leq y \leq z \leq t$ be real numbers such that $xy + xz + xt + yz + yt + zt = 1.$
[b]a)[/b] Prove that $xt < \frac 13,$
b) Find the least constant $C$ for which inequality $xt < C$ holds for all possible values $x$ and $t.$ | \frac{1}{3} |
1.1. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ with edge $a$. Find the angle and distance between the lines $A_{1} B$ and $A C_{1}$. | \frac{}{\sqrt{6}} |
5. As shown in Figure 4. In the right prism $A B C-A_{1} B_{1} C_{1}$, $\angle A C B=90^{\circ}, A C$ $=6, B C=C C_{1}=\sqrt{2}, P$ is a moving point on the line segment $B_{1} C$. Then the minimum value of $A P+P C_{1}$ is | 5 \sqrt{2} |
[ Cauchy's Inequality ]
The product of positive numbers $a_{1}, a_{2}, \ldots, a_{n}$ is equal to 1. Prove that $\left(1+a_{1}\right)\left(1+a_{2}\right) \ldots\left(1+a_{n}\right) \geq 2^{n}$. | 2^{n} |
Given the graph of the power function $y=f(x)$ passes through the point $(3, \sqrt{3})$, then $f(9)=$ （ ）
A: 3
B: -3
C: -$\sqrt{3}$
D: $\sqrt{3}$ | A |
1. If the line $l_{1}: a x+(1-a) y=3$ is perpendicular to $l_{2}:(a-1) x+(2 a+3) y=2$, then the value of $a$ is
A. -3
B. 1
C. 0 or $-\frac{3}{2}$
D. 1 or -3 | 1or-3 |
Dudeney, Amusements in Mathematics Problem 20 "A neighbour of mine," said Aunt Jane, "bought a certain quantity of beef a two shillings a pound, and the same quantity of sausages at eighteenpence a pound. I pointed out to her that if she had divided the same money equally between beef and sausages she would have gained... | £8,8s |
8.1. The real number a is such that $2a - 1/a = 3$. What is the value of $16a^4 + 1/a^4$? | 161 |
2. In each cell of a $2 \times 2$ table, one number is written. All numbers are distinct, the sum of the numbers in the first row is equal to the sum of the numbers in the second row, and the product of the numbers in the first column is equal to the product of the numbers in the second column. Find the sum of all four... | 0 |
1. (15th "Hope Cup" Grade 1 High School Competition Question) Given that the set $M$ consists of all functions $f(x)$ that satisfy the following conditions:
(1) When $x \in[0,+\infty)$, the function value is a non-negative real number;
(2) For any $s, t \in[0,+\infty)$, $f(s)+f(t) \leqslant f(s+t)$. Among the functions... | A |
The value of Avogadro's constant is represented by $N_A$. Which of the following statements is correct?
A: At standard temperature and pressure, a mixed gas of 5.6 g of ethylene and cyclopropane contains 0.4 $N_A$ carbon atoms.
B: When 1 mol of $Cl_2$ is introduced into an excess amount of water, the number of electron... | \text{A} |
Example 2 Try to find the minimum value of the function $f(x, y)=6\left(x^{2}+y^{2}\right) \cdot (x+y)-4\left(x^{2}+x y+y^{2}\right)-3(x+y)+5$ in the region $A=\{(x, y) \mid x>0, y>0\}$. | 2 |
Given the function $f\left(x\right)=x^{3}-x^{2}-x+1$.
$(1)$ Find the equation of the tangent line to the function $f\left(x\right)$ at the point $x=-1$.
$(2)$ Find the maximum and minimum values of the function $f\left(x\right)$ on the interval $\left[0,4\right]$. | 0 |
3A. Solve the equation
$$
\sqrt{1+3 \sin ^{3} x}=3-\sqrt{1-\cos ^{4} x}
$$ | \frac{\pi}{2}+2k\pi,k\in\mathbb{Z} |
Find the sides of parallelogram $ABCD$, in which the radii of the circumcircles of triangles $ABC$ and $ABD$ are 5 and $\sqrt{13}$, respectively, and the distance between the centers of these circles is 2. | AB=6,AD=9\sqrt{\frac{2}{5}} |
Given $2^{a}=3$, $2^{b}=5$, then $\log _{2}45=$____. (Express in terms of $a$ and $b$) | 2a+b |
Problem 3. a) Find the maximal value of the function $y=\left|4 x^{3}-3 x\right|$ in the interval $[-1,1]$. b)Let $a, b$ and $c$ be real numbers and $M$ be the maximal value of the function $y=$ $\left|4 x^{3}+a x^{2}+b x+c\right|$ in the interval $[-1,1]$. Prove that $M \geq 1$. For which $a, b, c$ is the equality re... | 1 |
Given the function f(x) = 2cos^2(x) - 2$\sqrt{3}$sin(x)cos(x).
(I) Find the monotonically decreasing interval of the function f(x);
(II) Find the sum of all the real roots of the equation f(x) = $- \frac{1}{3}$ in the interval [0, $\frac{\pi}{2}$]. | \frac{2\pi}{3} |
Example 8. For the cube $\mathrm{ABCD}-\mathrm{A}_{1} \mathrm{~B}_{1} \mathrm{C}_{1} \mathrm{D}_{1}$, with edge length $\mathrm{a}$, and the centers of the two bases being $\mathrm{O}, \mathrm{O}_{1}$, find the following distances:
(1) A to $\mathrm{B}_{1} \mathrm{D}_{1}$, (2) $\mathrm{A}$ to $\mathrm{A}_{1} \mathrm{C}... | \frac{\sqrt{6}}{2} \mathrm{a}, \frac{\sqrt{6}}{3} \mathrm{a}, \frac{\sqrt{2}}{2} \mathrm{a}, \frac{\sqrt{3}}{3} \mathrm{a}, \frac{\sqrt{3}}{3} \mathrm{a}, 90^{\circ}, 60^{\circ}, \operatorname{arctg} \ |
# 1. Problem 1.1*
The sum of ten natural numbers $a_{1}<\ldots<a_{10}$ is 300. Find the maximum possible sum of $a_{1}+a_{3}+\ldots+a_{9}$. Points for the problem: 8 .
# | 147 |
Factorization: $-3am^{2}+12an^{2}=$____. | -3a\left(m+2n\right)\left(m-2n\right) |
1. If the number $100^{10}$ is written as a sum of tens $(10+10+10+\ldots)$, how many addends will there be | 10^{19} |
2. The sum of the first term and the second term of a geometric sequence is 30, the sum of the third term and the fourth term is 120, find the sum of the fifth term and the sixth term. | 480 |
Example 3. Find $\int \arcsin x d x$. | x\arcsinx+\sqrt{1-x^{2}}+C |
3. As shown in Figure $1, \Omega$ is the geometric body obtained after the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$ is cut by the plane $E F G H$, removing the geometric body $E F G H B_{1} C_{1}$. Here, $E$ and $F$ are points on the line segments $A_{1} B_{1}$ and $B B_{1}$, respectively, different from $B_... | D |
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