problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Problem 2. $n$ mushroom pickers went to the forest and brought a total of 338 mushrooms (it is possible that some of them did not bring any mushrooms home). Boy Petya, upon learning this, said: "Some two of them must have brought the same number of mushrooms!" For what smallest $n$ will Petya definitely be right? Don't... | 27 |
Let the universal set $U=\{x\in \mathbb{N}^* | x < 6\}$, where the set $A=\{1,3\}$, and $B=\{3,5\}$. Then, the complement of $A \cup B$ with respect to $U$ is ( )
A: $\{1,4\}$
B: $\{2,4\}$
C: $\{2,5\}$
D: $\{1,5\}$ | B |
The Ebbinghaus forgetting curve describes the law of human brain forgetting new things. Based on this, a research team found that after learning course $A$, 20% of the memorized content is forgotten every week. In order to ensure that the memorized content does not fall below $\frac{1}{12}$, the content needs to be rev... | 10 |
On the number line, there are two points $A$ and $B$. The number represented by point $A$ is $-1$, and the distance between point $B$ and point $A$ is $4$ units. Therefore, the number represented by point $B$ is ______. | 3 \text{ or } -5 |
(10 points) The function $f(x) = \frac{x}{1+x^2}$ is defined on the interval $(-1,1)$ and is an odd function, and $f' = \frac{1-x^2}{(1+x^2)^2}$.
(1) Determine the analytical expression of the function $f(x)$.
(2) Prove that $f(x)$ is an increasing function on the interval $(-1,1)$ by definition.
(3) Solve the inequ... | t \in \left(0, \frac{1}{2}\right) |
Example 12 The sports meet lasted for $n$ days $(n>1)$, and a total of $m$ medals were awarded. On the first day, 1 medal was awarded, and then $\frac{1}{7}$ of the remaining $m-1$ medals were awarded. On the second day, 2 medals were awarded, and then $\frac{1}{7}$ of the remaining medals were awarded, and so on, unti... | 6 |
Find all pairs $ (m, n)$ of positive integers such that $ m^n \minus{} n^m \equal{} 3$. | (m, n) = (4, 1) |
Michel must write the numbers from 1 to 7 in each of the circles in the figure below, so that the results obtained by adding the 3 numbers at the vertices of triangles $X, Y$, and $Z$ are 3 consecutive numbers. Which number can be in the central circle? Find all possibilities.
(1+\sqrt{3})(2-\sqrt{3})$;$(2)\frac{\sqrt{36}×\sqrt{12}}{\sqrt{3}}$;$(3)\sqrt{18}-\sqrt{8}+\sqrt{\frac{1}{8}}$;$(4)(3\sqrt{18}+\frac{1}{5}\sqrt{50}-4\sqrt{\frac{1}{2}})÷\sqrt{32}$. | 2 |
10. For what values of the parameter $a$ does the equation $x^{4}-40 x^{2}+144=a\left(x^{2}+4 x-12\right)$ have exactly three distinct solutions? | 48 |
The line represented by the equation $(m+2)x + (m-1)y - 3 = 0$ ($m \in \mathbb{R}$) always passes through a fixed point ( )
A: $(1, -1)$
B: $( -2, 1)$
C: $(1, -2)$
D: $( -1, 1)$ | A |
Task 11.7. For what least natural $x$ is the expression
$$
\sqrt{29+\sqrt{x}}+\sqrt{29-\sqrt{x}}
$$
an integer? | 400 |
The basic inequality can be described as: If $a \geqslant 0$, $b \geqslant 0$, then $\_\_\_\_\_\_$, the equality holds if and only if $a=b$. | \frac{a+b}{2} \geqslant \sqrt{ab} |
For a triangle with angles $\alpha, \beta, \gamma$,
$$
\sin \alpha+\sin \beta=(\cos \alpha+\cos \beta) \sin \gamma
$$
is satisfied. What is the measure of angle $\gamma$? | 90 |
What is $\nu_{2}\left(2^{100}+2^{200}\right)$? | 100 |
Shift the graph of the function $f(x)=\sin 2x+ \sqrt {3}\cos 2x$ to the left by $\varphi (\varphi > 0)$ units. If the resulting graph is symmetric about the $y$-axis, then the minimum value of $\varphi$ is \_\_\_\_\_\_. | \frac {\pi}{12} |
109 The tangent line of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ intersects the $x$-axis at $A$ and the $y$-axis at $B$. Then the minimum value of $|A B|$ is ( )
A. $2 \sqrt{a^{2}+b^{2}}$
B. $a+b$
C. $\sqrt{2 a b}$
D. $4 \sqrt{a b}$ | +b |
6. (8 points) By expanding the expression $(1+\sqrt{7})^{211}$ using the binomial theorem, we obtain terms of the form $C_{211}^{k}(\sqrt{7})^{k}$. Find the value of $k$ for which such a term attains its maximum value. | 153 |
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... | 20 |
Factorize: $ab^2+10ab+25a$. | a(b+5)^2 |
Given that the slope of line $l_1$ is equal to the slope of line $l_2$: $x - y + 2 = 0$, the angle of inclination of line $l_1$ is ( )
A: 135°
B: 120°
C: 60°
D: 45° | D |
In $\triangle OAB$, $\overrightarrow{OA} = \overrightarrow{a}$, $\overrightarrow{OB} = \overrightarrow{b}$, $OD$ is the altitude from $O$ to side $AB$, and if $\overrightarrow{AD} = \lambda \overrightarrow{AB}$, then the real number $\lambda$ equals
A: $\dfrac{\overrightarrow{a} \cdot (\overrightarrow{b}-\overrightarr... | \text{B} |
Task 3. (15 points) The bases $AB$ and $CD$ of trapezoid $ABCD$ are equal to 101 and 20, respectively, and its diagonals are perpendicular to each other. Find the scalar product of vectors $\overrightarrow{AD}$ and $\overrightarrow{BC}$. | 2020 |
$i$ is the imaginary unit. Find the value of the complex number $$\frac {-3+2i }{1+i}$$. | - \frac {1}{2}+ \frac {5}{2}i |
The real part of the complex number $\dfrac{5i}{1+2i}$ is ______. | 2 |
1. Let the sequence $\left\{a_{n}\right\}$ have the sum of the first $n$ terms $S_{n}=2 a_{n}-1(n=1,2, \cdots)$, and the sequence $\left\{b_{n}\right\}$ satisfies $b_{1}=3$, $b_{k+1}=a_{k}+b_{k}(k=1,2, \cdots)$. Find the sum of the first $n$ terms of the sequence $\left\{b_{n}\right\}$. | 2^{n}+2n-1 |
2. Solve the system of equations
$$
\left\{\begin{array}{l}
11 x-y z=18 \\
11 y-z x=18 \\
11 z-x y=18
\end{array}\right.
$$ | (2,2,2),(9,9,9),(-11,-11,\frac{139}{11}),(-11,\frac{139}{11},-11),(\frac{139}{11},-11,-11) |
13.443 In a flask, there is a salt solution. From the flask, $1 / n$ part of the solution is poured into a test tube, and the solution remaining in the flask is evaporated until the percentage of salt content doubles. After that, the solution from the test tube is poured back into the flask. As a result, the salt conte... | \frac{n+1}{n-1}\cdotp;\frac{1}{3} |
The number of roots of the cubic function $f(x) = x^3 - 3x + 1$ is ( )
A: 0
B: 1
C: 2
D: 3 | D |
Let $(x+1)^2+(x+1)^{11}=a+a_{1}(x+2)+a_{2}(x+2)^2+\ldots+a_{10}(x+2)^{10}+a_{11}(x+2)^{11}$. Find the value of $a_{10}$ (express your answer as a numerical value). | -11 |
## Task 7
An animal has two right and two left legs, two legs in front and two at the back.
How many legs does it have? | 4 |
2. Given that the lengths of two altitudes of $\triangle A B C$ are 5 and 20. If the length of the third altitude is also an integer, then the maximum length of the third altitude is ( ).
(A) 5
(B) 6
(C) 7
(D) 8 | B |
The maximum possible number of right angles in a convex polygon is ( )
A: 3
B: 4
C: 5
D: 6 | \text{B} |
# 3.2. Solve the system of equations:
$$
\left\{\begin{array}{l}
2^{x+y}-2^{x-y}=4 \\
2^{x+y}-8 \cdot 2^{y-x}=6
\end{array}\right.
$$ | \frac{5}{2},\frac{1}{2} |
Example 2.2.3 In the sequence $1,9,81, \cdots, 9^{2005}$, delete the terms whose highest digit is 9. How many terms are left in the remaining sequence? | 1914 |
Given the proposition $p: \exists x \in \mathbb{R}, ax^2 + 2ax + 1 \leqslant 0$. If the proposition $\lnot p$ is true, then the range of values for the real number $a$ is. | [0, 1) |
Given $f\left(x\right)=|mx-1|-|x+2a^{2}|$.
$(1)$ When $a=1$ and $m=2$, find the set $M$ of values of $x$ for which $f\left(x\right) \gt 2$;
$(2)$ When $m=1$, if the inequality $f\left(x\right) \lt -3a$ holds for any real number $x$, find the range of real numbers for $a$. | a \in (-1, -\frac{1}{2}) |
Let $E F G H$ be a convex quadrilateral, and $K, L, M, N$ be the midpoints of segments $E F, F G, G H, H E$ respectively; $O$ be the intersection point of segments $K M$ and $L N$. It is known that $\angle L O M = 90^{\circ}$, $K M = 3 L N$, and the area of quadrilateral $K L M N$ is $S$. Find the diagonals of quadrila... | 2\sqrt{\frac{5S}{3}},2\sqrt{\frac{5S}{3}} |
99(1209). How many consecutive natural numbers, starting from 1, need to be added to get a three-digit number written with identical digits? | 36 |
# Task 1. (10 points)
How many natural numbers $n$ not exceeding 2017 exist such that the quadratic trinomial $x^{2}+x-n$ can be factored into linear factors with integer coefficients
# | 44 |
## Task A-1.3.
Two congruent squares with side lengths of $1+\sqrt{2}$ have the same center, and their intersection is a regular octagon. What is the area of this octagon? | 2+2\sqrt{2} |
A ray of light from point P(2, 3) is reflected off the line $x+y=-1$ and then passes through point Q(1, 1). Find the equation of the line of the reflected ray. | 4x-5y+1=0 |
Given the functions $f(x)=x^{2}-ax+a+3$ and $g(x)=ax-2a$. If there exists an $x\_0∈R$ such that $f(x\_0) < 0$ and $g(x\_0) < 0$ hold simultaneously, find the range of values for the real number $a$.
A: $(-∞,2)$
B: $(0,4)$
C: $(6,+∞)$
D: $(7,+∞)$ | D |
According to the standard convention for exponentiation,
\[2^{2^{2^{2}}} = 2^{(2^{(2^2)})} = 2^{16} = 65536.\]
If the order in which the exponentiations are performed is changed, how many other values are possible?
$\textbf{(A) } 0\qquad \textbf{(B) } 1\qquad \textbf{(C) } 2\qquad \textbf{(D) } 3\qquad \textbf{(E) } 4... | \textbf{(B)}1 |
Let the function $f(x) = \sin^2(x+\pi) - \cos^2(x- \frac{\pi}{3})$.
(1) Find the smallest positive period of $f(x)$ and the intervals where $f(x)$ is monotonically increasing;
(2) If $|f(x)-m| \leq 2$ holds for all $x \in \left[-\frac{\pi}{6}, \frac{\pi}{4}\right]$, find the range of the real number $m$. | \left[-\frac{7}{4}, \frac{3}{2}\right] |
In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If "$\frac{a}{b}=\frac{{\cos B}}{{\cos A}}$" is "$\triangle ABC$ is an isosceles triangle", then it is ( ).
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
... | D |
Example 5. When $b \neq c$, if the parabolas $y=x^{2}+b x+c$ and $y=-x^{2}+c x+b$ are tangent to each other, then $b, c$ satisfy the relation $\qquad$
(1992, Taiyuan City Junior High School Mathematics Competition) | b=c+8 |
7. Given $x, y, z \in \mathbf{R}$, then $\sum \frac{x^{2}}{(3 x-2 y-z)^{2}}$ has the minimum value of $\qquad$ ("sum" indicates cyclic sum).
| \frac{5}{49} |
Śárka poured juice into a glass and a mug and filled both containers with water. The mug had twice the volume of the glass. The ratio of juice to water in the glass was $2: 1$ and in the mug $4: 1$. Then she poured the contents of the glass and the mug into a jug. What was the ratio of juice to water in the jug?
(L. H... | 34:11 |
Example 11 Consider the polynomial $p(x)=x^{n}+n x^{n-1}+a_{2} x^{n-2}+\cdots+a_{n}$, let $v_{i}(1 \leqslant i \leqslant n)$ be all the roots of $p(x)$, and $\left.\left|\gamma_{1}\right|^{16}|+| \gamma_{2}\right|^{16}+\cdots+\left|\gamma_{n}\right|^{16}$ $=n$, find the roots of $p(x)$. | \gamma_{1}=\gamma_{2}=\cdots=\gamma_{n}=-1 |
1. (16 points) Solve the equation $x-7=\frac{4 \cdot|x-3|}{x-3}$. If the equation has multiple roots, write their sum in the answer. | 11 |
Two lighthouses A and B are a distance of $a$ nautical miles and $2a$ nautical miles away from the ocean observation station C, respectively. Lighthouse A is located at 35° north by east from the observation station, and lighthouse B is located at 25° south by east from the observation station. The distance between lig... | B |
A Louis Vuitton store in Shanghai had a number of pairs of sunglasses which cost an average of \$$900$ per pair. LiAngelo Ball stole a pair which cost \$$2000$. Afterwards, the average cost of sunglasses in the store dropped to \$$890$ per pair. How many pairs of sunglasses were in the store before LiAngelo Ball stole?... | 111 |
3. For a regular octagon $A_{1} A_{2} \cdots A_{8}$ with side length 1, take any two points $A_{i} 、 A_{j}$. Then the maximum value of $\overrightarrow{A_{i} A_{j}} \cdot \overrightarrow{A_{1} \vec{A}_{2}}$ is $\qquad$ . | \sqrt{2}+1 |
13. We use $A, B, C, D, E$ to represent the number of gold coins held by students 甲, 乙, 丙, 丁, 戊, respectively. The total number of these gold coins is denoted by $n$, which means $n=A+B+C+D+E$. It is known that:
(1) $A, B, C, D, E$ are all greater than or equal to 1;
(2) $A<B<C<D<E$;
(3) Each of the five students knows... | 19 |
3. An isosceles trapezoid $A B C D$ with bases $A B$ and $D C$ has a circle inscribed in it with the center at point $O$. Find the area of the trapezoid if $O B=b$ and $O C=c$. | 2bc |
3. Variant 1.
The height $A H$ and the bisector $C L$ of triangle $A B C$ intersect at point $O$. Find the angle $B A C$, if it is known that the difference between the angle $C O H$ and half the angle $A B C$ is $46^{\circ}$. | 92 |
2. A certain product at the same price underwent 2 price adjustments in 3 stores. Store A first decreased the price by a percentage of $a$, then increased it by a percentage of $b$; Store B first decreased the price by a percentage of $\frac{a+b}{2}$, then increased it by a percentage of $\frac{a+b}{2}$; Store C first ... | A |
Given any number $x$, $f(x)$ is an odd function, and $g(x)$ is an even function. Also, when $x>0$, $f'(x)>0$, and $g'(x)>0$, then for $x0$, $g'(x)>0$
B: $f'(x)>0$, $g'(x)0$
D: $f'(x)<0$, $g'(x)<0$ | \text{B} |
6. If in space three lines $a, b, c$ are pairwise skew lines, then the number of lines that intersect $a, b, c$ is
(A) 0
(B) 1
(C) more than 1 but finite
(D) infinitely many | D |
The coordinates of point Q, which is symmetric to point P(-3, 4) with respect to the line $x+y-2=0$, are ( )
A: (-2, 1)
B: (-2, 5)
C: (2, -5)
D: (4, -3) | \text{B} |
2. Kolya is twice as old as Olya was when Kolya was as old as Olya is now. And when Olya is as old as Kolya is now, their combined age will be 36 years. How old is Kolya now?
ANS: 16 years. | 16 |
Which of the following four characters can be considered as symmetrical figures?
A: 幸
B: 福
C: 惠
D: 州 | \text{A} |
Example 2. Solve the equation
$$
|x|=x^{2}+x-2
$$ | \sqrt{2},-1-\sqrt{3} |
Given that $\{a_n\}$ is a geometric sequence with all positive terms, and the common ratio $q \neq 1$, then the relationship between $a_1+a_4$ and $a_2+a_3$ is ( )
A: $a_1+a_4 > a_2+a_3$
B: $a_1+a_4 < a_2+a_3$
C: $a_1+a_4 = a_2+a_3$
D: It's uncertain | \text{A} |
Example 3. Solve the system of equations in integers:
$$
\left\{\begin{array}{l}
x^{2}-y^{2}-z^{2}=1 \\
y+z-x=3
\end{array}\right.
$$ | {(9;8;4),(-3;-2;2),(9;4;8),(-3;2;-2)} |
Among the following conclusions:
1. "$p \land q$" being true is a sufficient but not necessary condition for "$p \lor q$" to be true;
2. "$p \land q$" being false is a sufficient but not necessary condition for "$p \lor q$" to be true;
3. "$p \land q$" being true is a sufficient but not necessary condition for "$\neg p... | 1, 3, 4 |
Given a mapping $f: A \to B$ where $A = B = \{ (x, y) | x, y \in \mathbb{R} \}$, and $f: (x, y) \to (x-y, x+y)$, determine the element in $B$ that corresponds to the element $(1, 3)$ in $A$. | (-2, 4) |
3. Given $\frac{1}{a}-|a|=1$, what is the value of $\frac{1}{a}+|a|$?
(A) $\frac{\sqrt{5}}{2}$
(B) $-\frac{\sqrt{5}}{2}$
(C) $-\sqrt{5}$
(D) $\sqrt{5}$ | D |
Given $A=x^{2}+xy+2y-2$, $B=2x^{2}-2xy+x-1$. Find:
$(1) 2A-B$;
$(2)$ If the value of $2A-B$ is independent of $x$, find the value of $y$. | \frac{1}{4} |
The Tournament of Towns is held once per year. This time the year of its autumn round is divisible by the number of the tournament: $2021\div 43 = 47$. How many times more will the humanity witness such a wonderful event?
[i]Alexey Zaslavsky[/i] | 4 |
In $\triangle{ABC}, AB=10, \angle{A}=30^\circ$ , and $\angle{C=45^\circ}$. Let $H, D,$ and $M$ be points on the line $BC$ such that $AH\perp{BC}$, $\angle{BAD}=\angle{CAD}$, and $BM=CM$. Point $N$ is the midpoint of the segment $HM$, and point $P$ is on ray $AD$ such that $PN\perp{BC}$. Then $AP^2=\dfrac{m}{n}$, where ... | 77 |
1.149 Calculate the value of the following expression
$$\frac{\left(10^{4}+324\right)\left(22^{4}+324\right)\left(34^{4}+324\right)\left(46^{4}+324\right)\left(58^{4}+324\right)}{\left(4^{4}+324\right)\left(16^{4}+324\right)\left(28^{4}+324\right)\left(40^{4}+324\right)\left(52^{4}+324\right)}$$ | 373 |
4. Place 27 balls numbered $1 \sim 27$ into three bowls, Jia, Yi, and Bing, such that the average values of the ball numbers in bowls Jia, Yi, and Bing are $15$, $3$, and $18$, respectively, and each bowl must contain no fewer than 4 balls. Then the maximum value of the smallest ball number in bowl Jia is $\qquad$ | 10 |
Let the common ratio of the geometric sequence $\{a_n\}$ be $q$, and the sum of the first $n$ terms be $T_n$.
A: If $q > 1$, then the sequence $\{T_n\}$ is monotonically increasing.
B: If the sequence $\{T_n\}$ is monotonically increasing, then $q > 1$.
C: If $T_n > 0$, then the sequence $\{T_n\}$ is monotonically incr... | \text{D} |
Given that the line $l: y=kx+b$ is the common tangent line of the functions $f\left(x\right)=ax^{2}(a \gt 0)$ and $g\left(x\right)=e^{x}$. If $\left(1,f\left(1\right)\right)$ is the point of tangency between the line $l$ and the function $f\left(x\right)$, then $b=$____. | -\frac{1}{2}e^{\frac{3}{2}} |
G7 Let $A B C$ be an isosceles triangle with $A C=B C$. The point $D$ lies on the side $A B$ such that the semicircle with diameter $[B D]$ and center $O$ is tangent to the side $A C$ in the point $P$ and intersects the side $B C$ at the point $Q$. The radius $O P$ intersects the chord $D Q$ at the point $E$ such that... | \frac{6}{5} |
6. On an island, there live chameleons of five colors. When one chameleon bites another, the color of the bitten chameleon changes according to some rule, and the new color depends only on the color of the biter and the color of the bitten. It is known that 2023 red chameleons can agree on a sequence of bites, after wh... | 5 |
Which of the following statements is correct? ( )
A: If $a$, $b$, $c \in \mathbb{R}$, then a sufficient condition for "$\forall x \in \mathbb{R}, ax^{2}+bx+c \geqslant 0$" is "$b^{2}-4ac \leqslant 0$"
B: If $a$, $b$, $c \in \mathbb{R}$, then a necessary and sufficient condition for "$ab^{2} > cb^{2}$" is "$a > c$"
C: T... | D |
The inequality $X^{2}+2a < \frac{a}{b}+ \frac{16b}{a}$ holds for any $a$, $b \in (0,+\infty)$. Then, the range of the real number $x$ is ( ).
A: $(-2,0)$
B: $(-\infty,-2)\cup(0,+\infty)$
C: $(-4,2)$
D: $(-\infty,-4)\cup(2,+\infty)$ | \text{C} |
A pile of apples is known to be more than 50 but less than 70. They can be divided into two piles, three piles, and also five piles. How many apples are there in this pile? It can also be divided into 1, 2, 4, 6, 10, 12, 15, 20, 30, 60 piles. | 60 |
A circle is inscribed in a sector that is one sixth of a circle of radius 6 . (That is, the circle is tangent to both segments and the arc forming the sector.) Find, with proof, the radius of the small circle.
![](https://cdn.mathpix.com/cropped/2024_04_17_45b306ab1b407a5798d6g-2.jpg?height=352&width=396&top_left_y=24... | 2 |
Numbers like $45132$ are called "wave numbers," meaning the tens and thousands digits are greater than the digits adjacent to them. The number of distinct five-digit "wave numbers" that can be formed using $1$, $2$, $3$, $4$, $5$ without repeating any digit is __________. | 16 |
225. Find the condition under which the difference of the roots in the equation $x^{2}+p x+$ $+q=0$ would be equal to $a$. | ^{2}-p^{2}=-4q |
Example \ Solve the equation $3 x^{3}-[x]=3$ | \sqrt[3]{\frac{4}{3}} |
16. (2 marks) Each face and each vertex of a regular tetrahedron is coloured red or blue. How many different ways of colouring are there? (Two tetrahedrons are said to have the same colouring if we can rotate them suitably so that corresponding faces and vertices are of the same colour.
(2 分) 在正四面體的每一個頂點和每一個面都塗上紅色或藍色, ... | 36 |
Given that the sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n=2n^2-3n (n\in\mathbb{N}^*)$, find the value of $a_4$. | 11 |
## 213. Math Puzzle $2 / 83$
Michael wishes his uncle a happy birthday, hoping that he may live to be at least 3 Gs old. How many years does this wish correspond to?
(1 Gs $=1$ Gigasecond $=10^{9}$ ) | 95.13 |
With the continuous development of technology, "smartphones" have become an indispensable necessity in people's lives. The table below shows the annual total shipment of mobile phones in a certain city in Guangxi (unit: ten thousand units).
| Year | 2018 | 2019 | 2020 | 2021 | 2022 |
|------|------|------|------|-----... | 2.81 |
The coefficient of $x^7y^2$ in the expansion of $(x-y)(x+y)^8$ is __________ (fill in the blank with a number). | 20 |
In the geometric sequence $\{a_n\}$, $a_1=1$, the common ratio $q \in \mathbb{R}$ and $q \neq 1$. If $a_n = a_1a_2a_3…a_{10}$, then $n$ equals ( )
A: 44
B: 45
C: 46
D: 47 | C |
In the Cartesian coordinate system $xOy$, with $O$ as the pole and the positive half-axis of $x$ as the polar axis, the polar equations of circle $C_{1}$ and line $C_{2}$ are respectively $\rho=4\sin \theta$ and $\rho\cos (\theta- \frac {\pi}{4})=2 \sqrt {2}$.
(Ⅰ) Find the polar coordinates of the intersection points... | a=-1, b=2 |
Let $a$ be a real number, $a>1.$ Find the real numbers $b \ge 1$ such that $$\lim_{x \to \infty} \int\limits_0^x (1+t^a)^{-b} \mathrm{d}t=1.$$ | b = \frac{1}{a} + 1 |
Let $\{a_n\}$ be a sequence of positive terms, and its sum of the first $n$ terms $S_n$ satisfies: $4S_n = (a_n - 1)(a_n + 3)$. Find the value of $a_{1005}$. | 2011 |
Consider the multiplicative group $A=\{z\in\mathbb{C}|z^{2006^k}=1, 0<k\in\mathbb{Z}\}$ of all the roots of unity of degree $2006^k$ for all positive integers $k$.
Find the number of homomorphisms $f:A\to A$ that satisfy $f(f(x))=f(x)$ for all elements $x\in A$. | 8 |
Determine the smallest integer $n > 1$ with the property that $n^2(n - 1)$ is divisible by 2009. | 42 |
3. As shown in the figure, from the right-angle vertex $C$ of the right triangle $\triangle A B C$, lines $C E$ and $C F$ are drawn to the trisection points of the hypotenuse $A B$. It is known that $C E=\sin \alpha, C F=\cos \alpha$ (where $\alpha$ is an acute angle). Then $A B=$ | \frac{3 \sqrt{5}}{5} |
Given a function $f(x)$ defined on $\mathbb{R}$ that satisfies $f(x+y) = f(x) + f(y)$, and when $x > 0$, $f(x) > 0$.
(Ⅰ) Find $f(0)$;
(Ⅱ) Determine the parity of the function and prove it;
(Ⅲ) Solve the inequality $f(a-4) + f(2a+1) < 0$. | a < 1 |
Given variables $a$ and $b$ have been assigned values, the algorithm to swap the values of $a$ and $b$ is ( )
A: $a=b$, $b=a$
B: $a=c$, $b=a$, $c=b$
C: $a=c$, $b=a$, $c=a$
D: $c=a$, $a=b$, $b=c$ | D |
5.1. Find the smallest natural solution of the inequality $2^{x}+2^{x+1}+\ldots+2^{x+2000}>2^{2017}$. Answer. 17. | 17 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.