problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Given the functions $f(x)=x^{2}-2x$ and $g(x)=ax+2 (a > 0)$, if $\forall x_{1}\in[-1,2]$, $\exists x_{2}\in[-1,2]$, such that $f(x_{1})=g(x_{2})$, then the range of values for the real number $a$ is $(\quad\quad)$.
A: $(0, \frac {1}{2}]$
B: $[\frac {1}{2},3]$
C: $(0,3]$
D: $[3,+\infty)$ | D |
295. Find all natural numbers $n$ for which the number $7 n+5$ is divisible by the number $3 n-1$. | 1,4 |
Given an arithmetic sequence \\(\{a_n\}\) that satisfies \\(a_1+a_2=10\\) and \\(a_4-a_3=2\\), and a geometric sequence \\(\{b_n\}\) that satisfies \\(b_2=a_3\\) and \\(b_3=a_7\\), then \\(b_5=\\) \\(\\)
A: \\(32\\)
B: \\(64\\)
C: \\(128\\)
D: \\(256\\) | \text{B} |
3. In triangle $A B C$, point $M$ is the midpoint of $A C$, moreover, $B C=2 A C / 3$ and $\angle B M C=2 \angle A B M$. Find the ratio $A M / A B$. | \frac{3}{2\sqrt{5}} |
On the diagonal $B D$ of the rectangular trapezoid $A B C D\left(\angle D=90^{\circ}, B C \| A D\right)$, a point $Q$ is taken such that $B Q: Q D=1: 3$. A circle with center at point $Q$ touches the line $A D$ and intersects the line $B C$ at points $P$ and $M$. Find the length of the side $A B$, if $B C=9, A D=8, P M... | 3 |
36 . 18 If person A's height or weight is greater than person B's in at least one aspect, then we say A is not inferior to B. Among 100 young men, if someone is not inferior to the other 99, he is called a great young man. Then, the maximum number of great young men among the 100 young men could be
(A) 1.
(B) 2.
(C) 50... | 100 |
1. Given quadratic trinomials $f_{1}(x)=x^{2}+2 x+a, f_{2}(x)=x^{2}+b x-1, f_{3}(x)=2 x^{2}+(6-b) x+3 a+1$ and $f_{4}(x)=2 x^{2}+(3 b-2) x-a-3$. Let the differences of their roots be $A, B, C$ and $D$, respectively, and given that $|A| \neq|B|$. Find the ratio $\frac{C^{2}-D^{2}}{A^{2}-B^{2}}$. The values of $A, B, C, ... | 2 |
If the function $f(x) = 2\sin(2x + \varphi)$ ($|\varphi| < \frac{\pi}{2}$) is translated to the right by $\frac{\pi}{6}$ units and then passes through the point ($\frac{\pi}{12}$, $-\sqrt{2}$), then $\varphi$ equals ( )
A: $-\frac{\pi}{12}$
B: $-\frac{\pi}{6}$
C: 0
D: $\frac{\pi}{6}$ | A |
In the complex plane, the point corresponding to the complex number $\frac {1}{(1+i)^{2}+1} + i^4$ is located in ( )
A: The first quadrant
B: The second quadrant
C: The third quadrant
D: The fourth quadrant | \text{D} |
3. Given a sequence of numbers $a_{1}, a_{2}, \cdots, a_{2006}$, where $a_{1}$ $=1$, and the sum of each pair of consecutive terms is 3. Then $a_{1}-a_{2}+$ $a_{3}-a_{4}+\cdots+a_{2003}-a_{2004}+a_{2005}=$ $\qquad$ | -1001 |
During this year's May Day holiday, Xiangxi Prefecture received 1.603 million tourists and generated a tourism revenue of 1.673 billion yuan, showing a promising tourism recovery trend. Express 1.673 billion in scientific notation as ( ).
A: $16.73\times 10^{8}$
B: $1.673\times 10^{8}$
C: $1.673\times 10^{9}$
D: $1... | C |
Given that $\overrightarrow {a}$=$(2,0)$, $\overrightarrow {b}$=$(1,2)$, and the real number λ satisfies $|$$\overrightarrow {a}$$-λ \overrightarrow {b}$$|=$$\sqrt {5}$$, find the value(s) of λ. | λ=- \frac {1}{5} \text{ or } 1 |
## Task B-3.5.
A trapez with mutually perpendicular diagonals has bases of length $a=12$ and $c=4$, and the extensions of the legs of the trapezoid intersect at an angle $\alpha$. If $\cos \alpha=\frac{4}{5}$, calculate the area of this trapezoid. | 36 |
A function $f(x)=2\sin(\omega x-\frac{\pi}{6})$ is drawn on a piece of paper, where $\omega > 0$. The graph of one period of this function is shown on the paper. The paper is then folded along the $x$-axis to form a dihedral angle. The spatial distance between adjacent highest and lowest points on the original graph is... | D |
Given $f(x)$ satisfies for all $x \in \mathbb{R}$, $f(-x) + f(x) = 0$, and when $x \geq 0$, $f(x) = e^x + m$ (where $m$ is a constant), then the value of $f(-\ln 5)$ is ( )
A: 4
B: -4
C: 6
D: -6 | \text{B} |
Given that all terms of the geometric sequence $\{a_n\}$ are positive, with the first term $a_1 = 1$, and $a_4$, $3a_3$, $a_5$ form an arithmetic sequence.
(1) Find the general formula for the sequence $\{a_n\}$.
(2) The sum of the first $n$ terms of the sequence $\{a_{n+1} - \lambda a_n\}$ is $S_n$. If $S_n = 2^n - 1... | 1 |
3. From the 10 numbers $0,1,2,3,4,5,6,7,8,9$, choose 3 numbers such that their sum is an even number not less than 10.
The number of different ways to choose them is $\qquad$ . | 51 |
Given vectors $\overrightarrow{a}=(\cos α, \sin α)$ and $\overrightarrow{b}=(\cos β, \sin β)$ with $| \overrightarrow{a}- \overrightarrow{b}|= \frac{2 \sqrt{5}}{5}$.
1. Find the value of $\cos (α-β)$.
2. If $0 < α < \frac{\pi}{2}$, $-\frac{\pi}{2} < β < 0$, and $\sin β=- \frac{5}{13}$, find the value of $\sin α$. | \frac{33}{65} |
Given the distinct points $P(x_1, y_1), Q(x_2, y_2)$ and $R(x_1+x_2, y_1+y_2)$.
Line segments are drawn connecting these points to each other and to the origin $O$.
Of the three possibilities: (1) parallelogram (2) straight line (3) trapezoid, figure $OPRQ$,
depending upon the location of the points $P, Q$, and $R$,... | \textbf{(D)} |
In the geometric sequence $\{a_n\}$, $S_n$ represents the sum of the first $n$ terms. It is known that $a_5 = 2S_4 + 3$, and $a_6 = 2S_5 + 3$. Find the common ratio $q$ of this sequence. | 3 |
Example 4 Perform the following operations on a positive integer: if it is even, divide it by 2; if it is odd, add 1. Continue doing this until the number becomes 1 and the operation stops. Find how many numbers will become 1 after 9 operations. | F_9 |
8. Using 3 ones, 2 twos, and 1 three, the number of different 6-digit numbers that can be formed is ( ).
(A) 45
(B) 50
(C) 60
(D) 70 | C |
Example 3. Find $\lim _{x \rightarrow 0}\left(\frac{1}{4 \sin ^{2} x}-\frac{1}{\sin ^{2} 2 x}\right)$. | -\frac{1}{4} |
## Task 3 - 280923
In a triangle $ABC$, let the side lengths and angle measures be denoted as usual by $a, b, c$ and $\alpha, \beta, \gamma$.
The angle bisector of $\angle BAC$ intersects the side $BC$ at a point $D$. Here, $AD=b$.
Furthermore, it is assumed that one of the three angle measures $\alpha, \beta, \gamm... | \alpha=60,\beta=45,\gamma=75\quad |
Given an arithmetic sequence $\{a_n\}$ with a common difference of $5$, and the sum of the first $n$ terms is $S_n$, and $a_1$, $a_2$, $a_5$ form a geometric sequence, then $S_6$ = ( )
A: $80$
B: $85$
C: $90$
D: $95$ | C |
1-36 Find all such four-digit numbers, when 400 is written to their left, the result is a perfect square.
Find all such four-digit numbers, when 400 is written to their left, the result is a perfect square. | 4001 \text{ or } 8004 |
1. Find all values of $p$, for each of which the numbers $p-2$, $2 \cdot \sqrt{p}$, and $-3-p$ are respectively the first, second, and third terms of some geometric progression. | 1 |
Given a sequence $\{a\_n\}$ where the sum of its first $n$ terms is $S\_n$, and $S\_n = \frac{4}{3}(a\_n - 1)$, find the minimum value of $(4^{n-2} + 1)(\frac{16}{a\_n} + 1)$. | 4 |
A certain souvenir is very popular among tourists. The cost price of each souvenir is $40$ yuan, and the selling price is set to be no less than $44$ yuan and no more than $52 yuan. During the sales period, a merchant found that when the selling price is set at $44$ yuan, 300 souvenirs can be sold per day. For every in... | 50 \leqslant x \leqslant 52 |
5. The force with which the airflow acts on the sail can be calculated using the formula
$F=\frac{A S \rho\left(v_{0}-v\right)^{2}}{2}$, where $A-$ is the aerodynamic force coefficient, $S-$ is the area of the sail $S$ $=4 \mathrm{M}^{2} ; \rho-$ is the density of air, $v_{0}$ - is the wind speed $v_{0}=4.8 \mathrm{~m... | 1.6\mathrm{M}/\mathrm{} |
234. Write the number $z=-5 i$ in trigonometric and exponential forms. | 5(\cos\frac{3\pi}{2}+i\sin\frac{3\pi}{2}) |
Given the sets $M=\{-1, 1\}$ and $N=\{-1, 0, 2\}$, then $M \cap N$ is ( )
A: $\{-1, 1\}$
B: $\{-1\}$
C: $\{0\}$
D: $\{-1, 0\}$ | B |
2. let $n$ be a natural number. Determine the number of subsets $A \subset\{1,2, \ldots, 2 n\}$ such that $x+y=2 n+1$ does not hold for any two elements $x, y \in A$.
## Solution: | 3^n |
8.2. Losharik is going to visit Sovunya along the river at a speed of 4 km/h. Every half hour, he launches paper boats that float to Sovunya at a speed of 10 km/h. With what time interval do the boats arrive at Sovunya? | 18 |
Problem 8.6. In a $3 \times 3$ table, real numbers are arranged. It turns out that the product of the numbers in any row and any column is 10, and the product of the numbers in any $2 \times 2$ square is 3. Find the number in the central cell. | 0.00081 |
1. Find the number of all different solutions to the equation
$$
x_{1}+x_{2}+\ldots+x_{n}=k
$$
( $k$ and $n$ are given numbers) in the set $\mathbb{N}$. | \binom{k-1}{n-1} |
## Task 25/67
We are looking for all prime numbers $p$ for which $3 p+4=a^{2}$ (where $a$ is a natural number). | 7 |
Let $\theta$ be a constant number such that $0\leq \theta \leq \pi.$
Evaluate
\[ \int_0^{2\pi} \sin 8x|\sin (x-\theta)|\ dx. \] | -\frac{4}{63} \sin 8\theta |
1. Calculate: $\frac{2}{1} \times \frac{2}{3} \times \frac{4}{3} \times \frac{4}{5} \times \frac{6}{5} \times \frac{6}{7} \times \frac{8}{7}=$ (write in decimal form, rounded to two decimal places). | 1.67 |
3. It is known that for some positive coprime numbers $m$ and $n$, the numbers $m+1947 n$ and $n+1947 m$ have a common prime divisor $d>9$. Find the smallest possible value of the number $d$ under these conditions. | 139 |
3. The sequence $\left(a_{n}\right)$ is defined by $a_{1}=\frac{1}{2}, a_{n}=\frac{a_{n-1}}{2 n a_{n-1}+1}$ for $n>1$. Determine the sum $a_{1}+\ldots+a_{k}$, for any natural number $k$. | \frac{k}{k+1} |
Example 3. Let $x, y$ satisfy $3 x^{2}+2 y^{2}=6 x$, find the maximum value of $x^{2}+y^{2}$:
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | 4 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $a\cos B=(3c-b)\cos A$.
(1) Find $\sin A$;
(2) If $a=2 \sqrt{2}$, and the area of $\triangle ABC$ is $\sqrt{2}$, find the value of $b+c$. | 4 |
9. In the figure below, $A$ is the midpoint of $DE$, the areas of $\triangle D C B$ and $\triangle E B C$ satisfy $S_{\triangle D C B}+S_{\triangle E B C}=12$, then the area of $\triangle A B C$ is $\qquad$ . | 6 |
Example 4 Let the edge length of the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ be $1, P$ is a moving point on the face diagonal $B C_{1}$, and $Q$ is a moving point on the base $A B C D$. Try to find the minimum value of $D_{1} P+P Q$.
(1998 "Hope Cup" Competition Question) | 1+\frac{\sqrt{2}}{2} |
I3.1 若 $\mathrm{f}(x)=x^{4}+x^{3}+x^{2}+x+1$, 求 $\mathrm{f}\left(x^{5}\right)$ 除以 $\mathrm{f}(x)$ 的餘值 $a$ 。
If $\mathrm{f}(x)=x^{4}+x^{3}+x^{2}+x+1$, determine the remainder $a$ of $\mathrm{f}\left(x^{5}\right)$ divided by $\mathrm{f}(x)$. | 5 |
Let α, β, γ be three different planes, and let a, b be two different lines. Consider the following four propositions:
① If line a is parallel to plane α, and line b is parallel to plane α, then line a is parallel to line b;
② If line a is parallel to plane α, line b is parallel to plane β, and line a is parallel to lin... | A |
10.3. Lёsha colors cells inside a $6 \times 6$ square drawn on graph paper. Then he marks the nodes (intersections of the grid lines) to which the same number of colored and uncolored squares are adjacent. What is the maximum number of nodes that can be marked? | 45 |
A student used the "five-point method" to draw the graph of the function $f(x)=A\sin(\omega x+\varphi)$ ($\omega\ \gt 0$, $|\varphi|＜\frac{π}{2}$) within one period. The student listed and filled in some of the data in the table below:
| $\omega x+\varphi$ | $0$ | $\frac{π}{2}$ | $\pi$ | $\frac{{3π}}{2}$ | $2\pi$ |
|-... | \frac{7\pi}{24} |
Given that the distance between two parallel lines $3x+4y-4=0$ and $ax+4y+b=0$ ($b>0$) is 2, find the value of $a+b$. | 9 |
Let \\(k \in \mathbb{Z}\\), among the following, the angles that have the same terminal side are ( ).
A: \\((2k+1)180^{\circ}\\) and \\((4k\pm1)180^{\circ}\\)
B: \\(k \cdot 90^{\circ}\\) and \\(k \cdot 180^{\circ} + 90^{\circ}\\)
C: \\(k \cdot 180^{\circ} + 30^{\circ}\\) and \\(k \cdot 360^{\circ} \pm 30^{\circ}\\)
D: ... | A |
Bogdanov I.I.
Seryozha drew a triangle $ABC$ and drew the median $AD$ in it. Then he told Ilya the length of the median $AD$ and the length of the side $AC$ in this triangle. Ilya, based on these data, proved the statement: angle $CAB$ is obtuse, and angle $DAB$ is acute. Find the ratio $AD: AC$ (and prove for any tri... | 1:2 |
The solution set of the inequality system $\left\{{\begin{array}{l}{3x-1≥x+1}\\{x+4＞4x-2}\end{array}}\right.$ is ( ).
A: $1\leqslant x \lt 2$
B: $x\leqslant 1$
C: $x \gt 2$
D: $1 \lt x\leqslant 2$ | A |
108 The number of real roots of the equation $1-\lg \sin x=\cos x$ is
A. 0
B. 1
C. 2
D. Greater than 2 | A |
3. A middle school selects student council members from students in grades 7 to 12, and at least one of the following conditions must be met:
(1) At most 1 person from grade 7;
(2) At most 2 people from grade 8;
(3) At most 3 people from grade 9;
(4) At most 4 people from grade 10;
(5) At most 5 people from grade 11;
(... | C |
Example 4 Find the range of real numbers $m$ such that the equation $x^{2}+(m-2) x+m+6=0$ about $x$:
(1) has one real root greater than 2 and the other real root less than 2;
(2) has two real roots both greater than 1. | -\frac{5}{2}<\leqslant-2 |
Which of the following conclusions in space is valid by analogy with the three conclusions in the plane?
\\(①\\) Two lines parallel to the same line are parallel;
\\(②\\) If a line is perpendicular to one of two parallel lines, it must also be perpendicular to the other;
\\(③\\) If a line intersects one of two paral... | \text{A} |
The program obtained by using the bisection method to solve the equation $x^2-2=0$ is ( )
A: Organizational structure diagram
B: Process flow diagram
C: Knowledge structure diagram
D: Program flowchart | D |
3. The sides of a right-angled triangle are $a-b, a, a+b$, and both $a$ and $b$ are positive integers. Then one of the sides could be ( . .
(A) 61
(B) 71
(C) 81
(D) 91 | C |
6.67 Let $a_{1}, a_{2}, \cdots, a_{n}$ denote any permutation of the integers $1,2, \cdots, n$. Let $f(n)$ be the number of permutations that satisfy the conditions:
(1) $a_{1}=1$,
(2) $\left|a_{i}-a_{i+1}\right| \leqslant 2, i=1,2, \cdots, n-1$
Determine whether $f(1996)$ is divisible by 3. | 1 |
Given the function $f(x)= \begin{cases} \log_{\frac{1}{2}}x, & x\geqslant 1 \\ 1-2^{x}, & x < 1 \end{cases}$, then $f(f(2))=$ \_\_\_\_\_\_. | \dfrac{1}{2} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is known that vectors $\vec{m}=(\cos A-2\cos C,2c-a)$ and $\vec{n}=(\cos B,b)$ are parallel.
$(1)$ Find the value of $\frac{\sin C}{\sin A}$;
$(2)$ If $b\cos C+c\cos B=1$ and the perimeter of $\triangl... | b=2 |
7.5. One hundred non-zero integers are written in a circle such that each number is greater than the product of the two numbers following it in a clockwise direction. What is the maximum number of positive numbers that can be among these 100 written numbers? | 50 |
3. In $\triangle A B C$, $B C=a, C A=b, A B=c$. If $2 a^{2}+b^{2}+c^{2}=4$, then the maximum value of the area of $\triangle A B C$ is $\qquad$ (Provided by Li Hongchang) | \frac{\sqrt{5}}{5} |
6. The graphs of the functions $f(x)=2 x^{2}-2 x-1$ and $g(x)=$ $-5 x^{2}+2 x+3$ intersect at two points. The equation of the line passing through these two points is $y=a x+b$. Find the value of $a-b$. | -1 |
Given $x, y \in \mathbb{R}$, $x \gt 0$, $y \gt 0$, and $x+2y=xy$, the possible values of $e^{y}-\frac{8}{x}$ are $\left(\ \ \right)$ (reference data: $e^{1.1}\approx 3$, $e^{1.2}\approx 3.321$)
A: $\frac{5}{4}$
B: $\frac{3}{2}$
C: $e-1$
D: $e$ | D |
The number of elements in a finite set $S$ is denoted as $\text{card}(S)$. Suppose $A$ and $B$ are finite sets, and consider the following propositions:
① The necessary and sufficient condition for $A \cap B = \emptyset$ is $\text{card}(A \cup B) = \text{card}(A) + \text{card}(B)$;
② The necessary condition for $A ... | \text{B} |
What is the correct ascending order of the fractions $\frac{4}{3}, \frac{4}{5}, \frac{4}{6}, \frac{3}{5}, \frac{6}{5}$ and $\frac{2}{5} ?$
(a) $\frac{2}{5}<\frac{3}{5}<\frac{4}{6}<\frac{4}{5}<\frac{6}{5}<\frac{4}{3}$
(d) $\frac{2}{5}<\frac{3}{5}<\frac{4}{5}<\frac{4}{6}<\frac{6}{5}<\frac{4}{3}$
(b) $\frac{4}{3}<\frac{4}... | \frac{2}{5}<\frac{3}{5}<\frac{4}{6}<\frac{4}{5}<\frac{6}{5}<\frac{4}{3} |
3. $f(x)$ is an odd function, $g(x)$ is an even function, and $f(x)+g(x)+f(x) g(x)=\left(e^{x}+\sin x\right) \cdot \cos x$, then $f(x)$ has $\qquad$ zeros in $[0,4 \pi)$. | 5 |
Given the universal set $U=\{a, b, c, d, e\}$, $M=\{a, c, d\}$, $N=\{b, d, e\}$, then $(\complement_{U} M) \cap N$ equals to ( )
A: $\{b\}$
B: $\{d\}$
C: $\{b, e\}$
D: $\{b, d, e\}$ | C |
Given $\alpha \in \left( \frac{\pi}{2}, \pi \right)$, and $\sin \frac{\alpha}{2} + \cos \frac{\alpha}{2} = \frac{\sqrt{6}}{2}$, find the value of $\cos \alpha$. | -\frac{\sqrt{3}}{2} |
Let's determine the mass of the Earth's atmosphere! Assume that the temperature is constant! (See the article published in the September issue.)
Translating the text as requested, while retaining the original line breaks and formatting. | 5\cdot10^{18}\mathrm{~} |
## [ Quadrilaterals (extreme properties).]
For what value of the height does a rectangular trapezoid with an acute angle of $30^{\circ}$ and a perimeter of 6 have the maximum area
# | 1 |
Let $x_1$, $x_2$, and $x_3$ be the roots of the polynomial $x^3+3x+1$. There are relatively prime positive integers $m$ and $n$ such that $\tfrac{m}{n}=\tfrac{x_1^2}{(5x_2+1)(5x_3+1)}+\tfrac{x_2^2}{(5x_1+1)(5x_3+1)}+\tfrac{x_3^2}{(5x_1+1)(5x_2+1)}$. Find $m+n$. | 10 |
The domain of the power function $f(x) = x^{-2}$ is ( )
A: $\mathbb{R}$
B: $\{x | x \in \mathbb{R} \text{ and } x \neq 0\}$
C: $[0, +\infty)$
D: $(0, +\infty)$ | B |
Which of the following operations is correct?
A: $a^{2}+a^{3}=a^{5}$
B: $(ab)^{2}=ab^{2}$
C: $m^{2}\cdot m^{3}=m^{5}$
D: $(-a^{2})^{3}=a^{6}$ | C |
Given a function $f(x)$ satisfying $f\left( \frac{1}{x+1} \right) = 3x - 1$, find the explicit form of $f(x)$ (without stating the domain). | f(x) = \frac{3}{x} - 4 |
The vertex of the parabola $y = x^2 - 8x + c$ will be a point on the $x$-axis if the value of $c$ is:
$\textbf{(A)}\ - 16 \qquad \textbf{(B) }\ - 4 \qquad \textbf{(C) }\ 4 \qquad \textbf{(D) }\ 8 \qquad \textbf{(E) }\ 16$ | 16 |
Which of the following sets of three line segments can form a quadrilateral with a line segment of length $5$?
A: $2$, $2$, $2$
B: $1$, $1$, $8$
C: $1$, $2$, $2$
D: $1$, $1$, $1$ | A |
6.5. In the room, there are 10 people - liars and knights (liars always lie, and knights always tell the truth). The first said: "In this room, there is at least 1 liar." The second said: "In this room, there are at least 2 liars." The third said: "In this room, there are at least 3 liars." And so on,
up to the tenth,... | 5 |
Given the vectors $\overrightarrow{OA}=(k,12)$, $\overrightarrow{OB}=(4,5)$, $\overrightarrow{OC}=(-k,0)$, and points $A$, $B$, $C$ are collinear, find the value of $k$. | -24 |
## Problem Statement
Find the derivative.
$$
y=x\left(2 x^{2}+1\right) \sqrt{x^{2}+1}-\ln \left(x+\sqrt{x^{2}+1}\right)
$$ | 8x^{2}\sqrt{x^{2}+1} |
(2013•Guangdong) Let the integer $n \geq 4$, and the set $X = \{1, 2, 3, \ldots, n\}$. Define the set $S = \{(x, y, z) | x, y, z \in X, \text{ and exactly one of the conditions } x < y < z, y < z < x, z < x < y \text{ holds}\}$. If $(x, y, z)$ and $(z, w, x)$ are both in $S$, then the correct option is ( )
A: $(y, z,... | \text{B} |
Let $n\ge 2$ be a positive integer. Find the number of functions $f:\{1,2,\ldots ,n\}\rightarrow\{1,2,3,4,5 \}$ which have the following property: $|f(k+1)-f(k)|\ge 3$, for any $k=1,2,\ldots n-1$.
[i]Vasile Pop[/i] | x_n = x_{n-1} + x_{n-2} |
Find the maximum number of Permutation of set {$1,2,3,...,2014$} such that for every 2 different number $a$ and $b$ in this set at last in one of the permutation
$b$ comes exactly after $a$ | 1007 |
5. In the country of Lemonia, coins of denominations $3^{n}, 3^{n-1} \cdot 4, 3^{n-2} \cdot 4^{2}, 3^{n-3} \cdot 4^{3}, \ldots, 3 \cdot 4^{n-1}, 4^{n}$ piastres are in circulation, where $n-$ is a natural number. A resident of the country went to the bank without any cash on hand. What is the largest amount that the ba... | 2\cdot4^{n+1}-3^{n+2} |
G2.2 If $R$ is the remainder of $1^{6}+2^{6}+3^{6}+4^{6}+5^{6}+6^{6}$ divided by 7 , find the value of $R$. | 6 |
Let the constant $a>0$, and the expansion of $(x+ \frac {a}{ \sqrt {x}})^9$ includes a term with $x^6$ whose coefficient is 4. Find the value of $\lim_{n→∞}(a+a^2+…+a^n)$. | \frac{1}{2} |
Example 11 For real number $x$, find the maximum value of the function $f(x)=$ $\sqrt{8 x-x^{2}}-\sqrt{14 x-x^{2}-48}$. | 2\sqrt{3} |
The trunk of the sky-high tree branches into two at a height of one foot. In the following, we consider the part between two branches as a branch, on which there is no further branching. Every branch of the sky-high tree is straight and ends one foot higher than its end closer to the ground. We consider the children of... | \binom{2n+2}{n+1}-\binom{2n+2}{n} |
11.4. Find all natural numbers $x$ such that the product of all digits in the decimal representation of $x$ equals $x^{2}-10 x-22$ | 12 |
In a certain place, four people, $A$, $B$, $C$, and $D$, successively contracted the novel coronavirus, with only $A$ having visited an epidemic area.
1. If the probabilities of $B$, $C$, and $D$ being infected by $A$ are $\frac{1}{2}$ each, what is the probability that exactly one of $B$, $C$, and $D$ is infected with... | \frac{11}{6} |
Observe the following equations:
$$\frac {1}{2} + \frac { \sqrt {3}}{2}i = \cos \frac {\pi}{3} + i\sin \frac {\pi}{3}$$
$$(\frac {1}{2} + \frac { \sqrt {3}}{2}i)^2 = \cos \frac {2\pi}{3} + i\sin \frac {2\pi}{3}$$
$$(\frac {1}{2} + \frac { \sqrt {3}}{2}i)^3 = \cos\pi + i\sin\pi$$
$$(\frac {1}{2} + \frac { \sqrt ... | \cos \frac {n\pi}{3} + i\sin \frac {n\pi}{3} |
Given the function $f(x)=\cos 2x+a\sin x+b$ where $a \lt 0$.
$(1)$ If the maximum value of $f(x)$ is $\frac{9}{8}$ and the minimum value is $-2$ when $x\in R$, find the values of real numbers $a$ and $b$.
$(2)$ If $a=-2$ and $b=1$, and the function $g(x)=m\sin x+2m$ where $x\in[\frac{\pi}{6},\frac{2\pi}{3}]$, and $... | (-\infty, -\frac{2}{3}) |
Determine all positive integers $a,b,c$ such that $ab + ac + bc$ is a prime number and
$$\frac{a+b}{a+c}=\frac{b+c}{b+a}.$$ | (1, 1, 1) |
9. As shown in the figure, in the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 2, $E$ is the midpoint of $B C$, and point $P$ moves along the line segment $D_{1} E$. Then, the minimum distance from point $P$ to the line $C C_{1}$ is $\qquad$. | \frac{2\sqrt{5}}{5} |
In the Cartesian coordinate system $xOy$, there is a line $l$ with slope $1$ passing through a fixed point $(-2,-4)$. A polar coordinate system is established with $O$ as the pole and the non-negative half of the $x$-axis as the polar axis. The curve $C$ is given by the polar equation $\rho\sin^2\theta - 4\cos\theta=0$... | 12\sqrt{2} |
Given the sample $x\_1, x\_2, \ldots, x\_n$ with a standard deviation of $s = 8.5$, and another sample $3x\_1 + 5, 3x\_2 + 5, \ldots, 3x\_n + 5$ with a standard deviation of $s' = \_\_\_\_\_\_\_\_.$ | 25.5 |
8.044. $\sqrt{3} \sin 2 x+\cos 5 x-\cos 9 x=0$.
8.044. $\sqrt{3} \sin 2 x+\cos 5 x-\cos 9 x=0$. | x_{1}=\frac{\pin}{2};x_{2}=(-1)^{k+1}\frac{\pi}{21}+\frac{\pik}{7},n,k\inZ |
3. After teacher Mary Ivanovna moved Vovochka from the first row to the second, Vanechka from the second row to the third, and Mashenka from the third row to the first, the average age of students sitting in the first row increased by one week, those sitting in the second row increased by two weeks, and those sitting i... | 9 |
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