problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Determine all pairs $(m, n)$ of positive integers such that $2^{m}-1=3^{n}$ | (2,1) |
Given the function $f(x) = 2\sin \omega x\cos \omega x + 2 \sqrt {3}\sin^2 \omega x - \sqrt {3}$ ($\omega > 0$) has the smallest positive period of $\pi$.
(Ⅰ) Determine the intervals of increase for the function $f(x)$.
(Ⅱ) Translate the graph of the function $f(x)$ to the left by $\dfrac{\pi}{6}$ units and then upward... | \dfrac{59\pi}{12} |
6. In an accident, a maternity hospital lost all the labels of 8 babies born on that day. Without any markings, the probability that exactly 4 families take the right babies is $\qquad$ . | \frac{1}{64} |
1.8 Let $a$ be a real number not equal to 0. In which of the following cases does
$$
\frac{\frac{a}{a+y}+\frac{y}{a-y}}{\frac{y}{a+y}-\frac{a}{a-y}}=-1 .
$$
(A) All real numbers except two real values of $y$.
(B) Only two real values of $y$.
(C) All real values of $y$.
(D) 36.5 .
(E) 36 .
(14th American High School Mat... | A |
31. What is the remainder when the number $2^{99}$ is divided by 7?
95 More problems! | 1 |
11. Given the sequence $\left\{a_{n}\right\}$ satisfies:
$$
\begin{array}{l}
a_{1}=\frac{9}{4}, \\
2 a_{n+1} a_{n}-7 a_{n+1}-3 a_{n}+12=0(n \geqslant 1) .
\end{array}
$$
(1) Let $c_{n}=a_{n}-2$, find the general term formula for the sequence $\left\{c_{n}\right\}$;
(2) Let $[x]$ denote the greatest integer not exceedin... | 45 |
$$\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\dots+\frac{1}{2014\cdot2015}=\frac{m}{n},$$ where $\frac{m}{n}$ is irreducible.
a) Find $m+n.$
b) Find the remainder of division of $(m+3)^{1444}$ to $n{}$. | 16 |
## Problem Statement
Calculate the definite integral:
$$
\int_{\sqrt{2}}^{2} \frac{d x}{x \sqrt{x^{2}-1}}
$$ | \frac{\pi}{12} |
3. Real numbers $x, y, z$ satisfy $x^{2}+y^{2}-x+y=1$. Then the range of the function
$$
f(x, y, z)=(x+1) \sin z+(y-1) \cos z
$$
is . $\qquad$ | \left[-\frac{3 \sqrt{2}+\sqrt{6}}{2}, \frac{3 \sqrt{2}+\sqrt{6}}{2}\right] |
8. (10 points) Two small rulers form a set of rulers, and the small rulers can slide along the large ruler. Each unit on the large ruler is marked with a natural number. The first small ruler divides 11 units on the large ruler into 10, and the second small ruler divides 9 units on the large ruler into 10. The starting... | 7 |
Players A and B each take two dice with uniform density to play a game, with the following rules: if the sum of the points rolled is a multiple of $3$, the original roller continues to roll; if the sum of the points rolled is not a multiple of $3$, the turn passes to the other player. It is stipulated that player A rol... | \frac{14}{27} |
5. Misha painted all integers in several colors such that numbers whose difference is a prime number are painted in different colors. What is the smallest number of colors that Misha could have used? Justify your answer. | 4 |
The sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ is $S_n$. Given that $a_3=3$ and $a_{10}=10$, the value of $S_7$ is ( )
A: 30
B: 29
C: 28
D: 27 | \text{C} |
6. Let $a$, $b$, $c$ be non-zero real numbers, and satisfy $a^{2}-b^{2}=b c$ and $b^{2}-c^{2}=c a$.
Prove: $a^{2}-c^{2}=a b$. | a^{2}-c^{2}=a b |
4. If the equations $x^{2}+2 a x+b^{2}=0$ and $x^{2}+2 c x-$ $b^{2}=0$ have a common root, and $a, b, c$ are exactly the three sides of a triangle, then this triangle is ( ).
(A) Acute triangle
(B) Obtuse triangle
(C) Right triangle with $a$ as the hypotenuse
(D) Right triangle with $c$ as the hypotenuse | C |
If the line $mx+4y-2=0$ is perpendicular to the line $2x-5y+n=0$, and the foot of the perpendicular from the intersection point is $(1, p)$, find the value of $m-n+p$. | 20 |
2A. Let $A B C D$ be a cyclic quadrilateral such that: $\overline{A B}=\overline{B C}=\sqrt{3}$, $\overline{C D}=2$ and $\measuredangle B A D=30^{\circ}$. Calculate the area of the quadrilateral $A B C D$. | \frac{7\sqrt{3}}{4} |
2. The numbers 1 through 25 are each colored blue or red. Determine all possible colorings that satisfy the following rules:
- The number 5 is red.
- If the numbers $x$ and $y$ have different colors and $x+y \leqslant 25$, then $x+y$ is blue.
- If the numbers $x$ and $y$ have different colors and $x \cdot y \leqslant ... | 3 |
Given $\{a_{n}\}$ is a geometric sequence, which of the following conclusions is correct?
A: If $a_{2}=a_{4}$, then $a_{2}=a_{3}$
B: If $a_{3} \gt a_{1}$, then $a_{4} \gt a_{2}$
C: $\frac{{a}_{2}+{a}_{4}}{2}≥{a}_{3}$
D: $\frac{{a}_{2}^{2}+{a}_{4}^{2}}{2}≥{a}_{3}^{2}$ | D |
# Task 5. 20 points
A beginner economist-cryptographer received a cryptogram from the ruler, which contained another secret decree on the introduction of a commodity tax on a certain market. The cryptogram specified the amount of tax revenue to be collected. It was also emphasized that a larger amount of tax revenue c... | 8640 |
23. Form the equation of the line passing through two points $A\left(x_{1} ; y_{1}\right)$ and $B\left(x_{2} ; y_{2}\right)$ | (x-x_{1})(y_{2}-y_{1})=(y-y_{1})(x_{2}-x_{1}) |
In how many different ways can one choose 5 numbers from $\{1,2, \ldots, 18\}$, such that no two of these numbers are consecutive? | 2002 |
Three. (This question is worth 25 points) $a$ is a real number greater than zero. It is known that there exists a unique real number $k$ such that the quadratic equation $x^{2}+\left(k^{2}+a k\right) x+1999+k^{2}+a k=0$ has two roots that are both prime numbers. Find the value of $a$. | 2 \sqrt{502} |
Folklore
The function $f(x)$ is defined for all $x$, except 1, and satisfies the equation: $(x-1) f\left(\frac{x+1}{x-1}\right)=x+f(x)$. Find $f(-1)$. | -1 |
The maximum value of the function $f(x)=\begin{cases} \frac{1}{x}, & x\geqslant 1 \\ -x^{2}+2, & x < 1 \end{cases}$ is ______. | 2 |
Using the numbers from 1 to 22 exactly once each, Antoine writes 11 fractions: for example, he can write the fractions $\frac{10}{2}, \frac{4}{3}, \frac{15}{5}, \frac{7}{6}, \frac{8}{9}, \frac{11}{19}, \frac{12}{14}, \frac{13}{17}, \frac{22}{21}, \frac{18}{16}$, and $\frac{20}{1}$.
Antoine wishes to have as many fract... | 10 |
Given the line $l_{1}$: $\begin{cases} x=1-2t \\ y=2+kt \end{cases}(t\text{ is a parameter})$ and the line $l_{2}$: $\begin{cases} x=s \\ y=1-2s \end{cases}(s\text{ is a parameter})$, find the value of $k$ such that $l_{1}$ and $l_{2}$ are perpendicular. | k = -1 |
Calculate:$(1)2\sin60°-3\tan30°-{(-\frac{1}{3})}^{0}+(-1)^{2023}$;$(2)|1-\sqrt{2}|- \sqrt{12}+{(\frac{1}{3})}^{-1}-2\cos45°$. | 2-2\sqrt{3} |
## Task 14/90
Given an equilateral arc triangle, whose arcs touch each other at the corners. What is its area $A$ if the arc radius $r=1$ LE? | \sqrt{3}-\frac{\pi}{2} |
Xiaoqiang obtained the following conclusions when studying the graph and properties of the power function $y=x^a(a=1,2,3,\frac{1}{2},-1)$. Which of the following statements is correct?
A: The graph of the power function must pass through the fixed points $(0,0)$ and $(1,1)$.
B: The graph of the power function cannot ... | \text{B} |
What is the greatest integer $k$ which makes the statement "When we take any $6$ subsets with $5$ elements of the set $\{1,2,\dots, 9\}$, there exist $k$ of them having at least one common element." true?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ 5
$ | 4 |
## Problem Statement
Based on the definition of the derivative, find $f^{\prime}(0)$:
$$
f(x)=\left\{\begin{array}{c}
1-\cos \left(x \sin \frac{1}{x}\right), x \neq 0 \\
0, x=0
\end{array}\right.
$$ | 0 |
No matter what real number \\(k\\) is, the line \\((2k-1)x-(k+3)y-(k-11)=0\\) always passes through a fixed point. The coordinates of this fixed point are \_\_\_\_. | (2,3) |
Given an arithmetic sequence $\{a_n\}$, where $a_7 + a_9 = 16$ and $a_4 = 1$, find the value of $a_{12}$.
A: 15
B: 30
C: 31
D: 64 | A |
[Systems of nonlinear algebraic equations]
Solve the system of equations:
$$
\begin{aligned}
& 1 / x=y+z \\
& 1 / y=z+x \\
& 1 / z=x+y
\end{aligned}
$$ | (\frac{\sqrt{2}}{2};\frac{\sqrt{2}}{2};\frac{\sqrt{2}}{2});(-\frac{\sqrt{2}}{2};-\frac{\sqrt{2}}{2};-\frac{\sqrt{2}}{2}) |
Given the function $f(x)=(x^{3}-2x)e^{x}$, then the value of $\lim_{\Delta x \to 0} \frac{f(1+\Delta x)-f(1)}{\Delta x}$ is ( )
A: $-e$
B: $1$
C: $e$
D: $0$ | D |
Given that the opposite of $m$ is $2$, the absolute value of $n$ is $8$, and $m+n \gt 0$, find the value of $mn$. | -16 |
41. Under what condition do three lines, given by the equations: $A_{1} x+B_{1} y+C_{1}=0 ; \quad A_{2} x+B_{2} y+C_{2}=0 \quad$ and $\quad A_{3} x+B_{3} y+C_{3}=0$, pass through the same point? | |\begin{pmatrix}A_{1}&B_{1}&C_{1}\\A_{2}&B_{2}&C_{2}\\A_{3}&B_{3}&C_{3}\end{pmatrix}|=0 |
8.119. $\operatorname{tg}\left(\frac{5 \pi}{2}+x\right)-3 \operatorname{tg}^{2} x=(\cos 2 x-1) \cos ^{-2} x$.
8.119. $\tan\left(\frac{5 \pi}{2}+x\right)-3 \tan^{2} x=(\cos 2 x-1) \cos^{-2} x$. | \frac{\pi}{4}(4k-1),k\inZ |
Example 11 For non-negative integers $n, f(n)$ is determined by $f(0)=0, f(1)=1, f(n)$ $=f\left(\left[\frac{1}{2} n\right]\right)+n-2\left[\frac{1}{2} n\right]$, find the maximum value of $f(n)$ when $0 \leqslant n \leqslant 1997$. | 10 |
Three, (25 points) Given that $a, b, c$ are positive integers, and the parabola $y=$ $a x^{2}+b x+c$ intersects the $x$-axis at two distinct points $A, B$. If the distances from $A, B$ to the origin are both less than 1, find the minimum value of $a+b+c$.
Translate the above text into English, please retain the origin... | 11 |
Given vectors $\overrightarrow{a}=(2,-1)$ and $\overrightarrow{b}=(-1,2)$, find the value of $({2\overrightarrow{a}+\overrightarrow{b}})•\overrightarrow{a}$. | 6 |
10.5. Let $k$ be a positive integer. An infinite strictly increasing sequence of positive integers $\left\{a_{n}\right\}$ satisfies: for any positive integer $n$, we have $a_{n+1}-a_{n} \leqslant k$, and $a_{n}$ is a multiple of 1005 or 1006, but not a multiple of 97. Find the smallest possible value of $k$. | 2010 |
Given that $\{a_n\}$ is a geometric sequence with a common ratio not equal to 1, and the sum of its first $n$ terms is $S_n$. It is also given that $a_5$, $a_3$, $a_4$ form an arithmetic sequence.
(1) Find the common ratio of the sequence $\{a_n\}$.
(2) Prove that for any $k \in \mathbb{N}^{+}$, $S_{k+2}$, $S_k$, $... | -2 |
1. Let $m, n \in \mathbf{N}_{+}$, and $m>n$, sets $A, B, C$ satisfy
$$
\begin{array}{l}
A=\{1,2, \cdots, m\}, B=\{1,2, \cdots, n\}, \\
C \subseteq A, B \cap C \neq \varnothing .
\end{array}
$$
Then the number of sets $C$ that meet the conditions is $\qquad$ | 2^{m-n}\left(2^{n}-1\right) |
## Task 1.
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$,
$$
f\left(x^{2}+f(y)\right)=\left(f(x)+y^{2}\right)^{2}
$$ | f(x)=x^{2} |
Given the curve $$\begin{cases} \overset{x=2-t\sin30 ^\circ }{y=-1+t\sin30 ^\circ }\end{cases}$$ (where $t$ is a parameter) intersects with the curve $x^2+y^2=8$ at points B and C, the value of $|BC|$ is ( )
A: $2 \sqrt {7}$
B: $\sqrt {60}$
C: $7 \sqrt {2}$
D: $\sqrt {30}$ | D |
Given that $a$, $b$, $c$, and $d$ are all real numbers, and $a \gt b \gt 0 \gt c \gt d$, which of the following inequalities is true?
A: $a^{2} \gt c^{2}$
B: $\ln \left(ab\right) \gt \ln \left(cd\right)$
C: $\frac{c}{a} \gt \frac{d}{b}$
D: $2^{-c} \gt 2^{-d}$ | C |
Given the complex number $z$ satisfies $zi=-2+i$ ($i$ is the imaginary unit), the point corresponding to the complex number $z$ in the complex plane is located in ( )
A: the first quadrant
B: the second quadrant
C: the third quadrant
D: the fourth quadrant | A |
## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0}\left(2-e^{x^{2}}\right)^{1 / \ln \left(1+\operatorname{tg}^{2}\left(\frac{\pi x}{3}\right)\right)}$ | e^{-\frac{9}{\pi^{2}}} |
Let's divide every natural number with at least two digits by the sum of its digits! When will the quotient be the largest, and when will it be the smallest? | 1.9 |
Let $m \subset \alpha$, $n \subset \beta$, and $\alpha \perp \beta$. Then, "m is perpendicular to $\beta$" is a condition for "m is perpendicular to n" that is ( )
A: Sufficient but not necessary
B: Necessary but not sufficient
C: Necessary and sufficient
D: Neither sufficient nor necessary | \text{A} |
Given vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(-1,m)$, if $\overrightarrow{a} \perp \overrightarrow{b}$, then $m=$ ______. | \dfrac{1}{2} |
From a bag containing 2 red balls and 2 black balls, two balls are drawn. The mutually exclusive but not opposite events are ( )
A: At least one black ball and at least one red ball
B: At least one black ball and both are black balls
C: At least one black ball and both are red balls
D: Exactly one black ball and exactl... | \text{D} |
The monotonically decreasing interval of the function $y=\frac{1}{2}x^2-\ln x$ is ()
A: $(-1,1)$
B: $(0,1)$
C: $\left[1,+\infty \right)$
D: $\left[0,+\infty \right)$ | B |
4. As shown in Figure 1, in the tetrahedron $D-ABC$, it is known that $DA \perp$ plane $ABC$, and $\triangle ABC$ is an equilateral triangle with a side length of 2. Then, when the tangent value of the dihedral angle $A-BD-C$ is 2, $V_{D-ABC}=$ $\qquad$ | 2 |
A piece of clothing is marked at 132 yuan. If it is sold at a 10% discount, it still yields a 10% profit. What is the cost price of this piece of clothing?
A: 106 yuan
B: 105 yuan
C: 118 yuan
D: 108 yuan | \text{D} |
Simplify $\dfrac{ \sqrt{1-2\sin {40}^{\circ}\cos {40}^{\circ}}}{\cos {40}^{\circ}- \sqrt{1-{\sin }^{2}{50}^{\circ}}} =$\_\_\_\_\_\_\_\_． | 1 |
Task 15. (8 points)
Announcement: "Have free time and want to earn money? Write now and earn up to 2500 rubles a day working as a courier with the service 'Food.There-Here!'. Delivery of food from stores, cafes, and restaurants.
How much will a school student working as a courier with the service 'Food.There-Here!' e... | 17400 |
In the expansion of $(1-x)(1+x)^{10}$, the coefficient of the term containing $x^5$ is __________. | 42 |
5. Given is a triangle $\triangle A B C$ such that $|\angle B A C|+|\angle C B A|=135^{\circ}$. Point $N$ is the foot of the altitude from vertex $A$ such that $|N C|=6 \cdot|N B|$. Calculate the length of side $\overline{B C}$ if the area of triangle $\triangle A B C$ is $3024 \mathrm{~mm}^{2}$.
## COUNTY COMPETITION... | 84 |
3. A program for each four-digit number prints the product of its digits. Which numbers will be printed by the program exactly once?
## SOLUTION: | 1,625,2401,4096,6561 |
Given the curves $C_{1}$: $y=\sin x$, $C_{2}$: $y=\sin({2x+\frac{{2π}}{3}})$, which of the following conclusions is correct?
A: Translate curve $C_{1}$ to the left by $\frac{{2π}}{3}$ units, then stretch the horizontal coordinates of each point on the resulting curve by a factor of $2$, while keeping the vertical coor... | \text{B} |
How many positive integers, less than 2011, are multiples of 3 or 4, but not of 5? | 804 |
11. A. As shown in Figure 4, the parabola $C: y=a x^{2}+b x-3$, with vertex $E$, intersects the $x$-axis at points $A$ and $B$, and the $y$-axis at point $C$. It is given that $O B=O C=3 O A$. The line $y=-\frac{1}{3} x+1$ intersects the $y$-axis at point $D$. Find $\angle D B C-\angle C B E$. | 45^{\circ} |
16. Let $f(x)$ be a function defined on $\mathbf{R}$, satisfying
$$
\left|f(x)+\cos ^{2} x\right| \leqslant \frac{3}{4},\left|f(x)-\sin ^{2} x\right| \leqslant \frac{1}{4} \text {. }
$$
Then the function $f(x)=$ . $\qquad$ | \sin ^{2} x-\frac{1}{4} |
Described quadrilaterals Area of a trapezoid
The bases of an isosceles trapezoid circumscribed around a circle are 2 and 18. Find the area of the trapezoid.
# | 60 |
Let's divide 17 into two parts such that $3/4$ of one part is $a$ greater than $5/6$ of the other part, where $a>0$. What are the limits within which the value of $a$ can vary to ensure that the problem is always solvable? | <12\frac{3}{4} |
Given $f(x)=x^{2}+bx+c (b,c \in \mathbb{R}, b < 0)$.
(1) If the domain of $f(x)$ is $[0,1]$ and its range is also $[0,1]$, find the values of $b$ and $c$.
(2) If $b=-2$, find the range of the real number $c$ such that the function $g(x)=\frac{f(x)}{x}$ is always greater than $c$ for any $x \in [3,5]$. | c < \frac{3}{2} |
7. Draw a line $l$ through the point $P(1,1)$ such that the midpoint of the chord intercepted by the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$ is exactly $P$. Then the equation of the line $l$ is $\qquad$ . | 4 x+9 y=13 |
## Problem Statement
Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
$M_{1}(5 ; 2 ; 0)$
$M_{2}(2 ; 5 ; 0)$
$M_{3}(1 ; 2 ; 4)$
$M_{0}(-3 ;-6 ;-8)$ | 8\sqrt{3} |
(7) $\cos \left(\sqrt{1-\sqrt{x^{2}+5 x+7}}+\sqrt{x^{2}+5 x+6}\right)=$ | 1 |
The equations $ 2x \plus{} 7 \equal{} 3$ and $ bx\minus{}10 \equal{} \minus{}\!2$ have the same solution for $ x$. What is the value of $ b$?
$ \textbf{(A)}\minus{}\!8 \qquad
\textbf{(B)}\minus{}\!4 \qquad
\textbf{(C)}\minus{}\!2 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 8$ | -4 |
$A$ and $B$ together can do a job in $2$ days; $B$ and $C$ can do it in four days; and $A$ and $C$ in $2\frac{2}{5}$ days.
The number of days required for A to do the job alone is:
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 2.8$ | \textbf{(B)}\3 |
Problem 4. For an infinite sequence of numbers $x_{1}, x_{2}, x_{3}, \ldots$, for all natural $n \geq 4$, the relation $x_{n}=x_{n-1} \cdot x_{n-3}$ holds. It is known that $x_{1}=1, x_{2}=1, x_{3}=-1$. Find $x_{2022}$. | 1 |
A regular hexagonal prism has two body diagonals of lengths 12 and 13. What is the volume of the prism? | 539.53 |
[ Tournaments and tournament tables ] [ Problems on percentages and ratios ]
In a volleyball tournament, which was held in a round-robin format, $20 \%$ of all teams did not win a single game. How many teams were there
# | 5 |
Given sets $A=\{(x,y)\mid (x-1)^{2}+(y-2)^{2}\leqslant \frac{5}{4}\}$ and $B=\{(x,y)\mid |x-1|+2|y-2|\leqslant a\}$, find the range of values for $a$ such that $A \subseteq B$. | a \geqslant \frac{5}{2} |
The ratio of the surface area of a cube to the surface area of its inscribed sphere. | \frac{6}{\pi} |
4. 42 Find the positive root of the equation $x=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+x}}}}$, and prove that there is only one positive root. | 2 |
Given a complex number $z$ satisfying $\left(i-1\right)z=2i$, then ( )
A: $|z|=\sqrt{2}$
B: The imaginary part of $z$ is $-i$
C: The conjugate of $z$ is $\overline{z}=-1+i$
D: $z$ is a root of the equation $x^{2}-2x+2=0$ | AD |
11. Given $O$ is the circumcenter of $\triangle A B C$, $|A B|=2$, $|A C|=1, \angle B A C=\frac{2 \pi}{3}$. Let $\boldsymbol{A B}=\boldsymbol{a}, \boldsymbol{A} \boldsymbol{C}=\boldsymbol{b}$. If $\boldsymbol{A} \boldsymbol{O}=\lambda_{1} \boldsymbol{a}+\lambda_{2} \boldsymbol{b}$, then $\lambda_{1}+\lambda_{2}=$ | \frac{13}{6} |
3.1. The friends who came to visit Gavrila occupied all the three-legged stools and four-legged chairs in the room, but there was no place left for Gavrila himself. Gavrila counted that there were 45 legs in the room, including the "legs" of the stools and chairs, the legs of the visiting guests (two for each!), and Ga... | 9 |
If the terminal side of angle $\theta$ passes through point P(3, -4), then $\sin(\theta-\pi)=$ \_\_\_\_\_\_. | \frac{4}{5} |
14. Find the minimum and maximum values of the function $y=\frac{x-2}{x^{2}-4 x+5}$ on $x \in\left[\frac{5}{2}, 3\right]$. | \frac{1}{2};\frac{2}{5} |
The coefficient of $x^4$ in the expansion of the binomial $(x+1)^{10}$ is ______. | 210 |
If $x_0=x_1=1$, and for $n\geq1$
$x_{n+1}=\frac{x_n^2}{x_{n-1}+2x_n}$,
find a formula for $x_n$ as a function of $n$. | x_n = \frac{1}{(2n-1)!!} |
A manufacturer built a machine which will address $500$ envelopes in $8$ minutes. He wishes to build another machine so that when both are operating together they will address $500$ envelopes in $2$ minutes. The equation used to find how many minutes $x$ it would require the second machine to address $500$ envelopes al... | \textbf{(B)}\\frac{1}{8}+\frac{1}{x}=\frac{1}{2} |
Let there be a figure with $9$ disks and $11$ edges, as shown below.
We will write a real number in each and every disk. Then, for each edge, we will write the square of the difference between the two real numbers written in the two disks that the edge connects. We must write $0$ in disk $A$, and $1$ in disk $I$. Find... | \frac{20}{33} |
1. We define $a \star b=a \times a-b \times b$, then $3 \star 2+4 \star 3+5 \star 4+\cdots+20 \star 19=$ | 396 |
Given the inequality about $x$: $|2x-m| \leqslant 1$ has exactly one integer solution, which is $2$.
- (I) Find the value of the integer $m$;
- (II) Given $a$, $b$, $c \in \mathbb{R}$, if $4a^{4}+4b^{4}+4c^{4}=m$, find the maximum value of $a^{2}+b^{2}+c^{2}$. | \sqrt{3} |
Calculate the value of $$\sqrt[3]{(2-\pi )^{3}} + \sqrt {(3-\pi)^{2}}$$
Options:
A: 5
B: -1
C: 2π-5
D: 5-2π | \text{B} |
If acute angle $α$ satisfies $\sin \left(α+ \frac {π}{2}\right)= \frac {3}{5}$, then $\sin α=$ ()
A: $ \frac {2}{5}$
B: $ \frac {3}{5}$
C: $ \frac {3}{4}$
D: $ \frac {4}{5}$ | D |
5 Given the sequence $\left\{x_{n}\right\}, x_{1}=1$, E $x_{n+1}=\frac{\sqrt{3} x_{n}+1}{\sqrt{3}-x_{n}}$, then $\sum_{n=1}^{2008} x_{n}=(\quad)$.
(A) 0
(B) -1
(C) $2+\sqrt{3}$
(D) $-2+\sqrt{3}$ | 0 |
For the one-variable quadratic equation $x^{2}-2x+a=0$ with two distinct real roots, the range of $a$ is ______. | a < 1 |
A school requires each student to choose 5 out of 8 courses, where at most one of the two courses, A and B, can be chosen. How many different course selection schemes are there? (Answer with a number.) | 36 |
10.5. In triangle $A B C$, points $M$ and $N$ are the midpoints of sides $A C$ and $B C$ respectively. It is known that the point of intersection of the medians of triangle $A M N$ is the point of intersection of the altitudes of triangle $A B C$. Find the angle $A B C$. | 45 |
329. Construct an example of a polynomial $P(x)$ of the third degree with integer coefficients and the leading coefficient equal to 1, which is divisible by 3 for all integer values of $x$. | P(x)=x(x+1)(x+2) |
To satisfy the equation $\frac{a+b}{a}=\frac{b}{a+b}$, $a$ and $b$ must be:
$\textbf{(A)}\ \text{both rational}\qquad\textbf{(B)}\ \text{both real but not rational}\qquad\textbf{(C)}\ \text{both not real}\qquad$
$\textbf{(D)}\ \text{one real, one not real}\qquad\textbf{(E)}\ \text{one real, one not real or both not rea... | \textbf{(E)} |
5. The graph of the function $y=\sqrt{3} \cos x-\sin x$ is shifted to the left by $m$ units, and the resulting graph is symmetric about the $y$-axis. Then the smallest positive value of $m$ is ( ).
A. $\frac{\pi}{6}$
B. $\frac{\pi}{3}$
C. $\frac{2 \pi}{3}$
D. $\frac{5 \pi}{6}$ | \frac{5\pi}{6} |
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