problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
91. The base of the right prism $A B C D A_{1} B_{1} C_{1} D_{1}$ is an isosceles trapezoid $A B C D$, in which $A D$ is parallel to $B C, |A D| / |B C| = n, n > 1$. Planes are drawn through the edges $A A_{1}$ and $B C$ parallel to the diagonal $B_{1} D$; planes are drawn through the edges $D D_{1}$ and $B_{1} C_{1}$ ... | \frac{(5n+3)^{3}}{12(n+1)^{3}} |
4. $y=\sin \left(\frac{\pi}{3}+x\right)-\sin 3 x$ 的最大值为 $\qquad$
The maximum value of $y=\sin \left(\frac{\pi}{3}+x\right)-\sin 3 x$ is $\qquad$ | \frac{8\sqrt{3}}{9} |
1. Given the tetrahedron $S-ABC$ with the projection $H$ of vertex $S$ on the base being the orthocenter of $\triangle ABC$, $BC=2$, $SB=SC$, and the dihedral angle between the side face $SBC$ and the base is $60^{\circ}$, find the volume of the tetrahedron. | \frac{\sqrt{3}}{3} |
Given the parametric equations of curve $C_1$ as $\begin{cases} x = 1 + \frac{1}{2}t \\ y = \frac{\sqrt{3}}{2}t \end{cases}$ ($t$ is the parameter) and the polar coordinate equation of curve $C_2$ as $\rho^2 = \frac{12}{3 + \sin^2 \theta}$, establish a polar coordinate system with the origin $O$ of the rectangular coor... | \frac{4}{3} |
A forest farm has 30,000 saplings, including 4,000 pine saplings. To investigate the growth of the saplings, a stratified sampling method is used to draw a sample of 150 saplings. The number of pine saplings in the sample is ( )
A: 30
B: 25
C: 20
D: 15 | \text{C} |
[ Arithmetic operations. Numerical identities ] Joke problems
Calculate the product
$$
\left(100-1^{2}\right)\left(100-2^{2}\right)\left(100-3^{2}\right) \ldots\left(100-25^{2}\right)
$$
# | 0 |
7. Nestašni Darko, upon arriving at the amusement park, forgot about his savings. He spent 4 kuna on two scoops of ice cream, and then spent half of the remaining money on a ride in the bumper cars. Then he bought a chocolate banana for 2 kuna, and spent half of the remaining money on a ride on the carousel. After that... | 80 |
If $\vec{a} = (1, \lambda, 2)$ and $\vec{b} = (2, -1, 2)$, and the cosine of the angle between $\vec{a}$ and $\vec{b}$ is $\frac{8}{9}$, then $\lambda$ equals ( )
A: $-2$ or $\frac{2}{55}$
B: $2$ or $- \frac{2}{55}$
C: $2$
D: $-2$ | \text{A} |
Given the function $f(x)=\frac{1}{2}{x}^{2}-3x+2lnx$.
$(1)$ Find the equation of the tangent line to the curve $y=f\left(x\right)$ at the point $\left(1,f\left(1\right)\right)$.
$(2)$
$(ⅰ)$ If for any $x_{1}$, $x_{2}\in \left[1,3\right]$, $|f(x_{1})-f(x_{2})|\leqslant 2m-2$, find the range of real number $m$.
$... | {x_1}+{x_2} > \frac{7}{2} |
Given that $f(e^x) = x + 2$, find $f(1)$ and the analytical expression of $f(x)$. | f(x) = \ln{x} + 2 |
Given line $m$, line $n$, and planes $\alpha$ and $\beta$, which of the following propositions is correct?
(A) If $m \parallel \alpha$ and $n \parallel \beta$, then $\alpha \parallel \beta$.
(B) If $m \parallel \alpha$ and $m \parallel n$, then $n \parallel \alpha$.
(C) If $m \perp \alpha$ and $\alpha \perp \beta$, the... | D |
Which of the following sets of line segments $a$, $b$, and $c$ can form a right triangle?
A: $a=1$, $b=1$, $c=2$
B: $a=3$, $b=4$, $c=5$
C: $a=5$, $b=10$, $c=12$
D: $a=4$, $b=5$, $c=6$ | B |
In the geometric sequence $\\{a\_n\\}$, it is known that $a\_1=1$, $a\_k=243$, and $q=3$. Find the sum of the first $k$ terms of the sequence, denoted as $S\_k$. | 364 |
Let the moving line $x=m$ intersect with the graphs of the functions $f(x)=x^3$ and $g(x)=\ln x$ at points $M$ and $N$, respectively. The minimum value of $|MN|$ is __( ).
A: $ \dfrac{1}{3}(1+\ln 3)$
B: $ \dfrac{1}{3}\ln 3$
C: $1+\ln 3$
D: $\ln 3-1$ | \text{A} |
Given the conditions $a+acosC=\sqrt{3}csinA$, $\left(a+b+c\right)\left(a+b-c\right)=3ab$, $\left(a-b\right)\sin \left(B+C\right)+b\sin B=c\sin C$. Choose any one of these three conditions and complete the following question, then solve it. In triangle $\triangle ABC$, where the sides opposite angles $A$, $B$, and $C$ a... | 6 + 4\sqrt{2} |
Example 6 Given that $x, y, z$ are two non-negative rational numbers, and satisfy $3x+2y+z=5, x+y-z=2$. If $S=2x+y-z$, then what is the sum of the maximum and minimum values of $S$?
(1996, Tianjin City Junior High School Mathematics Competition) | 5 |
Two regular $2n$-sided pyramids $(n \geq 2)$ (with a regular polygon base and equal side edges), whose bases are congruent, are joined by placing their congruent bases in coincidence, such that a convex polyhedron composed entirely of triangular faces is formed. What is the maximum number of sides that a plane section ... | 2(n+1) |
In the arithmetic sequence $\{a_n\}$, if $a_4 + a_6 + a_8 + a_10 + a_12 = 120$, then the value of $3a_9 - a_{11}$ is ( )
A: 42
B: 45
C: 48
D: 51 | C |
4. The following is the time on the clock behind Pleasant Goat as seen in the mirror. The actual time at this moment is ( ).
A. 7:30
B. $6: 30$
C. 5:30
D. $4: 30$
E. 3:30 | 4:30 |
# Task 8. Number Systems in System Administration
Task text. In a workers' settlement of shift workers, a network failure occurred as a result of which one of the devices in the gas extraction complex began to return not the standard display of the device's IP address and subnet mask, but its binary representation. Th... | 178.176.230.128 |
The smallest positive integer in the set $A=\{x\in\mathbb{R} \,|\, |x-3|\geqslant 2\}$ is ______. | 1 |
Given $ab^2c^3d^4e^5<0$. Which of the following judgments is correct? ( )
A: $abcde<0$
B: $ab^2cd^4e<0$
C: $ab^2cde<0$
D: $abcd^4e<0$ | B |
9.5. For 2011 positive integers, write their $\mathrm{C}_{2011}^{2}$ pairwise sums on the blackboard. Question: Is it possible that exactly half of these $\mathrm{C}_{2011}^{2}$ numbers are multiples of 3, and at the same time, exactly $=$ one-third of them are congruent to 1 modulo 3? | a=b=670, c=671 |
The equation of the line that passes through the point (2, 0) and is tangent to the curve $$y= \frac {1}{x}$$ is ( )
A: x+4y+2=0
B: x+4y-2=0
C: x+y+2=0
D: x+y-2=0 | \text{D} |
20. There is a pentagon $A B C D E$. If the vertices $A$, $B$, $C$, $D$, $E$ are colored with one of the three colors: red, yellow, green, such that adjacent vertices are colored differently, then there are a total of different coloring methods. | 30 |
Given $m$ and $n$ are different lines, and $\alpha$ and $\beta$ are non-coincident planes, consider the following propositions:
1. If $\alpha \parallel \beta$, $m \subset \alpha$, $n \subset \beta$, then $m \parallel n$;
2. If $m \perp \alpha$, $n \perp \beta$, $m \parallel n$, then $\alpha \parallel \beta$;
3. If $\a... | 2, 3 |
The range of the inverse function of $y= \sqrt {1-2x}$ is ______. | (-\infty, \frac{1}{2}] |
9.026. $\log _{0.3}(3 x-8)>\log _{0.3}\left(x^{2}+4\right)$. | x\in(\frac{8}{3};\infty) |
In the triangular pyramid $P-ABC$, $PA \perp$ the base $ABC$, $AB=1$, $AC=2$, $\angle BAC=60^{\circ}$, the volume is $\frac{\sqrt{3}}{3}$, then the volume of the circumscribed sphere of the triangular pyramid is $\_\_\_\_\_\_\_\_\_\_.$ | \frac{8 \sqrt{2}}{3} \pi |
Find all ordered pairs $(a, b)$ of positive integers such that $a^2 + b^2 + 25 = 15ab$ and $a^2 + ab + b^2$ is prime. | (a, b) \in \{(1, 2), (2, 1)\} |
Problem 6. (Variant 2).
$C A K D$ is a square with a side length of 6. On side $C D$, a point $B(B D=2)$ is chosen, and on line $A D$, a point $E$ is chosen such that the perimeter of triangle $B E C$ is the smallest possible. Then, on line $D C$, a point $F$ is marked such that the perimeter of triangle $F E A$ is th... | 0.3\sqrt{34} |
Given $a\geqslant 0$, the function $f(x)=(-x^{2}+2ax)e^{x}$.
$(I)$ For what value of $x$ does $f(x)$ attain its maximum value? Prove your conclusion;
$(II)$ Suppose $f(x)$ is a monotonic function on $[-1,1]$, find the range of values for $a$;
$(III)$ Let $g(x)=(x-1)e^{2x}$, if $x\geqslant 1$, $f(x)\leqslant g(x)$... | 0\leqslant a\leqslant \dfrac {1}{2} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. If $\frac{c}{\sin B} + \frac{b}{\sin C} = 2a$ and $b = \sqrt{2}$, find the area of $\triangle ABC$. | 1 |
If $\cos (\alpha +\pi )=-\dfrac{2}{3}$, then $\sin (\alpha +\dfrac{3\pi }{2})=$ ()
A: $-\dfrac{2}{3}$
B: $\dfrac{2}{3}$
C: $-\dfrac{\sqrt{5}}{3}$
D: $\dfrac{\sqrt{5}}{3}$ | A |
Given that $a > 0$ and $a \neq 1$, among the following functions, the one that is both a monotonic function and an odd function in its domain is $(\ $ $\ $ $)$.
A: $y=\sin ax$
B: $y=\log _{a}x^{2}$
C: $y=a^{x}-a^{-x}$
D: $y=\tan ax$ | C |
Given the function $y=A\sin (\omega x+ \varphi )(A > 0, \omega > 0)$, the coordinates of a highest point on the graph is $(2,\sqrt{2})$, and the graph intersects the $x$-axis at point $(6,0)$ between this highest point and its adjacent lowest point. Determine the analytical expression of this function. | \sqrt{2}\sin(\frac{\pi}{8}x + \frac{\pi}{4}) |
The school bought 3 basketballs and 5 volleyballs. Each basketball costs 48 yuan, which is 18 yuan more than each volleyball. The total cost for both types of balls is \_\_\_\_\_\_ yuan. | 294 |
Let $\alpha$ and $\beta$ be two different planes, then a sufficient condition for $\alpha$ to be parallel to $\beta$ is:
A: There are infinitely many lines in $\alpha$ that are parallel to $\beta$.
B: $\alpha$ and $\beta$ are perpendicular to the same line.
C: $\alpha$ and $\beta$ are parallel to the same line.
D: ... | \text{B} |
Among the following expressions, the one that can be calculated using the formula for the difference of squares is:
A: $(-x-y)\left(x+y\right)$
B: $(2x+y)\left(y-2x\right)$
C: $(2x+y)\left(x-2y\right)$
D: $(-x+y)\left(x-y\right)$ | B |
Let $a_1\in (0,1)$ and $(a_n)_{n\ge 1}$ a sequence of real numbers defined by $a_{n+1}=a_n(1-a_n^2),\ (\forall)n\ge 1$. Evaluate $\lim_{n\to \infty} a_n\sqrt{n}$. | \frac{\sqrt{2}}{2} |
Example 5. If $x^{2}-1 \leqslant 0$, find the maximum and minimum values of the function $y=x^{2}-4 x$ +5. | y_{\text{minimum}}=2, y_{\text{maximum}}=10 |
In a theater troupe, there are 60 actors. Any two of them have at least once played in the same play. In each play, no more than 30 actors are involved.
What is the minimum number of plays the theater could have staged? | 6 |
28.3. On the sides $B C$ and $C D$ of the square $A B C D$, points $K$ and $M$ are chosen such that the perimeter of triangle $K M C$ is equal to twice the side length of the square $A B C D$. Find the angle $K A M$.
$$
(7-10 \text { grades })
$$ | 45 |
6.2. Masha and the Bear ate a basket of raspberries and 40 pies, starting and finishing at the same time. At first, Masha was eating raspberries, and the Bear was eating pies, then (at some point) they switched. The Bear ate both raspberries and pies 3 times faster than Masha. How many pies did Masha eat, if they ate t... | 4 |
Given the function $f(x)=2\cos^2x+\cos\left(\frac{\pi}{2}-2x\right)$, the smallest positive period of the function $f(x)$ is ______, and its range is ______. | [1- \sqrt{2},1+ \sqrt{2}] |
Given the function $f(x)= \frac {4^{x}-1}{4^{x}+1}$.
(1) Solve the inequality $f(x) < \frac {1}{3}$;
(2) Find the range of the function $f(x)$. | (-1, 1) |
2. Let $0 \leqslant x \leqslant 2$, the minimum value of the function $y=4^{x-\frac{1}{2}}-3 \cdot 2^{x}+5$ is
保留源文本的换行和格式，翻译结果如下：
2. Let $0 \leqslant x \leqslant 2$, the function $y=4^{x-\frac{1}{2}}-3 \cdot 2^{x}+5$'s minimum value is | \frac{1}{2} |
## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow-1} \frac{x^{3}+1}{\sin (x+1)}$ | 3 |
# Problem 7.3 (7 points)
A piece has fallen out of a dictionary, the first page of which is numbered 213, and the number of the last page is written with the same digits in some other order. How many pages are in the missing piece? | 100 |
Which of the following statements is correct regarding a prism? ( )
A: Only two faces are parallel.
B: All edges are parallel.
C: All faces are parallelograms.
D: The two base faces are parallel, and each lateral edge is also parallel to each other. | D |
Example 4 Find all $n \in \mathbf{N}^{*}$, such that the system of equations $\left\{\begin{array}{l}x_{1}+x_{2}+\cdots+x_{n}=9 \\ \frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots+\frac{1}{x_{n}}=1\end{array}\right.$ has positive solutions $x_{1}, x_{2}, \cdots, x_{n}$. For the obtained $n$, provide the corresponding solutions. | n=2:(\frac{9+3\sqrt{5}}{2},\frac{9-3\sqrt{5}}{2})or(\frac{9-3\sqrt{5}}{2},\frac{9+3\sqrt{5}}{2});n=3:(3,3,3) |
## Task B-2.8.
Two circles touch each other internally at point $F$. The diameter of one circle is $8 \mathrm{~cm}$, and the diameter of the other is half as large. From the boundary point $T$ of the diameter $\overline{T F}$ of the larger circle, we construct a tangent to the smaller circle. If point $E$ is the inter... | 6+\frac{2\sqrt{17}}{3}+\frac{16\sqrt{2}}{3}\mathrm{~} |
25) Which of the following numbers does NOT divide 100! (recall that 100! $=100 \cdot 99 \cdot$ $98 \cdots 3 \cdot 2 \cdot 1) ?$
(A) 1968
(B) 1988
(C) 1998
(D) 2008
(E) 2048 | D |
## Problem 3
Determine the sum of the first 30 terms of a geometric progression, knowing that the sum of the first two terms is equal to 8, and the difference between the second term and the first term is equal to 4. | 3^{30}-1 |
Given the function $f(x)=\begin{cases}
1+\log_{2}(2-x), & x < 1 \\
2^{x-1}, & x\geqslant 1
\end{cases}$, then $f(-6)+f(\log_{2}11)=$ _____. | \frac{19}{2} |
10. If $\sin \frac{\pi}{9}+\sin \frac{2 \pi}{9}+\cdots+\sin \frac{n \pi}{9}=\frac{1}{2} \tan \frac{4 \pi}{9}$, then the smallest positive integer $n$ is $\qquad$. | 4 |
\left.\begin{array}{l}{[\quad \text { Law of Sines }} \\ {\left[\begin{array}{l}\text { Angle between a tangent and a chord }]\end{array}\right]}\end{array}\right]
In triangle $ABC$, angle $\angle B$ is equal to $\frac{\pi}{6}$. A circle with a radius of 2 cm is drawn through points $A$ and $B$, tangent to line $AC$ a... | \sqrt{6} |
5. Six musicians gathered at a chamber music festival. . At each scheduled concert some of the musicians played while the others listened as members of the audience. What is the least number of such concerts which would need to be scheduled so that for every two musicians each must play for the other in some concert? | 4 |
Let $f(x)$ be a function defined on the set of integers. Given that $f(1) = 1$, $f(2) = 0$, and $f(-1) < 0$, and for any integers $x$ and $y$, $f(x+y) = f(x)f(1-y) + f(1-x)f(y)$. Which of the following statements are correct?
A: $f(0) = 0$
B: $f(x)$ is an odd function
C: $f(x)$ is symmetric about $x=2$
D: $f(1) + f... | ABD |
5. Let $a, b$ be real numbers, and the function $f(x) = ax + b$ satisfies: for any $x \in [0,1]$, we have $|f(x)| \leqslant 1$. Find
\[
S = (a+1)(b+1)
\]
the range of values.
(Yang Xiaoming, problem contributor) | S\in[-2,\frac{9}{4}] |
A pentagon has four interior angles each equal to $110^o$. Find the degree measure of the fifth interior angle. | 100^\circ |
Find all $n \in \mathbb{N}^{*}$ such that $n^{n}+1$ is a prime number less than $10^{18}$. | n=1,n=2,n=4 |
Xiaoli decides which subject among history, geography, or politics to review during tonight's self-study session based on the outcome of a mathematical game. The rules of the game are as follows: in the Cartesian coordinate system, starting from the origin $O$, and then ending at points $P_{1}(-1,0)$, $P_{2}(-1,1)$, $P... | \dfrac{3}{10} |
Problem 5.7. In a class, there are 31 students. Three of them have exactly three friends each, the next three have six each, the next three have nine each, ..., and the next three have thirty each. How many friends does the 31st student have? (Friendship between people is mutual.)
 . \\ \text { (A) } M>P>N \text { and } M>Q>N . \\ \text { (B) } N>P>M \text { and } N>Q>M .\end{array}$
$\begin{array}{l}\text { (C) } P>M>Q \text { and } P>N>Q \text {. } \... | A |
## Problem Statement
Calculate the definite integral:
$$
\int_{0}^{\frac{\pi}{2}} \frac{(1+\cos x) d x}{1+\sin x+\cos x}
$$ | \frac{1}{2}\cdot\ln2+\frac{\pi}{4} |
IMO 1985 Problem A1 A circle has center on the side AB of the cyclic quadrilateral ABCD. The other three sides are tangent to the circle. Prove that AD + BC = AB. | AD+BC=AB |
A way to check if a number is divisible by 7 is to subtract, from the number formed by the remaining digits after removing the units digit, twice the units digit, and check if this number is divisible by 7. For example, 336 is divisible by 7, because $33 - 2 \cdot 6 = 21$ is divisible by 7, but 418 is not because $41 -... | 13 |
3. The residents of the Immortal Island celebrate their birthdays every 3 years. Xiaohuaxian had her first birthday at the age of 1, and her second birthday at the age of 4. She receives a number of gifts equal to her age each birthday. By the time she finished celebrating her birthday this year, Xiaohuaxian had receiv... | 19 |
Read the following material and then answer the following questions: When simplifying algebraic expressions, sometimes we encounter expressions like $\frac{5}{\sqrt{3}},\frac{2}{\sqrt{3}+1}$, in fact, we can further simplify them as follows:
($1$) $\frac{5}{\sqrt{3}}=\frac{5×\sqrt{3}}{\sqrt{3}×\sqrt{3}}=\frac{5}{3}\s... | \sqrt{99}-1 |
Let $a = \left( \frac{7}{9} \right)^{-\frac{1}{4}}$, $b = \left( \frac{9}{7} \right)^{\frac{1}{5}}$, and $c = \log_{2}\frac{9}{7}$. Determine the order of $a$, $b$, and $c$.
A: $b < a < c$
B: $c < a < b$
C: $c < b < a$
D: $b < c < a$ | \text{C} |
\section*{Problem 3 - 031213}
Determine all real values of \(x_{1}, x_{2}, x_{3}\) that satisfy the equations
\[
\begin{aligned}
& x_{2}+x_{3}=p x_{1} \\
& x_{1}+x_{3}=p x_{2} \\
& x_{1}+x_{2}=p x_{3}
\end{aligned}
\]
and their dependence on the real number \(p\) (parameter)! | \begin{aligned}&Forp\neq-1,2:&(x_1,x_2,x_3)=(0,0,0)\\&Forp=2:&(x_1,x_2,x_3)=(x,x,x),wherexisanyreal\\&\text |
Given a four-digit natural number $N$, where each digit is not equal to $0$, and the sum of the thousands digit and hundreds digit is equal to the sum of the tens digit and units digit, we call this number a "sum symmetric number". Let $N'$ be the number obtained by swapping the thousands digit and hundreds digit, and ... | 3746, 4756, 6776, 5766, 7786, 8796 |
The sum of the intercepts of the line $x-2y+1=0$ on the two coordinate axes is ______. | -\frac{1}{2} |
## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 2} \frac{x^{3}-6 x^{2}+12 x-8}{x^{3}-3 x^{2}+4}$ | 0 |
Simplify $\frac{sin10°+sin20°cos30°}{cos10°-sin20°sin30°}$ to obtain ( ).
A: $\frac{\sqrt{3}}{3}$
B: $-\frac{\sqrt{3}}{3}$
C: $\sin 20^{\circ}$
D: $\sin 10^{\circ}$ | A |
A spinning turntable is rotating in a vertical plane with period $ 500 \, \text{ms} $. It has diameter 2 feet carries a ping-pong ball at the edge of its circumference. The ball is bolted on to the turntable but is released from its clutch at a moment in time when the ball makes a subtended angle of $\theta>0$ with the... | 12^\circ |
Brenda and Sally run in opposite directions on a circular track, starting at [diametrically](https://artofproblemsolving.com/wiki/index.php/Diameter) opposite points. They first meet after Brenda has run 100 meters. They next meet after Sally has run 150 meters past their first meeting point. Each girl runs at a con... | 350 |
Example 2. Form a differential equation for which the functions $y_{1}(x)=e^{x^{2}}, y_{2}(x)=e^{-x^{2}}$ form a fundamental system of solutions. | xy^{\\}-y^{\}-4x^{3}0 |
The opposite of $-2$ is
A: $2$
B: $-2$
C: $-\frac{1}{2}$
D: $\frac{1}{2}$ | \text{A} |
In a chocolate store, there are boxes with 8, 9, and 10 chocolates. Note that some quantities of chocolates cannot be bought exactly, such as, for example, 12 chocolates.
a) Find another quantity of chocolates that cannot be bought.
b) Verify that every number greater than 56 can be written in the form $8x + 9y$ with... | 31 |
Given vectors $a = (1, x - 1)$ and $b = (x + 1, 3)$, then "$x = 2$" is a(n) ( ) condition for "$a \parallel b$".
A: Sufficient but not necessary
B: Necessary but not sufficient
C: Necessary and sufficient
D: Neither sufficient nor necessary | \text{A} |
## Task 4
To become hard-boiled, 3 eggs need to cook for 5 minutes.
How long do you need to cook 6 eggs? | 5 |
The following is an example and solution process about polynomial operations, where $M$ and $N$ are two binomials in terms of $x$. A, B, and C respectively provide $M$, $N$, $P$, among which the correct one is ( ).
Example: First remove the parentheses, then combine like terms: $2(\underline{M})-3(\underline{N})$;
So... | D |
## Task 2 - 070622
Each of the two front wheels of a vehicle has a circumference of $210 \mathrm{~cm}$, and each of the two rear wheels has a circumference of $330 \mathrm{~cm}$.
Determine the shortest distance (in $\mathrm{m}$) that the vehicle must travel on a straight, flat road so that each of its wheels has comp... | 23.10\mathrm{~} |
In the sequence $\{a_n\}$, $a_{n+1} = ca_n$ (where $c$ is a non-zero constant), and the sum of the first $n$ terms is $S_n = 3^n + k$. The real number $k$ is ( )
A: 0
B: 1
C: -1
D: 2 | \text{C} |
5. Calculate: $: \frac{C_{11}^{11}}{1}+\frac{C_{11}^{1}}{2}+\frac{C_{11}^{2}}{3}+\cdots \cdots+\frac{C_{11}^{11}}{12}$ | \frac{1365}{4} |
Five monkeys are seated around a table. Their seats are labelled $P, Q, R, S$, and $T$, in clockwise order, as shown. The five monkeys are randomly numbered Monkey 1, Monkey 2, Monkey 3, Monkey 4, and Monkey 5. Monkey 1 remains in its seat. The remaining four monkeys then sit themselves in the remaining seats so that t... | \frac{3}{20} |
5. In a race, Lotar finished before Manfred, Victor finished after Jan, Manfred finished before Jan and Eddy finished before Victor. Who finished last of these five runners?
A Victor
B Manfred
C Lotar
D Jan
E Eddy | A |
15. A person's 5 trips to work (unit: $\mathrm{min}$) are $a, b, 8, 9, 10$. It is known that the average of this set of data is 9, and the variance is 2. Then the value of $|a-b|$ is $\qquad$. | 4 |
Find all pairs of integers $(x, y)$ for which $x^{2}+x y=y^{2}$. | (0,0) |
The graph of the function y = loga(x + 2) + 3 always passes through the fixed point $\boxed{\text{answer}}$. | (-1, 3) |
Given $i$ as the imaginary unit, and a complex number satisfies $(1+ \sqrt {3}i)z=1-i$, then $|\overline {z}|=$ （ ）
A: $\sqrt {2}$
B: $\frac {1}{2}$
C: $\frac { \sqrt {2}}{2}$
D: 2 | C |
Example 3. Calculate the value of the derivative of the implicit function $x y^{2}=4$ at the point $M(1,2)$. | -1 |
2. Place two equal-sized regular tetrahedra with one base overlapping, forming a triangular dipyramid. If a plane cuts through this triangular dipyramid, the cross-section can have at most ( ) sides.
A. $=$
B. Four
C. Five
D. Six | D |
2. The sum of the sum and the product of two positive integers is 2005, and one of them is a perfect square. Then the difference between the larger number and the smaller number is | 1001, 101 |
Given the function $f(x) = \begin{cases} 2^{x}, & x \leq 0 \\ f(x-1)-1, & x > 0 \end{cases}$, find the value of $f(\log_{2}9)$. | -\frac{55}{16} |
Example 1.1.4 Given 2005 points in the plane, where the distance between any two points is greater than $\sqrt{2}$, prove that there must be 223 points such that the distance between any two of them is not less than 2. | 223 |
If $3^{m}=5$ and $3^{n}=6$, then the value of $3^{m+n}$ is ______. | 30 |
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