problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
In the geometric sequence $\{a_n\}$, $a_n > 0$, and $a_3a_5 + a_2a_{10} + 2a_4a_6 = 100$, then the value of $a_4 + a_6$ is: ( )
A: 10
B: 20
C: 25
D: 30 | A |
If the curve $\dfrac{x^2}{k+4} + \dfrac{y^2}{k-1} = 1$ represents a hyperbola, then the range of $k$ is \_\_\_\_\_\_. | (-4,1) |
3. Let non-negative real numbers $x, y, z$ satisfy $x+y+z=1$. Then
$$
t=\sqrt{9+x^{2}}+\sqrt{4+y^{2}}+\sqrt{1+z^{2}}
$$
the minimum value of $t$ is $\qquad$. | \sqrt{37} |
21. Let $f$ be a function so that
$$
f(x)-\frac{1}{2} f\left(\frac{1}{x}\right)=\log x
$$
for all $x>0$, where log denotes logarithm base 10 . Find $f(1000)$. | 2 |
Given that vectors $\overrightarrow{{e}_{1}}$ and $\overrightarrow{{e}_{2}}$ are not collinear, $x$ and $y$ are real numbers, and $(3x-4y)\overrightarrow{{e}_{1}}+(2x-3y)\overrightarrow{{e}_{2}}=6\overrightarrow{{e}_{1}}+3\overrightarrow{{e}_{2}}$, then the value of $x-y$ is ( ).
A: $-3$
B: $3$
C: $0$
D: $2$ | B |
29. Given two positive numbers $x, y$ satisfying $x+y=4$, the range of real number $m$ that makes the inequality $\frac{1}{x}+\frac{4}{y} \geqslant m$ always true is $\qquad$. | \left(-\infty, \frac{9}{4}\right] |
$f(x)$ is a monotonically decreasing odd function defined on $(-2,2)$. When $f(2-a)+f(2a-3) < 0$, the range of $a$ is $(\ )$
A: $\left( 0,4 \right)$
B: $\left( 0,\dfrac{5}{2} \right)$
C: $\left( \dfrac{1}{2},\dfrac{5}{2} \right)$
D: $\left( 1,\dfrac{5}{2} \right)$ | D |
# 5. Plot the graph of the function
$$
y=\frac{\sqrt{\frac{1+x^{2}}{2 x}+1}+\sqrt{\frac{1+x^{2}}{2 x}-1}}{\sqrt{\frac{1+x^{2}}{2 x}+1}-\sqrt{\frac{1+x^{2}}{2 x}-1}}
$$ | {\begin{pmatrix}\frac{1}{x},&\text{for} |
Gabriela found an encyclopedia with $2023$ pages, numbered from $1$ to $2023$. She noticed that the pages formed only by even digits have a blue mark, and that every three pages since page two have a red mark. How many pages of the encyclopedia have both colors? | 44 |
6. Variant 1.
An artistic film with a duration of 192 minutes consists of four parts. It is known that the duration of any two parts differs by at least 6 minutes. What is the maximum duration that the shortest part can have? Express your answer in minutes. | 39 |
In $11!(11!=11 \times 10 \times \ldots \times 1)$, the largest factor that can be expressed as $6k+1$ (where $k$ is a natural number) is | 385 |
A convex polyhedron has faces that are either pentagons or hexagons, and exactly three edges extend from each vertex. How many pentagonal faces does the polyhedron have? Give an example of such a polyhedron. | 12 |
The bases of the parallelepiped are squares with side $b$, and all lateral faces are rhombuses. One of the vertices of the upper base is equidistant from all vertices of the lower base. Find the volume of the parallelepiped. | \frac{b^{3}}{\sqrt{2}} |
Given two lines $l_{1}$: $x-2y+1=0$ and $l_{2}$: $2x+ay-1=0$. If $l_{1}$ is perpendicular to $l_{2}$, then the value of the real number $a$ is:
A: $1$
B: $\frac{1}{2}$
C: $-\frac{1}{2}$
D: $-2$ | A |
If $\dfrac {\sin (\alpha- \dfrac {\pi}{4})}{\cos 2\alpha }=- \sqrt {2}$, then the value of $\sin \alpha+\cos \alpha$ is ( )
A: $- \dfrac { \sqrt {7}}{2}$
B: $- \dfrac {1}{2}$
C: $ \dfrac {1}{2}$
D: $ \dfrac { \sqrt {7}}{2}$ | C |
5. In a taxi company, there are currently 60 free cars: 10 black, 20 yellow, and 30 green. At a call, one of the cars, which happened to be the closest to the customer, drove out. Find the probability that a non-green taxi will arrive. | \frac{1}{2} |
Exercise 5. Let ABCD be a convex quadrilateral, with $\widehat{A B C}=90^{\circ}, \widehat{B A D}=\widehat{A D C}=80^{\circ}$. Let $M$ and $N$ be points on $[A D]$ and $[B C]$ such that $\widehat{C D N}=\widehat{A B M}=20^{\circ}$. Suppose finally $M D=A B$. What is $\widehat{M N B}$? | 70 |
Math problem from "Nine Chapters on the Mathematical Art": One acre of good farmland costs 300 yuan, and seven acres of bad farmland cost 500 yuan. A person bought a total of 100 acres of good and bad farmland, spending 10,000 yuan. How many acres of good and bad farmland did he buy? Let the number of acres of good far... | B |
Alice rolls a standard 6-sided die. Bob rolls a second standard 6 -sided die. Alice wins if the values shown differ by 1 . What is the probability that Alice wins?
(A) $\frac{1}{3}$
(B) $\frac{2}{9}$
(C) $\frac{5}{18}$
(D) $\frac{1}{6}$
(E) $\frac{5}{36}$ | \frac{5}{18} |
A high school math textbook contains $2$ required textbooks and $3$ selective required textbooks. Now, $3$ textbooks are to be selected from them, with the requirement that at least one textbook must be selected from each category. The total number of different ways to select the textbooks is ______. (Provide your answ... | 9 |
A positive integer $n$ is defined as a $\textit{stepstool number}$ if $n$ has one less positive divisor than $n + 1$. For example, $3$ is a stepstool number, as $3$ has $2$ divisors and $4$ has $2 + 1 = 3$ divisors. Find the sum of all stepstool numbers less than $300$.
[i]Proposed by [b]Th3Numb3rThr33[/b][/i] | 687 |
The ancient Chinese mathematical work "Jiuzhang Suanshu" contains the following problem: "There are plants named Pu (a type of aquatic plant) and Guan (commonly known as water onion or mat grass). Pu grows 3 feet in one day, and Guan grows 1 foot in one day. The growth of Pu halves each day, while the growth of Guan do... | C |
Simplify the following expressions:
$(1)$ $(0.027)^{-\frac{1}{3}} - \left(\frac{1}{7}\right)^{-2} + \left(2\frac{7}{9}\right)^{\frac{1}{2}} - \left(\sqrt{2} - 1\right)^{0}$
$(2)$ $\log_{3}\frac{\sqrt[4]{27}}{3} + \lg 25 + \lg 4 + 7^{\log_{7}2}$ | \frac{15}{4} |
17. As shown in the figure, a rectangular piece of paper 8 cm long and 6 cm wide is placed on the table along with another square piece of paper with a side length of 6 cm. The overlapping part between them is a square with a side length of 4 cm. The area covered on the table is $\qquad$ square centimeters. | 68 |
13. Let the sequence $\left\{a_{n}\right\}$ have the first term $a_{1}=1$, and the sum of the first $n$ terms $S_{n}$ and the general term $a_{n}$ satisfy $a_{n}=\frac{2 S_{n}^{2}}{2 S_{n}-1}$ $(n \geqslant 2)$, find: (1) the general formula for the sequence $\left\{S_{n}\right\}$; (2) the general formula for the seque... | S_{n}=\frac{1}{2n-1},\quada_{n}=\begin{cases}1,&n=1\\\frac{-2}{(2n-1)(2n-3)},&n\geqslant2\end{cases} |
4. A kilo of sausages was placed on a straight line between a dog in a kennel and a cat. The animals simultaneously rushed to the sausages. The cat runs twice as fast as the dog, but eats twice as slowly. Upon reaching the sausages, both ate without fighting and ate an equal amount. It is known that the cat could eat a... | 5x=7y |
Given the function $ f(x)= \frac {mx}{\ln x} $, the tangent to the curve $ y=f(x) $ at point $ (e^{2},f(e^{2})) $ is perpendicular to the line $ 2x+y=0 $ (where $ e $ is the base of the natural logarithm).
$(1)$ Find the analytic expression for $ f(x) $ and its interval of monotonic decrease.
$(2)$ Determine if there ... | k = 2 |
$\cos 30^{\circ}$ is equal to which of the following?
A: $\frac{{\sqrt{3}}}{3}$
B: $\frac{1}{2}$
C: $\frac{{\sqrt{2}}}{2}$
D: $\frac{{\sqrt{3}}}{2}$ | D |
9.5. Square $A B C D$ and isosceles right triangle $A E F$ $\left(\angle A E F=90^{\circ}\right)$ are positioned such that point $E$ lies on segment $B C$ (see figure). Find the angle $D C F$.
---
The square $A B C D$ and the isosceles right triangle $A E F$ with $\angle A E F = 90^{\circ}$ are arranged so that point... | 45 |
$30 \cdot 29$ Among the following numbers, the one that definitely cannot be the square of some natural number (where $n$ is a natural number) is
(A) $3 n^{2}-3 n+3$.
(B) $4 n^{2}+4 n+4$.
(C) $5 n^{2}-5 n-5$.
(D) $7 n^{2}-7 n+7$.
(E) $11 n^{2}+11 n-11$.
(Tianjin Junior High School Mathematics Competition; China Junior ... | B |
In a geometric sequence with positive terms $\{a_n\}$, if $a_2a_5=10$, then $\lg a_3+\lg a_4=$ ()
A: $-1$
B: $1$
C: $2$
D: $0$ | B |
Given the set $S=\{1,2,3,4,5,6,7,8\}$, for $x\in S$, if $x+1 \notin S$ and $x-1 \notin S$, then $x$ is a "good element" of $S$. Among all sets composed of $3$ elements from $S$, the number of sets that do not contain any "good elements" is ____. | 6 |
Let $ ABC$ be a triangle with circumradius $ R$, perimeter $ P$ and area $ K$. Determine the maximum value of: $ \frac{KP}{R^3}$. | \frac{27}{4} |
$22 \cdot 1$ If skew lines $a, b$ are in planes $\alpha$ and $\beta$ respectively, and $\alpha \cap \beta=c$, then line $c$
(A) intersects both $a$ and $b$.
(B) does not intersect either $a$ or $b$.
(C) intersects at least one of $a$ or $b$.
(D) intersects at most one of $a$ or $b$.
(High School Mathematics Competition... | C |
Given the circle $O: x^2 + y^2 = 4$ and the point $M(1, a)$.
(1) If there is exactly one line passing through point $M$ that is tangent to circle $O$, find the value of the real number $a$ and the equation of the tangent line;
(2) If $a = \sqrt{2}$, find the equations of the lines containing the shortest chord $AC$ and... | 4 |
## Task B-4.1.
In the set of complex numbers, solve the equation
$$
(x+5)^{5}=x^{5}+5^{5}
$$ | x_{1}=0,x_{2}=-5,x_{3}=\frac{-5+5\sqrt{3}i}{2},x_{4}=\frac{-5-5\sqrt{3}i}{2} |
When a die is thrown twice in a row, let the number of points facing up in the first throw be $p$, and in the second throw be $q$. Let $A=[\frac{p}{q}]$, where $\left[x\right]$ represents the largest integer not exceeding $x$. Then, which of the following statements is true?
A: $P(p+q=5)=\frac{1}{4}$
B: Events $p=6$ ... | BCD |
Given a sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$, $a_1 = a$, $a_2 = a^2$, and for $n \geq 1$, $a_{n+2} = a_{n+1} - a_n$. If $S_{56} = 6$, find the value of $a$. | a = 2 \quad \text{or} \quad a = -3 |
In the sequence $\{a_n\}$, we have $a_1 = 1$ and $a_{n+1} = \frac{a_n}{c \cdot a_n + 1}$ (where $c$ is a constant and $n \in \mathbb{N}^*)$, and $a_1, a_2, a_5$ form a geometric sequence with a common ratio not equal to 1.
(Ⅰ) Prove that the sequence $\left\{\frac{1}{a_n}\right\}$ is an arithmetic sequence;
(Ⅱ) Find ... | S_n = \frac{1}{2}\left(1 - \frac{1}{2n+1}\right) |
What is the ratio of the area of the shaded square to the area of the large square? (The figure is drawn to scale)
[asy] draw((0,0)--(0,4)--(4,4)--(4,0)--cycle); draw((0,0)--(4,4)); draw((0,4)--(3,1)--(3,3)); draw((1,1)--(2,0)--(4,2)); fill((1,1)--(2,0)--(3,1)--(2,2)--cycle,black); [/asy]
$\text{(A)}\ \dfrac{1}{6} \qq... | \frac{1}{8} |
3. A pentagram has five vertices. Draw straight lines through these five vertices so that each line must pass through two of them. Then, the total number of lines that can be drawn is ( ).
(A) 5
(B) 10
(C) 15
(D) 20 | B |
If $5a^{3}b^{n}$ and $-3a^{m}b^{2}$ are like terms, then the value of $mn$ is:
A: $3$
B: $4$
C: $5$
D: $6$ | D |
Given that $a>0$ and $b>0$, if $\frac{1}{2}$ is the arithmetic mean of $a$ and $b$, what is the minimum value of $\frac{1}{a}+\frac{1}{b}$?
A: 4
B: 2
C: 1
D: $\frac{1}{4}$ | A |
In a regular polygon, each exterior angle measures $60^{\circ}$. The sum of the interior angles of this polygon is ______ degrees. | 720^{\circ} |
10. Given $P_{0}(1, f(1))$ is a point on the curve $C: f(x)=\mathrm{e}^{x}$, the tangent line $l_{1}$ of $C$ at $P_{0}$ intersects the $x$-axis at point $Q_{1}\left(x_{1}, 0\right)$, a vertical line through $Q_{1}$ intersects $C$ at $P_{1}$, the tangent line $l_{2}$ of $C$ at $P_{1}$ intersects the $x$-axis at $Q_{2}\l... | -2022 |
2. Find the real solution to the equation $\sqrt[3]{x(3+\sqrt{8 x-3})-1}+\sqrt[3]{x(3-\sqrt{8 x-3})-1}=1$.
untranslated text:
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
translated text:
Find the real solution to the equation $\sqrt[3]{x(3+\sqrt{8 x-3})-1}+\sqrt[3]{x(3-\sqrt{8 x-3})-1}=1$.
Note: The note at the end is not... | x\geqslant\frac{3}{8} |
2. Let the complex numbers be
$$
\begin{array}{l}
z_{1}=(6-a)+(4-b) \mathrm{i}, \\
z_{2}=(3+2 a)+(2+3 b) \mathrm{i}, \\
z_{3}=(3-a)+(3-2 b) \mathrm{i},
\end{array}
$$
where, $a, b \in \mathbf{R}$.
When $\left|z_{1}\right|+\left|z_{2}\right|+\left|z_{3}\right|$ reaches its minimum value, $3 a+4 b$ $=$ | 12 |
Which of the following cannot be the number of positive integer divisors of the number $n^2+1$, where $n$ is an integer?
$
\textbf{(A)}\ 2
\qquad\textbf{(B)}\ 4
\qquad\textbf{(C)}\ 6
\qquad\textbf{(D)}\ 8
\qquad\textbf{(E)}\ \text{None of above}
$ | \text{E} |
Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0,b > 0$), with the left and right foci denoted as $F_1$ and $F_2$ respectively, a point $P$ is on the right branch of the hyperbola, and $|PF_1| = 4|PF_2|$. The maximum possible value for the eccentricity $e$ of this hyperbola is __( ).
A: $\frac{5}... | C |
6. The product $1!\cdot 2!\cdot 3!\cdot \cdots \cdot 99!\cdot 100!$ ends with $\qquad$ consecutive 0s.
(Hong Kong, China, provided) | 1124 |
10. (20 points) Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{1}=\frac{\pi}{6}, a_{n+1}=\arctan \left(\sec a_{n}\right)\left(n \in \mathbf{Z}_{+}\right) \text {. }
$$
Find the positive integer $m$ such that
$$
\sin a_{1} \cdot \sin a_{2} \cdots \cdot \sin a_{m}=\frac{1}{100} .
$$ | 3333 |
If the sum of the coefficients in the expansion of $(3x- \frac{1}{x})^{n}$ is $32$, then the coefficient of the term containing $x^{3}$ in this expansion is \_\_\_\_\_\_ . (Answer with a number) | -405 |
22. (14 points) Given the sequence $\left\{x_{n}\right\}$,
$$
x_{1}=a, x_{n+1}=\frac{2 x_{n}}{1+x_{n}^{2}} \text {. }
$$
(1) Let $a=\tan \theta\left(0<\theta<\frac{\pi}{2}\right)$, if $x_{3}<\frac{4}{5}$, find the range of $\theta$;
(2) Define the function $f(x)$ on $(-1,1)$, for any $x, y \in(-1,1)$, there is
$$
f(x)-... | 2^{n-2} |
In the expansion of $\left(2+x\right)^{4}$, the coefficient of $x^{2}$ is ( ).
A: $-24$
B: $24$
C: $-16$
D: $16$ | B |
Given a circle C with its center on the x-axis that passes through points (0, 0) and (-1, 1), and the equation of circle D is $(x-4)^2+y^2=4$
(1) Find the equation of circle C;
(2) A moving point P on circle D draws two tangents to circle C, intersecting the y-axis at points A and B, respectively. Find the range of... | \left[ \sqrt {2}, \frac {5 \sqrt {2}}{4} \right] |
First, draw $\angle AOB = 30^\circ$, then with $O$ as the vertex and ray $OB$ as one side, draw $\angle BOC = 15^\circ$. The degree measure of $\angle AOC$ is ( )
A: $45^\circ$
B: $15^\circ$
C: $45^\circ$ or $15^\circ$
D: $25^\circ$ | C |
5. For all positive real numbers $a, b, c, d$,
$$
\left(\frac{a^{3}}{a^{3}+15 b c d}\right)^{\frac{1}{2}} \geqslant \frac{a^{x}}{a^{x}+b^{x}+c^{x}+d^{x}}
$$
the real number $x=$ | \frac{15}{8} |
Suppose that $\mathbf{G}=10^{100}$. ( $\mathbf{G}$ is known as a googol.) How many times does the digit 9 occur in the integer equal to $\mathbf{G}-1009^{2}$ ? | 96 |
Given $a, b \in \mathbb{R}$, and $ab \neq 0$.
(I) If $ab > 0$, prove that: $$\frac{b}{a} + \frac{a}{b} \geq 2;$$
(II) If $ab < 0$, prove that: $$\left| \frac{b}{a} + \frac{a}{b} \right| \geq 2.$$ | \left| \frac{b}{a} + \frac{a}{b} \right| \geq 2 |
## Task 1.
Determine the smallest natural number $n$ for which there exist positive real numbers $x, y, z$ such that
$$
n=\lfloor x y\rfloor \cdot\lfloor y z\rfloor=(x+y+z)^{2}+2
$$
Note. For a real number $t$, we denote by $\lfloor t\rfloor$ the greatest integer less than or equal to $t$. | 70 |
Calculate: $\sqrt{18}÷\sqrt{8}-(\sqrt{5}-4)^{0}×(\frac{2}{3})^{-1}+\sqrt{(-3)^{2}}$. | 3 |
Given the function $f(x) = \begin{cases} \log_{4}x & x > 0 \\ 3^{x} & x\leqslant 0 \end{cases}$, find $f[f(\frac{1}{16})]$. | \frac{1}{9} |
Solve the equation: $5(y+2)+9=3(1-y)$. | y = -2 |
6.118. $\left\{\begin{array}{l}\sqrt{\frac{x+y}{2}}+\sqrt{\frac{x-y}{3}}=14, \\ \sqrt{\frac{x+y}{8}}-\sqrt{\frac{x-y}{12}}=3 .\end{array}\right.$ | (124;76) |
A triangle has a perimeter of $4$ [i]yards[/i] and an area of $6$ square [i]feet[/i]. If one of the angles of the triangle is right, what is the length of the largest side of the triangle, in feet?
[i]2016 CCA Math Bonanza Lightning #1.4[/i] | 5 |
2. Determine all positive integers $k$ for which there exist positive integers $r$ and $s$ that satisfy the equation
$$
\left(k^{2}-6 k+11\right)^{r-1}=(2 k-7)^{s}
$$ | 2,3,4,8 |
Simplify: $(1-\frac{2}{x-1})\cdot \frac{x^2-x}{x^2-6x+9}$ | \frac{x}{x-3} |
Given a circle C: $(x-a)^2+(y-b)^2=2$, with its center C on the curve $y= \frac{1}{x}$ (where $x \in [1,2]$). Find the value of $ab$ and the range of lengths of the segment intercepted by the line $l: x+2y=0$ on circle C. | [\frac{2\sqrt{5}}{5}, \frac{2\sqrt{10}}{5}] |
Given that the radius of circle O is 6cm, and the distance from the center O to line l is 5cm, the number of intersection points between line l and circle O is ( )
A: 0
B: l
C: 2
D: Cannot be determined | \text{C} |
1. Find the smallest 10-digit number, the sum of whose digits is greater than that of any smaller number. | 1999999999 |
The sum of the first three terms of a geometric sequence $\{a_n\}$ is equal to three times its first term. Then, the common ratio of this geometric sequence is __________. | -2 \text{ or } 1 |
1. (12 points) Solve the equation
$$
\sqrt[3]{(9-x)^{2}}-\sqrt[3]{(9-x)(7+x)}+\sqrt[3]{(7+x)^{2}}=4
$$ | 1 |
5. Solve the equation $\frac{\sqrt{(-x)^{2}}+(\sqrt{-x})^{2}}{x^{2}+(-x)^{2}}=\frac{1}{2016}$ ANSWER: $x=-2016$. | -2016 |
## Task A-1.8. (20 points)
Determine all natural numbers $n$ for which $n^{2}-440$ is a perfect square. | 111,57,27,21 |
Given that the terminal side of angle $\alpha$ passes through the point $(-1,6)$, then $\cos \alpha =\left(\ \ \right)$
A: $\frac{{6\sqrt{37}}}{{37}}$
B: $-\frac{{6\sqrt{37}}}{{37}}$
C: $\frac{{\sqrt{37}}}{{37}}$
D: $-\frac{{\sqrt{37}}}{{37}}$ | D |
## Task Condition
Calculate the area of the parallelogram constructed on vectors $a$ and $b$.
$a=10 p+q$
$b=3 p-2 q$
$|p|=4$
$|q|=1$
$(\widehat{p, q})=\frac{\pi}{6}$ | 46 |
4. The solution to the equation $x=\left(x^{2}-2\right)^{2}-2$ is | x_{1}=2, x_{2}=-1, x_{3,4}=\frac{-1 \pm \sqrt{5}}{2} |
6. Little Careless made a mistake while doing an addition problem, mistaking the tens digit 5 of one addend as 2, and the units digit 7 of the other addend as 1, resulting in a sum of 241. The correct sum should be $\qquad$ | 277 |
10、One day, Qiqi went to the zoo, he saw three types of animals: monkeys, pandas, and lions, the total number of these three types of animals is between $26-32$. The total number of monkeys and lions is more than the number of pandas. The total number of pandas and lions is more than twice the number of monkeys. The to... | 13 |
$4\cos 10^{\circ}-\tan 80^{\circ}=$ ( )
A: $-\sqrt{3}$
B: $-\sqrt{2}$
C: $-1$
D: $\sqrt{3}$ | A |
2. How many increasing finite sequences of natural numbers exist whose first element is equal to 1, the last element is equal to 25, and any two consecutive members differ by 2 or 3? | 351 |
If the domain of the function $f(x) = \sqrt{mx^2 - 6mx + m + 8}$ is $\mathbb{R}$, then the range of the real number $m$ is \_\_\_\_\_\_. | 0 \leq m \leq 1 |
$$
\begin{array}{c}
1.2013+2012-2011-2010+2009+2008- \\
2007-2006+\cdots+5+4-3-2+1=(\quad) .
\end{array}
$$
(A) 2013
(B) 2012
(C) 1
(D) 0 | A |
Example 8 A coach can travel from Nanjing to Shanghai on the Shanghai-Nanjing Expressway in $4 \mathrm{~h}$. It is known that a coach departs from Nanjing to Shanghai every half hour, and at the same time, a coach also departs from Shanghai to Nanjing. If a passenger takes a bus from Nanjing to Shanghai, how many coach... | 17 |
9) If $n$ is a natural number with 6 positive integer divisors, how many positive integer divisors does $n^{2}$ have? N.B.: among the divisors of a number, we also count 1 and the number itself.
(A) 11
(B) 12
(C) 15
(D) 36
(E) the answer depends on $n$ | E |
Each vertex of the tetrahedron $ABCD$ lies on the surface of the sphere $O$. $AB$, $AC$, and $AD$ are mutually perpendicular, and $AB = \sqrt{3}$, $AC = 2$, $AD = 3$. The surface area of the sphere $O$ is ( ).
A: $64\pi$
B: $16\pi$
C: $4\pi$
D: $\pi$ | B |
2. Define " $\triangle$ " as an operation on ordered pairs of real numbers, as shown below, $(a, b) \triangle(c, d)=(a c+b d, a d+b c)$. If for any real numbers $a, b$ we have $(a, b) \triangle(x, y)=(a, b)$, then $(x, y)$ is ( ).
(A) $(0,1)$
(B) $(1,0)$
(C) $(-1,0)$
(D) $(0,-1)$ | (B) |
Example 2 As shown in Figure 3, let $E$, $F$, and $G$ be the midpoints of the edges $AB$, $BC$, and $CD$ of a regular tetrahedron $ABCD$, respectively. Then the size of the dihedral angle $C-FG-E$ is ( ).
(A) $\arcsin \frac{\sqrt{6}}{3}$
(B) $\frac{\pi}{2}+\arccos \frac{\sqrt{3}}{3}$
(C) $\frac{\pi}{2}-\operatorname{ar... | D |
## Problem Statement
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty} \frac{(n+2)^{2}-(n-2)^{2}}{(n+3)^{2}}
$$ | 0 |
1. Given that the two roots of the equation $x^{2}+x-1=0$ are $\alpha, \beta$. Then the value of $\frac{\alpha^{3}}{\beta}+\frac{\beta^{3}}{\alpha}$ is $\qquad$ | -7 |
In a computer network with $n$ terminals, the probability of each terminal being used on any given day is $p$. The average number of terminals used in this network on any given day is ( ).
A: $np(1-p)$
B: $np$
C: $n$
D: $p(1-p)$ | B |
Given $\{a_{n}\}$ is an arithmetic sequence with the first term $1$ and a common difference $d \gt 0$, $\{b_{n}\}$ is a geometric sequence with the first term $2$, $a_{4}=b_{2}$, $a_{8}=b_{3}$.
$(1)$ Find the general formulas for $\{a_{n}\}$ and $\{b_{n}\}$;
$(2)$ If the $m$-th term $b_{m}$ of the sequence $\{b_{n}... | S_{20} = \frac{2^{41} - 2}{3} |
# Problem 10.1
For all real $x$ and $y$, the equality $f\left(x^{2}+y\right)=f(x)+f\left(y^{2}\right)$ holds. Find $f(-1)$.
## Number of points 7 | 0 |
In the arithmetic sequence \\(\{a_n\}\), it is known that \\(S_{15}=90\\), then \\(a_{8}=\\) \\(\\)
A: \\(3\\)
B: \\(4\\)
C: \\(6\\)
D: \\(12\\) | C |
10,11
[Application of trigonometric formulas (geometry) $]$
[ Identical transformations (trigonometry). ]
In a convex quadrilateral $A B C D: A C \perp B D, \angle B C A=10^{\circ}, \angle B D ... | 60 |
Let $\{a_n\}$ be a sequence where each term is positive, and let $S_n$ be the sum of the first $n$ terms of $\{a_n\}$. Given that $S_n$ satisfies $$2S_n^2 - (3n^2 + 3n - 2)S_n - 3(n^2+n) = 0$$ for $n \in \mathbb{N}^*$.
(i) Find the value of $a_1$.
(ii) Determine the general formula for the sequence $\{a_n\}$.
(ii... | T_n = \frac{3}{4} - \frac{2n + 3}{4 \cdot 3^n} |
1. Determine three complex numbers with a modulus of 1 such that both their sum and their product are equal to 1. | 1,i,-i |
If $3 \in \{1, m+2\}$, then $m=$ ? | 1 |
A bag contains 2 white balls and 3 black balls, all of the same size. If two balls are randomly drawn from the bag, let the random variable $\xi$ represent the number of white balls drawn. The value of $P(\xi=1)$ is $\_\_\_\_\_\_.$ | 0.6 |
2. Find the geometric locus of points $\left(x_{z} ; y_{z}\right)$, which are the vertices of parabolas $y=a x^{2}+z x+$ $c$, where $a$ and $c$ are fixed positive numbers, and $z$ takes all real values. | y_{z}=-x_{z}^{2}+ |
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