problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
For the hyperbola $\frac{{x}^{2}}{9}-\frac{{y}^{2}}{m}=1$, if the eccentricity $e=2$, then the real number $m=$____. | 27 |
1. Given integers $a, b, c, d, e$ satisfy $a \geqslant b \geqslant c \geqslant d \geqslant e$, and $a+b+c+d+e=a b c d e=1990$. Then, $a=$ $\qquad$ ,$b=$ $\qquad$ , $c=$ $\qquad$ , $d=$ $\qquad$ , $e=$ $\qquad$
(1990, Zu Chongzhi Cup Junior High School Mathematics Invitation Competition) | 1990, 1, 1, -1, -1 |
Example 10 Let $a, b, c$ be positive real numbers, find the minimum value of $\frac{a+3 c}{a+2 b+c}+\frac{4 b}{a+b+2 c}-\frac{8 c}{a+b+3 c}$. (3rd China Girls Mathematical Olympiad problem) | -17+12 \sqrt{2} |
How many pairs $(m,n)$ of integers satisfy the equation $m+n=mn$?
$\textbf{(A) }1\qquad \textbf{(B) }2\qquad \textbf{(C) }3\qquad \textbf{(D) }4\qquad \textbf{(E) }\text{more than }4$ | \textbf{(B)}2 |
The power function $f\left(x\right)=x^{\alpha }(\alpha \in R)$ satisfies: for any $x\in R$, $f\left(-x\right)=f\left(x\right)$, and $f\left(-1\right) \lt f\left(2\right) \lt 2$. Please write down a function $f\left(x\right)=$____ that meets the above conditions. | f(x) = x^{\frac{2}{3}} |
7. If a moving point $P(x, y)$ rotates counterclockwise on the unit circle with a constant angular velocity $\omega$, then the motion equation of point $Q\left(-2 x y, y^{2}-x^{2}\right)$ is
(A) rotates clockwise on the unit circle with angular velocity $\omega$;
(B) rotates counterclockwise on the unit circle with ang... | C |
Given the function $f(x)= \sqrt {3}\sin (2x- \frac {\pi}{3})-2\sin (x- \frac {\pi}{4})\sin (x+ \frac {\pi}{4})$.
$(1)$ Find the smallest positive period of the function $f(x)$ and the equation of the axis of symmetry for the graph;
$(2)$ Find the maximum and minimum values of the function $f(x)$ in the interval $\l... | - \frac { \sqrt {3}}{2} |
3B. In the set of real numbers, solve the system of equations
$$
\left\{\begin{array}{l}
x+y+z=\frac{13}{3} \\
\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{13}{3} \\
x y z=1
\end{array}\right.
$$ | (1,3,\frac{1}{3}) |
$2 \cdot 111$ two-digit number set $\{00,01, \cdots, 98,99\}$ has a subset $X$ with the following property: in any infinite sequence of digits, there are two adjacent digits that form an element of $X$. How many elements does $X$ have at minimum? | 55 |
11.1. Solve the inequality: $\sqrt{(x-2)^{2}\left(x-x^{2}\right)}<\sqrt{4 x-1-\left(x^{2}-3 x\right)^{2}}$. | 2 |
Xiao Ming plans to buy a total of 20 exercise books and notebooks, with a total cost not exceeding 60 yuan. It is known that each exercise book costs 2 yuan and each notebook costs 5 yuan. How many exercise books does Xiao Ming need to buy at least? | 14 |
Example 3 Suppose the lengths of the two legs of a right triangle are $a$ and $b$, and the length of the hypotenuse is $c$. If $a$, $b$, and $c$ are all positive integers, and $c=\frac{1}{3} a b-(a+b)$, find the number of right triangles that satisfy the condition.
(2010, National Junior High School Mathematics Competi... | 3 |
6. Solve the system
$$
\left\{\begin{array}{l}
\operatorname{tg}^{3} x+\operatorname{tg}^{3} y+\operatorname{tg}^{3} z=36 \\
\operatorname{tg}^{2} x+\operatorname{tg}^{2} y+\operatorname{tg}^{2} z=14 \\
\left(\operatorname{tg}^{2} x+\operatorname{tg} y\right)(\operatorname{tg} x+\operatorname{tg} z)(\operatorname{tg} ... | 4 |
G6.1 An $n$-sided convex polygon has 20 diagonals. Find $n$.
G6.2 Two dice are thrown. The probability of getting a total of $n$ is $\frac{k}{36}$. Find $k$.
G6.3 A man drives at $25 \mathrm{~km} / \mathrm{h}$ for 3 hours and then at $50 \mathrm{~km} / \mathrm{h}$ for 2 hours.
His average speed for the whole journey is... | 8,5,35,1 |
Given that the domain of $f(x)$ is $\left[-2,2\right]$, and the function $g(x)=\frac{f(x-1)}{\sqrt{2x+1}}$, what is the domain of $g(x)$?
A: $(-\frac{1}{2}$, $3]$
B: $(-1,+\infty )$
C: $(-\frac{1}{2}$, $0)\cup \left(0,3\right)$
D: $(-\frac{1}{2}$, $3)$ | A |
. A mason has bricks with dimensions $2 \times 5 \times 8$ and other bricks with dimensions $2 \times 3 \times 7$. She also has a box with dimensions $10 \times 11 \times 14$. The bricks and the box are all rectangular parallelepipeds. The mason wants to pack bricks into the box filling its entire volume and with no br... | 24 |
Given -- the equations of the two asymptotes of the hyperbola are $x+y=0$ and $x-y=0$, then its eccentricity is ( )
(A) $\frac{\sqrt{2}}{2}$
(B) $\sqrt{2}$
(C) $2 \sqrt{2}$
(D) $\sqrt{2}+1$ | \sqrt{2} |
2.120. $\frac{5 \sqrt[3]{4 \sqrt[3]{192}}+7 \sqrt[3]{18 \sqrt[3]{81}}}{\sqrt[3]{12 \sqrt[3]{24}+6 \sqrt[3]{375}}}$.
2.120. $\frac{5 \sqrt[3]{4 \sqrt[3]{192}}+7 \sqrt[3]{18 \sqrt[3]{81}}}{\sqrt[3]{12 \sqrt[3]{24}+6 \sqrt[3]{375}}}$.
The expression is already in a mathematical format and does not require translation. H... | \frac{31}{3} |
4. (SPA 2) In the triangle $A B C$, let $D, E$ be points on the side $B C$ such that $\angle B A D=\angle C A E$. If $M, N$ are, respectively, the points of tangency with $B C$ of the incircles of the triangles $A B D$ and $A C E$, show that
$$
\frac{1}{M B}+\frac{1}{M D}=\frac{1}{N C}+\frac{1}{N E} .
$$ | \frac{1}{MB}+\frac{1}{MD}=\frac{1}{NC}+\frac{1}{NE} |
Example 13. Calculate the circulation of the vector field given in spherical coordinates: $2=r \mathbf{e}_{r}+(R+r) \sin \theta \mathbf{e}_{\varphi}$, along the circle $L:\{r=$ $\left.R, \theta=\frac{\pi}{2}\right\}$ in the direction of increasing angle $\varphi$ directly and using Stokes' theorem. | 4\piR^{2} |
Let $P(z) = z^8 + \left(4\sqrt{3} + 6\right)z^4 - \left(4\sqrt{3} + 7\right)$. What is the minimum perimeter among all the $8$-sided polygons in the complex plane whose vertices are precisely the zeros of $P(z)$?
$\textbf{(A)}\ 4\sqrt{3} + 4 \qquad \textbf{(B)}\ 8\sqrt{2} \qquad \textbf{(C)}\ 3\sqrt{2} + 3\sqrt{6} \qq... | 8\sqrt{2} |
1. Solve the problem: octopuses with an even number of legs always lie, while octopuses with an odd number of legs always tell the truth. Five octopuses met, each having between 7 and 9 legs.
The first said: "We have 36 legs in total";
The second: "We have 37 legs in total";
The third: "We have 38 legs in total";
T... | 39 |
9. (16 points) Given the function
$$
f(x)=10 x^{2}+m x+n(m, n \in \mathbf{Z})
$$
has two distinct real roots in the interval $(1,3)$. Find the maximum possible value of $f(1) f(3)$. | 99 |
Due to a snow and ice disaster, a citrus orchard suffered severe damage. To address this, experts proposed a rescue plan for the fruit trees, which needs to be implemented over two years and is independent each year. The plan estimates that in the first year, the probability of the citrus yield recovering to 1.0, 0.9, ... | 0.2 |
3A. Given a triangle $ABC$ with area $P$. From its medians, another triangle is constructed, then from the medians of the second triangle, a third triangle is constructed, and so on. Generally, the $(n+1)$-th triangle is constructed from the medians of the $n$-th triangle. Calculate the sum of the areas of all triangle... | 4P |
Given the function $f(x)=a(x-\dfrac{1}{x})-\ln x$, where $a\in \mathbb{R}$.
(Ⅰ) If $a=1$, find the equation of the tangent line to the curve $y=f(x)$ at the point $P(1,f(1))$;
(Ⅱ) If for any $x\geqslant 1$, $f(x)\geqslant 0$ always holds, find the range of the real number $a$. | a\geqslant \dfrac{1}{2} |
The solution set of the inequality $\frac{{3x+4}}{{x-2}}\geqslant 4$ is ______. | (2,12] |
# 7.2. (7 points)
Two pedestrians set out at dawn. Each walked at a constant speed. One walked from $A$ to $B$, the other from $B$ to $A$. They met at noon and, without stopping, arrived: one - in $B$ at 4 PM, and the other - in $A$ at 9 PM. At what hour was dawn that day? | 6 |
For a positive real number $p$, find all real solutions to the equation
\[\sqrt{x^2 + 2px - p^2} -\sqrt{x^2 - 2px - p^2} =1.\] | x = \pm \sqrt{\frac{p^2 + \frac{1}{4}}{1 - 4p^2}} |
After a certain chip is localized, it undergoes two price reductions, and the unit price of each chip is reduced from $188 to $108. If the percentage of the two price reductions is the same, let the percentage of each price reduction be $x$. According to the question, the equation obtained is ( ).
A: $188(1-x^{2})=108... | D |
Given that point $M$ lies on the parabola $y^{2}=2px$, the abscissa of point $M$ is $4$, and the distance from point $M$ to the focus $F$ is $5$, then the value of $p$ is ______. | 2 |
## 69. Math Puzzle $2 / 71$
In a Siemens-Martin furnace, 20 t of steel with a $0.5 \%$ carbon content is melted together with 5 t of gray cast iron with a $5 \%$ carbon content.
What percent of carbon does the mixture contain? | 1.4 |
## Task Condition
Find the derivative.
$y=\left(2 x^{2}+6 x+5\right) \operatorname{arctg} \frac{x+1}{x+2}-x$ | (4x+6)\operatorname{arctg}\frac{x+1}{x+2} |
Given that $\sqrt {a}+ \frac{1}{ \sqrt {a}}=2$, find the value of $a^{2}+a^{-2}=$ \_\_\_\_\_\_. | 2 |
Given $a+b=8$ and $ab=c^2+16$, find the value of $a+2b+3c$. | 12 |
A slope has a gradient of 1.5:1. If a car's maximum climbing angle is 60°, then this car __ (fill in "can" or "cannot") drive on this slope. | \text{can} |
265 It is known that for all positive integers $n$,
$$
\prod_{i=1}^{n}\left(1+\frac{1}{3 i-1}\right) \geqslant \frac{k}{2} \sqrt[3]{19 n+8}
$$
always holds. Try to find the maximum value of $k$. | 1 |
7. Let $a, b>0$, and
$$
a^{2008}+b^{2008}=a^{2006}+b^{2006} \text {. }
$$
Then the maximum value of $a^{2}+b^{2}$ is ( ).
(A) 1
(B) 2
(C) 2006
(D) 2008 | B |
$(1)$ Put $6$ identical balls into $4$ identical boxes, each box is not empty, how many ways are there in total?$(2)$ Put $6$ identical balls into $4$ different boxes, each box is not empty, how many ways are there in total?$(3)$ Put $6$ different balls into $4$ identical boxes, each box is not empty, how many ways are... | 1560 |
8. $[a]$ represents the integer part of $a$, for example: [1.5]=1, [2]=2.
Calculate: $\left[\frac{17 \times 1}{23}\right]+\left[\frac{17 \times 2}{23}\right]+\cdots \cdots+\left[\frac{17 \times 67}{23}\right]+\left[\frac{17 \times 68}{23}\right]=$ | 1701 |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $B = \frac{\pi}{3}$, $\cos A = \frac{4}{5}$, and $b = \sqrt{3}$.
(1) Find the value of $\sin C$.
(2) Find the area of $\triangle ABC$. | \frac{9\sqrt{3} + 36}{50} |
If $8+6=n+8$, then $n$ equals
(A) 14
(B) 22
(C) 6
(D) -2
(E) 9 | 6 |
The positional relationship between two non-overlapping lines in the same plane is ( ).
A: There are two types: perpendicular or intersecting
B: There are three types: parallel, perpendicular, or intersecting
C: There are two types: parallel or intersecting
D: There are two types: parallel or perpendicular | C |
## Task 3 - 030713
How can one determine without performing the specified arithmetic operations whether the number
$$
\frac{378 \cdot 436-56}{378+436 \cdot 377}
$$
is greater or less than 1? | \frac{377\cdot436+380}{377\cdot436+378}>1 |
8,9
The area of triangle $ABC$ is $S, \angle BAC = \alpha, AC = b$. Find $BC$. | \sqrt{\frac{4S^{2}}{b^{2}\sin^{2}\alpha}+b^{2}-4S\cot\alpha} |
4. [4] Let $a, b$ be constants such that $\lim _{x \rightarrow 1} \frac{(\ln (2-x))^{2}}{x^{2}+a x+b}=1$. Determine the pair $(a, b)$. | (-2,1) |
11. In math class, the teacher gave the students in a certain class 2 questions, stipulating that getting one question right earns 10 points, half right and half wrong earns 5 points, and completely wrong or not doing it earns 0 points. After grading, the teacher found that all scoring situations were present in the cl... | 45 |
A random variable $\xi$ follows a normal distribution $N(10, 100)$. If $P(\xi > 11) = a$, then $P(9 < \xi \leq 11) = \ ?$ | 1-2a |
76. In the box, there are 100 white, 100 red, 100 blue, and 100 black balls. What is the smallest number of balls that need to be pulled out, without looking into the box, to ensure that among them there are at least 3 balls of the same color? | 9 |
Given the polar coordinate equations of two lines are $\sqrt {2}\rho= \frac {1}{\sin ( \frac {\pi}{4}+\theta)}$ and $\theta= \frac {\pi}{3}(\rho\in R)$, find the polar coordinates of their intersection point. | ( \sqrt {3}-1, \frac {\pi}{3}) |
Seven consecutive integers stood in a row, arranged from the smallest to the largest. After a while, the numbers got bored, so the first one swapped places with the last one, then the middle one moved to the very beginning of the row, and finally, the largest number stepped to the middle. To its great delight, it found... | -4,-3,-2,-1,0,1,2or-5,-4,-3,-2,-1,0,1 |
5. As shown in Figure $1, M$ is the midpoint of side $CD$ of the inscribed quadrilateral $ABCD$ in the semicircle $\odot O$ with diameter $AB$, $MN \perp AB$ at point $N$, $AB=10, AD=AN=$ 3. Then $BC=(\quad)$.
(A) 5
(B) 6
(C) 7
(D) 8 | C |
Given set A={-1, 1}, and set B={x |||x + 1/2| < 3/2, x ∈ Z}, find A∪B= ( )
A: {-1}
B: {-1, 1}
C: {-1, 0, 1}
D: {-1, 0, 1, 2} | \text{C} |
Which of the following transformations from left to right involves factorization?
A: $(x-4)(x+4)=x^{2}-16$
B: $x^{2}-y^{2}+2=(x+y)(x-y)+2$
C: $x^{2}+1=x(x+ \dfrac {1}{x})$
D: $a^{2}b+ab^{2}=ab(a+b)$ | D |
Given a complex number is a pure imaginary number \\(Z=\dfrac{a+i}{1+i}\\) (where \\(i\\) is the imaginary unit), then the real number \\(a=\\)_______ | -1 |
Example 6 Given $a>0, b>0$. Find the area of the triangle with side lengths $\sqrt{a^{2}+b^{2}}$, $\sqrt{a^{2}+4 b^{2}}$, and $\sqrt{4 a^{2}+b^{2}}$.
(The 12th "Hope Cup" National Mathematics Invitational (Junior $\Rightarrow)$ | \frac{3}{2} a b |
Among the following four judgments, the number of correct ones is ( )
① Proposition: "Let $a, b \in \mathbb{R}$, if $a+b \neq 6$, then $a \neq 3$ or $b \neq 3$" is a true proposition.
② If "p or q" is a true proposition, then both p and q are true propositions.
③ The negation of the proposition "For all $a, b \in... | \text{B} |
Given that point $A(x_{0},y_{0})(x_{0}{\neq }0)$ is the intersection point of one asymptote of the parabola $C_{1}$: $y^{2}{=}2px(p{ > }0)$ and the hyperbola $C_{2}$: ${\dfrac{x^{2}}{a^{2}}{-}}{\dfrac{y^{2}}{b^{2}}}{=}1(a{ > }0{,}b{ > }0)$, if the distance from point $A$ to the axis of symmetry of the parabola $C_{1}$ ... | 1 |
The maximum value of the function $f(x) = |x| - |x - 3|$ is ____. | 3 |
Problem 3. Masha has 4 pieces of red plasticine, 3 pieces of blue plasticine, and 5 pieces of yellow plasticine. First, she divided each non-red piece of plasticine in half, and then she divided each non-yellow piece of plasticine in half. How many pieces of plasticine did Masha get | 30 |
Given the proposition p: There exists an $n \in \mathbb{N}$ such that $2^n > 1000$, then the negation of p, $\neg p$, is ( )
A: For all $n \in \mathbb{N}$, $2^n \leq 1000$
B: For all $n \in \mathbb{N}$, $2^n > 1000$
C: There exists an $n \in \mathbb{N}$, $2^n \leq 1000$
D: There exists an $n \in \mathbb{N}$, $2^n < 100... | \text{A} |
1. During the physical education lesson, all students of grade 8 lined up. It turned out that boys and girls alternated in the line. It is known that exactly $52\%$ of the students in grade 8 are boys. Find the number of girls in grade 8. Don't forget to justify your answer. | 12 |
29. [18] Compute the remainder when
$$
\sum_{k=1}^{30303} k^{k}
$$
is divided by 101 . | 29 |
4. The diagram shows a triangle $A B C$ and two lines $A D$ and $B E$, where $D$ is the midpoint of $B C$ and $E$ lies on $C A$. The lines $A D$ and $B E$ meet at $Z$, the midpoint of $A D$.
What is the ratio of the length $C E$ to the length $E A$ ? | CE:EA=2:1 |
On Earth, which latitudinal circles are half as long as the Equator? | 60 |
Example 5 Find all positive integers $k$ such that for any positive numbers $a, b, c$ satisfying the inequality
$$
k(a b+b c+c a)>5\left(a^{2}+b^{2}+c^{2}\right)
$$
there must exist a triangle with side lengths $a, b, c$.
(First China Girls Mathematical Olympiad)
Analysis: To find $k$, we can first determine the upper... | 6 |
The function $f(x) = x^3 + ax^2 + 3x - 9$, it is known that $f(x)$ has an extreme value at $x = -3$, then $a =$ （ ）
A: 2
B: 3
C: 4
D: 5 | \text{D} |
Given that the sequence of positive numbers $\{a\_n\}$ satisfies $a_{n+1}=2a_{n}$, determine the type of sequence $\{a\_n\}$.
A: An increasing sequence
B: A decreasing sequence
C: A constant sequence
D: The monotonicity of the sequence cannot be determined | A |
Let $z=\frac{2-i}{1+i}$, then the conjugate of $z$ is ______. | \frac{1}{2} + \frac{3}{2}i |
Given a left focus of $F(-1,0)$ for an ellipse that passes through point $E(1, \frac{2 \sqrt{3}}{3})$. Point $P(1,1)$ forms chords $AB$ and $CD$ with slopes $k_{1}$ and $k_{2}$, respectively. Let $M$ and $N$ be the midpoints of line segments $AB$ and $CD$, respectively.
(1) Find the standard equation of the ellipse;
(... | (0,- \frac{2}{3}) |
The length of the chord obtained by intersecting the circle $x^2+y^2=4$ with the line $\sqrt {3}x+y-2\sqrt {3}=0$ is $\_\_\_\_\_\_\_.$ | 2 |
How many prime numbers are there between 20 and 30 ?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4 | 2 |
29. Jigsaw Puzzle. Everyone is familiar with the puzzle that involves cutting a piece of plywood with a picture drawn on it into small pieces, and then proposing to reassemble the original picture from these mixed pieces. Let the picture be cut into $n$ small pieces. We will call a "move" the joining together of two pi... | n-1 |
Given an ellipse $E$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with eccentricity $\frac{\sqrt{5}}{3}$, where the upper and lower vertices are $A$ and $B$ respectively, and the right vertex is $C$, and the area of $\triangle ABC$ is $6$. $(1)$ Find the equation of $E$;$(2)$ If point $P$ is a point on $E$ other than th... | (3,2) |
Task 5. Find all pairs of prime numbers $(p, q)$ for which
$$
2^{p}=2^{q-2}+q! .
$$ | (3,3)(7,5) |
Given that the domain of the function $f(x)$ is $D$, the function satisfies the following conditions:
1. $f(x)$ is a monotonic function within $D$;
2. If there exists an interval $[a, b] \subseteq D$ such that the range of $f(x)$ on $[a, b]$ is $[2a, 2b]$, then the function $f(x)$ is called a "successful function".
Det... | (0, \frac{1}{12}) |
Example 1. $P$ is a point inside an equilateral $\triangle A B C$, and the ratio of the sizes of $\angle A P B, \angle B P C, \angle C P A$ is $5: 6: 7$. Then the ratio of the sizes of the three interior angles of the triangle with sides $P A, P B, P C$ is ( ) (from smallest to largest).
(A) $2: 3: 4$
(B) $3: 4: 5$
(C)... | A |
Given a regular tetrahedron with a height of $H$, the radius of its inscribed sphere is $R$. Find the ratio of $R$ to $H$. | R:H=1:4 |
The solution set of the inequality $\frac{{x-1}}{{2x+1}}\leqslant 0$ is ( ).
A: $\left\{{x|x＜-\frac{1}{2}或x≥1}\right\}$
B: $\left\{{x|-\frac{1}{2}≤x≤1}\right\}$
C: $\left\{{x|-\frac{1}{2}＜x≤1}\right\}$
D: $\left\{{x|x≤-\frac{1}{2}或x≥1}\right\}$ | C |
Find all pairs of real numbers $(a,b)$ so that there exists a polynomial $P(x)$ with real coefficients and $P(P(x))=x^4-8x^3+ax^2+bx+40$. | (28, -48), (2, 56) |
The 6 members of an executive committee want to call a meeting. Each of them phones 6 different people, who in turn each calls 6 other people. If no one is called more than once, how many people will know about the meeting?
(A) 18
(B) 36
(C) 216
(D) 252
(E) 258 | 258 |
If the complex number $z$ satisfies $\left(2-i\right)z=i^{2022}$, then $z=\_\_\_\_\_\_.$ | -\frac{2}{5} - \frac{1}{5}i |
Rectangle $A B C D$ is made up of six squares. The areas of two of the squares are shown. The perimeter of rectangle $A B C D$, in centimetres, is
(A) 50
(B) 44
(C) 46
(D) 52
(E) 48
 | 48 |
The value of the binary number $111111_{(2)}$ when converted into a decimal number is ( )
A: 63
B: 62
C: 64
D: 61 | A |
Find the least positive integer $N$ which is both a multiple of 19 and whose digits add to 23. | 779 |
Let $f(x)$ be a function defined on $\mathbb{R}$ that is never zero, and for any real numbers $x$, $y \in \mathbb{R}$, it holds that $f(x) \cdot f(y) = f(x+y)$. If $a_1 = \frac{1}{2}, a_n = f(n) (n \in \mathbb{N}^*)$, then the range of the sum of the first $n$ terms $S_n$ of the sequence $\{a_n\}$ is
A: $\left[ \frac{... | C |
5. The 11 letters of 'IMOHKPRELIM' are written on 11 cards (one letter on each card). If three cards are randomly drawn, how many different combinations of the three letters are there? (Order is not considered. For instance 'IMO' and 'IOM' are regarded to be the same combination.)
現把「IMOHKPRELIM」這 11 個英文字母分別寫在 11 張卡紙上（... | 100 |
The function $f(x)=x^3-ax^2-bx+a^2$ has an extremum of $10$ at $x=1$. Then, the coordinates of the point $(a,b)$ are
A: $(3,-3)$
B: $(-4,11)$
C: $(3,-3)$ or $(-4,11)$
D: Does not exist | \text{B} |
Given an ellipse $C$ with its center at the origin and its foci on the $x$-axis, the left and right foci are $F_{1}$ and $F_{2}$, respectively, and $|F_{1}F_{2}|=2$. The point $(1, \frac {3}{2})$ is on the ellipse $C$.
(Ⅰ) Find the equation of the ellipse $C$;
(Ⅱ) A line $l$ passing through $F_{1}$ intersects the e... | (x-1)^{2}+y^{2}=2 |
If proposition p: "log2x < 0", and proposition q: "x < 1", then p is q's ______ condition. (Fill in "sufficient and necessary", "sufficient but not necessary", "necessary but not sufficient", or "neither sufficient nor necessary") | \text{sufficient but not necessary} |
Example 6 Given $A(-2,2)$, and $B$ is a moving point on the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$, $F$ is the left focus. When $\mid A B \mid +\frac{5}{3}|B F|$ takes the minimum value, find the coordinates of $B$.
(1999, National High School Mathematics Competition) | B\left(-\frac{5}{2} \sqrt{3}, 2\right) |
A function $f(x)$ is defined as an odd function on $\mathbb{R}$. When $x > 0$, $f(x)=x-2$.
1. Find the analytic expression of the function $f(x)$.
2. Solve the inequality $f(x) < 2$. | (-\infty, 4) |
25. If the 13 th of any particular month falls on a Friday, we call it Friday the 13 th. It is known that Friday the 13 th occurs at least once every calendar year. If the longest interval between two consecutive occurrences of Friday the 13 th is $x$ months, find $x$. | 14 |
18. Let the function $f(x)$ defined on the set of rational numbers satisfy: for any $a, b \in \mathbf{Q}_{+}$, we have
$$
f(a b)=f(a)+f(b),
$$
and for each prime number $p$, we have $f(p)=p$. Then when the value of $x$ is $(\quad)$, we have $f(x)<0$.
(A) $\frac{17}{32}$
(B) $\frac{11}{16}$
(C) $\frac{7}{9}$
(D) $\frac... | E |
A particle moves according to the law $s=2t^3$. Find its instantaneous velocity at $t=2$. | 24 |
Let $(m,n)$ be pair of positive integers. Julia has carefully planted $m$ rows of $n$ dandelions in an $m \times n$ array in her back garden. Now, Jana un Viviane decides to play a game with a lawnmower they just found. Taking alternating turns and starting with Jana, they can now mow down all the dandelions in a strai... | (m-1)(n-1) = 0 |
In trapezoid $K L M N$, the bases $K N$ and $L M$ are equal to 12 and 3, respectively. From point $Q$, lying on side $M N$, a perpendicular $Q P$ is dropped to side $K L$. It is known that $P$ is the midpoint of side $K L$, $P M=4$, and that the area of quadrilateral $P L M Q$ is four times smaller than the area of qua... | 16 |
A fair coin is tossed consecutively 3 times, and it lands on heads exactly 1 time. This is an example of:
A: A certain event
B: A random event
C: An impossible event
D: A determined event | \text{B} |
The Huawei Mate20 phone is equipped with the world's first 7-nanometer process chip, where 7 nanometers is equal to $0.000000007$ meters. The number $0.000000007$ in scientific notation is represented as ( ).
A: $0.7\times 10^{-8}$
B: $7\times 10^{-7}$
C: $7\times 10^{-8}$
D: $7\times 10^{-9}$ | D |
8.2. How many fractions $\frac{n}{m}$ exist with the following properties: $n$ and $m$ are two-digit natural numbers, and the value of the fraction does not change if 20 is added to $n$ and 19 is added to $m$? | 4 |
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