problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Given the set $U=\{0,1,2,3,4\}$, $M=\{0,4\}$, $N=\{2,4\}$, then $\complement_U(M\cup N)=$ \_\_\_\_\_\_\_\_. | \{1,3\} |
Given a sequence $A$, define the encryption algorithm $f_{i}$ on $A$: when $i$ is odd, increase the value of each odd element in $A$ by $i$ and decrease the value of each even element by $1$; when $i$ is even, increase the value of each even element in $A$ by $2i$ and decrease the value of each odd element by $2$, and ... | 9n^2 - 3n + 19 |
The slope of the tangent line to the curve $y=x^3-2x^2-4x+2$ at the point $(1, -3)$ is. | -5 |
2. Set $A$ is a subset consisting of 40 elements from $\{1,2,3, \cdots, 50\}$, and $S$ is the sum of all elements in set $A$. Then the number of possible values for $S$ is $\qquad$. | 401 |
## Aufgabe 4 $8258-4123 \quad 4831-1210 \quad 7725+5+38+756$
| 8524 |
Given the function $f(x) = a^x + x - b$, where its root $x_0 \in (n, n+1) (n \in \mathbb{Z})$, and the constants $a, b$ satisfy $2^a = 3, 3^b = 2$. Then, the value of $n$ is ( ).
A: $-2$
B: $-1$
C: $0$
D: $1$ | \text{B} |
18. Strange fractions. Can a fraction where the numerator is less than the denominator be equal to a fraction where the numerator is greater than the denominator? | \frac{-1}{2}=\frac{1}{-2} |
A $\emph{planar}$ graph is a connected graph that can be drawn on a sphere without edge crossings. Such a drawing will divide the sphere into a number of faces. Let $G$ be a planar graph with $11$ vertices of degree $2$, $5$ vertices of degree $3$, and $1$ vertex of degree $7$. Find the number of faces into which $G$ d... | 7 |
If the variance of a set of data $x_1, x_2, \ldots, x_n$ is 0.6, then the variance of another set of data $x_1-2013, x_2-2013, \ldots, x_n-2013$ is. | 0.6 |
Given the function $f(x)=\ln x-ax^{2}$, and the slope of the tangent line to the function $f(x)$ at the point $(2,f(2))$ is $-\frac{1}{2}$, find the value of $a$. | \frac{1}{4} |
If the complex numbers $z_1$ and $z_2$ correspond to points in the complex plane that are symmetric about the y-axis, and $z_1 = 2-i$, then the point corresponding to the complex number $\frac{z_1}{z_2}$ in the complex plane is located in ( )
A: The first quadrant
B: The second quadrant
C: The third quadrant
D:... | \text{B} |
In an isosceles triangle, if one angle is $94^{\circ}$, then the degree of one of its base angles is ______$^{\circ}$. | 43^{\circ} |
$9 \cdot 13$ If $\left(\log _{2} 3\right)^{x}-\left(\log _{5} 3\right)^{x} \geqslant\left(\log _{2} 3\right)^{-y}-\left(\log _{5} 3\right)^{-y}$, then
(A) $x-y \geqslant 0$
(B) $x+y \geqslant 0$
(C) $x-y \leqslant 0$
(D) $x+y \leqslant 0$
(China High School Mathematics League, 1999) | x+y\geqslant0 |
A6 A sequence $a_{n}, n \in \mathbb{N}$ is given, for which $a_{n+1}-a_{n}=5$ and $a_{1}=37$. Which of the following numbers is a term of this sequence?
(A) 500
(B) 1000
(C) 1500
(D) 2000
(E) none of the above | E |
Determine all natural numbers $n$ with two different positive divisors that are equally distant from $\frac{n}{3}$.
(Richard Henner) | allpositivemultiplesof6 |
Let $n$ be a positive integer, and let $x=\frac{\sqrt{n+2}-\sqrt{n}}{\sqrt{n+2}+\sqrt{n}}$ and $y=\frac{\sqrt{n+2}+\sqrt{n}}{\sqrt{n+2}-\sqrt{n}}$.
It is given that $14x^2+26xy+14y^2=2014$. Find the value of $n$. | 5 |
Example 2 In $\triangle A B C$, $A B=A C, \angle B A C=$ $80^{\circ}, O$ is a point inside the triangle, $\angle O B C=10^{\circ}, \angle O C B=$ $30^{\circ}$. Find the degree measure of $\angle B A O$. | 70^{\circ} |
4. On the plane $(x ; y)$, plot the set of points satisfying the equation $|3 x|+|4 y|+|48-3 x-4 y|=48$, and find the area of the resulting figure.
# | 96 |
Together, the products by four and by five of an integer use each digit from 1 to 9 once, and only once. What is this number? | 2469 |
1. (6 points) $\triangle=\bigcirc+\bigcirc+\bigcirc, \triangle+\bigcirc=40$, then $\bigcirc=$ $\qquad$ , $\triangle=$ $\qquad$ | 10;30 |
Given the function $f(x) = ax^2 - (2a+1)x + a + 1$.
1. If $a=2$, solve the inequality $f(x) \geq 0$ for $x$.
2. If for all $a \in [-2, 2]$, it holds that $f(x) < 0$, find the range of values for the real number $x$. | x \in (1, \frac{3}{2}) |
Example 16 Find all positive integer triples $(x, y, z)$ that satisfy the equation
$$
8^{x}+15^{y}=17^{z}
$$ | (x,y,z)=(2,2,2) |
32. In which cases does a month have the maximum number of Saturdays? What is this number? | 5 |
Problem 2. The probability of a student passing the credit is 0.8. If the credit is passed, the student is allowed to take the exam, the probability of passing which is 0.9. Find the probability that the student will pass the credit and the exam. | 0.72 |
Which of the following transformations, using the commutative property of addition, is correct?
A: $3+(-2)=2+3$
B: $4+(-6)+3=(-6)+4+3$
C: $[5+(-2)]+4=[5+(-4)]+2$
D: $\frac{1}{6}+(-1)+(\frac{5}{6})=(\frac{1}{6}+\frac{5}{6})+(1)$ | B |
Given two complex numbers $z_1 = -2 + i$ and $z_2 = a + 2i$, where $i$ is the imaginary unit and $a \in \mathbb{R}$ (i.e., $a$ is a real number). If the product $z_1z_2$ is a real number, find the value of $a$. | 4 |
The integer 42 is
(A) an odd number
(D) divisible by 7
(B) a prime number
(C) a perfect cube | D |
3. Given the equation about $x$: $x^{2}+a|x|+a^{2}-3=0(a \in \mathbf{R})$ has a unique real solution, then $a=$ | \sqrt{3} |
18. In a polynomial of $5^{\text{th}}$ degree $p(x)=a x^{5}+b x^{4}+c x^{3}+d x^{2}+e x+f$, each of the coefficients $a, b, c, d, e, f$ is 1 or -1. Knowing that $p(2)=11$, what is the value of $p(3)$?
(A) 178
(B) 244
(C) 126
(D) 142
(E) 196 | 142 |
A set containing three real numbers can be represented as $\{a, \frac{b}{a}, 1\}$, and also as $\{a^2, a+b, 0\}$. Find the value of $a + a^2 + a^3 + \ldots + a^{2011} + a^{2012}$. | 0 |
3+ [ Completing the square. Sums of squares $]$
What should the values of $a$ and $b$ be so that the polynomial $x^{4}+x^{3}+2 x^{2}+a x+b$ is a perfect square? | =\frac{7}{8},b=\frac{49}{64} |
The tangents at $B$ and $A$ to the circumcircle of an acute angled triangle $ABC$ meet the tangent at $C$ at $T$ and $U$ respectively. $AT$ meets $BC$ at $P$, and $Q$ is the midpoint of $AP$; $BU$ meets $CA$ at $R$, and $S$ is the midpoint of $BR$. Prove that $\angle ABQ=\angle BAS$. Determine, in terms of ratios of... | a : b : c = \sqrt{2} : \sqrt{2} : 1 |
8. In the diagram, the outer hexagon is regular and has an area of 216 . What is the shaded area?
A 108
В 96
С 90
D 84
E 72 | 72 |
## Problem Statement
Write the equation of the plane passing through point $A$ and perpendicular to vector $\overrightarrow{B C}$.
$A(-1 ; 3 ; 4)$
$B(-1 ; 5 ; 0)$
$C(2 ; 6 ; 1)$ | 3x+y+z-4=0 |
Example 7 Let $x=(15+\sqrt{220})^{19}+(15+\sqrt{200})^{22}$. Find the unit digit of the number $x$.
| 9 |
Among the following shapes, the one that can coincide with itself after rotating $180^{\circ}$ around a certain point is $ \left(\ \ \right)$
A: Parallelogram
B: Equilateral triangle
C: Isosceles right triangle
D: Regular pentagon | A |
Define $\triangle ABC$ with incenter $I$ and $AB=5$, $BC=12$, $CA=13$. A circle $\omega$ centered at $I$ intersects $ABC$ at $6$ points. The green marked angles sum to $180^\circ.$ Find $\omega$'s area divided by $\pi.$ | \frac{16}{3} |
nic$\kappa$y is drawing kappas in the cells of a square grid. However, he does not want to draw kappas in three consecutive cells (horizontally, vertically, or diagonally). Find all real numbers $d>0$ such that for every positive integer $n,$ nic$\kappa$y can label at least $dn^2$ cells of an $n\times n$ square.
[i]P... | d = \frac{2}{3} |
Given that $i$ is the imaginary unit, and the complex number $z$ satisfies $z(3+4i)=1+i$, then the point representing the conjugate of $z$ in the complex plane is in the
A: First quadrant
B: Second quadrant
C: Third quadrant
D: Fourth quadrant | \text{A} |
Let $f\in C^1[a,b]$, $f(a)=0$ and suppose that $\lambda\in\mathbb R$, $\lambda >0$, is such that
$$|f'(x)|\leq \lambda |f(x)|$$
for all $x\in [a,b]$. Is it true that $f(x)=0$ for all $x\in [a,b]$? | f(x) = 0 |
Use $ \log_{10} 2 \equal{} 0.301,\ \log_{10} 3 \equal{} 0.477,\ \log_{10} 7 \equal{} 0.845$ to find the value of $ \log_{10} (10!)$.
Note that you must answer according to the rules:fractional part of $ 0.5$ and higher is rounded up, and everything strictly less than $ 0.5$ is rounded down,
say $ 1.234\longrightarrow... | 22 |
Given the function $f(x) = \frac{e^x - e^{-x}}{2}$, where $x \in \mathbb{R}$, if for any $\theta \in (0, \frac{\pi}{2}]$, the inequality $f(m\sin\theta) + f(1-m) > 0$ holds, then the range of the real number $m$ is ( )
A: $(0, 1)$
B: $(0, 2)$
C: $(-\infty, 1)$
D: $(-\infty, 1]$ | \text{D} |
1. (3 points) Calculate: $\left[2 \frac{3}{5}-\left(8.5-2 \frac{2}{3}\right) \div 3.5\right] \times 7 \frac{1}{2}=$ | 7 |
Find all real solutions $(x,y,z)$ of the system of equations:
\[
\begin{cases}
\tan ^2x+2\cot^22y=1 \\
\tan^2y+2\cot^22z=1 \\
\tan^2z+2\cot^22x=1 \\
\end{cases}
\] | (x, y, z) = \left(\frac{\pi}{6}, \frac{\pi}{6}, \frac{\pi}{6}\right) \text{ or } \left(\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{4}\right) |
## Problem Statement
Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
$M_{1}(1 ; 0 ; 2)$
$M_{2}(1 ; 2 ;-1)$
$M_{3}(2 ;-2 ; 1)$
$M_{0}(-5 ;-9 ; 1)$ | \sqrt{77} |
If the graph of the function $y=ax^{2}-x+1(a$ is a constant) has only one intersection with the $x$-axis, then $a$ satisfies which of the following?
A: $a=\frac{1}{4}$
B: $a\leqslant \frac{1}{4}$
C: $a=0$ or $a=-\frac{1}{4}$
D: $a=0$ or $a=\frac{1}{4}$ | D |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$, and $\overrightarrow{a}-\overrightarrow{b}=(\sqrt{3},\sqrt{2})$, then $|2\overrightarrow{a}-\overrightarrow{b}|$ equals:
A: $\sqrt{15}$
B: $\sqrt{17}$
C: $2\sqrt{2}$
D: $2\sqrt{5}$ | C |
16. As shown in Figure 3, 13 small circles with a radius of 1 and 1 large circle, all their intersection points are tangent points. Then the area of the shaded part is
(A) $(3 \sqrt{3}+2) \pi$
(B) $10(\sqrt{3}-1) \pi$
(C) $4 \sqrt{3} \pi$
(D) $7 \pi$
(E) $(\sqrt{3}+6) \pi$ | D |
33. A certain natural number divides $25, 38$, and $43$, resulting in a sum of the three remainders being 18. Find this natural number. | 11 |
1. Calculate the value of the numerical expression: $36+64 \cdot 17-(502-352: 8+511)$. | 155 |
Call a pair of integers $(a,b)$ [i]primitive[/i] if there exists a positive integer $\ell$ such that $(a+bi)^\ell$ is real. Find the smallest positive integer $n$ such that less than $1\%$ of the pairs $(a, b)$ with $0 \le a, b \le n$ are primitive.
[i]Proposed by Mehtaab Sawhney[/i] | 299 |
Given a regression equation $y = 3 - 5x$, when the variable $x$ increases by one unit,
A. $y$ decreases by an average of 5 units.
B. $y$ increases by an average of 3 units.
C. $y$ decreases by an average of 3 units.
D. $y$ increases by an average of 5 units. | \text{A} |
A [i]massless[/i] spring with spring constant $k$ is vertically mounted so that bottom end is firmly attached to the ground, and the top end free. A ball with mass $m$ falls vertically down on the top end of the spring, becoming attached, so that the ball oscillates vertically on the spring. What equation describes the... | e |
The maximum value of the function $y = -\frac{1}{x+1}$ in the interval $[1, 2]$ is ( )
A: $-\frac{1}{3}$
B: $-\frac{1}{2}$
C: $-1$
D: Does not exist | A |
11. Suppose $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=\sqrt{2}$ and $\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=0$. Find
$$
\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}
$$ | 2 |
In the arithmetic sequence $\left\{a_n\right\}$, $a_3$ and $a_{4017}$ are two different extreme points of the function $f(x)=x^3-6x^2+4x-1$. Then, ${\log }_{ \frac{1}{4}}{a}_{2010}$ is $(\ )$.
A: $\dfrac{1}{2}$
B: $2$
C: $-2$
D: $-\dfrac{1}{2}$ | \text{D} |
6.12. Find the derivative of the complex three-link function
$$
f(x)=\arcsin \left(\sqrt{1-e^{x}}\right), x \leqslant 0
$$ | \frac{-e^{x/2}}{2\sqrt{1-e^{x}}},x<0 |
A certain shopping mall purchased a batch of daily necessities. If they are sold at a price of $5$ yuan per item, they can sell $30,000$ items per month. If they are sold at a price of $6$ yuan per item, they can sell $20,000$ items per month. It is assumed that the monthly sales quantity $y$ (items) and the price $x$ ... | 40000 |
B2. Graphically solve the equation $2^{x+1}=-x^{2}+2$. | x_{1}=0x_{2}=-1 |
## 25. School
In all the classes of one school, the same number of students studied. After a fire, six classes of the school became unsuitable for lessons, and therefore, five more students had to be added to each class where lessons were held. But then, due to water damage from the fire hoses, another ten classes wer... | 900 |
Given vectors $a=(-2,3)$ and $b=(3,m)$, and $a$ is perpendicular to $b$, find the value of $m$. | 2 |
A temperature measured in degrees Celsius $(C)$ can be converted to degrees Fahrenheit $(F)$ using the formula $F=\frac{9}{5} C+32$. If the temperature is 10 degrees Celsius, what is the temperature in degrees Fahrenheit?
(A) -26.4
(B) -12.2
(C) 75.6
(D) 50.0
(E) 43.8 | 50 |
Bi 4. For any real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying the inequality $x_{1}+x_{2}+\cdots+x_{n} \leqslant \frac{1}{2}$. Find the minimum value of $\left(1-x_{1}\right)\left(1-x_{2}\right) \cdots\left(1-x_{n}\right)$. | \frac{1}{2} |
Given $f(x) = \log_a x$, when $x > 1$, $f(x) > 0$. Then, when $0 < m < 1 < n$, the correct equation is ( )
A: $0 < f(m) < f(n)$
B: $f(m) < 0 < f(n)$
C: $f(n) < f(m) < 0$
D: $f(n) < 0 < f(m)$ | \text{B} |
1. 182 For 1 to 1000000000, find the sum of the digits of each number; then for the resulting 1 billion numbers, find the sum of the digits of each number, $\cdots$, until obtaining 1 billion single-digit numbers. Question: Among the resulting numbers, are there more 1s or 2s? | 1 |
3.137. $\frac{\cos 7 \alpha-\cos 8 \alpha-\cos 9 \alpha+\cos 10 \alpha}{\sin 7 \alpha-\sin 8 \alpha-\sin 9 \alpha+\sin 10 \alpha}$. | \operatorname{ctg}\frac{17\alpha}{2} |
7. Given the parabola $x^{2}=-8 y$ whose directrix intersects the $y$-axis at point $A$, a line through $A$ intersects the parabola at points $M$ and $N$. Point $B$ lies on the axis of symmetry of the parabola, and $\left(\overrightarrow{B M}+\frac{1}{2} \overrightarrow{M N}\right) \perp \overrightarrow{M N}$. Then the... | (6,+\infty) |
2. If the real number $\alpha$ satisfies $\cos \alpha=\tan \alpha$, then $\frac{1}{\sin \alpha}+\cos ^{4} \alpha=$ $\qquad$ | 2 |
Suppose the sum of the first $n$ terms of the sequence ${a_n}$ is $S_n$, with $a_1=1$ and $S_n + a_n = 2n$ for all $n \in \mathbb{N}^*$.
(1) Prove that the sequence ${a_n - 2}$ is a geometric sequence.
(2) Find the sum of the first $n$ terms of the sequence ${S_n}$, denoted as $T_n$. | T_n = n^2 - n + 2 - \left(\frac{1}{2}\right)^{n-1} |
2. Find the smallest positive integer $a$, such that there exists a positive odd integer $n$, satisfying
$$2001 \mid\left(55^{n}+a \cdot 32^{n}\right)$$ | 436 |
Given a function $f(x)=|x^{3}+ax+b|$ where $a,b\in\mathbb{R}$, if for any $x_1$, $x_2\in[0,1]$, the inequality $f(x_1)-f(x_2)\leqslant 2|x_1-x_2|$ always holds, then the range of values for $a$ is __________. | [-2, -1] |
11. The smallest positive period of the function $y=\sqrt[2 n]{\sin x}\left(n \in \mathbf{N}^{*}\right)$ is | 2\pi |
Let $P$ be a real polynomial such that $P(2)=5$ and $P(1)=3$. Determine the remainder of the Euclidean division of $P$ by $(X-2)(X-1)$. | 2X+1 |
PROBLEM 4. We say that a number of the form $\overline{0, a b c d e}$ has property $(P)$ if the digits $a, b, c, d, e$ belong to the set $\{4,6\}$.
a) Show that the solution to the equation $x+0,46646=1,1111$ has property $(P)$;
b) Determine how many different numbers of the form $\overline{0, a b c d e}$ have proper... | 32 |
Example 1. Segment $AB$ is divided by point $C(-3,0)$ in the ratio $\lambda=\frac{AC}{CB}=\frac{2}{3}$. Find the length of $AB$, if point $A(-5,-4)$ is given. | 5\sqrt{5} |
## Task B-4.6.
A square $K_{1}$ with an area of 1 is circumscribed by a circle, and to this circle, a square $K_{2}$ is circumscribed, whose sides are parallel to the sides of the initial square $K_{1}$. The area between the squares $K_{1}$ and $K_{2}$ is colored gray. Then, a circle is circumscribed around the square... | \frac{1}{3}(2^{4048}-1) |
# 6. Problem 6
Given a sequence starting with one, in which each subsequent term is equal to twice the sum of all previous terms. Find the smallest number so that the element under this number is divisible by $3^{2017}$. Points for the problem: 13.
# | 2019 |
5. The number of common terms (terms with the same value) in the arithmetic sequences $2,5,8, \cdots, 2021$ and $4,9, 14, \cdots, 2019$ is $\qquad$. | 134 |
Determine all pairs of positive integers $(x, y)$ satisfying the equation $p^x - y^3 = 1$, where $p$ is a given prime number. | (1, 1) |
In $\triangle ABC$, if $b=1$, $c= \sqrt {3}$, $\angle C= \dfrac {2\pi}{3}$, find the area of $\triangle ABC$. | \dfrac {\sqrt {3}}{4} |
4. If there exists a real number $\theta$ that satisfies $\cos \theta-\sin ^{2} \theta=\alpha$, then the range of real number $\alpha$ is | -\frac{5}{4} \leqslant \alpha \leqslant 1 |
A four-digit number has the following property: dividing this four-digit number by its last two digits results in a perfect square (if the tens digit is zero, then only the units digit is used for division), and this perfect square is exactly the square of the sum of the first two digits plus 1. For example, $4802 \div... | 1805 |
(5 points) (2015•Wanzhou District Mock) 8 people are seated in a row. Now, if we want to change the positions of 3 of these people while keeping the other 5 people's positions unchanged, then the number of different ways to do this is ( )
A． $C^8_3$ B． $C^8_3A^8_3$ C． $C^8_3A^3_2$ D． $3C^8_3$ | C |
5. Pinocchio's nose is very magical, every time he tells a lie, his nose grows 3 centimeters; every time he tells the truth, his nose shortens 1 centimeter. His nose started at 10 centimeters, after saying 10 sentences his nose became 28 centimeters. So, he told $\qquad$ true sentences. | 3 |
1.3. Let $f: \mathbb{R} \longrightarrow \mathbb{R}$ be a function from the set $\mathbb{R}$ of real numbers to itself. Find all such functions $f$ satisfying the two properties:
(a) $f(x+f(y))=y+f(x)$ for all $x, y \in \mathbb{R}$,
(b) the set $\left\{\frac{f(x)}{x}: x\right.$ is a nonzero real number $\}$ is finite. | f(x)=xorf(x)=-x |
In the Cartesian coordinate plane xOy, let A be a point on the semicircle O: x2+y2=2 (x≥0), the slope angle of the line OA is 45°, draw a vertical line from point A to the x-axis, with the foot of the perpendicular being H, draw a line parallel to OA passing through H, intersecting the semicircle at point B, then the e... | \sqrt {3}x + y - \sqrt {3} - 1 = 0 |
4. The matter is settled. A rower, passing under a bridge, lost his hat. After 15 minutes, he noticed the loss and caught the hat a kilometer from the bridge. What is the speed of the river's current? | 2\kappa |
81. In the box, there are pencils: 7 red and 5 blue. In the dark, pencils are taken. How many pencils need to be taken to ensure that there are at least two red and at least three blue among them? | 10 |
Determine the range of the function $y=2\cos^{2}x+3\sin x+3, x\in \left[\dfrac{\pi}{6}, \dfrac{2\pi}{3}\right]$. | \left[6, \dfrac{49}{8}\right] |
Given vectors $\overrightarrow{m}=(\sqrt{3}\sin \frac{x}{4},1)$ and $\overrightarrow{n}=(\cos \frac{x}{4},\cos^2 \frac{x}{4})$, let $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}$.
(Ⅰ) Find the interval of monotonic decrease for $f(x)$.
(Ⅱ) If $f(a)= \frac{3}{2}$, find the value of $\cos (\frac{2\pi}{3}-a)$.
(... | \left[0, \frac{3}{2}\right] |
Given $3^{a}=5^{b}=15$, the relationship between $a$ and $b$ is:
A: $\frac{1}{a}+\frac{1}{b}=1$
B: $ab \gt 4$
C: $a^{2}+b^{2} \lt 4$
D: $(a+1)^{2}+\left(b+1\right)^{2} \gt 16$ | ABD |
Given the hyperbola $\dfrac {x^{2}}{4}- \dfrac {y^{2}}{b^{2}}=1(b > 0)$, and a circle centered at the origin with the real semi-axis length of the hyperbola as its radius intersects the two asymptotes of the hyperbola at points $A$, $B$, $C$, and $D$. The area of quadrilateral $ABCD$ is $2b$. Then, the equation of the ... | D |
Problem 4. Given is $\triangle A B C$ with
$$
\measuredangle B A C=40^{\circ}, \measuredangle A B C=20^{\circ} \text { and } \overline{A B}-\overline{B C}=5 \text{ cm}
$$
If the angle bisector of $\measuredangle A C B$ intersects $A B$ at point $M$, find the length of $C M$. | 5\mathrm{~} |
For example, $5 x, y, z$ are positive real numbers, and satisfy $x+y+z=1$. Find the minimum value of $\frac{x^{4}}{y\left(1-y^{2}\right)}+\frac{y^{4}}{z\left(1-z^{2}\right)}+$ $\frac{z^{4}}{x\left(1-x^{2}\right)}$. (Mathematics Bulletin Problem) | \frac{1}{8} |
Given an ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, the length of the major axis is $\frac{3\sqrt{5}}{5}$ times the length of the minor axis. Let $A$ be the left vertex of the ellipse $C$, $F$ be the right focus of the ellipse $C$, and points $M(x_0, y_0) (x_0 > 0, y_0 > 0)$, $N$ are on the ellip... | \frac{5\sqrt{3}}{3} |
To obtain the graph of the function $y=3\cos (2x- \frac {\pi}{2})$, the graph of the function $y=3\sin (2x- \frac {\pi}{4})$ needs to be translated to the left along the $x$-axis by \_\_\_\_\_\_ units. | \frac {\pi}{8} |
A warehouse has six batches of goods, with weights of 150 tons, 160 tons, 180 tons, 190 tons, 200 tons, and 310 tons, respectively. The first shipment takes away two batches of goods, and the second shipment takes away three batches of goods. If the total weight of the goods taken away in the first shipment is half of ... | 200 |
Among the following sequences, which are both increasing and infinite sequences? (Fill in the question number)
(1) 1, 2, 3, …, 20;
(2) -1, -2, -3, …, -n, …;
(3) 1, 2, 3, 2, 5, 6, …;
(4) -1, 0, 1, 2, …, 100, … | 4 |
20. Each chocolate costs 1 dollar, each lioorice stick costs 50 cents and each lolly costs 40 cents. How many different combinations of these three items cost a total of 10 dollars? | 36 |
1. Given the set of real numbers $\{1,2,3, x\}$, the maximum element of the set is equal to the sum of all elements in the set, then the value of $x$ is $\qquad$ . | -3 |
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