problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
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Example 13 Given $a+b+c=0, a^{3}+b^{3}+c^{3}$ $=0$. Find the value of $a^{15}+b^{15}+c^{15}$. | 0 |
Problem 5. Find all two-digit numbers $x$ for which two of the following statements are true, and two are false:
1) $x$ is divisible by 5;
2) $x$ is divisible by 23;
3) $x+7$ is a square of some natural number;
4) $x-10$ is a square of some natural number. | 35,46,74 |
Given $5m+3n=2$, then $10m+6n-5=\_\_\_\_\_\_$. | -1 |
Given that the function $f(x+2)$ is an odd function, and it satisfies $f(6-x)=f(x)$, $f(3)=2$, then the value of $f(2008)+f(2009)$ is ( )
A: 0
B: 2
C: -2
D: 2009 | C |
2. (8 points) A certain hamburger in the supermarket costs 10 yuan each. This hamburger has recently launched a "buy two get one free" promotion, which means if you buy two hamburgers, you can get one hamburger for free. It is known that Dongdong and his friends need to buy 9 hamburgers. How much money do they need to ... | 60 |
3 . Let $n$ be a positive integer, $d$ be a digit among the 10 decimal digits, and suppose $\frac{n}{810}=0 . d 25 d 25 d 25 \cdots$, find $n$.
| 750 |
Convert radians to degrees: $$\frac {2}{3}\pi$$ = \_\_\_\_\_\_. | 120^\circ |
The shortest distance from a point on the curve $f(x)=\ln (2x-1)$ to the line $2x-y+3=0$ is ( )
A: $\sqrt{5}$
B: $2\sqrt{5}$
C: $3\sqrt{5}$
D: $0$ | A |
The complex number $(1+i)(1- i) =\_\_\_\_\_\_\_\_\_$ | 2 |
Given a complex number $z=a+(a+1)i$ ($a\in \mathbb{R}$) is a pure imaginary number, then the value of $z^{2010}$ is ( )
A: -1
B: 1
C: -i
D: i | \text{A} |
Given that in the complex plane, $i$ is the imaginary unit, the complex number $z= \frac {1+ai}{1-i}(a∈R)$ corresponds to a point on the line $x-y=1$. Find the value of $a=($ $)$.
A: $(-2)$
B: $(-1)$
C: $(1)$
D: $(2)$ | B |
Given the parabola $C_{1}: y^{2}=8x$ whose directrix intersects with the hyperbola $C_{2}: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ $(a > 0,b > 0)$ at points $A$ and $B$, and one of the asymptotes of the hyperbola $C_{2}$ is $y=\frac{4\sqrt{3}}{3}x$. If point $F$ is the focus of the parabola $C_{1}$, and $\triangle F... | D |
On a spherical surface with an area of $60\pi$, there are four points $S$, $A$, $B$, and $C$, and $\triangle ABC$ is an equilateral triangle. The distance from the center $O$ of the sphere to the plane $ABC$ is $\sqrt{3}$. If the plane $SAB$ is perpendicular to the plane $ABC$, then the maximum volume of the pyramid $S... | 27 |
If the abscissa of the points on the curve $x^{2}+y^{2}=1$ is changed to $\frac{1}{2}$ times the original value, and the ordinate is changed to $2$ times the original value, the equation of the curve $C$ obtained is ( ).
A: $2{x^2}+\frac{{{y^2}}}{2}=1$
B: $\frac{{{x^2}}}{2}+2{y^2}=1$
C: $\frac{{{x^2}}}{4}+4{y^2}=1$
... | D |
The function $y=f(x-1)$ is an odd function, and $y=f(x+1)$ is an even function (both defined on $\mathbb{R}$). If $0 \leq x < 1$, then $f(x) = 2^x$, find $f(10)$. | 1 |
If the solution to the linear equation $ax+b=0$ (where $a \neq 0$) is a positive number, then ( )
A: $a$ and $b$ have opposite signs
B: $b$ is greater than $0$
C: $a$ and $b$ have the same sign
D: $a$ is less than $0$ | \text{A} |
Among the four faces of a pyramid, the maximum number of right-angled triangles is ____. | 4 |
If two circles $x^2+y^2=r^2$ ($r>0$) and $(x-3)^2+(y+4)^2=49$ have a common point, then the range of $r$ is. | 2 \leq r \leq 12 |
If $a \in \mathbb{R}$, then "1 < a < 2" is a ( ) condition for "$a^2 - 3a \leq 0$".
A: Sufficient but not necessary
B: Necessary but not sufficient
C: Necessary and sufficient
D: Neither sufficient nor necessary | \text{A} |
N20 (20-1, Cuba) The last three digits of $1978^{n}$ and $1978^{m}$ are equal. Try to find positive integers $m$ and $n$, such that $m+n$ takes the minimum value (here $n>m \geqslant 1$). | 106 |
(This question is worth 10 points) Use the Euclidean algorithm to find the greatest common divisor (GCD) of 888 and 1147. | 37 |
4. Given real numbers $x, y$ satisfy $\frac{x^{2}}{3}+y^{2}=1$. Then
$$
P=|2 x+y-4|+|4-x-2 y|
$$
the range of values for $P$ is $\qquad$. | [2,14] |
Positive integers $a_0<a_1<\dots<a_n$, are to be chosen so that $a_j-a_i$ is not a prime for any $i,j$ with $0 \le i <j \le n$. For each $n \ge 1$, determine the smallest possible value of $a_n$. | 4n + 1 |
9. The three prime numbers $a, b, c$ satisfy $a<b<c<100$, and $(b-a) \times(c-b) \times(c-a)=240$, then the maximum sum of the three prime numbers $a, b, c$ is $\qquad$ _. | 251 |
* 6. Among $n(\geqslant 3)$ lines, exactly $p(\geqslant 2)$ are parallel, and no three lines among the $n$ lines intersect at the same point. Then the number of regions into which these $n$ lines divide the plane is | \frac{1}{2}\left(n^{2}-p^{2}+n+p+2\right) |
Paint three faces of an $8 \times 8 \times 8$ cube red and three faces blue (requiring that any three faces sharing a common vertex cannot all be the same color), then cut it into 512 $1 \times 1 \times 1$ small cubes. Among these 512 small cubes, there are $\qquad$ small cubes that have both red and blue faces. | 56 |
8.1. A truck and a car are moving in the same direction on adjacent lanes at speeds of 65 km/h and 85 km/h, respectively. At what distance from each other will they be 3 minutes after they are side by side? | 1 |
3. In parallelogram $A B C D$, the internal angle $\alpha$ is acute, side $B C$ measures $4.5 \mathrm{~cm}$. On side $A B$ lies point $G$ such that $|A G|:|G B|=2: 3$. The line through $C G$ intersects the line through $A D$ at point $P$. Calculate the length of segment $A P$. Draw a sketch. | 3 |
3. When $\frac{\pi}{4} \leqslant x \leqslant \frac{\pi}{2}$, the range of $y=\sin ^{2} x+\sqrt{3} \sin x \cos x$ is $\qquad$ | y\in[1,\frac{3}{2}] |
9.185. $x^{2}\left(x^{4}+36\right)-6 \sqrt{3}\left(x^{4}+4\right)<0$. | x\in(-\sqrt[4]{12};\sqrt[4]{12}) |
52. Xiao Pang's mother went to buy apples, wanting to buy 5 kilograms. When paying, she found she was short of 3 yuan and 5 jiao. As a result, she bought 4 kilograms and had 1 yuan and 5 jiao left. Xiao Pang's mother brought a total of $\qquad$ yuan $\qquad$ jiao. | 21,5 |
Example 1 Set $A=\left\{(x, y) \mid y=x^{2}+m x+2\right\}, B=\{(x, y) \mid x-y+1=0$ and $0 \leqslant x \leqslant 2\}$. If $A \cap B \neq \varnothing$, find the range of values for $m$. | \leqslant-1 |
Given a sequence \{a_n\} satisfying $a_1 = 1$ and $a_n = \log_n (n+1)$ for $n \geq 2$ and $n \in \mathbb{N}^*$. Define: $k \in \mathbb{N}^*$ that makes the product $a_1 \times a_2 \times \ldots \times a_k$ a positive integer to be called "easy integer". Find the sum of all "easy integers" in the interval [1, 2015]. | 2036 |
Problem 8.8.1. Different positive numbers $a, b, c$ are such that
$$
\left\{\begin{array}{l}
a^{2}+b c=115 \\
b^{2}+a c=127 \\
c^{2}+a b=115
\end{array}\right.
$$
Find $a+b+c$. | 22 |
G7.2 When the positive integer $N$ is divided by $6,5,4,3$ and 2, the remainders are $5,4,3,2$ and 1 respectively. Find the least value of $N$. | 59 |
$\underline{\text { Vasiliev N.B. }}$
How many maximum parts can the coordinate plane xOy be divided into by the graphs of 100 quadratic trinomials of the form
$y=a_{n} X^{2}+b_{n} x+c_{n}(n=1,2, \ldots, 100) ?$ | 10001 |
6.293
$$
\frac{\sqrt[7]{x-\sqrt{2}}}{2}-\frac{\sqrt[7]{x-\sqrt{2}}}{x^{2}}=\frac{x}{2} \cdot \sqrt[7]{\frac{x^{2}}{x+\sqrt{2}}}
$$ | 1 |
27.3. (СІІІА, 83). From the vertices of a regular $n$-gon $(n \geqslant 6)$, two different triples of points are chosen at random. What is the probability that the two triangles, whose vertices are the chosen triples, do not intersect? | 0.3 |
3. The gnomes dug a system of tunnels that formed a rectangular grid 7 by 2. The main entrance is located at the bottom-left vertex $(0,0)$, and the main exit is at the top-right vertex $(7,2)$. The tunnels are numbered as follows: the tunnel connecting vertices $(x, k)$ and $(x+1, k)$ has a number equal to $7 k + x + ... | 8 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. If $\sin (A-B)= \frac {a}{a+b}\sin A\cos B- \frac {b}{a+b}\sin B\cos A$.
$(1)$ Prove that: $A=B$;
$(2)$ If $A= \frac {7\pi}{24}$ and $a= \sqrt {6}$, find the area of $\triangle ABC$. | \frac {3( \sqrt {2}+ \sqrt {6})}{4} |
Given that $f''(x)$ is the derivative of $f(x)$, then $\lim_{t\to 0} \frac{f(3)-f(3-t)}{t}=$ ( ).
A: $f''(3)$
B: $f''(t)$
C: $-f''(3)$
D: $-f''(t)$ | A |
9. Given that the line $l$ with slope $k$ passing through the point $P(3,0)$ intersects the right branch of the hyperbola $C: x^{2}-\frac{y^{2}}{3}=1$ at points $A$ and $B$, and $F$ is the right focus of the hyperbola $C$, and $|A F|+|B F|=16$, find the value of $k$. | \3 |
2. Let planar vectors $\boldsymbol{a} 、 \boldsymbol{b}$ satisfy $|\boldsymbol{a}+\boldsymbol{b}|=3$. Then the maximum value of $a \cdot b$ is $\qquad$ | \frac{9}{4} |
Let $a$ be a real number, and the function $f(x) = x^3 + ax^2 + (a - 2)x$. The derivative of $f(x)$ is $f'(x)$, and $f'(x)$ is an even function. Then, the equation of the tangent line to the curve $y = f(x)$ at the origin is
A: $y = -2x$
B: $y = 3x$
C: $y = -3x$
D: $y = 4x$ | \text{A} |
Given $a$ and $b$ are lines, and $\alpha$, $\beta$, $\gamma$ are planes, which of the following statements is incorrect? ( )
A: If $a\parallel\alpha$, $a\parallel\beta$, and $\alpha\cap\beta=b$, then $a\parallel b$;
B: If $a\perp\alpha$, $b\perp\beta$, and $a\perp b$, then $\alpha\perp\beta$;
C: If $\alpha\perp\gamma$,... | \text{C} |
How many three-digit numbers with no repeated digits can be formed using the digits 0, 1, 2, 3, and 4? | 48 |
II. (25 points) A group purchases tickets to a park, with ticket prices as shown in Table 1:
Table 1
\begin{tabular}{|c|c|c|c|}
\hline Number of ticket buyers & $1 \sim 50$ & $51 \sim 100$ & Over 100 \\
\hline Price per ticket & 13 yuan & 11 yuan & 9 yuan \\
\hline
\end{tabular}
There are two tour groups, Group A and ... | 41, 71 |
Among the four numbers $0$, $-1$, $1$, $-5$, the smallest number is
A: $-5$
B: $-1$
C: $1$
D: $0$ | A |
Problem 9.1. Let $p$ be a real parameter such that the equation $x^{2}-3 p x-p=0$ has real and distinct roots $x_{1}$ and $x_{2}$.
a) Prove that $3 p x_{1}+x_{2}^{2}-p>0$.
b) Find the least possible value of
$$
A=\frac{p^{2}}{3 p x_{1}+x_{2}^{2}+3 p}+\frac{3 p x_{2}+x_{1}^{2}+3 p}{p^{2}}
$$
When does equality obta... | 2 |
1. Find a multiple of 2018 whose decimal expansion's first four digits are 2017. | 20171928 |
3. As shown in Figure 2, in $\triangle A B C$, $A B=A C$, $\angle B A C=78^{\circ}$, $P$ is a point inside $\triangle A B C$, and $\angle B C P=42^{\circ}$, $\angle A P C$ $=162^{\circ}$. Then the degree measure of $\angle P B C$ is | 21^{\circ} |
If $x \gt y \gt 0$, which of the following inequalities is correct?
A: $|x| \lt |y|$
B: $x^{2} \lt y^{2}$
C: $\frac{1}{x}＜\frac{1}{y}$
D: $\frac{x+y}{2}≤\sqrt{xy}$ | \text{C} |
Given $a$ is a constant, and the quadratic function $y=x^{2}-ax+a+3$.
$(1)$ If the graph of this quadratic function intersects the $x$-axis, find the range of values for $a$;
$(2)$ Given $y\geqslant 4$, find the solution set for $x$;
$(3)$ If there exists $x\in \left[2,4\right]$ such that $y=0$, find the range of... | [6,7] |
## SUBJECT III
Consider the adjacent angles $\mathrm{AOB}$ and $\mathrm{BOC}$ such that the measure of angle $\mathrm{BOC}$ is 5 times greater than the measure of angle $\mathrm{AOB}$. The bisector $\mathrm{OM}$ of angle BOC is constructed, $\mathrm{ON}$ is the opposite ray of ray $\mathrm{OM}$, and $\mathrm{DO}$ is p... | \mathrm{}(\nless\mathrm{AOB})=20,\mathrm{}(\nless\mathrm{BOC})=100,\mathrm{}(\nless\mathrm{COD})=40,\mathrm{}(\nless\mathrm{AON})=110 |
9. The Yangtze Farm has a pasture, where grass grows uniformly every day. If 24 cows are grazed on the pasture, they will finish the grass in 6 days; if only 21 cows are grazed, it will take 8 days to finish the grass. How many days will it take to finish the grass if 36 cows are grazed? | 3 |
Let's explain the following facts. The sequence of digits "1221" is first considered as a number in the base-13, then in the base-12, followed by the base-11, and finally in the base-10 number system. In each case, we convert our number to the number system with the base one less. In the first three cases, the result o... | 1221_{b}=1596_{b-1} |
8. Let the area of $\triangle A B C$ be $1, D$ be a point on side $B C$, and $\frac{B D}{D C}=\frac{1}{2}$. If a point $E$ is taken on side $A C$ such that the area of quadrilateral $A B D E$ is $\frac{4}{5}$, then the value of $\frac{A E}{E C}$ is $\qquad$. | \frac{7}{3} |
9. Teacher Wang and Teacher Li are walking around a circular path by the lake. They start from the same point at the same time, walking in opposite directions. After meeting, Teacher Wang continues walking in the original direction for 40 minutes to return to the starting point. It is known that Teacher Li takes 30 min... | 60 |
Given vectors $\overline{a}=(1,2,2)$ and $\overrightarrow{b}=(6,-3,2)$, which of the following conclusions is correct?
A: $\overrightarrow{a}+\overrightarrow{b}=(7,-1,4)$
B: $\overrightarrow{a}-\overline{b}=(-5,-1,0)$
C: $\overrightarrow{a}•\overrightarrow{b}=4$
D: $|\overrightarrow{a}|=\sqrt{5}$ | \text{AC} |
The front view of a cylinder is a rectangle with an area of $6$, its lateral area is ( ).
A: $8\pi$
B: $6\pi$
C: $4\pi$
D: $3\pi$ | B |
3. If a straight line $l$ passes through three different points $A(a, b)$, $B(b, a)$, and $C(a-b, b-a)$, then the line $l$ passes through ( ) quadrants.
(A) II, IV
(B) I, III
(C) II, III, IV
(D) I, III, IV | A |
1.2. Inside a rectangular parallelepiped $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$, there are two spheres such that the first one touches all three faces containing $A$, and the second one touches all three faces containing $C^{\prime}$. Additionally, the spheres touch each other. The radii of the spheres a... | 41 |
30th IMO 1989 shortlist Problem 15 Given a convex n-gon A 1 A 2 ... A n with area A and a point P, we rotate P through an angle x about A i to get the point P i . Find the area of the polygon P 1 P 2 ... P n . | 4\sin^2(\frac{x}{2})A |
[b]a)[/b] Among $9$ apparently identical coins, one is false and lighter than the others. How can you discover the fake coin by making $2$ weighing in a two-course balance?
[b]b)[/b] Find the least necessary number of weighing that must be done to cover a false currency between $27$ coins if all the others are true. | 3 |
Given that $a > 0$ and $b > 0$, $\sqrt {2}$ is the geometric mean of $a$ and $b$, and $\log _{a}x=\log _{b}y=3$, find the minimum value of $\frac {1}{x}+ \frac {1}{y}$. | \frac { \sqrt {2}}{2} |
Two students went to a company for a job interview together. Before the interview, the person in charge of the company told them, "We will hire 3 people from those interviewed, and the probability of both of you being hired together is $\frac{1}{70}$." Based on what this person said, we can infer that the number of peo... | A |
Let the sequence $\{a_n\}$ be such that $a_1 = 3$ and $a_n = a_{n-1} + 3^n$ where $n \in \mathbb{N}^*$ and $n \geq 2$. Find the general term $a_n$. | \frac{3}{2} (3^n - 1) |
Example 2 Draw three rays $O A, O B, O C$ from point $O$, with the angles between each pair being $60^{\circ}, 90^{\circ}, 120^{\circ}$, respectively. What is the minimum value of the angles between the angle bisectors of these three angles? | 45 |
7. Let $A_{1}$ and $A_{2}$ be the left and right vertices of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$. If there exists a point $P$ on the ellipse, different from $A_{1}$ and $A_{2}$, such that $\overrightarrow{P O} \cdot \overrightarrow{P A_{2}}=0$, where $O$ is the origin, then the range of the e... | D |
2. The arithmetic mean of several consecutive natural numbers is 5 times greater than the smallest of them. How many times is the arithmetic mean less than the largest of these numbers? | 1.8 |
Given that $m$ and $n$ are two roots of the quadratic equation $x^{2}+2x-5=0$, find the value of $m^{2}+mn+2m$. | 0 |
Given vectors $\overrightarrow{a}=(3,2)$ and $\overrightarrow{b}=(0,-1)$, find the coordinates of the vector $3\overrightarrow{b}-\overrightarrow{a}$. | (-3,-5) |
8,9
In a triangular pyramid $P A B C$, the lateral edge $P B$ is perpendicular to the plane of the base $A B C$, $P B=6, A B=B C$ $=\sqrt{15}, A C=2 \sqrt{3}$. A sphere with center $O$ lying on the face $A B P$ touches the planes of the other faces of the pyramid. Find the distance from the center $O$ of the sphere to... | \frac{24}{6+\sqrt{15}} |
Example 3. Calculate the circulation of the vector $\mathbf{a}=\sqrt{1+x^{2}+y^{2}} \mathbf{i}+y[x y+$ $\left.\ln \left(x+\sqrt{1+x^{2}+y^{2}}\right)\right]$ along the circle $x^{2}+y^{2}=R^{2}$. | \frac{\piR^{4}}{4} |
3. Given the function $f(x)=\sin \omega x+\cos \omega x(\omega>0), x \in \mathbf{R}$, if the function $f(x)$ is monotonically increasing in the interval $(-\omega, \omega)$, and the graph of the function $y=f(x)$ is symmetric about the line $x=\omega$, then the value of $\omega$ is | \frac{\sqrt{\pi}}{2} |
Which of the following sets of four line segments are in proportion?
A: $1cm$, $2cm$, $3cm$, $4cm$
B: $2cm$, $4cm$, $6cm$, $8cm$
C: $5cm$, $30cm$, $10cm$, $15cm$
D: $5cm$, $20cm$, $10cm$, $15cm$ | C |
123. Find the coordinates of the foci, the lengths of the axes, and the eccentricity of the ellipse given by the equation $2 x^{2}+y^{2}=32$. | F_{1}(0;4),F_{2}(0;-4),2b=8\sqrt{2},2a=8,\varepsilon=\frac{\sqrt{2}}{2} |
If the system of equations $\left\{\begin{array}{l}x+2y=4\\ 3x+ay=6\end{array}\right.$ has no solution, then the real number $a=$____. | a = 6 |
Solve the equation $y^3 = x^3 + 8x^2 - 6x +8$, for positive integers $x$ and $y$. | (x, y) = (9, 11) |
To obtain the graph of the function $$y=4\sin(2x+ \frac {\pi}{5}), x\in\mathbb{R}$$, it is only necessary to modify the graph of the function $$y=4\sin(x+ \frac {\pi}{5}), x\in\mathbb{R}$$ by ( )
A: Stretching the x-coordinates to twice their original length, keeping the y-coordinates unchanged
B: Stretching the y-coo... | C |
1. The number of non-negative integer solutions to $|x+1| \leqslant 3$ is ( ).
(A) 1
(B) 2
(C) 3
(D) 4 | C |
In the garden of Anya and Vitya, there were 2006 rose bushes. Vitya watered half of all the bushes, and Anya watered half of all the bushes. It turned out that exactly three bushes, the most beautiful ones, were watered by both Anya and Vitya. How many rose bushes remained unwatered? | 3 |
3. Let $a_{1}, a_{2}, \cdots, a_{2011}$ be positive real numbers, $S=\sum_{i=1}^{2011} a_{i}$, and
$$
\begin{array}{l}
(S+2011)\left(S+a_{1} a_{2}+a_{2} a_{3}+\cdots+a_{2011} a_{1}\right) \\
=4\left(\sqrt{a_{1} a_{2}}+\sqrt{a_{2} a_{3}}+\cdots+\sqrt{a_{2011} a_{1}}\right)^{2} .
\end{array}
$$
Then $S=$ . $\qquad$ | 2011 |
22. A worker takes a time ratio of $1: 2: 3$ to make 1 $\mathrm{A}$ part, 1 $\mathrm{B}$ part, and 1 $\mathrm{C}$ part, respectively. He can make 2 $\mathrm{A}$ parts, 3 $\mathrm{B}$ parts, and 4 $\mathrm{C}$ parts in 10 working hours. If he needs to make 14 $\mathrm{A}$ parts, 10 $\mathrm{B}$ parts, and 2 $\mathrm{C}$... | 20 |
Consider the function $f(x) = x^2 - 1$. Calculate the average rate of change of the function as the independent variable $x$ changes from 1 to 1.1. | 2.1 |
Example 4.38. Write the Fourier series for the function $f(x)$ defined by the equations
$$
\begin{gathered}
f(x)=\left\{\begin{array}{l}
-\frac{\pi}{4},-\pi<x<0 \\
0, x=0, x=-\pi \\
\frac{\pi}{4}, 0<x<\pi
\end{array}\right. \\
f(x)=f(x+2 \pi) .
\end{gathered}
$$ | f(x)\sim\sum_{n=1}^{\infty}\frac{\sin(2n-1)x}{2n-1} |
Given that $a$ and $b$ are opposite numbers, $c$ and $d$ are reciprocals, and $\left(x+3\right)^{2}$ and $|y-2|$ are opposite numbers, find the value of $2\left(a+b\right)-2\left(cd\right)^{4}+\left(x+y\right)^{2022}$. | -1 |
4. As shown in Figure 2, the edge length of the regular tetrahedron $ABCD$ is $6 \, \text{cm}$. There are points $E$ and $F$ on edges $AB$ and $CD$, respectively. If $AE = 1 \, \text{cm}$ and $CF = 2 \, \text{cm}$, then the length of line segment $EF$ is $\qquad$ cm. | \sqrt{23} |
Under standard temperature and atmospheric pressure, the product of the concentration of hydrogen ions (in moles per liter, represented as $[H^+]$) and the concentration of hydroxide ions (in moles per liter, represented as $[OH^-]$) in human blood is a constant equal to $10^{-14}$. Given the definition of the pH value... | C |
## Problem Statement
Find the cosine of the angle between vectors $\overrightarrow{A B}$ and $\overrightarrow{A C}$.
$A(3 ; 3 ;-1), B(1 ; 5 ;-2), C(4 ; 1 ; 1)$ | -\frac{8}{9} |
Given vectors $\overrightarrow{a}=(\cos θ,\sin θ)$ and $\overrightarrow{b}=(1,-2)$, if $\overrightarrow{a}//\overrightarrow{b}$, find the algebraic expression for $\frac {\sin θ-\cos θ}{\sin θ+\cos θ}=$ ___. | 3 |
Given $a=\log_2 0.3$, $b=3^{0.2}$, $c=0.3^2$, determine the correct order of these numbers.
A: $a<c<b$
B: $a<b<c$
C: $c<b<a$
D: $c<a<b$ | A |
Given that the binomial expansion of $(x-2)^n$ has 7 terms, the coefficient of the term with the largest binomial coefficient in the expansion is ( )
A: -280
B: -160
C: 160
D: 560 | \text{B} |
A company produces two types of barrel products, A and B. It is known that producing one barrel of product A requires 1 kg of material A and 2 kg of material B; producing one barrel of product B requires 2 kg of material A and 1 kg of material B. The profit from each barrel of product A is 300 yuan, and the profit from... | 2800 |
Given the operation defined as $\begin{vmatrix} \begin{matrix}a & b \\ c & d\end{matrix} \end{vmatrix} =ad-bc$, let $x=\frac{2-i}{3+i}$, and $y=\begin{vmatrix} \begin{matrix}4i & 3-xi \\ 1+i & x+i\end{matrix} \end{vmatrix}$, find the value of $y$. | -5 |
Given three fixed points in the Cartesian coordinate plane $A(1,0)$, $F_{1}(-2,0)$, $F_{2}(2,0)$, a moving point $P$ satisfies $|{\overrightarrow{PF_1}}|+|{\overrightarrow{PF_2}}|=6$. If $|{\overrightarrow{PA}}|≤\sqrt{6}$, then the range of the abscissa of point $P$ is ______. | [0,3] |
Given that $a$ and $b$ are positive real numbers, vector $\overrightarrow{m}=(a,4)$, vector $\overrightarrow{n}=(b,b-1)$, if $\overrightarrow{m}//\overrightarrow{n}$, then the minimum value of $a+b$ is _______. | 9 |
How many positive five-digit whole numbers are there for which the product of their five digits is 900?
(Karl Czakler)
Answer. 210 | 210 |
In the rectangular coordinate system xOy, the parametric equation of line l is $$\begin{cases} \overset{x=t}{y=1+t}\end{cases}$$ (t is the parameter), line m is parallel to line l and passes through the coordinate origin, and the parametric equation of circle C is $$\begin{cases} \overset{x=1+cos\phi }{y=2+sin\phi }\en... | 2+ \sqrt {2} |
18. (2005 National High School Mathematics League Additional Question) Let positive numbers $a, b, c, x, y, z$ satisfy $c y + b z = a$, $a z + c x = b$, $b x + a y = c$. Find the minimum value of the function $f(x, y, z) = \frac{x^{2}}{1+x} + \frac{y^{2}}{1+y} + \frac{z^{2}}{1+z}$. | \frac{1}{2} |
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