problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
2. Lily pours 296 litres of water into the top of the pipework shown in the diagram. Each time a pipe forks, half the water flows to one side and half to the other. How many litres of water will reach container $\mathrm{Y}$ ?
A 210
B 213
C 222
D 225
E 231 | 222 |
There exist two distinct positive integers, both of which are divisors of $10^{10}$, with sum equal to $157$. What are they? | 32, 125 |
$14 \cdot 75$ Find all natural numbers $a$ and
$$
b:\left[\frac{a^{2}}{b}\right]+\left[\frac{b^{2}}{a}\right]=\left[\frac{a^{2}+b^{2}}{a b}\right]+a b .
$$
(37th International Mathematical Olympiad Preliminary Question, 1996) | b=^{2}+1,\inN\text{or}=b^{2}+1,b\inN |
6.56 Given a function $f$ defined on the set of positive integers satisfies: $f(1)=1$, $f(2)=2, \quad f(n+2)=f(n+2-f(n+1))+f(n+1-f(n)) \quad(n \geqslant 1)$
(1) Prove: $0 \leqslant f(n+1)-f(n) \leqslant 1$, and when $f(n)$ is odd, $f(n+1)=f(n)+1$;
(2) Try to find: all values of $n$ that satisfy $f(n)=2^{10}+1$, and pro... | 2^{11} |
Point $A(5,-4)$ lies in which quadrant?
A: Quadrant I
B: Quadrant II
C: Quadrant III
D: Quadrant IV | D |
If $f(x)+\int_{0}^{1}f(x)dx=x$, then $\int_{0}^{1}f(x)dx=$ \_\_\_\_\_\_\_\_. | \frac{1}{4} |
Simplify $\frac{x^2-4x}{x^2-8x+16}=$____. | \frac{x}{x-4} |
In trapezoid $ABCD$, sides $AB$ and $CD$ are parallel, and diagonal $BD$ and side $AD$ have equal length. If $m\angle DCB=110^\circ$ and $m\angle CBD=30^\circ$, then $m\angle ADB=$
$\textbf{(A)}\ 80^\circ\qquad\textbf{(B)}\ 90^\circ\qquad\textbf{(C)}\ 100^\circ\qquad\textbf{(D)}\ 110^\circ\qquad\textbf{(E)}\ 120^\circ$ | 100 |
3. Find the minimum value of $x^{2}+4 y^{2}-2 x$, where $x$ and $y$ are real numbers that satisfy $2 x+8 y=3$. | -\frac{19}{20} |
3. The area of the closed figure enclosed by the curve $x^{2}+2 y^{2} \leqslant 4|y|$ is $\qquad$ | 2 \sqrt{2} \pi |
In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is $\begin{cases} x=3+ \frac {1}{2}t \\ y= \frac { \sqrt {3}}{2}t\end{cases}$ (where $t$ is the parameter), and the parametric equation of circle $⊙C$ is $\begin{cases} x= \sqrt {3}\cos \theta \\ y= \sqrt {3}(1+\sin \theta)\end{cases}$ (where... | (3,0) |
Let $1=d_1<d_2<d_3<\dots<d_k=n$ be the divisors of $n$. Find all values of $n$ such that $n=d_2^2+d_3^3$. | 68 |
Factorize the polynomial $a^2 - b^2 + 2a + 1$ to get ( )
A: $(a+b)(a-b)+(2a+1)$
B: $(a-b+1)(a+b-1)$
C: $(a-b+1)(a+b+1)$
D: $(a-b-1)(a+b+1)$ | \text{C} |
Let $$N = \sum^{512}_{i=0}i {512 \choose i}.$$ What is the greatest integer $a$ such that $2^a$ is a divisor of $N$? | 520 |
NT11 Determine the greatest number with $n$ digits in the decimal representation which is divisible by 429 and has the sum of all digits less than or equal to 11 .
| 30030000\ldots |
Zach has twelve identical-looking chocolate eggs. Exactly three of the eggs contain a special prize inside. Zach randomly gives three of the twelve eggs to each of Vince, Wendy, Xin, and Yolanda. What is the probability that only one child will receive an egg that contains a special prize (that is, that all three speci... | \frac{1}{55} |
Given the set M={x|-2≤x＜2} and the set N={x|x≥-2}, find M∩N=（ ）
A: {x|-2≤x＜2}
B: {x|x≥-2}
C: {x|x＜2}
D: {x|1＜x＜2} | \text{A} |
Given a sequence $\{a_n\}$ that satisfies: $a_1=1$, $a_{n+1}= \frac {1}{2}a_{n} (n\in\mathbb{N}^*)$, find $a_{n}=$ ______ ; and the sum of the first $n$ terms of the sequence $\{a_n\}$, $S_{n}=$ ______. | 2-2^{1-n} |
11 Given the ellipse $C: \frac{x^{2}}{4}+\frac{y^{2}}{2}=1$, a moving line $l$ passes through the point $P\left(\frac{\sqrt{2}}{3},-\frac{1}{3}\right)$ but does not pass through the point $Q(\sqrt{2}, 1)$, and intersects the ellipse $C$ at points $A$ and $B$.
(1) Find $\angle A Q B$;
(2) Let the area of $\triangle Q A ... | S<3 |
(The full score of this question is 14 points) Given the function $f(x) = -x^3 + 3x^2 + 9x + a$.
(I) Find the interval where $f(x)$ is monotonically decreasing;
(II) If the maximum value of $f(x)$ in the interval $[-2, 2]$ is 20, find its minimum value in this interval. | -7 |
12. For a natural number $n$, if there exist non-zero natural numbers $a$ and $b$ such that $n=a+b+a \times b$, then $n$ is called a "good number". For example, $3=1+1+1 \times 1$, so 3 is a "good number". Among the 100 natural numbers from 1 to 100, there are $\qquad$ "good numbers". | 74 |
1. Given that $a$ is a given real number, the number of subsets of the set $M=\left\{x \mid x^{2}-3 x-a^{2}+2=0, x \in \mathbf{R}\right\}$ is
(A) 1
(B) 2
(C) 4
(D) Uncertain | 4 |
4. Dina has a calculating machine, labelled $f$, that takes one number as input and calculates an output. The machine $f$ calculates its output by multiplying its input by 2 and then subtracting 3. For example, if Dina inputs 2.16 into $f$, the output is 1.32. If Dina inputs a number $x$ into $f$, she gets a first outp... | -\frac{13}{2} |
Problem 1. Let $a$ be a strictly positive real number. Determine the minimum value of the expression
$$
\left(\int_{0}^{1} f(x) \mathrm{d} x\right)^{2}-(a+1) \int_{0}^{1} x^{2 a} f(x) \mathrm{d} x
$$
as $f$ ranges over the set of concave functions $f:[0,1] \rightarrow \mathbb{R}$, with $f(0)=1$. | \frac{2a-1}{8a+4} |
If $x$, $y$, $z$ satisfy $x + y = 6$ and $z^2 = xy - 9$, then the value of $z$ is ( )
A: ±1
B: 0
C: 1
D: -1 | \text{B} |
If circle $C_{1}: x^{2}+y^{2}+ax=0$ and circle $C_{2}: x^{2}+y^{2}+2ax+y\tan \theta=0$ are both symmetric about the line $2x-y-1=0$, then $\sin \theta\cos \theta=$ ()
A: $ \dfrac {2}{5}$
B: $- \dfrac {6}{37}$
C: $- \dfrac {2}{5}$
D: $- \dfrac {2}{3}$ | C |
Given that all terms of the geometric sequence $\{a_n\}$ are positive, and the product of the first $n$ terms is $T_n$, if $T_5=1$, then it must be true that
A: $a_1=1$
B: $a_3=1$
C: $a_4=1$
D: $a_5=1$ | \text{B} |
One writes, initially, the numbers $1,2,3,\dots,10$ in a board. An operation is to delete the numbers $a, b$ and write the number $a+b+\frac{ab}{f(a,b)}$, where $f(a, b)$ is the sum of all numbers in the board excluding $a$ and $b$, one will make this until remain two numbers $x, y$ with $x\geq y$. Find the maximum val... | 1320 |
10.5. On a plane, there are eight different points. A numbering of these points with numbers from 1 to 8 is called good if the following condition is met:
there exists a line such that all points lie on one side of it and at different distances from it, and the distances from the points to this line increase with the ... | 56 |
# Problem No. 6 (10 points)
The density of a body is defined as the ratio of its mass to the volume it occupies. A homogeneous cube with a volume of \( V = 8 \, \text{m}^3 \) is given. As a result of heating, each of its edges increased by 4 mm. By what percentage did the density of this cube change?
Answer: decrease... | 6 |
Find the standard equations of the following circles:
(1) The center of the circle lies on the line $y=-x$ and passes through the points $(2,0)$ and $(0,-4)$;
(2) The center of the circle lies on the line $5x-3y=8$ and the circle is tangent to the coordinate axes. | 1 |
4. Given a square $A B C D$ and a point $P$ inside the square such that $\overline{A P}: \overline{B P}: \overline{C P}=1: 2: 3$. Calculate the angle $\measuredangle A P B$. | 135 |
5. In how many two-digit numbers is one digit twice the other?
A 6
B 8
C 10
D 12
E 14 | 8 |
7. Given $O$ is the circumcenter of acute $\triangle A B C$, $\angle B A C$ $=60^{\circ}$, extend $C O$ to intersect $A B$ at point $D$, extend $B O$ to intersect $A C$ at point $E$. Then $\frac{B D}{C E}=$ $\qquad$ . | 1 |
8. Let the sequence $\left\{a_{n}\right\}$ have 10 terms, $a_{i} \in\{1,-1\}$ $(i=1,2, \cdots, 10)$, and for any $k \leqslant 9$, we have
$$
\left|\sum_{i=1}^{k} a_{i}\right| \leqslant 2, \sum_{i=1}^{10} a_{i}=0 .
$$
Then the number of sequences $\left\{a_{n}\right\}$ that satisfy the conditions is | 162 |
Simplify or evaluate:
(1) $(1)( \dfrac {27}{8})^{- \frac {2}{3}}-( \dfrac {49}{9})^{0.5}+(0.008)^{- \frac {2}{3}}× \dfrac {2}{25}$
(2) Compute $\dfrac {\lg 5\cdot \lg 8000+(\lg 2^{ \sqrt {3}})^{2}}{\lg 600- \dfrac {1}{2}\lg 36- \dfrac {1}{2}\lg 0.01}$. | 1 |
2. Calculate the value of the expression $\sqrt{(\sqrt{3}-3)^{2}}-(\sqrt{3}-3)$. | 6-2\sqrt{3} |
The minimum value of the function f(x) = $$\begin{vmatrix} \cos{2x} & -\sin{x} \\ \cos{x} & \frac{\sqrt{3}}{2} \end{vmatrix}$$ in the interval [0, $\frac{\pi}{2}$] is _______. | -\frac{\sqrt{3}}{2} |
Problem 5. The probability of at least one hit with two shots is 0.96. Find the probability of four hits with five shots. | 0.4096 |
To reach the 10th step of a staircase, if it is stipulated that one can only step up one or two steps at a time, how many different ways are there to do so? | 89 |
# 7. Problem 7
A divisor of a natural number is called proper if it is different from 1 and the number itself. A number is called interesting if it has two proper divisors, one of which is prime, and the other is a perfect square, and the sum of which is also a perfect square (a perfect square is the square of an inte... | 70 |
Given $A=3x^{2}-x+2y-4xy$, $B=2x^{2}-3x-y+xy$.
$(1)$ Simplify $2A-3B$.
$(2)$ When $x+y=\frac{6}{7}$ and $xy=-1$, find the value of $2A-3B$.
$(3)$ If the value of $2A-3B$ is independent of the value of $y$, find the value of $2A-3B$. | \frac{49}{11} |
1. A compact disc has the shape of a circle of diameter 5 inches with a 1-inch-diameter circular hole in the center. Assuming the capacity of the CD is proportional to its area, how many inches would need to be added to the outer diameter to double the capacity? | 2 |
Given the function $f(x)=a^{x-1}$ ($x\geqslant 0$), its graph passes through the point $(2, \frac {1}{2})$, where $a > 0$ and $a\neq 1$.
(1) Find the value of $a$;
(2) Find the range of the function $y=f(x)$ ($x\geqslant 0$). | (0,2] |
Let $x_{1}, x_{2}, \ldots, x_{n}$ be a sequence where each term is 0, 1, or -2. If
$$
\left\{\begin{array}{l}
x_{1}+x_{2}+\cdots+x_{n}=-5 \\
x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=19
\end{array}\right.
$$
determine $x_{1}^{5}+x_{2}^{5}+\cdots+x_{n}^{5}$. | -125 |
1. Among the following four propositions: \\(①\\) All lateral edges of a regular pyramid are equal; \\(②\\) The lateral faces of a right prism are all congruent rectangles; \\(③\\) The generatrix of a cylinder is perpendicular to the base; \\(④\\) Cutting a cone with a plane passing through the axis of rotation always ... | 3 |
If the function $f(x)=\ln x$, then $f''(1)$ equals ( )
A: $2$
B: $e$
C: $1$
D: $0$ | C |
Find the domain of the function f(x) = $\sqrt{x+1}$ + ln(2-x). | [-1, 2) |
5. Given the hyperbola $C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)$ with left and right foci $F_{1}, F_{2}$, and eccentricity $\frac{5}{3}$, if a line $l$ passing through $F_{1}$ is tangent to the circle $x^{2}+y^{2}=a^{2}$ at point $T$, and $l$ intersects the right branch of the hyperbola $C$ at point $P$,... | 4 |
Among the natural numbers not exceeding 10,000, the number of odd numbers with distinct digits is ( )
A: 1260
B: 1530
C: 2260
D: 2605 | D |
Which of the following sets of three line segments can form a triangle?
A: $3$, $4$, $8$
B: $5$, $6$, $11$
C: $5$, $8$, $15$
D: $3$, $4$, $6$ | D |
There are $10$ different products, of which $3$ are defective.
$(1)$ If $3$ products are randomly selected for inspection, how many possible selections contain at least one defective product?
$(2)$ If each product is tested one by one until all $3$ defective products are found, and it turns out that the third defecti... | 1512 |
Given the function $f(x)=\log_{a}x^{2}+a^{|x|}$, if $f(-3) < f(4)$, then the solution set of the inequality $f(x^{2}-2x)\leqslant f(3)$ is $(\quad)$
A: $(-1,3)$
B: $[-1,3]$
C: $(-\infty,-1)\cup(3,+\infty)$
D: $[-1,0)\cup(0,2)\cup(2,3]$ | \text{D} |
Example 7 Find the maximum number of elements in a set $S$ that satisfies the following conditions:
(1) Each element in $S$ is a positive integer not exceeding 100;
(2) For any two different numbers $a, b$ in $S$, there exists a number $c$ in $S$ such that the greatest common divisor (gcd) of $a$ and $c$ is 1, and the ... | 72 |
Given triangle $\triangle ABC$ with angles $A$, $B$, $C$ and their respective opposite sides $a$, $b$, $c$, and $b \cos C = (2a - c) \cos B$.
1. Find the measure of angle $B$.
2. If $b = \sqrt{3}$, find the maximum value of $a + c$. | 2\sqrt{3} |
Example 2. Solve the equation
$$
\frac{|\operatorname{ctg} x y|}{\cos ^{2} x y}-2=\log _{\frac{1}{3}}\left(9 y^{2}-18 y+10\right)
$$ | \frac{\pi}{4}+\frac{\pik}{2},1,k\in\boldsymbol{Z} |
Which one of the following statements is not true?
(A) 25 is a perfect square.
(B) 31 is a prime number.
(C) 3 is the smallest prime number.
(D) 8 is a perfect cube.
(E) 15 is the product of two prime numbers. | C |
Given vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ with pairwise angles of $60^\circ$, and $|\overrightarrow{a}|=|\overrightarrow{b}|=|\overrightarrow{c}|=1$, find $|\overrightarrow{a}+\overrightarrow{b}-\overrightarrow{c}|$. | \sqrt{2} |
In a $m\times n$ chessboard ($m,n\ge 2$), some dominoes are placed (without overlap) with each domino covering exactly two adjacent cells. Show that if no more dominoes can be added to the grid, then at least $2/3$ of the chessboard is covered by dominoes.
[i]Proposed by DVDthe1st, mzy and jjax[/i] | \frac{2}{3} |
14.3.27 ** Find the sum of all divisors $d=2^{a} \cdot 3^{b}(a, b>0)$ of $N=19^{88}-1$. | 744 |
Example 2 Find all positive integers $n$ such that $n=d(n)^{2}$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | n=1 \text{ or } 9 |
Example 2. Solve the equation
$$
[x-1]=\left[\frac{x+2}{2}\right]
$$ | x\in[3,5) |
3. Find the number of natural numbers $k$, not exceeding 267000, such that $k^{2}-1$ is divisible by 267. Answer: 4000. | 4000 |
3. The average of five unequal natural numbers is 15, and the median is 18. Then the maximum value of the largest number among these five numbers is ( ).
(A) 35
(B) 36
(C) 37
(D) 38 | C |
## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow 1}\left(\ln ^{2} e x\right)^{\frac{1}{x^{2}+1}}
$$ | 1 |
13. Given the sequence $\{\sqrt[n]{n}\}\left(n \in \mathbf{Z}_{+}\right)$. Then the value of the largest term in the sequence is $\qquad$ . | \sqrt[3]{3} |
If $a < 0$, then the vertex of the parabola $y = 2x^2 + ax - 5$ is in ( )
A: The first quadrant
B: The second quadrant
C: The third quadrant
D: The fourth quadrant | D |
Example 9. Factorize $a b(a-b)+b c(b-c)+c a(c-a)$. | -(a-b)(b-c)(c-a) |
1. Let $x \neq y$, and the sequences $x, a_{1}, a_{2}, a_{3}, y$ and $b_{1}, x, b_{2}, b_{3}$, $y, b_{4}$ are both arithmetic sequences, then $\frac{b_{4}-b_{3}}{a_{2}-a_{1}}=$ $\qquad$ . | \frac{8}{3} |
Given that in $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $\frac{a-c}{a-b} = \frac{\sin A + \sin B}{\sin (A+B)}$.
(I) Find the value of angle $B$;
(II) If the circumradius of $\triangle ABC$ is $1$, find the maximum value of the area $S$ of $\triangle ABC$. | \frac{3\sqrt{3}}{4} |
In the book "经说" written during the Warring States period, it is recorded: "景:日之光,反蚀人,则景在日与人之间". This is the first study of plane mirror reflection by ancient Chinese people, reflecting the mathematical wisdom in traditional culture. In the plane rectangular coordinate system $xOy$, a ray of light is emitted from the p... | \text{A} |
2. How many of these four integers are prime?
$\begin{array}{llll}1 & 11 & 111 & 1111\end{array}$
A 0
B 1
C 2
D 3
E 4 | 1 |
[ Properties and characteristics of a parallelogram ]
One side of the parallelogram is three times larger than the other side. Find the sides of the parallelogram if its perimeter is 24.
# | 3,9,3,9 |
Given the function $f(x)=2\sqrt{2}\cos x\sin\left(x+\frac{\pi}{4}\right)-1$.
(I) Find the value of $f\left(\frac{\pi}{4}\right)$;
(II) Find the maximum and minimum values of $f(x)$ in the interval $\left[0, \frac{\pi}{2}\right]$. | -1 |
Calculate the value of the following expressions:
(1) $(\frac{2}{3})^{-2} + (1 - \sqrt{2})^{0} - (3\frac{3}{8})^{\frac{2}{3}}$;
(2) $\frac{2\lg 2 + \lg 3}{1 + \frac{1}{2}\lg 0.36 + \frac{1}{3}\lg 8}$. | 1 |
24. Let $a$ and $b$ be two real numbers. If the equation $a x+(b-3)=(5 a-1) x+3 b$ has more than one solution, what is the value of $100 a+4 b$ ? | 19 |
$9 \cdot 21$ Let the natural number $62 \alpha \beta 427$ be a multiple of 99, find $\alpha, \beta$.
(Shanghai Mathematical Competition, 1957) | \alpha=2,\beta=4 |
Given the function $f(x)=\lg (x+\sqrt{x^{2}+1})$, if for any $x\in (1,2]$, there is $f\left(\frac{x+1}{x-1}\right)+f\left[\frac{m}{{(x-1)}^{2}(x-6)}\right] > 0$, then the range of the real number $m$ is $(\quad)$
A: $[4,+\infty )$
B: $(12,+\infty )$
C: $(-\infty ,0)$
D: $(-\infty ,0]$ | \text{D} |
Given the function $f(x)=kx+b$, whose graph intersects the $x$ and $y$ axes at points A and B respectively, with $\overrightarrow{AB}=2\overrightarrow{i}+2\overrightarrow{j}$ ($\overrightarrow{i}$, $\overrightarrow{j}$ are unit vectors in the positive direction of the $x$ and $y$ axes). The function $g(x)=x^{2}-x-6$ is... | -3 |
The sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ is $S_n$, and it is given that $a_4 - a_2 = 8$, and $a_3 + a_5 = 26$. Let $T_n = \frac{S_n}{n^2}$. If there exists a positive integer $M$ such that for all positive integers $n$, $T_n \leq M$ holds, then the minimum value of $M$ is \_\_\_\_\_\_\_\_. | 2 |
7. What is $0.015 \%$ of 60 million?
A 900
B 9000
C 90000
D 900000
E 9000000 | 9000 |
Given that angle $\alpha = 2k\pi - \frac{\pi}{5} (k \in \mathbb{Z})$, if angle $\theta$ has the same terminal side as angle $\alpha$, then the value of $y= \frac{\sin \theta}{|\sin \theta|}+ \frac{\cos \theta}{|\cos \theta|}+ \frac{\tan \theta}{|\tan \theta|}$ is
A: $1$
B: $-1$
C: $3$
D: $-3$ | B |
The even function $f(x)$ and its derivative ${f'}(x)$ both have a domain of $R$. $f(x)$ is monotonically decreasing on the interval $[2023,2024]$. For any $x \in R$, the equation ${f'}(x) = {f'}(2-x)$ holds. If $a = 4^{-0.8}$, $b = \frac{1}{3}$, $c = \ln\frac{3}{2}$, then ()
A: $f(a) < f(b) < f(c)$
B: $f(a) < f(c) < ... | A |
It is known that $a+b+c=5$ and $ab+bc+ac=5$. What can $a^{2}+b^{2}+c^{2}$ be?
# | 15 |
Xiaoliang needs to calculate the variance $s_{1}^{2}$ of a set of data: $80$, $82$, $74$, $86$, $79$. In the process of calculating the average, each number in this set of data is subtracted by $80$, resulting in a new set of data: $0$, $2$, $-6$, $6$, $-1$. Let the variance of this new set of data be $s_{2}^{2}$. The ... | C |
Let $a=\frac{1}{2}\cos 7^{\circ}-\frac{\sqrt{3}}{2}\sin 7^{\circ}$, $b=\frac{2\tan 12^{\circ}}{1+\tan^2 12^{\circ}}$, $c=\sqrt{\frac{1-\cos 44^{\circ}}{2}}$. Which of the following statements is true?
A: $c \lt a \lt b$
B: $a \lt b \lt c$
C: $a \lt c \lt b$
D: $b \lt c \lt a$ | A |
12.288. The lateral face of a regular triangular pyramid forms an angle with the base plane, the tangent of which is equal to $k$. Find the tangent of the angle between the lateral edge and the apothem of the opposite face. | \frac{3k}{k^{2}-2} |
A certain fruit store sells a high-quality fruit at a cost price of $40$ yuan per kilogram. If the selling price is $50$ yuan per kilogram, then $500$ kilograms can be sold in a month. If the selling price increases by $1$ yuan based on $50$ yuan per kilogram, the monthly sales volume decreases by $10$ kilograms.
$(1... | 70 \text{ yuan} |
## Task B-4.2.
Let $z=x+2i$ and $w=3+yi$ be complex numbers, where $x, y \in \mathbb{R}$. Determine the smallest positive real number $x$ for which the fraction $\frac{z+w}{z-w}$ is an imaginary number. | \sqrt{5} |
2.84 Let the real number $x$ satisfy $|2-x|=2+|x|$, then $\sqrt{(x-1)^{2}}$ equals
(A) $x-1$.
(B) $1-x$.
(C) $\pm(x-1)$.
(D) 1 .
(Shanghai Junior High School Mathematics Competition, 1986) | 1-x |
Given a sample with a total capacity of 100 that is divided into several groups, if the frequency of a certain group is 0.3, then the frequency number of that group is ( ).
A: 3
B: 30
C: 10
D: 300 | B |
5.35 (1) Find the possible minimum value of the polynomial $P(x, y)=4+x^{2} y^{4}+x^{4} y^{2}-3 x^{2} y^{2}$.
(2) Prove that this polynomial cannot be expressed as a sum of squares of polynomials in variables $x, y$. | 3 |
380 The four planes in space are mutually parallel, and the distance between each adjacent pair of planes is $h$. A regular tetrahedron has its vertices on the four planes. Find the edge length of the regular tetrahedron. | \sqrt{10}h |
Given a sequence $\{a_n\}$ that satisfies: $$\frac {a_{n}+1}{a_{n+1}+1} = \frac {1}{2},$$ and $a_2=2$, then $a_4$ equals to ( )
A: $-\frac {1}{2}$
B: 23
C: 12
D: 11 | D |
13.1.1 $\star$ In the first quadrant of the Cartesian coordinate system, number the points (lattice points) with integer coordinates in the following manner: (1) $(0,0)$, (2) $(1,0)$, (3) $(1,1)$, (4) $(0,1)$, (5) $(0,2)$, (6) $(1,2)$, (7) $(2,2)$, (8) $(2,1)$, (9) $(2,0), \cdots \cdots$ (as shown in the order of the a... | (44,25) |
Given is a prime number $p$ and natural $n$ such that $p \geq n \geq 3$. Set $A$ is made of sequences of lenght $n$ with elements from the set $\{0,1,2,...,p-1\}$ and have the following property:
For arbitrary two sequence $(x_1,...,x_n)$ and $(y_1,...,y_n)$ from the set $A$ there exist three different numbers $k,l,... | p^{n-2} |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a=2\sqrt{2}$, $b=\sqrt{6}$, and $\cos A=\frac{1}{3}$, then which of the following statements is true?
A: The radius of the circumcircle of $\triangle ABC$ is $\frac{3}{2}$.
B: The radius of the circumci... | AC |
Given proposition $p: \exists x\in R$, such that $x^{2}+2x \lt 0$, then $\neg p$ is ( ).
A: $\exists x\in R$, such that $x^{2}+2x\geqslant 0$
B: $\exists x\in R$, such that $x^{2}+2x \gt 0$
C: $\forall x\in R$, such that $x^{2}+2x\geqslant 0$
D: $\forall x\in R$, such that $x^{2}+2x \lt 0$ | C |
Given that the sum of the first $n$ terms of the geometric sequence ${a_n}$ is $S_n$. If $S_6 = -7S_3$, then find the value of $$\frac {a_{4}+a_{3}}{a_{3}+a_{2}}$$. | -2 |
11.2. On the board, the expression $\frac{a}{b} \cdot \frac{c}{d} \cdot \frac{e}{f}$ is written, where $a, b, c, d, e, f$ are natural numbers. If the number $a$ is increased by 1, then the value of this expression increases by 3. If in the original expression the number $c$ is increased by 1, then its value increases b... | 60 |
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