problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
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A binary string is a string consisting of only 0βs and 1βs (for instance, 001010, 101, etc.). What is the probability that a randomly chosen binary string of length 10 has 2 consecutive 0βs? Express your answer as a fraction. | \frac{55}{64} |
The Tigers beat the Sharks 2 out of the 3 times they played. They then played $N$ more times, and the Sharks ended up winning at least 95% of all the games played. What is the minimum possible value for $N$?
$\textbf{(A)}\; 35 \qquad \textbf{(B)}\; 37 \qquad \textbf{(C)}\; 39 \qquad \textbf{(D)}\; 41 \qquad \textbf{(E... | 37 |
If $a, b \in \mathbb{R}$, then $a < b$ is a ( ) condition for $a^2 < b^2$.
A: Sufficient but not necessary
B: Necessary but not sufficient
C: Both sufficient and necessary
D: Neither sufficient nor necessary | D |
1. On the plane, $n$ circles intersect each other pairwise, the maximum number of intersection points is
$\qquad$. | n(n-1) |
Given the function $f(x) = e^{x} - ax^{2}$, where $a \in \mathbb{R}$, and ${f'}(x)$ is the derivative of the function $f(x)$.
$(1)$ Discuss the monotonicity of the function ${f'}(x)$.
$(2)$ If the equation $f(x) + {f'}(x) = 2 - ax^{2}$ has real roots in the interval $(0,1)$, find the range of values for $a$. | (1, e-1) |
5. Solve the equation $\sin \frac{\pi n}{12} \cdot \sin \frac{\pi k}{12} \cdot \sin \frac{\pi m}{12}=\frac{1}{8}$. Here $k, m, n-$ are natural numbers not exceeding 5. | (2;2;2),(1;2;5),(1;5;2),(2;1;5),(2;5;1),(5;1;2),(5;2;1) |
Example 1 (2003 National High School Competition Question) In tetrahedron $A B C D$, let $A B=1, C D=\sqrt{3}$, the distance between lines $A B$ and $C D$ is 2, and the angle between them is $\frac{\pi}{3}$. Then the volume of tetrahedron $A B C D$ is equal to ( ).
A. $\frac{\sqrt{3}}{2}$
B. $\frac{1}{2}$
C. $\frac{1}{... | \frac{1}{2} |
Given set A={-1, 0, 1, 2}, B={x | xΒ²>=1}, find Aβ©B (A intersect B).
A: {0}
B: {1}
C: {-1, 1}
D: {0, 1, 2} | \text{C} |
Given $a=\log _{0.2}0.05$, $b=0.5^{1.002}$, $c=4\cos 1$, which of the following statements is correct?
A: $a \lt b \lt c$
B: $b \lt c \lt a$
C: $c \lt b \lt a$
D: $b \lt a \lt c$ | D |
A tetrahedron inscribed in a parallelepiped cuts off four tetrahedra from the parallelepiped with its face planes. What fraction of the volume of the parallelepiped is the volume of the tetrahedron whose vertices are the centroids of the cut-off tetrahedra? | \frac{1}{24} |
943. A cylindrical tank must hold $V \Omega$ of water. What should its dimensions be so that the surface area (without the lid) is the smallest? | \sqrt[3]{\frac{V}{\pi}} |
Given the quadratic function $y=(x-1)^{2}+2$, when the function value $y$ increases as the value of $x$ increases, the range of $x$ is ______. | x > 1 |
67. In how many ways can $k$ items be chosen from $n$ items? | C_{n}^{k}=\frac{n!}{k!(n-k)!} |
Given $\sin\alpha - \sin\beta = 1 - \frac{\sqrt{3}}{2}$, $\cos\alpha - \cos\beta = \frac{1}{2}$, then $\cos(\alpha - \beta) =$ οΌγγοΌ
A: $-\frac{\sqrt{3}}{2}$
B: $-\frac{1}{2}$
C: $\frac{1}{2}$
D: $\frac{\sqrt{3}}{2}$ | \text{D} |
16. If the function $y=f(x)$ for every value $x_{1}$ in its domain, there exists a unique $x_{2}$ in its domain such that $f\left(x_{1}\right) f\left(x_{2}\right)=1$, then the function is called a "dependent function". Given the following propositions:
(1) $y=\frac{1}{x^{2}}$ is a dependent function;
(2) $y=\sqrt{2}+\s... | (2)(3) |
The increasing [sequence](https://artofproblemsolving.com/wiki/index.php/Sequence) $3, 15, 24, 48, \ldots\,$ consists of those [positive](https://artofproblemsolving.com/wiki/index.php/Positive) multiples of 3 that are one less than a [perfect square](https://artofproblemsolving.com/wiki/index.php/Perfect_square). Wha... | 63 |
35. In $\triangle A B C$, $\angle A C B=90^{\circ}, \angle A=31^{\circ}$, with $C$ as the center, $\triangle A B C$ is rotated by an angle $\theta$ to $\triangle A_{1} B_{1} C$ (the shape and size of $\triangle A B C$ remain unchanged during the rotation), and point $B$ exactly falls on $A_{1} B_{1}$. Then the size of ... | 62 |
On this year's "3.15", a newspaper conducted a survey on "spam text messages on mobile phones". The number of questionnaires collected from units A, B, C, and D formed an arithmetic sequence in that order, totaling 1000 questionnaires. For reporting purposes, a sample of 150 questionnaires was drawn from the collected ... | C |
3. What is the largest natural number that satisfies the following condition: any two adjacent digits in the same order form a two-digit number divisible by 23? | 46923 |
Given that $S_{n}$ is the sum of the first $n$ terms of the sequence $\{a_{n}\}$, and satisfies $S_{n}+n=2a_{n}$, $n\in N^{*}$.
$(1)$ Prove that the sequence $\{a_{n}+1\}$ is a geometric sequence.
$(2)$ If $b_{n}=\frac{2^{n}}{a_{n} \cdot a_{n+1}}$, and $T_{n}$ denotes the sum of the first $n$ terms of the sequence ... | 2 |
9. The ancient Greek mathematicians associated natural numbers with polygons in the following way,
Triangular numbers: $1,3,6,10,15$, $\qquad$
Square numbers: $1,4,9,16,25$, $\qquad$
Pentagonal numbers: 1, 5, 12, 22, 35, $\qquad$
Hexagonal numbers: $1,6,15,28,45$, $\qquad$
According to the above sequence, the 8th trian... | 36 |
If $1998=p^{s} q^{t} r^{u}$, where $p, q$ and $r$ are prime numbers, what is the value of $p+q+r$ ?
(A) 222
(B) 48
(C) 42
(D) 66
(E) 122 | 42 |
2.76 Determine for which values of $m$ one of the roots of the equation $z^{3}-\left(m^{2}-m+7\right) z-\left(3 m^{2}-3 m-6\right)=0$ is equal to -1. Find the two remaining roots of the equation for these values of $m$. | z_{2}=-3,z_{3}=4 |
Let $a$ be an irrational number, and $a$, $b$ satisfy $ab - a - b + 1 = 0$. Then $b$ is a(n) (γγ)
A: Negative rational number
B: Positive rational number
C: Negative irrational number
D: Positive irrational number | \text{B} |
Example 4. Find the analytic function $w=f(z)$, given its real part $u(x, y)=2 e^{x} \cos y$ and the additional condition $f(0)=2$. | f(z)=2e^{z} |
To understand the incidence rate of myopia among high school students in a certain school, a stratified sampling survey was conducted among the students. It is known that there are 300 students in the first year, 260 students in the second year, and 280 students in the third year. If 14 students were sampled from the t... | 15 |
A hollow glass sphere with uniform wall thickness, which is empty inside, has an outer diameter of $16 \mathrm{~cm}$ and floats in water such that $\frac{3}{8}$ of its surface remains dry. What is the wall thickness if the specific gravity of the glass is $s=2.523$? | 0.8\mathrm{~} |
1. In some cells of a $1 \times 2100$ strip, one chip is placed. In each of the empty cells, a number is written that is equal to the absolute difference between the number of chips to the left and to the right of this cell. It is known that all the written numbers are distinct and non-zero. What is the minimum number ... | 1400 |
Given sets $A=\{x\in \mathbb{R} | x\leqslant 0\}$, $B=\{x\in \mathbb{R} | -1\leqslant x\leqslant 1\}$, then $\complement _{\mathbb{R}}(A\cup B)=\left(\ \ \right)$
A: $(-\infty ,0)$
B: $\left[-1,0\right]$
C: $\left[0,1\right]$
D: $(1,+\infty )$ | D |
Adam wrote the numbers from 1 to 9 into a $3 \times 3$ table as shown in the image:
| 7 | 6 | 4 |
| :--- | :--- | :--- |
| 1 | 2 | 8 |
| 9 | 3 | 5 |
For this filling, the sum of the numbers in the three cells along each side is always the same. Adam found that the numbers can be filled into the table in other ways wi... | 12 |
The function $y=\log_{a}(2-ax)$ is decreasing on the interval $[0,1]$. The range of values for $a$ is __( ).
A: $(0,1)$
B: $(0,2)$
C: $(1,2)$
D: $(2,+\infty)$ | C |
Given the universal set $U=\{1,3,5,7,9\}$, and the complement of set $A$ in $U$, denoted as $C_UA=\{5,7\}$, find the set $A$. | \{1,3,9\} |
Example 8 (I) Given two identical equilateral triangular pieces of paper, it is required to cut and assemble one into a regular tetrahedron model and the other into a regular triangular prism model, such that their total surface areas are equal to the area of the original triangle. Please design a cutting and assembly ... | V_{\text{tetrahedron}}>V_{\text{prism}} |
Let $a \gt 0$, and the function $f\left(x\right)=x^{a}\ln x$.
$(1)$ Discuss the monotonicity of $f\left(x\right)$;
$(2)$ If $f\left(x\right)\leqslant x$, find the range of $a$;
$(3)$ If ${f'}\left(x\right)\leqslant 1$, find $a$. | a = \frac{1}{2} |
If $a:b=4:7$, then which of the following statements is definitely correct?
A: $7a=4b$
B: $(b-a):a=3:7$
C: $4a=7b$
D: $b-a=3$ | A |
For a positive integer $k$, let the set
$$
R_{n}=\left\{\begin{array}{l}
\{-k,-(k-1), \cdots,-1,1, \cdots, k-1, k\}, \quad n=2 k ; \\
\{-k,-(k-1), \cdots,-1,0,1, \cdots, k-1, k\}, \quad n=2 k+1 .
\end{array}\right.
$$
A device contains some balls and red or white strings connecting some pairs of balls. A "labeling" is... | 2n-1 |
Given that functions $f(x)$ and $g(x)$ are odd and even functions on $\mathbb{R}$ respectively, determine the symmetry of the graph of the function $h(x)=g(x)|f(x)|$.
A: Symmetric about the origin
B: Symmetric about the $x$-axis
C: Symmetric about the $y$-axis
D: Symmetric about the line $y=x$ | \text{C} |
5. Given a natural number $x=9^{n}-1$, where $n$ is a natural number. It is known that $x$ has exactly three distinct prime divisors, one of which is 13. Find $x$. | 728 |
1. Huanhuan and Lele play a game together. In the first round, they both get the same number of coins, and in the second round, they also get the same number of coins.
At the beginning, Huanhuan says: βI have 7 times as many coins as you.β
At the end of the first round, Lele says: βYou have 6 times as many coins as I d... | 70 |
A1. What is the solution to the equation $\sqrt[5]{9^{x-3}}-\frac{1}{\sqrt{3}}=0$ ?
(A) $x=-\frac{7}{4}$
(B) $x=-\frac{4}{7}$
(C) $x=\frac{4}{7}$
(D) $x=\frac{7}{4}$
(E) $x=-\frac{9}{4}$ | \frac{7}{4} |
Given the function $f(x)=\lg (1-x)+\lg (1+x)+x^{4}-2x^{2}$.
$(1)$ Find the domain of the function $f(x)$;
$(2)$ Determine the parity of the function $f(x)$;
$(3)$ Find the range of the function $f(x)$. | (-\infty, 0] |
4. Let $f(x)=\log _{2} \frac{1+2^{x}+4^{x} a}{3}$, where $a \in \mathbf{R}$. If $f(x)$ is defined for $x \in(-\infty, 1]$, then the range of values for $a$ is | a>-\frac{3}{4} |
Given the function $f(x)=x^{2}+1$
(1) Find $f(a)-f(a+1)$
(2) If $f(x)=x+3$, find the value of $x$. | x=-1 \text{ or } x=2 |
Calculate $\sin (\pi-\alpha)+\sin (\pi+\alpha)=$γγ()
A: $0$
B: $1$
C: $2\sin \alpha$
D: $-2\sin \alpha$ | A |
Given vectors $\overrightarrow{a}=(1,x),\overrightarrow{b}=(x-2,x)$, if $|\overrightarrow{a}+\overrightarrow{b}|=|\overrightarrow{a}-\overrightarrow{b}|$, then $x$ is equal to:
A: $0$
B: $-1$
C: $1$
D: $-2$ | C \text{ or } D |
19. If $\sqrt{9-8 \sin 50^{\circ}}=a+b \csc 50^{\circ}$ where $a, b$ are integers, find $a b$.
(2 marks)θ₯ $\sqrt{9-8 \sin 50^{\circ}}=a+b \csc 50^{\circ}$, ε
ΆδΈ $a γ b$ η²ζ΄ζΈ, ζ± $a b$ γ | -3 |
Example 8 Find the largest real number $\lambda$ such that for a polynomial with real coefficients $f(x)=x^{3}+a x^{2}+b x+c$ whose all roots are non-negative real numbers, if $x \geqslant 0$, then
$$
f(x) \geqslant \lambda(x-a)^{3},
$$
and ask when the equality holds in this inequality?
(1999, China Mathematical Olym... | -\frac{1}{27} |
Given a set of data $(x\_1)$, $(x\_2)$, $(x\_3)$, $(x\_4)$, $(x\_5)$ with a mean of $2$ and a variance of $\frac{1}{3}$, what are the mean and variance of another set of data $(3x\_1-2)$, $(3x\_2-2)$, $(3x\_3-2)$, $(3x\_4-2)$, $(3x\_5-2)$?
A: $2$, $\frac{1}{3}$
B: $4$, $3$
C: $4$, $\frac{2}{3}$
D: $2$, $1$ | \text{B} |
A body moves from point $A(1,1)$ to point $B(0,5)$ under the action of forces $\overrightarrow{F_{1}}=(3,-4)$, $\overrightarrow{F_{2}}=(2,-5)$, and $\overrightarrow{F_{3}}=(3,1)$. The work done by the resultant force of these three forces during this process is ______. | -40 |
If the function $y=\left(m+2\right)x^{m-1}$ is a direct proportion function, then the value of $m$ is ______. | 2 |
Given the equation of line $l$ is $x-y-a^2=0$ ($a\neq 0$), which of the following statements is correct? ( )
A: The line does not pass through the first quadrant.
B: The line does not pass through the second quadrant.
C: The line does not pass through the third quadrant.
D: The line does not pass through the fourth qua... | \text{B} |
Given that $\cos \alpha = -\frac{4}{5}$, and $\alpha$ is an angle in the third quadrant, find the values of $\sin \alpha$ and $\tan \alpha$. | \frac{3}{4} |
67. In triangle $ABC$, given are $\widehat{ABC}=100^{\circ}, \widehat{ACB}=$ $=65^{\circ}$. On $AB$, a point $M$ is taken such that $\widehat{MCB}=55^{\circ}$, and on $AC$ - a point $N$ such that $\widehat{NBC}=80^{\circ}$. Find $\widehat{NMC}$. | 25 |
Given vectors $\overrightarrow{a} = (\sin\theta, -2)$ and $\overrightarrow{b} = (1, \cos\theta)$ are perpendicular to each other, where $\theta \in (0, \frac{\pi}{2})$
(1) Find the values of $\sin\theta$ and $\cos\theta$;
(2) If $5\cos(\theta-\varphi)=3\sqrt{5}\cos\varphi$, $0<\varphi<\frac{\pi}{2}$, find the value ... | \frac{\pi}{4} |
8. $[\mathbf{6}]$ Let $A:=\mathbb{Q} \backslash\{0,1\}$ denote the set of all rationals other than 0 and 1. A function $f: A \rightarrow \mathbb{R}$ has the property that for all $x \in A$,
$$
f(x)+f\left(1-\frac{1}{x}\right)=\log |x| .
$$
Compute the value of $f(2007)$. | \log(2007/2006) |
## Task 13/78
It is to be proven: If $p$ and $q$ are the projections of two sides of a triangle onto the third side, and $\alpha$ and $\beta$ are the two angles adjacent to the third side $\left(\alpha ; \beta \neq 90^{\circ}\right)$, then the area $A$ of the triangle is given by $A=\frac{1}{2} p q(\tan \alpha+\tan \b... | \frac{1}{2}pq(\tan\alpha+\tan\beta) |
8. Given that the sum of the reciprocals of the roots of the equation $40 x^{2}+39 x-1=0$ is the $m$-th term of the arithmetic sequence $1,3,5, \cdots$, and the product of the reciprocals of the roots is the sum of the first $n$ terms of the geometric sequence $2,-6,18, \cdots$, then the value of $m+n$ is ( ).
A. 20
B.... | 24 |
If $f,g: \mathbb{R} \to \mathbb{R}$ are functions that satisfy $f(x+g(y)) = 2x+y $ $\forall x,y \in \mathbb{R}$, then determine $g(x+f(y))$. | \frac{x}{2} + y |
7.3. On side $B C$ of triangle $A B C$, a point $M$ is marked such that $A B=B M$ and $A M=M C$. It is known that angle $B$ is five times angle $C$. Find the angles of the triangle. | \angleA=60,\angleB=100,\angleC=20 |
For the equation $x^{2}+mx+6=0$ with one root being $-2$, the other root is ______. | -3 |
Example 7. Find the integral $\int e^{\frac{x}{2}} d x$. | 2e^{\frac{x}{2}}+C |
## Task A-1.3.
Each digit of the natural number $n$ (except the first) is strictly greater than the digit immediately to its left. Determine the sum of all digits of the number $9 n$. | 9 |
7. In square $ABCD$ with side length $4$, $E$ is the midpoint of $AB$, and $F$ is a moving point on side $BC$. If $\triangle ADE$ and $\triangle DCF$ are folded along $DE$ and $DF$ respectively, so that $A$ and $C$ coincide at $A'$, then the maximum distance from point $A'$ to the plane $DEF$ is $\qquad$. | \frac{4\sqrt{5}}{5} |
. Find all sequences $\left(a_{n}\right)_{n \geqslant 1}$ of strictly positive real numbers such that for every integer $n$, we have:
$$
\sum_{i=1}^{n} a_{i}^{3}=\left(\sum_{i=1}^{n} a_{i}\right)^{2}
$$ | a_{n}=n |
1. The oldest can be either Boris or Viktor. But Viktor could not take second place, as he lost to both Anton and Grigory. Therefore, Boris is second. This means Grigory is first, Anton is third, and Viktor is fourth. | Grigory,Boris,Anton,Viktor |
If the equation $1+x+x^{2}+x^{3}=a_{0}+a_{1}(1-x)+a_{2}(1-x)^{2}+a_{3}(1-x)^{3}$ holds for all $x\in R$, where $a_{0}$, $a_{1}$, $a_{2}$, $a_{3}$ are real constants, then the value of $a_{1}+a_{2}+a_{3}$ is ____. | -3 |
Given two sequences $\{a_n\}$ and $\{c_n\}$ satisfying the conditions: $a_1=1$, $a_{n+1}=2a_n+1$, $c_n= \frac {1}{(2n+1)(2n+3)}$.
(1) Prove that the sequence $\{a_n+1\}$ is a geometric sequence, and find the general formula for the sequence $\{a_n\}$.
(2) Find the sum of the first $n$ terms $T_n$ of the sequence $\... | 5 |
In the diagram, the equilateral triangle has a base of $8 \mathrm{~m}$. The perimeter of the equilateral triangle is
(A) $4 \mathrm{~m}$
(B) $16 \mathrm{~m}$
(C) $24 \mathrm{~m}$
(D) $32 \mathrm{~m}$
(E) $64 \mathrm{~m}$
+4 x=5$ are distinct real numbers greater than 1. | \in\langle-1,0\rangle |
Third question Given ten points in space, where no four points lie on the same plane. Connect some of the points with line segments. If the resulting figure contains no triangles and no spatial quadrilaterals, determine the maximum number of line segments that can be drawn. | 15 |
5. The sequence $a_{n}$ is defined as follows:
$a_{1}=2, a_{n+1}=a_{n}+\frac{2 a_{n}}{n}$, for $n \geq 1$. Find $a_{999}$. | 999000 |
9.6. Let \(a\) and \(b\) be positive numbers. Find the minimum value of the fraction \(\frac{(a+b)(a+2)(b+2)}{16ab}\). Justify your answer. | 1 |
Given that non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy the condition $| \overrightarrow{a}|=3| \overrightarrow{b}|=| \overrightarrow{a}+2 \overrightarrow{b}|$, find the cosine value of the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | -\dfrac{1}{3} |
(5) Let $\{a_{n}\}$ be a sequence of positive numbers, and let the sum of the first $n$ terms be $b_{n}$. The product of the first $n$ terms of the sequence $\{b_{n}\}$ is $c_{n}$, and $b_{n}+c_{n}=1$. Then, the number in the sequence $\left\{\frac{1}{a_{n}}\right\}$ that is closest to 2011 is ( ).
(A) 1980
(B) 2010
(C... | 1980 |
Define the set operation: $A\odot B = \{z | z = xy(x+y), x \in A, y \in B\}$. Given the sets $A=\{0,1\}$ and $B=\{2,3\}$, the sum of all elements of the set $A\odot B$ is ___. | 18 |
[ Decimal numeral system ]
Can you find two consecutive numbers, where the sum of the digits of the first one is 8, and the second one is divisible by 8?
# | 7172 |
3. In rectangle $A B C D$, there are two circles, the radius of the larger circle is 6, and the radius of the smaller circle is 4. The length of the rectangle $B C=18$, then the width of the rectangle $A B=$ $\qquad$ | 16 |
$1 \cdot 12$ Consider a 100-digit number consisting only of the digits 1 and 2. You are allowed to pick any 10 consecutive digits and swap the positions of the first 5 and the last 5 digits. If one 100-digit number can be obtained from another by performing this operation several times, then these two numbers are calle... | 21^5 |
The front wheel of Georgina's bicycle has a diameter of 0.75 metres. She cycled for 6 minutes at a speed of 24 kilometres per hour. The number of complete rotations that the wheel made during this time is closest to
(A) 610
(B) 1020
(C) 1360
(D) 1700
(E) 5430 | 1020 |
The probability of winning the jackpot in a certain lottery is one in ten million. If you buy five different tickets, the probability of winning is (γγ)
A: One in ten million
B: Five in ten million
C: Ten in ten million
D: Twenty in ten million | \text{B} |
5.2. Masha eats a bowl of porridge in 12 minutes, and Bear eats it twice as fast. How long will it take them to eat six bowls of porridge? | 24 |
Given a complex number $z$ satisfies $(z+2i)(3+i)=7-i$, then the point corresponding to the complex number $z$ in the complex plane is located in (γγ)
A: The first quadrant
B: The second quadrant
C: The third quadrant
D: The fourth quadrant | D |
The radii of circles $βO_1$ and $βO_2$ are 4 and 5, respectively, and the distance between their centers $O_1O_2=7$. The positional relationship between $βO_1$ and $βO_2$ is (γγ)
A: Externally disjoint
B: Intersecting
C: Externally tangent
D: Contained | \text{B} |
Given that $f(x)$ is an even function defined on $\mathbb{R}$ with a period of $3$, if $f(1) < 1$ and $f(5) = \frac{2a-3}{a+1}$, find the range of values for the real number $a$. | (-1, 4) |
8. Among the numbers $3,5,7,8$, the probability that the sum of any two selected numbers is odd is $\qquad$ $\%$.
In the numbers $3,5,7,8$, if any two numbers are selected and added together, the probability that their sum is odd is $\qquad$ $\%$. | 50 |
3. Given an integer $n \geqslant 2$. Find the smallest positive real number $c$, such that for any complex numbers $z_{1}, z_{2}, \cdots, z_{n}$, we have
$$
\left|\sum_{i=1}^{n} z_{i}\right|+c \sum_{1 \leqslant i<j \leqslant n}\left|z_{i}-z_{j}\right| \geqslant \sum_{i=1}^{n}\left|z_{i}\right| .
$$ | \frac{2}{n} |
9. Half-planes $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}$ have a common edge $l$. What is the sum of the dihedral angles $\alpha_{1} \widehat{l \alpha_{2}}, \alpha_{2} l \alpha_{3}, \ldots, \alpha_{n-1} l \alpha_{n}$, $\widehat{\alpha_{n} l} l \alpha_{1}$, which together fill the entire space? | 2\pi |
27. Let $0<p \leqslant x_{i} \leqslant q\left(i=1,2, \cdots, n, p, q\right.$ be given constants), try to find the maximum value of $F\left(x_{4}, x_{2}, \cdots\right.$, $\left.x_{n}\right)=\left(x_{1}+x_{2}+\cdots+x_{n}\right)\left(\frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots+\frac{1}{x_{n}}\right)$. $(n=5$ is from the 1977 ... | 25 + 6\left(\sqrt{\frac{p}{q}} - \sqrt{\frac{q}{p}}\right)^2 |
Let's determine the minimum and maximum value of the expression $\sqrt{x-2}+\sqrt{3-x}$. | 1 |
You are given a square $n \times n$. The centers of some of some $m$ of its $1\times 1$ cells are marked. It turned out that there is no convex quadrilateral with vertices at these marked points. For each positive integer $n \geq 3$, find the largest value of $m$ for which it is possible.
[i]Proposed by Oleksiy Masali... | 2n-1 |
The slope of the tangent line to the curve $y=x^2$ at point P is -3. The coordinates of point P are (γγ)
A: (3, 9)
B: (-3, 9)
C: $( \frac {3}{2}, \frac {9}{4})$
D: $(- \frac {3}{2}, \frac {9}{4})$ | \text{D} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, with $a= \sqrt {2}$, $A= \frac {\pi}{4}$, and $B= \frac {\pi}{3}$. The area of $\triangle ABC$ is $S=$ ______. | \frac {3+ \sqrt {3}}{4} |
Among the following functions, the variable $y$ that is an inverse proportion function of $x$ is:
A: $y=\frac{x}{2}$
B: $y=\frac{2}{{x+1}}$
C: $y=\frac{2}{x}$
D: $y=2x$ | C |
Ten identical crates each of dimensions $ 3$ ft $ \times$ $ 4$ ft $ \times$ $ 6$ ft. The first crate is placed flat on the floor. Each of the remaining nine crates is placed, in turn, flat on top of the previous crate, and the orientation of each crate is chosen at random. Let $ \frac{m}{n}$ be the probability that ... | 190 |
Which of the following conclusions are correct?
A: $P(A)=P(B)P(A|B)+P(\overline{B})P(A|\overline{B})$
B: $P(B)=P(A)P(B|A)+P(A)P(\overline{B}|A)$
C: $P(A|B)=\frac{{P(B)P(A|B)}}{{P(A)P(B|A)}}$
D: $P(A|B)=\frac{{P(A)P(B|A)}}{{P(A)P(B|A)+P(\overline{A})P(B|\overline{A})}}$ | AD |
6. Point $M$ lies on the edge $A B$ of the cube $A B C D A_{1} B_{1} C_{1} D_{1}$. A rectangle $M N L K$ is inscribed in the square $A B C D$ such that one of its vertices is point $M$, and the other three are located on different sides of the square base. The rectangle $M_{1} N_{1} L_{1} K_{1}$ is the orthogonal proje... | AM:MB=1:4 |
12.031. In the square $A B C D$, an isosceles triangle $A E F$ is inscribed; point $E$ lies on side $B C$, point $F$ - on side $C D$, and $A E=A F$. The tangent of angle $A E F$ is 3. Find the cosine of angle $F A D$. | \frac{2\sqrt{5}}{5} |
The solution to the equation $\frac{3}{{x-2}}=1$ is ____. | x=5 |
A certain company's cost for a type of children's toy is 40 yuan, and the factory price is 60 yuan. After market research, adjustments were made. If a distributor orders more than 100 units at a time, for every additional unit ordered, the factory price per toy decreases by 0.02 yuan, but it cannot be lower than 50 yua... | 6000 |
Find the equation of the line that passes through point A(2, 1) and has equal intercepts on the coordinate axes. | x + y - 3 = 0 |
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