problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
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16. Unified State Exam in Anchuria. (7-11 grades, 2 points). In Anchuria, the Unified State Exam is taking place. The probability of guessing the correct answer to each question on the exam is 0.25. In 2011, to receive a certificate, one needed to answer correctly[^0]
3 questions out of 20. In 2012, the School Adminis... | In2012,theprobabilityofobtainingcertificateishigher |
A circle of radius $1$ rolls around a circle of radius $\sqrt{2}$. Initially, the tangent point is colored red. Afterwards, the red points map from one circle to another by contact. How many red points will be on the bigger circle when the center of the smaller one has made $n$ circuits around the bigger one? | \lfloor n\sqrt{2} \rfloor + 1 |
The inequality $|x+3|+|x-1| \geq a$ holds for any real number $x$. Then, the range of the real number $a$ is ( )
A: $[1,3]$
B: $[-1,3]$
C: $(-\infty,4]$
D: $[4,+\infty)$ | C |
Given the function $f(x)=x^{2}-kx+(2k-3)$.
(1) If $k= \frac {3}{2}$, solve the inequality $f(x) > 0$;
(2) If $f(x) > 0$ holds for any $x∈R$, find the range of values for the real number $k$;
(3) If the two distinct zeros of the function $f(x)$ are both greater than $ \frac {5}{2}$, find the range of values for the real... | (6, \frac {13}{2}) |
Given the complex number $z= \frac {2i}{1-i}$, then $|z|=$ ______. | \sqrt {2} |
A city strengthens its investment in environmental protection construction in various townships with the focus on "key environmental sanitation improvement". The city plans to invest 15 million yuan in 2017 and 42.5 million yuan in 2019. If the average annual growth rate of the investment is $x$, which of the following... | \text{B} |
Find the coefficient of $x^3$ in the binomial expansion of $(2x+\frac{1}{x})^n$, given that the sum of the binomial coefficients in this expansion is $32$. (Fill in the answer with a number) | 80 |
Example 3 Let $S$ be a subset of $\{1,2,3, \cdots, 50\}$ with the following property: the sum of any two distinct elements of $S$ is not divisible by 7. Then, the maximum number of elements in $S$ is ( ).
(A) 6
(B) 7
(C) 14
(D) 22
(E) 23
(43rd American High School Mathematics Examination) | E |
4. Given a regular tetrahedron $P-ABC$ with a base edge length of 1 and a height of $\sqrt{2}$. Then its inscribed sphere radius is $\qquad$ . | \frac{\sqrt{2}}{6} |
Solve the equation: $3\left(x-2\right)+1=x-\left(2x-1\right)$. | \frac{3}{2} |
9. Let $A B C D$ be a regular tetrahedron with side length 1 . Let $X$ be the point in triangle $B C D$ such that $[X B C]=2[X B D]=4[X C D]$, where $[\varpi]$ denotes the area of figure $\varpi$. Let $Y$ lie on segment $A X$ such that $2 A Y=Y X$. Let $M$ be the midpoint of $B D$. Let $Z$ be a point on segment $A M$ s... | \frac{4}{7} |
12.4.3 ** Let $A B C D-A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ be a cube, $X$ be a moving point on line segment $A C$, and $Y$ be a moving point on line segment $B^{\prime} D^{\prime}$.
(1) Find the locus of the midpoint of line segment $X Y$;
(2) If $Z$ is a point on line segment $X Y$ such that $Z Y=2 X Z$, find... | (1)边长为\frac{} |
4. Determine the real values of $x, y, z$, for which
$$
\left\{\begin{array}{l}
\frac{2 x^{2}}{x^{2}+1}=y \\
\frac{3 y^{3}}{y^{4}+y^{2}+1}=z \\
\frac{4 z^{4}}{z^{6}+z^{4}+z^{2}+1}=x
\end{array}\right.
$$ | x_{1}=y_{1}=z_{1}=0x_{2}=y_{2}=z_{2}=1 |
Among the following functions, which one is an increasing function on the interval $(1, +\infty)$, and its graph is a symmetric figure about an axis? ( )
A: $y=-x^2+2x-1$
B: $y=\cos x$
C: $y=\lg|x-1|$
D: $y=x^3-3x^2+3x$ | \text{C} |
1. Given the lines $l_{1}: a x+2 y+6=0$, $l_{2}: x+(a-1) y+a^{2}-1=0$.
If $l_{1} \perp l_{2}$, then $a=$ $\qquad$ | \frac{2}{3} |
214. One of the angles of a triangle is $120^{\circ}$, and the lengths of the sides form an arithmetic progression. Find the ratio of the lengths of the sides of the triangle. | 3:5:7 |
3. A natural number $a$ is divisible by 35 and has 75 different divisors, including 1 and $a$. Find the smallest such $a$. | 490000 |
128. Digits and Squares. One of Professor Rackbrain's small Christmas puzzles reads as follows: what are the smallest and largest squares containing all ten digits from 0 to 9, with each digit appearing only once? | 10267538499814072356 |
Let the complex numbers $z_1 = 1 + 2i$ and $z_2 = 2 - i$, where $i$ is the imaginary unit. Then, the product $z_1z_2 =$ （ ）
A: $4 + 3i$
B: $4 - 3i$
C: $-3i$
D: $3i$ | A |
Given the function $f(x)=x^3+3x^2-9x-3$:
(I) If the tangent line to $f(x)$ at the point $(x_0, f(x_0))$ is perpendicular to the line $x - 9y + 1=0$, find the equation of the tangent line $l$;
(II) Determine the extreme values of the function $f(x)$. | -8 |
30th IMO 1989 shortlist Problem 18 Five points are placed on a sphere of radius 1. That is the largest possible value for the shortest distance between two of the points? Find all configurations for which the maximum is attained. Solution | \sqrt{2} |
If the point $(3,1)$ lies on the graph of the linear function $y=kx-2$ $(k\neq 0)$, then the value of $k$ is ____. | 1 |
2. Vasya's grandmother loves to pick berries in the forest and then make jam from them. Every weekend she goes by train, spending 200 rubles on tickets and collects 5 kilograms of berries. At the market, you can buy a kilogram of berries for 150 rubles, and sugar for 54 rubles per kilogram. From one kilogram of berries... | 1 |
3. The sum of 20 natural numbers is 2002. Find the greatest value that their GCD can take. | 91 |
Given the function $f(x) = ax^3 + bx + 8$, and $f(-2) = 10$, the value of $f(2)$ is ( )
A: $-10$
B: $-6$
C: $6$
D: $10$ | C |
Given vectors $\overrightarrow{a}=(x,-3)$, $\overrightarrow{b}=(2,-2)$, and $\overrightarrow{a} \perp \overrightarrow{b}$, then $x=$ ______. | -3 |
Factorize: $x^4-3x^3-28x^2$. | x^2(x-7)(x+4) |
【Question 4】Prime numbers are like the "building blocks" of natural numbers; every natural number can be written as the product of several primes (which may be the same), for example, $4=2 \times 2, 6=2 \times 3, 8=2 \times 2 \times 2, 9=3 \times 3, 10=2 \times 5$, etc. Therefore, $5 \times 13 \times 31-2$ written in t... | 3\times11\times61 |
6. Let $F_{1}$ and $F_{2}$ be the left and right foci of the hyperbola $C: \frac{x^{2}}{4}-\frac{y^{2}}{5}=1$, respectively. Point $P$ is on the right branch of the hyperbola $C$, and the excenter of $\triangle P F_{1} F_{2}$ opposite to $\angle P F_{1} F_{2}$ is $I$. The line $P I$ intersects the $x$-axis at point $Q$... | 4 |
3. Find the number of solutions in natural numbers for the equation $(x-4)^{2}-35=(y-3)^{2}$. | 3 |
Given the function $f(x)=2\cos x(\sin x-\cos x)$, $x\in R$, find $f(\frac{\pi}{4})=$ \_\_\_\_\_\_, and the maximum value of $f(x)$ is \_\_\_\_\_\_. | \sqrt{2}-1 |
In acute triangle ABC, the sides corresponding to angles A, B, and C are a, b, and c respectively. Given that sin(2A + $\frac {π}{2}$) - cos(B + C) = -1 + 2cosA.
1. Find the measure of angle A.
2. If the area of triangle ABC is S = 3$\sqrt {3}$, and b = 3, find the value of sinC. | sinC = \frac {2 \sqrt {39}}{13} |
Problem 3. For what least natural $k$ is the expression $2019 \cdot 2020 \cdot 2021 \cdot 2022 + k$ a square of a natural number? | 1 |
Given the function $f(x) = x\ln x - ax^{2} + a$.
(1) If $f(x) \leq a$, find the range of values for $a$.
(2) If $f(x)$ has a unique local minimum point $x_{0}$, find the range of values for $a$, and prove that $2a - 1 \lt f(x_{0}) \lt 0$. | (-\infty, \frac{1}{2}) |
In the positive arithmetic sequence $\{a_n\}$, $a_1$ and $a_4$ are the two roots of the equation $x^2-10x+16=0$. If the sum of the first $5$ terms of the sequence $\{\log_2 a_n\}$ is $S_5$ and $S_5 \in [n, n+1]$ where $n \in \mathbb{Z}$, then $n=$ ______. | 11 |
The function $f(x)$ is an even function defined on $R$. $f(x-1)$ is an odd function, and when $0 \lt x\leqslant 1$, $f(x)=\log_{2024}\frac{1}{x}$. Find $f(2025)+f(-\frac{1}{2024})$. | 1 |
Given a rectangular prism with a space diagonal length of $10$, the orthogonal projection of this diagonal onto one of the faces has a length of $8$. Determine the maximum volume of this rectangular prism. | 192 |
10.2. Find all values of the parameter $b$, for each of which the system of equations
$$
\left\{\begin{array}{l}
x^{2}-2 x+y^{2}=0 \\
a x+y=a b
\end{array}\right.
$$
has a solution for any $a$. | 0\leqb\leq2 |
The interval in which the function $f(x)=x^{3}-3x^{2}+1$ is decreasing is $(\ )$
A: $(2,+\infty)$
B: $(-\infty,2)$
C: $(-\infty,0)$
D: $(0,2)$ | D |
A new sequence formed by the arithmetic sequence with common difference $d$, $a_{1}, a_{2}, a_{3}, \ldots$, is $(a_{1}+a_{4}, a_{2}+a_{5}, a_{3}+a_{6}, \ldots)$. This new sequence is
A: An arithmetic sequence with common difference $d$
B: An arithmetic sequence with common difference $2d$
C: An arithmetic sequence with... | \text{B} |
Given that in $\triangle ABC$, $C=2A$, $\cos A= \frac{3}{4}$, and $2 \overrightarrow{BA}\cdot \overrightarrow{CB}=-27$.
(1) Find the value of $\cos B$;
(2) Find the length of $AC$. | 5 |
Given that the graphs of the linear functions $y=x+m$ and $y=2x-2$ intersect at the same point on the $y$-axis, find the value of $m$. | -2 |
15. Find the smallest constant $\lambda$, such that the inequality $x y\left(x^{2}+y^{2}\right)+y z\left(y^{2}+z^{2}\right)+z x\left(z^{2}+x^{2}\right) \leqslant \lambda(x+y+z)^{4}$ holds for all non-negative real numbers $x, y, z$. | \frac{1}{8} |
The domain of the function $y=\log_{2}(x-3)$ is ( )
A: $[3,+\infty)$
B: $(3,+\infty)$
C: $(-\infty,-3)$
D: $\mathbb{R}$ | \text{B} |
5. (ROM) Five points are given in the plane. Among the lines that connect these five points, no two coincide and no two are parallel or perpendicular. Through each point we construct an altitude to each of the other lines. What is the maximal number of intersection points of these altitudes (excluding the initial five ... | 310 |
What is the largest integer not exceeding $8x^3 +6x - 1$, where $x =\frac12 \left(\sqrt[3]{2+\sqrt5} + \sqrt[3]{2-\sqrt5}\right)$ ?
(A): $1$, (B): $2$, (C): $3$, (D): $4$, (E) None of the above. | 3 |
If the distance from the point $(2, 2)$ to the line $3x-4y+a=0$ is $a$, then $a=$ \_\_\_\_\_\_. | \frac {1}{3} |
Given $a=22^{55}$, $b=33^{44}$, $c=55^{33}$, the relationship among $a$, $b$, and $c$ is ( )
A: $a > b > c$
B: $b > a > c$
C: $c > a > b$
D: $c > b > a$ | A |
IMO 1981 Problem B3 The function f(x,y) satisfies: f(0,y) = y + 1, f(x+1,0) = f(x,1), f(x+1,y+1) = f(x,f(x+1,y)) for all non- negative integers x, y. Find f(4, 1981). | 2^{1984}-3 |
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which satisfy
$$
f(x+y)=f(x-y)+4 x y
$$
for all real numbers $x$ and $y$. | f(x)=x^{2}+ |
Given three points in space $P(-2,0,2)$, $M(-1,1,2)$, $N(-3,0,4)$, let $\overrightarrow{a}=\overrightarrow{PM}$, $\overrightarrow{b}=\overrightarrow{PN}$.
$(1)$ If $k\overrightarrow{a}+\overrightarrow{b}$ is perpendicular to $k\overrightarrow{a}-2\overrightarrow{b}$, find the value of $k$.
$(2)$ Find the distance f... | d = \frac{3\sqrt{2}}{2} |
Given that the center of a hyperbola is at the origin, its foci $F_1$ and $F_2$ lie on the coordinate axes, and one of its asymptotic lines is $y=x$. The hyperbola also passes through the point $(4, -\sqrt{10})$.
(1) Find the equation of the hyperbola;
(2) If point $M(3,m)$ is on this hyperbola, calculate the dot p... | \vec{MF_1} \cdot \vec{MF_2} = 0 |
12. The graph of the equation $|x|+|y|=2$ in the Cartesian coordinate system is
(A) a square
(B) two lines
(C) four lines
(D) a rectangle | A |
Which of the following propositions are true?
A: If $a \gt b \gt 0$, then $ac^{2} \gt bc^{2}$.
B: If $a \lt b$, then $\frac{1}{a} \gt \frac{1}{b}$.
C: If $a \gt b \gt 0$, then $a^{2} \gt ab \gt b^{2}$.
D: If $a \gt |b|$, then $a^{2} \gt b^{2}$. | C \text{ and } D |
If the equations of the asymptotes of the hyperbola passing through point P(5, -2) are $x-2y=0$ and $x+2y=0$, then the length of the real axis of this hyperbola is __________. | 6 |
8. Let $X_{n}=\{1,2, \cdots, n\}$, for any subset $A$ of $X_{n}$, denote $f(A)$ as the smallest element in $A$, then $\sum_{A \subseteq X_{n}} f(A)=$ $\qquad$ . | 2^{n+1}-2-n |
Example 6 Given $x, y, z \in \mathbf{R}_{+}$, and $x+y+z=1$. Find
$$
\frac{\sqrt{x}}{4 x+1}+\frac{\sqrt{y}}{4 y+1}+\frac{\sqrt{z}}{4 z+1}
$$
the maximum value. | \frac{3 \sqrt{3}}{7} |
9.2. What digit can the number $f(x)=[x]+[3 x]+[6 x]$ end with, where $x$ is an arbitrary positive real number? Here $[x]$ denotes the integer part of the number $x$, that is, the greatest integer not exceeding $x$.
| 0,1,3,4,6,7 |
In a unit circle, we draw the diameter $A C$, and let $B$ be the point on the arc $A C$ such that $\angle B O A = 30^{\circ}$ (where $O$ is the center of the circle). The tangent at $A$ intersects the line $C B$ at $D$. What is the difference between the length of the arc $A B$ and the segment $A D$? | 0.0123 |
Among the following statements:
① All equilateral triangles are congruent;
② Right-angled triangles with equal hypotenuses are congruent;
③ Isosceles triangles with equal vertex angles and equal sides are congruent;
④ Right-angled triangles with two equal acute angles are congruent.
The correct number of statements is ... | A |
If $(1+x)^4 = a + a_1x + a_2x^2 + a_3x^3 + a_4x^4$, then the value of $a_1 + a_3$ is. | 8 |
Let the function $f(x)=\ln (1+|x|)- \dfrac {1}{1+x^{2}}$, then the range of $x$ that makes $f(x) > f(2x-1)$ true is $(\quad)$
A: $(-\infty, \dfrac {1}{3})\cup(1,+∞)$
B: $( \dfrac {1}{3},1)$
C: $(- \dfrac {1}{3}, \dfrac {1}{3})$
D: $(-\infty,- \dfrac {1}{3})\cup( \dfrac {1}{3},+∞)$ | B |
4. Let $t \in \mathbf{R},(t+1)(1-|t|)>0$ holds if and only if ( ).
A. $|t|-1$ and $t \neq 1$
C. $t<1$ and $t \neq-1$
D. $t \neq \pm 1$ | \in(-\infty,-1)\cup(-1,1) |
In how many consecutive zeros does the decimal expansion of $\frac{26!}{35^3}$ end?
$\text{(A) }1\qquad\text{(B) }2\qquad\text{(C) }3\qquad\text{(D) }4\qquad\text{(E) }5$ | 3 |
12. Find all values of the coefficients $a, b$, and $c$ of the polynomial
$$
f(x)=x^{4}-x^{3}+a x^{2}+b x+c
$$
if it is known that it is divisible by the polynomial $\varphi(x)=x^{3}-2 x^{2}-5 x+6$ without a remainder. | =-7,b=1,=6 |
Task 2 - 200512 To transport a certain amount of gravel, a truck with a 5 t loading capacity would have had to make exactly 105 fully loaded trips. After 35 of these trips, it was replaced by another truck with a 7 t loading capacity.
Determine how many fully loaded trips this second truck still has to make to transpo... | 50 |
2. Jenny had to add 26 to a certain number. Instead she subtracted 26 and obtained -14 . What number should she have obtained?
A 28
B 32
C 36
D 38
E 42 | 38 |
$$\frac {3}{5} = \frac {12}{()} = 10 : \_\_\_\_\_\_ = \_\_\_\_\_\_ \div 100 = \_\_\_\_\_\_ \%$$ | 60 |
3. (3 points) The number 2017 is written on the board. Petya and Vasya are playing the following game: in one move, they can subtract any natural divisor of the number written on the board, except the number itself, and write the result of this subtraction on the board instead of the original number. Petya starts. The ... | Vasya |
Given $m, n \in \mathbb{R}$, and the sets $A = \{2, \log_{7}m\}$, $B = \{m, n\}$. If $A \cap B = \{0\}$, then $m + n =$ （ ）
A: 1
B: 2
C: 4
D: 8 | A |
Which of the following sequences is in the correct order according to the "syllogism" model?
① $y=\cos x$ ($x\in R$) is a trigonometric function;
② Trigonometric functions are periodic functions;
③ $y=\cos x$ ($x\in R$) is a periodic function.
A: ①②③
B: ②①③
C: ②③①
D: ③②① | \text{B} |
(Hungary 2004). A palace has the shape of a square divided into $2003 \times 2003$ rooms, like the squares of a large chessboard. There is a door between two rooms if and only if they share a wall. The main door allows, coming from outside the palace, to enter the palace through the room located at the northwest corner... | 99 |
18. The diagram shows a track layout for karting. The start and finish are at point $A$, and the kart driver can make as many laps as they like, returning to point $A$.
The young driver, Yu... | 34 |
5. I have thought of a two-digit number. I swapped its digits and added the number 15. I then halved the resulting number and swapped its digits. This gave me the number 62. What number did I think of?
Problems worth 10 points: | 73 |
Given the function $f(x)=\sin^{2}(x+\frac{π}{4})$, if $a=f(\log 5)$ and $b=f(\log \frac{1}{5})$, then which of the following is true?
A: $a+b=0$
B: $a-b=0$
C: $a+b=1$
D: $a-b=\sin \left(2\log 5\right)$ | C \text{ and } D |
23. $f(x)=\left[\cos ^{2}\left(x^{2}+\pi\right)+\sin ^{2} x^{2}\right] \cos \left(x^{3}+2 x\right)$ The derivative is ( ).
(A) $\left[8 x \cos x^{2} \cdot \sin x^{2}\right] \cos \left(x^{3}+2 x\right)-\left[\cos ^{2}\left(x^{2}+\pi\right)\right.$ $\left.+\sin ^{2} x^{2}\right] \sin \left(x^{3}+2 x\right) \cdot\left(3 x... | D |
Determine the largest number among the following three: $\log_{\frac{1}{2}}3$, $(\frac{1}{3})^{0.2}$, and $2^{\frac{1}{3}}$. | 2^{\frac{1}{3}} |
## Task 8
The area of a new playground is square and measures $1600 \mathrm{~m}^{2}$.
How long is one side of the playground?
How many meters of fence are needed for three sides? | 40\mathrm{~},120\mathrm{~} |
[ $[$ Divisibility of numbers. General properties $]$
Vovochka had more nuts than Pavlik. If Vovochka had given Pavlik as many nuts as Pavlik had, both boys would have had an equal number of nuts. But instead, Vovochka gave Pavlik a few nuts (no more than five), and divided the rest equally among three squirrels. How ... | 3 |
Given that the universal set is the set of real numbers $\mathbb{R}$, $A=\\left\\{ x\\left| \\dfrac{1-2x}{x-3}\\geqslant 0 \\right. \\right\\}$, and $B=\\left\\{ x\\left| {{x}^{2}}+a\\leqslant 0 \\right. \\right\\}$.
(1) Find $A\\cup B$ when $a=-4$;
(2) Find the range of real number values for $a$ if $\\left( {{C}\_{... | a > − \\dfrac{1}{4} |
6.2. How many minutes before $16:00$ was the angle between the hour and minute hands exactly the same in the previous instance | 21\frac{9}{11} |
Given the set $A={x|x^{2}-2ax-8a^{2}\leqslant 0}$.
(I) When $a=1$, find the set $∁\_RA$;
(II) If $a > 0$, and $(-1,1)\subseteq A$, find the range of values for the real number $a$. | a\geqslant \dfrac {1}{2} |
Let's determine those real numbers $p$ and $q$ for which $2^{p x+q}=p \cdot 2^{x}+q$ for all real $x$.
---
Determine the real numbers $p$ and $q$ such that $2^{p x+q}=p \cdot 2^{x}+q$ for all real $x$. | p=1,q=0 |
2. Consider the number $\mathrm{a}_{\mathrm{n}}=18 \underbrace{77 \ldots 7}_{\text {n times }} 889$, where $n$ is a natural number, and $\mathrm{C}_{\mathrm{n}}$ is the quotient of the division of the number $\mathrm{a}_{\mathrm{n}}$ by 13.
a) Show that $\mathrm{a}_{\mathrm{n}}$ is divisible by 13 for any $n$.
b) Det... | 8 |
The graph of the function $f(x)=\cos ^{2}x-\sin ^{2}x$ is shifted to the left by $\frac{π}{{12}}$ units to obtain the graph of the function $g(x)$. If $g(x)$ is decreasing on the interval $\left[0,a\right]$, then the maximum value of the real number $a$ is ( )
A: $\frac{{5π}}{{12}}$
B: $\frac{π}{3}$
C: $\frac{π}{6}$... | A |
8. Given the set $M=\left\{x \mid x^{2}-9>0\right\}, N=\left\{x \in \mathbf{Z} \mid x^{2}-8 x+a<0\right\}$. If the number of subsets of the set $M \cap N$ is 4, then the range of the real number $a$ is $\qquad$ . | [12,15) |
There are 3 numbers A, B, and C, such that $1001C - 2002A = 4004$, and $1001B + 3003A = 5005$. What is the average of A, B, and C?
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E) }\text{Not uniquely determined}$ | 3 |
If the retail price of a product in 2011 was 25% higher than in 2010, and it is desired to control the increase to only 10% in 2012 compared to 2010, then the price in 2012 should be reduced by ( )
A: 15%
B: 12%
C: 10%
D: 50% | \text{B} |
$21 \cdot 15$ As shown in the figure, an equilateral $\triangle ABC$ with side length 1, is folded along $EF$ so that $B$ lands on a point $B'$ on $AC$, and $FB' \perp AC$. Then the area of the quadrilateral $AEFC$ formed by the fold is
(A) $\frac{1}{6}(17 \sqrt{3}-27)$.
(B) $\frac{1}{6}(15 \sqrt{3}-27)$.
(C) $\frac{1}... | \frac{1}{8}(17\sqrt{3}-27) |
10. (20 points) Given $\angle A, \angle B, \angle C$ are the three interior angles of $\triangle ABC$, and the vector
$$
\boldsymbol{\alpha}=\left(\cos \frac{A-B}{2}, \sqrt{3} \sin \frac{A+B}{2}\right),|\boldsymbol{\alpha}|=\sqrt{2}.
$$
If when $\angle C$ is maximized, there exists a moving point $M$ such that $|\over... | \frac{2 \sqrt{3}+\sqrt{2}}{4} |
Task 16. Points $\boldsymbol{A}$ and $\boldsymbol{B}$ are located on a straight highway running from west to east. Point B is 9 km east of A. A car departs from point A heading east at a speed of 40 km/h. Simultaneously, a motorcycle departs from B in the same direction with a constant acceleration of 32 km/h².
 As shown in Figure 4, in the Cartesian coordinate system $x O y$, the vertices of the polygon $O A B C D E$ are
$$
\begin{array}{l}
O(0,0), A(0,6), B(4,6), \\
C(4,4), D(6,4), E(6,0) .
\end{array}
$$
It is known that the line $l$ passes through point $M$, intersects sides $O A$ and $D E$, and divides th... | y=-\frac{1}{3} x+\frac{11}{3} |
Find the derivative of the following functions:
(1) $y= \frac {x^{2}}{(2x+1)^{3}}$
(2) $y=e^{-x}\sin2x.$ | e^{-x}(2\cos2x-\sin2x) |
Given $a=\log_{2}3$, $b=\log_{3} \frac {1}{2}$, $c=(\frac {1}{2})^{3}$, determine the relationship among $a$, $b$, and $c$.
A: $a < b < c$
B: $c < b < a$
C: $b < c < a$
D: $a < c < b$ | C |
4. Among the natural numbers from 2021 to 9998, the numbers with the same tens and units digits are $\qquad$ in total. | 797 |
3. It is known that for functions $f(x), g(x), h(x)$ the following relation holds for any $x$ in the domain of definition:
$g(h(x))-f(g(x))=f(h(x))$, where $g(x)=(5 x+3) /(x+1)$ and $h(x)=x^{2}+x+2$. Find $f(4)$. | 2.3 |
Kozhevnikov P.A.
What is the minimum number of connections required to organize a wired communication network of 10 nodes, so that the communication remains possible between any two remaining nodes (even through a chain of other nodes) in the event of any two nodes failing?
# | 15 |
. The points $M$ and $N$ are chosen on the angle bisector $A L$ of a triangle $A B C$ such that $\angle A B M=\angle A C N=23^{\circ} . X$ is a point inside the triangle such that $B X=C X$ and $\angle B X C=2 \angle B M L$. Find $\angle M X N$. | 46 |
3. In the range $1 \sim 2000$, if a number is randomly selected, the probability that the integer can be divided by 6 but not by 4 is ).
(A) $\frac{1}{4}$
(B) $\frac{83}{1000}$
(C) $\frac{167}{2000}$
(D) $\frac{3}{4}$ | C |
Students A, B, and C were asked if they have visited cities A, B, and C.
Student A said: I have visited more cities than B, but I have not visited city B;
Student B said: I have not visited city C;
Student C said: The three of us have visited the same city.
From this, we can determine that the city visited by B... | A |
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