problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
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Task 2. On the island, every resident is either a knight (always tells the truth), a liar (always lies), or a commoner (can either tell the truth or lie). Knights are considered to be of the highest rank, commoners are of medium rank, and liars are of the lowest rank. A, B, and C are residents of this island. One of th... | B |
4. Confused Teacher is teaching the students long division, and he gave the students a question: The smallest four-digit number without repeated digits that can be divided by 13 is $\qquad$ . | 1027 |
Let $\mathbb{N}$ be the set of all positive integers. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that the number $(f(m)+n)(m+f(n))$ is a square for all $m, n \in \mathbb{N}$. Answer. All functions of the form $f(n)=n+c$, where $c \in \mathbb{N} \cup\{0\}$. | f(n) = n + c |
Problems 14, 15 and 16 involve Mrs. Reed's English assignment.
A Novel Assignment
The students in Mrs. Reed's English class are reading the same 760-page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 3... | 456 |
If the product of the eccentricities of the hyperbola \\(\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1\\) and the ellipse \\(\dfrac{x^2}{m^2} + \dfrac{y^2}{b^2} = 1 (a > 0, m > b > 0)\\) is greater than \\(1\\), then the triangle with sides \\(a, b, m\\) must be \\
A: Isosceles triangle \\
B: Acute triangle \\
C: Right trian... | \text{D} |
Prove by mathematical induction that for $n\in \mathbb{N}^*$, the product $(n+1)(n+2)\ldots(n+n)$ equals $2^n \times 1 \times 2 \times \ldots \times (2n - 1)$. During the induction step from $k$ to $k+1$, identify the algebraic expression that must be multiplied on the left side to achieve this equality. | 2(2k+1) |
Given $f(x) = e^x - ax^2$, the equation of the tangent line to the curve $y = f(x)$ at $(1, f(1))$ is $y = bx + 1$.
(1) Find the values of $a$ and $b$;
(2) Find the maximum value of $y = f(x)$ on $[0, 1]$;
(3) Prove that when $x > 0$, $e^x + (1-e)x - x\ln x - 1 \geqslant 0$. | e - 1 |
6. Find all such triples of distinct prime numbers $p, q, r$ and a natural number $k$, such that $p q - k$ is divisible by $r$, $q r - k$ is divisible by $p$, $r p - k$ is divisible by $q$, and the number $p q - k$ is positive. | 2,3,5allpermutations,k=1 |
13. How many five-digit numbers are exactly divisible by $6$, $7$, $8$, and $9$? | 179 |
11. (20 points) Let the sequence $\left\{a_{n}\right\}$ satisfy
$$
a_{1}=1, a_{n+1}=\frac{a_{n}}{n}+\frac{n}{a_{n}}\left(n \in \mathbf{N}_{+}\right) \text {. }
$$
Try to find $\left[a_{2009}^{2}\right]$. | 2009 |
338. Find the derivative of the function $y=\operatorname{tg} x^{5}$. | \frac{5x^{4}}{\cos^{2}x^{5}} |
4. In the maze diagram in Fig. 13, each segment (link) is a corridor, and each circle is a small room. Some rooms have beacons that hum, each with its own voice. When in any room, the robot hears the signal from each beacon and determines the distance to it by the attenuation, i.e., the number of links on the shortest ... | 3 |
21.16. All integers are written in a row, starting from one. What digit stands at the 206,788th place?
## 21.6. Periods of Decimal Fractions and Repunits
| Let $p$ be a prime number different from 2 and 5. The length of the period of the number $p$ is the number of digits in the period of the decimal representation o... | 7 |
127 Elements are non-empty sets $S$ of natural numbers that satisfy “if $x \in S$, then $14-x \in S$”. The number of such sets $S$ is $\qquad$ . | 127 |
1. There were 21 consecutive natural numbers written on the board. When one of the numbers was erased, the sum of the remaining numbers became 2017. Which number was erased? | 104 |
A certain supermarket in Beishan sells teapots and teacups. Each teapot is priced at $20$ yuan, and each teacup is priced at $6$ yuan. During the "Double Eleven" promotion period, the supermarket offers two discount options to customers:① Buy one teapot and get one teacup for free; ② Pay $90\%$ of the original price fo... | 181 \text{ yuan} |
Given vectors $\overrightarrow{a}=(4,3)$ and $\overrightarrow{b}=(1,-1)$.
(I) Find the cosine value of the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$;
(II) If vector $3\overrightarrow{a}+4\overrightarrow{b}$ is parallel to vector $λ\overrightarrow{a}-\overrightarrow{b}$, find the value of real number $... | -\frac{3}{4} |
Example 6 Find all pairs of primes $(p, q)$ such that
$$
p q \mid\left(5^{p}+5^{q}\right) .{ }^{[2]}
$$
(2009, China Mathematical Olympiad) | (2,3),(3,2),(2,5),(5,2),(5,5),(5,313),(313,5) |
Given that the function $y=f\left(x\right)$ is differentiable on $R$ and $f\left(0\right)=1$, when $x\neq -1$, its derivative satisfies $\left(x+1\right)\left[{f'}\left(x\right)-f\left(x\right)\right] \gt 0$. For the function $g(x)=\frac{{f(x)}}{{e^x}}$, which of the following conclusions is correct?
A: The function $... | AC |
Given that the variance of $x_1, x_2, x_3, \ldots, x_{2013}$ is 3, the variance of $3(x_1-2), 3(x_2-2), 3(x_3-2), \ldots, 3(x_{2013}-2)$ is ( )
A: 3
B: 9
C: 18
D: 27 | \text{D} |
4・184 For any three positive integers $x, y, z$, let
$$f(x, y, z)=[1+2+3+\cdots+(x+y-2)]-z$$
Find all positive integer quadruples $(a, b, c, d)$, such that
$$f(a, b, c)=f(c, d, a)=1993$$ | (23,42,23,42) |
Task 1. A vacuum robot is programmed to move on the floor according to the law:
$\left\{\begin{array}{l}x=(t-6)^{2} \\ y=0,0 \leq t \leq 7 ; y=(t-7)^{2}, t \geq 7\end{array}\right.$
where the axes are parallel to the walls. Time $t$ is measured in minutes, and coordinates in meters.
Find the path traveled by the rob... | 37;2\sqrt{2} |
To obtain the graph of $y=\sin (2x- \frac {\pi}{3})$, it is only necessary to shift the graph of $y=\sin 2x$ by
A: shifting it to the left by $\frac {\pi}{3}$ units
B: shifting it to the right by $\frac {\pi}{3}$ units
C: shifting it to the left by $\frac {\pi}{6}$ units
D: shifting it to the right by $\frac {\pi}{6}$ ... | D |
3. Find the smallest $n>2016$, such that $1^{n}+2^{n}+3^{n}+4^{n}$ is not divisible by 10. | 2020 |
Which of the following calculations is correct?
A: $3a+b=3ab$
B: $a^{2}+a^{2}=a^{4}$
C: $(a-b)^{2}=a^{2}-b^{2}$
D: $(-3a)^{2}=9a^{2}$ | D |
12. As shown in the figure, rectangle $ABCD$ is a cargo yard with a length of 1000 meters and a width of 600 meters, with entrances at $A$ and $D$. A toll booth $P$ is to be built inside the cargo yard, and a loading platform $H$ is to be built on the railway segment $BC$. The minimum sum of the lengths of the three ro... | 500\sqrt{3}+600 |
3. For which $y$ does the equation $x^{2}+2 y^{2}+8 z^{2}-2 x y z-9=0$ have no solutions for any $z$? | \frac{3}{\sqrt{2}}<|y|\leq2\sqrt{2} |
2. Find all digits $n$ and 2018-digit natural numbers $x, x=\overline{a_{2017} \ldots a_{2} a_{1} a_{0}}$, such that
$$
n \cdot x=\overline{\left(a_{2017}+n\right) \ldots\left(a_{2}+n\right)\left(a_{1}+n\right)\left(a_{0}+n\right)}
$$ | 2 |
Given the function $f(x)=ae^{x}+(2-e)x$ (where $a$ is a real number, and $e$ is the base of the natural logarithm), the tangent line of the curve $y=f(x)$ at $x=0$ is parallel to the line $(3-e)x-y+10=0$.
$(1)$ Find the value of the real number $a$, and determine the number of zeros of the function $f(x)$ in the inte... | 0 |
2. There are $n$ circles on a plane, each pair of which intersects at two points, and no three circles have a common point. They divide the plane into $f(n)$ regions, such as $f(1)=2, f(2)=4$, etc. The expression for $f(n)$ is ( ).
A. $2^{n}$
B. $n^{2}-n+2$
C. $2^{n}-(n-1)(n-2)(n-3)$
D. $n^{3}-5 n^{2}+10 n-4$ | f(n)=n^{2}-n+2 |
Example 3 Let the sequence $a_{0}, a_{1}, a_{2}, \cdots$ satisfy $a_{0}=a_{1}=11, a_{m+n}=\frac{1}{2}\left(a_{2 m}+a_{2 n}\right)-(m-n)^{2}$, $m, n \geqslant 0$, find $a_{45}$.
(1991 Soviet Ministry of Education Recommended Problem) | 1991 |
Let $S_{n}$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_{n}\}$, with $a_{2}=3$ and $S_{4}=16$. Find the value of $S_{9}$. | 81 |
SI. 4 The solution of the inequality $x^{2}+5 x-2 c \leq 0$ is $d \leq x \leq 1$. Find $d$. | -6 |
Given $(1+x) + (1+x)^2 + (1+x)^3 + \ldots + (1+x)^n = a_0 + a_1x + a_2x^2 + \ldots + a_nx^n$, and $a_0 + a_1 + a_2 + \ldots + a_n = 126$, then the constant term in the expansion of $\left( \sqrt {x}- \frac {1}{ \sqrt {x}} \right)^{n}$ is ( )
A: -15
B: 15
C: 20
D: -20 | \text{D} |
Given that the production cost $C$ (in yuan) of a certain shoe factory per day and the production quantity $n$ (in pairs) have the function relationship $C = 4000 + 50n$.
(1) If the production cost one day is 36,000 yuan, how many pairs of shoes were produced that day?
(2) If the selling price of each pair of shoes ... | 313 |
Fill in the corresponding braces with the following numbers: $2021$, $-1.7$, $\frac{2}{5}$, $0$, $-6$, $\frac{23}{8}$, $\frac{\pi}{2}$.
Set of positive numbers: $\{ \_\_\_\_\_\_\ldots \}$;
Set of integers: $\{ \_\_\_\_\_\_\ldots \}$;
Set of negative fractions: $\{ \_\_\_\_\_\_\ldots \}$;
Set of positive rationa... | \{2021, \frac{2}{5}, \frac{23}{8}\} |
1. In the empty fields of the given table, write numbers so that the sum of any three consecutive fields, both horizontally and vertically, is equal to 12.
| | 5 | | | | | | |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| | | | | | 1 | | |
| 6 | | | | | | | |
| | | | 2 | | | | ... | \begin{pmatrix}\hline2&5&5&2&5&5&2&5\\\hline4&7&1&4&7&1&4&7\\\hline6&0&6&6&0&6& |
69(1003). A cyclist and a motorcyclist are moving towards an intersection along two mutually perpendicular roads. At a certain moment, the cyclist is 8 km away, and the motorcyclist is 15 km away from the intersection. After how many minutes will the distance between them be 5 km, if the cyclist's speed is $\frac{1}{3}... | 12 |
Which of the following statements about propositions is correct? ( )
A: The negation of the proposition "If \\(x^{2}=1\\), then \\(x=1\\)" is "If \\(x^{2}=1\\), then \\(x\\neq 1\\)"
B: "\\(x=-1\\)" is a necessary but not sufficient condition for "\\(x^{2}-5x-6=0\\)"
C: The negation of the proposition "There exists \... | \text{D} |
## problem statement
Find the cosine of the angle between vectors $\overrightarrow{A B}$ and $\overrightarrow{A C}$.
$A(-2 ; 1 ; 1), B(2 ; 3 ;-2), C(0 ; 0 ; 3)$ | 0 |
# 4. Problem 4
In the younger group of a kindergarten, there are two (small) Christmas trees and five children. The teachers want to divide the children into two circles around each of the trees, with at least one child in each circle. The teachers can distinguish the children but not the trees: two such divisions int... | 50 |
\section*{Exercise 5 - 261035}
Prove the following statement:
If \(n\) is a natural number with \(1 \leq n \leq 5\), then:
When a sphere is cut by \(n\) planes, at most 22 regions are formed on the surface of the sphere. | 22 |
# Task 5. 20 points
A beginner economist-cryptographer received a cryptogram from the ruler, which contained another secret decree on the introduction of a commodity tax on a certain market. The cryptogram specified the amount of tax revenue to be collected. It was also emphasized that a larger amount of tax revenue c... | 8640 |
1. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x \in \mathbb{R}$, $f(x) \leqslant x$ and for all $x, y \in \mathbb{R}$, $f(x+y) \leqslant f(x)+f(y)$. | f(x)=x |
10. As shown in Figure 6, in two concentric circles with center 0, the chord $AB$ of the larger circle is tangent to the smaller circle at point $C$, and $AB=8 \text{ cm}$. Then the area of the shaded part in the figure is $\qquad$ | 16 \pi |
5. There are 19 children with ages ranging from $1 \sim 19$ years old standing in a circle. Record the difference in ages between each pair of adjacent children. The maximum possible sum of these 19 differences is . $\qquad$ | 180 |
Given $\left\{\begin{array}{l}{x=-3}\\{y=2}\end{array}\right.$ is a solution of the equation $2x+ky=0$, then $k$ is equal to:
A: $0$
B: $1$
C: $2$
D: $3$ | D |
2. (7 points) Cheburashka and Gena ate a cake. Cheburashka ate twice as slowly as Gena, but started eating a minute earlier. In the end, they each got an equal amount of cake. How long would it take Cheburashka to eat the cake alone? | 4 |
15. Find the sum: $1+2+3+\ldots+1999$. | 1999000 |
253. Find a number which, when divided by 17, 13, and 10, gives remainders of 15, 11, and 3, respectively.
From the treatise "De triangulis omnimodis libri quinque". | 1103 |
In a bag, there are 14 balls, including 4 white balls and 10 red balls. Now, if 6 balls are drawn from the bag such that there are at least 2 white balls and no more than 3 red balls, then the number of ways to draw the balls is ( )
A: 1
B: 2
C: 3
D: 4 | B |
## Task Condition
Find the derivative.
$$
y=\frac{1}{2} \cdot \operatorname{arctan} \frac{e^{x}-3}{2}
$$ | \frac{e^{x}}{e^{2x}-6e^{x}+13} |
Determine the number of pairs of integers $(m, n)$ such that
$$
\sqrt{n+\sqrt{2016}}+\sqrt{m-\sqrt{2016}} \in \mathbb{Q}
$$ | 1 |
## problem statement
Calculate the volume of the tetrahedron with vertices at points $A_{1}, A_{2}, A_{3}, A_{4_{\text {and }}}$ its height dropped from vertex $A_{4 \text { to the face }} A_{1} A_{2} A_{3}$.
$A_{1}(4 ;-1 ; 3)$
$A_{2}(-2 ; 1 ; 0)$
$A_{3}(0 ;-5 ; 1)$
$A_{4}(3 ; 2 ;-6)$ | \frac{17}{\sqrt{5}} |
## Problem Statement
Calculate the indefinite integral:
$$
\int \frac{\arccos ^{3} x-1}{\sqrt{1-x^{2}}} d x
$$ | -\frac{1}{4}\cdot(\arccosx)^{4}+\arccosx+C |
If $C_{x}^{12} = C_{x}^{18}$, then $x= \_\_\_\_\_\_$. | 30 |
6. In the Cartesian coordinate system, the curve represented by the equation $\frac{|x+y|}{2 a}+\frac{|x-y|}{2 b}=1$ (where $a, b$ are two distinct positive numbers) is
(A) Triangle
(B) Square
(C) Non-square rectangle
(D) Non-square rhombus | D |
ABC is a triangle with coordinates A =(2, 6), B =(0, 0), and C =(14, 0).
(a) Let P be the midpoint of AB. Determine the equation of the line perpendicular to AB passing through P.
(b) Let Q be the point on line BC for which PQ is perpendicular to AB. Determine the length of AQ.
(c) There is a (unique) circle passin... | 5\sqrt{2} |
What are the prime numbers $p$ such that $p \mid 29^{p}+1$? | 2,3,5 |
2.1. Vasya bought a ticket to a compartment carriage for the first time. Thinking that each compartment has 6 seats, Vasya decided that his compartment has number 5. In which compartment can Vasya's seat actually be, if it is known that each compartment has 4 seats? List all possible options. | 7,8 |
(Ⅰ) Find the range of $y=4^x-2^{x+1}$;
(Ⅱ) If the equation $4^x-2^{x+1}+a=0$ about $x$ has solutions, find the range of the real number $a$. | a\leq1 |
A mirror MN is hung on a vertical wall, and a person of height AB is standing upright in front of it. The full body of the person can be seen in the mirror if the size of MN and AB satisfies ( ).
A: MN=AB
B: MN=$\frac{1}{2}$AB
C: MN=$\frac{1}{3}$AB
D: MN=$\frac{1}{4}$AB | B |
Given that $φ∈[0,π)$, the function $f(x)=\\cos 2x+\\cos (x+φ)$ is an even function. Find the value of $φ$ and the minimum value of $f(x)$. | - \\dfrac {9}{8} |
Given the arithmetic sequence $\{a\_n\}$, where $a\_1=1$ and $a\_2=2$, find $a\_3=$ _____ . | 3 |
The sequences $3,20,37,54,71, \ldots$ and $16,27,38,49,60,71, \ldots$ each have 71 as a common term. The next term that these sequences have in common is
(A) 115
(B) 187
(C) 258
(D) 445
(E) 1006 | 258 |
Given that $a \in \mathbb{R}$, the function $f(x) = e^{x} - ae^{-x}$ has a derivative $f'(x)$, which is an odd function. Find the value of $a$.
A: $0$
B: $1$
C: $2$
D: $-1$ | D |
In the reaction $A(g) + 3B(g) \rightleftharpoons 2C(g)$, if the chemical reaction rate represented by substance A is $0.2 \, \text{mol}/(\text{L} \cdot \text{min})$, then the chemical reaction rate represented by substance B is $\_\_\_ \, \text{mol}/(\text{L} \cdot \text{min})$. | 0.6 |
Given a triangle with sides $a$, $b$, and $c$ satisfying $9 \geq a \geq 8 \geq b \geq 4 \geq c \geq 3$, the maximum area of the triangle is $ \boxed{16}$. | 16 |
Given that $a$, $b$, $c$ are all positive real numbers, let $M=\max\left\{\frac{1}{ac}+b, \frac{1}{a}+bc, \frac{a}{b}+c\right\}$, then the minimum value of $M$ is \_\_\_\_\_\_. | 2 |
2. Find all pairs of real numbers $x, y$, satisfying the equation
$$
\frac{2 x}{1+x^{2}}=\frac{1+y^{2}}{2 y}
$$ | 1-1 |
Problem 2. Zoki cycled $64 \%$ of the way, and walked the remaining $9 \mathrm{~km}$. How many kilometers did Zoki cycle? | 16\mathrm{~} |
Consider a regular $2 n$-gon $P, A_{1} A_{2} \ldots A_{2 n}$ in the plane, where $n$ is a positive integer. We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$, if the line segment $S E$ contains no other points that lie on the sides of $P$ except $S$. We color the si... | 6n |
Raskinna I.V.
Winnie-the-Poo, Piglet, Rabbit, and Donkey Eeyore emptied a barrel of honey. Piglet ate half of what Winnie-the-Poo ate, Rabbit ate half of what Winnie-the-Poo did not eat, and Donkey Eeyore got only a tenth of the barrel. What part of the barrel did Rabbit get? | \frac{3}{10} |
In a certain country, there are 21 cities and the government plans to build $n$ roads (all two-way), each connecting exactly 2 of the country's cities. What is the smallest value of $n$ such that, regardless of how the roads are built, it is possible to travel between any 2 cities (possibly passing through intermediate... | 191 |
Let $P(x,y)$ be a moving point on the graph of the function $y=x^{2}-1(x > \sqrt {3})$, and let $m= \frac {3x+y-5}{x-1}+ \frac {x+3y-7}{y-2}$. Find the minimum value of $m$. | 8 |
8) In the rectangle $A B C D$ (vertices indicated in a counterclockwise direction), $E$ and $F$ are the midpoints of the longer sides $A D$ and $B C$ respectively. Knowing that $A B F E$ is similar to $A B C D$, what is the value of $A D / A B$?
(A) $\frac{7}{5}$
(B) $\frac{3}{2}$
(C) $\sqrt{3}$
(D) $2 \sqrt{2}$
(E) th... | \sqrt{2} |
The roots of the equation $2\sqrt {x} + 2x^{ - \frac {1}{2}} = 5$ can be found by solving:
$\textbf{(A)}\ 16x^2-92x+1 = 0\qquad\textbf{(B)}\ 4x^2-25x+4 = 0\qquad\textbf{(C)}\ 4x^2-17x+4 = 0\\ \textbf{(D)}\ 2x^2-21x+2 = 0\qquad\textbf{(E)}\ 4x^2-25x-4 = 0$ | 4x^2-17x+4=0 |
In the sequence $\{a_{n}\}$, ${a_n}=1+\frac{1}{{2n-9}}$ ($n∈{N^*},a∈R,a≠0$), the values of its maximum term and minimum term are the values of $b_{2}-1$ and $b_{3}-9$ in the geometric sequence $\{b_{n}\}$, respectively.
$(1)$ Find the general formula of the sequence $\{b_{n}\}$;
$(2)$ Given the sequence $\{c_{n}\}$... | M_{n}=\frac{3^{n}(2n-3)+3}{4} |
8.133. $\sin ^{2} 3 x+\sin ^{2} 4 x=\sin ^{2} 5 x+\sin ^{2} 6 x$.
8.133. $\sin ^{2} 3 x+\sin ^{2} 4 x=\sin ^{2} 5 x+\sin ^{2} 6 x$.
The equation remains the same in English as it is a mathematical expression. | x_{1}=\frac{\pi}{2};x_{2}=\frac{\pin}{9},,n\inZ |
15. Let $n$ be a positive integer not exceeding 2014 with the property that $x^{2}+x+1$ is a factor of $x^{2 n}+x^{n}+1$. Find the sum of all possible values of $n$.
(2 marks)
Let $n$ be a positive integer not exceeding 2014 with the property that $x^{2}+x+1$ is a factor of $x^{2 n}+x^{n}+1$. Find the sum of all possib... | 1352737 |
6.133. Without solving the equation $3 x^{2}-5 x-2=0$, find the sum of the cubes of its roots. | \frac{215}{27} |
I4.1 If $f(t)=2-\frac{t}{3}$, and $f(a)=-4$, find $a$.
I4.2 If $a+9=12 Q+r$, where $Q, r$ are integers and $0<r<12$, find $r$.
I4.3 $x, y$ are real numbers. If $x+y=r$ and $M$ is the maximum value of $x y$, find $M$.
I4.4 If $w$ is a real number and $2^{2 w}-2^{w}-\frac{8}{9} M=0$, find $w$. | 1 |
## Task 3
Increase 1000 by 90. Decrease 1000 by 90. Divide 240 into six equal parts. Increase 7000 by 3000. | 10000 |
A3. A square is inscribed in a circle with radius $r$. The length of the side of the square is:
(A) $2 r \sqrt{2}$
(B) $r \sqrt[3]{3}$
(C) $\frac{r}{\sqrt{2}}$
(D) $\frac{r}{2}$
(E) None of the above answers is correct.
$, $\overrightarrow{b}=(1,0)$, and $\overrightarrow{c}=(4,-3)$. If $\lambda$ is a real number and $(\overrightarrow{a}+\lambda \overrightarrow{b}) \perp \overrightarrow{c}$, then $\lambda=($ $)$.
A: $\dfrac {1}{4}$
B: $\dfrac {1}{2}$
C: $1$
D: $2$ | B |
8. (13 points) On the coordinate plane, all points whose coordinates satisfy the conditions
$$
\left\{\begin{array}{l}
|2 x+3 y|+|3 x-2 y| \leq 13 \\
2 x^{2}-3 x y-2 y^{2} \leq 0
\end{array}\right.
$$
are shaded. Find the area of the resulting figure.
# | 13 |
Given that $f(x)$ is an even function with a domain of $R$, and when $x \geq 0$, $f(x) = x^2 - 4x$. Then, the solution set of the inequality $f(x+2) < 5$ is ____. | (-7,3) |
5. Given $A\left(\frac{1}{3}, \frac{1}{a}\right), B\left(\frac{1}{4}, \frac{1}{b}\right), C\left(\frac{1}{5}, \frac{1}{c}\right)$ satisfy $\frac{a}{b+c}=\frac{1}{3}, \frac{b}{a+c}=\frac{1}{2}$. Then the positions of points $A, B, C$ are suitable for
(A) on the same straight line
(B) forming an acute triangle
(C) formin... | A |
Given the general term of the sequence $\left\{a_{n}\right\}$ as ${a_{n}}=\begin{cases} n+\frac{15}{n},n\leqslant 5 \\ a\ln n-\frac{1}{4},n > 5 \end{cases}$, if the minimum value of $\left\{a_{n}\right\}$ is $\frac{31}{4}$, then the range of the real number $a$ is ________. | a \geqslant \frac{8}{\ln 6} |
3.1. The friends who came to visit Gavrila occupied all the three-legged stools and four-legged chairs in the room, but there was no place left for Gavrila himself. Gavrila counted that there were 45 legs in the room, including the "legs" of the stools and chairs, the legs of the visiting guests (two for each!), and Ga... | 9 |
2.054. $\frac{3 a^{2}+2 a x-x^{2}}{(3 x+a)(a+x)}-2+10 \cdot \frac{a x-3 x^{2}}{a^{2}-9 x^{2}}$. | 1 |
6. (1988 Mathematics Retest Question for Science Experimental Class Admission) Let $S=[\sqrt{1}]+[\sqrt{2}]+\cdots+[\sqrt{1988}]$, find $[\sqrt{S}]$.
保留源文本的换行和格式，翻译结果如下：
6. (1988 Mathematics Retest Question for Science Experimental Class Admission) Let $S=[\sqrt{1}]+[\sqrt{2}]+\cdots+[\sqrt{1988}]$, find $[\sqrt{S}]$... | 241 |
Example 9 For what value of $a$ does the inequality
$\log _{\frac{1}{a}}\left(\sqrt{x^{2}+a x+5}+1\right) \cdot \log _{5}\left(x^{2}+a x+6\right)+\log _{a} 3 \geqslant 0$ have exactly one solution. | 2 |
Given that $\{a\_n\}$ is a geometric sequence, and $S\_n$ is the sum of its first $n$ terms. If $a\_3a\_5= \frac {1}{4}a\_1$, and the arithmetic mean of $a\_4$ and $a\_7$ is $\frac {9}{8}$, then the value of $S\_5$ is ___. | 31 |
1. Set $A=\{0,1,2,3,4,5,6,7\}$, how many mappings $f$ from $A$ to $A$ satisfy the following conditions (1) and (2)?
(1) $i, j \in A, i \neq j$, then $f(i) \neq f(j)$;
(2) $i, j \in A, i+j=7$, then $f(i)+f(j)=7$. | 384 |
There is a $40\%$ chance of rain on Saturday and a $30\%$ of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive intege... | 107 |
Given sets $A=\{x|x+1)(x-2) \lt 0\}$ and $B=\{x|2^{x-1}\geq 1\}$, then $A$⋂$B=\underline{\hspace{1cm}}$
A: $(1,2)$
B: $[1,2)$
C: $(-1,2)$
D: $(0,2)$ | B |
Given points $A(-1,1)$, $B(1,2)$, $C(-2,-1)$, $D(2,2)$, the projection of vector $\overrightarrow{AB}$ in the direction of $\overrightarrow{CD}$ is ______. | \dfrac{11}{5} |
In the Cartesian coordinate system xOy, with the origin as the pole and the positive x-axis as the polar axis, a polar coordinate system is established. It is known that the parametric equation of curve C1 is $$\begin{cases} x=2\cos\alpha \\ y=2+2\sin\alpha\end{cases}$$ (where $\alpha$ is the parameter), and the equati... | 2\sqrt{3} |
4. The diagonals of the inscribed quadrilateral $A B C D$ intersect at point $P$, and triangle $A P D$ is acute-angled. Points $E$ and $F$ are the midpoints of sides $A B$ and $C D$ respectively. A perpendicular is drawn from point $E$ to line $A C$, and a perpendicular is drawn from point $F$ to line $B D$, these perp... | 90 |
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