problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
10.21 Write the complex number $z=\sin 36^{\circ}+i \cos 54^{\circ}$ in trigonometric form. | \sqrt{2}\sin36(\cos45+i\sin45) |
7. As shown in Figure 2, in plane $\alpha$, $\triangle A B C$ and $\triangle A_{1} B_{1} C_{1}$ are on opposite sides of line $l$, with no common points with $l$, and are symmetric about line $l$. Now, if plane $\alpha$ is folded along line $l$ to form a right dihedral angle, then the six points $A, B, C, A_{1}, B_{1},... | 11 |
We placed 8 balls of radius $r$ on a horizontal plane such that their centers are at the vertices of a regular octagon, and adjacent balls touch each other. What is the radius of the sphere that touches the plane and the balls? | \frac{(2+\sqrt{2})r}{2} |
$PA$ is perpendicular to the plane containing $\triangle ABC$, with $AB = AC = 13$, $BC = 10$, and $PA = 5$. The distance from point $P$ to line $BC$ is ______. | 13 |
2. The expression $\eta(x)-\eta(x-7 \pi)=\left\{\begin{array}{l}1, x \in[0,7 \pi), \\ 0, x \in(-\infty ; 0) \cup[7 ;+\infty) .\end{array}\right.$
Case 1. $x \in[0,7 \pi)$. The equation takes the form
$$
\sin x=1+\cos x
$$
Using the method of introducing an auxiliary argument, we get
$$
\sin \left(x-\frac{\pi}{4}\ri... | \frac{\pi}{2}+2\pik,k=0,1,2,3;\pi(2+1),\inZ |
Given the function $f(x)=2\cos ^{2}x+\cos (2x+ \frac {π}{3})$.
(1) If $f(α)= \frac { \sqrt {3}}{3}+1$, $0 < a < \frac {π}{6}$, find the value of $\sin 2α$;
(2) In acute triangle $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively; if $f(A)=- \frac {1}{2}$, $c=3$, the area of $\tri... | \sqrt {13} |
Throwing two fair cubic dice, let the sum of the numbers facing up on the two dice be denoted as $S$. In a single throw, given that $S$ is an odd number, the probability that $S=9$ is ( ).
A: $\boxed{\dfrac {1}{6}}$
B: $\boxed{\dfrac {2}{9}}$
C: $\boxed{\dfrac {1}{9}}$
D: $\boxed{\dfrac {1}{5}}$ | B |
Exercise 2. Let $A B C$ be a triangle such that $\widehat{C A B}=20^{\circ}$. Let $D$ be the midpoint of the segment $[A B]$. Suppose that $\widehat{C D B}=40^{\circ}$. What is the value of the angle $\widehat{A B C}$? | 70 |
Given the sets $M={x|x^{2} < 4}$ and $N={x|x < 1}$, determine the set $M∩N=( \ \ )$.
A: ${x|-2 < x < 1}$
B: ${x|x < -2}$
C: ${x|x < 1}$
D: ${x|x < 2}$ | A |
Example 7 Given the function $f(x)=\frac{x-5}{2 x+m}$ whose graph is symmetric with respect to the line $y=x$, find the real number $m$.
Translating the above text into English, while preserving the original text's line breaks and format, the result is as follows:
Example 7 Given the function $f(x)=\frac{x-5}{2 x+m}$... | -1 |
11. The smallest positive integer $x$ that satisfies $(x-1)^{200}>3^{300}$ is | 7 |
Given that the sequence $\{a_{n}\}$ is an arithmetic sequence, and $\{b_{n}\}$ is a geometric sequence with a common ratio of $2$, and $a_{2}-b_{2}=a_{3}-b_{3}=b_{4}-a_{4}$.
$(1)$ Prove that $a_{1}=b_{1}$;
$(2)$ If the set $M=\{k|b_{k}=a_{m}+a_{1}$, $1\leqslant m\leqslant 50\}$, find the number of elements in the s... | 6 |
Find the equation of the tangent line to the curve $y=xe^x-2x^2+1$ at the point $(0,1)$. | y=x+1 |
Find all positive integers $m, n$ such that $3^{m}-2^{n}=1$. | (1,1),(2,3) |
Given a point $(x, y)$, under the mapping $f$, the corresponding point is $(x+y, xy)$. Then, the corresponding point of $(-2, 3)$ under the mapping $f$ is. | (1, -6) |
Given that $\vec{a}$ and $\vec{b}$ are both unit vectors, and their angle is $60^{\circ}$, then $|\vec{a}+ \vec{b}|=$ ()
$A.1$ $B.\sqrt{2}$ $C.\sqrt{3}$ $D.2$ | C |
Find all strictly positive numbers $a, p, n$ such that $p^{a}-1=$ $2^{n}(p-1)$, where $p$ is a prime number. | 2 |
Given the lines $l_{1}$: $x+ay-1=0$ and $l_{2}$: $(a^{2}-2)x+y+2=0$, if $l_{1}\bot l_{2}$, then $a=\_\_\_\_\_\_$. | -2 \text{ or } 1 |
Given the function $f(x) = \log_{\frac{2}{3}}(x^2 - 2x - 3)$, determine the interval $E$ such that for any $x\_1$, $x\_2 \in E$, if $x\_1 < x\_2$, then $f(x\_1) < f(x\_2)$. The possible interval(s) for $E$ is/are:
A: $(-3, -1)$
B: $(-1, 0)$
C: $(1, 2)$
D: $(3, 6)$ | A |
6. In the Cartesian coordinate system, the curve represented by the equation $\frac{|x+y|}{2 a}+\frac{|x-y|}{2 b}=1(a, b$ are two unequal positive numbers) is
A. Triangle
B. Square
C. Non-square rectangle
D. Non-square rhombus | D |
Problem 1. Find $x$ for which
$$
\log _{2}\left(x^{2}+4\right)-\log _{2} x+x^{2}-4 x+2=0
$$ | 2 |
【Question 30】
There are two piles of stones, one with 11 stones and the other with 13 stones, denoted as $(11,13)$. Two players, A and B, take turns to remove stones, following these rules: one can take any number of stones from one pile or the same number of stones from both piles, but must take at least one stone. Th... | (3,5)or(8,13) |
An [integer](https://artofproblemsolving.com/wiki/index.php/Integer) between $1000$ and $9999$, inclusive, is called balanced if the sum of its two leftmost [digits](https://artofproblemsolving.com/wiki/index.php/Digit) equals the sum of its two rightmost digits. How many balanced integers are there? | 615 |
If the complex number $z = (m-1) + (m+2)i$ corresponds to a point on the line $2x-y=0$, then the value of the real number $m$ is. | 4 |
The largest three-digit number in base 7, when converted to decimal, is ( )
A: 322；
B: 332；
C: 342；
D: 352 | \text{C} |
16. Lego Explorer Guguji arrives at a medieval kingdom, which is ruled by an evil wizard. With the help of Guguji and his friends, the prince defeats the wizard and returns to his castle. The castle is built with many cubes with an edge length of 1 meter, arranged from top to bottom in a pattern as shown in the figure.... | 4901 |
Highways offer great convenience for traveling, but due to the high speeds on highways, accidents often lead to significant loss of life or property. In my country, the maximum speed limit on highways is 120 km/h, while the minimum is 60 km/h.
(Ⅰ) When a driver is driving at a speed of 120 kilometers per hour and dis... | 100 \text{ km/h} |
Given that the sum of the first $n$ terms of a geometric sequence ${a_{n}}$ is $S_{n}$, and $8a_{2}+a_{5}=0$, find the value of $\frac{S_{6}}{S_{3}}$. | \frac{S_{6}}{S_{3}}=-7 |
Throw a dice twice, and define the random variable $X$ as follows:
$$
X = \begin{cases}
0 & (\text{if the outcome of the first throw is different from the second throw}) \\
1 & (\text{if the outcome of the first throw is the same as the second throw})
\end{cases}
$$
The variance of the random variable $X$, denoted as ... | \text{C} |
65. Li's family contracted 100 mu of land to grow corn, with a yield of 600 pounds per mu. Liu's family contracted 20 mu less than Li's family, and as a result, the total production of both families was the same.
(1) Liu's family's total corn production is $\qquad$ pounds.
(2) Li's family's corn yield per mu is $\qquad... | 60000,150 |
2.33. In a regular quadrilateral pyramid, the dihedral angle at the lateral edge is $120^{\circ}$. Find the lateral surface area of the pyramid if the area of its diagonal section is $S$. | 4S |
11. Three circles $\Gamma, \Gamma_{1}, \Gamma_{2}$ have radii 6, 3, 2, respectively. $\Gamma_{1}$ and $\Gamma_{2}$ are externally tangent at $A$, while $\Gamma$ is internally tangent to both other circles, respectively at $A_{1}$ and $A_{2}$. Determine the radius of the circumcircle of $A A_{1} A_{2}$.
(A) $2 \sqrt{6}$... | 6 |
Problem 2. The quadratic function $f(x)=-x^{2}+4 p x-p+1$ is given. Let $S$ be the area of the triangle with vertices at the intersection points of the parabola $y=f(x)$ with the $x$-axis and the vertex of the same parabola. Find all rational $p$, for which $S$ is an integer.
| 0,1,\frac{1}{4},-\frac{3}{4} |
The curve $f(x)=\ln x-\frac{2}{x}$ has a slope of $\alpha$ at $x=1$. Then $\frac{\cos \alpha}{\sin \alpha-4\cos \alpha}=$
A: $\frac{1}{5}$
B: $-\frac{1}{5}$
C: $1$
D: $-1$ | \text{D} |
13. Equation
$$
1-3 \sin \theta+5 \cos 3 \theta=0(0<\theta<2 \pi)
$$
has ( ) different solutions.
(A) 2
(B) 4
(C) 5
(D) 6
(E) 8 | 6 |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}-\overrightarrow{b}|=3$ and $\overrightarrow{b}=(0,-1)$. If the projection of vector $\overrightarrow{a}$ in the direction of vector $\overrightarrow{b}$ is $-2$, then $|\overrightarrow{a}|=$ ()
A: $2$
B: $2\sqrt{3}$
C: $4$
D: $12$ | A |
Convert the binary number 11001 to a decimal number. | 25 |
9. In $\triangle A B C$, $\overrightarrow{G A}+\overrightarrow{G B}+\overrightarrow{G C}=0, \overrightarrow{G A} \cdot \overrightarrow{G B}=0$. $\frac{(\tan A+\tan B) \tan C}{\tan A \tan B}=$ $\qquad$ . | \frac{1}{2} |
3. Nils has a goose farm. Nils calculated that if he sells 75 geese, the feed will last 20 days longer than if he doesn't sell any. If he buys an additional 100 geese, the feed will run out 15 days earlier than if he doesn't make such a purchase. How many geese does Nils have? | 300 |
## 151. Math Puzzle 12/77
An ant is sitting on a cubic paving stone with an edge length of $9 \mathrm{~cm}$, at point $A$. Let's assume it wants to reach point $B$. There are many ways to get to $B$! But which is the shortest path?
## Shortest path sought!
 In the rectangular coordinate system $xOy$, the parametric equations of line $C$ are given by $\begin{cases} x = 2 + t \ y = t + 1 \end{cases}$ ($t$ is a parameter), and the equation of curve $P$ in the polar coordinate system with the origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis is gi... | \left(\frac{3}{2}, +\infty\right) |
$29 \cdot 62$ When the polynomial $x^{8}$ is divided by $x+\frac{1}{2}$, let $q_{1}(x)$ and $r_{1}$ represent the quotient and remainder, respectively. When $q_{1}(x)$ is again divided by $x+\frac{1}{2}$, let $q_{2}(x)$ and $r_{2}$ represent the quotient and remainder, respectively. Then $r_{2}$ equals
(A) $\frac{1}{25... | -\frac{1}{16} |
The graphs of the functions $y=x\ln (-x)$ and $y=x\ln x$ are symmetric about
A: the line $y=x$
B: the $x$-axis
C: the $y$-axis
D: the origin | \text{D} |
## Problem Statement
Find the indefinite integral:
$$
\int \frac{x^{3}+6 x^{2}+9 x+6}{(x+1)^{2}\left(x^{2}+2 x+2\right)} d x
$$ | -\frac{2}{x+1}+\frac{1}{2}\cdot\ln(x^{2}+2x+2)+\operatorname{arctg}(x+1)+C |
The standard deviation of the data 99, 100, 102, 99, 100, 100 is ( )
A: 0
B: 1
C: $$\sqrt {2}$$
D: $$\sqrt {6}$$ | \text{B} |
Given the equation of a circle $(x-1)^{2}+y^{2}=4$, there is only one line that is tangent to the circle passing through the point $(-1,a)$. Find the value of $a$.
A: $-3$
B: $3$
C: $1$
D: $0$ | D |
Given that the domain of an odd function is $\{-1, 2, a, b\}$, then $a+b=$ （ ）
A: $-1$
B: $1$
C: $0$
D: $2$ | A |
In the ancient Chinese mathematical masterpiece "The Arithmetic Classic of Zhang Qiujian," it is recorded: "One dou of clear wine is worth ten dou of grain, one dou of turbid wine is worth three dou of grain. Now, with 30 dou of grain, a total of 5 dou of wine is obtained. How many dou of clear and turbid wine are ther... | \text{A} |
9. If a three-digit positive integer $n$ makes the sum of the first $n$ positive integers not a factor of the product of the first $n$ positive integers. Then the maximum value of $n$ is ( ).
(A) 995
( B ) 996
( C) 997
(D) 998
(E) 999 | 996 |
2. Given the line $L: x+y-9=0$ and the circle $M: 2 x^{2}+2 y^{2}-8 x-8 y-1=0$, point $A$ is on line $L$, $B, C$ are two points on circle $M$, in $\triangle A B C$, $\angle B A C=45^{\circ}, A B$ passes through the center of circle $M$, then the range of the x-coordinate of point $A$ is $\qquad$ | 3\leqslantx\leqslant6 |
In the arithmetic sequence $\{a_n\}$, $S_4=4$, $S_8=12$, then $S_{12}=$ ______. | 24 |
## Problem Statement
Find the angle between the planes:
$2 x+2 y+z-1=0$
$x+z-1=0$ | \frac{\pi}{4} |
\section*{Problem 4 - 071034}
We are looking for all triples of natural numbers \(a_{i}(i=1,2,3)\) that satisfy the equation
\[
\sqrt{2 a_{1}^{2}-2 a_{2}^{2}}=a_{3}
\]
and for which \(1 \leq a_{i} \leq 10\) also holds! | (3,1,4),(6,2,8),(9,7,8) |
Given sets $M=\{-2,-1,0,1,2\}$ and $N=\{x|x^{2}+2x-3\leqslant 0\}$, then $M\cap N=\left(\ \ \right)$
A: $\{-2,-1,0,1\}$
B: $\{0,1,2\}$
C: $\{-2\}$
D: $\{2\}$ | A |
Example 8 List the positive integers that satisfy the condition "contain at least one digit 0 and are multiples of 4" in ascending order: $20,40,60,80,100,104, \cdots$ Then the 158th number in this sequence is ( ).
(A) 2000
(B) 2004
(C)2 0008
(D) 2012 | C |
Suppose the random variable $X$ follows a normal distribution $N(1, \sigma^2)$, and $P(X \leq a^2 - 1) = P(X > a - 3)$. Find the value of the positive number $a$. | a = 2 |
5.A. Given nine balls labeled with numbers $2,3, \cdots, 10$, two balls are randomly drawn, and their labels are noted. The probability that the larger label is divisible by the smaller label is ( ).
(A) $\frac{1}{4}$
(B) $\frac{2}{9}$
(C) $\frac{5}{18}$
(D) $\frac{7}{36}$ | B |
1. (a) Decompose 1 into unit fractions. | \frac{1}{2}+\frac{1}{3}+\frac{1}{6} |
3. Rectangle $A B C D$ consists of 6 shapes similar to the letter "L", and each "L" shape consists of 4 equal squares. If the sum of the perimeters of all the "L" shapes is $1200 \mathrm{~mm}$, what is the perimeter of rectangle $A B C D$?
 The vertex of the parabola is $O$ and the focus is $F$. When the moving point $P$ moves on the parabola, find the maximum value of the distance ratio $\left|\frac{P O}{P F}\right|$. | \frac{2 \sqrt{3}}{3} |
Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$. If $S_{13}=104$ and $a_6=5$, then the common difference of the sequence $\{a_n\}$ is ( ).
A: $2$
B: $3$
C: $4$
D: $5$ | \text{B} |
Two porters, A and B, are transporting a certain type of cargo. It is known that B transports 600 kg more per hour than A. The time it takes for A to transport 5,000 kg is equal to the time it takes for B to transport 8,000 kg. Determine how many kilograms of cargo A and B can each transport per hour. Let the amount of... | \text{B} |
G4.3 If $c=2 \sqrt{3} \times \sqrt[3]{1.5} \times \sqrt[6]{12}$, find the value of $c$. | 6 |
3. Let $s$ be a non-real root of the equation $x^{7}=1$. Then
\[
\begin{array}{l}
s(s+1)\left(s^{2}-s+1\right)\left(s^{2}+s+1\right) \\
=
\end{array}
\] | -1 |
5. Find the number of pairs of integers $(a ; b)$ such that $1 \leqslant a \leqslant 70, 1 \leqslant b \leqslant 50$, and at the same time,
the area $S$ of the figure defined by the system of inequalities
$$
\left\{\begin{array}{l}
\frac{x}{a}+\frac{y}{b} \geqslant 1 \\
x \leqslant a \\
y \leqslant b
\end{array}\right... | 1260 |
If a sequence $\{a_n\}$ satisfies $\frac{1}{a_{n+1}} - \frac{1}{a_n} = d$ ($n \in \mathbb{N}^*, d$ is a constant), then the sequence $\{a_n\}$ is called a "harmonic sequence". Given that the positive term sequence $\left\{ \frac{1}{b_n} \right\}$ is a "harmonic sequence", and $b_1 + b_2 + \ldots + b_9 = 90$, then the m... | B |
Among the following sets of line segments, which can form a right triangle?
A: $1$, $2$, $3$
B: $5$, $11$, $12$
C: $5$, $12$, $13$
D: $6$, $8$, $9$ | C |
6・94 Determine the largest real number $z$, such that
$$x+y+z=5, \quad xy+yz+zx=3,$$
and $x, y$ are also real numbers. | \frac{13}{3} |
If the equation $\left(a-2\right)x^{|a-1|}+3=9$ is a linear equation in $x$, then $a=\_\_\_\_\_\_$. | 0 |
Given $m$ and $n$ are two different lines, and $\alpha$ and $\beta$ are two different planes, then among the following statements, the correct one is ( )
A: If $m \parallel \alpha$, $n \perp \beta$ and $\alpha \perp \beta$, then $m \perp n$
B: If $m \perp \alpha$, $n \perp \beta$ and $m \perp n$, then $\alpha \perp... | \text{B} |
A rectangular sign that has dimensions $9 \mathrm{~m}$ by $16 \mathrm{~m}$ has a square advertisement painted on it. The border around the square is required to be at least $1.5 \mathrm{~m}$ wide. The area of the largest square advertisement that can be painted on the sign is
(A) $78 \mathrm{~m}^{2}$
(B) $144 \mathrm{~... | 36\mathrm{~}^{2} |
Let $ABC$ be a right triangle with $\angle ACB = 90^{\circ}$ and centroid $G$. The circumcircle $k_1$ of triangle $AGC$ and the circumcircle $k_2$ of triangle $BGC$ intersect $AB$ at $P$ and $Q$, respectively. The perpendiculars from $P$ and $Q$ respectively to $AC$ and $BC$ intersect $k_1$ and $k_2$ at $X$ and $Y$. De... | \frac{4}{9} |
The three sides of triangle $\triangle ABC$ are $a$, $b$, $c$, with the following conditions: ① $\angle A = \angle B - \angle C$; ② $\angle A : \angle B : \angle C = 3 : 4 : 5$; ③ $a^{2} = (b+c)(b-c)$; ④ $a : b : c = 5 : 12 : 13$. How many times can it be determined that $\triangle ABC$ is a right triangle?
A: $1$ tim... | C |
In triangle $\triangle ABC$, $\angle C=90^{\circ}$, $\angle B=30^{\circ}$, the angle bisector of $\angle BAC$ intersects $BC$ at point $D$. If $\overrightarrow{AD}=λ\overrightarrow{AB}+μ\overrightarrow{AC}$ ($\lambda, \mu \in \mathbb{R}$), then $\frac{λ}{μ}=$
A: $\frac{1}{3}$
B: $\frac{1}{2}$
C: $2$
D: $3$ | B |
Define a new operation: $a\oplus b=a\left(a-b\right)$. For example, $3\oplus 2=3\times \left(3-2\right)=3$, $-1\oplus 4=-1\times \left(-1-4\right)=5$.
1. Find all positive integer solutions to $3\oplus a=b$.
2. Given $2\oplus a=5b-2m$ and $3\oplus b=5a+m$, prove that the value of $12a+11b$ is independent of $m$.
3. Giv... | M \geq N |
$a$,$b$,$c$ are real. What is the highest value of $a+b+c$ if $a^2+4b^2+9c^2-2a-12b+6c+2=0$ | \frac{17}{3} |
13. As shown in the figure, a cube with an edge length of 12 cm is cut with one slice. The cut is made along $I J$ and exits at $L K$, such that $A I=D L=4$ cm, $J F=K G=3$ cm, and the cross-section $I J K L$ is a rectangle. The cube is cut into two parts, and the sum of the surface areas of these two parts is $\qquad$... | 1176 |
One. (20 points) Equations
$$
x^{2}+a x+b=0 \text { and } x^{2}+b x+a=0
$$
have a common root, and let the other two roots be $x_{1} 、 x_{2}$; Equations
$$
x^{2}-c x+d=0 \text { and } x^{2}-d x+c=0
$$
have a common root, and let the other two roots be $x_{3} 、 x_{4}$. Find the range of $x_{1} x_{2} x_{3} x_{4}$ $(a, ... | 0 < x_{1} x_{2} x_{3} x_{4} < \frac{1}{16} |
5. (Problem 1500 from "Mathematics Bulletin") Given positive integers $x_{1}<x_{2}<\cdots<x_{n}, x_{1}+x_{2}+\cdots+$ $x_{n}=2003, n \geqslant 2$. Find the minimum value of $f(n)=n\left(x_{1}+x_{n}\right)$. | 2254 |
In right triangle $ABC$, where $\angle C = 90^{\circ}$ and $\angle B = 35^{\circ}$, with $AB = 7$, the length of $BC$ is ( ).
A: $7\sin 35^{\circ}$
B: $7\cos 35^{\circ}$
C: $7\tan 35^{\circ}$
D: $\dfrac{7}{\cos 35^{\circ}}$ | B |
$A B$ and $A C$ are tangents to a circle with center $O, M$ is the intersection point of line $A O$ with the circle; $D E$ is a segment of the tangent line through point $M$, between $A B$ and $A C$. Find $D E$, if the radius of the circle is
15, and $A O=39$. | 20 |
The area of the enclosed shape formed by the curve $y= \frac {1}{x}$, the line $y=4x$, $x=1$, and the x-axis is ( )
A: $\ln2- \frac {1}{2}$
B: $\frac {1}{3}+\ln2$
C: $\ln2+ \frac {1}{2}$
D: $1+\ln2$ | C |
3. How many of the integers from $2^{10}$ to $2^{18}$ inclusive are divisible by $2^{9}$ ? | 511 |
9. Let $0 \leqslant x_{1} \leqslant x_{2}$, and the sequence $\left\{x_{n}\right\}$ satisfies
$$
x_{n+2}=x_{n+1}+x_{n}(n \geqslant 1) \text {. }
$$
If $1 \leqslant x_{7} \leqslant 2$, then the range of values for $x_{8}$ is
$\qquad$ | [\frac{21}{13},\frac{13}{4}] |
Let $f(x)$ be a function defined on $\mathbb{R}$ with a period of 2, and for any real number $x$, it always holds that $f(x) - f(-x) = 0$. When $x \in [0, 1]$, $f(x) = -\sqrt{1-x^2}$. Then, the number of zeros of the function $g(x) = f(x) - e^x + 1$ in the interval $[-2017, 2017]$ is ( )
A: 2016
B: 2017
C: 4032 ... | B |
Which of the following statements is incorrect?
A: The diagonals of a parallelogram bisect each other.
B: The diagonals of a rectangle are equal and bisect each other.
C: The diagonals of a rhombus are equal and bisect each other.
D: The diagonals of a square are equal and perpendicular bisectors of each other. | C |
Given the set $A=\{x|(x-1)(a-x)>0\}$, and the set $B=\{x||x+1|+|x-2|\leq 3\}$, and $(C_RA) \cup B = \mathbb{R}$, then the range of the real number $a$ is
( )
A: $(-\infty, 2)$
B: $(-1, +\infty)$
C: $[-1, 2]$
D: $[-1, 1) \cup (1, 2]$ | C |
3. How many four-digit numbers exist where the digit in the thousands place is greater than the digit in the hundreds place? | 4500 |
5. Let the cost of a pound of rice be x coins, a pound of brown sugar be y coins, and a pound of refined sugar be z coins. We obtain the system:
$$
\left\{\begin{array}{l}
4 x+\frac{9}{2} y+12 z=6 ; \\
12 x+6 y+6 z=8
\end{array}\right. \text { Subtract the first equation from twice the second equation and }
$$
expres... | 4 |
A1. Which digit of the 7-digit number 2345678 must be deleted so that the resulting 6-digit number is divisible by 9?
(A) 8
(B) 7
(C) 6
(D) 5
(E) 4 | 8 |
The value range of the quadratic function $y=x^2-4x+3$ in the interval $(1,4]$ is ( )
A: $[-1,+\infty)$
B: $(0,3]$
C: $[-1,3]$
D: $(-1,3]$ | C |
In the process of deriving a contradiction using the method of proof by contradiction, which of the following should be used as conditions? ( )
① The judgment opposite to the conclusion, i.e., the assumption
② The conditions of the original proposition
③ Axioms, theorems, definitions, etc.
④ The original concl... | C |
3. If $x_{i}=\frac{i}{101}$, then $S=\sum_{i=0}^{101} \frac{x_{i}^{3}}{3 x_{i}^{2}-3 x_{i}+1}$ is
A. 51
B. 52
C. 53
D. 54 | 51 |
$\left.\begin{array}{l}\text { [Inscribed angle, subtending the diameter]} \\ {\left[\begin{array}{l}\text { Auxiliary similar triangles }\end{array}\right]} \\ {[\text { Pythagorean Theorem (direct and inverse). }} \\ {[\text { Trapezoids (other). }}\end{array}\right]$
Given trapezoid $ABCD$, where angle $BAD$ is a r... | \arcsin\frac{1}{\sqrt{10}} |
## Task 3 - 200613
Determine from the set of all natural numbers from 20 to 39 those which are divisible by the product of their two digits! | 24,36 |
The proposition "The points on the perpendicular bisector of a segment are equidistant from the two endpoints of the segment" and its inverse proposition, negation, and contrapositive proposition, among them, the true propositions are ( )
A: 4
B: 3
C: 2
D: 1 | \text{B} |
Determine the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f(x-f(y))=1-x-y
$$ | f(x)=\frac{1}{2}-x |
9. (10 points) The figure is composed of 12 squares, each with an area of 1. How many rectangles in the figure have the same area as the shaded trapezoid?
Looking horizontally, the first three small squares in each row can form 1 rectangle with an area of 3, and the last three small squares can also form a rectangle wi... | 10 |
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