problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
8. Given the sequence $\left\{a_{n}\right\}$, the odd terms form an arithmetic sequence with a common difference of $d_{1}$, and the even terms form an arithmetic sequence with a common difference of $d_{2}$. For any $n \in \mathbf{N}^{*}$, it is true that $a_{n}<a_{n+1}$. If $a_{1}=1, a_{2}=2$, and the sum of the firs... | 11 |
Given the equation of a circle as $x^2+y^2+2x=0$, the radius of this circle is ( )
A: 1
B: 2
C: 3
D: 4 | A |
Which of the following can be calculated using the square difference formula?
A: $(-x+y)\left(x+y\right)$
B: $(-x+y)\left(x-y\right)$
C: $(x+2)\left(2+x\right)$
D: $(2x+3)\left(3x-2\right)$ | A |
9th Mexico 1995 Problem A1 N students are seated at desks in an m x n array, where m, n ≥ 3. Each student shakes hands with the students who are adjacent horizontally, vertically or diagonally. If there are 1020 handshakes, what is N? | 280 |
3. For which values of $a$, for any $b$ there exists at least one $c$, such that the system
$$
\left\{\begin{array}{l}
2 x+b y=a c^{2}+c \\
b x+2 y=c-1
\end{array}\right.
$$
has at least one solution? | \in[-1,0] |
5.3.1 * Given two vertices of a parallelogram $A(0,0), B(4,-3)$, point $P$ internally divides diagonal $A C$ in the ratio $2: 1$. When point $D$ moves on the circumference of a circle centered at $A$ with a radius of 3, find the locus of point $P$. | |z_{P}-(\frac{8}{3}-2\mathrm{i})|=2 |
Given: $y= \sqrt{x-2} + \sqrt{2-x} + 1$, find the value of the expression $(\sqrt{48y} + \sqrt{8x})(4\sqrt{3y} - 2\sqrt{2x}) - xy$. | 30 |
【Question 11】
Two trains depart from two cities at the same time heading towards each other, and they meet after 3.3 hours. If Train A departs 24 minutes earlier, then 3 hours after Train B departs, the two trains still need to travel 14 kilometers to meet; if Train B departs 36 minutes earlier, then 3 hours after Trai... | 660 |
## Task 3 - 050523
For each of the 600,000 inhabitants of Leipzig, 125 kg of potatoes are stored.
a) Calculate the amount to be provided in tons!
b) What is the maximum number of freight cars with a loading capacity of 15 t each that can be fully loaded with this amount?
c) How many tons on average are delivered ea... | 2679 |
Which of the following calculations is correct?
A: $\sqrt {25}=\pm 5$
B: $ \sqrt {\left(-9\right)^{2}}=-9$
C: $ \sqrt [3] {-8}=-2$
D: $ \sqrt {2}+\sqrt {3}=\sqrt {5}$ | \text{C} |
Given \\(\alpha\\) and \\(\beta\\) satisfy \\(\alpha+\beta=5\\) and \\(\alpha\beta=6\\), then the quadratic equation with \\(\alpha\\) and \\(\beta\\) as its roots is \\((\quad)\)
A: \\(x^{2}+5x+6=0\\)
B: \\(x^{2}-5x+6=0\\)
C: \\(x^{2}-5x-6=0\\)
D: \\(x^{2}+5x-6=0\\) | \text{B} |
13.303. Two wheels are connected by an infinite belt; the smaller one makes 300 more revolutions per minute than the second. The larger wheel makes 10 revolutions in a time interval that is 1 second longer than the time for the same number of revolutions of the smaller wheel. How many revolutions per minute does each w... | 300600 |
Boris has an incredible coin-changing machine. When he puts in a quarter, it returns five nickels; when he puts in a nickel, it returns five pennies; and when he puts in a penny, it returns five quarters. Boris starts with just one penny. Which of the following amounts could Boris have after using the machine repeatedl... | 7.45 |
Given $p$: "$\left(x-m\right)^{2} \gt 3\left(x-m\right)$" is a necessary but not sufficient condition for $q$: "$x^{2}-3x-4\leqslant 0$" to hold, then the range of real number $m$ is ( ).
A: $(-\infty ,-4]\cup \left[4,+\infty \right)$
B: $(-\infty ,-4)\cup \left(4,+\infty \right)$
C: $(-4,4)$
D: $\left[-4,4\right]$ | \text{B} |
Problem 2. Let $\left\{a_{n}\right\}_{n=1}^{\infty}$ be a sequence of integer numbers such that
$$
(n-1) a_{n+1}=(n+1) a_{n}-2(n-1)
$$
for any $n \geq 1$. If 2000 divides $a_{1999}$, find the smallest $n \geq 2$ such that 2000 divides $a_{n}$. Oleg Mushkarov, Nikolai Nikolov
| 249 |
5. (10 points) There is a special calculator. When a number is input, the calculator multiplies this number by 2, then reverses the order of the digits of the result. After adding 2, it displays the final result. If a two-digit number is input, and the final result displayed is 27, then the number initially input was .... | 26 |
8. The centroid of $\triangle A B C$ is $G$, the midpoints of each side are $D, E, F, \overrightarrow{G D}+\overrightarrow{G E}+\overrightarrow{G F}=$ | \overrightarrow{0} |
The line $l$ passes through points $A(2,1)$ and $B(1,m^2)$ ($m \in \mathbb{R}$). The range of the slope angle of line $l$ is ( )
A: $[0，\pi)$
B: $[0，\pi/2] \cup [\pi/2，\pi)$
C: $[0，\pi/2]$
D: $[0，\pi/2) \cup (\pi/2，\pi)$ | \text{D} |
## Subject III. (20 points)
A bathtub would be filled by the cold water tap in 5 minutes, and by the hot water tap in 7 minutes. The bathtub is equipped with a drain hole, through which, when the bathtub is full, the water can drain in 3 minutes. If we open both taps and do not close the drain hole, will the bathtub f... | 105 |
Problem 5.6. Several stones are arranged in 5 piles. It is known that
- in the fifth pile, there are six times more stones than in the third;
- in the second pile, there are twice as many stones as in the third and fifth combined;
- in the first pile, there are three times fewer stones than in the fifth, and ten fewer... | 60 |
2.2. How many 9-digit numbers divisible by 2 can be formed by rearranging the digits of the number 131152152? | 840 |
Given the function $f(x)=\sin (x+\theta)+\cos (x+\theta)$, if for any real number $x$, we have $f(x)=f(-x)$, then the possible value(s) of $\theta$ is(are) $\text{( }\,\,\text{)}$.
A: $\dfrac {\pi}{6}$
B: $\dfrac {\pi}{4}$
C: $\dfrac {\pi}{2}$
D: $\pi$ | \text{B} |
7.3. Let $A B C$ be an acute-angled triangle. The perpendicular from $A$ to $A C$ and the perpendicular from $B$ to $A B$ intersect at point $D$. On the side $A D$, an interior point $E$ is taken such that triangles $A B D$ and $C A E$ are congruent. The length of side $A B$ is $10 \mathrm{~cm}$, and $E B \perp B C$. C... | 30\mathrm{~} |
A wholesaler who deals in household chemicals and paper products has a container with a volume of $12 \mathrm{~m}^{3}$ and can accommodate 5 tons of goods. The chemical products occupy $1 \mathrm{~m}^{3}$ per ton, while the paper products occupy $3 \mathrm{~m}^{3}$ per ton. The profit is one hundred thousand forints pe... | 850000 |
11. The image shows the calendar that comes with the Windows operating system. There is a very magical flower that blooms every Wednesday and Friday in odd-numbered months, and every Tuesday and Thursday in even-numbered months. For example, October 1, Tuesday, is a time when it blooms. So, from November 1, 2013, to De... | 18 |
8. Let $P_{n}(k)$ denote the number of permutations of $\{1,2, \cdots, n\}$ with $k$ fixed points. Let $a_{t}=\sum_{k=0}^{n} k^{t} P_{n}(k)$. Then
$$
\begin{array}{l}
a_{5}-10 a_{4}+35 a_{3}-50 a_{2}+25 a_{1}-2 a_{0} \\
=
\end{array}
$$ | 0 |
11. Large Cube. A cube is assembled from 27 playing dice. a) (6 - 11 grades, 1 point) Find the probability that exactly 25 sixes are on the surface of the cube.
b) (7 - 11 grades, 1 point) Find the probability that at least one one is on the surface of the cube.
c) (8 - 11 grades, **1** point) Find the expected numbe... | )\frac{31}{2^{13}\cdot3^{18}}\approx9.77\cdot10^{-12};b)1-\frac{5^{6}}{2^{2}\cdot3^{18}}\approx0.99998992;)9;)189;e)6-\frac{5^{6} |
## Problem Statement
Write the equation of the plane passing through point $A$ and perpendicular to vector $\overrightarrow{B C}$.
$A(1 ; 0 ; -2)$
$B(2 ; -1 ; 3)$
$C(0 ; -3 ; 2)$ | 2x+2y+z=0 |
A function $f$, defined for the set of integers, is such that $f(x)=x-10$ if $x>100$ and $f(x)=f(f(x+11))$ if $x \leq 100$.
Determine, justifying your answer, the set of all possible values for $f$. | f(\mathbb{Z}) = \mathbb{Z} \cap [91, +\infty) |
Ex. 140. A circle is inscribed in a trapezoid with integer sides. The midline of the trapezoid divides it into two parts, the areas of which are 15 and 30. Find the radius of the inscribed circle.
## Several problems | \frac{5}{2} |
Given the sets $A=\{x|(x+4)(x-1)<0\}$ and $B=\{x|x^2-2x=0\}$, then $A\cap B=$ （ ）
A: $\{0\}$
B: $\{2\}$
C: $\{0,2\}$
D: $\{x|-4<x<1\}$ | A |
2. Find the solution to the system
$$
\left\{\begin{array}{l}
5 x^{7}+3 y^{2}+5 u+4 v^{4}=-2 \\
2 x^{7}+8 y^{2}+7 u+4 v^{4}=\frac{6^{5}}{3^{4} \cdot 4^{2}} \\
8 x^{7}+2 y^{2}+3 u+6 v^{4}=-6 \\
5 x^{7}+7 y^{2}+7 u+8 v^{4}=\frac{8^{3}}{2^{6} \cdot 4}
\end{array}\right.
$$ | {-1,\1,0,0} |
Example 4 Find the integer solution of the equation $\left[\frac{x}{1!}\right]+\left[\frac{x}{2!}\right]+\left[\frac{x}{3!}\right]+\cdots+$ $\left[\frac{x}{10!}\right]=1001$. | 584 |
Convert the polar coordinates $A\left(2, \dfrac{2\pi}{3}\right)$ into Cartesian coordinates.
A: $\left(1, \sqrt{3}\right)$
B: $\left(-1, \sqrt{3}\right)$
C: $\left(1, -\sqrt{3}\right)$
D: $\left(-1, -\sqrt{3}\right)$ | \text{B} |
A hobby group consists of 2 boys and 3 girls. Now, 2 students are randomly selected to participate in an activity. The probability of exactly selecting 2 girls is ______. | 0.3 |
Among the 10 natural numbers arranged in ascending order: 1, 4, 8, 10, 16, 19, 21, 25, 30, 43, how many groups of consecutive numbers have a sum that is a multiple of 11? Please explain your reasoning. | 7 |
Given that $k$ is a composite number, and $1 < k < 100$, when the sum of the digits of $k$ is a prime number, this prime number is called the "derived prime" of $k$.
(1) If the "derived prime" of $k$ is 2, then $k=$
(2) Let set $A=\{P(k) | P(k)$ is the "derived prime" of $k\}$, $B=\{k | P(k)$ is the "derived prime"... | 30 |
Given $x > 1$, find the minimum value of $\log_{x}9 + \log_{27}x$. | \frac{2\sqrt{6}}{3} |
## Task 11/73
A train passes a kilometer post with a two-digit kilometer number. After the time $\Delta t_{1}$, it passes another kilometer post with the same digits but in reversed order.
Finally, after the additional time $\Delta t_{2}=\Delta t_{2}$, it encounters a third kilometer post, whose number is the same as... | 16,61,106 |
8. Let $a \in \mathbf{R}$, complex numbers $z_{1}=a+\mathrm{i}, z_{2}=2 a+2 \mathrm{i}, z_{3}=3 a+4 \mathrm{i}$, where $\mathrm{i}$ is the imaginary unit. If $\left|z_{1}\right|,\left|z_{2}\right|,\left|z_{3}\right|$ form a geometric sequence, then the value of the real number $a$ is $\qquad$. | 0 |
Among the following functions, the one with the incorrectly chosen domain for the independent variable is ( )
A: $y=2x^2$, where $x$ is all real numbers
B: $y=\frac{1}{x+1}$, where $x$ is real numbers except $x \neq -1$
C: $y=\sqrt{x-2}$, where $x$ is real numbers with $x \geq 2$
D: $y=\frac{1}{x+3}$, where $x$ is... | B |
If the tangent line at point $P$ on the curve $y=e^{-x}$ is parallel to the line $2x+y+1=0$, find the coordinates of point $P$. | (-\ln 2,2) |
5. (8 points) On the radius $A O$ of a circle with center $O$, a point $M$ is chosen. On one side of $A O$ on the circle, points $B$ and $C$ are chosen such that $\angle A M B = \angle O M C = \alpha$. Find the length of $B C$ if the radius of the circle is $12$, and $\cos \alpha = \frac{1}{4}$? | 6 |
11.195. The radius of the base of the cone is $R$, and the angle of the sector of its lateral surface is $90^{\circ}$. Determine the volume of the cone. | \frac{\piR^3\sqrt{15}}{3} |
10. Xiaocuxin made a mistake when calculating the division, writing the dividend 2002 as 2020, resulting in a quotient of 106 and a remainder of 6. The correct quotient should be $\qquad$ . | 105 |
Which of the following statements is incorrect? ( )
A: The zero vector has no direction
B: The zero vector is only equal to the zero vector
C: The magnitude of the zero vector is 0
D: The zero vector is collinear with any vector | \text{A} |
Let $a$ be a real number, and $z=\frac{1+i}{1+ai}$ (where $i$ is the imaginary unit). If $z=\overline{z}$ (where $\overline{z}$ is the conjugate of the complex number $z$), then $a=$ ______. | 1 |
23. Jiji and Taotao take turns to take stones from the table, the total number of stones is more than 1 and less than 100. Each person takes a number of stones that is a factor of the number of stones on the table, but cannot take all of them. When a person cannot take any stones, the other wins. Jiji takes the stones ... | 49 |
Example 1 The license plate numbers issued by a city are all composed of 6 digits (from 0 to 9), but it is required that any two license plates must differ in at least 2 positions (for example, license plates 038471 and 030471 cannot be used simultaneously). Try to find the maximum number of different license plates th... | 100000 |
Given that \\(f(x)\\) is an odd function defined on \\(\mathbb{R}\\), and when \\(x \in (-\infty, 0]\), \\(f(x) = -x\lg(3-x)\\), then \\(f(1)=\\)
A: \\(0\\)
B: \\(\lg 3\\)
C: \\(-\lg 3\\)
D: \\(-\lg 4\\) | \text{D} |
If the point $\left(m,n\right)$ lies on the graph of the linear function $y=3-5x$, then the value of the algebraic expression $10m+2n-3$ is ______. | 3 |
1. Find all five-digit numbers where the second digit is five times the first, and the product of all five digits is 1000. | 15855,15585,15558 |
Variant 2. (2008 National Exam I for Science) If the line $\frac{x}{a}+\frac{y}{b}$ $=1$ passes through the point $M(\cos \alpha, \sin \alpha)$, then $(\quad)$
A. $a^{2}+b^{2} \leqslant 1$
B. $a^{2}+b^{2} \geqslant 1$
C. $\frac{1}{a^{2}}+\frac{1}{b^{2}} \leqslant 1$
D. $\frac{1}{a^{2}}+\frac{1}{b^{2}} \geqslant 1$ | D |
4. Let $F_{1}$ and $F_{2}$ be the left and right foci of the hyperbola $C: x^{2}-\frac{y^{2}}{24}=1$, respectively, and let $P$ be a point on the hyperbola $C$ in the first quadrant. If $\frac{\left|P F_{1}\right|}{\left|P F_{2}\right|}=\frac{4}{3}$, then the inradius of $\triangle P F_{1} F_{2}$ is $\qquad$. | 2 |
Consider the sequence $\left(a_{n}\right)$ with the first term $a_{1}=1$ and defined by the following relation: For any positive integer $n$,
$$
a_{n}=a_{\left\lfloor\frac{n}{2}\right\rfloor}+a_{\left\lceil\frac{n}{2}\right\rceil}
$$
Express $a_{n}$ as a function of $n$. | a_{n}=n |
The range of the data 0, -1, 3, 2, 4 is. | 5 |
The positive numbers $a, b, c,d,e$ are such that the following identity hold for all real number $x$:
$(x + a)(x + b)(x + c) = x^3 + 3dx^2 + 3x + e^3$. Find the smallest value of $d$. | 1 |
Solve the inequalities (system):$(1)\frac{1}{3}x-1≤\frac{1}{2}x+1$;$(2)\left\{\begin{array}{l}3x-(x-2)≥6\\ x+1＞\frac{4x-1}{3}\end{array}\right.$. | 2 \leq x < 4 |
A direction vector of the line $2x - 3y + 1 = 0$ is
A: $(2, -3)$
B: $(2, 3)$
C: $(-3, 2)$
D: $(3, 2)$ | \text{D} |
7.31 $\log _{3}(x-3)^{2}+\log _{3}|x-3|=3$.
Translate the above text into English, keeping the original text's line breaks and format, and output the translation result directly.
7.31 $\log _{3}(x-3)^{2}+\log _{3}|x-3|=3$. | x_{1}=0,x_{2}=6 |
$\triangle P Q R$ has an area of $27 \mathrm{~cm}^{2}$ and a base measuring $6 \mathrm{~cm}$. What is the height, $h$, of $\triangle P Q R$ ?
(A) $9 \mathrm{~cm}$
(B) $18 \mathrm{~cm}$
(C) $4.5 \mathrm{~cm}$
(D) $2.25 \mathrm{~cm}$
(E) $7 \mathrm{~cm}$ | 9\mathrm{~} |
1684. The average service life of the motor is 4 years. Estimate from below the probability that this motor will not serve for more than 20 years. | 0.8 |
Given the following propositions:
\\(①a > b \Rightarrow ac^{2} > bc^{2}\\);
\\(②a > |b| \Rightarrow a^{2} > b^{2}\\);
\\(③|a| > b \Rightarrow a^{2} > b^{2}\\);
\\(④a > b \Rightarrow a^{3} > b^{3}\\)
Among these, the correct propositions are \\((\quad)\)
A: \\(①②\\)
B: \\(②③\\)
C: \\(③④\\)
D: \\(②④\\) | D |
5. Point $E$ is the midpoint of edge $B B_{1}$ of the cube $A B C D A_{1} B_{1} C_{1} D_{1}$. Find the tangent of the angle between the lines $A E$ and $C A_{1}$.
- Let the edge of the cube be 1. Then $C A_{1}=\sqrt{3}$. Draw a line through point $A_{1}$ parallel to $A E$. It intersects the extension of edge $B B_{1}$... | \sqrt{14} |
Which of the following calculations is correct?
A: $\sqrt{3}+\sqrt{6}=3$
B: $\sqrt{6}-\sqrt{3}=\sqrt{3}$
C: $\sqrt{3}\times \sqrt{6}=3\sqrt{2}$
D: $\sqrt{3}+\sqrt{6}=\frac{1}{2}$ | C |
The domain of the function $f(x)= \frac{ \sqrt{2-x}}{\ln (x+1)}$ is ( )
A: $(-1,0)\cup(0,2]$
B: $[-1,0)\cup(0,2)$
C: $(-1,2)$
D: $(-1,2]$ | \text{A} |
4.1. (14 points) Find all integer values that the fraction $\frac{8 n+157}{4 n+7}$ can take for natural $n$. In your answer, write the sum of the found values. | 18 |
Calculate the result of $(\frac{1}{2}x^{3})^{2}$ correctly as:
A: $x^{6}$
B: $\frac{1}{4}x^{6}$
C: $\frac{1}{4}x^{5}$
D: $x^{9}$ | B |
$(1)$ Given $a^{m}=10$ and $a^{n}=2$, find $a^{m-2n}$;$(2)$ Given $2x+5y-3=0$, find the value of $4^{x}\cdot 32^{y}$. | 8 |
A line $l$ passing through the focus $F$ of the parabola $y^{2}=4x$ intersects the parabola at points $A$ and $B$. Given that the sum of the x-coordinates of points $A$ and $B$ is $\frac {10}{3}$, then the length of segment $AB$ is _______. | \frac{16}{3} |
9.5. Find the maximum value of the expression $(\sqrt{36-4 \sqrt{5}} \sin x-\sqrt{2(1+\cos 2 x)}-2) \cdot(3+2 \sqrt{10-\sqrt{5}} \cos y-\cos 2 y) \cdot$ If the answer is not an integer, round it to the nearest integer. | 27 |
Given vectors $\overrightarrow{m}$ and $\overrightarrow{n}$ satisfy $|\overrightarrow{m}|=2$, $|\overrightarrow{n}|=3$, and $|\overrightarrow{m} + \overrightarrow{n}|= \sqrt{17}$, then $|\overrightarrow{m}+ \overrightarrow{n}|=$ ()
A: $\sqrt {7}$
B: $3$
C: $\sqrt {11}$
D: $\sqrt {13}$ | B |
Given a sequence $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n$, if $S\_n = 3a\_n + 1$, find the value of $a\_{10}$:
A: $- \dfrac {3^{9}}{2^{10}}$
B: $- \dfrac {3^{10}}{2^{10}}$
C: $ \dfrac {3^{9}}{2^{10}}$
D: $ \dfrac {3^{10}}{2^{10}}$ | A |
1. Given non-empty sets
$$
\begin{array}{l}
A=\{x \mid m+1 \leqslant x \leqslant 2 m-1\}, \\
B=\left\{x \mid x^{2}-2 x-15 \leqslant 0\right\},
\end{array}
$$
and $A \subseteq B$.
Then the range of real number $m$ is | [2,3] |
1. (7 points) Winnie-the-Pooh eats 3 cans of condensed milk and a jar of honey in 25 minutes, while Piglet takes 55 minutes. One can of condensed milk and 3 jars of honey, Pooh eats in 35 minutes, while Piglet takes 1 hour 25 minutes. How long will it take them to eat 6 cans of condensed milk together? | 20 |
The base of an oblique parallelepiped is a rhombus with a side length of 60. The plane of the diagonal section passing through the larger diagonal of the base is perpendicular to the plane of the base. The area of this section is 7200. Find the smaller diagonal of the base, if the lateral edge is 80 and forms an angle ... | 60 |
(3) If a regular quadrilateral pyramid with a base edge length of 2 has an inscribed sphere with a radius of $\frac{1}{2}$, then the volume of this regular quadrilateral pyramid is $\qquad$ . | \frac{16}{9} |
Among the real numbers $-1$, $\sqrt{3}$, $\frac{1}{2}$, $3.14$, the irrational number is ( )
A: $-1$
B: $\sqrt{3}$
C: $\frac{1}{2}$
D: $3.14$ | B |
If the function $y=\left(m-3\right)x^{4-|m|}+m+7$ is a linear function, then $m=\_\_\_\_\_\_.$ | -3 |
A sufficient and necessary condition for $(x-2)(x+2) \gt 0$ is:
A: $x\leqslant 0$
B: $x\geqslant 0$
C: $x\geqslant 3$
D: $x \gt 2$ or $x \lt -2$ | D |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ with magnitudes $|\overrightarrow{a}| = 2$ and $|\overrightarrow{b}| = 1$ and an angle of $120^{\circ}$ between them, find the magnitude of the sum $|\overrightarrow{a} + \overrightarrow{b}|$. | \sqrt{3} |
A furniture store set the sticker price of a table $40$ percent higher than the wholesale price that the store paid for the table. During a special sale, the table sold for $35$ percent less than this sticker price. Find the percent the final sale price was of the original wholesale price of the table. | 91\% |
10. (20 points) The function $f(x)$ defined on $[0,1]$ satisfies: $f(0)=f(1)$, and for any $x, y \in [0,1]$ $(x \neq y)$, we have $|f(x)-f(y)|<|x-y|$. Find the smallest real number $m$, such that for any $x, y \in [0,1]$, we have
$$
|f(x)-f(y)|<m .
$$ | \frac{1}{2} |
If the random variable $X$ represents the number on the upper face when a fair cube with faces numbered $1$, $2$, $3$, $4$, $5$, $6$ is tossed, then the mean of the random variable $X$ is ( )
A: $2.5$
B: $3$
C: $3.5$
D: $4$ | C |
A new operation "$\*$" is defined in the set of real numbers $\mathbb{R}$ with the following properties:
1. For any $a, b \in \mathbb{R}$, $a \* b = b \* a$;
2. For any $a \in \mathbb{R}$, $a \* 0 = a$;
3. For any $a, b, c \in \mathbb{R}$, $(a \* b) \* c = c \* (ab) + (a \* c) + (b \* c) - 2c$
The monotonically decre... | \text{D} |
In the book "Mathematical Treatise in Nine Sections" volume five, "Field Mathematics", written by Qin Jiushao, a famous mathematician from the Southern Song Dynasty, there is a problem: "There is a piece of triangular sand field with three sides measuring 13 li, 14 li, and 15 li (1 li equals 500 meters). What is the ar... | 21 |
3. Solve the system $\left\{\begin{array}{l}2 x+y+8 \leq 0, \\ x^{4}+2 x^{2} y^{2}+y^{4}+9-10 x^{2}-10 y^{2}=8 x y .\end{array}\right.$ | (-3,-2) |
Let $a,b,c$ be positive real numbers. Find all real solutions $(x,y,z)$ of the system:
\[ ax+by=(x-y)^{2}
\\ by+cz=(y-z)^{2}
\\ cz+ax=(z-x)^{2}\] | (0, 0, 0), (0, 0, c), (a, 0, 0), (0, b, 0) |
In the Cartesian coordinate system, the coordinates of the point $A(2,3)$ with respect to the $x$-axis are ( ).
A: $(-2,-3)$
B: $(2,-3)$
C: $(-2,3)$
D: $(-3,-2)$ | B |
Define the operation $\oplus$ on the set of real numbers $\mathbb{R}$: $a \oplus b = a + b + 4$. Also define: if there exists an element $e$ in $\mathbb{R}$ such that for all $a \in \mathbb{R}$, $e \oplus a = a$, then $e$ is called the identity element of $\mathbb{R}$. What is the value of the identity element $e$ in $... | -4 |
4. Let the three sides of a triangle be $a, b, c$, and the radius of its circumcircle be $R, a \sqrt{bc} \geqslant (b+c) R$. Then the degree measure of the largest interior angle of the triangle is ( ).
(A) $150^{\circ}$
(B) $120^{\circ}$
(C) $90^{\circ}$
(D) $135^{\circ}$ | C |
When solving the equation system $\left\{\begin{array}{l}2ax+y=5\\ 2x-by=13\end{array}\right.$, due to carelessness, person A misread the variable $a$ in the equation system and obtained the solution $\left\{\begin{array}{l}x=\frac{7}{2}\\ y=-2\end{array}\right.$; person B misread the variable $b$ in the equation syste... | x=2, y=-3 |
There is a small island B in the sea, surrounded by reefs 3.8 nautical miles away. A warship travels from west to east to point A, where it sees the island to the north-northeast at an angle of 75°. After sailing 8 nautical miles to point C, it sees island B to the north-northeast at an angle of 60°. If the warship con... | \text{B} |
Example 3. Factorize: $x^{5}+x^{4}+1$.
| (x^{2}+x+1)(x^{3}-x+1) |
\section*{Problem 2 - 161012}
Give all real numbers \(x(x \neq-3)\) that satisfy the following inequality!
\[
\frac{2}{x+3}-\frac{1}{2} \geq \frac{5}{x+3}-\frac{1}{10}
\] | -10.5\leqx<-3 |
We inscribe a circle $\omega$ in equilateral triangle $ABC$ with radius $1$. What is the area of the region inside the triangle but outside the circle? | 3\sqrt{3} - \pi |
Let $F_{1}$ and $F_{2}$ be the left and right foci of the hyperbola $C$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a>0,b>0)$. The line $l$ passing through the left focus $F_{1}$ intersects $C$ at a point $P$ in the first quadrant. If $|F_{1}P|=|F_{1}F_{2}|$ and the cosine value of the slope of line $l$ is $\frac{7}{8}$, then ... | 2 |
Given that point $P$ is any point on circle $G$: ${{(x+\\sqrt{6})}^{2}}+{{y}^{2}}=32$, and the perpendicular bisector of the line segment connecting point $P$ and point $A(\\sqrt{6},0)$ intersects with $GP$ at point $Q$.
(I) Find the equation of the trajectory $C$ of point $Q$;
(II) A line $l$ passing through the coo... | -\\dfrac{1}{4} |
1. The equation $x^{2}+m x+1=0$ and the equation $x^{2}-x$ $-m=0$ have one common root, then $m=$ $\qquad$ | 2 |
2. Solve the system:
$$
\begin{aligned}
& \frac{x y}{x+y}=\frac{2}{3} \\
& \frac{y z}{y+z}=\frac{6}{5} \\
& \frac{z x}{z+x}=\frac{3}{4}
\end{aligned}
$$ | 1,2,3 |
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