problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Let $y = x\cos x$, then $y' =$ ? | \cos x - x\sin x |
1. Subtract the second equation from the first:
$x^{2}-2 x+y^{4}-8 y^{3}+24 y^{2}-32 y=-17 \rightarrow(x-1)^{2}+(y-2)^{4}=0 \rightarrow\left\{\begin{array}{l}x=1 \\ y=2\end{array}\right.$
Then $z=x^{2}+y^{4}-8 y^{3}=1+16-64=-47$ | -47 |
24. The construction cost of a project must not exceed 1.9 million yuan. If the project is undertaken by Company A, it will take 20 days, with a daily fee of 100,000 yuan; if undertaken by Company B, it will take 30 days, with a daily fee of 60,000 yuan. To shorten the construction period, it is decided that Company A ... | 10, 25 |
We mean a traingle in $\mathbb Q^{n}$, 3 points that are not collinear in $\mathbb Q^{n}$
a) Suppose that $ABC$ is triangle in $\mathbb Q^{n}$. Prove that there is a triangle $A'B'C'$ in $\mathbb Q^{5}$ that $\angle B'A'C'=\angle BAC$.
b) Find a natural $m$ that for each traingle that can be embedded in $\mathbb Q^... | 6 |
The sum of the maximum and minimum values of the function $y=2^x+\log_2(x+1)$ in the interval $[0,1]$ is (γγ)
A: 2
B: 3
C: 4
D: 5 | C |
To obtain the graph of the function $y=-\sin 2x$, the graph of the function $y=\sin (2x- \frac {\pi}{3})$ needs to be (γ).
A: Shifted to the left by $\frac {\pi}{3}$ units
B: Shifted to the left by $\frac {\pi}{6}$ units
C: Shifted to the right by $\frac {\pi}{3}$ units
D: Shifted to the right by $\frac {2\pi}{3... | C |
4. Given $\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma=1, \alpha, \beta, \gamma$ are all acute angles, then the maximum value of $\cos \alpha \cos \beta \cos \gamma$ is | \frac{2\sqrt{6}}{9} |
7. Let $G$ and $O$ be the centroid and circumcenter of $\triangle A B C$, respectively, and $A B=\sqrt{3}, A C=3$. Then $\overrightarrow{A G} \cdot \overrightarrow{A O}=$ | 2 |
B. As shown in Figure 2, in quadrilateral $A B C D$, $A C$ and $B D$ are diagonals, $\triangle A B C$ is an equilateral triangle, $\angle A D C=$ $30^{\circ}, A D=3, B D=5$. Then the length of $C D$ is ( ).
(A) $3 \sqrt{2}$
(B) 4
(C) $2 \sqrt{5}$
(D) 4.5 | B |
Given the function $f(x)=2\sin (2x+ \frac {\pi}{3})+1$, the smallest positive period of $f(x)$ is ______, and the maximum value of $f(x)$ is ______. | 3 |
9.6. Find all three-digit numbers that are five times the product of their digits. Answer: 175. | 175 |
The sum of a certain infinite geometric progression is greater than the sum of the infinite geometric progression $\frac{5}{12}, \frac{5}{16} \ldots$ by its own constant ratio; the first term of the former progression is equal to the first term of the given progression. Calculate the ratio and sum of both progressions. | q=\frac{5}{6},S=2.5 |
## Task 13/63
A mathematician had his bicycle stolen. When asked for his bicycle number, he replied: "You can calculate the number from the following information:
a) If you add the square of the first digit to the square of the second digit, you get the square of the third digit.
b) If you subtract the second digit ... | 4353055 |
14. Let $a$ be a prime number, $b$ be a positive integer, and $9(2 a+b)^{2}=509(4 a+511 b)$, find the values of $a, b$. | =251,b=7 |
11) A parallelepiped with a square base is inscribed in a sphere. If the side of the base is $\frac{1}{4}$ of the height, what is the ratio between the surface area of the sphere and the total surface area of the parallelepiped?
(A) $\pi$
(B) $\frac{\pi}{4}$
(C) $2 \pi$
(D) $\frac{\pi}{2}$
(E) depends on the radius of ... | \pi |
4.3. In a school, there are $n$ students who attend several clubs, and in each club, there are at least two students. It turned out that if there are any two common students in two clubs, then the number of students in these clubs is different. Prove that the number of clubs does not exceed $(n-1)^{2}$. | (n-1)^{2} |
10.5. Five athletes came to training with their balls, and when leaving, each took someone else's. In how many ways is this possible. | 44 |
5. On the extension of side $A C$ of triangle $A B C$ beyond point $A$, point $T$ is marked such that $\angle B A C=2 \angle B T C$. Find the area of triangle $A B C$, given that $A B=A C, B T=70$, $A T=37$. | 420 |
A steel bar with a total length of 14.8 m is used to make a rectangular container frame. If one side of the bottom of the container is 0.5 m longer than the other side, what height will maximize the volume of the container? | 1.2 \text{ m} |
For every integer $n \geq 2$, we denote by $f(n)$ the largest integer $m$ for which the following statement is true: "In the plane, no matter how $n$ lines are given, one can select $m$ of them such that among these $m$, any two are either parallel or any two intersect each other."
Determine the function $f(n)$! | f(n)=k |
Define the determinant operation $\left| \begin{matrix} a_1 & a_2 \\ a_3 & a_4 \end{matrix} \right|=a_1a_4-a_2a_3.$ The graph of the function $f(x)=\left| \begin{matrix} \sqrt{3} & \sin x \\ 1 & \cos x \end{matrix} \right|$ is shifted to the left by $n(n > 0)$ units, and the resulting graph corresponds to an even funct... | C |
$20 \cdot 74$ Let $AB$ be a chord of $\odot O$, and $CD$ be a diameter of $\odot O$ that intersects the chord $AB$. Let $M=|S_{\triangle CAB}-S_{\triangle DAB}|, N=2S_{\triangle OAB}$, then
(A) $M>N$.
(B) $M=N$.
(C) $M<N$.
(D) The relationship between $M$ and $N$ is uncertain.
(China Junior High School Mathematics Leag... | B |
Given the function f(x) = $\frac{1}{2}$x^2^ - (a+1)x + alnx.
(1) When a=-4, find the monotonic intervals of f(x);
(2) Given aβ(1,2], bβR, the function g(x) = x^3^ + bx^2^ - (2b+4)x + lnx. Prove that if the minimum value point of f(x) is equal to the minimum value point of g(x), then the maximum value of g(x) is not gre... | \frac{5}{4} |
When using the method of contradiction to prove the proposition "A triangle has at most one obtuse angle," the correct assumption is (γγ)
A: Assume there is at least one obtuse angle
B: Assume there are at least two obtuse angles
C: Assume there are no obtuse angles
D: Assume there are no obtuse angles or at least two ... | B |
Let $S$ be the set of sides and diagonals of a regular pentagon. A pair of elements of $S$ are selected at random without replacement. What is the probability that the two chosen segments have the same length?
$\textbf{(A) }\frac{2}5\qquad\textbf{(B) }\frac{4}9\qquad\textbf{(C) }\frac{1}2\qquad\textbf{(D) }\frac{5}9\qq... | \frac{4}{9} |
1. (2 points) In trapezoid $A B C D$ with bases $A D=20$ and $B C=10$, circles constructed on sides $A B, B C$, and $C D$ as diameters intersect at one point. The length of diagonal $A C$ is 18. Find the length of $B D$. | 24 |
4.34. All edges (including the base sides) of a triangular pyramid are equal. Find the ratio of the radius of the sphere inscribed in the pyramid to its height. | 1:4 |
Let $a_{1}, a_{2}, \cdots, a_{n}$ represent any permutation of the integers 1, 2, $\cdots, n$. Let $f(n)$ be the number of such permutations that satisfy: (1) $a_{1}=1$; (2) $\left|a_{i}-a_{i+1}\right| \leqslant 2, i=1$, $2, \cdots, n-1$. Determine whether $f(1996)$ is divisible by 3. | 1 |
In triangle $ABC$, it is known that $A=60^{\circ}$, $b=1$, and its area is $\sqrt{3}$. Find the value of $\frac{a+b+c}{\sin A+\sin B+\sin C}$. | \frac{2\sqrt{39}}{3} |
Let $ P_n\equal{}\sqrt[n]{\frac{(3n)!}{(2n)!}}\ (n\equal{}1,\ 2,\ \cdots)$.
Find the following limits.
(1) $ \lim_{n\to\infty} \frac{P_n}{n}$.
(2) $ \lim_{n\to\infty} \left(\frac{n\plus{}2}{n}\right)^{P_n}$. | \frac{27}{4e} |
If point $A(a,a-2)$ lies on the $x$-axis, then the coordinates of point $B(a+2,a-1)$ are ( ).
A: $(4,2)$
B: $(2,4)$
C: $(4,1)$
D: $(1,4)$ | \text{C} |
$26 \cdot 9$ Some points $P$ are within the region enclosed by a circle of radius 1, and the sum of the squares of the distances from these $P$ points to the endpoints of a certain diameter is 3. Let $n$ be the number of $P$ points, then $n$ is
(A) 0.
(B) 1.
(C) 2.
(D) 4.
(E) Infinite.
(20th American High School Mathem... | E |
6. At Matthijs' table tennis club, they always place their ping pong balls on a table with cylindrical ball holders. Below is a side view of a ping pong ball on top of a ball holder. The bottom of the ball touches the table exactly. It is known that the ball holder is 4 centimeters wide and 1 centimeter high.
How many... | 2\frac{1}{2} |
The 3rd problem of the mathematical class competition:
Find the six-digit square numbers such that, when the first three digits are erased, the square root of the number is obtained. | 376^2=141376625^2=390625 |
Example 5 Let $a_{1}=1, a_{2}=\frac{5}{3}, a_{n+2}=\frac{5}{3} a_{n+1}-\frac{2}{3} a_{n},(n=1,2, \cdots)$.
(1) Let $b_{n}=a_{n+1}-a_{n}(n=1,2, \cdots)$, find the general term formula of the sequence $\left\{b_{n}\right\}$;
(2) Find the sum of the first $n$ terms $S_{n}$ of the sequence $\left\{n a_{n}\right\}$. | S_{n}=\frac{3}{2}n(n+1)+\frac{(3+n)2^{n+1}}{3^{n-1}}-18 |
Let $S_0=0$ and let $S_k$ equal $a_1+2a_2+\ldots+ka_k$ for $k\geq 1$. Define $a_i$ to be $1$ if $S_{i-1}<i$ and $-1$ if $S_{i-1}\geq i$. What is the largest $k\leq 2010$ such that $S_k=0$? | 1092 |
If set $M=\{a, b, c\}$, then (γγ)
A: $\{a\} \in M$
B: $c \in M$
C: $b \subset M$
D: $c = \{c\}$ | B |
Given the sets $A={x|-3\leqslant x\leqslant 4}$ and $B={x|2m-1 < x < m+1}$,
(1) Find $A\cap B$ when $m=1$;
(2) Find the range of real number $m$ values such that $B\subseteq A$. | [-1,+\infty) |
Kim earned scores of 87,83, and 88 on her first three mathematics examinations. If Kim receives a score of 90 on the fourth exam, then her average will
$\mathrm{(A) \ \text{remain the same} } \qquad \mathrm{(B) \ \text{increase by 1} } \qquad \mathrm{(C) \ \text{increase by 2} } \qquad \mathrm{(D) \ \text{increase by... | B |
$(\lg 2)^{2}+\lg 2\times \lg 50-\lg 4=\left(\ \ \right)$
A: $2$
B: $0$
C: $\lg\frac{1}{2}$
D: $\frac{1}{4}$ | B |
Let the monotonic function $f(x)$ be defined on $(0,+\infty)$, and for any $x \in (0,+\infty)$, it satisfies $f[f(x)-\log_{2}x]=6$. If $x_{0}$ is a solution to the equation $f(x)-f'(x)=4$, and $x_{0} \in (a,a+1) (a \in \mathbb{N}^{*})$, then $a=$ γγ.
A: $4$
B: $3$
C: $2$
D: $1$ | D |
5. The area of a regular quadrilateral pyramid is $5 a^{2}$, where $a$ is the length of the side of the base of the pyramid.
a) Express the volume of this pyramid as a function of $a$.
b) Calculate the volume of the pyramid if $a=6 dm$. | 36\sqrt{15}\, |
```
(25 points) Let $x, y, z, r$ be real numbers. Find the minimum value of the following algebraic expressions:
$$
\begin{array}{l}
\text { (1) } x^{2}+y^{2}+x y+x+y \text {; } \\
\text { (2) } x^{2}+y^{2}+z^{2}+x y+y z+z x+x+y+z \text {; } \\
\text { (3) } x^{2}+y^{2}+z^{2}+r^{2}+x y+x z+x r+y z+y r+ \\
z r+x+y+z+r
\... | -\frac{1}{3},-\frac{3}{8},-\frac{2}{5} |
Given the function f(x) = sin(2x + Ο), where Ο is a real number, find the value of Ο such that f(x) β€ |f($$\frac{Ο}{6}$$)| holds true for all x β R. (Write out one value of Ο that satisfies the condition) | Ο = $$\frac{Ο}{6}$$ |
Exercise 3. Find the integer triplets $(x, y, n)$ such that $n^{2}=17 x^{4}-32 x^{2} y^{2}+41 y^{4}$. | (0,0,0) |
The distribution table of random variable $X$ is as follows. Find $E\left(2X-1\right)=\_\_\_\_\_\_$ and $D\left(2X-1\right)=\_\_\_\_\_\_$.
| $X$ | $0$ | $1$ | $2$ |
|-----|-----|-----|-----|
| $P$ | $0.3$ | $p$ | $0.3$ | | 2.4 |
a) Given six natural numbers. All of them are distinct and their sum is 22. Find these numbers and prove that there are no others.
b) The same question for 100 numbers, which sum up to 5051. | 1,2,3,4,5,7 |
## Problem Statement
Calculate the definite integral:
$$
\int_{0}^{1} e^{\sqrt{(1-x) /(1+x)}} \cdot \frac{d x}{(1+x) \sqrt{1-x^{2}}}
$$ | e-1 |
Example 44. In triangle $A B C$, the ratio $|B C|:|A C|=3$ and $\widehat{A C B}=\alpha$. Points $D$ and $K$ are taken on side $A B$ such that $\widehat{A C D}=\widehat{D C K}=\widehat{K C B}$. Find: $|C D|:|C K|$. | \frac{2\cos\frac{\alpha}{3}+3}{1+6\cos\frac{\alpha}{3}} |
Example 1. The numbers $a, b, c$ belong to the open interval $\left(0, \frac{\pi}{2}\right)$ and satisfy the following equations
$$
\begin{array}{l}
\cos a=a, \\
\sin \cos b=b, \\
\cos \sin c=c .
\end{array}
$$
Arrange these three numbers in ascending order. | b<a<c |
γQuestion 16γ
A team 50 meters long, moving from east to west at a speed of 3 meters per second, and another team 60 meters long, moving from west to east at a speed of 2 meters per second. The time required from when the leaders of the two teams meet until the last members of the teams part ways is $\qquad$ seconds. | 22 |
Given sets $A=\{\sin 90^\circ, \cos 180^\circ\}$ and $B=\{x|x^2+x=0\}$, then $A \cap B=$ ? | \{-1\} |
The negation of the proposition "$\forall x\in R$, $e^{x} \gt 0$" is ( ).
A: $\exists x\in R$, $e^{x}\leqslant 0$
B: $\exists x\in R$, $e^{x} \lt 0$
C: $\forall x\in R$, $e^{x}\leqslant 0$
D: $\forall x\in R$, $e^{x} \lt 0$ | A |
8. Famous skater Tony Hawk is riding a skateboard (segment $A B$) in a ramp, which is a semicircle with diameter $P Q$. Point $M$ is the midpoint of the skateboard, $C$ is the foot of the perpendicular dropped from point $A$ to the diameter $P Q$. What values can the angle $\angle A C M$ take if it is known that the an... | 12 |
An equilateral triangle ABC has its vertices on the surface of a sphere with center O. G is the centroid of triangle ABC, and OG = $\frac{\sqrt{3}}{3}$. The area of the circumcircle of triangle ABC is $\frac{2\pi}{3}$. Find the volume of the sphere. | \frac{4\pi}{3} |
Evaluate $\frac{i}{\sqrt{7}+3i}$.
A: $\frac{3}{16}-\frac{\sqrt{7}}{16}i$
B: $\frac{3}{16}+\frac{\sqrt{7}}{16}i$
C: $-\frac{3}{16}+\frac{\sqrt{7}}{16}i$
D: $-\frac{3}{16}-\frac{\sqrt{7}}{16}i$ | B |
The surface area of a sphere with a diameter of 4 is equal to \_\_\_\_\_\_. | 16\pi |
$\left.\begin{array}{ll}\text { [Mean proportionals in a right triangle] } \\ {[\quad \text { Pythagorean Theorem (direct and inverse). }}\end{array}\right]$
In a right triangle $ABC$, a height $CD$ is dropped from the vertex of the right angle $C$. The projection of segment $BD$ onto the leg $BC$ is $l$, and the proj... | (^{\frac{2}{3}}+^{\frac{2}{3}})\sqrt{^{\frac{2}{3}}+^{\frac{2}{3}}} |
An even function $f(x)$ defined on $\mathbb{R}$ satisfies $f(x+1) = -f(x)$, and it is increasing on the interval $[-1, 0]$. Consider the following statements about $f(x)$:
1. $f(x)$ is a periodic function;
2. $f(x)$ is increasing on the interval $[0, 1]$;
3. $f(x)$ is decreasing on the interval $[1, 2]$;
4. $f(... | 1, 4 |
γExample 7γA newly built city, to solve the city's traffic problems, has purchased a large number of buses. They plan to set up bus stops at 1985 different locations, connecting them with several routes, with the following wishes:
(1) To open as many routes as possible;
(2) Each pair of routes should have at least one ... | 63 |
30. (5 points)
On Lie Island, half of the people only lie on Wednesday, Friday, and Saturday, while the other half only lie on Tuesday, Thursday, and Sunday. One day, everyone on the island said, βI will tell the truth tomorrow.β So, this day is ( ).
A. Tuesday
B. Wednesday
C. Friday
D. Saturday
E. Sunday | C |
Exercise 3. We have 100 light bulbs, numbered from 1 to 100, each of which can be either on or off. These light bulbs are connected to three switches A, B, and C.
Pressing A changes the state of all the light bulbs: those that were on turn off, and those that were off turn on.
Pressing B changes the state only of the... | 34 |
Problem 9.2. A triangle $A B C$ with centroid $G$ and incenter $I$ is given. If $A B=42, G I=2$ and $A B \| G I$, find $A C$ and $B C$.
Ivailo Kortezov | AC=48,BC=36 |
In [trapezoid](https://artofproblemsolving.com/wiki/index.php/Trapezoid) $ABCD$, leg $\overline{BC}$ is [perpendicular](https://artofproblemsolving.com/wiki/index.php/Perpendicular) to bases $\overline{AB}$ and $\overline{CD}$, and diagonals $\overline{AC}$ and $\overline{BD}$ are perpendicular. Given that $AB=\sqrt{11... | 110 |
12. (5 points) Two people, A and B, start from points $A$ and $B$, which are 10 kilometers apart, and walk towards each other. If they start at the same time, they will meet 1 kilometer away from the midpoint of $A$ and $B$. If A starts 5 minutes later, they will meet at the midpoint of $A$ and $B$. At this time, A has... | 10 |
The domain of the function $f(x)= \sqrt {x-1}+\lg (x+1)$ is $(\quad)$
A: $(-1,1]$
B: $(-1,1)$
C: $[-1,1]$
D: $[1,+\infty)$ | D |
Calculate the result of the integral $$\int_{0}^{\frac{\pi}{2}} \sin x \, dx$$. | 1 |
Which of the following calculations is correct?
A: $x^{4}+x^{2}=x^{6}$
B: $x^{3}-x^{2}=x$
C: $(x^{3})^{2}=x^{6}$
D: $x^{6}\div x^{3}=x^{2}$ | C |
The starting fare for taxis in Wenzhou city is 10 yuan, plus a fuel surcharge of 1 yuan (i.e., for travel within 4 kilometers, the fare is 10 yuan plus 1 yuan for fuel, totaling 11 yuan). Beyond 4 kilometers, an additional 1.5 yuan is charged for each kilometer traveled (less than 1 kilometer is counted as 1 kilometer)... | B |
In the complex plane, the coordinates of the point corresponding to the complex number z = $$\frac {2+3i}{i}$$ are ( )
A: (3, 2)
B: (2, 3)
C: (-2, 3)
D: (3, -2) | \text{D} |
15. A total of 99 people participated in a mathematics competition, which was divided into three sessions, testing the contestants' abilities in geometry, number theory, and combinatorics. Xiao Ming ranked 16th in the number theory exam, 30th in the combinatorics exam, and 23rd in the geometry exam, and he did not tie ... | 167 |
[Tournaments and tournament tables]
A school table tennis championship was held according to the Olympic system. The winner won six matches. How many participants in the tournament won more games than they lost? (In a tournament according to the Olympic system, participants are divided into pairs. Those who lost a gam... | 16 |
For positive integers $n$ and $k$, let $f(n, k)$ be the remainder when $n$ is divided by $k$, and for $n > 1$ let $F(n) = \max_{\substack{1\le k\le \frac{n}{2}}} f(n, k)$. Find the remainder when $\sum\limits_{n=20}^{100} F(n)$ is divided by $1000$. | 512 |
4. The first team of workers is paving asphalt on one section of the road, while the second team, which has 6 more workers, is working on another section, which is three times larger. The productivity of all workers is the same. What is the smallest number of workers that could have been in the first team if they compl... | 4 |
11. Let complex numbers $z_{1}, z_{2}$ satisfy $z_{2}=z_{1} i-2$, and $z_{1}$ corresponds to point $Z_{1}$ on the curve $|z-2|+|z+2|=10$ in the complex plane. Then the trajectory equation of $z_{2}$ in the complex plane, corresponding to point $Z_{2}$, is $\qquad$
(in the original Cartesian coordinate system of the com... | \frac{(x+2)^{2}}{21}+\frac{y^{2}}{25}=1 |
38. Find the smallest natural number that, when increased by 1, would be divisible by $4, 6, 10, 12$. | 59 |
3.111.
$$
\frac{\cos ^{2}\left(\frac{5}{2} \pi-2 \alpha\right)+4 \cos ^{2}\left(\frac{7}{2} \pi-\alpha\right)-4}{1+\cos (4 \alpha-\pi)-8 \sin ^{2}(5 \pi-\alpha)}
$$ | \frac{1}{2}\operatorname{ctg}^{4}\alpha |
Example 3 Find the integer solutions of the equation $x^{2}-5 x y+6 y^{2}-3 x+5 y-$ $25=0$. | (x,y)=(-50,-25),(4,-1),(-26,-9),(-16,-9),(-6,-1),(50,15),(-72,-25) |
XXIX OM - II - Problem 4
From the vertices of a regular $2n$-gon, 3 different points are chosen randomly. Let $p_n$ be the probability that the triangle with vertices at the chosen points is acute. Calculate $\lim_{n\to \infty} p_n$.
Note. We assume that all choices of three different points are equally probable. | \frac{1}{4} |
3. Given two equations about $x$: $x^{2}+2 b x+a$ $=0$ and $x^{2}+a x+2 b=0$ have exactly one common root. Then the minimum value of $a^{2}+b^{2}$ is $(\quad)$.
(A) $\frac{1}{5}$
(B) $\frac{2}{5}$
(C) $\frac{3}{5}$
(D) $\frac{4}{5}$ | A |
Let $G$ be any convex region tightly sandwiched between parallel lines $l_{1}$ and $l_{2}$ (i.e., the line segment connecting any two points on its boundary is contained within its region), and its boundary $C$ intersects with $l_{1}$ and $l_{2}$.
Parallel to $l_{1}$, the line $l$ divides $G$ into two parts $A$ and $B$... | \frac{a^{2}+2 a b}{b^{2}} |
6. How many positive integers are a power of 4 and can be written in base 3 using only the digits 0 and 1, with 0 as many times as desired (even none) and 1 at most twice?
(A) 4
(B) 2
(C) 1
(D) 0
(E) Infinite. | 2 |
(IMO 2016, problem 5)
We write on the board the equality:
$$
(x-1)(x-2) \ldots(x-2016)=(x-1)(x-2) \ldots(x-2016)
$$
We want to erase some of the 4032 factors in such a way that the equation on the board has no real solutions. What is the minimum number of factors that must be erased to achieve this?
## - Solutions ... | 2016 |
Given the sets $A=\{x|x>1\}$ and $B=\{x|a<x<a+1\}$.
1. If $B \subseteq A$, find the range of the real number $a$.
2. If $A \cap B \neq \emptyset$, find the range of the real number $a$. | (0, +\infty) |
3. Simplify the fraction $\frac{2 x^{6}+5 x^{4}-3 x^{3}+2 x^{2}-12 x-14}{4 x^{6}-4 x^{4}-6 x^{3}-3 x^{2}+25 x-28}$. After simplification, the degrees of the polynomials in the numerator and the denominator should be reduced.
# | \frac{x^{3}+2x+2}{2x^{3}-3x+4} |
The Tasty Candy Company always puts the same number of pieces of candy into each one-pound bag of candy they sell. Mike bought 4 one-pound bags and gave each person in his class 15 pieces of candy. Mike had 23 pieces of candy left over. Betsy bought 5 one-pound bags and gave 23 pieces of candy to each teacher in her sc... | 302 |
526 ** The 10 complex roots of the equation $x^{10}+(13 x-1)^{10}=0$ are $r_{1}, \overline{r_{1}}, r_{2}, \overline{r_{2}}$, $r_{3}, \overline{r_{3}}, r_{4}, \overline{r_{4}}, r_{5}, \overline{r_{5}}$. Find the value of the expression
$$
\frac{1}{r_{1} \overline{r_{1}}}+\frac{1}{r_{2} \overline{r_{2}}}+\cdots+\frac{1}{... | 850 |
Let $N$ be the number of ways of distributing $8$ chocolates of different brands among $3$ children such that each child gets at least one chocolate, and no two children get the same number of chocolates. Find the sum of the digits of $N$. | 24 |
If $|z-1|=|z+1|$, then the point corresponding to the complex number $z$ is on ( )
A: the real axis
B: the imaginary axis
C: the first quadrant
D: the second quadrant | \text{B} |
Plan to hold volleyball, basketball, and soccer competitions in 4 candidate venues, with no more than 2 events in the same venue. The total number of arrangements is ( )
A: 24 kinds
B: 36 kinds
C: 42 kinds
D: 60 kinds | \text{D} |
The subway project in Jinan is progressing rapidly, and after the subway is opened, it will bring convenience to the citizens. It is known that after a certain route is opened, the train departure time interval $t$ (unit: minutes) satisfies $2 \leq t \leq 20$. According to market research, the passenger capacity of the... | \$132 |
In the arithmetic sequence $\{a_n\}$, it is known that $a_2 + a_{10} = 16$, then $a_4 + a_8 =$ οΌγγοΌ
γγγγγAοΌ 12γγγγγBοΌγγγγγ16γγγγγCοΌγγγγγ20γγγγγDοΌγγγγγ24 | \text{B} |
Solve the equations:$(1) 4x+3=5x-1$;$(2) 4\left(x-1\right)=1-x$. | x = 1 |
3. Once, a detective had to interrogate three witnesses of a robbery: John White, Sam Gray, and Bob Black. John insisted that all of Sam's statements were lies, while Sam kept repeating that Bob was lying. Bob, for his part, tried to convince the detective not to believe either White or, especially, Gray. The detective... | SamGray |
A company has 3 factories producing the same type of electronic product, with the production ratio of the first, second, and third factories being 1:2:1. Using stratified sampling method (each factory's products as one layer), 100 electronic products are sampled from the 3 factories for a lifespan test. Based on the te... | 1013 |
Given a positive integer $n(n \geqslant 2)$. Find the largest positive real number $C$, such that for any $x_{1}, x_{2}, \cdots, x_{n}$, we have
$$
\sum_{i=1}^{n} x_{i}^{2} \geqslant \frac{1}{n+1}\left(\sum_{i=1}^{n} x_{i}\right)^{2}+C\left(\sum_{i=1}^{n} i x_{i}\right)^{2} .
$$ | \frac{12}{n(n+1)(n+2)(3n+1)} |
## Task 3 - 260613
The connecting roads of three places $A, B, C$ form a triangle. In the middle of the road from $B$ to $C$ lies another place $D$. The distance from $A$ via $B$ to $C$ is $25 \mathrm{~km}$, from $B$ via $C$ to $A$ is $27 \mathrm{~km}$, and from $C$ via $A$ to $B$ is $28 \mathrm{~km}$.
A pioneer grou... | 19\mathrm{~} |
If $9x^{2}-kxy+4y^{2}$ is a perfect square trinomial, then the value of $k$ is:
A: $6$
B: $\pm 6$
C: $12$
D: $\pm 12$ | D |
In triangle $ABC$, point $D$ is on side $BC$, with $BC=3BD$. If $AB=1$, and $AC=2$, then the maximum value of $AD \cdot BD$ is \_\_\_\_\_\_. | \frac{\sqrt{2}}{2} |
Mobile payment has become one of the most common payment methods for people. A large supermarket randomly selected $100$ customers to investigate their payment methods, and the results are recorded and organized in the table below. If one person is randomly selected from these $100$ customers, the probability that the ... | D |
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