problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
## Task 1 - 050731
What digits does the product end with?
$$
z=345926476^{3} \cdot 125399676^{2} \cdot 2100933776^{3}
$$ | 76 |
Example 8 Given a graph $G$ with $e$ edges and $n$ vertices with degrees $d_{1}, d_{2}, \cdots, d_{n}$, and an integer $k$ less than
$$
\min \left\{d_{1}, d_{2}, \cdots, d_{n}\right\} .
$$
Prove: Graph $G$ contains an induced subgraph $H$ (i.e., if two vertices in $H$ are connected by an edge in graph $G$, then they a... | \frac{k n^{2}}{2 e+n} |
If $5^{\lg x} = 25$, then the value of $x$ is ( )
A: 10
B: 0.01
C: 100
D: 1000 | C |
Which of the following statements is correct? ( )
A: Three points in space can determine a plane.
B: A triangle is always a planar figure.
C: If points A, B, C, and D are both on plane α and plane β, then plane α coincides with plane β.
D: A quadrilateral with all four sides equal is a planar figure. | B |
How many positive integers between 1 and 1988 can be expressed in the form
$$
[x]+[2 x]+[4 x]+[8 x]
$$
for some real number $x$? | 1061 |
[Example 5.2.2] Solve the equation:
$$
x^{4}-x^{2}+8 x-16=0 .
$$ | x_{1,2}=\frac{-1\\sqrt{17}}{2},x_{3,4}=\frac{1\\sqrt{15}\mathrm{i}}{2} |
6. The base of the pyramid $S A B C$ is an isosceles triangle $A B C$, with $A B = B C = 3$, and $A C = 5$. The height of the pyramid $S A B C$ is the segment $S O$, where $O$ is the intersection point of the line passing through vertex $B$ parallel to side $A C$ and the line passing through $C$ perpendicular to side $... | \frac{35\sqrt{11}}{396} |
11. Given $0 \leqslant a_{k} \leqslant 1(k=1,2, \cdots, 2002)$, let $a_{2003}=a_{1}, a_{2004}=a_{2}$. The maximum value of $\sum_{k=1}^{20002}\left(a_{k}-a_{k+1} a_{k+2}\right)$ is $\qquad$ . | 1001 |
Observe the equation:
$$
\begin{aligned}
(1+2+3+4)^{2} & =(1+2+3+4)(1+2+3+4) \\
& =1 \cdot 1+1 \cdot 2+1 \cdot 3+1 \cdot 4+2 \cdot 1+2 \cdot 2+2 \cdot 3+2 \cdot 4+ \\
& +3 \cdot 1+3 \cdot 2+3 \cdot 3+3 \cdot 4+4 \cdot 1+4 \cdot 2+4 \cdot 3+4 \cdot 4
\end{aligned}
$$
Note that $4 \times 4=16$ products are formed when ... | 10000 |
1. Let $a, b$ be rational numbers, and $|a|>0$, the equation $||x-a|-b|=5$ has three distinct roots. Then $b=(\quad)$.
(A) 1
(B) 3
(C) 5
(D) 6 | C |
If 554 is the base $b$ representation of the square of the number whose base $b$ representation is 24, then $b$, when written in base 10, equals
$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 16$ | \textbf{(C)}\12 |
The equation of the axis of symmetry for the graph of the function $y=\cos(2x+\frac{\pi}{6})$ could be ( )
A: $x=-\frac{\pi}{6}$
B: $x=-\frac{\pi}{12}$
C: $x=\frac{\pi}{6}$
D: $x=\frac{\pi}{12}$ | \text{B} |
In a large box, 10 smaller boxes were placed. In each of the nested boxes, either 10 even smaller ones were placed, or nothing was placed. In each of the smaller ones, either 10 or none were placed again, and so on. After this, it turned out that there were exactly 2006 boxes with contents. How many are empty? # | 18055 |
6. In how many different ways can a chess king move from square $e 1$ to square $h 5$, if it is allowed to move only one square to the right, up, or diagonally to the right-up? | 129 |
Adding the same number to both sides of an inequality or multiplying both sides by the same number does not change the direction of the inequality. ( )
A: True
B: False
C:
D: | \text{B} |
(10) (15 points) Given the sequence of positive numbers $\left\{a_{n}\right\}(n \geqslant 0)$ satisfies $a_{n}=\frac{a_{n-1}}{m a_{n-2}}, n=2$,
$3, \cdots$, where $m$ is a real parameter. If $a_{2009}=\frac{a_{0}}{a_{1}}$, find the value of $m$. | 1 |
The point $M$ moves such that the sum of squares of the lengths from $M$ to faces of a cube, is fixed. Find the locus of $M.$ | \text{The locus of } M \text{ is a sphere centered at } \left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right). |
## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 3} \frac{x^{3}-4 x^{2}-3 x+18}{x^{3}-5 x^{2}+3 x+9}$ | \frac{5}{4} |
Evaluate $2 \times 0+1 \times 4$. | 4 |
2. Let $(a_{n})$ be a geometric sequence with positive terms. Denote
$$
S_{n}=\log a_{1}+\log a_{2}+\ldots+\log a_{n-1}+\log a_{n}
$$
Prove: if for some $m \neq n$ it holds that $S_{n}=S_{m}$, then $S_{n+m}=0$. | 0 |
Let $x_1,x_2,\cdots,x_n$ be non-negative real numbers such that $x_ix_j\le 4^{-|i-j|}$ $(1\le i,j\le n)$. Prove that\[x_1+x_2+\cdots+x_n\le \frac{5}{3}.\] | \frac{5}{3} |
Given in \\(\triangle ABC\\), the sides opposite to angles \\(A\\), \\(B\\), and \\(C\\) are \\(a\\), \\(b\\), and \\(c\\) respectively, \\(\overrightarrow{AD} = \lambda \overrightarrow{AB} (0 < \lambda < 1)\\), \\(b = 2\sqrt{5}\\), \\(CD = 4\\), and \\(B = \frac{\pi}{6}\\).
\\((\\)I\\()\\) If the product \\(ac\\) rea... | \sin \angle BDC = \frac{16\sqrt{5}}{5} |
220. Differentiate the function $y=2 x^{3}\left(x^{6}-1\right)$. | 6x^{2}(3x^{6}-1) |
Find the equation of line $l$:$(1)$ Find the equation of the line passing through point $P(1,2)$ and parallel to the line $3x-2y+5=0$;$(2)$ Find the equation of the line passing through point $P(1,-1)$ and perpendicular to the line $2x+3y+1=0$. | 3x-2y-5=0 |
Find the number of integers $n$ for which $\sqrt{\frac{(2020 - n)^2}{2020 - n^2}}$ is a real number. | 90 |
Use mathematical induction to prove that "5n - 2n is divisible by 3". In the second step of the proof, when n=k+1, in order to use the induction hypothesis, 5k+1 - 2k+1 should be transformed into. | 5(5^{k} - 2^{k}) + 3×2^{k} |
Given that the function $y=f(x)$ has a tangent line equation $y=\frac{1}{2}x+2$ at the point $M(1,f(1))$, determine the value of $f(1)+f'(1)$. | 3 |
Due to the rainstorms, our basement filled with water. To remove the water, three pumps were set up. The first pump alone could pump out the water in 3 hours, the second in 4 hours, and the third in 6 hours. After 30 minutes of working together, the second pump broke down. The remaining water was pumped out by the firs... | 1\frac{3}{4} |
6. Find all non-zero polynomials $P(x)$ that satisfy $P\left(x^{2}-2 x\right) \equiv(P(x-2))^{2}, x \in \mathbf{R}$. | P(y)=(y+1)^{n},n\in{Z}^{+} |
Given an arithmetic sequence $\{a_n\}$ satisfies $a_1+a_2+a_3=9$ and $a_2+a_8=18$. The sum of the first $n$ terms of sequence $\{b_n\}$ is $S_n$, and it satisfies $S_n=2b_n-2$.
$(1)$ Find the general formula for sequences $\{a_n\}$ and $\{b_n\}$;
$(2)$ Sequence $\{c_n\}$ satisfies $c_n= \frac {a_n}{b_n}$, find the sum ... | T_n=3- \frac {2n+3}{2^n} |
Problem 5. In quadrilateral $A B C D$, it is known that $A B=$ $=B C=C D, \angle A=70^{\circ}$ and $\angle B=100^{\circ}$. What can the angles $C$ and $D$ be equal to? $\quad[8$ points] (M.A.Volchkevich) | 60 |
Compute the $100^{\text{th}}$ smallest positive integer $n$ that satisfies the three congruences \[\begin{aligned} \left\lfloor \dfrac{n}{8} \right\rfloor &\equiv 3 \pmod{4}, \\ \left\lfloor \dfrac{n}{32} \right\rfloor &\equiv 2 \pmod{4}, \\ \left\lfloor \dfrac{n}{256} \right\rfloor &\equiv 1 \pmod{4}. \end{aligned}\] ... | 6491 |
3. (5 points) Using the digits $1, 2, 3, 5, 6, 7, 8, 9$, the maximum number of prime numbers that can be formed is $\qquad$ (each digit can only be used once, and must be used). | 6 |
5. Solve the inequality $\quad \frac{|x|+21-7 \sqrt{|x|+9}}{x^{2}-8|x|}>0$ | x\in(-\infty;-8)\cup(-7;0)\cup(0;7)\cup(8;+\infty) |
4. The height of a ball thrown vertically upwards from the ground is a quadratic function of its motion time. Xiao Hong throws two balls vertically upwards, 1 second apart. Assuming the two balls are thrown from the same height above the ground, and they reach the same maximum height above the ground 1.1 seconds after ... | 1.6 |
There are $270$ students at Colfax Middle School, where the ratio of boys to girls is $5 : 4$. There are $180$ students at Winthrop Middle School, where the ratio of boys to girls is $4 : 5$. The two schools hold a dance and all students from both schools attend. What fraction of the students at the dance are girls?
$\... | \frac{22}{45} |
Let $A$ be the area of the locus of points $z$ in the complex plane that satisfy $|z+12+9i| \leq 15$. Compute $\lfloor A\rfloor$. | 706 |
Given that "There exists an $x \in \mathbb{R}$ such that $ax^{2}+2ax+1 \leqslant 0$" is a false statement, the range of values for the real number $a$ is $\_\_\_\_\_\_$. | [0,1) |
The lines $y=x$, $y=2x$, and $y=3x$ are the three medians of a triangle with perimeter $1$. Find the length of the longest side of the triangle. | \frac{\sqrt{58}}{\sqrt{34} + 2 + \sqrt{58}} |
33.68 If the octal representation of a perfect square is $a b 3 c$ (i.e., $n^{2}=$ $\left.(a b 3 c)_{8}=8^{3} a+8^{2} b+8 \cdot 3+c\right)$, where $a \neq 0$, then $c$ is
(A) 0.
(B) 1.
(C) 3.
(D) 4.
(E) cannot be uniquely determined.
(33rd American High School Mathematics Examination, 1982) | 1 |
Given the complex number $z$ that satisfies the equation $\frac{z+i}{z}=i$ ($i$ is the imaginary unit), find $\overline{z}=($ $)$.
A: $\frac{1}{2}+\frac{1}{2}i$
B: $\frac{1}{2}-\frac{1}{2}i$
C: $-\frac{1}{2}+\frac{1}{2}i$
D: $-\frac{1}{2}-\frac{1}{2}i$ | A |
There are two arithmetic sequences: 2, 6, 10, ..., 190 and 2, 8, 14, ..., 200. Create a new sequence from the common terms of these two sequences, arranged in ascending order. Find the sum of all terms in the new sequence. | 1472 |
22. Mowgli asked a snake and a tiger what day it was. The snake always lies on Monday, Tuesday and Wednesday but tells the truth otherwise. The tiger always lies on Thursday, Friday and Saturday but tells the truth otherwise. The snake said "Yesterday was one of my lying days". The tiger also said "Yesterday was one of... | Thursday |
The parabola $y=x^{2}-2x-3$ and the coordinate axes intersect at points that lie on the same circle. The equation of the circle determined by these intersection points is __________. | (x-1)^{2}+(y+1)^{2}=5 |
In the plane rectangular coordinate system $xOy$, it is known that the parametric equations of curve $C_{1}$ are $\left\{\begin{array}{l}x=2t\\ y={t}^{2}\end{array}\right.$ (where $t$ is the parameter). Establish a polar coordinate system with the coordinate origin $O$ as the pole and the non-negative $x$-axis as the p... | \frac{\sqrt{3}}{2} |
How many perfect squares are there between 2 and 150 ? | 11 |
10. The sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=\frac{\pi}{6}, a_{n+1}=\arctan \left(\sec a_{n}\right)\left(n \in \mathbf{N}^{*}\right)$. Find the positive integer $m$, such that
$$
\sin a_{1} \cdot \sin a_{2} \cdots \sin a_{m}=\frac{1}{100} .
$$ | 3333 |
## Problem Statement
Calculate the volumes of the bodies bounded by the surfaces.
$$
\frac{x^{2}}{16}+\frac{y^{2}}{9}+\frac{z^{2}}{16}=1, z=2, z=0
$$ | 22\pi |
Ancient China's "The Mathematical Classic of Sunzi" recorded a mathematical problem as follows: "Three people share a car, and two cars are empty; two people share a car, and nine people walk. How many cars are there?" This means "Every 3 people share a car, leaving 2 cars in the end; every 2 people share a car, leavin... | 15 |
1. (17 points) When walking uphill, the tourist walks 3 km/h slower, and downhill 3 km/h faster, than when walking on flat ground. Climbing the mountain takes the tourist 8 hours, while descending the mountain takes 4 hours. What is the tourist's speed on flat ground? | 9 |
4. The number 2090 is written as the product of six different integers. Determine the smallest value of the sum of these numbers. | -33 |
In the syllogism above, the diagonals of a rhombus are equal, a square is a rhombus, so the diagonals of a square are equal. Among the reasoning in the syllogism ( )
A: The major premise is wrong
B: The minor premise is wrong
C: The form of reasoning is wrong
D: The conclusion is wrong | A |
Among the following lines, one that is parallel to the line $x - 2y + 1 = 0$ is ( )
A: $2x - y + 1 = 0$
B: $2x - 4y + 2 = 0$
C: $2x + 4y + 1 = 0$
D: $2x - 4y + 1 = 0$ | D |
Given $|\overrightarrow{a}|=4$, $|\overrightarrow{b}|=3$, and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $120^{\circ}$, calculate:
$(1) (2\overrightarrow{a}-3\overrightarrow{b}) \cdot (2\overrightarrow{a}+\overrightarrow{b})$;
$(2) |\overrightarrow{a}+\overrightarrow{b}|$. | \sqrt{13} |
Let's simplify the following expressions:
$$
\frac{\frac{2 x+1}{x+1}-\frac{2 x-1}{x}}{\frac{2 x-1}{x-1}-\frac{2 x+1}{x}}
$$
$$
\frac{x^{2}+x}{x^{2}+3 x+2}-\frac{x^{2}-3 x+2}{x^{2}-x}+\frac{x^{2}-7 x+12}{x^{2}-5 x+6}
$$ | 1-2(\frac{1}{x+2}-\frac{1}{x}+\frac{1}{x-2}) |
Assume $m$ and $n$ are two different lines in plane $\alpha$; $l_1$ and $l_2$ are two intersecting lines in plane $\beta$. A sufficient but not necessary condition for $\alpha$ to be parallel to $\beta$ is ( )
A. $m \parallel \beta$ and $l_1 \parallel \alpha$
B. $m \parallel l_1$ and $n \parallel l_2$
C. $m \parall... | B |
If the graph of the proportional function $y=-2x$ is shifted up by $3$ units, then the analytical expression of the graph after the shift is ______. | y = -2x + 3 |
23. (USA 4) Find all natural numbers $n$ for which $2^{5}+2^{11}+2^{n}$ is a perfect square. | 12 |
7. In the Cartesian coordinate system $x o y$, the graph of the function $f(x)=a \sin a s+\cos a s(a>0)$ over an interval of the smallest positive period length and the graph of the function $g(x)=\sqrt{a^{2}+1}$ enclose a closed figure. The area of this figure is
保留源文本的换行和格式，翻译结果如下：
7. In the Cartesian coordinate sy... | \frac{2\pi}{}\sqrt{^{2}+1} |
10. Distribute a pile of candies entirely among three children, Jia, Yi, and Bing. The original plan was for Jia, Yi, and Bing to receive candies in the ratio of $5: 4: 3$. In reality, the ratio of candies received by Jia, Yi, and Bing was $7: 6: 5$. One of the children received 15 more candies than originally planned.... | Bing,150 |
1. A and B take turns shooting at a target, with A starting first. Let the hit rates of the two people be $a, b \in (0,1)$. It is known that the probability of both hitting the target for the first time is the same, and once A hits the target, he stops shooting. Then the equation that the real numbers $a, b$ need to sa... | \frac{a}{1-a}=b |
\section*{Task 1 - 041231}
In a mathematical circle, six participants agree on a real number \(a\), which the seventh participant, who had left the room beforehand, is supposed to determine. Upon his return, he receives the following information:
1. \(a\) is a rational number.
2. \(a\) is an integer that is divisible ... | 7 |
13. As shown in Figure 3, in $\triangle A B C$, $A B=1, A C=2$,
$D$ is the midpoint of $B C$, $A E$
bisects $\angle B A C$ and intersects $B C$ at
point $E$, and $D F / / A E$. Find the length of $C F$. | \frac{3}{2} |
# Problem 4.
In-1 The angle at the vertex in the axial section of a cone is $60^{\circ}$. Outside this cone, there are 11 spheres of radius 3, each touching two adjacent spheres, the lateral surface of the cone, and the plane of its base. Find the radius of the base of the cone. | \frac{3}{\sin\frac{\pi}{11}}-\sqrt{3} |
Determine the range of the real number $m$ for which the equation $4\cos x - \cos^2 x + m - 3 = 0$ always has real solutions for $x$. | [0, 8] |
Triangles $OAB,OBC,OCD$ are isoceles triangles with $\angle OAB=\angle OBC=\angle OCD=\angle 90^o$. Find the area of the triangle $OAB$ if the area of the triangle $OCD$ is 12. | 3 |
Example 2.10. $I=\int_{0}^{\pi / 2} \sin ^{3} x \sin 2 x d x$.
| \frac{2}{5} |
5. The number $N$ is written as the product of consecutive natural numbers from 2019 to 4036: $N=2019 \cdot 2020 \cdot 2021 \cdot \ldots \cdot 4034 \cdot 4035 \cdot 4036$. Determine the power of two in the prime factorization of the number $N$.
(points) | 2018 |
6. For any positive integer $n$, define $Z(n)$ as the smallest positive integer $m$ such that $1+2+\cdots+m$ is a multiple of $n$. Regarding the following three propositions:
(1) If $p$ is an odd prime, then $Z(p)=p-1$;
(2) For any positive integer $a$, $Z\left(2^{a}\right)>2^{a}$;
(3) For any positive integer $a$, $Z\... | D |
Example 13. When measuring a part, random errors occur that follow a normal distribution with the parameter $\sigma=10$ mm. Find the probability that the measurement is made with an error not exceeding $15 \mathrm{MM}$. | 0.866386 |
The eleventh question: A certain meeting has 30 participants, each of whom has at most 5 acquaintances among the others, and in any group of 5 people, there exist two who are not acquaintances. Find the maximum positive integer $\mathrm{k}$, such that in the 30 people satisfying the above conditions, there always exist... | 6 |
Determine all real numbers $ a$ such that the inequality $ |x^2 \plus{} 2ax \plus{} 3a|\le2$ has exactly one solution in $ x$. | 1, 2 |
7.50 Given that $n$ planes divide a cube into 300 parts, find the minimum value of $n$.
| 13 |
二元一次方程$2x+3y=8$的正整数解有（ ）
A: $1$个
B: $2$个
C: $3$个
D: 无穷多个 | \text{A} |
## Task 2.
Let $N$ be a natural number. We call a staircase a part of a square plate of dimensions $N \times N$ that consists of the first $K$ fields in the $K$-th row for $K=1,2, \ldots, N$. In how many ways can the staircase be cut into rectangles of different areas that consist of the fields of the given plate? | 2^{N-1} |
22. The numbers $x$ and $y$ satisfy both of the equations
$$
23 x+977 y=2023 \quad \text { and } \quad 977 x+23 y=2977 .
$$
What is the value of $x^{2}-y^{2}$ ?
A 1
B 2
C 3
D 4
E 5 | 5 |
32. If rational numbers $a, b$ satisfy $a+2 b=0$, then $\left|\frac{a}{|b|}-1\right|+\left|\frac{|a|}{b}-2\right|+|| \frac{a}{b}|-3|=$ | 4or6 |
5.090 In a chess match between two teams of 8 people, the participants of the games and the color of the pieces for each participant are determined by a draw. What is the number of different outcomes of the draw? | 2^{8}\cdot8! |
23. How many matches can be obtained from a wooden cube with an edge of 1 m, if each match must have a length of 5 cm, a width of 2 mm, and a height of 2 mm? (Assume that no material is lost to sawing.) | 5000000 |
Given an arithmetic sequence $\left\{ a_n \right\}$ and a geometric sequence with positive terms $\left\{ b_n \right\}$, where $a_1=b_1=1$ and $a_3+a_7=10$, $b_3=a_4$
$(1)$ Find the general formula for sequences $\left\{ a_n \right\}$ and $\left\{ b_n \right\}$
$(2)$ If $c_n=a_n \cdot b_n$, find the sum of the first ... | T_n=(n-1)\times 2^n+1 |
In square $P Q R S, M$ is the midpoint of $P S$ and $N$ is the midpoint of $S R$. If the area of $\triangle S M N$ is 18 , then the area of $\triangle Q M N$ is
(A) 36
(B) 72
(C) 90
(D) 48
(E) 54
$ is ( )
A: $y=|x-1|$
B: $y=\log_{2}x$
C: $y=(x+1)^{2}$
D: $y=\left(\frac{1}{2}\right)^{x}$ | D |
If two lines \\({{l}\_{1}}:ax+2y+6=0,{{l}\_{2}}:x+(a-1)y+3=0\\) are parallel, then \\(a\\) equals \\((\\) \\()\\).
A: \\(\dfrac{2}{3}\\)
B: \\(1\\) or \\(2\\)
C: \\(2\\) or \\(-1\\)
D: \\(-1\\) | \text{D} |
If $|\cos \theta| = \cos \theta$ and $|\tan \theta| = -\tan \theta$, then the terminal side of $\frac{\theta}{2}$ is in ( )
A: The first and third quadrants
B: The second and fourth quadrants
C: The first and third quadrants or on the positive $x$-axis
D: The second and fourth quadrants or on the $x$-axis | \text{D} |
5. There is an unlimited number of chips in six colors. What is the smallest number of chips that need to be arranged in a row so that for any two different colors, there are two adjacent chips of these colors in the row. | 18 |
Given vectors $\overrightarrow{a}=(3,m)$ and $\overrightarrow{b}=(1,-3)$, if vector $\overrightarrow{a} \perp \overrightarrow{b}$, then $|\overrightarrow{a}|=$ _________. | \sqrt{10} |
357. Find \( f^{\prime}(1 / 5) \) if \( f(x)=\operatorname{arctg} 5 x+x^{2} \). | 2.9 |
Example 11 Let real numbers $x, y, z$ satisfy
$$
\left\{\begin{array}{l}
x+y=z-1, \\
x y=z^{2}-7 z+14 .
\end{array}\right.
$$
Question: What is the maximum value of $x^{2}+y^{2}$? For what value of $z$ does $x^{2}+y^{2}$ achieve its maximum value? | 8 |
# Task 4.
In modern conditions, digitalization - the conversion of all information into digital code - is considered relevant. Each letter of the alphabet can be assigned a non-negative integer, called the code of the letter. Then, the weight of a word can be defined as the sum of the codes of all the letters in that ... | 100 |
11. The sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=1$, and $a_{n+1}=\frac{\left(n^{2}+n\right) a_{n}}{3 a_{n}+n^{2}+n}\left(n \in \mathbf{N}^{*}\right)$, then its general term formula $a_{n}=$ | a_{n}=\frac{n}{4n-3}(n\in{N}^{*}) |
## Problem Statement
Find the derivative.
$y=\frac{x^{6}+x^{3}-2}{\sqrt{1-x^{3}}}$ | \frac{9x^{5}}{2\sqrt{1-x^{3}}} |
In [triangle](https://artofproblemsolving.com/wiki/index.php/Triangle) $ABC$, $AB = 13$, $BC = 14$, $AC = 15$. Let $D$ denote the [midpoint](https://artofproblemsolving.com/wiki/index.php/Midpoint) of $\overline{BC}$ and let $E$ denote the intersection of $\overline{BC}$ with the [bisector](https://artofproblemsolving.... | 3 |
## Problem Statement
Find the indefinite integral:
$$
\int \frac{-3 x^{3}+13 x^{2}-13 x+1}{(x-2)^{2}\left(x^{2}-x+1\right)} d x
$$ | -\frac{1}{x-2}-\frac{3}{2}\cdot\ln(x^{2}-x+1)-\sqrt{3}\cdot\operatorname{arctg}(\frac{2x-1}{\sqrt{3}})+C |
Determine the range of the function $f(x)=\arcsin x+\arctan x$. | \left[-\dfrac{3\pi}{4}, \dfrac{3\pi}{4}\right] |
Given the function $f(x)=2(x-1)e^{x}$.
$(1)$ If the function $f(x)$ is monotonically increasing in the interval $(a,+\infty)$, find the range of values for $f(a)$;
$(2)$ Let the function $g(x)=e^{x}-x+p$, if there exists $x_{0} \in [1,e]$ such that the inequality $g(x_{0}) \geqslant f(x_{0})-x_{0}$ holds, find the rang... | [-e,+\infty) |
7. Given that the domain of the function $f(x)$ is $(a, b)$, and $b-a>2$. Then the domain of $F(x)=f(3 x-1)-f(3 x+1)$ is ( ).
(A) $\left(\frac{a-1}{3}, \frac{b+1}{3}\right)$
(B) $\left(\frac{a+1}{3}, \frac{b-1}{3}\right)$
(C) $\left(\frac{a-1}{3}, \frac{b-1}{3}\right)$
(D) $\left(\frac{a+1}{3}, \frac{b+1}{3}\right)$ | B |
6. In tetrahedron $ABCD$, $\angle ABC = \angle BAD = 90^{\circ}$, $AB = 1$, $CD = 2$, and the angle between $AD$ and $BC$ is $30^{\circ}$. Then the volume of the circumscribed sphere of the tetrahedron is $\qquad$. | \frac{13\sqrt{13}}{6}\pi |
Given points $(3,y_1),(\sqrt{10},y_2),(\pi,y_{3})$ all lie on the line $y=-3x+b$, the correct order of $y_{1}$, $y_{2}$, $y_{3}$ is:
A: $y_{2} \lt y_{1} \lt y_{3}$
B: $y_{3} \lt y_{1} \lt y_{2}$
C: $y_{3} \lt y_{2} \lt y_{1}$
D: $y_{2} \lt y_{3} \lt y_{1}$ | \text{D} |
Given a hyperbola $C$: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$, satisfying: (1) The foci are $F_{1}(-5, 0)$ and $F_{2}(5, 0)$; (2) The eccentricity is $\frac{5}{3}$. Find the equation of the hyperbola $C$, $f(x, y) = 0$. If condition (2) is removed and replaced with another condition that still l... | \text{B} |
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