problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
## Task Condition
Find the derivative.
$y=\ln ^{3}(1+\cos x)$ | -\frac{3\cdot\sinx\cdot\ln^{2}(1+\cosx)}{1+\cosx} |
A multi-digit number is called optimistic if its digits increase from left to right. If the digits of the number decrease from left to right, we say that it is a pessimistic number. Find all optimistic numbers for which the following is true:
$\triangleright$ the arithmetic mean of the digits of the given number is 7,... | 59,68,489,579,678 |
Example 5 Let $a, b, c$ be positive real numbers, find the minimum value of
$$\frac{a+3 c}{a+2 b+c}+\frac{4 b}{a+b+2 c}-\frac{8 c}{a+b+3 c}$$ | -17+12 \sqrt{2} |
Among the following functions, the one that is an odd function and has the smallest positive period of $\pi$ is ( )
A: $y=\cos \left( \frac {3\pi}{2}-2x\right)$
B: $y=|\cos x|$
C: $y=\sin \left( \frac {\pi}{2}+2x\right)$
D: $y=|\sin x|$ | A |
Source: 2017 Canadian Open Math Challenge, Problem A2
-----
An equilateral triangle has sides of length $4$cm. At each vertex, a circle with radius $2$cm is drawn, as shown in the figure below. The total area of the shaded regions of the three circles is $a\cdot \pi \text{cm}^2$. Determine $a$.
[center][asy]
size(2.5c... | 2 |
In $2005$ Tycoon Tammy invested $100$ dollars for two years. During the first year
her investment suffered a $15\%$ loss, but during the second year the remaining
investment showed a $20\%$ gain. Over the two-year period, what was the change
in Tammy's investment?
$\textbf{(A)}\ 5\%\text{ loss}\qquad \textbf{(B)}\ 2\%... | 2 |
(1) Calculate the values of $a + a^{-1}$ and $a^2 + a^{-2}$ given that $a^{\frac{1}{2}} + a^{-\frac{1}{2}} = 2$.
(2) Find the range of real numbers $x$ for which $0.2^x < 25$. | (-2, +\infty) |
Given the function $f(x) = \sin(2x + \frac{\pi}{6}) + \cos 2x$, then an interval where $f(x)$ is monotonically decreasing is ( )
A: \[ \frac{\pi}{12}, \frac{7\pi}{12}\]
B: \[- \frac{5\pi}{12}, \frac{\pi}{12}\]
C: \[- \frac{\pi}{3}, \frac{2\pi}{3}\]
D: \[- \frac{\pi}{6}, \frac{5\pi}{6}\] | \text{A} |
Given the function $f(x) = \cos \omega x - 1$ ($\omega > 0$) has exactly $4$ zeros in the interval $[0, 2\pi]$, the range of values for $\omega$ is ______. | [3, 4) |
863. The product of three prime numbers is 5 times their sum. Find all such triples of numbers. | 2,5,7 |
11.088. A sphere, a cylinder with a square axial section, and a cone are given. The cylinder and the cone have the same bases, and their heights are equal to the diameter of the sphere. How do the volumes of the cylinder, the sphere, and the cone relate? | 3:2:1 |
An object moves $8$ cm in a straight [line](https://artofproblemsolving.com/wiki/index.php/Line) from $A$ to $B$, turns at an angle $\alpha$, measured in radians and chosen at random from the interval $(0,\pi)$, and moves $5$ cm in a straight line to $C$. What is the [probability](https://artofproblemsolving.com/wiki/i... | \frac{1}{3} |
## Task 18
$a=9, b=628 ; \quad$ To the sixfold of $a$, add half of $b$. | 368 |
6. If the fraction $\frac{p}{q}\left(p, q \in \mathbf{Z}_{+}\right)$ is converted to a decimal as
$$
\frac{p}{q}=0.198 \cdots,
$$
then when $q$ takes the minimum value, $p+q=$ . $\qquad$ | 121 |
6. Given that a line with slope $k$ is drawn through a focus of the ellipse $x^{2}+2 y^{2}=3$, intersecting the ellipse at points $A$ and $B$. If $AB=2$, then $|k|=$ $\qquad$ | \sqrt{1+\sqrt{3}} |
9. Given real numbers $x, y$ satisfy
$$
(2 x+1)^{2}+y^{2}+(y-2 x)^{2}=\frac{1}{3} \text {. }
$$
Then the value of $x+y$ is . $\qquad$ | -\frac{2}{3} |
4. Find all polynomials with real coefficients such that for every real number $x$ the following holds
$$
(1+2 x) P(2 x)=\left(1+2^{1000} x\right) P(x)
$$ | P(x)=a_{0}(x+\frac{1}{2})(x+\frac{1}{2^{2}})\cdot\ldots\cdot(x+\frac{1}{2^{999}}) |
8.5. Given a point $A$ and a circle $S$. Draw a line through point $A$ such that the chord cut off by the circle $S$ on this line has a given length $d$. | R^{2}-^{2}/4 |
In a trapezoid $ABCD$ whose parallel sides $AB$ and $CD$ are in ratio $\frac{AB}{CD}=\frac32$, the points $ N$ and $M$ are marked on the sides $BC$ and $AB$ respectively, in such a way that $BN = 3NC$ and $AM = 2MB$ and segments $AN$ and $DM$ are drawn that intersect at point $P$, find the ratio between the areas of t... | \frac{4}{25} |
Find all triplets of positive integers $a$, $b$ and $c$ such that $a \geq b \geq c$ and
$\left(1+\frac{1}{a}\right)\left(1+\frac{1}{b}\right)\left(1+\frac{1}{c}\right)=2$ | \{(7, 6, 2), (9, 5, 2), (15, 4, 2), (8, 3, 3), (5, 4, 3)\} |
12 Let $A$ be an $m \times m$ matrix. Let $X_{i}$ be the set of entries in the $i$th row and $Y_{j}$ the set of entries in the $j$th column, $1 \leq i, j \leq m$. $A$ is called cool if the sets $X_{1}, \ldots, X_{m}, Y_{1}, \ldots, Y_{m}$ are all different. Determine the smallest value for $n$ so that a cool $2005 \tim... | 13 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $c^{2}=(a-b)^{2}+6$ and $C= \frac {\pi}{3}$, find the area of $\triangle ABC$. | \frac {3 \sqrt {3}}{2} |
Given lines $m$, $n$ and plane $α$, if $n⊥α$, then "$m⊂α$" is a "($\quad\quad$)" of "$n⊥m$".
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Necessary and sufficient condition
D: Neither necessary nor sufficient condition | A |
# 5. Option 1.
In the district, there are three villages A, B, and C connected by dirt roads, with any two villages being connected by several (more than one) roads. Traffic on the roads is two-way. We will call a path from one village to another either a road connecting them or a chain of two roads passing through a ... | 26 |
The domain of the function $f(x) = \log_{3}(2x^{2}-8x+m)$ is $\mathbb{R}$, then the range of $m$ is. | m > 8 |
Let $f(x)= \frac {4^{x}}{4^{x}+2}$. Then, $f\left( \frac {1}{11}\right)+f\left( \frac {2}{11}\right)+f\left( \frac {3}{11}\right)+\ldots+f\left( \frac {10}{11}\right)=$ ()
A: $4$
B: $5$
C: $6$
D: $10$ | B |
If natural numbers $a$, $b$, $c$ satisfy the equation $a^2 + b^2 = c^2$, then there are at least $\boxed{1}$ even number among $a$, $b$, $c$. | 1 |
## Problem Statement
Calculate the indefinite integral:
$$
\int \frac{2 x^{3}-6 x^{2}+7 x-4}{(x-2)(x-1)^{3}} d x
$$ | 2\cdot\ln|x-2|-\frac{1}{2(x-1)^{2}}+C |
12. In 2019, the Mathematics Olympiad underwent a trial reform: A city held 5 joint competitions in the second year of high school. A student who ranks in the top 20 of the city in 2 out of these 5 competitions can enter the provincial team training and does not need to participate in the remaining competitions. Each s... | \frac{269}{64} |
The ordinary equation of the curve corresponding to the equation $$\begin{cases} \overset{x=t+1}{y=3-t^{2}}\end{cases}$$ (t is a parameter, t∈R) is ________ | y = -x^2 + 2x + 2 |
5. Use $0,1, \cdots, 9$ to form a ten-digit number without repeated digits that is divisible by 11. Find the largest and smallest numbers among them. | 9876524130, 1024375869 |
Among the following propositions, the correct one is\\((\\) \\()\\)
A: For categorical variables \\(X\\) and \\(Y\\), the larger the observed value \\(k_{0}\\) of the random variable \\(K^{2}\\), the lower the confidence in judging that "\\(X\\) and \\(Y\\) are related".
B: Proposition \\(p\\): \\(\exists x_{0} > 0\\)... | B |
5. Find the smallest positive integer $n$ that satisfies $\tau(n)=6$.
| 12 |
1. Given $a b c=1$, let
$$
\begin{aligned}
M & =\frac{1}{a+a b+1}+\frac{1}{b+b c+1}+\frac{1}{c+c a+1}, \\
N & =\frac{a}{a+a b+1}+\frac{b}{b+b c+1}+\frac{c}{c+c a+1} .
\end{aligned}
$$
Then the relationship between $M$ and $N$ is ( ).
(A) $M>N$
(B) $M=N$
(C) $M<N$
(D) Cannot be determined | B |
$32 \cdot 64$ A triangle has integer side lengths and area, one side length is 21, and the perimeter is 48, then the length of the shortest side is
(A) 8.
(B) 10.
(C) 12.
(D) 14.
(E) 16.
(14th American High School Mathematics Examination, 1963) | 10 |
In an isosceles trapezoid $ABCD$, the base $AB$ is twice as large as the base $CD$. Point $M$ is the midpoint of $AB$. It is known that the center of the circle inscribed in the triangle $MCB$ lies on the circle circumscribed around the triangle $MDC$. Find the angle $\angle MBC$.
[img]https://cdn.artofproblemsolving.c... | 60^\circ |
A strandon néhány doboz mokka kockacukor kiömlött a homokba. Jóska egy olyan téglatestet rakott össze kis cukorkockákból, melynek alapján az egyik irányban 1-gyel, a másikban 2-vel több sor volt, mint ahány réteg volt egymás fölött; Laci pedig kockát rakott össze, így minden cukrot fölhasználtak. Jóska kérte Lacit, adj... | 504216 |
The base plane, axis, and apex of two circular cones are common, the ratio of their aperture angles is $1: 2$. The two generators of the larger aperture angle cone form an angle $\alpha$. Project these generators perpendicularly onto the base plane, the angle of the projections is $\beta$. Determine the aperture angle ... | \cosx=\frac{1}{2}(1+\cos\beta+\sqrt{(1-\cos\beta)(\cos\alpha-\cos\beta)}) |
A factory's product qualification rate is 98%. The inspection unit randomly selects 10 pieces with replacement. Calculate the variance of the number of qualified products among the 10 pieces selected. | 0.196 |
Given a sequence $ \{a_n\} $ whose sum of the first $n$ terms is $S_n=5n^2+kn-19 (n\in \mathbf{N}^*)$, and $a_{10}=100$, then $k=$
A: $5$
B: $6$
C: $-5$
D: $-6$ | \text{A} |
4. Ministers in Anchuria. In the government cabinet of ministers in Anchuria, there are 100 ministers. Among them, there are crooks and honest ministers. It is known that among any ten ministers, at least one minister is a crook. What is the smallest number of crook ministers that can be in the cabinet? | 91 |
The minimum value of the function $f(x)=2\sin^{2}\left(\frac{\pi}{4}+x\right)-\sqrt{3}\cos 2x \left(\frac{\pi}{4} \leqslant x \leqslant \frac{\pi}{2}\right)$ is ______. | 2 |
315. Two of the angles of a trapezoid are $60^{\circ}$ and $130^{\circ}$. What can be said about its other angles? | 120 |
What is the value of $(2^0-1+5^2-0)^{-1}\times5?$
$\textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25$ | \frac{1}{5} |
A2. For which values of the parameter $a \in \mathbb{R}$ is the function $f(x)=\log \left(x^{2}+(a+4) x+9\right)$ defined on the set of all real numbers?
(A) $a-10$
(C) $-2<a<10$
(D) $-10<a<2$
(E) $a=-4$ | -10<<2 |
On the leg $A C$ of the right triangle $A B C$, a circle is constructed with $A C$ as its diameter, which intersects the hypotenuse $A B$ at point $K$. Find the area of triangle $C K B$, if the leg $A C$ is equal to $b$, and the angle $A B C$ is equal to $\beta$. | \frac{1}{2}b^2\cos^2\beta\cot\beta |
Example 8 As shown in Figure 8, given that $O$ and $I$ are the circumcenter and incenter of $\triangle ABC$ respectively, $AD$ is the altitude on side $BC$, and point $I$ lies on segment $OD$. Prove: the circumradius $R$ of $\triangle ABC$ equals the exradius $r_{a}$ on side $BC$.
(1998, National High School Mathematic... | R = r_{a} |
6. In the quadrilateral pyramid $P-ABCD$, it is known that $AB // CD$, $AB \perp AD$, $AB=4$, $AD=2\sqrt{2}$, $CD=2$, $PA \perp$ plane $ABCD$, $PA=4$. Let $Q$ be a point on the line segment $PB$, and the sine of the angle formed by line $QC$ and plane $PAC$ is $\frac{\sqrt{3}}{3}$. Then
$\frac{PQ}{PB}$ is $\qquad$ | \frac{7}{12} |
Given an ellipse C: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$$ (a>b>0) with left and right foci $F_1$ and $F_2$, respectively. A line $l$ passing through point $F_2$ intersects C at points A and B. The perimeter of △$ABF_1$ is 20, and the eccentricity of C is $$e= \frac {4}{5}$$.
(I) Find the equation of C.
(II)... | \frac {25}{4} |
5. (10 points) If $\frac{O}{11}<\frac{7}{\square}<\frac{4}{5}$, then the maximum sum of the non-zero natural numbers that can be filled in “ $O$ ” and “ $\square$ ” is
保留源文本的换行和格式，翻译结果如下：
5. (10 points) If $\frac{O}{11}<\frac{7}{\square}<\frac{4}{5}$, then the maximum sum of the non-zero natural numbers that can be f... | 77 |
Problem 3.3. Several letters A and several letters B were sitting on a pipe. After some A fell and some B disappeared, only three letters remained on the pipe, and the following dialogue occurred between them:
First letter: “I am the only one of my kind here.”
Second letter: “There are definitely fewer than two A let... | B,A,B |
5. What is the least number greater than 1992 that gives a remainder of $7$ when divided by $9$? | 1996 |
34. A bag of fruit contains 10 fruits, with an even number of apples, at most two oranges, a multiple of 3 bananas, and at most one pear. How many types of these bagged fruits are there? | 11 |
If $$- \frac {\pi}{8} < \theta < 0$$, then the order of magnitude for $\sin\theta$, $\cos\theta$, $\tan\theta$ is ( )
A: $\sin\theta < \cos\theta < \tan\theta$
B: $\sin\theta < \tan\theta < \cos\theta$
C: $\tan\theta < \sin\theta < \cos\theta$
D: None of the above | \text{C} |
Example 2 Let $S=\{1,2, \cdots, n\}, A$ be an arithmetic sequence with at least two terms and a positive common difference, all of whose terms are in $S$, and adding any other element of $S$ to $A$ does not form an arithmetic sequence with the same common difference as $A$. Find the number of such $A$ (here, a sequence... | [\frac{n^{2}}{4}] |
## Task 1
In 5 windows of a residential building, there are a total of 40 glass panes.
How many panes are in 3 windows? | 24 |
Given a circle $C: (x+1)^2 + (y-2)^2 = 2$, a point $P(x_1, y_1)$ outside the circle is connected to the circle by a tangent line at point $M$. $O$ is the origin of the coordinates, and $|PM| = |PO|$. Find the coordinates of point $P$ when $|PM|$ reaches its minimum value. | \left(-\frac{3}{10}, \frac{3}{5}\right) |
In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $\triangle ABC$ is an obtuse triangle and $a=6$, $b=8$, then the impossible value for $c$ is:
A: $3$
B: $4$
C: $9$
D: $12$ | C |
1. Calculate: $143 \cdot 91 \cdot 77-\sqrt{143} \cdot \sqrt{91} \cdot \sqrt{77}$. | 1001000 |
25.2 The graphs of the algebraic expressions $2 x+3 y-6=0, 4 x-3 y-6=0, x=2$ and $y=$ $\frac{2}{3}$ intersect at
(A) 6 points.
(B) 1 point.
(C) 2 points.
(D) no points.
(E) an unlimited number of points.
(6th American High School Mathematics Examination, 1955) | B |
Evaluate the sum \[ 11^2 - 1^1 + 12^2 - 2^2 + 13^2 - 3^2 + \cdots + 20^2 - 10^2. \] | 2100 |
## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0}\left(\frac{\ln (1+x)}{6 x}\right)^{\frac{x}{x+2}}
$$ | 1 |
3. If the function $y=k x(k>0)$ intersects with the function $y=\frac{1}{x}$ at points $A$ and $C$, and $A B$ is perpendicular to the $x$-axis at point $B$. Then the area of $\triangle A B C$ is ( ).
(A) 1
(B) 2
(C) $k$
(D) $k^{2}$ | A |
If $x \lt y$, which of the following inequalities is true?
A: $x+5 \lt y+5$
B: $-2x \lt -2y$
C: $a^{2}x \gt a^{2}y$
D: $x-1 \gt y-1$ | A |
Example 5. Find $\lim _{x \rightarrow-1} \frac{3 x^{4}+2 x^{3}-x^{2}+5 x+5}{x^{3}+1}$. | \frac{1}{3} |
# Task 7. (14 points)
Several businessmen decided to open a company and divide all the profits into equal parts. One of the businessmen was appointed as the director. One day, this director of the company transferred part of the profit from the company's account to his own personal account. This amount of money was th... | 19 |
Given that $-1$ is a root of the equation $(a+1)x^2 + x - 1 = 0$, find the value of $a$ and the other root. | \frac{1}{2} |
Given that $a > 2b$ ($a, b \in \mathbb{R}$), the range of the function $f(x) = ax^2 + x + 2b$ is $[0, +\infty)$. Determine the minimum value of $$\frac{a^2 + 4b^2}{a - 2b}$$. | \sqrt{2} |
1. Find the sum of all natural numbers $n$ that are multiples of three and for which the number of divisors (including 1 and $n$ itself) is equal to $\frac{n}{3}$. (For example, the number 12 has 6 divisors: $1,2,3,4,6,12$.) | 51 |
Given the function $f(x)=\sin(\omega x+\frac{π}{4})$ where $\omega \in \mathbb{N}$ has exactly one zero in the interval $\left[0,\pi \right)$, find the value of $\omega$. | 1 |
Given the inequality $x^{2}-ax+a\geqslant 0$ holds for any real number $x$, find the maximum value of $a$. | 4 |
In the complex plane, the distance between the point corresponding to the complex number $\frac{2}{1+i}i$ (where $i$ is the imaginary unit) and the origin is $\_\_\_\_\_\_\_.$ | \sqrt{2} |
## Problem Statement
Find the point of intersection of the line and the plane.
$\frac{x-3}{1}=\frac{y-1}{-1}=\frac{z+5}{0}$
$x+7 y+3 z+11=0$ | (4;0;-5) |
2. Solve the equation $\left(\frac{x}{400}\right)^{\log _{5}\left(\frac{x}{8}\right)}=\frac{1024}{x^{3}}$. | \frac{8}{5},16 |
Given the function $$f(x)=\sin(x+ \frac {\pi}{2})\cos(x- \frac {\pi}{3})$$.
(1) Find the smallest positive period of the function $f(x)$.
(2) The graph of the function $y=f(x)$ is translated downwards by $$\frac {1}{4}$$ units, and then the vertical coordinates of each point on the graph are stretched to 4 times their ... | 2 |
The number of zeros of the function $f(x) = \ln x + 2x - 1$ is ( )
A: 4
B: 3
C: 2
D: 1 | \text{D} |
2. The gnomes have gone to work, and Snow White is feeling lonely. She laid out a pile of fifteen stones on the table. Every minute, Snow White splits one pile into two non-empty piles and adds a stone to one of them. How can Snow White use these actions to get seven identical piles? | 3333333 |
In the expansion of the binomial ${(\sqrt{3}x+\frac{1}{{\sqrt[3]{x}}})^n}$, only the binomial coefficient of the $11$th term is the largest. The number of terms in the expansion where the exponent of $x$ is an integer is ( ).
A: $3$
B: $5$
C: $6$
D: $7$ | \text{D} |
8・133 Let $x_{n}=(1+\sqrt{2}+\sqrt{3})^{n}, n=1,2,3, \cdots$ If $x_{n}$ is expressed as $x_{n}=q_{n}+r_{n} \sqrt{2}+s_{n} \sqrt{3}+t_{n} \sqrt{6}$, where $q_{n}, r_{n}, s_{n}$, and $t_{n}$ are all integers, find the following limits:
$$
\lim _{n \rightarrow \infty} \frac{r_{n}}{q_{n}}, \lim _{n \rightarrow \infty} \fra... | \frac{1}{\sqrt{2}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{6}} |
The coefficient of $x^2$ in the expansion of $(2x+1)^6$ is ( )
A: 30
B: 120
C: 60
D: 15 | C |
Given that the domain of functions $f(x)$ and $g(x)$ is $\{1, 2, 3\}$, and it satisfies $f(1)=f(3)=1$, $f(2)=3$, $g(x)+x=4$, find the value(s) of $x$ for which $f[g(x)]>g[f(x)]$. | 2 |
9.41 Let $a_{1}, a_{2}, \cdots, a_{n}$ be given real numbers, not all zero. If real numbers $r_{1}$, $r_{2}, \cdots, r_{n}$ satisfy the inequality
$$
\sum_{k=1}^{n} r_{k}\left(x_{k}-a_{k}\right) \leqslant\left(\sum_{k=1}^{n} x_{k}^{2}\right)^{\frac{1}{2}}-\left(\sum_{k=1}^{n} a_{k}^{2}\right)^{\frac{1}{2}}
$$
for any ... | r_{i}=\frac{a_{i}}{\sqrt{\sum_{k=1}^{n}a_{k}^{2}}},i=1,2,\cdots,n |
A truck started from city $A$ to city $B$ at midnight, and a car started from city $B$ to city $A$ (on the same route) at $t_{1}$. They met at $t_{2}$. The car arrived at its destination $r$ hours later than the truck. - After finishing their business, they started back, and met again at $t_{3}$, and finally arrived ho... | 16\frac{2}{5} |
Given that the radius of circle $\odot O_{1}$ is $1$, the radius of circle $\odot O_{2}$ is $r$, and the distance between their centers is $5$. If there exists a point $P$ on circle $\odot O_{2}$ such that $PO_{1}=2$, then the range of $r$ is ______. | 3\leqslant r\leqslant 7 |
If $|a-1|+|b+2|=0$, then $(a+b)^{2013}+|b|=$ ? | 1 |
The one-variable quadratic equation about $x$ is $x^{2}+\left(2k+1\right)x+k^{2}+1=0$ with two distinct real roots $x_{1}$ and $x_{2}$.
$(1)$ Find the range of real number $k$.
$(2)$ If the two real roots $x_{1}$ and $x_{2}$ of the equation satisfy $x_{1}+x_{2}=-x_{1}\cdot x_{2}$, find the value of $k$. | k = 2 |
6. As shown in Figure 1, in the tetrahedron $A B C D$, $D A=D B=D C$ $=2, D A \perp D B, D A \perp$ $D C$, and the cosine of the angle between $D A$ and the plane $A B C$ is $\frac{\sqrt{6}}{3}$. Then the radius $R$ of the circumscribed sphere of the tetrahedron is . $\qquad$ | \sqrt{3} |
7. In tetrahedron $ABCD$, $AD > AB$, $AD \perp AB$, $AD \perp AC$, $\angle BAC = \frac{\pi}{3}$. Let the areas of $\triangle ADB$, $\triangle ADC$, $\triangle ABC$, and $\triangle BCD$ be $S_{1}$, $S_{2}$, $S_{3}$, and $S_{4}$, respectively, and they satisfy $S_{1} + S_{2} = S_{3} + S_{4}$. Then $\frac{S_{3}}{S_{1}} + ... | \frac{3}{2} |
## Task Condition
Find the derivative.
$$
y=x\left(2 x^{2}+5\right) \sqrt{x^{2}+1}+3 \ln \left(x+\sqrt{x^{2}+1}\right)
$$ | 8\sqrt{(x^{2}+1)^{3}} |
In the arithmetic sequence $\{a\_n\}$, $a\_2=4$, $a\_4+a\_7=15$,
(1) Find the general term formula $a\_n$ of the sequence;
(2) If $\{b\_n=\frac{a\_1+a\_2+ \cdots +a\_n}{n}\}$, let $c\_n=a\_n+b\_n$, find the sum of the first $n$ terms $T\_n$ of the sequence $\{c\_n\}$. | \frac{3n^2+21n}{4} |
Given that $F\_1$ and $F\_2$ are the left and right foci of the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$, and the moving point $P$ is on the ellipse, find the maximum value of $|PF\_1| \cdot |PF\_2|$. | 25 |
The solution set of the inequality $-x^2+5x>6$ is ______. | (2, 3) |
11. (5 points)
Xiao Huhu encountered a problem: Compare the sizes of $\frac{99}{100}, \frac{100}{101}, \frac{199}{201}$. He is very confused, please help him find the correct answer.
A. $\frac{99}{100}>\frac{100}{101}>\frac{199}{201}$
B. $\frac{199}{201}>\frac{100}{101}>\frac{99}{100}$
C. $\frac{99}{100}>\frac{199}{201... | D |
In a cylindrical measuring cup placed horizontally with a base radius of $\sqrt{3}$ cm, there is a certain amount of water. Now, a solid iron ball with a radius of $R$ cm is placed into the cup. The ball is completely submerged in the water without any water spilling out. If the water level rises exactly by $R$ cm, the... | \frac{3}{2} |
3.335.
$$
\frac{\cos 64^{\circ} \cos 4^{\circ}-\cos 86^{\circ} \cos 26^{\circ}}{\cos 71^{\circ} \cos 41^{\circ}-\cos 49^{\circ} \cos 19^{\circ}}=-1
$$ | -1 |
## SUBJECT I
Calculate:
$$
\mathrm{a}=\left\{\left[2^{3} \cdot 5^{2}+\left(25^{50}: 5^{99}+2^{2} \cdot 3\right) \cdot 5^{2}\right]: 5^{3}+2^{7}+11^{1991}:\left(11^{2}\right)^{995}\right\}:\left(3^{3}+3^{2}\right)
$$ | 4 |
3. In a dark room, a drawer contains socks of two colors and two sizes, 4 pairs in total (one pair of each color and size). How many socks must be taken out to ensure that there are two pairs of different colors and different sizes? | 7 |
1. Find the locus of the intersection point of two perpendicular intersecting tangents to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$. | x^{2}+y^{2}=a^{2}+b^{2} |
74. A school uses a drug fumigation method to disinfect classrooms. It is known that when the drug is burning, the content of the drug $y(\mathrm{mg})$ per cubic meter of air is directly proportional to the time $x$ (min); after the drug burns out, $y$ is inversely proportional to $x$ (as shown in the figure). It is me... | 30 |
The value of $\sin 960^\circ$ is ________. | -\frac{\sqrt{3}}{2} |
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