problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Given an odd function $f$ with domain $\mathbb{R}$ satisfying $f(x+4)=f(x)$, and for $x \in (0,2)$, $f(x)=\ln(x^2+a)$ where $a>0$. If the function $f(x)$ has 9 zeros in the interval $[-4,4]$, find the range of values for the real number $a$. | (0,1) |
The ratio of the volume of a cube's circumscribed sphere to the volume of its inscribed sphere is \(\quad\)
A: \(\sqrt{3} : 1\)
B: \(3 : 1\)
C: \(3\sqrt{3} : 1\)
D: \(9 : 1\) | C |
Given the general term formula of the sequence $\{a\_n\}$, which is $a\_n=(-1)^{n}+n$, write out the first 5 terms of the sequence $\{a\_n\}$. | 4 |
Example 1 Let $a, b, c > 0$, and $a+b+c=1$. Prove:
$$
\sqrt{a^{1-a} b^{1-b} c^{1-c}} \leqslant \frac{1}{3} \text {. }
$$
(2008, Australian Mathematical Olympiad) | \frac{1}{3} |
If the function $y=f(x)$ is the inverse function of $y=2^x$, then the value of $f(2)$ is ( )
A: 4
B: 2
C: 1
D: 0 | \text{C} |
6.172. $\left(x+\sqrt{x^{2}-1}\right)^{5} \cdot\left(x-\sqrt{x^{2}-1}\right)^{3}=1$. | x_{1}=1,x_{2}=-1 |
Let $D$ be a point in the plane of $\triangle ABC$, and $\overrightarrow {BC}=5 \overrightarrow {CD}$. If $\overrightarrow {AB}=x \overrightarrow {AC}+y \overrightarrow {AD}$, then $x+2y=$ \_\_\_\_\_\_. | -4 |
One square is inscribed in a circle, and another square is circumscribed around the same circle so that its vertices lie on the extensions of the sides of the first (see figure). Find the angle between the sides of these squares.
[img]https://3.bp.blogspot.com/-8eLBgJF9CoA/XTodHmW87BI/AAAAAAAAKY0/xsHTx71XneIZ8JTn0iDMHu... | 15^\circ |
If the variance of a set of data $a_1, a_2, a_3, a_4, a_5, a_6$ is 2, then the variance of another set of data $2a_1, 2a_2, 2a_3, 2a_4, 2a_5, 2a_6$ is ( )
A: 2
B: 6
C: 8
D: -2 | \text{C} |
4. The sequence $\left\{x_{n}\right\}$ is defined as follows: $x_{1}=\frac{1}{2}, x_{k+1}=x_{k}^{2}+x_{k}, k=1,2, \cdots$. Find the integer part of the following sum:
$$
\frac{1}{1+x_{1}}+\frac{1}{1+x_{2}}+\cdots+\frac{1}{1+x_{2005}}
$$ | 1 |
Given the complex number $z=-2i+\frac{3-i}{i}$, then the point corresponding to the conjugate complex number $\bar{z}$ in the complex plane is in
A: The first quadrant
B: The second quadrant
C: The third quadrant
D: The fourth quadrant | \text{B} |
The zero point of the function $f\left(x\right)=\ln x+2x-5$ is $x_{0}$, and $x_{0}\in \left(k,k+1\right)$, where $k\in Z$. What is the value of $k$?
A: $0$
B: $1$
C: $2$
D: $3$ | C |
If $\tan(\pi -\alpha)=-2$, then $\frac{cos(2\pi-\alpha)+2cos(\frac{3\pi}{2}-\alpha)}{sin(\pi-\alpha)-sin(-\frac{\pi}{2}-\alpha)}=$____. | -1 |
2. Given a regular triangular prism with a base edge length of 1, and the diagonals of the two non-adjacent side faces are perpendicular to each other. Then the length of its lateral edge is ( ).
(A) $\sqrt{2}$
(B) $\frac{\sqrt{2}}{2}$
(C) 2
(D) $\frac{\sqrt{3}}{2}$ | B |
Due to a traffic accident, a certain liquid on a truck leaked into a pond. In order to decontaminate, following the environmental protection department's suggestion, a chemical agent that reacts with the polluted liquid will be released into the pond. It is known that for every $a$ (where $1 \leq a \leq 4$, $a \in \mat... | a_{\text{min}} = 24-16 \sqrt {2} |
In the three-dimensional Cartesian coordinate system $O-xyz$, the set of all points for which $z=1$ forms what kind of figure? Find the distance from point $P(2, 3, 5)$ to the plane $xOy$. | 5 |
Example 2. Find the integral $\int \sqrt{x+4} d x$. | \frac{2}{3}(x+4)\sqrt{x+4}+C |
89. Calculate the value of the expression $\frac{2 a-b}{3 a-b}+\frac{5 b-a}{3 a+b}$, given that $10 a^{2}-3 b^{2}+5 a b=0$ and $9 a^{2}-b^{2} \neq 0$. | -3 |
The coordinates of the point symmetric to the origin (0, 0) with respect to the line $x-2y+2=0$ are \_\_\_\_\_\_. | \left(-\frac {4}{5}, \frac {8}{5}\right) |
What is the value of $123456^{2}-123455 \times 123457$ ? | 1 |
In the arithmetic sequence $\left\{a_{n}\right\}$ and the geometric sequence $\left\{b_{n}\right\}$, $a_{1}=b_{1}=1$, $a_{2}=b_{2}$, and $2+a_{4}=b_{3}$.
1. Find the general formulas for $\left\{a_{n}\right\}$ and $\left\{b_{n}\right}$.
2. Find the sum $S_{n}$ of the first $n$ terms of the sequence $\left\{a_{n}+b_{n}... | S_{n} = n^{2} + \frac{3^{n} - 1}{2} |
Given the function $f(x)=2x^{2}-ax+\ln x$ is not monotonic in its domain, find the range of values for the real number $a$. | (4,+\infty) |
## Task 3 - 180823
Determine all two-digit natural numbers with the following property:
If you add $2$ to the desired number, you get three times the number that results from swapping the digits of the original number. | 82 |
Given that the equation of circle $C$ is $x^{2}+y^{2}=1$, and $P$ is a point on the hyperbola $\frac{x^{2}}{4}-\frac{y^{2}}{9}=1$. Draw two tangent lines from $P$ to the circle, with the points of tangency being $A$ and $B$. Determine the range of values for $\overrightarrow{PA}\cdot\overrightarrow{PB}$.
A: $[0,\frac{... | \text{B} |
Convert the logarithmic equation $\log_{5}20=a$ into an exponential equation. | 5^{a}=20 |
$$
\text { 1. Given } \sqrt[3]{a}+a=\sqrt{6}, b^{3}+b=\sqrt{6} \text {. Then } a+b
$$ | a+b=\sqrt{6} |
8. Generate a random integer $x \in\{1,2, \cdots, 2016\}$ with equal probability, then the probability that the sum of the digits of $x$ in binary is no more than 8 is . $\qquad$ | \frac{655}{672} |
Find all integers $a$ and $b$ such that $7^{a}-3 \times 2^{b}=1$. | (1,1)(2,4) |
In the rectangular coordinate system $xOy$, if the distance from the focus of the hyperbola $x^{2}- \frac{y^{2}}{b^{2}}=1(b > 0)$ to its asymptote is equal to the distance from the point $M(1,2)$ on the parabola $y^{2}=2px$ to its focus, then the real number $b=$ _____ . | 2 |
Let $S$ be a subset of $\{1, 2, . . . , 500\}$ such that no two distinct elements of S have a
product that is a perfect square. Find, with proof, the maximum possible number of elements
in $S$. | 306 |
10. Let $F_{1}$ and $F_{2}$ be the left and right foci of the hyperbola
$$
\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)
$$
respectively. If there exists a point $P$ on the right branch of the hyperbola such that
$$
\left(\overrightarrow{O P}+\overrightarrow{O F_{2}}\right) \cdot \overrightarrow{F_{2} P}=0 \text... | B |
# Problem 10.
Set $A$ on the plane $O x y$ is defined by the equation $x^{2}+y^{2}=2 x+2 y+23$. Set $B$ on the same plane is defined by the equation $|x-1|+|y-1|=5$. Set $C$ is the intersection of sets $A$ and $B$. What is the maximum value that the product of the lengths of $n$ segments $X Y_{1} \cdot X Y_{2} \cdot X... | 1250 |
Farmer James has three types of cows on his farm. A cow with zero legs is called a $\textit{ground beef}$, a cow with one leg is called a $\textit{steak}$, and a cow with two legs is called a $\textit{lean beef}$. Farmer James counts a total of $20$ cows and $18$ legs on his farm. How many more $\textit{ground beef}$s ... | 2 |
$\sin (-600^{\circ})=$ ()
A: $\dfrac {1}{2}$
B: $\dfrac { \sqrt {3}}{2}$
C: $- \dfrac {1}{2}$
D: $- \dfrac { \sqrt {3}}{2}$ | B |
2. On the coordinate plane, lines of the form $y=a x+b$ are drawn, where $a$ and $b$ are natural numbers from 1 to 8. Among all the intersection points of these lines, select the point with the greatest sum of coordinates. Write the answer in the form ( $x ; y$ ). | (7,57) |
8.2. The work was divided equally among the workers in the team. After the first day, they counted how many people had completed no less than 30 percent of their share; it turned out that 70 percent of all workers had done so. When they counted only those who had completed no less than 70 percent of their share, it tur... | No |
Read the example and answer the question:Example: Let $a$ and $b$ be rational numbers, satisfying $a+\sqrt{2}b=3-2\sqrt{2}$. Find the value of $a+b$.Solution: From the given conditions, we have $a-3+\left(b+2\right)\sqrt{2}=0$. Since $a$ and $b$ are rational numbers, $a-3$ and $b+2$ are also rational numbers. Therefore... | 64 \text{ or } -64 |
## Task 4 - 170824
Dieter tells his classmates:
"My brother Fritz is only half as old as I am. If you multiply the number of his years by itself, you get my father's age. My mother is three years younger than my father. All together, we are 87 years old."
Determine the age of all 4 people! (Only the completed years ... | 36,6,12,33 |
6.1. (14 points) In a hut, several residents of the island gathered, some from the Ah tribe, and the rest from the Ukh tribe. The residents of the Ah tribe always tell the truth, while the residents of the Ukh tribe always lie. One of the residents said: "There are no more than 16 of us in the hut," and then added: "We... | 15 |
Find the smallest positive integer $k$ such that \[(16a^2 + 36b^2 + 81c^2)(81a^2 + 36b^2 + 16c^2) < k(a^2 + b^2 + c^2)^2,\] for some ordered triple of positive integers $(a,b,c)$. | k = 1297 |
## Task Condition
Calculate the area of the parallelogram constructed on vectors $a$ and $b$.
\[
\begin{aligned}
& a=p+2 q \\
& b=3 p-q \\
& |p|=1 \\
& |q|=2 \\
& (\widehat{p, q})=\frac{\pi}{6}
\end{aligned}
\] | 7 |
The vertex of angle $\alpha$ is at the origin, its initial side coincides with the positive x-axis, and its terminal side passes through the point $(-1, -2)$. Then, $\sin\alpha$ is \_\_\_\_\_\_. | -\frac{2\sqrt{5}}{5} |
Given $0<x<\frac{1}{3}$, the maximum value of $x(1-3x)$ is ______. | \frac{1}{12} |
(R. Henner) Determine all triples $(a, b, c)$ of positive integers for which the conditions
$$
\begin{aligned}
\operatorname{gcd}(a, 20) & =b, \\
\operatorname{gcd}(b, 15) & =c \quad \text { and } \\
\operatorname{gcd}(a, c) & =5
\end{aligned}
$$
hold. | {(20,20,5),(20-10,10,5),(10-5,5,5)\mid |
Find the smallest possible $\alpha\in \mathbb{R}$ such that if $P(x)=ax^2+bx+c$ satisfies $|P(x)|\leq1 $ for $x\in [0,1]$ , then we also have $|P'(0)|\leq \alpha$. | \alpha = 8 |
Given that a non-zero common difference arithmetic sequence $\{a_n\}$ has a sum of its first 4 terms equal to 10, and $a_2$, $a_3$, and $a_7$ form a geometric sequence,
(I) Find the general term $a_n$ formula.
(II) Let $b_n = 2^{a_n}$, determine the sum of the first $n$ terms of the sequence $\{b_n\}$, $S_n$. | \frac{8^n - 1}{28} |
4.070 It is known that for any $n$ the sum of the first $n$ terms of a certain numerical sequence is expressed by the formula $S_{n}=2 n^{2}+3 n$. Find the tenth term of this sequence and prove that this sequence is an arithmetic progression. | 41 |
Problem 9.5. Let $A B C D$ be a cyclic quadrilateral with circumcircle $k$. The rays $\overrightarrow{D A}$ and $\overrightarrow{C B}$ meet at point $N$ and the line $N T$ is tangent to $k, T \in k$. The diagonals $A C$ and $B D$ meet at the centroid $P$ of $\triangle N T D$. Find the ratio $N T: A P$.
Ivailo Kortezov | 3 |
Let $p$ be a prime number and let $\mathbb{F}_p$ be the finite field with $p$ elements. Consider an automorphism $\tau$ of the polynomial ring $\mathbb{F}_p[x]$ given by
\[\tau(f)(x)=f(x+1).\]
Let $R$ denote the subring of $\mathbb{F}_p[x]$ consisting of those polynomials $f$ with $\tau(f)=f$. Find a polynomial $g \in ... | g = x^{p-1} |
Given $\frac{2}{1-i} = 1 + ni$, where $n \in \mathbb{R}$, $i$ is the imaginary unit, find $n =$ _____. | n = 1 |
$15 \cdot 26$ Let the natural number $n(n>3)$ have the following property: when the set $S_{n}=\{3,4, \cdots, n\}$ is arbitrarily divided into two groups, there is always a group containing three numbers $a, b, c$ (allowing $a=b$), such that $a b=c$. Find the smallest value of such $n$.
(China National Training Team Te... | 243 |
A four-digit number is divisible by 9. If the last digit is removed, the resulting three-digit number is exactly a multiple of 4. The last digit of the largest such four-digit number is ( )
A: 6
B: 4
C: 3
D: 2 | C |
4. (6 points) Xiaogang, while calculating division, wrote the divisor 74 as 47, resulting in a quotient of 5 and a remainder of 8. The correct quotient is $\qquad$ and the correct remainder is $\qquad$.
| 3,21 |
## 46. Who is Murdoch?
- A spy named Murdoch has infiltrated the country. He was arrested along with two indigenous residents (we will denote the defendants with Latin letters $A, B$, and $C$). One of the indigenous residents, mistakenly arrested along with the spy, was a knight, and the other was a liar. Of the three... | A |
Given the function $f(x)=x^{3}+x$, for any $m\in[-2,2]$, the inequality $f(mx-2)+f(x) < 0$ always holds true. Determine the range of values for $x$. | -2, \dfrac{2}{3} |
5. In $\triangle A B C$, $A C=8, B C=10,32 \cos (\mathrm{A}-\mathrm{B})=31$, then the area of $\triangle A B C$ is $\qquad$ $\_$ | 15\sqrt{7} |
Given two equations about $x$: $3\left(2x-1\right)=k+2x$ and $\frac{x-k}{2}=x+2k$.$(1)$ If the solution to the equation $3\left(2x-1\right)=k+2x$ is $x=4$, find the solution to the equation $\frac{x-k}{2}=x+2k$;$(2)$ If the solutions to the equations $3\left(2x-1\right)=k+2x$ and $\frac{x-k}{2}=x+2k$ are the same, find... | k=-\frac{1}{7} |
Given a complex number $z$ satisfies $z + |z| = 2 + 8i$, where $i$ is the imaginary unit, find $|z|$. | |z|= 17 |
For any integer $n$, the polynomial $(n+7)^{2}-n^{2}$ is divisible by
A: $2$
B: $n$
C: $7$
D: $n+7$ | C |
56. Given that $x$ divided by 3 leaves a remainder of 2, divided by 5 leaves a remainder of 4, and divided by 7 leaves a remainder of 4, the largest three-digit number $x$ that satisfies these conditions is | 914 |
6. Given $a_{1}, a_{2}, b_{1}, b_{2}, \cdots, b_{k}$ are vectors in a plane, no two of which are parallel, satisfying $\left|a_{1}-a_{2}\right|=1$, and $\left|a_{i}-b_{j}\right| \in\{1,2,3\}(i=1,2 ; j=1,2, \cdots, k)$. Then the maximum value of $k$ is $\qquad$ | 10 |
Given $f(x)=2\cos^2 \omega x+\sqrt{3}\sin 2\omega x-2$ ($\omega>0$) with the smallest positive period of $\pi$, which of the following statements are correct?
A: $\omega =2$
B: The function $f(x)$ is increasing on $[0,\frac{\pi}{6}]$
C: The line $x=\frac{\pi}{3}$ is an axis of symmetry for the graph of $y=f(x)$
D: ... | BD |
10. Given the sequence $\left\{a_{n}\right\}$ satisfies: $a_{1}=2 t-3(t \in \mathbf{R}$ and $t \neq \pm 1)$,
$$
a_{n+1}=\frac{\left(2 t^{n+1}-3\right) a_{n}+2(t-1) t^{n}-1}{a_{n}+2 t^{n}-1}\left(n \in \mathbf{N}^{*}\right) \text {. }
$$
(1) Find the general term formula of the sequence $\left\{a_{n}\right\}$;
(2) If $t... | a_{n}=\frac{2(^{n}-1)}{n}-1(n\in{N}^{*}) |
Square $ABCD$ has side length $13$, and points $E$ and $F$ are exterior to the square such that $BE=DF=5$ and $AE=CF=12$. Find $EF^{2}$.
[asy]
size(200);
defaultpen(fontsize(10));
real x=22.61986495;
pair A=(0,26), B=(26,26), C=(26,0), D=origin, E=A+24*dir(x), F=C+24*dir(180+x);
draw(B--C--F--D--C^^D--A--E--B--A, linew... | 578 |
Find all positive integers $x, y$ for which:
$$
1+x+y+xy=2008 \text{. }
$$ | (1,1003),(1003,1),(3,501),(501,3),(7,250),(250,7) |
The equation of the line that is parallel to the line $y= \frac {1}{2}x+3$ and passes through the point $(0,-1)$ is ( )
A: $2x+y+1=0$
B: $x+2y+2=0$
C: $x-2y-2=0$
D: $2x-y-1=0$ | C |
1.026. $\frac{(3.4-1.275) \cdot \frac{16}{17}}{\frac{5}{18} \cdot\left(1 \frac{7}{85}+6 \frac{2}{17}\right)}+0.5\left(2+\frac{12.5}{5.75+\frac{1}{2}}\right)$. | 3 |
Given the definition of a matrix transformation from vector $\overrightarrow{OP\_n} = (x\_n, y\_n)$ to vector $\overrightarrow{OP_{n+1}} = (x_{n+1}, y_{n+1})$ as follows:
$$\begin{pmatrix} x_{n+1} \\ y_{n+1} \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} x_n \\ y_n \end{pmatrix},$$
where... | (2, 4030) |
31 - 44 The system of equations $\left\{\begin{array}{l}x+2 y=n \\ 3 x-y=m\end{array}\right.$ (where $n, m$ are both odd numbers) has integer solutions, then the solution $\left\{\begin{array}{l}x=p \\ y=q\end{array}\right.$ has the parity situation:
(A) $p, q$ are both even.
(B) $p, q$ are both odd.
(C) $p$ is even, $... | D |
23. Represent the number 231 as the sum of several natural numbers so that the product of these addends also equals 231. | 231=3+7+11+\underbrace{1+1+\ldots+1}_{210\text{ones}} |
1. The range of the function $y=\frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1-x^{2}}+1}$ is | [1,\sqrt{2}] |
Simplify $f(α)= \dfrac {\sin (π-α)\cos (2π-α)\sin (-α+ \dfrac {3}{2}π)}{\cos (-π-α)\cos (-α+ \dfrac {3}{2}π)}= $ \_\_\_\_\_\_ . | -\cos α |
## Task Condition
Calculate the areas of figures bounded by lines given in polar coordinates.
$$
r=\sin 6 \phi
$$ | \frac{\pi}{2} |
3. The product of the areas of the six faces of a rectangular prism is 14641, then the volume of the rectangular prism is
保留源文本的换行和格式，直接输出翻译结果。 | 11 |
Given the function f(x) = e^|x| + ax^2, and f'(1) = e + 1, find f'(-1). | -e - 1 |
1. Pete and Vova want to cross a river 160 (m) wide. They have a small rubber boat, in which only one person can paddle at a speed of 32 (m/min). They started the crossing simultaneously: Pete in the boat, Vova - swimming. At some point, Pete left the boat and continued swimming on his own, while Vova, catching up with... | 120\mathrm{} |
Which of the following equations has a solution of $x=2$?
A: $4x=2$
B: $3x+6=0$
C: $\frac{1}{2}x=0$
D: $7x-14=0$ | D |
1. Verify that the number $7^{20}-1$ is divisible by $10^{3}$ and determine the last three digits of the number $7^{2015}$. | 943 |
In the expansion of $\left( x+\frac{3}{\sqrt[3]{x}} \right)^{n}$, the ratio of the sum of the coefficients of the terms to the sum of the binomial coefficients is $32$. Then, the coefficient of $x$ is
A: $45$
B: $90$
C: $135$
D: $270$ | D |
At the New Year's school party in the city of Lzheretsark, 301 students came. Some of them always tell the truth, while the rest always lie. Each of the 200 students said: "If I leave the hall, then among the remaining students, the majority will be liars." Each of the other students stated: "If I leave the hall, then ... | 151 |
Example 8 Color each vertex of a quadrilateral pyramid with one color, and make the endpoints of the same edge have different colors. If only 5 colors are available, how many different coloring methods are there? | 420 |
74(1184). Find all pairs of natural numbers that satisfy the equation: a) $x^{2}-y^{2}=105$; b) $2 x^{2}+5 x y-12 y^{2}=28$. | (53,52),(19,16),(13,8),(11,4) |
1. Vasya thought of a four-digit number and for each pair of its adjacent digits, he wrote down their product on the board. After that, he erased one product, and the numbers 20 and 21 remained on the board. What is the smallest number Vasya could have thought of? | 3745 |
11. The lengths of the three altitudes of a triangle are in the ratio $3: 4: 6$. If the radius of the inscribed circle of the triangle is 1 , find the radius of the circumcircle of the triangle.
某三角形三條高的長度之比為 $3: 4: 6$ 。若該三角形內切圓的半徑是 1 , 求它的外接圓半徑。 | \frac{16}{5} |
Example 2 For a finite set $A$, there exists a function $f: N \rightarrow$ $A$, with the following property: if $i, j \in N$, and $|i-j|$ is a prime number, then $f(i) \neq f(j)$. How many elements must the set $A$ have at least? | 4 |
8.354. $\operatorname{tg}\left(\frac{3 \pi}{2}-x\right)+\frac{\cos \left(\frac{7 \pi}{2}+x\right)}{1+\cos x}=2$. | (-1)^{\}\frac{\pi}{6}+\pi,\inZ |
67. $\int\left(x^{3}+2^{x}\right) d x$ | \frac{1}{4}x^{4}+\frac{2^{x}}{\ln2}+C |
If one external angle of an isosceles triangle is $80^{\circ}$, then the measure of its base angle is $\_\_\_\_\_\_$ $(^{\circ}).$ | 40^{\circ} |
Given the function $f(x) = \left(\frac{1}{2}\right)^{\sqrt{x^2 - 4ax + 8}}$ is monotonic on $[2, 6]$, determine the range of values for $a$.
A: $(-\infty, 1] \cup [3, +\infty)$
B: $(-\infty, 1]$
C: $[3, +\infty)$
D: $[\frac{3}{2}, \frac{11}{6}]$ | \text{B} |
Determine the smallest integer $n$ such that there exist $n$ real numbers $x_{1}, \ldots, x_{n}$ all belonging to the interval $(-1,1)$ and for which
$$
x_{1}+\cdots+x_{n}=0 \quad \text { and } \quad x_{1}^{2}+\cdots+x_{n}^{2}=2020
$$ | 2022 |
3. A family of raccoons has settled in a zoo. The number of raccoons $N$ after $t$ years $(t \geq 0)$ is given by the function $N=10 \cdot e^{\frac{2}{5} t}$.
(a) How many raccoons are in the family at the time of settlement?
(b) How many years would it take for the family of raccoons to number 100 members? Round the... | 6 |
Given the function $f(x)=\ln x+a(x^{2}-3x+2)$, where $a$ is a parameter.
$(1)$ When $a=0$, find the equation of the tangent line to the function $f(x)$ at $x=1$;
$(2)$ Discuss the number of extreme points of the function $f(x)$, and explain the reason;
$(3)$ If for any $x\in[1,+\infty)$, $f(x)\geqslant 0$ always ... | 0\leqslant a\leqslant 1 |
4.22. Let's introduce a coordinate system with the origin $O$ at the center of the Earth, with the axes $O x$ and $O y$ passing through the points on the equator with longitudes $0^{\circ}$ and $90^{\circ}$ respectively, and the axis $\mathrm{Oz}$ passing through the North Pole. What are the coordinates of a point on t... | (R\cos\varphi\cos\psi,R\cos\varphi\sin\psi,R\sin\varphi) |
Given that $a$, $b$, and $c$ are real numbers, which of the following statements are correct?
A: If $a \gt b$, then $\frac{1}{a}＜\frac{1}{b}$
B: If $ac^{2} \gt bc^{2}$, then $a \gt b$
C: If $a \gt b$ and $k\in N^{*}$, then $a^{k} \gt b^{k}$
D: If $c \gt a \gt b \gt 0$, then $\frac{a}{c-a}＞\frac{b}{c-b}$ | B \text{ and } D |
Example 5 Delete all perfect squares from the sequence of positive integers $1,2,3, \cdots$, to get a new sequence $\left\{a_{n}\right\}$. Then the 2003rd term of $\left\{a_{n}\right\}$ is ( ).
(A) 2046
(B) 2047
(C) 2048
(D) 2049
(2003, National High School Mathematics Competition) | C |
3. Let $f_{1}(x)=\sqrt{2}, f_{2}(x)=\sin x+\cos \sqrt{2} x$, $f_{3}(x)=\sin \frac{x}{\sqrt{2}}+\cos \sqrt{2} x, f_{4}(x)=\sin x^{2}$. Among the above functions, the number of periodic functions is ( ).
(A) 1
(B) 2
(C) 3
(D) 4 | B |
4. Given that $AB$ is a chord of the circle $\odot O$ with radius 1, and $AB = a < 1$. A regular $\triangle ABC$ is constructed inside $\odot O$ with $AB$ as one side. $D$ is a point on $\odot O$ different from point $A$, and $DB = AB = a$. The extension of $DC$ intersects $\odot O$ at point $E$. Then the length of $AE... | B |
Let the function $f(x)=\begin{cases}3x-1, & x < 1 \\ 2^{x}, & x\geqslant 1\end{cases}$, then the range of $a$ that satisfies $f(f(a))=2^{f(a)}$ is
A: $\left[ \frac{2}{3} ,+\infty\right)$
B: $\left[\frac{2}{3},1\right]$
C: $\left[1,+\infty\right)$
D: $\left[0,1\right]$ | A |
Let $a$,$b$,$c$ be positive real numbers satisfying $abc=1$. Prove that inequality $\dfrac{a^k}{a+b}+ \dfrac{b^k}{b+c}+\dfrac{c^k}{c+a}\ge \dfrac{3}{2}$ holds for all integer $k$ ($k \ge 2$). | \frac{3}{2} |
On the graph of the function $y=-3x+1$, there are two points $A(1$,$y_{1})$ and $B(3$,$y_{2})$. The relationship between $y_{1}$ and $y_{2}$ is ( ).
A: $y_{1} \gt y_{2}$
B: $y_{1} \lt y_{2}$
C: $y_{1}=y_{2}$
D: Unable to determine | A |
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