problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
If $f(x)$ is an odd function with a period of 5 on $\mathbb{R}$, and it satisfies $f(1)=1$ and $f(2)=2$, then $f(3)-f(4)=$ ( )
A: 1
B: 2
C: -2
D: -1 | \text{D} |
## Task Condition
Compose the equation of the normal to the given curve at the point with abscissa $x_{0}$.
$y=\frac{x^{2}-3 x+6}{x^{2}}, x_{0}=3$ | 9x-26\frac{1}{3} |
Given the function $f(x) = x^2 + a\ln x$,
(Ⅰ) When $a = -2e$, determine the intervals of monotonicity and the extremal values of the function $f(x)$;
(Ⅱ) If the function $f(x)$ is decreasing on the interval $[1, 4]$, determine the range of the real number $a$. | (-\infty, -32] |
3. (25 points) $a$, $b$, $c$ are positive integers, and $a^{2}+b^{3}=$ $c^{4}$. Find the minimum value of $c$.
保留了源文本的换行和格式。 | 6 |
6. Team A and Team B each draw 7 players to participate in a Go relay match according to a pre-arranged order. Both sides start with their No. 1 players competing. The loser is eliminated, and the winner then competes with the No. 2 player of the losing side, $\cdots$, until all players of one side are eliminated, and ... | 3432 |
## Problem 3
$3^{1}+3^{2}+3^{3}+\ldots+3^{2012}=\left(3^{1}+3^{2}+3^{3}+3^{4}\right)+3^{4}\left(3^{1}+3^{2}+3^{3}+3^{4}\right)+\ldots+3^{2008}\left(3^{1}+3^{2}+3^{3}+3^{4}\right)=$ (1p) $120+120 \cdot 3^{4}+\ldots 120 \cdot 3^{2008}=(\mathbf{1 p})$
$120\left(1+3^{4}+\ldots+3^{2008}\right)$ (1p)
It is sufficient to f... | 60 |
What is the smallest number of non-overlapping tetrahedra into which a cube can be divided?
# | 5 |
To investigate the distribution of heights among high school students in your school, assume that the sample size drawn by your deskmate is the same as yours and the sampling method is reasonable. Which of the following statements is correct?
A: The histogram of sample frequency distribution between you and your deskma... | \text{D} |
Example 1 Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$, such that for any real numbers $x, y$, we have
$$
f(x) f(y)=f\left(x^{2}+y^{2}\right) .
$$ | f(x)=0 |
Example 1 Let $P$ and $Q$ be two moving points on the circumference of the circle $x^{2}+y^{2}=1$, and they satisfy the condition that with a fixed point $A\left(0, \frac{1}{2}\right)$ inside the circle, $\angle P A Q = \frac{\pi}{2}$. Find the equation of the locus of point $M$, the intersection of the two tangents pa... | 3 x^{2}+3 y^{2}+4 y-8=0 |
8. In $\triangle A B C$, $\angle C=\frac{\pi}{3}$, let $\angle B A C=\theta$. If there exists a point $M$ on line segment $B C$ (different from points $B$ and $C$), such that when $\triangle B A M$ is folded along line $A M$ to a certain position to get $\triangle B^{\prime} A M$, it satisfies $A B^{\prime} \perp C M$,... | (\frac{\pi}{6},\frac{2\pi}{3}) |
Which of the following analyses of a dataset is incorrect?
A: The smaller the range of the data, the more concentrated and stable the sample data distribution.
B: The smaller the mean of the data, the more concentrated and stable the sample data distribution.
C: The smaller the standard deviation of the data, the more ... | \text{B} |
Biologists have discovered a virus with a length of approximately $0.00000032$ mm. The correct scientific notation for $0.00000032$ is:
A: $3.2\times 10^{7}$
B: $3.2\times 10^{8}$
C: $3.2\times 10^{-7}$
D: $3.2\times 10^{-8}$ | C |
The function $f(x) = x^3 + mx^2 + 2x + 5$ has a derivative denoted as $f'(x)$. The value of $f'(2) + f'(-2)$ is ( )
A: $28+4m$
B: $38+4m$
C: $28$
D: $38$ | C |
5. Find all pairs of positive integers $(m, n)$ such that $6^{m} + 2^{n} + 2$ is a perfect square. | (0,0),(1,0),(1,3) |
11. For a positive integer $n$, select several positive divisors of $n$ such that these divisors do not have an integer multiple relationship with each other. Let the maximum number of divisors that can be selected be $f(n)$, then $f\left(2^{2020} \times 2020\right)+f\left(3^{2021} \times 2021\right)+f\left(5^{2022} \t... | 16 |
7.127. $\log _{2}\left(25^{x+3}-1\right)=2+\log _{2}\left(5^{x+3}+1\right)$ | -2 |
6. Calculate:
$$
C_{n}^{0} \times \frac{1}{2}+\frac{1}{2} C_{n}^{1} \times\left(\frac{1}{2}\right)^{2}+\frac{1}{3} C_{n}^{2} \times\left(\frac{1}{2}\right)^{3}+\cdots+\frac{1}{n+1} C_{n}^{n}\left(\frac{1}{2}\right)^{n+1}=
$$
$\qquad$ | \frac{(\frac{3}{2})^{n+1}-1}{n+1} |
Example 6 In the right triangle $\triangle ABC$, $\angle C=90^{\circ}$, the incircle $\odot O$ touches $AC$, $BC$ at points $E$, $F$, respectively. The rays $BO$, $AO$ intersect the line $EF$ at points $M$, $N$. Prove: $\frac{1}{5}<\frac{S_{\triangle OMN}}{S_{\triangle ABC}}<\frac{1}{4}$. | \frac{1}{5} < \frac{S_{\triangle OMN}}{S_{\triangle ABC}} < \frac{1}{4} |
9. Let $f: \mathbf{R} \rightarrow \mathbf{R}$ satisfy: for any real numbers $x, y$, we have $f(2 x)+f(2 y)=f(x+y) f(x-y)$, and $f(\pi)=0$, and $f(x)$ is not identically zero, then the period of $f(x)$ is $\qquad$. | 4\pi |
21. (1999 National Team Selection Test) For non-negative real numbers $x_{1}, x_{2}$, $\cdots, x_{n}$ satisfying the condition $x_{1}+x_{2}+\cdots+x_{n}=1$, find the maximum value of $\sum_{j=1}^{n}\left(x_{j}^{4}-x_{j}^{5}\right)$. | \frac{1}{12} |
A2. Of the two drawn lines in the coordinate system, one of the lines has the equation $y=a x+b$ for some different real numbers $a$ and $b$ (see figure). Which of the following equations could be the equation of the other line?
(A) $y=a x-b$
(B) $y=b x+a$
(C) $y=\frac{b}{a} x+b$
(D) $y=-b x+a$
(E) $y=\frac{a}{b} x+a$ | \frac{}{b}x+ |
2. Given the system of inequalities about $x$
$$
\left\{\begin{array}{l}
2 x-1>3, \\
4 x-2 \geqslant a
\end{array}\right.
$$
the solution set is $x>b$. Then the value of the expression
$$
|a-3 b|-2|a-4 b|
$$
is ( ).
(A) $3 a-11 b$
(B) $5 b-a$
(C) $22-3 a$
(D) $a-10$ | D |
4. Let $f(x)=x^{3}+3 x^{2}+5 x+7$. Find the polynomial $g(x)$ of the smallest degree such that
$$
f(3)=g(3), \quad f(3-\sqrt{3})=g(3-\sqrt{3}), \quad f(3+\sqrt{3})=g(3+\sqrt{3}) .
$$ | 12x^{2}-19x+25 |
82. A passenger fell asleep after traveling half the distance. When he woke up, he still had to travel half the distance he traveled while asleep. So, the distance he traveled while asleep is $\qquad$ of the total distance. (Fill in the fraction) | \frac{1}{3} |
389. Given the density function $f(x)=\frac{1}{\sqrt{2 \pi}} \mathrm{e}^{-x^{2} / 2},(-\infty<x<\infty)$ of a normally distributed random variable $X$. Find the density function $g(y)$ of the random variable $Y=X^{2}$. | (y)=\frac{1}{\sqrt{2\piy}}e^{-y/2} |
Example 5 Given the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ and an inscribed parallelogram with one pair of opposite sides passing through the foci $F_{1}$ and $F_{2}$ of the ellipse. Find the maximum area of the parallelogram. ${ }^{[4]}$
(2013, National High School Mathematics League Shandong Province Preliminary... | 6 |
One, (40 points) Given positive real numbers $a, b, c, d$ satisfying $a+b+c+d=4$. Prove:
$$
\frac{b}{\sqrt{a+2 c}}+\frac{c}{\sqrt{b+2 d}}+\frac{d}{\sqrt{c+2 a}}+\frac{a}{\sqrt{d+2 b}} \geqslant \frac{4 \sqrt{3}}{3} .
$$ | \frac{4\sqrt{3}}{3} |
3. Point $O$ is the center of a certain circle, $A$ is a point outside the circle, $B$ is a point on the circle such that $AB$ is a tangent. $AO=6$. Find the greatest possible value of the area of triangle $AOB$. | 9 |
The function $f$, defined on the set of ordered pairs of positive integers, satisfies the following properties:
\[f(x, x) = x,\; f(x, y) = f(y, x), {\rm \ and\ } (x+y)f(x, y) = yf(x, x+y).\]
Calculate $f(14,52)$. | 364 |
Find the equation of the circle that passes through points $A(1,-1)$, $B(-1,1)$ and has its center on the line $x+y-2=0$. | (x-1)^2 + (y-1)^2 = 4 |
Simplify and find the value of:$(1)\tan 10^{\circ}\tan 20^{\circ}+\sqrt{3}(\tan10^{\circ}+\tan20^{\circ})$.$(2)\tan 23^{\circ}+\tan 37^{\circ}+\sqrt{3}\tan 23^{\circ}\tan 37^{\circ}$. | \sqrt{3} |
4. Points $A, B, C, D, E$ are sequentially located on a line, such that $A B=B C=2, C D=1, D E=3$. Circles $\Omega$ and $\omega$, touching each other, are such that $\Omega$ passes through points $A$ and $E$, and $\omega$ passes through points $B$ and $C$. Find the radii of circles $\Omega$ and $\omega$, given that the... | R=8\sqrt{\frac{3}{11}},r=5\sqrt{\frac{3}{11}} |
Factorization: $a^{2}-9=$____. | (a + 3)(a - 3) |
Given sets $A=\{x||x| \lt 3\}$ and $B=\{x|x^{2}-4x+3 \lt 0\}$, then $A\cap B$ is equal to:
A: $\{x\left|\right.1 \lt x \lt 3\}$
B: $\{x\left|\right.-2 \lt x \lt 1\}$
C: $\{x\left|\right.2 \lt x \lt 3\}$
D: $\{x\left|\right.-2 \lt x \lt 3\}$ | \text{A} |
\section*{Exercise 3 - V11223}
Discuss the function
\[
f(x)= \begin{cases}\frac{|x|}{x} & \text { for }|x|>1 \\ x^{3} & \text { for }|x| \leq 1\end{cases}
\]
Calculate the area of the region bounded by the graph of the function, the x-axis, and the lines \(x=-2\) and \(x=2\)! | \frac{5}{2} |
Calculate: $-3 + 2 = \quad$; $\quad (-3) \times 2 = \quad$. | -6 |
There are 18 ping-pong balls on the table. Two people take turns picking up balls and putting them into their pockets. The person who picks up the 18th ping-pong ball wins. The rules are that each time a player must pick up at least 1 ball, but no more than 4 balls. In this game, the player who goes first has a winning... | \text{C} |
10.235. In an isosceles triangle, the angle at the base is $72^{\circ}$, and the bisector of this angle has a length of $m$. Find the lengths of the sides of the triangle. | ;\frac{(1+\sqrt{5})}{2} |
1.108 Let $A B C D E F$ be a regular hexagon. A frog starts at vertex $A$ and can randomly jump to one of the two adjacent vertices each time. If it reaches point $D$ within 5 jumps, it stops jumping; if it does not reach point $D$ within 5 jumps, it stops after 5 jumps. How many different ways can the frog jump from t... | 26 |
It is known that there are 12 items on a shelf, with 4 on the upper shelf and 8 on the lower shelf. Now, if 2 items are to be moved from the lower shelf to the upper shelf, and the relative order of the other items remains unchanged, then the number of different ways to adjust the items is
A．420 B．... | C |
20. As shown in the figure, the base of the pyramid $P-ABCD$ is a square with side length 4, and $PD \perp$ the base $ABCD$. Given $PD=6$, and $M, N$ are the midpoints of $PB, AB$ respectively.
(1) Find the volume of the tetrahedron $P-DMN$;
(2) Find the tangent value of the dihedral angle $M-DN-C$.
| 4,\frac{3\sqrt{5}}{2} |
1. 计算下列 Legendre 符号:
$$\begin{array}{c}
\left(\frac{13}{47}\right),\left(\frac{30}{53}\right),\left(\frac{71}{73}\right),\left(\frac{-35}{97}\right),\left(\frac{-23}{131}\right),\left(\frac{7}{223}\right), \\
\left(\frac{-105}{223}\right),\left(\frac{91}{563}\right),\left(\frac{-70}{571}\right),\left(\frac{-286}{647}\r... | -1,-1,1,1,1,1,-1,1,-1,1 |
11. Let $f(x), g(x)$ be odd and even functions defined on $\mathbf{R}$, respectively, and $f(x)+g(x)=2^{x}$. If for $x \in[1,2]$, the inequality $a f(x)+g(2 x) \geqslant 0$ always holds, then the range of the real number $a$ is | [-\frac{17}{6},+\infty) |
Given $$\sin\theta + \cos\theta = \frac{1}{3}$$, then $\sin2\theta=$ （ ）
A: $$\frac{8}{9}$$
B: $$- \frac{8}{9}$$
C: $$\frac{4}{9}$$
D: $$- \frac{4}{9}$$ | \text{B} |
Given $\tan x=\frac{1}{3}$, then $\sin x\cos x+1$ equals to ( )
A: $\frac{13}{10}$
B: $-\frac{13}{10}$
C: $\frac{10}{13}$
D: $-\frac{10}{13}$ | A |
Given a complex number $z$ with an imaginary part of $2$, and $z^{2}+3$ is a pure imaginary number, then $|z|=$____. | \sqrt{5} |
If the direction vector of line $l$ is $\overrightarrow{d}=(1,\sqrt{3})$, then the inclination angle of line $l$ is ______. | \frac{\pi}{3} |
Let $y=f''(x)$ be the derivative of $y=f'(x)$. A student discovered that any cubic function $f(x)=ax^{3}+bx^{2}+cx+d(a\neq 0)$ has a center of symmetry $(x_{0},f(x_{0}))$, where $x_{0}$ satisfies $f''(x_{0})=0$. Given $f(x)= \frac {1}{3}x^{3}- \frac {1}{2}x^{2}+3x- \frac {5}{12}$, then $f( \frac {1}{2017})+f( \frac {2}... | D |
Given a line $y= \frac { \sqrt {3}}{3}x$ and a circle $C$ with its center on the positive x-axis and a radius of 2, intersecting at points $A$ and $B$, and the length of chord $AB$ is $2 \sqrt {3}$.
(Ⅰ) Find the equation of circle $C$;
(Ⅱ) If point $P(m, n)$ is on circle $C$, find the maximum value of $\sqrt {3}m+n... | 2 \sqrt {3}+4 |
Let $a$, $b$, $c$ be three distinct non-zero real numbers satisfying $a+\frac{2}{b}=b+\frac{2}{c}=c+\frac{2}{a}$. Find the value of $(a+\frac{2}{b})^{2}+(b+\frac{2}{c})^{2}+(c+\frac{2}{a})^{2}$. | 6 |
2.058. $\left(\left(\frac{1}{a}+\frac{1}{b+c}\right):\left(\frac{1}{a}-\frac{1}{b+c}\right)\right):\left(1+\frac{b^{2}+c^{2}-a^{2}}{2 b c}\right)$;
$$
a=1 \frac{33}{40} ; b=0.625 ; c=3.2
$$ | 1 |
In the complex plane, the point corresponding to the complex number $\left(1-4i\right)\left(2+3i\right)$ is located in which quadrant?
A: First quadrant
B: Second quadrant
C: Third quadrant
D: Fourth quadrant | D |
10.4. Let x be some natural number. Among the statements:
$2 \times$ greater than 70
x less than 100
$3 x$ greater than 25
x not less than 10
x greater than 5
three are true and two are false. What is x | 9 |
I3.1 A rectangular piece of paper is folded into the following figure. If the area of $\triangle A B C$ is $\frac{1}{3}$ of the area of the original rectangular piece of paper, find the value of $P$.
I3.2 If $Q$ is the positive integral solution of the equation $\frac{P}{2}\left(4^{x}+4^{-x}\right)-35\left(2^{x}+2^{-x... | 60 |
Simplify and then evaluate the expression: $(3m^2-4mn)-2(m^2+2mn)$, where $m$ and $n$ satisfy the condition that the sum of the monomial $-x^{m+1}y^3$ and $\frac{3}{2}y^nx^2$ is still a monomial. | -23 |
Given the function $f(x)=2\sin ^{2}x+\cos ( \frac {π}{3}-2x)$.
(1) Find the interval where $f(x)$ is decreasing on $[0,π]$;
(2) In triangle $ABC$, let the sides opposite to angles $A$, $B$, $C$ be $a$, $b$, $c$ respectively. If $f(A)=2$, and vector $\overrightarrow{m}=(1,2)$ is collinear with vector $\overrightarrow{n}... | \frac {a}{b}= \sqrt {3} |
22. Given ten 0's and ten 1's, how many 0-1 binary sequences can be formed such that no three or more than three 0's are together? For example, 01001001010011101011 is such a sequence, but the sequence 01001000101001110111 does not satisfy this condition. | 24068 |
1.40. Point $M$ lies inside an equilateral triangle $A B C$. Calculate the area of this triangle, given that $A M=$ $=B M=2 \text{ cm}, \text{a} C M=1 \text{cm}$. | \frac{9\sqrt{3}+3\sqrt{15}}{8} |
Let \(\triangle ABC\) have side lengths \(AB=30\), \(BC=32\), and \(AC=34\). Point \(X\) lies in the interior of \(\overline{BC}\), and points \(I_1\) and \(I_2\) are the incenters of \(\triangle ABX\) and \(\triangle ACX\), respectively. Find the minimum possible area of \(\triangle AI_1I_2\) as \( X\) varies along \(... | 126 |
14. The diagram shows a square $P Q R S$ of side-length 1 . $W$ is the centre of the square and $U$ is the midpoint of $R S$. Line segments $T W, U W$ and $V W$ split the square into three regions of equal area. What is the length of $S V$ ?
A $\frac{1}{2}$
B $\frac{2}{3}$
C $\frac{3}{4}$
D $\frac{4}{5}$
E $\frac{5}{6}... | \frac{5}{6} |
In how many ways can one tile a $2 \times n$ rectangle with $2 \times 1$ dominoes? | F_{n+1} |
Given that $F$ is the focus of the parabola $C$: $y^{2}=2px (p > 0)$, a line passing through $F$ with a slope angle of $60^{\circ}$ intersects the curve $C$ at points $A$ and $B$ (point $B$ is in the first quadrant, and point $A$ is in the fourth quadrant). $O$ is the coordinate origin, and $M$ is the foot of the perpe... | C |
The sum of the first $n$ terms of the sequence $\{a_n\}$ is denoted as $S_n$, with $a_1=t$. The point $(S_n, a_{n+1})$ lies on the line $y=2x+1$, where $n\in\mathbb{N}^*$.
(1) If the sequence $\{a_n\}$ is a geometric sequence, find the value of the real number $t$;
(2) Let $b_n=n\cdot a_n$, under the condition of (... | 1 |
Suppose the coordinate system and parameter equation $4-4$:
In the rectangular coordinate system, the parameter equation of line $l$ is $\begin{cases} x = 6 + \frac{\sqrt{2}}{2}t \\ y = \frac{\sqrt{2}}{2}t \end{cases}$ (where $t$ is a parameter). Establish a polar coordinate system with the coordinate origin as the po... | 2 |
## Task 1
Two cars, a "Trabant" and a "Wartburg," drove from Rostock to Berlin at the same time.
The "Trabant" drove 60 km per hour, and the "Wartburg" drove 80 km per hour.
How many km did the "Trabant" drive in 3 hours?
How many km did the "Wartburg" drive in 3 hours?
How many km is the "Wartburg" ahead of the "... | 180\mathrm{~} |
3. Let the set $M=\left\{(x, y)\left|\log _{x}\right| y\left|=\log _{|y|}\right| x, x, y \in\right.$ $R\}, N=\left\{(x, y) \mid y=\sqrt{\left|2-x^{2}\right|}\right\}$. Then the number of elements in $M \cap N$ is ( ).
(A) 1
(B) 2
(C) 3
(D) 4 | A |
Among the real numbers $-2$, $2$, $0$, $1$, the smallest real number is ______. | -2 |
Given the proposition $p$: $\forall x \in \mathbb{R}$, the function $f(x) = \lg (ax^{2}-x+ \frac{1}{16}a)$ is defined, and the proposition $q$: $\forall x > 0$, the inequality $\sqrt {2x+1} < 1+ax$ always holds. If the proposition "$p$ or $q$" is true and the proposition "$p$ and $q$" is false, find the range of values... | a \in [1, 2] |
Given the following functions:
1. $f(x) = \lg(|x-2|+1)$,
2. $f(x) = (x-2)^2$,
3. $f(x) = \cos(x+2)$.
Consider the three propositions:
A. $f(x+2)$ is an even function;
B. $f(x)$ is a decreasing function on the interval $(-\infty, 2)$ and an increasing function on the interval $(2, +\infty)$;
C. $f(x+2) - f(x)$ is an i... | 2 |
For any real numbers $x$ and $y$, find the minimum value of $|x-1|+|x|+|y-1|+|y+1|$. | 3 |
40. Given numbers: one and nine zeros. It is allowed to choose two numbers and replace each with their arithmetic mean. What is the smallest number that can end up in the place of one after a series of such operations? | \frac{1}{512} |
In an opaque bag, there is $1$ white ball and $2$ red balls, which only differ in color. A ball is randomly drawn from the bag, the color is noted, and then returned. Another ball is then randomly drawn. The probability of drawing two white balls in succession is ( ).
A: $\frac{1}{9}$
B: $\frac{1}{6}$
C: $\frac{1}{3... | A |
Given the function $f(x)=2\sin x\cos x+2 \sqrt {3}\cos ^{2}x$.
$(1)$ Find the intervals of monotonicity for the function $f(x)$.
$(2)$ When $x\in\left[- \dfrac {\pi}{3}, \dfrac {\pi}{3}\right]$, find the maximum and minimum values of the function $f(x)$. | 0 |
4. Let $n>0, n=2^{k} m$ (where $m$ is odd). Then the greatest common divisor of $\mathrm{C}_{2 n}^{1}, \mathrm{C}_{2 n}^{3}$, $\cdots, \mathrm{C}_{2 n}^{2 n-1}$ is $\qquad$ | 2^{k+1} |
55. The students of Class 1 (1) are arranged in a square formation. In the row where Xiaoming is, there is 1 student to his left and 4 students to his right. In the column where Xiaoming is, there are 3 students in front of him and 2 students behind him. Class 1 (1) has $\qquad$ students in total. | 36 |
(1) Calculate: $$(2 \frac {1}{4})^{ \frac {1}{2}}-(- \frac {1}{2})^{0}-(3 \frac {3}{8})^{- \frac {2}{3}}+( \frac {3}{2})^{-2}+(0.125)^{ \frac {1}{3}}$$
(2) $$\log_{ \sqrt {3}}9+2^{ \frac {1}{\log_{3}2}}$$． | 7 |
The median to a 10 cm side of a triangle has length 9 cm and is perpendicular to a second median of the triangle. Find the exact value in centimeters of the length of the third median. | 3\sqrt{13} |
4.4. A smooth sphere with a radius of 1 cm was dipped in blue paint and launched between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm, respectively (this sphere ended up outside the smaller sphere but inside the larger one). Upon touching both spheres, the sphere leaves a blue trail. During its m... | 38.25 |
2. (10 points) The clock struck the year 2013, and Brother Xiaoming sighed: This is the first year in my life that I will spend without any repeated digits. It is known that Brother Xiaoming was born in a year that is a multiple of 19. Therefore, in 2013, Brother Xiaoming's age is ( ) years old.
A. 16
B. 18
C. 20
D. 22 | 18 |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $\sin (A+B)= \frac {1}{3}$, $a=3$, and $c=4$, then $\sin A=$ \_\_\_\_\_\_. | \sin A = \frac{1}{4} |
Given the parabola $C: y^2 = -4x$ with focus $F$, and point $A(-2, 1)$. Let $P$ be a moving point on the parabola $C$. The minimum value of $|PF| + |PA|$ is \_\_\_\_\_\_. | 3 |
In a shopping mall's promotional event, for every 100 yuan spent, a customer receives one lottery ticket, with more purchases yielding more tickets. A drawing unit consists of 1000 lottery tickets, with 1 grand prize, 10 first prizes, and 50 second prizes available. Let the events of winning the grand prize, first priz... | \frac {989}{1000} |
$4 \cdot 145$ Solve the system of equations
$$
\left\{\begin{array}{l}
x_{1}+x_{2}+x_{3}=6, \\
x_{2}+x_{3}+x_{4}=9, \\
x_{3}+x_{4}+x_{5}=3, \\
x_{4}+x_{5}+x_{6}=-3, \\
x_{5}+x_{6}+x_{7}=-9, \\
x_{6}+x_{7}+x_{8}=-6, \\
x_{7}+x_{8}+x_{1}=-2, \\
x_{8}+x_{1}+x_{2}=2 .
\end{array}\right.
$$ | x_{1}=1,x_{2}=2,x_{3}=3,x_{4}=4,x_{5}=-4,x_{6}=-3,x_{7}=-2,x_{8}=-1 |
$20 \cdot 73$ In a right triangle with hypotenuse of length $h$, the radius of the inscribed circle is $r$. The ratio of the area of the inscribed circle to the area of the triangle is
(A) $\frac{\pi r}{h+2 r}$.
(B) $\frac{\pi r}{h+r}$.
(C) $\frac{\pi r}{2 h+r}$.
(D) $\frac{\pi r^{2}}{h^{2}+r^{2}}$.
(E) None of these.
... | \frac{\pir}{+r} |
$36 \cdot 28$ There are 7 boys and 13 girls standing in a row, let $S$ be the number of positions where a boy and a girl are adjacent, for example, the arrangement $G B B G G G B G B G G G B G B G G B G G$ has $S=12$, then the average value of $S$ is closest to (considering all 20 positions)
(A) 9 .
(B) 10 .
(C) 11 .
(... | 9 |
If the line $y=kx$ is tangent to the circle $x^2+y^2-6x+8=0$, and the tangent point is in the fourth quadrant, then $k=$ ______. | -\frac {\sqrt{2}}{4} |
Let $P(x) = x^2 - 3x - 7$, and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1$. David computes each of the three sums $P + Q$, $P + R$, and $Q + R$ and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If $Q... | 71 |
Given $f(x) = \log_{3}(x+3)$, then $f^{-1}(2) =$ ( )
A: $\log_{3}^{5}$
B: $32$
C: $6$
D: $243$ | C |
p1. A fraction is called Toba-$n$ if the fraction has a numerator of $1$ and the denominator of $n$. If $A$ is the sum of all the fractions of Toba-$101$, Toba-$102$, Toba-$103$, to Toba-$200$, show that $\frac{7}{12} <A <\frac56$.
p2. If $a, b$, and $c$ satisfy the system of equations
$$ \frac{ab}{a+b}=\frac12$$
$... | 6 |
3. In a convex polygon, a quadrilateral has two diagonals, a pentagon has 5 diagonals. Observe and explore how many diagonals a convex decagon has ( ).
(A) 29
(B) 32
(C) 35
(D) 38 | C |
Given that the positive numbers $x$ and $y$ satisfy $x + y = 1$, determine the range of values for $x - y$ and the minimum value for $\frac{1}{x} + \frac{x}{y}$. | 3 |
11. Find the range of real number $a$ such that
$$
\sin 2 \theta-2 \sqrt{2} a \cos \left(\theta-\frac{\pi}{4}\right)-\frac{\sqrt{2} a}{\sin \left(\theta+\frac{\pi}{4}\right)}>-3-a^{2}
$$
holds for $\theta \in\left[0, \frac{\pi}{2}\right]$. | \in(-\infty,1)\cup(3,+\infty) |
On a $6 \times 6$ toy board, each cell represents a light button. When someone presses a button, it turns on if it is off and turns off if it is on. In addition, all buttons that share a side with a pressed button also change their state: from on to off or from off to on. Starting with all buttons off and pressing each... | 20 |
If $(x-2008) + (x+2007)i$ is a pure imaginary number, then $x=$ . | 2008 |
119. Extracting Roots. Once, in a conversation with Professor Simon Greathead, a man of rather eccentric mind, I mentioned the extraction of cube roots.
- Amazing, - said the professor, - what ignorance people show in such a simple matter! It seems that in the extraction of roots, since the only roots were those extra... | 5832,17576,19683 |
Problem: Place the 2004 positive integers $1, 2, \cdots, 2004$ randomly on a circle. By counting the parity of all adjacent 3 numbers, it is known that there are 600 groups where all 3 numbers are odd, and 500 groups where exactly 2 numbers are odd. How many groups have exactly 1 odd number? How many groups have no odd... | 206, 698 |
A line $l$ passes through the focus $F$ of the parabola $C: y^2 = 4x$ and intersects the parabola $C$ at points $A$ and $B$. The range of values for $$\frac {1}{|AF|}+ \frac {1}{|BF|}$$ is ( )
A: $\{1\}$
B: $(0,1]$
C: $[1,+\infty)$
D: $\left[ \frac {1}{2},1\right]$ | A |
Simplify first, then evaluate: $(1-\frac{1}{x})\div \frac{{x}^{2}-2x+1}{{x}^{2}}$, where $x=\sqrt{2}+1$. | 1+\frac{\sqrt{2}}{2} |
1. Paul is filling the cells of a rectangular table alternately with crosses and circles (he starts with a cross). When the table is filled in completely, he determines his score as $X-O$ where $X$ is the sum of squares of the numbers of crosses in all the rows and columns, and $O$ is the sum of squares of the numbers... | 134 |
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