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10. Let $a, b, c$ be distinct positive numbers. Prove that the equations
$$
\begin{array}{l}
a x+y z+z=0, z x+b y+z=0, \\
y z+z x+c=0
\end{array}
$$
yield three distinct sets of real roots for $x, y, z$; and that the ratio of the product of the three values of $x$ to the product of the three values of $y$ is
$$
b(b-c)... | Proof: From the third equation, $z=-\frac{c}{x+y}$. Substituting into the first two equations yields $a x(x+y)=c(y+1)$, $b y(x+y)=c(x+1)$. Clearly, $y=\frac{a x^{2}-c}{c-a x}$. Adding $x$ to both sides gives $x+y=\frac{c(x-1)}{c-a x}$. Substituting into the second equation results in $b\left(a x^{2}-c\right)(x-1)-(x+1)... | \frac{b(b-c)}{a(c-a)} | Algebra | proof | Yes | Yes | cn_contest | false | 703,380 |
Example 1. Find the minimum value of $|x-1|+|x-3|+|x-5|$. | To solve this type of problem, we can use the method of "fixing points, dividing segments, and discussing."
Points. Let $x-1=0, x-3=0, x-5=0$, to determine $x_{1}=1, x_{2}=3, x_{3}=5$ three points.
Dividing segments. The above three points divide the number line into four segments: $(-\infty, 1]$, $(1,3]$, $(3,5]$, $... | 4 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,381 |
Example 2. Solve the system of equations
$$
\left\{\begin{array}{l}
|x|+|y|=2, \\
|x+y|=1
\end{array}\right.
$$ | Solve when $x+y>0$,
$$
\Rightarrow\left\{\begin{array}{l}
|x|+|y|=2, \\
|x+y|=1
\end{array}\right]\left\{\begin{array}{l}
|x|+|1-x|=2, \\
x+y=1 .
\end{array}\right.
$$
The points $x=0, x=1$ divide the number line into three segments: $(-\infty, 0)$, $[0,1)$, $[1,+\infty)$, yielding the solutions of the system of equat... | null | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,382 |
Example 4. Draw the graph of the function $y=\sqrt{(x-2)^{2}}+|x+3|$.
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly.
Example 4. Draw the graph of the function $y=\sqrt{(x-2)^{2}}+|x+3|$. | $$
\begin{aligned}
y & =\sqrt{(x-2)^{2}}+|x+3| \\
& =|x-2|+|x+3|
\end{aligned}
$$
The graph is as above.
(Author's affiliation: Tianjin Nankiao Xiaozhan No.1 High School) | null | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,385 |
Example 1.
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | Let $y=\sqrt[3]{1+3 x}, x=\frac{1}{3}\left(y^{3}-1\right)$, when $x \rightarrow 0$, $y \rightarrow 1$.
Original expression $=\frac{2}{3} \lim _{y \rightarrow 1} \frac{y^{3}-2 y^{3}+1}{y^{3}-3 y+2}$ is still of the form $\frac{0}{0}$. Let $z=y-1$, then,
Original expression $=\frac{2}{3} \lim _{z \rightarrow 0} \frac{\l... | 2 | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 703,386 |
Example 2. Find $\lim _{x \rightarrow \pi / 3} \frac{1-2 \cos x}{\sqrt{3}-2 \sin x} \cdot\left(\frac{0}{0}\right.$ indeterminate form) | Let $u=\operatorname{tg} \frac{x}{2}, x \rightarrow \frac{\pi}{3}$, then $u \rightarrow \sqrt{\frac{3}{3}}$,
$$
\begin{array}{c}
\sin x=\frac{2 u}{1+u^{2}}, \quad \cos x=\frac{1-u^{2}}{1+u^{2}} . \\
\text { Original expression }=\lim _{u \rightarrow \sqrt{3} / 3 \sqrt{3} u^{2}-4 u+\sqrt{3}} \\
u=u-\sqrt{3} / 3 \lim _{u... | -\sqrt{3} | Calculus | math-word-problem | Yes | Yes | cn_contest | false | 703,387 |
Example 1. Prove: $C_{n}^{1}+2 C_{n}^{2}+3 C_{n}^{3}+\cdots$ $+n C_{n}^{n}=n \cdot 2^{n-1}$ | Prove that $(1+x)^{n}=C_{n}^{0}+C_{n}^{1} x+C_{n}^{2} x^{2} +\cdots+C_{n}^{n} x$ "Differentiate with respect to $x$ and then set $x=1$"
This can also be used to prove: $1^{2} C_{n}^{1}+2^{2} C_{n}^{2}+\cdots+n^{2} C_{n}^{n} =n(n+1) 2^{n-2}$.
Two, using integration
Translate the above text into English, please retain ... | null | Combinatorics | proof | Yes | Yes | cn_contest | false | 703,388 |
Example 2. Prove: $\quad \frac{1}{1 \cdot 2} C_{n}^{0}+\frac{1}{2 \cdot 3} C_{n}^{1}+\cdots$
$$
+\frac{1}{(n+1)(n+2)} C_{n}^{n}=\frac{2^{n+2}-n-3}{(n+1)(n+2)} .
$$ | Prove that integrating both sides of $(1+x)^{n}=C_{n}^{0}+C_{n}^{1} x+C_{n}^{2} x^{2} +\cdots+C_{n}^{n} x^{n}$ yields:
$$
\begin{array}{c}
\frac{(1+x)^{n+1}}{n+1}=C_{n}^{0} x+C_{\frac{n}{2}}^{\frac{1}{2}} x^{2}+\cdots \\
+\frac{C_{n}^{n}}{n+1} x^{n+1}+C .
\end{array}
$$
Let $x=0$, we find $C=\frac{1}{n+1}$. Substituti... | proof | Combinatorics | proof | Yes | Yes | cn_contest | false | 703,389 |
Example 1. For $x, y \in R$, prove:
(1) $|\sin x \sin y| \leqslant|x-y|$;
(2) $|\operatorname{arctg} x-\operatorname{arctg} y| \leqslant|x-y|$. | Proof: Without loss of generality, assume $x \geqslant y$, then $\left|\int_{y}^{x} \cos t d t\right|$ $\leqslant \int_{y}^{x}|\cos t| d t \leqslant \int_{y}^{x} d t=\left|\int_{y}^{x} d t\right|$, which means $|\sin x-\sin y| \leqslant|x-y|$. Similarly, we can prove (2). | proof | Calculus | proof | Yes | Yes | cn_contest | false | 703,390 |
Question $\mathrm{A}-1$. Find and prove the maximum value of $f(x)=x^{3}-3 x$, where $x$ is any real number, satisfying $x^{4}+36$ $\leqslant 13 x^{2}$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | $$
\begin{array}{l}
\quad \text { Solve } x^{4}+36 \leqslant 13 x^{2} \Rightarrow\left(x^{2}-4\right)\left(x^{2}-9\right) \\
\leqslant 0 \Rightarrow(x+3)(x+2)(x-2)(x-3) \leqslant 0 \\
\Rightarrow-3 \leqslant x \leqslant-2 \text { or } 2 \leqslant x \leqslant 3 \text {. Therefore, } f(x) \cdots f(y) \\
=\left(x^{8}-y^{3... | null | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,392 |
Question $\mathrm{A}-2$. Find the unit digit of $\left[\frac{10^{2} 0000}{10^{100}+3}\right]$, where $[x]$ is the greatest integer not greater than $x$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | Let $10^{100}=n$, then
$$
\begin{aligned}
A & =\left[\frac{10^{20000}}{10^{100}+3}\right]=\left[\frac{n^{200}}{n+3}\right] \\
& =\left[\frac{n^{200}-3^{200}}{n+3}+\frac{3^{200}}{n+3}\right] .
\end{aligned}
$$
Since $n+3$ divides $n^{25}+3^{25}$, it also divides $n^{50}-3^{50}$, and thus divides $n^{200}-3^{200}$. At t... | null | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 703,393 |
Example 27. Regular $\triangle \mathrm{ABC}$, regular: $\triangle C D E$ and regular $\triangle \mathrm{EHK}$ (vertices are arranged in a clockwise direction) share common vertices $\mathrm{C}$ and $\mathrm{E}$, and $\mathrm{AD}=\mathrm{DK}$.
Prove: $\triangle B H D$ is also an equilateral triangle. | $\begin{array}{l}\text { Prove } \triangle \mathrm{ADC} \xrightarrow{\mathrm{R}\left(\mathrm{C}, 60^{\circ}\right)} \rightarrow \triangle \mathrm{BCE} \text {, } \\ \therefore \mathrm{BE}=\mathrm{AD}=\mathrm{DK} \text {, } \\ \text { (BE, AD) }=60^{\circ} \text {, } \\ \mathrm{E} \xrightarrow{\mathrm{R}\left(\mathrm{H}... | proof | Geometry | proof | Yes | Yes | cn_contest | false | 703,394 |
Question B-2. Proof: The set of triples $T=(x-y, y-z, z-x)$ (where $x, y, z$ are complex numbers) that satisfy the system of equations
$$
\begin{array}{l}
x(x-1)+2 y z=y(y-1)+2 z x \\
=z(z-1)+2 x y
\end{array}
$$
is finite. Find all such triples. | Prove H1 $x(x-1)+2 y z=y(y-1)+2 z x$
we get
$$
\begin{array}{l}
x(x-1)-y(y-1)=2 z x-2 y z, \\
(x-y)(x+y-1)=2 z(x-y), \\
(x-y)(x+y-2 z-1)=0 . \\
\text { Let } x-y=a, y-z=b, z-x=c, \text { then the original }
\end{array}
$$
equation system can be written as
$$
\left\lvert\, \begin{array}{l}
a(b-c-1)=0, \\
b(c-a-1)=0, \\... | (0,0,0), (0,-1,1), (1,0,-1), (-1,1,0) | Algebra | proof | Yes | Yes | cn_contest | false | 703,396 |
1. There are $n$ points $(n>1)$, sequentially denoted as $P_{1}, P_{2}, \cdots, P_{n}$. Connecting these $n$ points, find the number of ways to connect them such that the polyline $P_{1} P_{2} \cdots P_{n}$ does not intersect itself. | Let's first consider the situation where the position of point $P_{n}$ is determined. In this case, there are $2^{n-2}$ ways to connect the points.
When $n=2$, if we fix one point as $P_{2}$, the other point must be $P_{1}$, giving 1 way (when $n=2$, $2^{n-2}=1$). Now, we need to connect $P_{1}$ and $P_{n}$: either by... | n \cdot 2^{n-2} | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false | 703,397 |
4. Sequence $a_{1}, a_{2}, a_{3}, \cdots, H_{1} \cdot a_{1}=1$, $a_{n+1}=\frac{1}{16}\left(1+4 a_{n}+\sqrt{1+24 a_{n}}\right)$, find the general term formula. | Let $b_{n}=\sqrt{1+24 a_{n}}$, then $b_{1}=5$,
| not found | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,400 |
Example 29. Given an equilateral triangle $\mathrm{ABC}$, a line parallel to $\mathrm{AC}$ intersects $A B$ and $B C$ at $M$ and $P$, respectively. $D$ is the centroid of $\triangle P M B$, and $E$ is the midpoint of line segment $\mathrm{AP}$. Calculate the angles of triangle $\mathrm{DEC}$. | $$
\mathrm{B}\left(\mathrm{D},-60^{\circ}, \frac{1}{2}\right)
$$
K.
$$
\begin{array}{l}
P \xrightarrow{S\left(D,-60^{\circ}, \underset{2}{\frac{1}{2}}\right)} \mathrm{H}, \\
\mathrm{S} \xrightarrow{\mathrm{S}\left(\mathrm{D},-60^{\circ}, \frac{1}{2}\right)} \mathrm{R} \text {, } \\
\mathrm{S}\left(\mathrm{D},-60^{\circ... | \angle \mathrm{EDC}=60^{\circ}, \angle \mathrm{DEC}=90^{\circ}, \angle \mathrm{DCE}=30^{\circ} | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 703,402 |
Example 1. (1986 Jilin Province Eight Cities Junior High School Mathematics Competition Question) In Figure 1, the degree of $\angle A+\angle B+\angle C+\angle D$ $+\angle E+\angle F$ is $\qquad$ . | Solve $\begin{aligned} & \angle A+(\angle B+\angle C)+(\angle D+\angle E \\ & +\angle F) \\ = & \angle A+\left(180^{\circ}-\angle B P C\right)+\left(360^{\circ}\right. \\ & -\angle F L D) \\ = & {\left[\angle A+\left(180^{\circ}-\angle B P C\right)\right.} \\ & \left.+\left(180^{\circ}-\angle F L D\right)\right]+180^{\... | 360^{\circ} | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 703,405 |
Example 6. (Problem from the 24th American College Mathematics Competition) Let $UV$ be a chord of circle $O$, and let $M$ be the midpoint of $UV$. $AB$ and $CD$ are two other chords passing through $M$, and $AC$ and $BD$ intersect $UV$ at $P, Q$ respectively. Prove that $M$ is the midpoint of $PQ$.
保留源文本的换行和格式，直接输出翻译... | Proof As shown in Figure 6, let $\angle A=\theta, \angle B=\varphi$, $\angle A M U=\alpha, \quad \angle U M C=\beta, \quad M U=a, \quad M P$ $=p, \quad M Q=q$, then $\angle D=\theta, \angle C=\varphi$, $\angle B M V=\alpha, \angle V M D=\beta, M V=a$. Applying the Law of Sines to $\triangle A M P, \triangle C M P, \tri... | proof | Geometry | proof | Yes | Yes | cn_contest | false | 703,406 |
Example 1. Factorize: $6 x^{2}-x-15$.
| $\begin{array}{l}\text { Solution: Original expression }=6\left(x^{2}-\frac{1}{6} x-\frac{5}{2}\right) \\ =6\left[\left(x^{2}-\frac{1}{6} x+\frac{1}{144}\right)-\left(\begin{array}{l}5 \\ 2\end{array}\right. \\ \left.\left.+\frac{1}{144}\right)\right] \\ =(2 x+3)(3 x-5) .\end{array}$ | (2 x+3)(3 x-5) | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,411 |
Example 2. Factorize in the set of real numbers $R$:
$$
a^{4}-14 a^{2}+25
$$ | \begin{array}{l}\text { Sol } \text { Original expression }=\left(a^{4}-10 a^{2}+25\right)-4 a^{2} \\ =\left(a^{2}-5\right)^{2}-(2 a)^{2}=\left(a^{2}+2 a-5\right) \\ -\cdot\left(a^{2}-2 a-5\right)=(a+1+\sqrt{6})(a+1 \\ -\sqrt{6})(a-1+\sqrt{6})(a-1-\overline{\sqrt{6}}) .\end{array} | (a+1+\sqrt{6})(a+1-\sqrt{6})(a-1+\sqrt{6})(a-1-\sqrt{6}) | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,412 |
Example 3. Factorize: $a^{5}+a+1$.
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | Solve the equation $\begin{aligned} & \left(a^{5}-a^{2}\right)+\left(a^{2}+a+1\right) \\ & =\left(a^{2}+a+1\right)\left(a^{3}-a^{2}+1\right) .\end{aligned}$ | \left(a^{2} | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,413 |
Lake center. Factorize in the set of real numbers $k$:
$$
3 x^{3}-4 x+1
$$ | \begin{aligned} \text { Original expression }= & \left(3 x^{3}-3 x\right)-(x-1) \\ = & (x-1)\left(3 x^{2}+3 x-1\right) \\ = & 3(x-1)\left(x+\frac{3-\sqrt{21}}{6}\right) \\ & \cdot\left(x+\frac{3+\sqrt{21}}{6}\right)\end{aligned} | 3(x-1)\left(x+\frac{3-\sqrt{21}}{6}\right)\left(x+\frac{3+\sqrt{21}}{6}\right) | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,414 |
Example 5. Factorize in the set of real numbers $R$:
$$
(x-1)(x-2)(x-3)(x-4)-24
$$ | \begin{aligned} \text { Sol } & \text { Original expression }=[(x-1)(x-4)][(x-2) \\ & \cdot(x-3)]-24 \\ = & \left(x^{2}-5 x+4\right)\left[\left(x^{2}-5 x+4\right)+2\right]-24 \\ = & \left(x^{2}-5 x+4\right)^{2}+2\left(x^{2}-5 x-4\right)-24 \\ = & {\left[\left(x^{2}-5 x+4\right)+6\right]\left[\left(x^{2}-5 x+4\right)-4\... | x\left(x^{2}-5 x+10\right)(x-5) | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,415 |
Example 6. Factorize in the set of real numbers $R$:
$$
(x+1)(x+2)(x+3)(x+6)-3 x^{2}
$$ | \[
\begin{aligned}
& \text { Original expression }=[(x+1)(x+6)][(x+2) \\
& \cdot(x+3)]-3 x^{2} \\
= & {\left[\left(x^{2}+6\right)+7 x\right]\left[\left(x^{2}+6\right)+5 x\right]-3 x^{2} } \\
= & \left(x^{2}+6\right)^{2}+12 x\left(x^{2}+6\right)+32 x^{2} \\
= & \left(x^{2}+4 x+6\right)\left(x^{2}+8 x+6\right) \\
= & \le... | \left(x^{2}+4 x+6\right)(x+4-\sqrt{10})(x+4+\sqrt{10}) | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,416 |
For example. (Beauty Giant: 26th Y Students Mathematics Competition Questions) In $\triangle A B C$, $\angle A<\angle C<90^{\circ}<\angle B$, construct the external angle bisectors of $\angle A$ and $\angle B$. If the lengths of these two external angle bisectors (from the vertex to the intersection point with the exte... | As shown in Figure 10, let $A D$ and $B E$ be the external angle bisectors of $\angle A$ and $\angle B$ respectively, with $A D = A B = B E$. Let $\angle A = \alpha$. Then, from $B E = A B$, we get $\angle B E A = \alpha$, so $\angle X B E = 2 \alpha$, and $\angle D B A = \angle X B C = 2 \angle X B E = 4 \alpha$. Also... | 12^{\circ} | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 703,417 |
Example 7. Factorize:
$$
x^{4}+y^{4}+z^{4}-2 x^{2} y^{2}-2 x^{2} z^{2}-2 y^{2} z^{2}
$$ | \[
\begin{array}{l}
\text { Solve the original expression }=\left(x^{4}+2 x^{2} y^{2}+y^{4}\right) \\
-2\left(x^{2}+y^{2}\right) z^{2}+z^{4}-4 x^{2} y^{2} \\
=\left(x^{2}+y^{2}-z^{2}\right)^{2}-(2 x y)^{2} \\
=\left[\left(x^{2}+2 x y+y^{2}\right)-z^{2}\right] \\
\text { - }\left[\left(x^{2}-2 x y+y^{2}\right)-z^{2}\rig... | (x+y+z)(x+y-z)(x-y+z)-(x-y-z) | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,418 |
Factorize: $x^{4}+y^{4}+(x+y)^{4}$
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.
---
Factorize: $x^{4}+y^{4}+(x+y)^{4}$ | \begin{aligned} \text { Sol } & \text { Original expression }=\left(x^{4}+2 x^{2} y^{2}+y^{4}\right) \\ + & (x+y)^{4}-2 x^{2} y^{2} \\ = & \left(x^{2}+y^{2}\right)^{2}+(x+y)^{4}-2 x^{2} y^{2} \\ = & {\left[(x+y)^{2}-2 x y\right]^{2}+(x+y)^{4} } \\ & -2 x^{2} y^{2} \\ = & 2(x+y)^{4}-4 x y(x+y)^{2}+2 x^{2} y^{2} \\ = & 2... | null | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,420 |
Example 11. Factorize:
$$
3 x^{2}-7 x y-6 y^{2}-10 x+8 y+8 \text{. }
$$ | Solve $3 x^{2}-7 x y-6 y^{2}$
$$
\begin{array}{l}
=(x-3 y)(3 x+2 y), \\
3 x^{2}-10 x+8=(x-2)(3 x-4), \\
-6 y^{2}+8 y+8=(-3 y-2)(2 y-4) .
\end{array}
$$
The factorization of the above three quadratic trinomials can be expressed in the following form:
$$
\begin{array}{l}
\text { Therefore } 3 x^{2}-7 x y-6 y^{2}-10 x+8 ... | (x-3 y-2)(3 x+2 y-4) | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,422 |
Example 12. Factorize:
$$
\left(3 x^{2}-2 x+1\right)\left(3 x^{2}-2 x-7\right)+12
$$ | Let $y=3 x^{2}-2 x+1$, then
$$
\begin{array}{l}
\text { original expression }=y(y-8)+12 \\
=(y-2) \cdot(y-6) \\
=\left[\left(3 x^{2}-2 x+1\right)-2\right] \\
\cdot\left[\left(3 x^{2}-2 x+1\right)-6\right] \\
=(x-1)(3 x+1)(x+1)(3 x-5),
\end{array}
$$ | (x-1)(3 x+1)(x+1)(3 x-5) | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,423 |
Example 13. Factorize:
$$
(x+y)^{3}+2xy(1-x-y)-1 .
$$ | Let $x+y=u, x y=v$, then
$$
\begin{aligned}
& (x+y)^{3}+2 x y(1-x-y)-1 \\
= & u^{3}+2 v(1-u)-1 \\
= & \left(u^{3}-1\right)-2 v(u-1) \\
= & (u-1)\left(u^{2}+u+1-2 v\right) .
\end{aligned}
$$
Therefore, the original expression $=(x+y-1)\left(x^{2}+y^{2}+x+y+1\right)$. | (x+y-1)\left(x^{2}+y^{2}+x+y+1\right) | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,424 |
Example 15. Factorize:
$$
(a-b)^{3}+(b-c)^{3}+(c-a)^{3} .
$$ | Solution 1 Using term splitting and substitution
$$
\begin{array}{l}
\because a-b=-(b-c)-(c-a), \\
\therefore \text { original expression }=-[(b-c)+(c-a)]^{3} \\
\quad+(b-c)^{3}+(c-a)^{3} \\
=-3(b-c)^{2}(c-a)-3(b-c) \\
\cdot(c-a)^{2} \\
= 3(a-b)(b-c)(c-a) .
\end{array}
$$
Solution 2 Using factor theorem.
$\because$ W... | 3(a-b)(b-c)(c-a) | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,426 |
Example 1. Some people stand in a row, among them, A does not stand at the head, and B does not stand at the end. How many ways are there to arrange them? | (1) 䒴
$P_{4}^{4}$
etc. transmission
(2) one of the middle three positions $P_{3}^{\frac{1}{3}} P \frac{1}{3} P_{3}^{3}$, so the number of ways for not standing at the head and not standing at the tail is $P_{4}^{4}+P{ }_{3}^{1} P_{3}^{1} P_{3}^{3}=78$ (benefit).
Solution two (using the properties 2 and 3 mentioned ea... | 78 | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false | 703,429 |
Example 3. Five people stand in a row, requiring that A does not stand at the head, B does not stand at the end, and C and D do not stand together. How many ways are there to arrange them? | Let $E=\{$ permutations of five people standing in a row $\}$,
$A_{1}=\{$ permutations where A stands at the head $\}$,
$A_{2}=\{$ permutations where B stands at the tail $\}$,
$A_{3}=\{$ permutations where C and D stand together $\}$.
Then the number of permutations required in the problem is
$$
\begin{align... | 50 | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false | 703,431 |
Example 4. Five people stand in a row. When they reline up, none of them stands in their original position. How many different formations are there? | Let the original formation of the five people be
$$
a_{1} a_{2} a_{3} a_{4} a_{5}
$$
$E=\{$ all permutations of five people standing in a line $\}$,
$A_{i}=\{a$ standing in his original position in the permutations $\}$
$$
\text { ( } i=1,2,3,4,5) \text {, }
$$
Then the number of desired permutations is
$$
\begin{arra... | 44 | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false | 703,432 |
Example 1. For the inequality
$$
x^{2}+4 x y+4 y^{2}+10 x+a y+b>0
$$
to hold for all real numbers $x, y$, determine the conditions that the constants $a, b$ must satisfy. | To make the inequality always hold, we need:
$$
x^{2}+4 x y+4 y^{2}+10 x+a y+b
$$
to be a real quadratic form plus a small positive number $\varepsilon$. We can let
$$
\begin{aligned}
& x^{2}+4 x y+4 y^{2}+10 x+a y+b \\
= & (x+2 y+m)^{2}+\varepsilon, \quad(m \in R)
\end{aligned}
$$
which means
$$
\begin{aligned}
& x^... | a=20, b>25 | Inequalities | math-word-problem | Yes | Yes | cn_contest | false | 703,433 |
Example 4. On the sides $AB$, $BC$, and $CA$ of an equilateral triangle $ABC$, there are moving points $D$, $E$, and $F$ respectively, such that $|AD| + |BE| + |CF| = |AB|$. If $|AB| = 1$, when does the area of $\triangle DEF$ reach its maximum value? What is this maximum value?
---
Translating the text as requested,... | Let $|A D|=x,|B E|=y,|C F|=z$, then $|D B|=1-x,|E C|=1-y$, $|F A|=1-z$.
$\therefore S_{\triangle D B P}=S_{\triangle A B C}-\left(S_{\triangle A D P}+\tilde{S}_{\triangle E B D}+S_{\triangle C F I}\right)$.
$S_{\triangle D R}=\sqrt{3}-\frac{\sqrt{3}}{4}[x(1-z)+y(1$
$$
\begin{array}{c}
-x)+z(1-y) \\
=\frac{\sqrt{3}}{4}(... | \frac{\sqrt{3}}{12} | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 703,435 |
Example 1. Given: $F_{1}$, $F_{2}$ are the foci of an ellipse, $P$ is a point on the ellipse,
$$
\begin{array}{l}
\angle F_{1} F_{2} P=\alpha, \\
\angle F_{2} F_{1} P=\beta .
\end{array}
$$
Prove: The eccentricity of this ellipse $e=\frac{\cos \frac{\alpha+\beta}{2}}{\cos \frac{\alpha-\beta}{2}}$. | Prove: As shown in the figure, in $\triangle P F_{1} F_{2}$, according to the Law of Sines, we have
$$
\frac{\left|P F_{1}\right|}{\sin \alpha}=\frac{\left|P F_{2}\right|}{\sin \beta}=\frac{\left|F_{1} F_{2}\right|}{\sin F_{1} P F_{2}} .
$$
By the properties of proportional segments, the sum of the interior angles of ... | proof | Geometry | proof | Yes | Yes | cn_contest | false | 703,436 |
Example 2. Given a point $M$ (different from $C$) on a line $l$ passing through the center $C$ of a circle, and two points $A, B$ on the circle such that $A, B$ are on the same side of $l$, and the lines $AM, BM$ make equal angles with the line $l$. $AB$ intersects $l$ at $N$. Prove: The position of point $N$ is indepe... | Proof As shown in the figure, establish a polar coordinate system with origin $O$ -
Let the radius of $\bigcirc C$ be $\boldsymbol{R}, |O C|=a$, $\angle B M x=\angle A M C=\alpha$, then the equation of $\bigcirc C$ is $a^{2}+\rho^{2}-2 a \rho \cos (\pi-\theta)=R^{2}$, i.e., $\rho^{2}+2 a \cos \theta \cdot \rho+\left(a^... | proof | Geometry | proof | Yes | Yes | cn_contest | false | 703,437 |
Example 2. (IMO-12-6)
In the plane, 100 points are given, no three of which are collinear. We consider all possible triangles formed by these points: Prove that at most $70 \%$ of the triangles are acute-angled. | First, prove: Given any five points, with no three points being collinear, then among the triangles formed by these points, at least three are non-acute triangles. We consider three cases:
(1) The convex hull of the given 5 points is a pentagon $A B C D E$. Then at least two of its interior angles are obtuse. They cou... | proof | Geometry | proof | Yes | Yes | cn_contest | false | 703,439 |
Example 2. Given: $f(x)=2 x+3,|x| \leqslant 2$, and $x \in Z$. Find the range of the function. | $$
\begin{array}{c}
\text { Solve } \because|x| \leqslant 2, \text { and } x \in Z, \\
\therefore x \text { can only be }-2,-1,0,1,2 . \\
\text { Also } \because f(-2)=-1, f(-1)=1, f(0) \\
=3, f(1)=5, f(2)=7,
\end{array}
$$
$\therefore$ the range of the function is $\{-1,1,3,5,7\}$.
$$ | \{-1,1,3,5,7\} | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,440 |
Example 3. Find the range of $y=\frac{3 x-1}{3 x-2}$.
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly.
Example 3. Find the range of $y=\frac{3 x-1}{3 x-2}$. | Solve: From $y=\frac{3 x-1}{3 x-2}$ we get $x=\frac{2 y-1}{3 y-3}$.
To make $x$ meaningful, it is necessary that $3 y-3 \neq 0$, i.e., $y \neq 1$.
Therefore, the range of the function is all real numbers except $y \neq 1$.
Four, Using a quadratic function to find the range
If $y=a x^{2}+b x+c \quad(a \neq 0)$, then the... | y \neq 1 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,441 |
Column 4. Find the range of $y=2 x^{2}-x-3$.
| \begin{array}{l}\because a=2>0, \\ \therefore y \geqslant \frac{4 \times 2 \times(-3)-(-1)^{2}}{4 \times 2}=-3 \frac{1}{8} .\end{array} | -3 \frac{1}{8} | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,442 |
Example 5. Find the range of $y=\frac{1}{x^{2}+x+1}$. | Solve: The original equation can be transformed into $y x^{2}+y x+y-1=0$.
$$
\begin{array}{l}
\because x \in R, \\
\therefore y^{2}-4 \times y \times(y-1) \geqslant 0,
\end{array}
$$
which is $3 y^{2}-4 y \leqslant 0$.
Since $y_{1}=0, y_{2}=\frac{4}{3}$ are the two roots of $3 y^{2}-4 y=0$, and $a=3>0$,
$\therefore$ t... | 0 \leqslant y \leqslant \frac{4}{3} | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,443 |
Example 6. Find the range of $y=x+\sqrt{1-2 x}$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | Solve: The original expression can be transformed into
$$
y=-\frac{1}{2}(1-2 x)+\sqrt{1-2 x}+\frac{1}{2} \text {. }
$$
Let $u=\sqrt{1-2 x}$, then $u \geqslant 0$, and
$$
\begin{aligned}
y & =-\frac{1}{2} u^{2}+u+\frac{1}{2} \\
& =-\frac{1}{2}(u-1)^{2}+1 .
\end{aligned}
$$
$\therefore y \leqslant 1$, i.e., the value ra... | null | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,444 |
Example 1. If $|x-1|+|y-1|=1$, find the extremum of $x^{2}+y^{2}$.
untranslated text remains in its original format and line breaks are preserved. | Solve $|x-1|+|y-1|=1$ whose graph is shown in Figure 1. Among the family of circles centered at the origin that have common points with the graph of $|x-1|+|y-1|=1$, the radius of the circle passing through point $A$ is $\frac{\sqrt{2}}{2}$, which is the minimum, and the radius of the circle passing through point $B$ i... | \frac{1}{2} \text{ and } 5 | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 703,446 |
Example 2. If $2 x^{2}+3 y^{2}=1$, find the extremum of $x+y$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.
However, it seems you might have wanted the translation of the problem statement itself, which is provided below:... | Solve $2 x^{2}+3 y^{2}=1$; the graph is an ellipse (see Figure 2). For the family of lines $x+y=c$ that have common points with the ellipse $2 x^{2}+3 y^{2}=1$, the lines for which $c$ takes extreme values must be tangent to the ellipse.
From $x+y=c$, we get $y=c-x$.
Substituting $y=c-x$ into $2 x^{2}+3 y^{2}=1$, we ge... | c=\pm \frac{5 \sqrt{6}}{6} | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,447 |
Example 4. Given $(x-2)^{2}+(y-2)^{2}=1$, find the extremum of $x \cdot y$.
| Solve: The graph of $(x-2)^{2}+(y-2)^{2}=1$ is a circle with center at point $(2,2)$ and radius 1. This circle is symmetrical about the line $y=x$, so the hyperbola family $y=\frac{c}{x}$ that intersects the circle $(x-2)^{2}+(y-2)^{2}=1$ has extremal values of the coefficient at the hyperbolas passing through points $... | \frac{9-4 \sqrt{2}}{2} \text{ and } \frac{9+4 \sqrt{2}}{2} | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,449 |
Example 3. (IMO-21-6)
$A, E$ are opposite vertices of a regular octagon. A frog starts jumping from point $A$. If the frog is at any vertex other than $E$, it can jump to either of the two adjacent vertices. When it reaches $E$, it stops there. Let $e_{n}$ be the number of different paths that reach $E$ in $n$ steps. P... | Proof As shown in the figure, let the regular octagon be $A B C D E F G H$. The number of paths from $A$ to $B, C, D, A$ after $n$ steps (the meaning is as in the original note, just change $E$ to $B, C, D, A$) are denoted as $b_{n}, c_{n}, d_{n}, a_{n}$ respectively. Due to symmetry, the number of paths from $A$ to $H... | proof | Combinatorics | proof | Yes | Yes | cn_contest | false | 703,450 |
Example 5. If $x \cdot y=4(x>0, y>0)$, find the extremum of $2 x+3 y$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | Solve: As shown in the figure, $5, x \cdot y=4$ is a hyperbola passing through the point $(2,2)$.
Let $2 x+3 y=c$,
Obviously, when the line $2 x+3 y=c$ is tangent to $x \cdot y=4$, the y-intercept $\frac{c}{3}$ is the smallest, yielding $\left\{\begin{array}{l}2 x+3 y=c, \\ x \cdot y=4 .\end{array}\right.$
Solving thi... | null | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,451 |
甽6. If $x+3y=10$, find the extremum of $x^{2}+3y^{2$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
6. If $x+3y=10$, find the extremum of $x^{2}+3y^{2}$. | Solve $x+3 y$
$=10$ as the line in Figure 6. Let $x^{2}+3 y^{2}=c$, then
$$
\frac{x^{2}}{(\sqrt{c})^{2}}+\frac{y^{2}}{\left(\sqrt{\frac{c}{3}}\right)^{2}}=1 \text {. }
$$
From Figure 6, we know that when the line $x+3 y=10$ is tangent to the ellipse $x^{2}+3 y^{2}=c$, the semi-axes of the ellipse are the shortest. We ... | 25 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,452 |
11. A three-digit number in base seven, when expressed in base nine, has its digits in reverse order of the original number. Find this number. | Let the three digits of this number in base 7 be $x$, $y$, and $z$. According to the problem, we have
$$
49 x + 7 y + z = 81 z + 9 y + x,
$$
which simplifies to $24 x - y - 40 z = 0$,
$$
y = 8(3 x - 5 z).
$$
Since $y$ must be less than 7 and $3 x - 5 z$ is an integer, $y = 0$ and $3 x - 5 z = 0$ must hold for the equ... | 503 | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 703,453 |
i2. Proof: The cube of an integer can be expressed as the difference of two squares. The cube of an odd integer can be expressed in two ways, and the difference of the cubes of any two consecutive integers can also be expressed as the difference of two squares.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
i2. Proof: The cu... | Prove that $\quad n^{3}=n^{2} \cdot n=n^{2}\left\{\left(\frac{n+1}{2}\right)^{2}\right.$ $\left.-\left(\frac{n-1}{2}\right)^{2}\right\}=\frac{n^{2}(n+1)^{2}}{4}-\frac{n^{2}(n-1)^{2}}{4}$ holds for both odd and even $n$.
If $n$ is odd, $n^{3}$ can also be expressed as $\left(\frac{n^{3}+1}{2}\right)^{2}-\left(\frac{n^{... | proof | Number Theory | proof | Yes | Yes | cn_contest | false | 703,454 |
13. To complete a certain project, the number of days required for $A$ to work alone is $m$ times that of $B$ and $C$ working together, the number of days required for $B$ to work alone is $n$ times that of $A$ and $C$ working together, and the number of days required for $C$ to work alone is $p$ times that of $A$ and ... | Let $x$, $y$, $z$ be the number of days required for $A$, $B$, $C$ to complete the work individually, respectively, then we have
$$
\frac{m}{x}=\frac{1}{y}+\frac{1}{z}
$$
Therefore, $\frac{m+1}{x}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$.
Similarly, $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{n+1}{y}=\frac{p+1}{z}$.
Thus, ... | 2 | Algebra | proof | Yes | Yes | cn_contest | false | 703,455 |
14. A number is the product of three prime factors. The squares of these three prime factors; the sum is 2331, the number 7560 is less than this number and is coprime with it, and the sum of its divisors (excluding 1 itself) is 10560. Find this number. | Let $c, b, c$ be the prime factors of a number, then
$$
a^{2}+b^{2}+c^{2}=2331.
$$
In addition, we know that the number of integers less than this number and coprime to it is
$$
\begin{array}{l}
a b c\left(1-\frac{1}{a}\right)\left(1-\frac{1}{b}\right)\left(1-\frac{1}{c}\right). \\
\text { That is, }(a-1)(b-1)(c-1)=75... | 8987 | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 703,456 |
15. Between two stations $A$ and $B$ on a double-track railway, there are two down trains departing at 6:00 and 6:45, and two up trains departing at 7:15 and 8:30. If these four trains pass each other simultaneously, find the equations formed by their speeds $x_{1}, x_{2}, x_{3}, x_{4}$ as follows:
$$
\frac{3 x_{2}}{x_... | Let $T_{1}, T_{2}$ be the trains going downhill, and $T_{3}, T_{4}$ be the trains going uphill. They meet at 6 o'clock in the morning. Then, the distances traveled by $T_{1}, T_{2}, T_{3}, T_{4}$ when they meet are $x_{1} y, x_{2}\left(y-\frac{3}{4}\right), x_{3}\left(y-\frac{5}{4}\right)$, $z_{1}\left(y-\frac{5}{2}\ri... | 4 y=\frac{3 x_{2}}{x_{2}-x_{1}}=\frac{4 m+5 x_{3}}{x_{1}+x_{3}}=\frac{4 m+10 x_{4}}{x_{1}+x_{4}} | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,457 |
16. $A, B$ together hunted 10 birds, the product of the squares of the number of bullets used by the two is 2880, and the product of the number of bullets used is 48 times the product of the number of birds they hunted. If the number of bullets used by the two were exchanged, then $B$ would get 5 more than $A$, find ho... | Let $x, y$ represent the number of bullets used by $A, B$ respectively. Let $A$ use $u$ bullets to hit 1 bird, and $B$ use $v$ bullets to hit 1 bird. Then $\frac{x}{u}$ and $\frac{y}{v}$ represent the number of birds obtained. Thus, we have
$$
\begin{array}{l}
x^{2}+y^{2}=2880, \\
x y=\frac{48 x y}{u v}, \text { i.e., ... | A \text{ obtains 6 birds, and } B \text{ obtains 4 birds.} | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,458 |
17. Find the real roots of the following equations:
$$
\begin{array}{l}
x^{2}+v^{2}+w^{2}=a^{2}, v w+u(y+z)=b c, \\
y^{2}+w^{2}+u^{2}=b^{2}, w u+v(z+x)=c a, \\
z^{2}+u^{2}+v^{2}=c^{2}, u v+w(x+y)=a b .
\end{array}
$$ | $$
\text { Solve } \begin{array}{c}
\left(y^{2}+w^{2}+u^{2}\right)\left(z^{2}+u^{2}+v^{2}\right) \\
=b^{2} c^{2}=(v w+u y+u z)^{2} 。
\end{array}
$$
Simplifying, we get
$$
\left(u^{2}-y z\right)^{2}+(w u-v y)^{2}+(u v-w z)^{2}=0 \text { . }
$$
Since the roots are real, it is necessary that
$$
u^{2}-y z=0, w u-v y=0, u... | x= \pm \frac{a^{2}}{\sqrt{a^{2}+b^{2}+c^{2}}}, \quad u= \pm \frac{b c}{\sqrt{a^{2}+b^{2}+c^{2}}} | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,459 |
18. Solve the system of equations: $\left\{\begin{array}{l}x+y+z-u=12, \\ x^{2}+y^{2}+z^{2}-u^{2}=6, \\ x^{3}+y^{3}+z^{3}+u^{3}=218, \\ x y+z u=45 .\end{array}\right.$ | From the second and fourth equations, we can obtain
$$
(x+y)^{2}+(z-u)^{2}=96 \text {. }
$$
But $(x+y)+(z-u)=12$,
thus $(x+y)-(z-u)=8$.
It is easy to get $x+y=10, z-u=2$.
Therefore, $x^{\circ}+y^{\prime \prime}=(x+y)^{3}-3 x y(x+y)$
$$
\begin{array}{l}
=1000-30 x y, \\
z^{3}-u^{3}=(z-u)^{3}+3 z u(z-u)=8+6 z u
\end{arr... | x+y=10, x y=21, z-u=2, z u=24 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,460 |
Example 4. (IMO-14-2)
Prove that for $n \geqslant 4$, any cyclic quadrilateral can be divided into $n$ cyclic quadrilaterals. | Prove that if a cyclic quadrilateral is an isosceles trapezoid, it can be divided into any number of isosceles trapezoids by a line parallel to the bases, hence the proposition holds for isosceles trapezoids.
Assume that the cyclic quadrilateral \(ABCD\) is not an isosceles trapezoid, and without loss of generality, l... | proof | Geometry | proof | Yes | Yes | cn_contest | false | 703,461 |
19. From the equations $x^{-1}+y^{-1}+z^{-1}=a^{-1}$, $x+y+z=b$, $x^{2}+y^{2}+z^{2}=c^{2}$, $x^{3}+y^{8}+z^{3}=d^{3}$, eliminate $x, y, z$, and prove that if $x, y$ are absolutely unequal finite numbers, then $b$ cannot equal $d$. | From the first equation, we get
$$
a(y z+z x+x y)=x y z .
$$
From the second and third equations, we get
$$
\begin{array}{l}
2(y z+z x+x y) \\
=(x+y+z)^{2}-\left(x^{2}+y^{2}+z^{2}\right)=b^{2}-c^{2} .
\end{array}
$$
From the above, we can get $d^{3}=x^{3}+y^{3}+z^{3}=(x+y+z)^{3}$
$$
\begin{array}{l}
-3(x+y+z)(y z+z x... | b \neq d | Algebra | proof | Yes | Yes | cn_contest | false | 703,462 |
20. Prove that the roots of the equation $x^{4}-p x^{3}+q x^{2}-r x+\frac{r^{2}}{p^{2}}$ $=0$ are in proportion, and thus solve the equation
$$
x^{4}-12 x^{3}+47 x^{2}-72 x+36=0
$$ | To prove that for the equation $x^{4}-p x^{3}+q x^{2}-r x+s=0$, the roots are in proportion such that $s=\frac{r^{2}}{p^{2}}$.
Let $a, b, c, d$ be the roots of the equation, then $\frac{a}{b}=\frac{c}{d}=k$. And $a+b+c+d=p, a b c+a b d+a c d+b c d=r, a b c d=s$,
i.e., $(b+d)(1+k) \times p, b d k(b+d)(1+k)=r, b^{2} d^{... | a=6, b=2, c=3, d=1 | Algebra | proof | Yes | Yes | cn_contest | false | 703,463 |
Example 2. Find the domain of the function
$$
y=\frac{\sqrt{(4-x)(x+3)}\left(\sqrt{x^{2}-1}\right)^{0}}{(x-3) \lg (x+2)}
$$
The domain of this example is selected from the book "Comprehensive Application of Middle School Mathematics" (edited by Wei Jingnong and Zhang Wenxiong, Zhejiang Science and Technology Press). O... | To make the function defined, it is necessary and sufficient that
$$
\begin{array}{l}
\left\{\begin{array}{l}
(4-x)(x+3) \geqslant 0, \\
x^{2}-1>0, \\
(x-3) \lg (x+2) \neq 0, \\
x+2>0
\end{array}\right. \\
\Leftrightarrow\left\{\begin{array}{l}
-3 \leqslant x \leqslant 4, \\
x \neq \pm 1, \\
x \neq -1 \text { and } x \... | \{x \mid -2<x<-1 \text{ or } 1<x \leqslant 4 \text{ and } x \neq 3\} | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,466 |
Example 3, if $f(2)$ has a domain of $[-1, 0]$, try to find the domains of the functions $j(-x)$, $f\left(x^{2}\right)$, $f\left(2^{x}\right)$, $f\left(\lg x^{2}\right)$, $f(20 \arcsin x)$. | Solve $-1 \leqslant -x \leqslant 10, -10 \leqslant x \leqslant 1$ for the domain of $f(-x)$, the range of $x^{2}$ is $0 \leqslant x^{2} \leqslant 10$, the domain of $f\left(x^{2}\right)$ is $-\sqrt{10} \leqslant x \leqslant \sqrt{10}$. Similarly, the domains of the next three functions are $\left(-\infty, \log _{2} 10\... | not found | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,467 |
Example 2. Find the range of the function $y=\frac{x+1}{x^{2}+x+1}$. | Solve the inequality system $-3 y^{2}+2 y+1 \geqslant 0, y \neq 0$. The solution set is $-\frac{1}{3} \leqslant y \leqslant 1, y \neq 0$. However, when $x=-1$, $y=0$. By including $\{0\}$, the range becomes $-\frac{1}{3} \leqslant y \leqslant 1$.
2. $b^{2}-4 a c \geqslant 0$. In this case, $a x^{2}+b x+c$ has roots $x_... | -\frac{1}{3} \leqslant y \leqslant 1 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,469 |
Example 3. Find the range of the function $y=\frac{x^{2}+x+1}{x+2}$. | Solving $x+2$ is not a factor of $x^{2}+x+1$. Therefore, solving the inequality $(1-y)^{2}-4(1-2 y) \geqslant 0$ yields the range $(-\infty,-3-4 \sqrt{3}] \cup[3+4 \sqrt{3}$, $+\infty$).
(Author's affiliation: Tianjin Hebei District Jiaoshi Jinjin School) | (-\infty,-3-4 \sqrt{3}] \cup [3+4 \sqrt{3}, +\infty) | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,470 |
Example 2. On the parabola $y=2 x^{2}$, there is a moving point $A$ ( $m$, $\left.2 m^{2}\right), m \in\left[\frac{1}{2}, 1\right]$ and $B\left(n, 2 n^{2}\right)$. The tangent line at point $A$ is perpendicular to $A B$. Find the maximum and minimum values of $n$. | Solve: The slope of the tangent line to the parabola $y=2 x^{2}$ at $A\left(m, 2 m^{2}\right)$ is $4 m, A B$'s slope is $\frac{2 n^{2}-2 m^{2}}{n-m}$ $=2(n+m)$. From the condition $4 m \cdot 2(n+m)$ $=-1, n=-\left(m+\frac{1}{8 m}\right)$. Therefore, $n$ on $\left[-\frac{1}{2}, 1\right]$
has the maximum and minimum val... | -\frac{3}{4} \text{ and } -\frac{5}{8} | Calculus | math-word-problem | Yes | Yes | cn_contest | false | 703,475 |
Example 1. The function $y=\log _{0.5}\left(x^{2}+4 x+4\right)$ is increasing in what interval? | \[
\begin{array}{l}
\quad \text { Solve } y=\log _{0.5}\left(x^{2}+4 x+4\right) \\
=\log _{0.5}(x+2)^{2} \text { The domain is } x \neq-2 . \\
f_{2}(x)=(x+2)^{2} \text { is decreasing in }(-\infty,-2) , \\
f_{1}(x)=\log _{0.5} x \text { is decreasing in } f_{2}[(-\infty,-2)] \\
=(0,+\infty) \text { . Therefore, } y=\lo... | (-\infty,-2) | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,476 |
The equation about $x$, $7 x^{2}-(k+13) x + k^{2}-k-2=0$, has two real roots $x_{1}, x_{2}$, and $00$.
Since $0<x_{1}<1<x_{2}<2$, we have
$$
1<x_{1}+x_{2}=\frac{k+13}{7}<3,
$$
$$
0<x_{1} x_{2}=\frac{k^{2}-k-2}{7}<2 \text {. }
$$ | Solve (1), (2), and (3) to get:
$$
\frac{3-2 \sqrt{21}}{3}0,
\end{array}
$$
The graph opens upwards, because
$$
f\left(x_{1}\right)=f\left(x_{2}\right)=0
$$
(Figure 2), hence \( f(0) > 0 \) and \( f(1) < 0 \), which leads to the system of inequalities:
$$
\left\{\begin{array}{l}
k^{2}-k-2>0, \\
k^{2}-2k-80 < 0.
\end{a... | \frac{3-2 \sqrt{21}}{3} < k < \frac{3+2 \sqrt{21}}{3} | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,478 |
Example 1. Given $f\left(1-x^{2}\right)=\frac{1-x^{2}}{x^{2}}$. Find $f\left(\frac{1}{2}\right)$. | $\begin{array}{l}\text { Solve } f\left(1-x^{2}\right)=\frac{1-x^{2}}{1-\left(1-x^{2}\right)}, \\ \text { hence } f(x)=\frac{x}{1-x}, f\left(\frac{1}{2}\right)=1.\end{array}$ | 1 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,480 |
Example 2. Let $f(x+1)=2 x^{3}-5 x+1$, find $f(x)$.
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | Let $t=x+1, x=t-1$,
$$
\begin{array}{l}
f(t)=2 t^{3}-6 t^{2}+t+4 \\
f(x)=2 x^{3}-6 x^{2}+x+4
\end{array}
$$ | null | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,481 |
Example 4. Let $f(x)$ be a linear function of $x$, and $f[f(x)]=9 x+8$, find $f(x)$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | Solve for $f(x)=a x+b$, substitute into the original equation: $a= \pm 3, b=2$ or -4. | f(x) = 3x + 2 \text{ or } f(x) = -3x - | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,484 |
Example 2. Prove: $\cos \frac{\pi}{2 n}+\cos \frac{3 \pi}{2 n}$
$$
\begin{array}{l}
+\cos \frac{5 \pi}{2 n}+\cdots+\cos \frac{(4 n-1) \pi}{2 n} \\
=0 \\
\quad \sin \frac{\pi}{2 n}+\sin \frac{3 \pi}{2 n}+\sin \frac{5 \pi}{2 n} \\
+\cdots+\sin \frac{(4 n-1) \pi}{2 n}=0 .
\end{array}
$$ | Proof Consider the binomial equation $x^{2 n}+1=0$. From $x^{2 n}=-1=\cos \pi+i \sin \pi$, we get
$$
\begin{array}{l}
x_{k}=\cos \frac{(2 k+1) \pi}{2 n} \\
+i \sin \frac{(2 k+1) \pi}{2 n} \\
(k=0,1,2, \cdots, 2 n-1)
\end{array}
$$
That is $\square$
$$
\begin{array}{l}
x_{0}=\cos \frac{\pi}{2 n}+i \sin \frac{\pi}{2 n},... | proof | Algebra | proof | Yes | Yes | cn_contest | false | 703,486 |
Example 3. Draw a chord $AB$ of the curve $\left\{\begin{array}{l}x=4 \cos \theta, \\ y=2 \sin \theta\end{array}\right.$ (where $\theta$ is a parameter) through the point $M(2,1)$, and $M$ is the point that trisects $AB$. Find the equation of $AB$. | Here, the curve should use the ordinary equation: $x^{2}+4 y$ $=16, AB$ should use the standard parametric form (as it involves chord length): $\left\{\begin{array}{l}x=2+t \cos a, \\ y=1+t \sin \alpha .\end{array}\right.$
( $t$ is the parameter) Substituting it into the curve equation and simplifying:
$$
\begin{array}... | y-1=-\frac{4 \pm \sqrt{7}}{6}(x-2) | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 703,487 |
1. Find all positive integers $m, n$, such that $(m+n)^{m}=n^{m}+1413$ | Given that positive integers $m, n$ satisfy $(m+n)^{m}=n^{m} +1413$, since $(m+n)^{m} \geqslant m^{m}+n^{m}$, it must be that $m^{m} \leqslant 1413$.
Thus, $m \leqslant 4$. At most, when the positive integer $n, n$ satisfies $(n +n)^{m}=n^{m}+1413$, $m$ cannot be an even number. Otherwise, regardless of whether $n$ is... | m=3, n=11 | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 703,488 |
2. Let $p$ be a prime number greater than 3, and for some natural number $n$, the number $p^{n}$ is exactly a 20-digit number. Prove: This number contains at least three identical digits. | Proof by contradiction. Assume that among the 20 digits of $p^{n}$, no three or more are the same, then at most two digits are the same. Thus, the 10 digits $\{0,1, 2, \cdots, 9\}$, in the 20 positions of $\boldsymbol{p}^{n}$, each digit appears exactly twice (otherwise, $p^{n}$ would have fewer than 20 digits). Theref... | proof | Number Theory | proof | Yes | Yes | cn_contest | false | 703,489 |
3. Given a
convex pentagon with
all interior angles being obtuse. Prove: it is possible to find two diagonals of this pentagon, such that the two circles with these diagonals as diameters can completely cover this pentagon. | Consider the diagonals $A C, A D$ as diameters, with $\mathrm{O}_{1}, \mathrm{O}_{2}$ (see figure) as the centers of $\odot \mathrm{O}_{1}, \odot \mathrm{O}_{2}$. It is sufficient to prove that $\odot O_{1}, \odot O_{2}$ can cover the convex pentagon.
Since $\angle B>90^{\circ}$, $\triangle A B C$ can be covered by $\... | proof | Geometry | proof | Yes | Yes | cn_contest | false | 703,490 |
4. There are 1987 glass pieces, each coated with one of three colors: red, yellow, or blue. Perform the following operation: wipe two pieces of different colors clean, then coat them with the third color (for example, wipe the red and blue off a blue piece and a red piece, then coat both pieces with yellow). Prove (1) ... | Let the number of red, yellow, and blue chips be $x$, $y$, and $z$ respectively. Among the remainders when $x$, $y$, and $z$ are divided by 3, there must be two that are equal.
In fact, without loss of generality, let $x=3a+1$, $y=3b+2$, $z=3c$, where $a$, $b$, and $c$ are integers. Then,
$$
x+y+z=3(a+b+c+1) \neq 1987... | proof | Logic and Puzzles | proof | Yes | Yes | cn_contest | false | 703,491 |
5. On the sides $AB$ and $BC$ of the square $ABCD$, take points $P$ and $Q$ respectively, and connect $DP$, $DQ$, and $PQ$. Let the areas of $\triangle DPQ$, $\triangle DAP$, $\triangle DQC$, and $\triangle PBQ$ be $S_{1}$, $S_{2}$, $S_{3}$, and $S_{4}$ respectively. When do the points $P$ and $Q$ achieve the minimum v... | Let's assume the side length of the square is 1, and establish a coordinate system as shown. Let $P(0, b), Q(a, 0)$, then,
$$
\begin{array}{l}
S_{2}=\frac{1}{2}(1-b), \\
S_{3}=\frac{1}{2}(1-a), \\
S_{4}=\frac{1}{2} a b, \\
S_{1}=1-S_{2}-S_{3}-S_{4} . \\
\quad=\frac{1}{2}(a+b-a b) .
\end{array}
$$
Thus,
$$
\begin{align... | \frac{9}{32} | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 703,492 |
$6 . a_{1}, a_{2}, \cdots, a_{2} n$ are $2 n$ distinct integers. If the equation
$$
\begin{array}{c}
\left(x-a_{1}\right)\left(x-a_{2}\right) \cdots\left(x-a_{2 n}\right) \\
+(-1)^{n-1}(n!)^{2}=0 \text { has an integer solution } r \text {, prove that: } \\
r=\frac{a_{1}+a_{2}+\cdots+a_{2 n}}{2 n} .
\end{array}
$$ | From the given, we know
$$
\begin{array}{l}
\left(r-a_{1}\right)\left(r-a_{2}\right) \cdots\left(r-a_{2 n}\right) \\
=(-1)^{n}(n!)^{2} .
\end{array}
$$
Note: \( r-a_{1}, r-a_{2}, \cdots, r-a_{2 n} \) are distinct integers. Taking the absolute value on both sides, we get \( \left|r-a_{1}\right|\left|r-a_{2}\right| \cdo... | r=\frac{a_{1}+a_{2}+\cdots+a_{2 n}}{2 n} | Algebra | proof | Yes | Yes | cn_contest | false | 703,493 |
Example 1. Five points in a square
In a square with a side length of 1. Regardless of how five points are placed, prove: there must be two points among them, the distance between which is no more than $\sqrt{2}$. | Prove that connecting the midpoints of two pairs of opposite sides of a square divides it into four equal smaller squares. Placing any five points inside the larger square is equivalent to placing these five points in any determined way into these four smaller squares. Here, we consider each smaller square as a "drawer... | proof | Geometry | proof | Yes | Yes | cn_contest | false | 703,495 |
Example 2. How Many Friends Do You Know?
In any given gathering, there must be two people who have the same number of friends. | Proof: Let the number of people attending this gathering be $n$. For any one of these people, it is possible that they have no friends at all, meaning the other $n-1$ people are all strangers to them; however, they can have at most $n-1$ friends, which means they know everyone else at the gathering. We denote the numbe... | proof | Combinatorics | proof | Yes | Yes | cn_contest | false | 703,496 |
2. A doctor is trying a new drug. He gives 48 tablets to one of his patients, who must finish these tablets within thirty days, taking at least one tablet per day. As long as the patient follows the doctor's instructions, for any integer $k(1 \leqslant k \leqslant 11)$, there will certainly be a consecutive period of $... | (Let $p_i$ represent the number of pills the patient takes from day 1 to day $i$, under the condition, it is easy to know that
$$
1 \leqslant p_{1}<p_{2}<p_{3}<\cdots<p_{30}=48 \text {. }
$$
By adding 11 to each term in the above sequence, we get
$$
\begin{array}{l}
12 \leqslant p_{1}+11<p_{2}+11<\cdots<p_{30}+11 \\
=... | proof | Combinatorics | proof | Yes | Yes | cn_contest | false | 703,497 |
Example 1. Find the integer solutions for $4 x+7 y+14 z=40$. | $$
\begin{array}{l}
\text { Solve } x=\frac{1}{4}(40-7 y-14 z) \\
=10-y-3 z-\frac{1}{4}\left(3 y+2 z\right) . \\
\text { Let } \frac{3 y+2 z}{4}=u \quad(u \in Z) \text {, then } \\
2 z=4 u - 3 y \text {, } \\
z=2 u-y-\frac{y}{2} \text {. } \\
\text { Let } y=2 t \quad(t \in Z) \text {. } \\
\end{array}
$$
Substitute b... | \begin{array}{l}
x=10-7 u+7 t, \\
y=2 t, \\
z=2 u-3 t .
\end{array} | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 703,498 |
Example 2. Given an integer $n$, if $n$ plus 38 is a perfect square, prove that $n$ plus 38 is a perfect square, find $n$.
The text seems to have some typographical errors or unclear phrasing. Here is a corrected version for clarity:
Example 2. Given an integer $n$, if $n$ plus 38 is a perfect square, prove that $n$ ... | Let $n-51=x^{2},(x, y \in Z)$
$$
n+38=y^{2} .
$$
Subtracting the equations yields
$$
\begin{array}{l}
(y-x)(y+x)=89=1 \cdot 89 \text { ? } \\
=(-1)(-89) .
\end{array}
$$
Thus, $\left\{\begin{array}{l}y+x=1,89,-1,-89, \\ y-x=89,1,-89,-1 .\end{array}\right.$
Therefore, $\left\{\begin{array}{l}x=-44,41,41, \quad 41, \\ ... | 1987 | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 703,499 |
Example 3. Let $x$ and $y$ both be natural numbers, 11. $x<y, xy$ is a four-digit natural number, the first digit of which is 2. If this first digit 2 is removed, the remaining number is exactly $x+y$. Try to find these two numbers. (From a 1983 competition question) | Solve according to the problem statement:
$$
x y=2000+x+y \text{. }
$$
Decomposing, we get: $(x-1)(y-1)=2001=3 \times 23 \times 29$.
Thus, $(x-1)(y-1)=29 \times 69$ (or $23 \times 87$).
The rest can be easily completed by the reader. | null | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 703,500 |
Example 5. Find the integer solutions of the equation $1!+2!+3!+\cdots+x!=y^{2}$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | Solve $\because 1!=1,1!+2!=3$, $1!+2!+3!=9, \quad 1!+2!+3!+4!=33$.
When $n \geqslant 5$, the last digit of $n!$ is 0,
$\therefore$ the last digit of $1!+2!+3!+\cdots+n!$ is 3.
According to the property of square numbers (1). It must be $n<5$, and it is evident that the integer
solutions of the equation are $\left\{\beg... | null | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 703,502 |
Example 7. Find the integer solutions of $x^{2}-6 y^{2}=1$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | Solve $x^{2}=1+6 y^{2}$, substituting $y=1,2,3, \cdots$ to make $1+6 y^{2}$ a perfect square, the smallest positive value is $y=2$, yielding the particular solution $\left\{\begin{array}{l}x=5, \\ y=2 \text {. }\end{array}\right.$ Then the other integer solutions can be given by $x+y \sqrt{6}= \pm(5 +2 \sqrt{6})^{n}$. ... | \left\{\begin{array}{l}x=5, \\ y=2 \text {. }\end{array}\right.} | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 703,504 |
Example 8. There are two four-digit numbers, Jia and Yi. The common logarithm of Yi is $A+\lg B$, where $A$ and $B$ are natural numbers. The sum of the thousand's and hundred's digits of Jia equals $5B$, and the sum of the unit's and ten's digits of Jia equals the difference between Yi and Jia plus $B$. Find the two nu... | Let the number A be $x y z u$, (where $x, y, z, u$ are all single-digit natural numbers).
From $x+y=5 B$, using the digit domain: $0<x+y \leqslant 18$, i.e., $53 \leqslant 18$. It can be known that $1 \leqslant B \leqslant 3$, thus the number is $B \times 10^{4}$ and $A=3$.
Since the difference between the second num... | 1987, 2000 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,505 |
Example 9. Find all integer values of $x$ that make $y=\frac{x^{2}-2 x+4}{x^{2}-3 x+3}$ an integer. | Solve $y=1+\frac{x+1}{x^{2}-3 x+3}$.
$$
\text { Let } \frac{x+1}{x^{2}-3 x+3}=k \text {, }
$$
i.e., $k x^{2}-(3 k+1) x+3 k-1=0, x$ is a real number.
$\Delta(k) \geqslant 0$,
i.e., $(3 k+1)^{2}-4 k(3 k-1) \geqslant 0$.
Thus, $3 k^{2}-10 k-1 \leqslant 0$.
$\frac{5-\sqrt{28}}{3} \leqslant k \leqslant \frac{5+\sqrt{28}}{... | x = -1, 1, 2 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,506 |
Example 3. Six points in space
There are six points in space, where no three points are collinear; no four points are coplanar. After connecting every two points with a line segment, each line segment is either painted red or blue. Prove that regardless of how the coloring is done, there must exist a triangle whose thr... | Prove that from any point, to the other five points, a total of
five line segments can be connected. If these five line segments are colored with red and blue, then at least two of these five line segments must be in the same color group, which means we can always find three line segments of the same color (for example... | proof | Combinatorics | proof | Yes | Yes | cn_contest | false | 703,507 |
Example 10. In $\triangle A B C$, $A B$ $=36$ cm, $A C=21$ cm, $B C$ $=m$ cm, $m$ is an integer. Also, there is a point $D$ on $A B$ and a point $E$ on $A C$ such that $A D=D E=E C$ $=n$ cm, $n$ is an integer. Find the integer solutions for $m, n$. | In $\triangle A B C$ and $\triangle A D E$, according to the cosine rule, we have
$$
\begin{array}{l}
\frac{33^{2}+21^{2}-m^{2}}{2 \times 33 \times 21}=\cos A \\
=\frac{n^{2}+(21-n)^{2}-n^{2}}{2 \times n \times(21-n)}, \\
\frac{1530-m^{2}}{33 \times 21}=\frac{21-n}{n} . \\
n\left(2223-m^{2}\right)=3^{3} \times 7^{2} \t... | m=30, n=11 | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 703,508 |
Example 11. Find the positive integer solutions of the system of equations
$$
\left\{\begin{array}{l}
6 x-y-z=18, \\
x^{2}+y^{2}+z^{2}=1987
\end{array}\right.
$$ | $$
\begin{array}{l}
\text { Sol } \because x, y, \quad z \in N, \\
\therefore(y+z)^{2} \leqslant 2\left(y^{2}+z^{2}\right)=2\left(1987-x^{2}\right) \\
y^{2}+z^{2}=1987-x^{2}>1987-14^{2} \\
=1791 \geqslant 42^{2}, \\
\quad y+z \geqslant 42 .
\end{array}
$$
- From (1), $6 x=18+(y+z) \geqslant 60$, we have $x \geqslant 10... | \left\{\begin{array}{l}
x=13, \\
y=27,33 \\
z=33,27
\end{array}\right.} | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,509 |
Example 13. Find the integer solutions of the equation $y^{2}+y=x^{4}+x^{8}+x^{2}+x$. | The original equation is transformed into
$$
(2 y+1)^{2}=4 x^{4}+4 x^{3}+4 x^{2}+4 x+1 \text {. }
$$
Taking $2 x^{2}+x$ and $2 x^{2}+x+1$, for all integer $x$ we have $\left(2 x^{2}+x\right)^{2}0, \\ x^{2}-2 x>0 .\end{array}\right.$
Therefore, there is no solution within $x2$, which should be eliminated.
Thus, $x=-2,-... | x=-1,-1,0,0,2,2, \\ y=-1,0,-1,0,-6,5 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,511 |
Example 14. Find the positive integer solutions of the equation $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$. | Solution: Clearly, $x, y, z$ are all greater than 1, and at least one of $x, y, z$ is less than 4. Otherwise, if $x \geqslant 4, y \geqslant 4, z \geqslant 4$, then $\frac{1}{x}+\frac{1}{y}+\frac{1}{z} \leqslant \frac{3}{4}$, which clearly contradicts the problem. Now let $x \leqslant y \leqslant z$, then $1<x<4$, so $... | \left\{\begin{array}{l}x=2,2,3 \\ y=3,4,3 \\ z=6,4,3\end{array}\right.} | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 703,512 |
1. The page numbers of a book range from 1 to $n$. When all these page numbers are added together, one of the page numbers was mistakenly added twice. The incorrect sum obtained is 1987. What is the page number that was added twice? | Answer: 34 | 34 | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 703,513 |
2. A person writes the number $2^{x} 9^{y}$ as a four-digit number $2 x 9 y$, by substituting $x$ and $y$ with positive integers less than 10, and it turns out that $2 x 9 y=2^{x} 9^{y}$. What are the values of $x$ and $y$? | 答: $\left\{\begin{array}{l}x=5, \\ y=2 .\end{array}\right.$
Answer: $\left\{\begin{array}{l}x=5, \\ y=2 .\end{array}\right.$ | x=5, y=2 | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 703,514 |
4. Let the equation be $m x^{2}+(m-2) x+(m-3)=0$. Find all numerical solutions of the equation.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | $m=1, x=2,-1 ; m=3, x=0$. | m=1, x=2,-1 ; m=3, x=0 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,516 |
6. Find the integer solutions of the indeterminate equation $x^{2}+18 y^{2}=1987$.
untranslated text:
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
translation:
Find the integer solutions of the indeterminate equation $x^{2}+18 y^{2}=1987$.
Note: The note at the end is not part of the original text and is provided for clari... | 答: $\left\{\begin{array}{l}x=23, \\ y=9\end{array}\right.$
Answer: $\left\{\begin{array}{l}x=23, \\ y=9\end{array}\right.$ | \left\{\begin{array}{l}x=23, \\ y=9\end{array}\right.} | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 703,519 |
10. Given the cubic equation $2 x^{3}-3 a x^{2}+2(a+7) x+a^{2}-9 a+8=0$ whose roots are all natural numbers, find these three roots and the integer value of $a$.
| Answer: $1,2,3 ; a=4$ | 1,2,3 ; a=4 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 703,523 |
Example 7. (IMO-9-1)
In parallelogram $A B C D$, $A B=a ; A D=1$, $\angle A B C=\alpha, \triangle A B C$ is an acute triangle. Prove that when and only when $a \leqslant \cos \alpha + \sqrt{8} \sin \alpha$, four circles $K_{\mathrm{A}}, K_{\mathrm{B}}, K_{\mathrm{C}}, K_{\mathrm{D}}$ with radius 1 and centers at $A, B,... | Prove (i) First, let $\alpha \geqslant 60^{\circ}$. If $E$ is the intersection of $K_{\mathrm{B}}$ and $K_{\mathrm{C}}$, then $\angle C B E=60^{\circ}$. If we establish a Cartesian coordinate system with $B$ as the origin and $B C$ as the $x$-axis, and $B C=1$, then the coordinates of $A$ are $(a \cos \alpha, a \sin \a... | a \leqslant \cos \alpha + \sqrt{3} \sin \alpha | Geometry | proof | Yes | Yes | cn_contest | false | 703,524 |
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