problem stringlengths 1 13.6k | solution stringlengths 0 18.5k ⌀ | answer stringlengths 0 575 ⌀ | problem_type stringclasses 8
values | question_type stringclasses 4
values | problem_is_valid stringclasses 1
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class | __index_level_0__ int64 0 742k |
|---|---|---|---|---|---|---|---|---|---|
Example 11. Given real numbers $x, y, z$ satisfy $x=6-y$, $z^{2}=x y-9$. Prove: $x=y$ (84 Tianjin Mathematics Competition Question) | Given that $x+y=6, x y=z^{2}+9$. Therefore, the quadratic equation with roots $x, y$ is
$$
t^{2}-6 t+\left(z^{2}+9\right)=0 \text {. }
$$
$\because x, y$ are real numbers. $\therefore \Delta \geqslant 0$,
i.e., $36-3 z^{2}-36 \geqslant 0 \Rightarrow z^{2} \leqslant 0 . \because z$ is a real number,
$\therefore z^{2} \g... | proof | Algebra | proof | Yes | Yes | cn_contest | false | 704,492 |
Example 12. In $\triangle ABC$, points $D, E$ are on sides $BC, AB$ respectively, and $\angle 1=\angle 2=\angle 3$. If the perimeters of $\triangle ABC$, $\triangle EBD$, and $\triangle ADC$ are $m$, $m_1$, and $m_2$ respectively, prove that $\frac{m_1 + m_2}{m} \leqslant \frac{5}{4}$.
(89 National League Question) | Proof: Let $BC=a, AC=b, AB=c$, then $m=a+b+c$. Since $\angle 1=\angle 2=\angle 3$, we have $\triangle ABC \sim \triangle EBD \sim \triangle DAC$. Therefore, $DC=\frac{b}{a}, \frac{ED}{b}=\frac{BD}{a}=\frac{a-DC}{a}=\frac{a^2-b^2}{a^2}$.
Let $t=\frac{m_1+m_2}{m}=\frac{m_1}{m}+\frac{m_2}{m}=\frac{ED}{b}+\frac{DC}{b}$, t... | \frac{m_1+m_2}{m} \leqslant \frac{5}{4} | Geometry | proof | Yes | Yes | cn_contest | false | 704,493 |
Example 13. Prove: For any rectangle $A$, there always exists a rectangle $B$, such that the ratio of the perimeters and the ratio of the areas of rectangle $A$ and $B$ are both equal to a constant $k(k \geqslant 1)$. (83 Fujian Mathematics Competition Question) | Proof: Let the length and width of rectangles $A, B$ be $a, b$ and $x, y$, respectively, then we have
$$
x+y=k(a+b), \quad x y=k a b
$$
($k, a, b$ are known numbers). By the converse of Vieta's theorem, $x$, $y$ are the two roots of the equation $z^{2}-k(a+b) z+k a b=0$.
To prove the existence of rectangle $B$, we nee... | proof | Geometry | proof | Yes | Yes | cn_contest | false | 704,494 |
Example 14. Given $a+b=c+d, a^{8}+b^{3}=c^{8}+d^{3}$. Prove: $a^{1980}+b^{1980}=c^{1980}+d^{1080}$. (89 Beijing Mathematics Competition Question) | $$
\begin{array}{l}
\text { Prove that from } a^{3}+b^{3}=c^{3}+d^{3} \text { we can get } \\
(a+b)\left[(a+b)^{2}-3 a b\right] \\
=(c+d)\left[(c+d)^{2}-3 c d\right] \text {. } \\
\text { Also, } a+b=c+d \text {, } \\
\text { 1) If } a+b=c+d=0 \Rightarrow a=-b, c=-d \text {, } \\
\text { then } a^{1989}+b^{1989}=a^{198... | proof | Algebra | proof | Yes | Yes | cn_contest | false | 704,495 |
Example 15. Let $a, b \in N$, when $a^{2}+b^{2}$ is divided by $a+b$, the quotient is $q$, and the remainder is $r$. Find all pairs of numbers $(a, b)$ such that $q^{2}+r=1977$. (19th 1MO problem) | Given $a^{2}+b^{2}=q(a+b)+r$,
$$
q^{2}+r=1977 \quad(0 \leqslant r < q^2),
$$
if $q \geq 45$, then $q^2 \geq 2025 > 1977$, a contradiction. If $q \leq 43$, then $r > 128$, also a contradiction. Hence $q=44$, $r=41$, and $a^{2}+b^{2}=44(a+b)+41$.
This simplifies to $(a-22)^{2}+(b-22)^{2}=1009$. The solutions are
$(a, b)=... | (a, b)=(50,37),(50,7),(37,50),(7,50) | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 704,496 |
Example 16. In the system of equations
$$
\left\{\begin{array}{l}
a x_{1}^{4}+b x_{1}^{3}=x_{2}^{2}, \\
a x_{2}^{4}+b x_{2}^{3}=x_{3}^{2}, \\
a x_{3}^{4}+b x_{3}^{3}=x_{4}^{2}, \\
a x_{4}^{4}+b x_{4}^{3}=x_{1}^{2}
\end{array}\right.
$$
where $a, b$ are real numbers, and $a \neq 0$. Prove:
1) When $b^{2}+4 a \geqslant ... | 1) First, consider the special case where
$$
x_{1}=x_{2}=x_{3}=x_{4}=x \neq 0
$$
In this case, each equation becomes
$$
\begin{array}{l}
x^{2}\left(a x^{2}+b x-1\right)=0 . \\
\because x \neq 0, \\
\therefore a x^{2}+b x-1=0 .
\end{array}
$$
Since \( a \neq 0 \) and \( b^{2}+4 a \geqslant 0 \), this quadratic equatio... | proof | Algebra | proof | Yes | Yes | cn_contest | false | 704,497 |
2. Let $\alpha, \beta$ be the roots of the quadratic equation $x^{2}+(p-2) x+1$ $=0$ and. $[1+\alpha(p+\alpha)-\beta][1+\beta(p+\beta)$ $-\alpha]=-\frac{7}{2}$, find the value of $p$.
| (Answer: $-\frac{1}{2}$ or $\frac{9}{2}$ ) | -\frac{1}{2} \text{ or } \frac{9}{2} | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,499 |
3. If the two roots $\alpha, \beta$ of the quadratic equation $x^{2}-x+1-m=0$ satisfy $|\alpha|+|\beta| \leqslant 5$. What is the range of the real number $m$? | (笭: $\frac{3}{4} \leqslant m \leqslant 7$ )
(Reed: $\frac{3}{4} \leqslant m \leqslant 7$ ) | \frac{3}{4} \leqslant m \leqslant 7 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,500 |
5. Let the quadratic equation $x^{2}-3 x+a+4=0$ have two roots that are both integers. (1) Prove: (1) the roots must be one odd and one even. (2) $a$ is even. (2) When the roots have the same sign, find the roots and $a$.
Translate the above text into English, please retain the original text's line breaks and format, ... | (Answer: $a=-2$ ) | a=-2 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,501 |
Example 3. The equation $x^{2}+2(1+a) x+\left(3 a^{2}+4 a b\right.$ $\left.+4 b^{2}+2\right)=0$ has real roots, find the values of $a, b$. (87 National League Question)
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | Given that the equation has real roots, then
$$
\begin{array}{l}
\Delta=4(1+a)^{2}-4\left(3 a^{2}+4 a b+4 b^{2}+2\right) \\
\geqslant 0, \\
\therefore \quad 2 a^{2}-2 a+2 a b+4 b^{2}+1 \leqslant 0 .
\end{array}
$$
which implies $(a-1)^{2}+(a+2 b)^{2} \leqslant 0$.
$$
\begin{array}{l}
\text { Also, }(a-1)^{2}+(a+2 b)^... | null | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,502 |
Example 1. Let $A, B, C$ be real numbers. Try to find the sufficient and necessary conditions for the inequality $A(x-y)(x-z)+B(y-z)(y-x)$ $+C(z-x)(z-y) \geqslant 0$
to hold for any real numbers $x, y, z$. | Solve: Rewrite (1) as
$$
\begin{array}{l}
A(x-y)^{2}-(B-A-C)(x-y)(y-z) \\
+C(y-z)^{2} \geqslant 0 .
\end{array}
$$
Let $t=\frac{x-y}{y-z}(y \neq z)$ in the above equation, we get
$$
A t^{2}-(B-A-C) t+C \geqslant 0
$$
which holds for any real number $t$ if and only if
$$
\begin{array}{l}
\left\{\begin{array}{l}
A>0, \... | A \geqslant 0, B \geqslant 0, C \geqslant 0, A^{2}+B^{2}+C^{2} \leqslant 2(A B+B C+C A) | Inequalities | math-word-problem | Yes | Yes | cn_contest | false | 704,503 |
Example 3. Let $a_{1}, a_{2}, \cdots, a_{n}$ and $b_{1}, b_{2}, \cdots, b_{n}$ be real numbers. Prove that the necessary and sufficient condition for the inequality
$$
\sum_{i=1}^{n} a_{i} x_{i} \leqslant \sum_{i=1}^{n} b_{i} x_{i}
$$
to hold for any real numbers satisfying $x_{1} \leqslant x_{2} \leqslant \cdots \leqs... | To prove the necessity, we can use special values to verify. In equation (3), let \( x_{1}=x_{2}=\cdots=x_{\mathbf{k}}=-1, x_{\mathbf{k}+1}=\cdots=x_{0}=0 \) (where \( 1 \leqslant k \leqslant n-1 \)), we get
$$
\begin{array}{l}
-\sum_{1=1}^{k} a_{1} \leqslant-\sum_{1=1}^{\mathrm{k}} b_{1}, \\
\therefore \sum_{1=1}^{k} ... | proof | Inequalities | proof | Yes | Yes | cn_contest | false | 704,505 |
Example 5. Prove: The inequality
$$
\begin{aligned}
& x^{2}+y^{2}+z^{2} \geqslant 2 x y \cos \alpha+2 y z \cos \beta \\
+ & 2 z x \cos \gamma
\end{aligned}
$$
holds for all real numbers $x, y, z$ if and only if $\alpha \pm \beta+\gamma$ equals an odd multiple of $\pi$. | To prove (5), we have:
$$
\begin{aligned}
& x^{2}-2(y \cos \alpha+z \cos \gamma) x+y^{2}+z^{2} \\
- & 2 y z \cos \beta \geqslant 0 \\
\Leftrightarrow & (x-y \cos \alpha-z \cos \gamma)^{2}+y^{2} \sin ^{2} \alpha \\
& +z^{2} \sin ^{2} \gamma-2 y z(\cos \beta+\cos \alpha \cos \gamma) \geqslant 0 \\
\Leftrightarrow & (x-y ... | proof | Inequalities | proof | Yes | Yes | cn_contest | false | 704,507 |
Example 9. Let $a, b, A, B$ be known constants, and for all real numbers $x$ it is always true that
$$
\begin{aligned}
f(x) & =1-a \cos x-b \sin x-A \cos 2 x \\
-B \sin 2 x & \geqslant 0 .
\end{aligned}
$$
Prove: $a^{2}+b^{2} \leqslant 2, A^{2}+B^{2} \leqslant 1$. (19th IMO Problem) | Let $r=\sqrt{a^{2}+b^{2}}, R=\sqrt{A^{2}+B^{2}}$, $a=r \cos \alpha, \quad b=r \sin \alpha, A=R \cos 2 \beta$, $B=R \sin 2 \beta$, then
$$
f(x)=1-r \cos (x-\alpha)-R \cos 2(x-\beta) .
$$
Assume there exist $a, b, a^{2}+b^{2}>2$, then $r>\sqrt{2}$, 1) when $\sin 2(\alpha-\beta) \geqslant 0$,
$$
\begin{array}{l}
f\left(\... | proof | Inequalities | proof | Yes | Yes | cn_contest | false | 704,511 |
Example 4 - For a $4n+2$-sided polygon $A_{1} A_{2} A_{3} \cdots A_{4 n+2}$ (where $n$ is a natural number), each interior angle is an integer multiple of $30^{\circ}$.
Given the quadratic equations in $x$
$$
\begin{array}{l}
x^{2}+2 x \sin A_{1}+\sin A_{2}=0, \\
x^{2}+2 x \sin A_{2}+\sin A_{3}=0, \\
x^{2}+2 x \sin A_{... | Since each interior angle is an integer multiple of $30^{\circ}$, the possible angles are $30^{\circ}, 60^{\circ}, 90^{\circ}, 120^{\circ}, 150^{\circ}$, with their respective sine values being $\frac{1}{2}, \frac{\sqrt{3}}{2}, 1$.
If $\sin A_{1}=\frac{1}{2}$, by $\sin A_{2} \geqslant \frac{1}{2}, \sin A_{3} \geqslant ... | A_{1}=A_{2}=A_{3}=90^{\circ}, A_{4}=A_{5}=A_{6}=150^{\circ} | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,513 |
2 . Let for all real numbers $x$, the inequality $x^{4}+(a-2) x^{2}+a \geqslant 0$ always holds, find the range of values for $a$.
| $(a \geqslant 4-2 \sqrt{3})$ | a \geqslant 4-2 \sqrt{3} | Inequalities | math-word-problem | Yes | Yes | cn_contest | false | 704,516 |
5. Let $x, y$ be positive numbers, and $x+y=k$. Try to find the range of $k$ such that the inequality
$$
\left(x+\frac{1}{x}\right)\left(y+\frac{1}{y}\right) \geqslant\left(\frac{k}{2}+\frac{2}{k}\right)^{2}
$$
always holds. | $(0<k \leqslant 2 \sqrt{2+\sqrt{5}})$ | (0<k \leqslant 2 \sqrt{2+\sqrt{5}}) | Inequalities | math-word-problem | Yes | Yes | cn_contest | false | 704,517 |
11. For which real numbers $\lambda$, the inequality $\frac{1}{2}\left(e^{x}+\right.$ $\left.e^{-\mathrm{x}}\right) \leqslant e^{\lambda \mathrm{x}^{2}}$ holds for all real numbers $x$. | $\left(\lambda \geqslant \frac{1}{2}\right)$ | \lambda \geqslant \frac{1}{2} | Inequalities | math-word-problem | Yes | Yes | cn_contest | false | 704,520 |
Example 1. (4th Ling Ling Camp Test Question) Given 1989 points, no three of which are collinear. Divide them into 30 groups with different numbers of points. From any three different groups, take one point from each to form a triangle. To maximize the total number of such triangles, how many points should each group h... | Let the number of points in each group be $x_{\mathrm{K}} (k=1,2, \cdots, 30)$, all $x_{\mathbf{k}}$ are distinct, $\sum_{\mathrm{k}=1}^{30} x_{\mathrm{k}}=1989$, the total number of triangles is $F\left(x_{1}, x_{2}, \cdots, x_{2}, 0\right)$
$$
=\sum_{1 \leqslant i<j<k \leqslant 30} x_{1} x_{1} x_{k},
$$
Obviously, t... | 51,52, \cdots, 56,58,59, \ldots, 81 | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false | 704,521 |
$$
\begin{array}{c}
\text { Example 2. Let } \sum_{k=1}^{11} x_{k}=100, x_{k} \in N(k=1,2, \\
\cdots, 11) \text {, find the } F=\prod_{1 \leqslant i<j \leqslant 11}\left(x_{1}+x_{j}\right)!\text { }
\end{array}
$$
minimum value. | To find the minimum value of $F$ of type $N(100,0)$.
First, prove the reverse inequality in (4): Note
$$
\begin{array}{c}
F\left(a, b, c_{3}, \cdots, c_{11}\right) \\
=(a+b)!\cdot \prod_{3 \leqslant j \leqslant 11}\left(a+c_{j}\right)!\left(b+c_{j}\right)! \\
\cdot \prod_{3 \leqslant i < j \leqslant 11}\left(a+c_{i}+c_... | F(9,9, \ldots, 9,10) | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 704,522 |
Example 3. Let the sum of 10 distinct positive odd numbers be 1990, find the maximum value of the product of these 10 numbers. | Let the positive odd number be $2 x_{k}-1(k=1,2, \cdots$, 10 ), where all $x$ are unequal, then
$$
\sum_{k=1}^{10}\left(2 x_{k}-1\right)=1990,
$$
It is easy to know $\sum_{k=1}^{10} x_{k}=\frac{1}{2}(1990+10)=1000$.
Let $\varphi(x)=2 x-1$,
$$
\text { This problem is to find } F=\prod_{k=1}^{10} \varphi\left(x_{k}\righ... | 189 \cdot 191 \cdots 197 \cdot 201 \cdot 203 \cdots 209 | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 704,523 |
Example 5. For the equation $(1989 x)^{2}-1988 \cdot 1990 x$ $-1=0$, the larger root is $r$. For the equation $x^{2}+1989 x-1990=0$, the smaller root is $s$. Find the value of $r-s$. (Adapted from the 1984 Beijing Mathematics Competition) | Solve the equation $(1989 x)^{2}-1988 \cdot 1990 x-1$ $=0$, we get
$(1989 x)^{2}-(1989-1)(1989+1) x-1=0$.
Obviously, $1989^{2}-(1989-1)(1989+1)-1$ $=0$, so the equation has one root $x_{1}=1$, then the other root is $\boldsymbol{x}_{2}=-\frac{1}{1989^{2}}$, thus the larger root $r=1$.
Similarly, for the equation $x^{2... | 1991 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,524 |
Example 1. Let the two roots of the equation $x^{2}+p x+q=0$ be consecutive integers, prove: $p^{2}-4 q-1=0$. | Proof: Let the two roots of the original equation be $a$ and $a+1$, then
$$
\begin{aligned}
q & =a(a+1), \\
-p & =a+(a+1)=2 a+1,
\end{aligned}
$$
$\therefore q$ is the product of two consecutive integers. By the corollary, we have
$$
4 q+1=(2 a+1)^{2},
$$
and
$$
\begin{array}{l}
p^{2}=(2 a+1)^{2}, \\
\therefore \quad ... | p^{2}-4 q-1=0 | Algebra | proof | Yes | Yes | cn_contest | false | 704,526 |
Example 2. Prove: For any natural number $n$, the algebraic expression $n_{2}+3 n+3$ is never equal to the product of two consecutive natural numbers. | Proof: Let \( M=n^{2}+3 n+3 \), then
\[
\begin{array}{l}
4 M+1=4\left(n^{2}+3 n+3\right)+1 \\
=4 n^{2}+12 n+13 \\
\because \quad 4 n^{2}+12 n+9<4 n^{2}+12 n+13 \\
<4 n^{2}+16 n+16,
\end{array}
\]
i.e., \((2 n+3)^{2}<4 n^{2}+12 n+13\)
\[
<(2 n+4)^{2}.
\]
\(\therefore 4 n^{2}+12 n+13\) cannot be a perfect square, which ... | proof | Algebra | proof | Yes | Yes | cn_contest | false | 704,527 |
Example 3. If $n$ is a natural number, and $9 n^{2}+5 n+26$ equals the product of two consecutive natural numbers, find $n$. (1985 Shanghai Junior High School Mathematics Competition Question) | Let $M=9 n^{2}+5 n+26$, then
$$
\begin{aligned}
4 M+1 & =4\left(9 n^{2}+5 n+26\right)+1 \\
& =36 n^{2}+20 n+105 .
\end{aligned}
$$
Let $4 M+1=(6 n+k)^{2}$, where $k$ is a natural number, we get
$$
36 n^{2}+20 n+105=36 n^{2}+12 k n+k^{2},
$$
Solving for $n$ gives $n=\frac{k^{2}-105}{20-12 k}$.
From $n>0$ we get $\frac... | n=6 \text{ or } n=2 | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 704,528 |
Example 4. Prove: Each term in the sequence 49, 4489, 444889, ... is a perfect square.
(1964 American Competition Problem) | Prove that the general term formula of the sequence is
$$
\begin{aligned}
a_{\mathrm{n}} & =44 \cdots 488 \cdots 89=\underbrace{44 \cdots 488 \cdots 8}_{n \uparrow}+1 \\
& =4(\underbrace{11 \cdots 122 \cdots 2}_{n \uparrow})+\underbrace{1}_{n \uparrow} .
\end{aligned}
$$
Let \( M=\underbrace{11 \cdots 122 \cdots 2}_{\... | proof | Number Theory | proof | Yes | Yes | cn_contest | false | 704,529 |
Example 5. If $f(x)=x^{2}+x$, prove that the equation $4 f(a)=f(b)$ has no positive integer solutions for $a$ and $b$. (Canada 9th Grade High School Mathematics Competition Question) | Proof $\because f(x)=x^{2}+x=x(x+1)$,
$\therefore$ if $a, b$ are positive integers, then $f(a), f(b)$ are both the product of two consecutive natural numbers.
Therefore, to prove that the equation $4 f(a)=f(b)$ has no positive integer solutions for $a$ and $b$, we need to prove that if $f(a)$ is the product of two con... | proof | Algebra | proof | Yes | Yes | cn_contest | false | 704,530 |
For example, prove that $x_{1}^{2}+x_{2}^{2}+x_{3}^{2} \geqslant \frac{1}{3}\left(x_{1}+x_{2}\right.$ $\left.+x_{3}\right)^{2}$. | Consider the function $y=x^{2}$ at: $x_{1}, x_{2}, x_{3}$ and $\frac{1}{3}\left(x_{1}+\right.$ $\left.x_{2}+x_{3}\right)$. Since $G\left(\frac{1}{3}\left(x_{1}+x_{2}+x_{3}\right)\right.$, $\left.\frac{1}{3}\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\right)\right)$ is the centroid of $\triangle A B C_{1}$, where $A\left(x:, x_{... | proof | Inequalities | proof | Yes | Yes | cn_contest | false | 704,531 |
In triangle $ABC$, $AD$ is the altitude on the hypotenuse, connecting the incenters $M, N$ of $\triangle ABD$ and $\triangle ACD$ respectively intersect $AB, AC$ at $K, L$. Let the areas of $\triangle ABC$ and $\triangle AKL$ be $S, T$ respectively, then $S \geqslant 2T$. | Prove as follows:
Connect $D M, \dot{D} N, A N$.
From $\triangle A B D \sim \triangle C A D$ we know
$\frac{D M}{D N}=\frac{A B}{A C}$.
Also, $\angle M D N=90^{\circ}$,
thus $\triangle M D N \sim \triangle B A C$,
$$
\therefore \angle M N D=\angle C \text {. }
$$
Therefore, $N, L, C, D$ are concyclic.
Hence $\angle A... | S \geqslant 2T | Geometry | proof | Yes | Yes | cn_contest | false | 704,533 |
Example 6. If $k$ is an integer, the quadratic equation $(k-1) x^{2}-p x+k=0$ has two positive integer roots, find $k^{k 0}\left(p^{p}+k^{k}\right)$. (84 Beijing Mathematics Competition Question) | Let the two roots be $x_{1}, x_{2}$, then $x_{1}, x_{2}$ are both positive integers. By Vieta's formulas, $x_{1} x_{2}=\frac{k}{k-1}$ is also a positive integer. Since $k$ is a positive integer, if $k-1 \neq 1$, then $k-1$ and $k$ are coprime. In this case, $\frac{k}{k-1}$ cannot be an integer, so $k-1=1$, i.e., $k=2$ ... | 1984 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,535 |
Example 7. Given the quadratic equation $2 x^{2}+a^{2} x+6 a=0$ has one root greater than 0, and the other root less than -2, find the range of $a$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | Slightly explained: Let the function $y=2 x^{2}+a^{2} x+6 a$, the graph is as follows. Suppose the roots of the equation are $x_{1}, x_{2}$, then $x_{1}<0$. From the graph, we know that when $x=-2$, $y>4$.
When $x=0$, $y<0$, so $a<0$.
Combining (1) and (2), the range of values for $a$ is $a<-1$. | null | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,536 |
Example 8. Let $a, b$ be the two real roots of the equation $x^{2} + p x + 1 = 0$, and $c, d$ be the two real roots of the equation $x^{2} + q x + 1 = 0$. Prove: $(a-c)(b-c)(a+d)(b+d) = q^{2} - p^{2}. \quad$ (85 Hefei Mathematics Competition Question) | Prove that from $a+b=-p, ab=1, c+d=-q$, $cd=1$, we have $(a-c)(b-c)=c^{2}-(a+b)c+ab=c^{2}+qc+1+(p-q)c=(p-q)c_{3}$, $(a+d)(b+d)=d^{2}+(a+b)d+ab=d^{2}+qd+1-(p+q)d=-(p+q)d$. Therefore, $(a-c)(b-c)(a+d)(b+d)=\left(q^{2}-p^{2}\right)cd=q^{2}-p^{2}$. | q^{2} - p^{2} | Algebra | proof | Yes | Yes | cn_contest | false | 704,537 |
Example 9. Given that $p, q, r$ are all positive numbers. Prove: Among the three equations $x^{2}-\sqrt{p} x+\frac{q}{8}=0$, $x^{2}-\sqrt{q} x+\frac{r}{8}=0, x^{2}-\sqrt{r} x+\frac{p}{8}=0$ in terms of $x$, at least one equation has two distinct positive roots. (Fuzhou Mathematical Competition, 1981) | Proof by contradiction. Suppose that all three equations have no unequal positive roots, then the discriminants of the three equations are respectively
$$
\begin{array}{l}
\Delta_{1}=p-\frac{q}{2}<0, \quad \Delta_{2}=q-\frac{r}{2}<0, \\
\Delta_{3}=r-\frac{p}{2}<0 .
\end{array}
$$
Thus, $\Delta_{1}+\Delta_{2}+\Delta_{3... | proof | Algebra | proof | Yes | Yes | cn_contest | false | 704,538 |
Example 10. Prove that the equation $x^{2}+10 m x-5 n+3=0$ has no integer roots ($m, n$ are both integers). (84 Kunming Mathematics Competition Question) | Prove that by the quadratic formula we get
$x_{1,2}=-5 m \pm \sqrt{25 m^{2}+5 n-3}$ - hence it is sufficient to prove that $25 m^{2}+5 n-3$ is not a perfect square.
$\because 25 m^{2}+5 n=5\left(5 m^{2}+n\right)$ is a multiple of 5, its last digit is 0 or 5,
$\therefore 25 m^{2}+5 n - 3$ can only end in 2 or 7. However... | proof | Algebra | proof | Yes | Yes | cn_contest | false | 704,539 |
2. A two-digit number, the sum of its digits multiplied by 3, when added to the number itself, equals the two-digit number obtained by swapping the positions of its digits, the two-digit numbers that satisfy this condition ( ).
(A) There is one.
(B) There are four.
() There are five.
(D) None exist. | $B$ | B | Number Theory | MCQ | Yes | Yes | cn_contest | false | 704,541 |
1. The points corresponding to real numbers $a, b$ on the number line are shown in the figure, with 0 being the origin, then the value of $\frac{a+b}{a-b}$ is ( ).
(A) Less than zero.
(B) Greater than zero.
(C) Equal to zero.
(D) Cannot be determined. | B
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | B | Algebra | MCQ | Yes | Yes | cn_contest | false | 704,543 |
3. Below are four questions solved by a student,
(1) $6 a^{\frac{2}{3}} \times 7 a^{\frac{1}{2}}=42 a^{1 \frac{1}{6}}$,
(2) $(-a x)^{0}+\left(-a x^{8}\right)=a^{5} x^{8} 1$
(3) $\left(-1989^{\circ}\right)^{1980}=-1$,
(4) $\left[(-3)^{m}\right]^{2}=3^{m^{2}}$.
The number of questions the student solved correctly is ().... | B
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | null | Algebra | MCQ | Yes | Yes | cn_contest | false | 704,545 |
$$
\begin{array}{l}
\text { 4. As shown in the figure, in } \triangle A B C, \angle A: \angle B: \angle C \\
=3: 5: 10 \text {, and } \triangle A^{\prime} B^{\prime} C \cong \triangle A B C \text {, then } \\
\angle B C A^{\prime}: \\
\angle B C B^{\prime} \text { is equal to } \\
(\quad) .
\end{array}
$$
(A) $1: 2$.
(... | C
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | null | Geometry | MCQ | Yes | Yes | cn_contest | false | 704,546 |
5. If the projections of the two legs of a right triangle onto the hypotenuse are in the ratio $1: 4$, then the ratio of these two legs is ( ).
(A) $1: \sqrt{2}$.
(B) $1: 2$.
(C) $1: 3$.
(D) $1: 4$. | B
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | B | Geometry | MCQ | Yes | Yes | cn_contest | false | 704,547 |
6. Let $x$ be an irrational number, but $(x-2)(x+6)$ is a rational number, then which of the following conclusions is correct? ( )
(A) $x^{2}$ is a rational number.
(B) $(x+6)^{2}$ is a rational number.
(C) $(x+2)(x-6)$ is an irrational number.
(D) $(x+2)^{2}$ is an irrational number. | C
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | null | Algebra | MCQ | Yes | Yes | cn_contest | false | 704,548 |
Three, someone buys 13 chicken eggs, 5 duck eggs, and 9 goose eggs, spending a total of 9.25 yuan; if they buy 2 chicken eggs, 4 duck eggs, and 3 goose eggs, they spend a total of 3.20 yuan. Try to find: how much would it cost to buy 1 chicken egg, 1 duck egg, and 1 goose egg each? | Three, Solution: Let the cost of 1 chicken egg be $x$ yuan, 1 duck egg be $y$ yuan, and 1 goose egg be $z$ yuan. According to the problem, we have
$$
\left\{\begin{array}{l}
13 x+5 y+9 z=9.25, \\
2 x+4 y+3 z=3.20 .
\end{array}\right.
$$
Solving, we get $x+y+z=1.05$. | x+y+z=1.05 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,556 |
Given the equation $x^{2}+(a-6) x+a=0$ $(a \neq 0)$ has two integer roots, find the integer value of $a$.
---
The equation $x^{2}+(a-6) x+a=0$ $(a \neq 0)$ is known to have two integer roots. Try to find the integer value of $a$. | Let the two integer roots of the equation be \(x_{1}, x_{2}\), and \(x_{1} \geqslant x_{2}\). By Vieta's formulas, we have:
\[
\left\{\begin{array}{l}
x_{1}+x_{2}=6-a, \\
x_{1} x_{2}=a .
\end{array}\right.
\]
(1)
\[
\begin{array}{l}
+ \text { (2): } x_{1} x_{2}+x_{1}+x_{2}=6, \\
\therefore \quad\left(x_{1}+1\right)\le... | 16 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,557 |
Three, let $A B C D$ be a square cardboard, and divide it into three rectangles with lines $P Q$ and $R S$ parallel to $B C$,
as shown in the figure. Fold the cardboard so that point $C$ lands on point $C^{\prime}$ on $A B$, and point $S$ lands on point $S^{\prime}$ on $P Q$, and
$B C^{\prime}=1$. Try to find
the lengt... | Three, Hint: Let $A C^{\prime}=x, B T=y$, and note that $C^{\prime} T=C T$. Apply the Pythagorean theorem to Rt $\triangle B C^{\prime} T$ to get an expression for $y$ in terms of $x$, and then consider $\triangle P C^{\prime} S^{\prime} \sim \triangle B T C^{\prime}$ and $C^{\prime} S^{\prime}=C S$, which can give ano... | 3 \sqrt{2} | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,558 |
Four, there is a rectangular prism composed of $m n k$ unit cubes, with its length, width, and height being $m, n$, and $k$ units respectively, and $m \geqslant n \geqslant k$. The three faces of such a rectangular prism containing vertex $A$ are colored, such that exactly half of the unit cubes have none of their face... | Four, Hint: The key to this problem lies in listing the relationship between $m, n, k$: $\frac{m n k}{2}=(m-1)(n-1)(k-1)$, and then discussing for $k=3,4,5, \cdots$. The number of them is
$$
\frac{m n k}{2}=60,72,84,90,120 .
$$ | 60,72,84,90,120 | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false | 704,559 |
2. Let $M=\sqrt{a^{2}+b^{2}+c^{2}}$, where $a, b$ are consecutive integers, and $c=a \cdot b$, then $M($ ).
(A) must be even.
(B) must be odd.
(C) is irrational.
(D) may be odd or even. | B
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | B | Number Theory | MCQ | Yes | Yes | cn_contest | false | 704,561 |
3. The three sides of $\triangle A B C$ are $a=2$, $b=2 \sqrt{2}, c=\sqrt{2}+\sqrt{6}$, then the area of the smallest circular paper piece that can cover this triangle is ( ).
(1) $2 \pi$.
(B) $4 \pi$.
(C) $2 \sqrt{3} \pi$.
(D) $(2+\sqrt{3}) \pi$. | D
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | null | Geometry | MCQ | Yes | Yes | cn_contest | false | 704,562 |
5. In a room, there are several red, green, blue, yellow, and white socks. To ensure that 10 pairs of socks (two socks of the same color make a pair) are taken out of the room, the minimum number of socks that should be taken out is ( ).
(A) 23. (B)24. (C)25. (D) 26. | B
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | B | Combinatorics | MCQ | Yes | Yes | cn_contest | false | 704,565 |
1. If $x=\sqrt{19-8 \sqrt{3}}$, then the value of the fraction $\frac{x^{4}-6 x^{3}-2 x^{2}+18 x+23}{x^{3}-7 x^{2}+5 x+15}$ is $\qquad$ | $5$ | 5 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,566 |
4. The equation $6 x+\sqrt{3 x-2}+2 k-3=0$ has real roots, the range of values for $k$ is $\qquad$ - | $k \leqslant -\frac{1}{2}$ | k \leqslant -\frac{1}{2} | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,569 |
5. A person is walking along the tram route, and a tram catches up from behind every 12 minutes, while a tram comes from the opposite direction every 4 minutes. Assuming both the person and the trams are moving at a constant speed, then the trams are dispatched from the starting station every $\qquad$ minutes. | $6$ | 6 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,570 |
6. In Rt $\triangle A B C$, $\angle B=90^{\circ}, A B=$ $4, B C=2, D$ is any point inside $\mathrm{R} t \triangle A B C$, and through $D$ are drawn lines $E F, M N$, $P T$ parallel to the three sides of the triangle. Let $S$ be the sum of the areas of $\triangle D E P, \triangle D M F$ and $\triangle D N T$, then the m... | $\frac{4}{3}$ | \frac{4}{3} | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,571 |
Four, as shown in the figure, in $\triangle A B C$, $\angle A=90^{\circ}$, $A D \perp B C$ at $D, P$ is the midpoint of $A D$, $B P$ intersects $A C$ at $E, E F \perp B C$ at $F, A E=3, E C=12$. Find the length of $E F$. | Extend $F E, B A$ to intersect at point $H$,
$\left.\begin{array}{l}A D \| H F \\ P \text { is the midpoint of } A D\end{array}\right\} \Rightarrow H E=E F$.
Also, $\angle H A C=90^{\circ}$ ?
$\left.\begin{array}{l}\angle E F C=90^{\circ} \\ \text { and on the same side of } H C\end{array}\right\} \Rightarrow H, A, F, ... | 6 | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,573 |
Five, Proof: For any positive integer $k$, at least one of the two numbers $2k-1$ and $2k+1$ cannot be equal to the sum of two integers squared.
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | If $k$ is odd, let $k=2m+1$, then $2k+1=4m+3$; if $k$ is even, let $k=2m$, then $2k-1=4m-1=4(m-1)+3$.
That is, among the two odd numbers $2k-1$ and $2k+1$, at least one is congruent to 3 modulo 4. We will prove that an integer of the form $4m+3$ cannot be the sum of two integer squares.
Assume $4m+3=a^2+b^2$ (where $a... | null | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 704,574 |
1. The arithmetic square root of the square of a real number $m$ equals ( ). (A) m. (B) $|m| \cdot$ (C) $-m_{0}$ (D) $(\sqrt{m})^{2}$. | B
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | B | Algebra | MCQ | Yes | Yes | cn_contest | false | 704,576 |
2. Among the following statements, the correct one is ( ).
(A) Natural numbers can be divided into prime numbers and composite numbers.
(B) Decimals all belong to rational numbers.
(C) Any rational number can be expressed in the form $\frac{n}{m}$ (where $m, n$ are integers, $m \neq 0$).
(D) Infinite decimals all belon... | C
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | C | Number Theory | MCQ | Yes | Yes | cn_contest | false | 704,577 |
3. The solution situation of the equation $\frac{x+1}{x^{2}+2}=\frac{x-1}{x^{2}+3}$ is
(A) one solution.
(B) two solutions
(C) infinitely many solutions.
(D) no solution. | A
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | A | Algebra | MCQ | Yes | Yes | cn_contest | false | 704,578 |
5. Given that the mantissa of the common logarithm of a positive number $N$ is $\frac{1}{a}$, then the mantissa of the common logarithm of $\frac{1}{N}$ is ( ).
(A) $\frac{1}{a}$.
(B) $\frac{a-1}{a}$.
(C) $-\frac{1}{a}$.
(D) $a$. | B
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | null | Algebra | MCQ | Yes | Yes | cn_contest | false | 704,580 |
7. The ratio of the areas of two similar triangles is $1: 3$, then the ratio of the corresponding angle bisectors is ( ).
(A) $1: \sqrt{3}$.
(B) $1: 3$.
(C) $1: 9$.
(D) Cannot be determined. | A
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | A | Geometry | MCQ | Yes | Yes | cn_contest | false | 704,582 |
8. Given $\log _{10} x=k$, then $\log _{2} \sqrt{ } x=(\quad)$.
(A) $\frac{k}{2}$.
(B) $\frac{k}{4}$.
(C) $2 k$.
(D) $4 k$. | C
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | C | Algebra | MCQ | Yes | Yes | cn_contest | false | 704,583 |
$\begin{array}{l}\text { 1. Calculate } \frac{x^{2}-9}{x^{2}-5 x+6} \div \frac{x^{2}+6 x+9}{x^{2}-4 x+4} \\ \times \frac{x^{3}+27}{x^{3}-8}= \\\end{array}$ | 1. $\frac{x^{2}-3 x+9}{x^{2}+2 x+4}$ | \frac{x^{2}-3 x+9}{x^{2}+2 x+4} | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,584 |
In $\triangle A B C$, $\angle C=30^{\circ}, B M$ is a median, $A C=2 a$. If $\triangle A B C$ is folded along $B M$, the overlapping part of the two smaller triangles $A B M$ and $B C M$ is exactly one quarter of the area of $\triangle A B C$. Find the area of $\triangle A B C$. | iii) Solve by discussing two scenarios.
i) If in the original triangle, $\angle A B M >$ $\angle C B M$, after folding as shown in Figure (1), where $C^{\prime}$ is the position where point $C$ lands after folding. Then solve to get $S \triangle A_{B C}=\frac{\sqrt{3}}{2} a^{2}$.
ii) If
in the original triangle,
$\ang... | S \triangle A B C=\frac{\sqrt{3}}{2} a^{2} \text{ or } S \triangle A B C=\frac{a}{2} | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,586 |
3. In an equilateral triangle with side length 1, there is a point $P$, the distances from point $P$ to the three sides are $a, b, c$, respectively, then $a+b+c=$ $\qquad$ - | 3. $\frac{\sqrt{3}}{2}$, | \frac{\sqrt{3}}{2} | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,587 |
1. If $(3 x+1)^{5}=a x^{5}+b x^{4}+c x^{3}+d x^{2}$ $+e x+f$, then the value of $a-b \div c-d+e-f$ is ( ).
(A) -32 .
(B) 32 .
(C) 1024 .
(D) -1024 . | B
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | B | Algebra | MCQ | Yes | Yes | cn_contest | false | 704,593 |
2. For real numbers $x, y$, define the operation ※: $x ※ y=a x^{y} +b+1$. If $1 \% 2=869, 2 \% 3=883$, then $2 \times 9$ $=$ ( ).
(A) 1988.
(B) 1989.
(C) 1990.
(D) 1991. | D
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | null | Algebra | MCQ | Yes | Yes | cn_contest | false | 704,594 |
3. As shown in the figure, in isosceles right $\triangle ABC$, $AD$ is the median on the hypotenuse $BC$, $BE \perp AD$, intersecting $AC$ at $E$, $EF \perp BC$. If $AB = BC = a$, then $EF$ equals ( ).
(A) $\frac{1}{3} a$.
(B) $\frac{1}{2} a$.
(C) $\frac{2}{3} a$.
(D) $\frac{2}{5} a$. | A
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | null | Geometry | MCQ | Yes | Yes | cn_contest | false | 704,595 |
4. If $47^{100}$ is a 168-digit number, then $47^{17}$ is a ( )-digit number.
(A)27. (B)28. (C)29. (D) 30 | C
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | C | Number Theory | MCQ | Yes | Yes | cn_contest | false | 704,596 |
There are three piles of small stones, with the numbers being 19, 8, and 9. Now, the following operation is performed: each time, one stone is taken from any two of these three piles, and then these two stones are added to the other pile. Can the following conditions be met after several such operations?
(1) One pile ... | (1) It can be done, with a minimum of six operations.
(2) It is impossible to do. | 6 | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 704,597 |
$6 . \sqrt{x}+\sqrt{y}=\sqrt{2009}$ The positive integer solutions $(x, y)$ of, the maximum value of $x+y$ is ().
(A) 1189.
(B) 1517.
(C) 1657.
(D) 1749. | B
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | B | Algebra | MCQ | Yes | Yes | cn_contest | false | 704,599 |
6, The solution to the system of equations $\left\{\begin{array}{l}x(x+1)(3 x+5 y)=144, \\ x^{2}+4 x+5 y=24\end{array}\right.$ is $\qquad$ $\cdot$ | $\begin{array}{l}\left\{\begin{array}{l}x_{1}=-4 \\ y_{1}=4.8\end{array}\right. \\ \left\{\begin{array}{l}x_{2}=3 \\ y_{2}=0.6\end{array}\right.\end{array}$ | null | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,605 |
1. As shown in the figure, $AB$ is the diameter of $\odot O$, $AB=4$, chord $BC=3$, the angle bisector of $\angle ABC$ intersects the semicircle at $D$, the extensions of $AD$ and $BC$ meet at $E$. Then the area of quadrilateral $ABCD$ is ( ) times the area of $\triangle DCE$.
(A) 7 times.
(B) 8 times.
(C) 3 times.
(D)... | A
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | null | Geometry | MCQ | Yes | Yes | cn_contest | false | 704,607 |
3. If for any allowed value of $x$ within a certain specific range, $P=|1-2 x|+|1-3 x|+|1-4 x|$ $+\cdots+|1-9 x|+|1-10 x|$ is a constant, then this constant is ( ).
(A) 2 .
(B) 3 .
(C) 4 .
(D) 5 . | $B$ | B | Algebra | MCQ | Yes | Yes | cn_contest | false | 704,610 |
5. Let $x, y$ be positive integers, and $x<y$. The characteristic of $\lg x$ is $a$, and the mantissa is $\alpha$; the characteristic of $\lg y$ is $b$, and the mantissa is $\beta$. They satisfy the conditions: $a^{2}+b^{2}=5, \alpha+\beta=1$. Then the maximum value of $x$ is ( ).
(A) 50. (B) 80. (C) 100. (D) 200 | B
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | null | Algebra | MCQ | Yes | Yes | cn_contest | false | 704,612 |
6. Let $A D$ be the altitude on the base $B C$ of isosceles $\triangle A B C$, and $B M_{1}, B M_{2}, \cdots, B M_{\mathrm{n}}$ be the $n$ angle bisectors of $\angle A B C$, intersecting $A D$ at points (from $A$ to $D$) $M_{1}, M_{2}, \cdots, M_{\mathrm{n}}$, respectively. Extend $C M_{2}$ to intersect $A B$ at $E$. T... | A
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | null | Geometry | MCQ | Yes | Yes | cn_contest | false | 704,613 |
3. For an equilateral $\triangle ABC$ with side length $1$, $P$ is a point on side $AB$. $PQ \perp BC, \quad QR \perp AC, \quad RS \perp AB$ (where $Q, R, S$ are the feet of the perpendiculars). If $SP=\frac{1}{4}$, then $AP=$ | 3. $\frac{1}{9}$ or $\frac{5}{9}$, | \frac{1}{9} \text{ or } \frac{5}{9} | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,616 |
4. Let the incircle of $\triangle A B C$ touch side $B C$ at point $D$, dividing $B C$ into segments of lengths $m, n$. Also, $\angle B A C$ $=60^{\circ}$, then the area of $\triangle A B C$ is $\qquad$ | 4. $\sqrt{3} m n$, | \sqrt{3} m n | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,617 |
2. Among the following four equations:
$$
\begin{array}{l}
3 \sqrt{\frac{3}{8}}=\sqrt{3+\frac{3}{8}}, \\
\sqrt{19-6 \sqrt{10}}=\sqrt{10}-3, \\
1 \lg^{2} 15=2 \lg 15, \frac{2^{\mathrm{n}+4}-2\left(2^{\mathrm{n}}\right)}{2\left(2^{\mathrm{n}}+3\right)}=\frac{7}{8}
\end{array}
$$
The number of incorrect ones is ().
(A) O... | $A$ | A | Algebra | MCQ | Yes | Yes | cn_contest | false | 704,622 |
3. The integer $m$ that allows $2 m-1, 4 m+5, 20-m$ to be the three sides of a triangle has (.) options.
(A) two.
(B) six.
(C) twelve.
(D) eighteen. | $\mathrm{A}$ | A | Algebra | MCQ | Yes | Yes | cn_contest | false | 704,623 |
4. In isosceles $\triangle A B C$, the base $B C$ is $\frac{1}{4}$ of the perimeter. A line parallel to the two legs is drawn from any point $P$ on the base, intersecting the two legs at $E$ and $F$. Then the ratio of the perimeter of quadrilateral $A E P F$ to the perimeter of $\triangle A B C$ is ( ).
(A) equal to $\... | C
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | null | Geometry | MCQ | Yes | Yes | cn_contest | false | 704,624 |
$5 . a, b$ are two non-zero real numbers, $a^{2}, b^{2}$ are the roots of the quadratic equation $x^{2}-c x+c=0$, then the algebraic expression
(A) 0 .
(B) 2 .
(C) does not exist .
(D) none of the above. | D
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | D | Algebra | MCQ | Yes | Yes | cn_contest | false | 704,625 |
6. On side $AB$ of $\triangle ABC$, take points $D, E$, and on side $CA$, take points $F, G$. Connect $BF, FE, EG, GD$. Divide $\triangle ABC$ into five smaller triangles of equal area, $=S_{\triangle DGA}$. Then the ratio $\frac{F}{C} \frac{G}{A}$ equals ( ).
(A) $\frac{1}{3}$.
(B) $4$.
(C) $\frac{2}{5}$.
(D) $\frac{7... | B
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | null | Geometry | MCQ | Yes | Yes | cn_contest | false | 704,626 |
In $\triangle A B C$. 1, the side length opposite to $\angle A$, $BC$, is denoted as $a$, the radius of the excircle $\odot O$ is $R$, and it touches $BC$ and the extensions of $AB$, $AC$ at $D$, $E$, $F$ respectively.
Prove: $R \leqslant a \cdot (1 + \sin \frac{A}{2}) \cdot 2 \cos \frac{A}{2}^{\circ}$ | $$
\begin{array}{l}
\quad G E=B C \\
=a \text {. }
\end{array}
$$
Construct the incircle $O^{\prime}$ of $\triangle A B C$, which touches the three sides at $G$, $H$, and $K$, respectively. Let the radius of $\odot O^{\prime}$ be $r$, and $R+r \leqslant O O^{\prime}$ (equality holds when $A B = A C$). Then,
$$
R+r \le... | R \leqslant a \cdot \frac{1+\sin \frac{A}{2}}{\cos \frac{A}{2}} | Geometry | proof | Yes | Yes | cn_contest | false | 704,635 |
Five, find all integer solutions of $a$, $b$ that satisfy the inequality $\frac{a+b}{(a-b)|a b|}>1$.
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | Five, discuss in three cases. The integer solutions that satisfy the conditions are only two pairs: $\left\{\begin{array}{l}a=2 \text {, } \\ b=1 ｝\end{array}\right.$ $\left\{\begin{array}{l}a=-2 \\ b=-1\end{array}\right.$ | null | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,636 |
2. If $x<-2$, then $y=|1-| 1+x \mid$ is equal to ( ).
(A) $2+x$.
(B) $-2-x$.
(C) $x_{0}$
(D) $-x$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | B
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | B | Number Theory | proof | Yes | Yes | cn_contest | false | 704,638 |
3. In $\triangle A B C$, if $2 \sin A \cos B=\sin C$, then this triangle must not be ().
(A) Isosceles triangle.
(B) Right triangle.
(C) Scalene triangle.
(D) Equilateral triangle. | C
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | C | Geometry | MCQ | Yes | Yes | cn_contest | false | 704,639 |
4. Let the equation $2 x^{2}-(a+1) x+a+3=0$ have two roots whose difference is 1, then the value of $a$ is ( ).
(A) 9 and -3.
(B) 9 and 3.
(C) -9 and 3.
(D) -9 and -3. | A
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | A | Algebra | MCQ | Yes | Yes | cn_contest | false | 704,640 |
5. On a plane, there are two congruent equilateral triangles, one of which has one vertex at the center of the other, as shown in the figure. Then the ratio of the maximum area to the minimum area of the overlapping part is ( ).
(A) 1 .
(B) 2 .
(C) 3 .
(D) 4 . | B
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | B | Geometry | MCQ | Yes | Yes | cn_contest | false | 704,642 |
1. In $\triangle A B C$, $A B=3, A C=4$, $B C=5$, now fold it so that point $B$ coincides with point $C$, then the length of the fold is $\qquad$ | $\frac{15}{8}$ | \frac{15}{8} | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,643 |
Five, given that $a, b, c$ are all real numbers, and $a>0$, $b^{2}-a c<0$, $\alpha, \beta$ are the two roots of the quadratic equation $x^{2}-(a+c) x -b^{2}+a c=0$. Prove that
(1) $\alpha$ and $\beta$ are both positive;
(2) if $\alpha \geqslant \beta$, then $\alpha \geqslant a, \alpha \geqslant c$.
| (1) Given $a>0, b^{2}-a c<0$, we have $-b^{2}+a c>0$.
Also,
$$
\begin{array}{l}
\alpha+\beta=a+c>0, \alpha \beta=-b^{2}+a c>0 \\
\Rightarrow a>0, \beta>0 .
\end{array}
$$
(2) Let $f(x)=(x-\alpha)(x-\beta)$
$$
=x^{2}-(a+c) x-b^{2}+a c \text {, }
$$
Then,
$$
\begin{array}{l}
f(a)=(a-\alpha)(a-\beta)=-b^{2} \leqslant 0, ... | proof | Algebra | proof | Yes | Yes | cn_contest | false | 704,648 |
2. Let $t$ be the integer closest to $\sqrt{\sqrt[3]{2}-1}+\sqrt[3]{2}$, then $\sqrt{3-2 \sqrt{t}}$ equals $\qquad$ | 2. $\sqrt{2}-1$; | \sqrt{2}-1 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,650 |
3. If $\left|3^{x}-\log _{3} x\right|=3^{x}+\log _{3} x=k$, then the value of $k$ is ( ).
(A) 3,
(B) 1.
(C) 0.
(D) Cannot be determined. | A
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | A | Algebra | MCQ | Yes | Yes | cn_contest | false | 704,652 |
3. When $x+2 y=10$, the maximum value of $\lg x+\lg y$ is ().
(A) $1 \mathrm{~g} 12$.
(B) $\lg \frac{15}{2}$.
(C) $\lg 15$.
(D) $\lg \frac{25}{2}$. | 3. D.
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | D | Algebra | MCQ | Yes | Yes | cn_contest | false | 704,658 |
4. The three sides of $\triangle A B C$ are $a, b, c$, and the quadratic equation in $x$ is $b^{2} x^{2}-\left(b^{2}+c^{2}-a^{2}\right) x+c^{2}=0$. The situation of the roots of this equation is ().
(A) It has real roots.
(B) It has no real roots.
(C) It has two equal real roots.
(D) It has two unequal real roots. | 4. B.
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | null | Algebra | MCQ | Yes | Yes | cn_contest | false | 704,659 |
Four, a certain project, if contracted by Team A and Team B, can be completed in $2 \frac{2}{5}$ days, with a payment of 1800 yuan; if contracted by Team B and Team C, it can be completed in $3 \frac{3}{4}$ days, with a payment of 1500 yuan; if contracted by Team C and Team A, it can be completed in $2 \frac{6}{7}$ day... | Four, Solution List as follows:
From the table above, "on the premise of completing this work within a week," it is clear that choosing Team B would incur the least cost. | B | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,661 |
1. If | $x-y+2$ | and $(x+y-1)^{2}$ are opposite numbers, then $x=$ $\square$ | -, 1. $x=-\frac{1}{2}, y=\frac{3}{2}$ | x=-\frac{1}{2} | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,663 |
3. Calculate: $a \cdot a^{2} \cdot a^{3} \cdot \cdots \cdot a^{100}=$ | 3. $a^{5050}=$
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly.
However, since the provided text is already in a mathematical form which is universal and does not require translation, the output remains the same:
3. $a^{5050}=... | a^{5050} | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,666 |
4. If the two real roots of $x^{2}+a x+b=0$ are $\alpha$, $\beta(\alpha \beta \neq 0)$, then the two real roots of $b x^{2}+a x+1=0$ | 4. $\frac{1}{\alpha}, \frac{1}{\beta}$ | \frac{1}{\alpha}, \frac{1}{\beta} | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,667 |
10. Consider the following sequences:
$$
\begin{array}{l}
3,7,11,15,19,23,27,31, \cdots \\
2,5,8,11,14,17,20,23, \cdots
\end{array}
$$
The 20th pair of identical numbers in them is $\qquad$ | 10. 239 . | 239 | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 704,673 |
1. If $\frac{1}{x-y}$ is meaningful, then ( ).
(A) $x \div 1$.
(B) $y \neq 0$.
(C) $x \neq 1$ or $y \neq 0$.
(D) $x \neq 1$ and $y \neq 0$. | D
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | D | Algebra | MCQ | Yes | Yes | cn_contest | false | 704,674 |
7. In square $A B C D$, $E$ is on $B C$, $B E$ $=2, C E=1, P$ is on $B D$, then the minimum possible value of the sum of the lengths of $P E$ and $P C$ is $\qquad$ . | 7. $\sqrt{13}$; | \sqrt{13} | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,675 |
3. The solution situation of the equation $\sqrt{x-1} \cdot \sqrt{x+1}=-\sqrt{x^{2}-1}$ is ().
() Unique solution.
(B) Two solutions.
(C) Infinitely many solutions.
(D) No real solutions. | A
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | A | Algebra | MCQ | Yes | Yes | cn_contest | false | 704,677 |
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