problem stringlengths 1 13.6k | solution stringlengths 0 18.5k ⌀ | answer stringlengths 0 575 ⌀ | problem_type stringclasses 8
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4. Given the three sides of an obtuse triangle are 3, $4, x$, then the range of values for $x$ is ( ).
(A) $1<x<7$.
(B) $5 \ll x<7$.
(C) $1<x<\sqrt{7}$.
(D) $5<x<7$ or $1<x<\sqrt{7}$. | $D$ | D | Geometry | MCQ | Yes | Yes | cn_contest | false | 704,678 |
5. Given that $\alpha$ is a root of the equation $x^{2}-5 x+1=0$, then the last digit of $\alpha^{4}+\alpha^{-4}$ is ().
(A) 3.
(B) 5.
(C) 7.
(D) 9. | 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 | 704,679 |
6. The sum of the natural numbers 1, $2,3,4, \cdots, 1988,1989$ is an odd number. If $t$ numbers are prefixed with "-", then the sum of these 1989 numbers ( ).
(A) is always odd.
(B) is always even.
(C) is odd when $t$ is odd.
(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. | null | Number Theory | MCQ | Yes | Yes | cn_contest | false | 704,680 |
8. (1) From $x^{2}-y^{2}=0$, we can get $\left\{\begin{array}{l}x+y=0, \\ x-y=0,\end{array}\right.$
(2) If there are two points on a line that are equidistant from another line, then the two lines are parallel,
(3) If two triangles have two sides corresponding equal, and the areas are also equal, then they must be cong... | A
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 | 704,682 |
9. $\{x\}$ represents the smallest integer not less than $x$, $[x]$ represents the largest integer not greater than $x$. For example, $\{\pi\}=4,[\pi]=3$, then the solution to the equation $\{x\}^{2}+4([x]+1)+4=0$ is ( ) .
(A) $x=-2$.
(B) $x-3$.
(D) $-3<x<-2$.
Translating the given text into English while preserving t... | 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 | 704,683 |
Three, solve the equation, $\sqrt{1-\sqrt{1+x}}=x$.
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | Three, $x=0$ is a root of the original equation. | x=0 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,685 |
8. If $12^{x}=3,12^{y}=2$, then $8^{\frac{1-2 x}{1-x+y}}$ $=$ | 8. $\frac{4}{3}$ | \frac{4}{3} | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,686 |
1. The number of real roots of the equation $\left|x^{2}-1\right|=\frac{1}{10}\left(x+\frac{11}{10}\right)$ is ( ).
(A) 0 .
(B) 2 .
(C) 3 .
(D) 4 . | 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 | 704,688 |
2. A triangle and a trapezoid have equal areas, the height of one side of the triangle is equal to the height of the trapezoid, and the length of that side is 10. Then the length of the midline of the trapezoid is ( ).
(A) 5 .
(B) 10 .
(C) 20 .
(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 | 704,689 |
3. Let $0<a<1$, and $a$ is a constant, then the graph of $y=(\lg a) x^{2}-\left(\lg \frac{1}{a}\right) x-(\lg a)^{3}$ is ( ).
Translate the text above 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. | null | Combinatorics | MCQ | Yes | Yes | cn_contest | false | 704,690 |
4. Given that the ratio of the squares of the three sides of a triangle is $1: 2: 3$, which of the following conclusions is correct? ( )
(A) There are two altitudes that are perpendicular to each other, but no medians that are perpendicular to each other.
(B) There are two medians that are perpendicular to each other, ... | 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 | 704,691 |
5. In $\triangle A B C$, the sides opposite to $\angle A, \angle B, \angle C$ are $m, m+1, \sqrt{2 m+1}$, respectively. Then the value of $\sqrt{ }(\lg \sin A+\lg \cos C)^{2}$ is
(A) $\lg \frac{m}{m+1}$.
(B) $2 \lg \frac{m}{m+1}$.
(C) $\lg \frac{m \sqrt{2 m+1}}{(m+1)^{2}}$.
(D) $2 \lg \frac{m+1}{m}$. | 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 | 704,692 |
6. In the obtuse triangle $\triangle A B C$ with centroid $G$, $B C$ $=1, \angle A=30^{\circ}, D$ is the midpoint of $B C$, then the range of $G D$ is ().
(A) $\frac{1}{6}<D G<\frac{\sqrt{13}}{6}$.
(B) $0<D G<\frac{\sqrt{3}+2}{6}$.
(C) $\frac{1}{6} \leqslant D G \leqslant \frac{\sqrt{13}}{6}$.
(D) $0<D G \leqslant \fra... | 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 | 704,693 |
7 ・Let $f_{k}=k^{2}+(k+1)^{2}+\cdots+(3 k)^{2}$, then $f_{4}-f_{3}$ is ( ).
(A) $(12 k)^{2}-(9 k)^{2}$.
(B) $12^{2}-9^{2}$.
(C) $10^{2}+11^{2}+12^{2}-0$.
(D) $12^{2}-9^{2}-3^{2}$. | C
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | null | Algebra | MCQ | Yes | Yes | cn_contest | false | 704,694 |
8. Given $\lg 2=0.30103$, then the sum of the first and last digits of $2^{1000}$ is ().
(A) 3.
(B) 5.
(C) 8.
(D) 10. | 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 | 704,695 |
1. For the equation $x^{2}+b x+c=0$ with roots $r$ and $s$, and the equation $x^{2}+p x+q=0$ with roots $r^{2}$ and $s^{2}$, then $p$ equals ( ).
(A) $4 c-b^{2}$.
(B) $b^{2}-4 c$.
(C) $b^{2}-2 c$.
(D) $2 c-b^{2}$. | 1. 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 | 704,697 |
3. Let the integer part of $\frac{\sqrt{13}+3}{\sqrt{13}-3}$ be $m$, and the decimal part be $n$, then the value of $198 m+9 n+n^{2}+1$ is $\qquad$ | 3. 1790 , | 1790 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,699 |
Given the parabola $y=a x^{2}+b x+c$ passes through $(c, 2)$, and $a|a|+b|b|=0$, also $a x^{2}+b x+c$ $-2>0$ has no solution. Try to find the values of $a, b, c$. | 一、Prompt: The parabola is below $y=2$ and is tangent to $y$ $=2$ at the point $(c, 2)$. $(a=-6, b=6, c=$ $\frac{1}{2}$ ) | a=-6, b=6, c=\frac{1}{2} | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,701 |
Three, take $m$ points randomly inside a convex $n$-sided polygon, and connect these $m+n$ points (the $m$ points taken and the original $n$ vertices) in such a way that no two connecting lines intersect, and all regions in the figure are triangles. How many triangles are there in the figure? | Three, Prompt: The angles in the figure consist of two parts, one part of the angle has the vertex at the vertex of an $n$-sided polygon, and the other part of the angle has the vertex at $m$ points. According to this, calculate the degree of the angle, and argue that there are $(2m+n-2)$ triangles in the figure. | 2m+n-2 | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false | 704,702 |
1. Prove that there do not exist integers $a$ and $b$, such that
$$
a^{2}+b^{2}=1 \times 2 \times 3 \times \cdots \times(n-1) \times n \text { . }
$$
(Where $7 \leqslant n<14$ ) | Proof: Since $7 \leqslant n < 14$, then $1 \times 2 \times 3 \times \cdots \times n$ is divisible by 7, i.e., $a^{2} + b^{2}$ is divisible by 7. If $a$ is not a multiple of 7, let $a = 7t + r \quad (1 = 1, 2, \cdots, 6)$, then $a^{2}$ is of the form $7k + 1$, $7k + 2$, or $7k + 4$.
Thus, $a^{2} + b^{2}$ is of the form... | proof | Number Theory | proof | Yes | Yes | cn_contest | false | 704,703 |
3. Prove:
(1) There does not exist an integer $n$, such that $n^{2}+n+1$ is divisible by 5,
(2) There exist infinitely many integers $n$, such that $n^{2}+n+1$ is divisible by 343. | (1) Let $n=5k+\tau, k$ be an integer, $r$ $=0, 1, 2, 3, 4$. Substitute $n$ into $n^{2}+n+1$, and discuss for $r$.
(2) Since $343=7^{3}$. We set $n=7k+r$, $r=0, \pm 1, \pm 2, \pm 3$.
$n^{2}+n+1=49k^{2}+14kr+7k+r^{2}+r+1$.
When $r=2$ or -3, $r^{2}+r+1$ is a multiple of 7. Let's take $r=2$, at this point
$$
n^{2}+n+1=7\le... | proof | Number Theory | proof | Yes | Yes | cn_contest | false | 704,705 |
4. Find the largest odd number that cannot be expressed as the sum of three distinct composite numbers.
untranslated text:
4.求一个不能用三个不相等的合数之和表示的最大奇数。 | The smallest sum of three unequal composite numbers is
$$
4+6+8=18 \text{.}
$$
We prove that 17 is the largest odd number that cannot be expressed as the sum of three unequal composite numbers.
In fact, it is only necessary to prove that any odd number $2k-1$ not less than 19 can always be expressed as the sum of thr... | 17 | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 704,706 |
5. Does there exist a sequence of 14 consecutive positive integers, each of which is divisible by at least one prime number not less than 2 and not greater than 11? | There do not exist 14 consecutive positive integers that meet the requirements.
Assume they exist. Let these 14 consecutive positive integers be $N$, $N+1, N+2, \cdots, N+13$.
By symmetry, assume $N$ is even, thus $N$, $N+2, \cdots, N+12$ are all divisible by 2.
The remaining seven odd numbers $N+1, N+3, N+5$, $N+7,... | proof | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 704,707 |
7. Suppose point $O$ is inside a 1000-sided polygon $A_{1} A_{2} \cdots A_{1000}$, and the vertices of the 1000-sided polygon are numbered arbitrarily, with the numbers being $1, 2, \cdots, 1000$. The line segments $O A_{1}$, $O A_{2}, \cdots, O A_{1000}$ are also numbered arbitrarily, with the numbers being $1, 2, \cd... | Let the sum of the areas of these triangles and the segments be $S$, then
$$
\begin{aligned}
1000 S & =S_{1}+S_{2}+\cdots+S_{1} 00 \\
& =\frac{3}{2} \times 1000 \times 1001
\end{aligned}
$$
However, $\frac{3}{2} \times 1000 \times 1001$ is not an integer,
so the requirement of the problem is impossible.
8. Prove that ... | proof | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false | 704,710 |
9. From the given number
$$
1234567891011121314 \cdots 9899100
$$
erase 100 digits to obtain the largest possible remaining number. | (The numbers obtained are $9999978596061 \cdots 9899100$ ) | 9999978596061 \cdots 9899100 | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 704,711 |
3. Let $m=\left(\frac{1}{3}\right)^{-\frac{1}{5}}, n=\left(\frac{1}{4}\right)^{\frac{1}{3}}$, $p=\left(\frac{1}{5}\right)^{\frac{1}{4}}$. Then the size relationship is ( ).
(A) $m<n<p$.
(B) $m<p<n$.
(C) $n<p<m$.
(D) $p<n<m$. | 3. C.
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | null | Algebra | MCQ | Yes | Yes | cn_contest | false | 704,712 |
Four, $E, F$ are
on the sides $B C$ and $C D$
of rectangle $A B C D$,
if the areas of $\triangle C E F$,
$\triangle A B E$, $\triangle A D F$
are 3,
4, 5 respectively. Find the area $S$ of $\triangle A E F$. | Let $AB = a, BC = b$, then $BE = \frac{8}{a}$,
$$
\begin{array}{l}
CE = b - \frac{8}{a}, DF = \frac{10}{b}, FC = a - \frac{10}{b}. \\
\left\{\begin{array}{l}
\frac{1}{2}\left(b - \frac{8}{a}\right) \times \left(a - \frac{10}{b}\right) = 3, \\
ab = 3 + 4 + 5 + S .
\end{array}\right. \\
S = \sqrt{144 - 80} = 8 .
\end{arr... | 8 | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,715 |
Fear, from the seven digits $0,1,2,3,4,5,6$, many seven-digit numbers without repeated digits can be formed, among which some are multiples of 55. Among these multiples of 55, find the largest and the smallest. (Write out the reasoning process) | Five, Solution: Let the sum of the four digits in the odd positions of such a seven-digit number be $A$, and the sum of the three digits in the even positions be $B$, then $|A-B|=11k$ (where $k$ is a natural number or 0).
Also, $A+B=21$,
Since $A, B$ are positive numbers less than 21, so $A-B=5$, thus $A=16, B=5$.
Amo... | 1042635, 6431205 | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 704,717 |
Six, as shown in the figure, the lengths of the three sides of $\triangle A B C$ are $B C$ $=17, C A=18, A B=19$. Through any point $P$ inside $\triangle A B C$, perpendicular lines $P D, P E$, $P F$ are drawn to the three sides of $\triangle A B C$ (with $D, E, F$ being the feet of the perpendiculars), and $B D+C E$ $... | Six、 $B D+B F=18$.
переведено как:
Six、 $B D+B F=18$.
However, the correct English translation should be:
Six. $B D+B F=18$.
Note: The original text uses a full-width comma and the number "Six" followed by a full-width period, which is preserved in the translation. If the format is to be strictly followed, the tr... | B D+B F=18 | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,718 |
4. If the difference between the two roots of the quadratic equation $x^{2}-p x+q=0$ is 1, express $q$ in terms of $p$, then $q=$ | $4 \cdot \frac{1}{4}\left(p^{2}-1\right)$ | \frac{1}{4}\left(p^{2}-1\right) | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,719 |
7. If the sum of the number of sides of two convex polygons is 17, and the sum of the number of diagonals is 47, then the number of sides of these two convex polygons is | 7. 8 and 9, | 8 \text{ and } 9 | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,722 |
8. Given $\triangle A B C$ with two sides $B C=a, A C$ $=b$, and the medians $A D, B E$ on sides $B C, A C$ are perpendicular to each other, then the length of the third side $A B$ expressed in terms of $a, b$ is | 8. $\frac{1}{5} \sqrt{5\left(a^{2}+b^{2}\right)}$ | \frac{1}{5} \sqrt{5\left(a^{2}+b^{2}\right)} | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,723 |
12. When programming a computer to print out 10000 numbers, there is a bug, every time it prints the number 3, it prints “X” instead, how many numbers are printed incorrectly.
保留了源文本的换行和格式。 | 12. 3439 . | 3439 | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false | 704,728 |
Two people, A and B, participated in the same exam and left the examination room at 10:00 AM, then had a meal at the same time. A said: "I left the examination room at the earlier of 2 hours before lunch and 1.5 hours after the exam started." B said: "I left the examination room at the later of 2.5 hours before lunch a... | II. The exam start time and lunch start time are 9 AM, 12 PM or 8:30 AM, 12:30 PM. | 9 \text{ AM}, 12 \text{ PM} \text{ or } 8:30 \text{ AM}, 12:30 \text{ PM} | Logic and Puzzles | math-word-problem | Yes | Yes | cn_contest | false | 704,729 |
Four, as shown in the figure, $P$ is a point inside the square $ABCD$, $PA=5$, $PB=8$, $PC=13$. Find the area of square $ABCD$.
---
The translation maintains the original text's line breaks and format. | Four, Hint: Like tears, draw $P E \perp$ $A B, P F \perp B C$, and let the side length of the square be a, $\quad P_{E}=x, \quad P F$ $=y$, solve to get the area of square $A B C D$ is approximately 153. | 153 | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,731 |
1. The solution to the equation $\left(1+x^{2}\right)^{2}=4 x\left(1-x^{2}\right)$ is $x=$ | 1. $-1 \pm \sqrt{2}$, | -1 \pm \sqrt{2} | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,732 |
6. The number of natural numbers $n$ that make $2 n(n+1)(n+2)(n+3)+12$ expressible as the sum of the squares of two natural numbers is ( ).
(A) 0.
(B) 1.
(C) Finite (but more than 1).
(D) Infinitely many. | $A$ | A | Number Theory | MCQ | Yes | Yes | cn_contest | false | 704,738 |
1. Let $a, b$ be integers, and consider the following four propositions:
(1) If $a+5 b$ is even, then $a-3 b$ is also even.
(2) If $a+b$ is divisible by 3, then $a, b$ are both divisible by 3.
(3) If $a+b$ is a prime number, then $a-b$ is definitely not a prime number.
(4) If $c=a+b \neq 0$, then
$$
\frac{a^{3}-b^{3}}{... | B
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | null | Number Theory | MCQ | Yes | Yes | cn_contest | false | 704,741 |
2. Person A and Person B are running on a circular track at their respective constant speeds. If both start from the same point and run in opposite directions, after their first meeting, B runs for another 8 minutes to reach the original starting point. It is known that A takes 6 minutes to complete one lap. Therefore,... | 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 | 704,742 |
3. Let the two legs of a right triangle be $a$, $b$, the hypotenuse be $c$, and the altitude to the hypotenuse be $h$. Then, the shape of the triangle formed by the sides $c+h$, $a+b$, $h$ is ().
(A) Right triangle.
(B) Acute triangle.
(C) Obtuse triangle.
(D) Cannot be determined, the shape depends on the sizes of $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 | 704,743 |
4. Given the quadratic function $y=a x^{2}+b x+c(a>0)$ has the axis of symmetry $x=2$, and when $x_{1}=\sqrt{2}, x_{2}=\pi$, $x_{3}=0$, the values of the quadratic function $y$ are $y_{1}, y_{2}$, $y_{3}$, respectively. Then the size relationship of $y_{1}, y_{2}, y_{3}$ is ().
(A) $y_{1}>y_{2}>y_{3}$.
(B) $y_{1}y_{1}>... | 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 | 704,744 |
5. For $\triangle A B C$ with side lengths $B C=a$, $C A=b, A B=c$, take any point $P$ inside $\triangle A B C$, and draw lines parallel to the three sides, intersecting the three sides (as shown in the figure). If $D E=a^{\prime}, F G=a^{\prime}, H I=c^{\prime}$, then the value of $\frac{a^{\prime}}{a}+\frac{b^{\prime... | 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 | 704,745 |
3. Divide each face (a total of six faces) of a $3 \times 3 \times 3$ cube into 9 smaller squares of equal size (a total of 54 smaller squares). Now, use red, yellow, and blue to color these smaller squares, such that adjacent squares sharing an edge cannot be the same color. Then, the maximum number of smaller squares... | C
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | null | Combinatorics | MCQ | Yes | Yes | cn_contest | false | 704,746 |
1. Given two integers $A, B$ whose sum is 192, and the least common multiple is 660, find these two numbers $A=$ $\qquad$ $B=$ $\qquad$ | 1. $A=132, B=60$ | A=132, B=60 | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 704,747 |
1. $(x+y)(x-y)+4(y-1)$ The result of factoring out is $\qquad$ .
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | 1. $(x+y-2)(x-y+2)$ | (x+y-2)(x-y+2) | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,749 |
$$
\begin{array}{c}
\text { 3. Given } x+y+z=3 a \quad(a \neq 0) \text {, then } \\
\frac{(x-a)(y-a)+(y-a)(z-a)+(z-a)(x-a)}{(x-a)^{2}+(y-a)^{2}+(z-a)^{2}}
\end{array}
$$
The value is $\qquad$ | 3. $-\frac{1}{2}$, | -\frac{1}{2} | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,750 |
5. The solution set of the inequality $\left|x^{2}+3 x-8\right|>x^{2}+3 x-8$ is | 5. $\frac{-3-\sqrt{41}}{2}<x<\frac{-3+\sqrt{41}}{2}$. | \frac{-3-\sqrt{41}}{2}<x<\frac{-3+\sqrt{41}}{2} | Inequalities | math-word-problem | Yes | Yes | cn_contest | false | 704,765 |
2. As shown in the figure, the radius of circle $A$ is $r$, and the radius of circle $O$ is $4r$. Circle $A$ starts from the position shown in the figure and rolls without slipping around circle $O$. To make the center of circle $A$ return to its original position, the number of times circle $A$ rolls is ( ).
(A) 3.
(B... | 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 | 704,767 |
3. If $M=3 x^{2}-8 x y+9 y^{2}-4 x+6 y$ +13, then the following must be true ( ).
(A) $M \geqslant 0$.
(B) $M0$.
(D) $M \leqslant 0$. | 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 | 704,768 |
5. As shown in the figure, in $\triangle A B C$, $A B=A C$. $\angle A=36^{\circ}$. To make $\triangle A B C \sim \triangle B C D$, the additional condition needed is ().
(A) $B C=C D$.
(B) $\frac{B D}{A C}=\frac{C D}{B C}$.
(C) $A D=D C$.
(D) $\frac{B C}{\sin A}=\sin 18^{\circ}$. | 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 | 704,770 |
2. There is a car, the circumference of its front wheel is $5 \frac{5}{12}$ meters, and the circumference of its rear wheel is $6 \frac{1}{3}$ meters. Then it must travel $\qquad$ meters to make the number of revolutions of the front wheel 99 more than that of the rear wheel. | 2. 3705 ; | 3705 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,778 |
1. $\sqrt{ }(\lg 3)^{2}-\lg 9+1$ equals $(\quad)$. (A) $\pm(\lg 3-1)$.
(B) $\lg 3-1$
(C) $1-\lg 3$.
(D) 1. | 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 | 704,780 |
$\begin{array}{l}\text { 2. The unit digit of } 1988^{1^{089}}+1989^{1088} \text { is } \\ \text { ( ). } \\ \text { (A) 9. (B) 7. (C) 5. (D) } 3 . \\\end{array}$ | A
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | null | Number Theory | MCQ | Yes | Yes | cn_contest | false | 704,781 |
3. The roots of $2 x^{2}-18 x+36=0$ and $x^{2}-20 x$ $+75=0$ are the lengths of four line segments, then these segments ( ).
(A) can form a quadrilateral with a perimeter of 38.
(B) can form a quadrilateral with a perimeter of 29.
(C) cannot form a quadrilateral.
(D) none of the above answers is correct. | 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 | 704,783 |
4. As
shown, hexagon
$A B C D E F$
is composed of
five squares,
with the side
length of each
square being
1 cm. A line through $A$ intersects $E D, C D$ at $M, N$ respectively. If the areas on both sides of line $M N$ in this hexagon are equal, then the length of $E M$ is: $\qquad$ cm. | 4. $\frac{\sqrt{5}-1}{2}$ | \frac{\sqrt{5}-1}{2} | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,790 |
1. (Australia 3) The integer 9 can be expressed as the sum of two consecutive integers
$$
9=4+5=2+3+4 \text {. }
$$
Is there a positive integer that can be expressed as the sum of 1990 consecutive integers and can also be expressed in exactly 1990 ways as the sum of consecutive integers? | Let $N$ be the sum of 1990 consecutive positive integers $m$, $m+1, \cdots, m+1989$, then
$$
N=995 (2 m+1989) \text{. }
$$
Suppose $N$ can be expressed as the sum of at least two consecutive positive integers in exactly 1990 ways, i.e., there are exactly 1990 pairs of positive integers $n, k \quad(k \geqslant 2)$, such... | N=5^{10} 199^{180} \text{ or } N=5^{180} 199^{10} | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 704,793 |
2. (Addition to the problem $1^{*}$ ) $n$ countries, each country's team of 3 representatives, form $m$ representative meetings $A_{0}(1), A_{0}(2), \cdots, A_{0}(m)$ called a meeting circle. If
1 each meeting has $n$ representatives, exactly 1,
() no two meetings have the same representatives;
(3) $A_{0}(i)$ and $A_{0... | We use $(m, n)$ to denote a conference circle with $n$ countries and $m$ meetings, and use $\left(A_{\mathrm{n}}(i), j\right)$ to denote a concept formed by all the representatives and the $j$-th representative of the $n+1$-th country in the meeting $A_{n}(i)$. If $A,(0), \cdots, A \sim(m)$ is an $(m, n)$ assembly, the... | 1990 | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false | 704,794 |
11. (Mexico 2) Determine all positive integers $k$ such that the set
$$
X=\{1990,1990+1, \cdots, 1990+k\}
$$
can be partitioned into two disjoint subsets $A$ and $B$ such that the sum of the elements in $A$ is equal to the sum of the elements in $B$. | Let the positive integer $k$ be such that
$$
X=\{1990,1990+1, \cdots, 1990+k\}
$$
can be divided into subsets $A$ and $B$ that satisfy the problem's requirements, then the sum of all elements in $X$ must be even, i.e.,
$$
\begin{array}{l}
\text { I, }(1990+n) \\
=1990(k+1), \quad h(k+1) \\
2
\end{array}
$$
This implie... | k \equiv 3 \pmod{4} \text{ or } k \equiv 0 \pmod{4} \text{ and } k \geq 92 | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false | 704,795 |
6. Given a polyhedron with an even number of edges, prove: it is possible to label each edge with an arrow such that an even number of arrows point to each vertex of the polyhedron. | To the edges of a polyhedron, the number of odd vertices (i.e., vertices with an odd number of arrows pointing to them) can be reduced by 2 each time, as long as the direction of the arrows is changed along any route connecting two odd vertices. Ultimately, it is impossible to leave a single vertex, because the total n... | proof | Combinatorics | proof | Yes | Yes | cn_contest | false | 704,797 |
3. In a dark room, a drawer contains socks of two colors and two sizes, 4 pairs in total (one pair of each color and size). How many socks must be taken out to ensure that there are two pairs of different colors and different sizes? | New $A_{1}, A, B, B$ each two. Period out 6 as $A ., A_{1}, 1, B_{1}, B_{1}, B_{2}$, does not meet the question's requirements. But when any 7 are taken out, there must be one color (let's say A) with 4 pieces all taken out, and among the 3 pieces of another color, there must be two. | 7 | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false | 704,802 |
5. Find all natural numbers $x$ that satisfy the following condition: the product of the digits of $x$ is equal to $44x - 86868$, and the sum of the digits is a perfect square. | In fact, $x \geqslant[85363+43]=1975$. If the number of digits of $x$, $k \geqslant 5$, then
$44 x-86868^{\circ}>4 \times 10^{k}-10^{2} \geqslant 3 \times 10^{k}$, so $x$ must be a four-digit number. $1 \leqslant S(x) \leqslant 36$. Hence $S(x)=1$, 8 or 27. Clearly, $S(x)=1$ does not meet the requirements.
Also, $0<P(x... | not found | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 704,804 |
12. (Netherlands 3) Consider a "bead" as a unit cube with a hole along its diagonal, strung together with a flexible thread, ensuring that adjacent cubes share at least one vertex through which the thread passes. Let $A$ be the starting vertex where the thread enters the first cube, and $B$ be the terminal vertex where... | To make the description easier, we refer to a rectangular prism with edge lengths $p, q, v$ as a $p \times q \times r$ rectangular prism.
When $q=r=1$, it is clear that for any positive integer $p$, $p$ unit cubes can be arranged into a $p \times 1 \times 1$ rectangular prism. You can consider; when $p$ is odd, $A, B$... | proof | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false | 704,806 |
4. For the set $\{00,01, \cdots, 98,99\}$, a subset $X$ satisfies: in any infinite sequence of digits, there are two adjacent digits that form an element of $X$. What is the minimum number of elements that $X$ should contain? | 4. For any $i, j \in\{0,1, \cdots, 9\}, \lambda$ should include $i j$ or $i i$ (otherwise the sequence $i j i j i \ldots$ would satisfy the problem's requirements). There are $10+C_{10}^{2}=55$ such unordered pairs $(i, j)$, hence $|X| \geqslant 55$.
On the other hand, if we take $X=\{i j: 0 \leqslant i \leqslant j \l... | 55 | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false | 704,809 |
5. Prove: Any group of people can be divided into two groups, such that the total number of acquaintances within the same group is less than the number of acquaintances between different groups. | 5. This problem is equivalent to: For any graph with $n$ points $(n \geqslant 2)$, if it has at least one edge, then its vertices can be colored with red and blue such that the number of edges with the same color at both ends is equal to the number of edges with different colors at the ends. Induction on $n$, for $n=2$... | proof | Combinatorics | proof | Yes | Yes | cn_contest | false | 704,810 |
1. In space, there are four lines, two of which are colored red, and two are colored blue, and any red line is perpendicular to any blue line. Prove: either the two red lines are parallel, or the two blue lines are parallel. | 1. If the two red lines are not parallel, then they are coplanar or skew. At this time, there is a plane parallel to them, each passing line is perpendicular to two intersecting lines on this plane, thus being perpendicular to these two planes, therefore the two blue lines are parallel. | proof | Geometry | proof | Yes | Yes | cn_contest | false | 704,812 |
2. In $\triangle A B C$, points $M, K, L$ are taken on sides $A B, B C, C A$ respectively, such that $M K\|A C, M L\| B C$. Let $B L \cap M K=P, A K \cap M L=Q$. Prove that $P Q \| A B$. | $$
\begin{aligned}
\because & \frac{K P}{P M}=\frac{B P}{P L}=\frac{B K}{K C} \\
= & \frac{B M}{M A}=\frac{K Q}{Q A} \\
& \therefore P Q \| A B .
\end{aligned}
$$
The translation is as follows:
$$
\begin{aligned}
\because & \frac{K P}{P M}=\frac{B P}{P L}=\frac{B K}{K C} \\
= & \frac{B M}{M A}=\frac{K Q}{Q A} \\
& \t... | proof | Geometry | proof | Yes | Yes | cn_contest | false | 704,813 |
3. Given $A_{1}, A_{2}, \cdots ; B_{1}, B_{2}, \cdots$ are geometric sequences. Can $A_{1}+B_{1}, A_{2}+B_{2}, A_{3}+B_{3}$ and $A_{1}+B_{4}$ determine $A_{5}+B_{5}$? | 3. Let $A_{1}=a p^{i-1}, B_{1}=b q^{i-1}(1-i-5)$,
$S_{1}=A_{1}+B_{1}$, then
$$
S_{i+1}(p+q)=S_{i+2}+S_{i} p q(1 \leqslant i \leqslant 3) .
$$
If $S_{1} S_{3} \neq S_{2}^{2}$, then from
$$
\left\{\begin{array}{l}
S_{2}(p+q)-S_{1} p q=S_{3}, \\
S_{3}(p+q)-S_{2} p q=S_{4},
\end{array}\right.
$$
$p+q$ and $p q$ can be un... | S_{5}=\frac{S_{3}^{2}}{S_{1}} | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,814 |
4. The street plan of a city is a $5 \times 5$ square grid, (as shown in the figure) and there is a sweeper at point $A$. i. Find the total length of all streets.
保留了原文的换行和格式，翻译结果如上。 | 4. This graph has 16 odd vertices (the four inner points on each boundary) and can be divided into eight pairs of adjacent ones. Therefore, the length of the shortest cleaning route is $60+8=68$. | 68 | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,815 |
13. (Norway 1) Let $a, b$ be integers, $1 \leqslant a \leqslant b, M=\left[\frac{a+b}{2}\right]$. Define $f: Z \rightarrow Z$ as $f(n)=\left\{\begin{array}{ll}n+a, & n<M \\ n-b, & n \geqslant M .\end{array}\right.$ Let $f^{1}(n)=f(n) ; f^{1+1}(n)=f\left(f^{1}(n)\right), i=1$, $2, \cdots$. Find the smallest positive int... | Let $S$ be the set of integers $n$ satisfying $M-b \leqslant n \leqslant M+a-1$, then $f(S) \subseteq S$. For any $k$, if $f^{\mathrm{k}}(0)=0$, according to the definition of $f$, we can assume that there are $r$ times where $f(n)=n+a$ and $s$ times where $f(n)=n-b$, here $r, s \in \mathbb{Z}$. Thus, $k=r+s$, and
$$
a... | k=\frac{a+b}{(a, b)} | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 704,817 |
2. Is there a function that exists: its graph on the coordinate plane t intersects with any straight line at least at one point? | 2. For any odd number $n \geqslant 3, n$-degree polynomial function $y$ $=P(x)$ satisfies the problem requirements. For any line $y=a x$ $+b$, the equation $P(x)-a x-b=0$ has at least one real root $x_{0}$, so they have an intersection point $\left(x_{0}, a x_{0}+b\right)$, and for any line $x=c$, they also have an int... | proof | Calculus | math-word-problem | Yes | Yes | cn_contest | false | 704,819 |
Can you make all the cells in the Light Bulb Square show three adjacent identical symbols?
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 3. Prompt: Odd rows are arranged as
$$
++00++00++\cdots \text {; }
$$
Even rows are arranged as
$$
00++00++00 \cdots .
$$ | null | Logic and Puzzles | math-word-problem | Yes | Yes | cn_contest | false | 704,820 |
14. (Poland 1) Let $P$ be any point inside a regular tetrahedron $T$. Draw 4 planes through $P$ parallel to the faces of $T$, dividing $T$ into 14 regions. From these 14 regions, remove the tetrahedra and parallelepipeds, and denote the remaining regions as $f(P)$. These regions are adjacent to edges but not to the sam... | Solution 1: Let the distances from $P$ to the four faces of $T$ be $d_{1}, d_{2}, d_{3}, d_{4}$, and the height of $T$ be $h$. Let $x_{1}=d_{1} / h$, then we have
$$
x_{1}+x_{2}+x_{3}+x_{4}=1 \text {. }
$$
Among the 14 blocks formed by the divisions, there are 4 tetrahedra, whose volumes are $x_{1}^{3}, x_{2}^{3}, x_{... | \frac{3}{4} \text{ and } 0 | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,821 |
15. (Poland 3) Prove that for every integer $k>1$, there exists a multiple of it that is less than $k^{4}$, and the decimal representation of this multiple contains at most 4 different digits (these digits may repeat). | Prove that for an integer $n$, given $2^{0}-1 \leqslant k$, there exist different $x, y$ in $A_{0}$ such that
$$
x \equiv y \quad(\bmod k).
$$
Let $p=|x-y|$, then $p$ is a multiple of $k$, and the decimal representation of $p$ can only contain the digits 0, 1, 8, or 9.
Furthermore, from $2^{n-1} \leqslant k$ we get $... | proof | Number Theory | proof | Yes | Yes | cn_contest | false | 704,822 |
20. (USSR 1) Find all natural numbers $n$, such that every natural number represented in decimal notation with $n$ - 1 digits of 1 and one digit of 7 is a prime number. | Solve: A natural number $N$ composed of $n-1$ digits 1 and one digit 7 can be expressed as
$$
N=A_{\mathrm{a}}+6 \times 10^{\mathrm{k}},
$$
where $A_{0}$ is a natural number consisting of $n$ ones, and $0 \leqslant k \leqslant n$. It is a prime number. For any positive integer 1, we have $10=1$ (mod7). Therefore,
$$
\... | n=1, 2 | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 704,827 |
21. (Couplet 3) Try to prove that on a standard plane, it is impossible to construct a closed broken line with rational points as vertices, passing through an odd number of edges of length 1. | Let the rational points $\left(\frac{a}{b}, \frac{c}{d}\right)$ and $\left(\frac{a^{\prime}}{b^{\prime}}, \frac{c^{\prime}}{d^{\prime}}\right)$ be the endpoints of a line segment of a broken line, with the length of the segment being 1. Suppose $\frac{a}{b}, \frac{c}{d}, \frac{a^{\prime}}{b^{\prime}}, \frac{c^{\prime}}... | proof | Number Theory | proof | Yes | Yes | cn_contest | false | 704,829 |
22. (Japan) Let $m, n$ be odd numbers, and consider a rectangle in the coordinate plane with vertices at $(0,0), (0, m), (n, 0)$. A side of the rectangle is called "good" if it lies on a line of the form $x=j$ or $y=k$, where $j, k$ are integers; otherwise, it is called "bad". Suppose the given rectangle is divided int... | To prove that the triangles obtained from the partition, if only one side is a "boundary edge," then the triangle is called a "good triangle"; otherwise, it is called a "ring triangle." Let $V$ be the set of midpoints of the "sides" of all triangles formed by the partition, where each side is parallel to one of the coo... | proof | Geometry | proof | Yes | Yes | cn_contest | false | 704,830 |
4. Given that the sides of triangle $T^{\prime}$ are respectively equal to the three medians of triangle $T$ and that the two triangles have one pair of angles equal, prove that these two triangles are similar.
保留源文本的换行和格式，直接输出翻译结果如下：
4. Given that the sides of triangle $T^{\prime}$ are respectively equal to the thre... | Proof: Let $A D, B E$, $C F$ be the medians of $\triangle A B C$, and $M$ be the centroid. Extend $A D$ to $G$ such that $D G=M D$, and connect $B G, C G, F D$, $E D$ (Figure 4).
Obviously, to prove the conclusion of this problem, it is only necessary to prove that $\triangle A B C \sim \triangle C M G$ (regardless of... | proof | Geometry | proof | Yes | Yes | cn_contest | false | 704,834 |
5. Color some of the squares in a $2 \times n$ grid such that no $2 \times 2$ subgrid is completely colored, and let $p_n$ denote the number of all such distinct coloring methods. Prove that $p_{1089}$ is divisible by 1089. | Let $a_{0}$ represent the $2 \times n$ grid meeting the given conditions:
$$
a_{11}=b_{1-1}, \quad b_{11}=3(a, \ldots \vdots b,-1) .
$$
From (1), we get
$$
\begin{array}{l}
a_{n}=3\left(a_{n-1}+a_{n-2}\right), \\
b_{n}=3\left(b_{n-1}+b_{n-2}\right) .
\end{array}
$$
From (2) and $p_{n}=a_{4}+b_{n}$, we have
$$
p_{\mat... | proof | Combinatorics | proof | Yes | Yes | cn_contest | false | 704,835 |
6. Let $a_{1}, a_{2}, \cdots, a_{k}$ be a finite sequence of positive integers not exceeding $n$, where each term is different from its two adjacent terms and there do not exist any four indices $p<q<r<s$, such that $a_{p}=a_{s} \neq a_{q}=a_{r}$. Find the maximum value of the number of terms $k$.
Translate the above ... | $$
\begin{array}{l}
n, n, n-1, n-1, \cdots, 2,2,1,1, \\
2,2, \cdots, n-1, n-1, n, n
\end{array}
$$
To meet the requirements, it has $k=4n-2$ terms. We need to prove that the number of terms is no less than $4n-2$. Any three consecutive terms cannot be the same number. Moreover, if a term can be adjacent to (or the fir... | 4n-2 | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false | 704,836 |
5. i will arrange $1 \times 1$ square tiles into a strip with a diagonal length of 100. Find the approximate minimum value of the total length of the cuts, not exceeding 2. | For $a>$. $a^{2}+b^{2}=100^{2}, ab=1$,
$$
\therefore c+b=\sqrt{100^{2}+2}>100 .
$$
The perimeter of the rectangle $2(a+b)$ is due to the original square's perimeter.
Given its two uses, $2 L+4 \geqslant 2(a+b)$, $L \geqslant a+b-2>98$.
On the other hand,
Any two parallelograms with the same base and height can be:
t... | 99 | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,838 |
1. The sequence $2,3,5,6,7,10,11, \cdots$ contains all positive integers that are neither perfect squares nor perfect cubes, find the 500th term of this sequence. | 1. Answer: 528. | 528 | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 704,841 |
4. Solve the equation: $\frac{1}{x^{2}-10 x-29}=$ $+\frac{1}{x^{2}-10 x-45}-\frac{2}{x^{2}-10 x-69}=0$. | 4. Let $x^{2}-10 x=y$ and simplify, the original equation becomes $(y-29) \frac{-40}{(y-69)}=\frac{24}{(y-69)(y-45)}$. This equation has a unique solution $y=39$, $\therefore x^{2}-10 x=39$, when $x=13$ it meets the requirement, | x=13 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,844 |
5. $n$ is the smallest positive integer satisfying the following condition:
(2) $n$ has exactly 75 positive divisors (including 1 and itself). Find $\frac{n}{75}$. | 5. To ensure that $n$ is divisible by 75 and the obtained $n$ is the smallest, we can set $n=2^{\gamma_{13}} r_{2} \gamma_{3}$ and
$(r_{1}+1)(r_{2}+1)(r_{3}+1)=75$ $(r_{2} \geqslant 1, r_{3} \geqslant 2)$.
It is easy to prove that when $r_{1}=r_{2}=4, r_{3}=2$, $n$ has the minimum value. At this time, $\frac{n}{75}=2^... | 432 | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 704,845 |
6. A biologist wants to estimate the number of fish in a lake. On May 1, he randomly caught 60 fish and marked them, then released them back into the lake. On September 1, he found that some of the fish were no longer in the lake (due to death or migration), and on September 1, 40% of the fish in the lake were not ther... | 6. Let the number of fish in the lake on May 1 be $x$, and the number of fish in the lake on September 1 be $y$. According to the problem, we have
$$
y=0.75 x+0.40 y \text {. }
$$
On September 1, the number of tagged fish in the lake is $0.75 \times 60 = 45$. Assuming that the tagged fish on September 1 can represent ... | 1050 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,846 |
7. The vertices of a triangle are $P(-8$, 5), $Q(-15,-19), R(1,-7), \angle P$ bisector can be expressed as $a x+2 y+c=0$, find $a+c$,
| 7. Extend $P R$ to $T$, such that $P Q=P T$. Since $P Q=25, P R=15$, then the coordinates of $T$ are $(7,-15)$, and the angle bisector intersects $Q T$ at the midpoint $(-4,-17)$.
Thus, the slope of the angle bisector is $-\frac{11}{2}$, and the equation is
$$
\begin{array}{r}
11 x+2 y+78=0 . \\
\text { Hence } a+c=11+... | 89 | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,847 |
Find $i \div j$.
Translate the text above into English, keep the original text's line breaks and format, and output the translation result directly. | 9. Tossing a coin 10 times has $2^{10}$ outcomes. Let $A(n)$ be the number of outcomes where no heads appear consecutively. By enumeration, it is easy to see that $A(1)=2, A(2)=3, A(3)=5$. It can also be proven by induction that there is a recurrence relation:
$$
A(n+2)=A(n+1) + A(n) \text {. }
$$
This is because $A(n... | 73 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,849 |
5. (Greece 2) Let $f(0)=f(1)=0$, and
$$
\begin{aligned}
f(n+2)= & 4^{n+2} f(n+1)-16^{0+1} f(n) \\
& +n \cdot 2^{n^{2}},
\end{aligned}
$$
where $n=0,1,2, \cdots$. Prove that $f(1989), f(1990)$, and $f(1991)$ are all divisible by 13. | Let $f(n)=g(n) 2^{\mathrm{n}^{2}}$, then the recurrence relation in the problem can be written as
$$
g(n+2)-2 g(n+1)+g(n)=n \cdot 16^{-n-1}.
$$
That is,
$$
\begin{array}{l}
{[g(n+2)-g(n+1)]-[g(n+1)} \\
-g(n)]=n \cdot 16^{-n-1}.
\end{array}
$$
For the above equation, taking $n$ as $0,1, \cdots, n-1$, we get
$$
g(n+1)-... | proof | Algebra | proof | Yes | Yes | cn_contest | false | 704,850 |
11. We notice that $6!=8 \cdot 9 \cdot 10$. Try to find the largest positive integer $n$ such that $n!$ can be expressed as the product of $n-3$ consecutive natural numbers. | 11. If $n!$ can be expressed as the product of $(n-3)$ consecutive integers, then there exists an integer $k$, such that
$$
\begin{array}{l}
n!=(n+k)(n+k-1) \cdots(k+4) \\
=-(n+k)! \\
(k+3)!
\end{array}
$$
Originally,
$$
\begin{array}{l}
\frac{n+k}{k+3} \cdot \frac{n+k-1}{k+2} \cdot \ldots \cdot \frac{n+2}{5} \cdot \... | 23 | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 704,852 |
The sum of all side lengths and diagonal lengths of a 12-sided polygon can be written in the form $a+b \sqrt{2}+c \sqrt{3}+d \sqrt{6}$, where $a, b, c$, $d$ are positive integers. Find $a+b+c+d$.
The text above is translated into English, preserving the original text's line breaks and format. | Coinciding with the origin, one vertex is at $(12,0)$, and the coordinates of the other vertices are $(12 \cos k x, 12 \sin k x)$, where $x=30^{\circ}$, $k=1,2, \cdots, 11$. The length of the line segment connecting $(12,0)$ and $(12 \cos k x, 12 \sin k x)$ is $24 \sin \frac{k x}{2}$. Therefore, from this,
$$
\begin{ar... | 720 | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,853 |
13. Let $T=\left\{9^{4} \mid k\right.$ be an integer, $0 \leqslant k$ $\leqslant 4000\}$. It is known that $9^{1000}$ has 3817 digits, and its most significant digit is 9. How many elements in $T$ have 9 as their most significant digit? | 13. Notice that $9^{k}$ has one more digit than $9^{k-1}$, unless $9^{1}$ starts with the digit 9. In the latter case, by long division, $9^{k-1}$ starts with the digit 1, and $9^{k}$ has the same number of digits. Therefore, from $9^{0}$ to $9^{4000}$, there are 3816 digit increases, so there must be 184 times when th... | 184 | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 704,854 |
14. $A B C D$ is a rectangle, $A B=12 \sqrt{3}, B C=$ $13 \sqrt{3}$, diagonals $A C, B D$ intersect at $P$. If the triangle $A B P$ is cut off and $A P, B P$ are joined, find its volume.
| 14. As shown in the figure, $N$ is the intersection of the altitude from $P$ to $BCD$.
It is easy to prove that $N$ is the circumcenter of $\angle BCD$, and
\[
\begin{array}{l}
r=\frac{n^{2}}{\sqrt{4 n^{2}-m^{2}}} . \\
\because P B=\frac{1}{2} \sqrt{m^{2}+n^{2}}, \text { by the Pythagorean theorem, we have } \\
P N=\f... | 594 | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,855 |
$\begin{array}{l}2 \cdot\left(\frac{1}{4}\right)^{-\frac{1}{4}}=(\quad) \text {. } \\ \text { (A) }-16 \text {. } \\ \text { (B) }-\sqrt{2} \text {. } \\ \text { (C) }-\frac{1}{16} \text {. } \\ \text { (D) } \frac{1}{256} \\ \text { (E) } \sqrt{2} \text {. } \\\end{array}$ | E
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 | 704,858 |
3. The four interior angles of a quadrilateral form an arithmetic sequence. If the smallest angle is $75^{\circ}$, then the largest angle is ( ).
(A) $95^{\circ}$.
(E) $100^{\circ}$.
(C) $105^{\circ}$.
(D) $110^{\circ}$.
(E) $115^{\circ}$. | 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 | 704,859 |
4. In parallelogram $ABCD$, $\angle ABC = 120^{\circ}, AB = 16, BC = 10$. Extend $CD$ to $E$ such that $DE = 4$. If $BE$ intersects $AD$ at $F$, then $FD$ equals ( ).
(A) 1.
(B) 2.
(C) 3.
(D) 4.
(E) 5. | 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 | 704,860 |
$$
f_{u:-1}(k)=f_{1}\left(f_{u}(k)\right) .
$$
Find $f_{1991}\left(2^{1000}\right)$.
| Let $k$ be much smaller, and we can estimate the value range of $f_{1}(k)$ as follows:
Suppose the positive integer $a$ has $m$ digits, then by changing all the digits of $a$ to 9, we have
$$
f_{1}(a) \leqslant 9^{2} m^{2}=81 m^{2}.
$$
If $a \leqslant b$, then $m$ is no greater than the integer part of $\lg b$ plus 1... | 256 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,861 |
9. For a cube, either one of its edges is painted red, or all are painted black, with each face having at least one edge painted black. The minimum number of black edges is ( ).
(A) 2.
(B) 3.
(C) 4.
(D) 5.
(E) 6. | 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 | 704,866 |
10. A $11 \times 11 \times 11$ cube consists of $11^{3}$ unit cubes. How many unit cubes are visible? ().
(A) 328 .
(B) 329 .
(C) 330 .
(D) 331 .
(E) 332 . | I)
Translate the text above into English, please retain the line breaks and format of the source text, and output the translation result directly. | not found | Geometry | MCQ | Yes | Yes | cn_contest | false | 704,867 |
11. Among the positive integers less than 50, the number of positive integers that have an odd number of positive divisors is ( ).
(A) 3 .
(B) $\overline{5}$.
(C) 7 .
(D) 9 .
(E) 11 . | 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 | 704,868 |
12. $f$ is given by $f(x)=a x^{2}-\sqrt{2}, a$ is… $a=$ ( ).
(A) $\frac{2-\sqrt{2}}{2}$.
(B) $\frac{1}{2}$.
(C) $2-\sqrt{2}$.
(D) $\frac{\sqrt{2}}{2}$.
(E) $\frac{2+\sqrt{2}}{2}$. | 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 | 704,869 |
7. (Hungary 3) Let $k$ be the incenter of $\triangle A B C$, $G_{1}, B_{1}$ be the midpoints of sides $A B, A C$ respectively, and let $A C$ intersect $C_{1} K$ at point $B_{2}$, and line $A B$ intersect $B_{1} K$ at point $C_{2}$. If the area of $\triangle A B_{2} C_{2}$ is equal to the area of $\triangle A B C$, find... | Solution 1: Let $a=BC, b=CA, c=AB$, $b^{*}=AC_{2}, \quad c^{*}=AB_{2}, \quad s=\frac{1}{2}(a+b+c)$.
Let $r$ be the inradius of $\triangle ABC$. Since
$$
\begin{array}{l}
S_{\triangle C_{1} B_{2}}=\frac{1}{2} AC_{1} \cdot AB_{2} \sin \angle A, \\
S_{\triangle B C}=\frac{1}{2} AC \cdot AB \sin \angle A,
\end{array}
$$
t... | 60^{\circ} | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,872 |
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