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 |
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15. Among four numbers, the sums of any three of them are 180, 197, 208, and 222, respectively. Then the largest of these four numbers is ( ).
(A) 77 .
(B) 83 .
(C) 89 .
(D) 95 .
(E) Uncertain. | 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 | 704,873 |
18. Among $\{1,2,3, \cdots, 99,100\} \pitchfork$ $3^{a}+7^{b}$, the probability that the last digit is 8 is ( ).
(A) $\frac{1}{16}$.
(B) $\frac{1}{8}$.
(C) $\frac{3}{16}$.
(D) $\frac{1}{5}$.
(E) $\frac{1}{4}$. | C
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 | 704,876 |
21. Let the base $ABCD$ of the cone $P-ABCD$ be a square, and the vertex $P$ is equidistant from $A, E, C, D$. If $AB = 1, \angle APB = 2\theta$, then the volume of the cone is ().
(A) $\frac{\sin \theta}{6}$
(B) $\frac{\operatorname{ctg} \theta}{6}$.
(C) 1
(D) $\frac{1-\sin 2\theta}{6}$.
(F) $\frac{\sqrt{\cos 2\theta}... | E
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 | 704,879 |
8. (Iceland 2) A plane passing through the midpoint of a circular cone's height and tangent to the circumference of the base circle divides the cone into two parts. Find the ratio of the smaller part to the entire cone.
保留源文本的换行和格式如下：
8. (Iceland 2) A plane passing through the midpoint of a circular cone's height and... | Let's assume
the base of the cone is 1,
the height PN = h.
(Figure 5). Denote
$A, C$, and let $A$
Let $B$ be the antipodal point of $A$ on the circumference of the base circle, $M$ be the midpoint of the height $PN$, $O$ be the midpoint of $AC$, i.e., the center of the circle. Also, let $E$ be a point on the ellipse s... | \frac{\sqrt{3}}{9} | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,883 |
25. 9 congruent spheres are placed inside a unit cube, with the center of one sphere located at the center of the cube, and the other 8 spheres each touching this sphere and the three faces of the cube, then the radius of each sphere is ( ).
(A) $1-\frac{\sqrt{3}}{2}$.
(E) $\frac{2 \sqrt{3}-3}{2}$.
(C) $\sqrt{2}$.
(D) ... | P
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 | 704,884 |
1. Let $d_{1}, d_{2}, \cdots, d_{k}$ be all the divisors of the positive integer $n$, $1=d_{1}<d_{2}<d_{3}<\cdots<d_{k}=n$. Find all $n$ such that $k \geqslant 4$ and $d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}=n$. | Number, but the sum of the squares of four odd numbers is even, which cannot equal $n$. Contradiction.
Thus, $n$ is even, $d_{2}=2$.
If $n$ is a multiple of 4, then $4 \in\left\{d_{3}, d_{4}\right\}$. Among the squares of the 4 factors, there are already two $\left(2^{2}\right.$ and $\left.4^{2}\right)$ that are multip... | 130 | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 704,890 |
3. Line $l$ intersects side $AB$ of $\triangle ABC$ at $B_{1}$, and side $AC$ at $C_{1}$. The centroid $G$ of $\triangle ABC$ and point $A$ are on the same side of $l$. Prove:
$S_{B_{1} C_{1}} + S_{C C_{1} C B} \geqslant \frac{4}{9} S_{\triangle ABC}$.
When does equality hold? | Let $D$ be the midpoint of $B C$. Connect $D B_{1}, D C_{1}$, then
$$
\begin{array}{l}
=\frac{2}{3}\left(S_{A_{1}} \mathrm{n}+S_{A_{1-C}^{1}}\right) \text {. } \\
\end{array}
$$
Draw a line through $G$ parallel to $l$, intersecting $A B, A C$ at $B_{2}, C_{2}$ respectively. Since $G$ and $A$ are on the same side of $l... | proof | Geometry | proof | Yes | Yes | cn_contest | false | 704,892 |
9. (Ireland 1) Let $l$ be a line passing through point $C$ and parallel to side $AB$ of $\triangle ABC$. The angle bisector of $\angle A$ intersects $BC$ at $D$ and $l$ at $E$, and the angle bisector of $\angle B$ intersects $AC$ at $F$ and $l$ at $G$. If $GF=DE$, prove: $AC=BC$. | Let $BC = a$, $CA = b$, $AB = c$, $\angle A = 2\alpha$, $\angle B = 2\beta$. As shown in Figure 7, by the Angle Bisector Theorem, we have $\frac{CF}{FA} = \frac{a}{c}$.
Thus, $CF = \frac{ak}{a+c}$.
Similarly, we get $CD = \frac{ab}{b+c}$.
In $\triangle CFG$, $\angle CGF = \rho$, $\angle FCG = 2\alpha$, so by the Law o... | AC=BC | Geometry | proof | Yes | Yes | cn_contest | false | 704,894 |
4. A natural number $N$ has exactly 12 divisors (including 1 and $N$), and these divisors are numbered in increasing order: $d_{1}<$ $d_{2}<\cdots<d_{12}$. The divisor with the index $d{ }_{4}-1$ equals $\left(d_{1}+d_{2}+d_{1}\right) \times d_{8}$, find $N$.
| 4. First, use the relationship
$$
d_{d_{4}-1}=\left(d_{1}+d_{2}+d_{4}\right) \times d_{8} \geqslant d_{5} \times d_{8}
$$
$=N$, we get $d_{4}=13, d_{5}=d_{2}+14, \quad N$ $=\left(d_{2}+14\right) \times d_{8}$. Then, using the enumeration method to take $d_{2}=2,3$, $5, 7, 11$ to get the only possible solution $d_{2}=3,... | 1989 | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 704,898 |
5. On a chessboard, there are 8 pieces placed, with one on each horizontal and vertical line. Prove: Among all the black squares on the chessboard, there is an even number of pieces. | ‥，If we number the rows and columns from 1 to 8, then each square on the chessboard has a coordinate. For white squares, the sum of these two coordinates is even. The sum of the coordinates of the 8 squares occupied by the 8 pieces is $2(1+\cdots+8)$, which is even. Therefore, among these 8 squares, there must be an ev... | proof | Combinatorics | proof | Yes | Yes | cn_contest | false | 704,899 |
4. If a $23 \times 23$ square is composed of 1000 squares of sizes $1 \times 1, 2 \times 2, 3 \times 3$, how many $1 \times 1$ squares are needed at minimum? | 4. At least 1 $1 \times 1$ square is needed. First, it needs to be shown that a $23 \times 23$ square can be composed of the following: $2 \times 2, 3 \times 3$ squares, and exactly 1 $1 \times 1$ square. Then, it also needs to be proven that a $1 \times 1$ square is indispensable. | 1 | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false | 704,907 |
$A^{\prime} D, B C$ and $B^{\prime} C$ midpoints lead to an isomorphism:
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 7. Consider $\triangle E F G$, which is obtained by translating each point of $\triangle B C B^{\prime}$ by the vector $D \vec{C}$ (4). Then, under a homothety with center $D$ and a coefficient of 2, the points $B C$, $B^{\prime} C$, and the intersection point are transformed into points $E$, (, and $A^{\prime}$, respe... | null | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,910 |
Example 1. As shown in the figure, in quadrilateral $ABCD$, points $E$, $F$ are on $BC$, $CD$ respectively, and $\frac{DF}{FC}=1$, $\frac{E}{B}=2$. If the area of $\triangle ADF$ is $m$, and the area of quadrilateral $AECF$ is $n(n>m)$, find the area of quadrilateral $ABCD$. (1989, National Junior High School League) | Connect $A C$.
Since $\triangle A D F$ and $\triangle A F C$ have the same height and base, we have $S \triangle \triangle \triangle^{\mathrm{B}}=S \triangle \triangle D F=m$.
Then $S_{\triangle \triangle P}=S_{\triangle E C E}-S \triangle \triangle F C=n-m$. Also, $\triangle A B E$ and $\triangle A C_{E}$ have the sam... | \frac{1}{2}(3n+m) | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,916 |
Example 2. As shown in the figure,
$2, P$ is a point inside the triangle,
where the areas of the four
small triangles are marked in the figure. Find the area of $A B C$. (Third A1ME)
Note: The translation keeps the original text's line breaks and format as requested. However, the mathematical notation and the figure r... | Solve $-\frac{S \triangle \triangle P D}{S \triangle C D}=\frac{B D}{C D}$,
X
$\frac{S \therefore \operatorname{LA} 1}{S \triangle \mathrm{CAD}}=\frac{B D}{C D}$.
Let $S \triangle \mathrm{BPF}=x, S: A P E=y$,
Then
$$
\begin{array}{l}
S \frac{\mathrm{BAD}}{S-\mathrm{CAD}}=-\frac{S P C D}{S \triangle \mathrm{CPD}}=\frac... | 315 | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,917 |
Inverting 5. A sequence whose terms are all 0 or 1 is called a $0, 1$ sequence. Let $A$ be a finite $0,1$ sequence. Denote $f(A)$ as the $0,1$ sequence obtained by changing every 1 in $A$ to 0,1 and every 0 to 1,0. For example, $f((1,0,0,1))=(0,1,1,0$, $\cdots)$ (n times).
Question: In $f^{4}((1))$, how many terms are... | Let $f^{\mathrm{n}}((1))=f^{\mu}$. The number of pairs of consecutive terms in $f^{n}$ that are 0,0 or 0,1 are denoted as $g_{\mathrm{n}}, h_{\mathrm{n}}$.
According to the problem, the 0,0 pairs in $f^{2}$ must be derived from the 0,1 pairs in $f^{n-1}$ after the transformation $f$.
The 0,1 pairs in $f^{\mathrm{n}-1}... | g_{u}=\frac{1}{3}\left(2^{\mathrm{n}-1}-(-1)^{y-1}\right) | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false | 704,918 |
Example 6. $N$ is a positive integer, in the square $R$ with vertices at $(N, 0)$, $(0, N)$, $(-N, 0)$, $(0, -N)$ (including the boundary), how many integer points are there? | Let the number of integer points in the square $R$ (including the boundary) be $a \times$.
When $N=1$, the integer points are (1,0), (0,1), (-1,0), (0,-1), and (0,0), totaling 5 points, so $a_{1}=5$.
When $N$ increases to $N+1$, in the first quadrant and the positive half-axis of the $x$-axis, the added integer point... | 2 N^{2}+2 N+1 | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false | 704,919 |
Example 7. For any non-negative integer $n$, prove that
$$
\left[(1+\sqrt{3})^{2 n+1}\right]
$$
is divisible by $2^{n+1}$.
Here, the notation $[x]$ denotes the greatest integer less than or equal to $x$. | Let $u=1+\sqrt{3}$, $v=1-\sqrt{3}$. Then $u$, $v$ are the two roots of the quadratic equation
$$
x^{2}-2 x-2=0
$$
Construct the sequence $\left\{T_{n}\right\}$, such that
$$
T_{n}=u^{n}+v^{n}.
$$
Then $T_{0}=2, T_{1}=u+v=2$.
For any positive integer $k$, since $u, v$ are roots of the equation, we have
$$
\begin{array... | proof | Number Theory | proof | Yes | Yes | cn_contest | false | 704,920 |
Question 3. Through the endpoints of the diameter $AB$ of $\odot O$, construct two parallel rays $AD$ and $BC$. Take any point $E$ on the arc between $AD$ and $BC$, and draw the tangent line to $\odot O$ through $E$, intersecting $AD$ at $D$ and $BC$ at $C$.
Prove: (1) The circle with $CD$ as its diameter is tangent t... | Proof (1) As shown in the figure, let $O^{\prime}$ be the midpoint of $CD$, and draw $O^{\prime} F \perp AB$ at $F$. Find $O E$, $O^{\prime} A$, $O^{\prime} O$, $O D$.
$$
\because AD \| BC \text {, }
$$
$O^{\prime} O$ is the midline of trapezoid $ABCD$, $\therefore O^{\prime} O \| AD$.
$$
\begin{array}{l}
S \triangle D... | proof | Geometry | proof | Yes | Yes | cn_contest | false | 704,921 |
Prove: If $a, b, c$ are the side lengths of a triangle, and $2 s=a+b+c$, then $\frac{a^{\mathrm{n}}}{b+c}+\frac{b^{\mathrm{n}}}{c+a}+\frac{c^{\mathrm{n}}}{a+b}$ $\geqslant\left(\frac{2}{3}\right)^{\mathrm{D}-2} s^{\mathrm{D}-1}, n \geqslant 1$. | The text provides the following generalization:
Let $a_{1}, a_{2}, \cdots, a_{N} \in \mathbb{R}^{+}$, and $2 s=\sum_{k=1}^{N} a_{k}$, $n \geqslant m \geqslant 1$, then
$$
\sum_{k=1}^{N}\left(\frac{a_{k}^{n}}{\left(2 s-a_{k}\right)^{m}} \geqslant \frac{(2 s)^{n-m}}{N^{n-m-1}(N-1)^{m}} .\right.
$$
Proof: Without loss of... | proof | Inequalities | proof | Yes | Yes | cn_contest | false | 704,922 |
Example 1. Calculate: $\sqrt{31 \cdot 30 \cdot 29 \cdot 28+1}$. (7th American Invitational Mathematics Examination) | ```
\begin{array}{l}
\because \sqrt{ }(x+1) x(x-1)(x-2)+1 \\
= \sqrt{ }\left(x^{2}-x-2\right)\left(x^{2}-x\right)+1 \\
= \sqrt{ }\left(x^{2}-x-1\right)^{2} \\
=\left|x^{2}-x-1\right| . \\
\therefore \text { when } x=30 \text {, the original expression }=\left|30^{2}-30-1\right| \\
=869 .
\end{array}
``` | 869 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,923 |
Example 2. Compare $\frac{23^{1088}+1}{23^{1880}+1}$ with $\frac{23^{1089}+1}{23^{1880}+1}$. | Solve by first comparing the quotient of $-\frac{x+1}{23}$ and $\frac{23 x+1}{23^{2} x+1}$.
$$
\begin{array}{l}
23 \frac{x+1}{x+1} \div \frac{23 x+1}{23^{2} x+1} \\
=\frac{(x+1)\left(23^{2} x+1\right)}{(23 x+1)^{2}} \\
=\frac{23^{2} x^{2}+\left(23^{2}+1\right) x+1}{23^{2} x^{2}+2 \times 23 x+1}>1 \text {. } \\
\end{arr... | \frac{23^{1988}+1}{23^{1989}+1}>\frac{23^{1989}+1}{23^{1980}+1} | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 704,924 |
Example 3. Prove: $13^{25}>25!$. | In this example, all given numbers are specific figures. The first thought of the students is as follows:
Analysis: To prove the original inequality, it is equivalent to proving
$$
13 \cdot 13 \cdots 13 > 1 \cdot 2 \cdots 25 \text{. }
$$
25 times
That is, to prove $13 \cdot 13 \cdots 13 > (13-12)(13-11) \cdots (13+12)^... | proof | Inequalities | proof | Yes | Yes | cn_contest | false | 704,925 |
Example 5. Using the three vertices of a triangle and nine points inside it (a total of 12 points) as vertices, the number of small triangles that the original triangle can be divided into is ( ).
( A ) 15. ( B ) 19 .
( C ) 22. ( D ) Cannot be determined.
( 1988, Jiangsu Junior High School Competition) | Solve: First, consider the case of placing $n$ points inside a triangle. Suppose the $n-1$ points inside $\triangle ABC$ can divide the original triangle into $a_{n-1}$ small triangles. Next, we examine the situation after adding the $n$-th internal point $P_{a}$. (1) If the point is inside a small triangle, the three ... | B | Geometry | MCQ | Yes | Yes | cn_contest | false | 704,927 |
Example 3. Take a point $P$ inside $\triangle ABC$, and draw three lines through $P$ parallel to the three sides of $\triangle ABC$,
thus forming three triangles
$t_{1}, t_{2}, t_{3}$ with areas
4, 9, 49 (as
shown in Figure 3).
Find the area of $\triangle ABC$.
(From the 2nd $\triangle I M E$) | $$
\begin{array}{l}
\text{Given that } \triangle t_{1}, \triangle t_{2}, \triangle t_{3} \text{ and } \triangle A B C \text{ are all similar triangles.} \\
\text{Let the areas of } \triangle A B C, \triangle t_{1}, \triangle t_{2}, \triangle t_{3} \text{ be } S, S_{1}, S_{2}, S_{3} \text{, respectively.} \\
\text{It is... | 144 | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,928 |
Example 6. A football invitational tournament has 16 cities participating, each city sending Team A and Team B. According to the competition rules, each pair of teams plays at most one match, and the two teams from the same city do not play against each other. If, after two days of the tournament, it is found that exce... | Solve the problem with a general discussion. Suppose there are $n$ cities participating in the competition, all of which meet the conditions of the problem. Let the number of matches played by Team B of City $A$ be $a_{a}$, obviously $a_{1}=0$.
In the case of $n$ cities, according to the competition rules, each team c... | 15 | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false | 704,929 |
Example 1. (1), Example 3) There is a square chessboard, with a total of $2 k \times 2 k=4 k^{2}(k \in N)$ small squares, each with an area of $1 \mathrm{~cm} \mathrm{~m}^{2}$. Now, if any two small squares are removed, can the remaining small squares be covered by $2 k^{2}-1$ pieces of $1 \times 2 \mathrm{~cm}^{2}$ ti... | Since $1 \times 2 \times\left(2 k^{2}-1\right)=4 k^{2}-2$, if the small cards overlap, they will definitely not cover the area.
Interleave the arches and color them in black and white (Figure 1), so that the number of black and white squares is equal, and each $1 \times 2$ small card must cover one black and one white... | proof | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false | 704,930 |
Example 3 - What is the maximum number of knights that can be placed on an $8 \times 8$ chessboard so that no two knights attack each other (assuming there are enough knights)? | We will alternately color the chessboard in black and white, so there will be 32 black squares and 32 white squares. According to the knight's move (see Figure 1), a knight on a black square can only capture a knight on a white square. Therefore, placing knights on all black squares means they will not capture each oth... | 32 | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false | 704,932 |
Example 4. (26th IMO Shortlist) Super chess is played on a $12 \times 12$ square board. Its knight moves from one corner of a $3 \times 4$ rectangle to the opposite corner. Is it possible for a knight to jump to every square exactly once and then return to the starting square? | This kind of route does not exist. The proof is given below using proof by contradiction.
Assume there exists a route that meets the requirements, and the chessboard is alternately colored in black and white (as shown in Figure 7). Clearly, the knight must move from a black square to a white square in one step, and fr... | proof | Combinatorics | proof | Yes | Yes | cn_contest | false | 704,933 |
Example 5. Given the sequence $\left\{x_{0}\right\}: x_{n+1}=$ $\frac{x_{\mathrm{n}}+(2-\sqrt{3})}{1-x_{n}(2-\sqrt{3})}$. Find the value of $x_{1001}-x_{401}$. | Given the shape of the recurrence relation, we can set $x_{\mathrm{n}}=\operatorname{tg} \alpha_{\mathrm{n}}$, and since $\operatorname{tg} \frac{\pi}{12}=2-\sqrt{3}$, we know that $x_{n+1}=\operatorname{tg} \alpha_{n+1}$
$$
=\frac{\operatorname{tg} \alpha_{\mathrm{a}}+\operatorname{tg} \frac{\pi}{12}}{1-\operatorname{... | 0 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,934 |
Material Journal No. 86, Issue 4 introduces the proof of the following inequality:
Let the incircle of $\triangle A B C$ touch the sides at $A^{\prime}$, $B^{\prime}$, $C^{\prime}$, then $S \triangle A^{\prime} B^{\prime} C^{\prime} \leqslant \frac{1}{4} S \triangle A B C$.
(Equality holds if and only if $a=b=c$)
Thi... | Lemma: In $\triangle ABC$, there are points $D, E, F$ on the sides $BC, CA, AB$ respectively, such that $BD: DC = \lambda_1$, $CE: EA = \lambda_2$, and $AF: FB = \lambda_3$. Then,
- $S \triangle AEC$.
Proof: As shown in the figure, $\frac{S \triangle E F}{S \triangle B C} = \frac{A F \cdot A E}{A B \cdot A C} = \frac{... | S \triangle A'B'C' \leq \frac{1}{4} S \triangle ABC | Inequalities | proof | Yes | Yes | cn_contest | false | 704,935 |
Example 1. Solve the system of equations
$$
\left\{\begin{array}{l}
3 x+2 y-6=0, \\
2(x+2 y)+5(x-3)=0 .
\end{array}\right.
$$ | Solve the following system of equations:
$$
A:\left\{\begin{array}{l}
(x+2 y)+2(x-3)=0, \\
2(x+2 y)+5(x-3)=0 .
\end{array}\right.
$$
$A$ can be transformed into a homogeneous linear system of equations with $x+2 y, x-3$ as unknowns, whose coefficient determinant $C=1 \neq 0$. Therefore, $A$ has only the trivial solutio... | \left\{\begin{array}{l}x=3, \\ y=-\frac{3}{2} .\end{array}\right.} | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,936 |
Example 2. Solve the system of equations
$$
\left\{\begin{array}{l}
x^{2}+5 y+2 z+1=0, \\
7\left(x^{2}+4 y\right)-2 y-7 z+1=0, \\
5\left(x^{2}+4 y\right)-4(y+3 z)+3(z-1)=0 .
\end{array}\right.
$$ | Solve: This system of equations can be viewed as a set of linear equations in terms of $x^{2}+4 y$; $y+3 z$, $z-1$. Let the unknowns form a linear system of equations, whose coefficient determinant $D:=-18 \neq 0$. Therefore, $x^{2}+4 y=0$, $y+3 z=0$, $z-1=0$, solving we get
$$
\left\{\begin{array} { l }
{ x _ { 1 } =... | null | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,937 |
Theorem: For the sequence $\left\{x_{\mathrm{n}}\right\}$ determined by the recurrence relation $x_{\mathrm{n}+2}=p x_{\mathrm{n}+1}+q x_{\mathrm{n}}$ $(p, q \in R)$ and initial conditions $x_{1}, x_{2}$, if the characteristic equation has complex roots $\alpha, \beta$, then $\left\{x_{\mathrm{n}}\right\}$ is a periodi... | Proof: Let $x_{n}=A \alpha^{n-1}+B \beta^{n-1}(A, B$ be constants not both zero determined by $x_{1}, x_{2})$. Suppose $\alpha=\cos \frac{2 k \pi}{T}+i \sin \frac{2 k \pi}{T}(k<T, \quad k, \quad T \in \mathbb{N})$, then $\alpha^{T}=1$, but $\beta=\bar{\alpha}$, hence $\beta^{T}=1$. Therefore, for any $n \in \mathbb{N}$... | proof | Algebra | proof | Yes | Yes | cn_contest | false | 704,938 |
Example 4. Given $\triangle A B C$ and a line $D E$ parallel to $B C$ intersecting at points $B D E$, the area of $\triangle B D E$ equals $k^{2}$. Then, what is the relationship between $k^{2}$ and the area $S$ of $\triangle A B C$ for the problem to have a solution? How many solutions are there?
(1987, Shanghai) | Solve as shown in Figure 4, let $AD: AB = x$.
Given $DE \| BC$, we know $\triangle ADE \sim \triangle ABC$.
Then $\frac{S_{\triangle DRE}}{S_{\triangle ABC}} = \frac{AD^2}{AB^2} = x^2$,
Thus $S_{\triangle DE} = x^2 \cdot S$.
Also, $\frac{S_{\triangle ABE}}{S_{\triangle ADB}} = \frac{AB}{AD} = \frac{1}{x}$,
Therefore, $... | k^2 \leq \frac{S}{4} | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,939 |
Question: Determine the positive rational number $m$ that makes $\frac{1}{\pi} \operatorname{arctg} \sqrt{m}$ always a rational number. | If $\sqrt{ } \bar{m}=\operatorname{tg} r \pi$, where $r$ and $m$ are rational numbers, then $2 \cos 2 r \pi=2\left(\frac{2}{\sec ^{2} r \pi}-1\right)$
$$
=\frac{4}{m+1}-2 \text {. }
$$
Since $2 \cos 2 r \pi=e^{2 \mathrm{r} \pi i}+e^{-2 \mathrm{r} \pi i}$ is both an algebraic integer and a rational number, it must be a... | m=3,1,\frac{1}{3} | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 704,940 |
MIC test question, there is a unique small question: If $\frac{a+b}{a-b}=\frac{b+c}{b-c}=\frac{c+a}{c-a}$, then
$$
\begin{aligned}
a+b+c & =\ldots, a^{2}+b^{2}+c^{2}= \\
\frac{1}{a}+\frac{1}{b}+\frac{1}{c} & =\ldots
\end{aligned}
$$ | Let $\frac{c+a}{c-a}=k$, then $a=\left(\frac{k+1}{k-1}\right) b$, $b=\left(\frac{k+1}{k-1}\right) c, c=\left(\frac{k+1}{k-1}\right) a$. It follows that $\frac{k+1}{k-1}=-\frac{1}{2} \pm \frac{\sqrt{3}}{2} i$, hence all three answers are zero. | 0 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,941 |
Question 1. Let $z \in C, |z|=1$ but $z \neq -1$, then $z=\frac{1+t i}{1-t i}(t$ is a real number related to $z)$. | Given $|z|=1, z \neq-1$, hence $z=\cos \theta+i \sin \theta$, $\theta \neq k \pi$ (where $k$ is $\pm 1, \pm 3, \cdots$). Therefore, if we let $t=\operatorname{tg} \frac{\theta}{2}$, then we have
$$
\begin{aligned}
z & =\frac{1-\operatorname{tg}^{2} \frac{\theta}{2}}{1+\operatorname{tg}^{2} \frac{\theta}{2}}+i \frac{2 \... | z=\frac{1+t i}{1-t i} | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,942 |
Question 2. Let $z=x+y i$, prove that $\frac{|x|+|y|}{\sqrt{2}} \leqslant|z| \leqslant|x|+|y|$. | \begin{aligned}|z| & =\sqrt{|x|^{2}+|y|^{2}} \leqslant \sqrt{(|x|+|y|)^{2}} \\ & =|x|+|y| .(\text { right inequality holds) } \\ \text { Also } & 2|z|^{2}=2\left(|x|^{2}+|y|^{2}\right) \\ & \geqslant|x|^{2}+2|x y|+|y|^{2} \\ & =(|x|+|y|)^{2} \text {.(left inequality holds) }\end{aligned} | proof | Inequalities | proof | Yes | Yes | cn_contest | false | 704,943 |
Question 3. Let $Z_{1}$ and $Z_{2}$ correspond to the complex numbers $y_{1}$,
$z_{2}$, then the necessary and sufficient condition for $O Z_{1} \perp O Z_{2}$ is
$$
z_{1} \bar{z}_{2}+\bar{z}_{1} z_{2}=0
$$ | Prove $\overrightarrow{O Z_{1}} \perp \overrightarrow{O Z_{2}} \Leftrightarrow z_{1}=k i \cdot z_{2}(k \neq 0$ is a real number). If $z_{1}=k i \cdot z_{2}$, then $z_{1} z_{2}+\bar{z}_{1} z_{2}$
$$
\begin{array}{l}
=k i \cdot z_{2} \cdot \bar{z}_{2}+k i z_{2} \cdot z_{2}=k i \bar{z}_{2} \bar{z}_{2} \\
-k i \bar{z}_{2} ... | proof | Algebra | proof | Yes | Yes | cn_contest | false | 704,944 |
Example 1. Prove that $(\sqrt{ } 26-5)$ "wears a decimal expression with at least $n$ consecutive teeth after the decimal point!":
Preserve the original text's line breaks and format, output the translation result directly. | $$
\begin{array}{l}
\text { Prove that } 0 \leq \sqrt{ } 26-5=\frac{1}{\sqrt{ } 26}+5<\frac{1}{10}, \\
\therefore 0<(\sqrt{ } 26-5)^{\prime \prime}<\frac{1}{10^{D}},
\end{array}
$$
The conclusion holds. | proof | Number Theory | proof | Yes | Yes | cn_contest | false | 704,945 |
Example 1. Let $a>0, b>0, 0<x<\pi$, $x \neq \frac{\pi}{2}$. Find the extremum of the function $y=a \operatorname{tg} x+b \operatorname{ctg} x$. | Solve $|y| \geqslant 2 \sqrt{a b}$, when $x=\operatorname{arctg} \sqrt{\frac{b}{a}}$ or $\pi-\operatorname{arctg} \sqrt{\frac{b}{a}}$ the equation holds.
Thus $y_{\max }=-2 \sqrt{a} \bar{b} , y_{\mathrm{m} / \mathrm{a}}=2 \sqrt{ } a b$. | y_{\max }=2 \sqrt{a b}, y_{\min }=-2 \sqrt{a b} | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,948 |
Example 5. As shown in Figure 5, in $\triangle ABC$, $AB=2$, $AC=3$, I, II, and III respectively denote the squares constructed on $AB$, $BC$, and $CA$. What is the maximum value of the sum of the areas of the three shaded regions? (1988, National Junior High School League)
---
The translation preserves the original ... | Solve in $\triangle A D E$ and $\triangle A B C$ !! $\angle B A C + \angle D A E = 180^{\circ}$.
Also, $A C = A D, A B = A E$,
thus $S \triangle A D B = S \triangle A B C$.
Similarly, the areas of the other two shaded triangles are also equal to $S \triangle \triangle B C$.
$$
\begin{array}{r}
\therefore S \text { shad... | 9 | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,950 |
Example 6. Let $P$, $Q$ be two fixed points on line segment $B C$, and $A P = C Q$, with $A$ being a moving point outside $B C$. When point $A$ moves to make $\angle B A P$
$$
=\angle C A Q \text {, }
$$
what kind of triangle is $\triangle A B C$? Prove your conclusion. (1986, National Junior High School Mathematics Co... | Solve As shown in Figure 6, since $\triangle A B P$ and $\triangle A C Q$ have equal heights and bases, hence $S \triangle \triangle B P=S, \triangle C=$
$\triangle A B P$ and $\triangle A C Q$ have the same vertex angle $\triangle B A P$ $=\angle C A Q$,
$$
\begin{array}{ll}
X & B Q=B P+P Q, \quad C P=C Q+P Q, \\
\the... | proof | Geometry | proof | Yes | Yes | cn_contest | false | 704,951 |
Example 1. A person earns a monthly salary of 200 yuan and receives a bonus of 20 yuan at the end of the month. After receiving the salary each month, they arrange the monthly expenses according to $\frac{4}{5}$ of the current savings. How much will be the remaining balance in the twelfth month? | Let the balance at the end of the $n$-th month be $a_{\mathrm{n}}$ yuan. According to the problem,
$$
\begin{array}{l}
a_{1}=200 \times\left(1-\frac{4}{5}\right)+20=60, \\
a_{2}=260 \times\left(1-\frac{4}{5}\right)+20=72 .
\end{array}
$$
In general, $a_{0}=\left(a_{0-1}+200\right) \times\left(1-\frac{4}{5}\right)+20$,... | 75 | Algebra | math-word-problem | Yes | Yes | cn_contest | false | 704,952 |
Example 2. The sides of an $n$-sided polygon are sequentially denoted as $a_{1}, a_{2}, \cdots$, 20. Each side is painted one of three colors: red, yellow, or green, such that no two adjacent sides have the same color. How many different coloring methods are there? | Solve: First, color the edge $a_{1}$, there are 3 ways to do so.
Then color the edge $a_{2}$, since $a_{2}$ must be a different color from $a_{1}$, there are 2 ways to color $a_{2}$.
Similarly, for $a_{3}, a_{4}, \cdots, a_{n-1}, a_{n}$, there are 2 ways to color each.
Thus, there are $3 \cdot 2^{n-1}$ ways to color in... | 2^{n}+2(-1)^{n} | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false | 704,953 |
Example 3. How many $n$-digit numbers can be formed using $1,2,3,4$ that contain an even number of 1s. | Let $a_{0}$ be the number of $n$-digit numbers with an even number of 1s.
We discuss in two cases:
(i) 1 is in the first position, at this time the remaining $n-1$ digits have an odd number of 1s.
Each of the $n-1$ digits has 4 choices, so there are $4^{n-1}$ different cases. However, these $4^{n-1}$ cases include bot... | \frac{1}{4}\left(4^{n}+2^{n}\right) | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false | 704,954 |
Example 4. How many $n$-digit numbers can be formed using $0,1,2, 3, 4$ such that exactly one digit is 1. | Let it be called “qualified number”, and the total number of all “qualified numbers” is denoted as $x_{\mathrm{n}}$.
The number of “qualified numbers” starting from 0, 1, 2, 3, 4 are denoted as $y_{\mathrm{n}}, z_{\mathrm{n}}, u_{\mathrm{n}}, v_{\mathrm{n}}, w_{\mathrm{n}}$ respectively.
By symmetry, 1 and 3 are both... | x_{2n} = 8 \cdot 3^{n-1} \text{ and } x_{2n+1} = 14 \cdot 3^{n-1} | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false | 704,955 |
Example 1. The lengths of the three sides $a, b, c$ of a triangle are all integers, and $a \leqslant b \leqslant c$. If $b=10$, then the number of such triangles is ( ).
(A) 10.
(B) 55.
(C) $10^{2}$.
(D) Infinitely many.
(1990, Suzhou High School Competition) | When $b=n$, take $a=k(1 \leqslant k \leqslant n)$, and $b \leqslant c<a+b$, then $n \leqslant c<n+k$, at this time the value of $c$ has exactly $k$ possibilities, i.e., $c=n, n+1, \cdots, n+k-1$. The table is as follows:
\begin{tabular}{c|c|c|c}
\hline$a$ & $b$ & $c$ & Number of triangles \\
\hline 1 & $n$ & $n$ & 1 \\... | 55 | Number Theory | MCQ | Yes | Yes | cn_contest | false | 704,956 |
1. (1987, National Junior High School League) Among the triangles with three consecutive natural numbers as sides and a perimeter not exceeding 100, the number of acute triangles is $\qquad$ | (29)
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | 29 | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,958 |
2. (1988, Jiangsu) Let $m$, $n$, $5$ all be natural numbers, satisfying $m \leqslant n \leqslant p$ and $m+n+p=15$. The number of triangles with side lengths $m$, $n$, $p$ is $\qquad$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result dir... | ( 7 )
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 | proof | Yes | Yes | cn_contest | false | 704,959 |
5. (1986, Shanghai) The three sides of a triangle are all positive integers, one of which has a length of 4, but it is not the shortest side. How many different triangles are there? ...
The translation is provided while retaining the original text's line breaks and format. | (8).
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | 8 | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,962 |
Example 1. Let $0<p \leqslant x \leqslant y \leqslant q$. Prove:
$$
\frac{x}{y}+\frac{y}{x} \leqslant \frac{p}{q}+\frac{q}{p} .
$$ | Prove that if $A_{1}=x q-y p, A_{2}=y p-x q$, then $A_{1} \geqslant 0, A_{1}+A_{2}=0$. Let $B_{1}=x p$, $B_{2}=y q$, then $B_{1} \leqslant B_{2}$. By Theorem (1), $T_{2}=A_{1} B_{1}+A_{2} B_{2} \leqslant 0$, i.e., $x p(x q-y p)+y q(y p-x q) \leqslant 0$. Dividing both sides of the above inequality by $x y p q$ yields t... | proof | Inequalities | proof | Yes | Yes | cn_contest | false | 704,965 |
Example 2. Let $a_{1}, a_{2}, \cdots, a_{n}$ be pairwise distinct positive integers. Prove that:
$$
\sum_{k=1}^{n} \frac{a_{k}}{k^{2}} \geqslant \sum_{k=1}^{n} \frac{1}{k} \text {. }
$$
(20th IMO) | Prove that for $b_{1} \leqslant b_{2} \leqslant \cdots b_{n}, b_{1}, \quad b_{2}, \cdots, b_{0}$ being a permutation of $a_{1}, a_{2}, \cdots, a_{\mathrm{n}}$, $A_{\mathrm{k}}=a_{\mathrm{k}} - b_{k}, B_{k}=\frac{1}{k^{2}} (k=1,2, \cdots, n)$, it is known from (I) that $\left\{A_{\mathrm{k}}\right\}(1 \leqslant k \leqsl... | proof | Inequalities | proof | Yes | Yes | cn_contest | false | 704,966 |
Example 3. In the right trapezoid $A B C D$, $A B = 7, A D = 2, B C = 3$. Find the point $P$ on $B A$ such that the triangle with vertices $P, A, D$ is similar to the triangle with vertices $P, B, C$. How many such points $P$ are there?
(A) 1.
(B) 2.
(C) 3.
(D) 4.
(198, Junior High School Competition) | Let $A P=x$, then $P B=7-x$,
(1) If $\triangle P A D \sim \triangle A P B$, then $\frac{x}{7-x}=\frac{2}{3}$.
Solving it, we get $x=\frac{14}{5}<7$. Suitable.
(2) If $\triangle P A D \sim \triangle C B P$, then
$$
\begin{array}{l}
\frac{x}{3}=\frac{2}{7-x}, \\
\therefore x_{1}=6, x_{2}=1
\end{array}
$$
Both satisfy t... | C | Geometry | MCQ | Yes | Yes | cn_contest | false | 704,968 |
Example 4. Let $0<p \leqslant a_{1} \leqslant q(i=1,2, \cdots, n)$, and $b_{1}, b_{2}, \cdots, b_{\mathrm{n}}$ be any permutation of $a_{1}, a_{2}, \cdots, a_{\mathrm{n}}$. Prove:
$$
\begin{aligned}
n & \leqslant \frac{a_{1}}{b_{1}}+\frac{a_{2}}{b_{2}}+\cdots+\frac{a_{\mathrm{n}}}{b_{\mathrm{n}}} \\
& \leqslant n+\left... | Suppose $0<p \leqslant a_{1} \leqslant a_{2} \leqslant \cdots \leqslant a_{0}$ $\leqslant q$, then the sequence $\left\{\frac{1}{a_{k}}\right\}(1 \leqslant k \leqslant n)$ is monotonically decreasing. Let $A_{\mathrm{x}}=\frac{1}{a_{\mathrm{k}}}-\frac{1}{b_{\mathrm{k}}}(1 \leqslant k \leqslant n)$, by (II) we know, the... | proof | Inequalities | proof | Yes | Yes | cn_contest | false | 704,969 |
Example 6. (Chebyshev's Inequality) Let $a_{1} \leqslant a_{2} \leqslant \cdots \leqslant a_{n}, b_{1} \leqslant b_{2} \leqslant \cdots \leqslant b_{n}$, then
$$
\begin{array}{l}
\sum_{k=1}^{n} a_{k} b_{n-k+1} \leqslant \frac{1}{n}\left(\sum_{k=1}^{n} a_{k}\right) \\
\cdot\left(\sum_{k=1}^{n} b_{k}\right) \leqslant \su... | Proof: Let $\{t_{r}\}$ be a periodic sequence with period $n$. When $1 \leqslant k \leqslant n$, $t_{k}=b_{k}$. Let $A_{k}=t_{k+m}-b_{k}$ $(1 \leqslant k \leqslant n, m=0,1,2, \cdots, n-1)$. By (I), the sequence $\{A_{k}\}$ satisfies $S_{n}=0, S_{k} \geqslant 0$ ($1 \leqslant k \leqslant n-1$). Therefore, by Theorem (1... | proof | Inequalities | proof | Yes | Yes | cn_contest | false | 704,971 |
Example 1. A pile of toothpicks 1000 in number, two people take turns to pick any number from it, but the number of toothpicks taken each time must not exceed 7. The one who gets the last toothpick loses. How many toothpicks should the first player take on the first turn to ensure victory? (New York Math Competition) | From $1000=125 \times 8$, we know that one should first dare to take 7 moves, so that the latter can achieve a balanced state: $124 \times(7+1)+1$. | 7 | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false | 704,972 |
Theorem 2 The root numbers of two piles of matches are $m, n$, and two players take turns to take any number from any pile.
1 ) It is stipulated that the one who takes the last match wins. If $m \neq n$, the first taker has a winning strategy;
2 ) It is stipulated that the one who takes the last match loses. If $m \neq... | Prove 1) Let the state where both piles of matches have 2 left be a 2-balanced state (denoted as $(i, 2)$), and $(i, 0)$ be a quasi-winning state. Clearly:
a) Appropriately taking once from a 2-unbalanced state can either let the opponent get a 2-balanced state or win directly;
b) Taking once from a 2-balanced state wi... | proof | Combinatorics | proof | Yes | Yes | cn_contest | false | 704,973 |
Theorem 3 There are three piles of matches with root numbers $n_{1}, n_{2}, n_{3}$, and two players take turns to take any number from any pile.
1 ) It is stipulated that the one who takes the last match wins. If ( $n_{1}$, $n_{2}, n_{3}$ ) is not a 3-balanced state, then the first player has a winning strategy.
2 ) It... | 1) Clearly, (1, 1, 1) is a pre-winning state. By Theorem 2, ( $i, j, 0$ ) will be a pre-winning state ( $i, j \in N$ and $i \neq j$ ). It is evident that:
a) Appropriately taking once from a 3-unbalanced state will inevitably lead to the opponent getting a 3-balanced state or oneself winning,
b) Taking once from a 3-ba... | proof | Combinatorics | proof | Yes | Yes | cn_contest | false | 704,975 |
$P A \perp B C, P A=B C=l, P A, B C$ have a common perpendicular line $E D=h$. Prove: the volume of the pyramid $P-A B C_{i}$
Translate the text into English, please keep the original text's line breaks and format, and output the translation result directly. | Proof 1: Draw $B L^{\prime} \therefore A P, C C^{\prime} \mathbb{\|} A P$, then the pyramid $P-A B C$ is supplemented to form a triangular prism $A B C-P B^{\prime} C^{\prime}$ (as shown in Figure 5). Given
$\left.\begin{array}{l}P A \perp D E \\ P A \perp B C\end{array}\right\} \Rightarrow P A \perp$ section $E B C$.
... | \frac{1}{6} l^{2} h | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,978 |
Example 4. The perimeter of a hexagon is 20, each side is an integer, and no three sides can form a triangle. Then, such a hexagon ( ).
(A) does not exist.
(B) is unique.
(C) has a finite number,
but more than one.
(D) has an infinite number. | If we can find six integers $a_{1}, a_{2}, \cdots, a^{\theta}$, satisfying: (1) $a_{1}+a_{2}+\cdots+a_{8}=20$; (2) $a_{1} \leqslant a_{2}, \quad a_{1}+a_{2} \leqslant a_{3}, \quad a_{2}+a_{3} \leqslant a_{4}$, $a_{3}+a_{4} \leqslant a_{5}, a_{4}+a_{5} \leqslant a_{63}$ (3) $a_{1}+a_{2}+a_{3}+a_{4}+a_{5}>a_{8}$. Then th... | D | Geometry | MCQ | Yes | Yes | cn_contest | false | 704,979 |
$P$ is the midpoint of edge $A B$, $Q$ is the midpoint of edge $C D$, then the length of line $P Q$ is ( ).
(A ) $\frac{1}{2}$.
(B) $\frac{\sqrt{3}}{2}$.
(C ) $\frac{3}{4}$.
(D) $\frac{\sqrt{2}}{2}$.
(1987, North High School Grade 1) | Solve for the Chengxi body $\triangle B C$ D
a regular hexahedron (cube)
cube, as shown in Figure 7, PQ
the length is the diagonal of the cube's face, which
distance, that is, the side length of the cube is $\frac{\sqrt{2}}{2}$. Therefore, the answer is (D). | D | Geometry | MCQ | Yes | Yes | cn_contest | false | 704,980 |
Example 2. Find the area of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$. | $+\left(\frac{y}{b}\right)^{2}<1$ represents $S$. If $S_{0}$ denotes $x^{2}+y^{2}$ and $S_{0}=\pi$, then $S=\pi a b$.
| \pi a b | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,982 |
Example 1. Prove: A $5 \times$ 5 chessboard is 2-good. | Proof According to the Pigeonhole Principle, in a $5 \times 5$ chessboard with 25 small squares, at least 13 of the small squares are of the same color, let's assume they are black. Using Theorem 2, we have
$$
\begin{array}{l}
m=n=5, \quad a=13, \quad q=2, \quad r=3, \\
r C_{4+1}^{2}+(n-r) C_{4}^{2}=11, \\
C_{\mathrm{m... | proof | Combinatorics | proof | Yes | Yes | cn_contest | false | 704,983 |
Example 2. On a $4 \times n$ chessboard, if any cells are colored red, white, or blue, then the necessary and sufficient condition for the existence of a monochromatic rectangle is $n=19$. | If $n=19$, then in a $4 \times 19$ chessboard, there must be $\left[\frac{4 \times 19-1}{3}\right]+1=26$ small squares of the same color, let's assume they are red. At this point,
$$
\begin{array}{l}
m=4, n=19, a=26, q=1, r=7, \\
r C_{4+1}^{2}+(n-r) C_{4}^{2}=7, \\
C_{\mathrm{m}}^{2}=6,
\end{array}
$$
The conditions o... | 19 | Combinatorics | proof | Yes | Yes | cn_contest | false | 704,984 |
Example 3. For a $10 \times 11$ chessboard with a three-coloring, prove that there must exist a monochromatic rectangle.
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | Analysis Suppose there are at least
$$
\left[\frac{10 \times 11-1}{3}\right]+1=37 \text { red small squares, } m=10, n=11 \text {, }
$$
$q=3, r=4$, at this time,
$$
\begin{array}{l}
r C_{\mathrm{Q}+1}^{2}+(n-r) C_{\mathrm{q}}^{2}=45, \\
C_{\mathrm{m}}^{2}=45 .
\end{array}
$$
According to Theorem 3, there is either a r... | null | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false | 704,985 |
Example 4. There is a sufficiently large grid paper, with each small cell colored using $t(t \geqslant 2)$ different colors. Prove: There exists a rectangle, the four corners of which are of the same color (1986, All-Russian Mathematical Olympiad). | To prove that it is sufficient to take a grid paper with $t+1$ rows and $\frac{t^{2}(t+1)}{2}+1$ columns.
We have $m=t+1, n=\frac{t^{2}(t+1)}{2}+1$,
$$
\begin{aligned}
a & \left.=\left[\frac{t^{2}(t+1)}{2}+1\right)(t+1)\right]+1 \\
& =t(t+1)^{2}+2 \\
& =\left(\frac{t^{2}(t+1)}{2}+1\right)+\left(\frac{t(t+1)}{2}+1\right... | proof | Combinatorics | proof | Yes | Yes | cn_contest | false | 704,986 |
Example 2 - (1975, Soviet University Student Mathematics Competition)
Prove the inequality
$$
\frac{x^{\mathrm{n}}+y^{\mathrm{n}}}{2} \geqslant\left(\frac{x+y}{2}\right)^{\mathrm{n}} \text {. }
$$
where $x \geqslant 0, y \geqslant 0, n$ is a natural number. | Prove that when $x=0$ or $y=0$, inequality (12) obviously holds. When $x>0$ and $y>0$, since the sequence $\left\{(x / y)^{\mathrm{R}-1}\right\}$ is a convex sequence, by property 2, we get
$$
\begin{array}{l}
\sum_{1=0}^{\mathrm{n}}(x / y)^{\mathrm{i}} C_{\mathrm{n}}^{1} \leqslant 2^{\mathrm{n}-1}\left[1+(x / y)^{\mat... | proof | Inequalities | proof | Yes | Yes | cn_contest | false | 704,988 |
Example 3. Let $x>0, x \neq 1, n$ be a natural number, then
we have $(n+1)(1+x)^{n} \leqslant 2^{n} \frac{1-x^{n+1}}{1-x}$. | Prove that since $\left\{x^{0-1}\right\}$ is a convex sequence, by property
3 we have $\frac{\sum_{1=0}^{n} x C_{\mathrm{n}}^{1}}{2^{\mathrm{n}}} \leqslant \frac{S_{\mathrm{n}+1}}{n+1}$.
But $S_{\mathrm{n}+1}=\frac{1-x^{n+1}}{1-\bar{x}}, \sum_{\mathrm{n}=0}^{\mathrm{n}} x^{1} C_{\mathrm{n}}^{1}=(1+x)^{\mathrm{n}}$,
thu... | proof | Inequalities | proof | Yes | Yes | cn_contest | false | 704,989 |
Example 1. (IMO-29-5) In the right triangle $\triangle ABC$, $AD$ is the altitude on the hypotenuse $BC$. The line connecting the incenter of $\triangle ABD$ and the incenter of $\triangle ACD$ intersects sides $AB$ and $AC$ at $K$ and $L$, respectively. Let $S_{\triangle B C}=S, S_{\triangle K L}=T$. Prove: $S \geqsla... | Prove that if $AB$ is the $y$-axis and $AC$ is the $x$-axis, and we denote $B(0, c), C(0, b)$, then $D\left(\frac{b c^{2}}{a^{2}}, \frac{b^{2} c}{a^{2}}\right)$. Therefore, $A D=\frac{b c}{a}, C D=\frac{b^{2}}{a}, B D=\frac{c^{2}}{a}$. The incenter of $\triangle A C D$ and $\triangle A B D$ are (denote
$$
\begin{aligne... | S \geqslant 2 T | Geometry | proof | Yes | Yes | cn_contest | false | 704,993 |
Example 2. In $\triangle A B C$, $\angle C=30^{\circ}, O$ is the circumcenter, $I$ is the incenter, point $D$ on side $A C$ and point $E$ on side $B C$ are such that $A D=B E=A B$. Prove: $O I=D E$.
保留了源文本的换行和格式，这是翻译结果。 | Prove that if $C B$ is taken as the $x$-axis to establish a coordinate system, then
$$
\begin{array}{l}
C(0,0), B(a, 0), A\left(\frac{\sqrt{3}}{2} b, \frac{1}{2} b\right), \\
D E^{2}=(b-c)^{2}+(a-c)^{2}-\sqrt{3}(b-c) \\
\cdot(a-c),
\end{array}
$$
The incenter $I\left(\frac{(2+\sqrt{3}) a b}{2(a+b+c)}, \frac{a b}{2(a+b... | proof | Geometry | proof | Yes | Yes | cn_contest | false | 704,994 |
1. A competition has $n \geqslant 2$ participants, lasting $k$ days. Each day, the participants' scores are $1,2, \cdots, n$, and all participants' total scores are 26 points. Find all possible pairs $(n, k)$. | 1. It is easy to know that the total sum of $k$ days is
$$
k \times \frac{1}{2} n(n+1)=26 n .
$$
Therefore, $k(n+1)=52$.
Thus, $(n, k)=(51,1),(25,2),(12,4)$, $(3,13)$. Only $(51,1)$ is impossible to achieve.
For $(25,2)$, we have
$$
\begin{array}{l}
(26,26, \cdots, 26)=(1,2, \cdots, 24,25) \\
+(25,24, \cdots, 2,1) .
\... | (25,2),(12,4),(3,13) | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false | 704,995 |
2. $\frac{1}{2} n(n+1)$ distinct numbers are randomly arranged in a triangle:
Let $M_{\mathrm{k}}$ be the maximum number in the $k$-th row (counting from the top), find the probability that $M_{1}<M_{2}<M_{3}<\cdots<M_{\mathrm{n}}$ holds. | 2. Let the required probability be $p_{\mathrm{n}}$, obviously, $p_{1}=1$, $p_{2}=2 / 3$.
Since the largest number must appear in the last row to satisfy the inequality described in the problem. The probability of this situation occurring is
$$
\frac{n}{\frac{1}{2} n(n+1)}=\frac{2}{n+1} .
$$
Therefore,
$$
p_{n}=\frac... | \frac{2^{n}}{(n+1)!} | Combinatorics | math-word-problem | Yes | Yes | cn_contest | false | 704,996 |
3. Let $A B C D$ be a cyclic quadrilateral, and the diagonals $A C$ and $B D$ intersect at $X$. The feet of the perpendiculars from $X$ to the sides $A B, B C, C D$, and $D A$ are $A^{\prime}, B^{\prime}, C^{\prime}, D^{\prime}$, respectively. Prove: $A^{\prime} B^{\prime}+C^{\prime} D^{\prime}=A^{\prime} D^{\prime}+B^... | 3. By the sine theorem,
$$
\begin{array}{l}
A^{\prime} B^{\prime}=X B \sin \angle A B C, \\
C^{\prime} D^{\prime}=X D \sin \angle A D C .
\end{array}
$$
Since $\angle A B C+\angle A D C=\pi$, therefore,
$$
\begin{array}{l}
A^{\prime} B^{\prime}+C^{\prime} D^{\prime}=(X B+X D) \text { sin } \angle A B C \\
=B D \sin \a... | proof | Geometry | proof | Yes | Yes | cn_contest | false | 704,997 |
4. A particle can move along the $x$-axis at a speed of 2 meters per second, and move at a speed of 1 meter per second in other places on the plane. The particle starts from the origin, find the region it can reach.
| 4. Due to symmetry, we only consider the first quadrant. Suppose a particle moves along the $x$-axis to $(t, 0)$, and then to the point $(x, y)$, then $(x-t)^{2}+y^{2} \leqslant\left(\frac{2-t}{2}\right)^{2}$, i.e., $y^{2} \leqslant \frac{3}{4}\left(\frac{4}{9}(2-x)^{2}-\left(t-\frac{2(2 x-1)}{3}\right)^{2}\right)$. Wh... | not found | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 704,998 |
5. Let the function $f$ be defined on the positive integers, satisfying
$$
\begin{aligned}
f(1)=1, & f(2)=2, \\
f(n+2) & =f[n+2-f(n+1)] \\
& +f\left[n+1-f^{n}(n)\right](n \geqslant 1) .
\end{aligned}
$$
(a) Prove:
(i) $0 \leqslant f(n+1)-f(n) \leqslant 1$,
(ii) If $f(n)$ is odd, then $f(n+1)=f(n)+1$
(b) Determine (and ... | 5. (a) (i) Let for $n=1,2, \cdots, n-1$,
$f(n+1) \cdots f(n)=0$ or 1.
Then for each $n \in\{1,2, \cdots, m-1\}$,
$$
\begin{array}{l}
{[(n+2)-f(n+1)]-[(n+1)-f(n)]} \\
=1-[f(n+1)-f(n)] \in\{0,1\} .
\end{array}
$$
Therefore, by induction hypothesis,
$$
\begin{aligned}
& f[n+2-f(n+1)]-f[n+1-f(n)] \\
& \in\{0,1\} . \\
& \t... | proof | Algebra | proof | Yes | Yes | cn_contest | false | 704,999 |
1. In $\triangle A B C$, $D, E, F$ are the midpoints of $B C$, $C A, A B$ respectively, and $G$ is the centroid. For each value of $\angle B A C$, how many non-similar $\triangle A B C$ are there such that $A E G F$ is a cyclic quadrilateral? | 1. From $A, E, G, F$ being concyclic, we get
$$
\begin{aligned}
\angle C G_{E} & =\angle B A C \\
= & \angle C E D .
\end{aligned}
$$
Let $C G$ intersect $D E$ at $M$. Then from (1),
it is easy to derive
$$
C M \times C G=C E^{2},
$$
i.e., $\frac{1}{2} m_{\mathrm{e}} \times \frac{2}{3} m_{\mathrm{c}}=\left(\frac{1}{2... | 1 | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 705,000 |
Example 6. There is a rectangular prism with length, width, and height being positive integers $m, n, r (m \leqslant n \leqslant r)$, respectively. The surface of the prism is painted red and then it is cut into unit cubes. It is known that the number of unit cubes without any red color is equal to the number of unit c... | The number is $(n-2)(r-2)$.
According to the problem,
$$
(n-2)(r-2)=1985 \text {. }
$$
Obviously, when $n, r<3$, (1) has no solution.
When $n, r \geqslant 3$,
$$
\because \quad 1 \leqslant n-2 \leqslant r-2 \text {, }
$$
and $1985=1 \times 1985=5 \times 397$,
solving it, we get $n=3, r=1987$ or $n=7, r=399$.
$2^{\ci... | 1, 3, 1987; 1, 7, 399; 5, 5, 1981; 5, 7, 663; 3, 3, 1081 | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 705,001 |
2. $a_{1}, a_{2}, \cdots, a_{n}$ are positive real numbers, $S_{4}$ is the sum of the products of $k$ numbers taken from $a_{1}$, $a_{2}, \cdots, a_{\mathrm{n}}$. Prove:
$$
\begin{array}{l}
S_{\mathrm{k}} S_{\mathrm{n}-3} \geqslant\left(C_{\mathrm{n}}^{\mathrm{k}}\right)^{2} a_{1} a_{2} \cdots a_{\mathrm{n}} \\
(k=1,2,... | $\begin{array}{l}\text { 2. By the arithmetic-geometric mean inequality } \\ S_{k} S_{n-k} \\ \geqslant C_{n}^{k} \cdot C_{n}^{n-k}\left(\prod a_{i_{1} i_{2} \cdots i_{k}}\right)^{\frac{1}{C_{n}^{k}}} \\ \text { - }\left(\prod_{1_{1}, 1_{2}, \ldots-1}\right)_{n}^{1 u-\bar{k}} \\ =\left(C_{n}^{k}\right)^{2}\left(a_{1} a... | proof | Inequalities | proof | Yes | Yes | cn_contest | false | 705,002 |
3. Consider all bases $AB$ fixed, and point $C$ drawn such that the height of $\triangle ABC$ is a constant $h$. When is the product of its heights maximized?
---
The translation maintains the original text's format and line breaks as requested. | 3. Since $h_{\mathrm{a}} \cdot a=h_{\mathrm{b}} \cdot b=h \cdot c$ is a constant, we have
$$
k_{\mathrm{a}} h_{\mathrm{b}} h_{h}=\frac{h^{3} \cdot c^{2}}{a b},
$$
it is only necessary to minimize $a b$.
When $a=b$, $\triangle A B C$ is an isosceles triangle. Let this point be $C$ and $C_{0}$, and the tangent line of t... | a=b | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 705,003 |
4.199J individuals are divided into several mutually disjoint subsets, such that
(a) in each subset, no one knows everyone else.
(b) in each subset, among any three people, at least two do not know each other.
(c) in each subset, for any two people who do not know each other, there is exactly one person in the subset w... | 4. (i) Consider only one group. Let $y_{1}$ and $y_{2}$ be unknown to each other within the same group. By (c), there exists $x$ who knows both $y_{1}$ and $y_{2}$. Suppose, apart from $x$ and themselves, $y_{1}$ knows $z_{11}, z_{12}, \cdots, z_{11}$. $y_{2}$ knows $z_{21}$, $z_{22}, \cdots, z_{2 \mathrm{k}}$.
By (b)... | 398 | Combinatorics | proof | Yes | Yes | cn_contest | false | 705,004 |
Example 7. A triangle with all sides as integers, and the longest side being 11, has ——. (1983, National Junior High School Competition)
A triangle with all sides as integers, and the longest side being 11, has ——. (1983, National Junior High School Competition) | Let's study a more general case, that is, how many triangles are there with the maximum side length of $n$? We will discuss this using the lattice point method.
Suppose the lengths of the two sides of the triangle are $x, y, x, l, n$, which are natural numbers, and $x \leqslant y \leqslant n, x+y>n$.
Since the side l... | null | Geometry | math-word-problem | Yes | Yes | cn_contest | false | 705,006 |
Example 10: There are 100 points on a plane, where the distance between any two points is no less than 3. Now, connect every two points that are exactly 3 units apart with a line. Prove: the number of these line segments will not exceed 300. (1984, Beijing Junior High School Competition) | First, consider the maximum number of points whose distance from a certain point $A$ is: 3. Let $B_{1} A=B_{2} A=\cdots=B_{1} A$
$$
\begin{array}{l}
=3, \text { and } B_{1} B_{2} \geqslant 3, B_{2} B_{3} \geqslant 3, \cdots, B_{\mathrm{k}-1} B_{1} \\
\geqslant 3, B_{\mathrm{k}} B_{1} \geqslant 3 . \text { Then } \angle... | 300 | Combinatorics | proof | Yes | Yes | cn_contest | false | 705,009 |
Example 1. $T$ is the set of all integer points in the coordinate plane (points with both integer horizontal and vertical coordinates are called integer points). If two integer points $(x, y)(u, v)$ satisfy $|x-u|+|y-v|=1$, then these two points are called adjacent points. Prove: there exists a set $S \subseteq T$, suc... | Notice that for every integer point $(u, v)$, there are 4 adjacent points: $(u+1, v), (u-1, v)$, $(u, v+1), (u, v-1)$. We aim to construct a mapping that transforms these 5 points into 5 different numbers, thereby converting the problem of point sets into the problem of number sets.
In fact, let $f: (u, v) \rightarrow... | proof | Combinatorics | proof | Yes | Yes | cn_contest | false | 705,010 |
Example 4. (Question 24 TMO attached j added question) A 990th degree polynomial $P(x)$ satisfies $P(k)=f_{k}, k=992, 993, \cdots, 1782$, where $f_{s}$ is the Fibonacci sequence.
Prove: $P(1983)=f_{19 \text{ 193 }}-1 .{}^{\circ}$ | Ming |Newton-Gregory interpolation polynomial, we get
$$
P(x)=\sum_{j=0}^{990}\binom{x-992}{j} \Delta f_{\theta 82} \text {. (*) }
$$
Here, $\Delta f_{\theta 82}$ is the $j$-th order forward difference of the sequence $f_{\theta \theta_{2}}, f_{8 \theta_{3}}, \cdots, f_{1082}$ at the term with index 992.
Note that $\D... | P(1983)=f_{1983}-1 | Algebra | proof | Yes | Yes | cn_contest | false | 705,012 |
3. $\left(4^{-1}-3^{-1}\right)^{-1}=(\quad)$.
(A) -12 .
(B) -1 .
(C) $-\frac{1}{12}$.
(D) 1. (E) 12 . | 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 | 705,015 |
4. The following triangle that cannot exist is ( ).
(A) Acute equilateral triangle.
(B) Isosceles right triangle.
(C) Obtuse right triangle.
(D) Scalene right triangle.
(E) Scalene obtuse 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 | 705,016 |
7. If $x=\frac{a}{b}$ and $a \neq b, b \neq 0$, then $\frac{a+b}{a-b}$ $=$ ( ).
(A) $\frac{x}{x+1}$.
(B) $\frac{x+1}{x-1}$.
(C) 1 .
(D) $x-\frac{1}{x}$.
(1:) $x+\frac{1}{x}$ | 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 | 705,019 |
10. A point $P$ is 9 units away from the center of a circle with a radius of 15, and the number of chords passing through $P$ and having integer lengths is ( ).
(A) 11.
(B) 12.
(C) 13.
(D) 14.
(E) 29. | 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 | 705,022 |
Example 5. (22nd IMO selected problems) Let $\left\{f_{\mathrm{n}}\right\}$ be the Fibonacci sequence.
(a) Find all pairs of real numbers $(a, b)$ such that for each $n, a f_{0}+b f_{\mathrm{n}-1}$ is a term in the sequence $\left\{f_{\mathrm{n}}\right\}$.
(b) Find all pairs of positive real numbers $(u, v)$ such that ... | (a) Let real numbers $a, b$ be such that for all integers $n, a f_{n}+b f_{n+1}$ is a term in the Fibonacci sequence. Then for $n=1, 2, 3$, we can set
$$
\begin{array}{l}
a+b=f_{m}, a+2 b=f_{m^{\prime}}, \\
2 a+3 b=f_{m^{\prime \prime}} .
\end{array}
$$
where $m, m^{\prime}, m^{\prime \prime}$ are some three natural n... | \begin{array}{l}
a=0, \\
b=1,
\end{array} \left\{\begin{array}{l}
a=1, \quad a=f_{m-2}, \\
b=0, \quad b=f_{m-1} .
\end{array} \quad(m \geqslant 3)\right.
} | Number Theory | math-word-problem | Yes | Yes | cn_contest | false | 705,023 |
11. $A$ and $B$ are running a 10-kilometer long-distance race, starting from the same point. They first run 5 kilometers uphill, then return along the same route to the starting point. $A$ starts 10 minutes earlier, with an uphill speed of 15 kilometers per hour and a downhill speed of 20 kilometers per hour. $B$ has 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 | 705,024 |
13. Players X, Y, Z participate in a game with no ties. If the win-loss ratio of X and Y is $2: 3$, then the win-loss ratio of Z is ( ). The probability is $\% \frac{q}{p+q}$.
(A) $3: 20$.
(B) $5: 6$.
(C) $3: 5$.
(D) $17: 3$.
(E) $20: 3$. | 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 | 705,026 |
14. Let $x$ be the cube of a positive integer, and let $d$ be the number of positive divisors of $x$. Then $d$ can be ( ).
(A) 200.
(B) 201.
(C) 202.
(D) 203.
(E) 204. | 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 | 705,027 |
15. A round table has exactly 60 chairs, and $N$ people sit in such a way that when a new person arrives, they will inevitably sit next to someone. The minimum possible value of $N$ is ( ).
(A) 15.
(B) 20.
(C) 30.
(D) 40.
(E) 58. | 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 | 705,028 |
17. If the digits of a positive integer $N$ are reversed and it remains the same, then $N$ is called a palindrome. The year 1991 is the only year in this century with the following two properties:
(a) It is a palindrome;
(b) It can be factored into the product of a two-digit prime palindrome and a three-digit prime pal... | $\mathrm{D}$ | D | Number Theory | MCQ | Yes | Yes | cn_contest | false | 705,030 |
19. In $\triangle A B C$, $\angle C=90^{\circ}, A C=3$, $B C=4$. In $\triangle A B D$, $\angle A=90^{\circ}, A D=12$. Points $C$ and $D$ are on opposite sides of $A B$. A line through $D$ parallel to $A C$ intersects the extension of $C B$ at $E$. If $\frac{D E}{D B}=\frac{m}{n}$, where $m, n$ are coprime positive inte... | $\mathrm{B}$ | 128 | Geometry | MCQ | Yes | Yes | cn_contest | false | 705,032 |
22. Two circles are externally tangent, $P A B$ and $P A^{\prime} B^{\prime}$ are common external tangents to the two circles, touching the smaller circle at points $A$ and $A^{\prime}$, and the larger circle at points $B$ and $B^{\prime}$. If $P A = A B = 4$, then the area of the smaller circle is ( ).
(A) $1.44 \pi$.... | 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 | 705,036 |
$$
\begin{array}{l}
\text { 25. If } T_{\mathrm{n}}=1+2+3+\cdots+n, \\
P_{\mathrm{u}}=\frac{T_{2}}{T_{2}-1} \cdot T_{3}-1 \cdot \frac{T_{4}}{T_{4}-1} \cdots \\
\cdot \frac{T_{\mathrm{n}}}{T_{\mathrm{n}}-1}, n=2,3,4, \cdots,
\end{array}
$$
Then $P_{1091}$ is closest to which of the following numbers?
(A) 2.0 .
(B) 2.3 ... | $\mathrm{D}$ | D | Algebra | MCQ | Yes | Yes | cn_contest | false | 705,039 |
26. If an $n$-digit positive integer, its $n$ digits are a permutation of the elements of the set $\{1,2, \cdots, n\}$, and the first $k$ digits form an integer that is divisible by $k$, for $k=1,2, \cdots, n$, then we call this $n$-digit positive integer a “good number”. For example, 321 is a three-digit “good 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 | Number Theory | MCQ | Yes | Yes | cn_contest | false | 705,040 |
29. As shown in the figure, fold the equilateral
triangle $\triangle A B C$ so that the original
vertex lands on side $B C$,
if $B A^{\prime}=1, A^{\prime} C=2$,
then the length of the fold $P Q$ is
(A) $\frac{8}{5}$.
(B) $\frac{7}{20} \sqrt{21}$.
(C) $\frac{1+\sqrt{5}}{2}$.
(D) $\frac{13}{8}$. (E) $\sqrt{3}$. | B
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | null | Geometry | MCQ | Yes | Yes | cn_contest | false | 705,043 |
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