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62,245 | ## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0}\left(\frac{1+x^{2} \cdot 2^{x}}{1+x^{2} \cdot 5^{x}}\right)^{\frac{1}{\sin ^{3} x}}
$$ | \frac{2}{5} |
62,251 | In the diagram, the cube has a volume of $8 \mathrm{~cm}^{3}$. What is the value of $x$ ?
(A) 2
(B) 8
(C) 4
(D) 6
(E) 3
 | 2 |
62,252 | Points $P$ and $Q$ are located on side $B C$ of triangle $A B C$, such that $B P: P Q: Q C=1: 2: 3$. Point $R$ divides side $A C$ of this triangle so that
$A R: R C=1: 2$. What is the ratio of the area of quadrilateral $P Q S T$ to the area of triangle $A B C$, if $S$ and $T$ are the points of intersection of line $B R$ with lines $A Q$ and $A P$ respectively? | \frac{5}{24} |
62,255 | 13. In $\triangle A B C$, $\angle A B C=40^{\circ}, \angle A C B=20^{\circ}, N$ is a point inside the triangle, $\angle N B C=30^{\circ}, \angle N A B=20^{\circ}$, find the degree measure of $\angle N C B$.
(Mathematical Bulletin Problem 1023) | 10 |
62,283 | Ex. 165. Inside an angle of $90^{\circ}$, a point is given, located at distances of 8 and 1 from the sides of the angle. What is the minimum length of the segment passing through this point, with its ends lying on the sides of the angle? | 5\sqrt{5} |
62,305 | 5. Given real numbers $x, y$ satisfy $x+y=1$. Then, the maximum value of $\left(x^{3}+1\right)\left(y^{3}+1\right)$ is $\qquad$ . | 4 |
62,378 | 2A. Given points $A(-2,0)$ and $B(2,0)$. Points $C$ and $D$ lie on the normals to the segment $AB$ at points $A$ and $B$ respectively, such that $\measuredangle COD$ is a right angle. Determine the geometric locus of the points of intersection of lines $AD$ and $BC$. | \frac{x^{2}}{4}+y^{2}=1 |
62,380 | Problem 4. Let $n \geq 2$ be a natural number. Determine the set of values that the sum
$$
S=\left[x_{2}-x_{1}\right]+\left[x_{3}-x_{2}\right]+\ldots+\left[x_{n}-x_{n-1}\right]
$$
can take, where $x_{1}, x_{2}, \ldots, x_{n}$ are real numbers with integer parts $1,2, \ldots, n$. | {0,1,2,\ldots,n-1} |
62,400 | ## Problem Statement
Calculate the definite integral:
$$
\int_{0}^{2 \operatorname{arctan} \frac{1}{2}} \frac{(1-\sin x) d x}{\cos x(1+\cos x)}
$$ | -\frac{1}{2}+2\ln\frac{3}{2} |
62,406 | 11. (20 points) Given non-zero complex numbers $x, y$ satisfy
$$
\begin{array}{l}
y^{2}\left(x^{2}-x y+y^{2}\right)+x^{3}(x-y)=0 . \\
\text { Find } \sum_{m=0}^{29} \sum_{n=0}^{29} x^{18 m n} y^{-18 m n} \text { . }
\end{array}
$$ | 180 |
62,419 | 1. In a certain electronic device, there are three components, with probabilities of failure being $0.1$, $0.2$, and $0.3$, respectively. If one, two, or three components fail, the probabilities of the device malfunctioning are $0.25$, $0.6$, and $0.9$, respectively. Find the probability that the device malfunctions. | 0.1601 |
62,427 | 1. Jarris the triangle is playing in the $(x, y)$ plane. Let his maximum $y$ coordinate be $k$. Given that he has side lengths 6,8 , and 10 and that no part of him is below the $x$-axis, find the minimum possible value of $k$. | \frac{24}{5} |
62,435 | 5. Find the number of such sequences: of length $n$, each term is 0, 1, or 2, and 0 is neither the predecessor nor the successor of 2. | \frac{1}{2}[(1+\sqrt{2})^{n+1}+(1-\sqrt{2})^{n+1}] |
62,467 | 1.1. January first of a certain non-leap year fell on a Saturday. And how many Fridays are there in this year? | 52 |
62,470 | 4. Let the carrying capacity of the truck be $a$ tons, and for transporting 60 tons of sand, $n$ trips were required. Then there are two possible situations:
a) $n-1$ full trucks and no more than half a truck: then to transport 120 tons, $2(n-1)+1=2n-1$ trips will be needed. According to the condition, $2n-1-n=5$, so $n=6$. This situation is realized if $5a<60$ and $5.5a \geqslant 60$, that is, $\left[10 \frac{10}{11} ; 12\right)$.
b) $n-1$ full trucks and more than half a truck: then to transport 120 tons, $2(n-1)+2=2n$ trips will be needed. According to the condition, $2n-n=5$, so $n=5$. Then: $4.5a<60$ and $5a \geqslant 60$, that is, $\left[12 ; 13 \frac{1}{3}\right)$. | [10\frac{10}{11};13\frac{1}{3}) |
62,474 | 3. Let the complex number $z=x+y \mathrm{i}$ have real and imaginary parts $x, y$ that form the point $(x, y)$ on the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{16}=1$. If $\frac{z-1-\mathrm{i}}{z-\mathrm{i}}$ is a real number, then the complex number $z=$ $\qquad$. | \\frac{3\sqrt{15}}{4}+\mathrm{i} |
62,489 | [ Counting in two ways ] [ Different tasks on cutting ]
Inside a square, 100 points are marked. The square is divided into triangles in such a way that the vertices of the triangles are only the 100 marked points and the vertices of the square, and for each triangle in the partition, each marked point either lies outside this triangle or is its vertex (such partitions are called triangulations). Find the number of triangles in the partition. | 202 |
62,506 | Task B-2.3. Andrea and Mirela are preparing for a math competition. When their teacher asked them how many problems they solved yesterday, they did not answer directly. He only found out that Andrea solved fewer problems than Mirela, and each of them solved at least one problem. They also said that the product of the number of problems solved by Andrea and the number of problems solved by Mirela, increased by their sum, is equal to 59. How many problems did Andrea solve? How many different answers are there to this question? | 5 |
62,518 | 8. For any real numbers $A, B, C$, the maximum value of the trigonometric expression $\sin ^{2} A \cos ^{2} B+\sin ^{2} B \cos ^{2} C+\sin ^{2} C \cos ^{2} A$ is
$\qquad$ . | 1 |
62,580 | 22. Find the number of $n$-permutations with repetition, $N_{n}$, formed by $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$ such that $a_{1}$ and $a_{3}$ both appear an even number of times. | \frac{6^{n}+2\cdot4^{n}+2^{n}}{4} |
62,613 | Gabriel draws squares divided into nine cells and writes the natural numbers from 1 to 9, one in each cell. He then calculates the sum of the numbers in each row and each column. The figure shows one of Gabriel's squares; note that the sum of the numbers in the third row is $5+8+2=15$ and the sum of the numbers in the second column is $9+7+8=24$. In this example, the six sums are $6, 12, 15, 15, 18$, and 24.
| $\mathbf{6}$ | $\mathbf{9}$ | $\mathbf{3}$ | 18 |
| :--- | :--- | :--- | :--- |
| $\mathbf{4}$ | $\mathbf{7}$ | $\mathbf{1}$ | 12 |
| $\mathbf{5}$ | $\mathbf{8}$ | $\mathbf{2}$ | 15 |
| 15 | 24 | 6 | |
| | | | |
a) Gabriel filled in a square and made only five sums: 9, 13, 14, 17, and 18. What is the missing sum?
b) Explain why it is not possible for all the sums in one of Gabriel's squares to be even numbers.
c) Fill in the square so that the sums are 7, 13, 14, 16, 18, and 22.
## Combinatorics
# | 19 |
62,614 | 6.275 $\frac{2+x}{2-x}+\sqrt{x}=1+x$ | 0 |
62,618 | 1. It is known that $\operatorname{tg}(2 \alpha-\beta)-4 \operatorname{tg} 2 \alpha+4 \operatorname{tg} \beta=0, \operatorname{tg} \alpha=-3$. Find $\operatorname{ctg} \beta$. | -1 |
62,642 | 4. In square $A B C D$ with side 2, point $A_{1}$ lies on $A B$, point $B_{1}$ lies on $B C$, point $C_{1}$ lies on $C D$, point $D_{1}$ lies on $D A$. Points $A_{1}, B_{1}, C_{1}, D_{1}$ are the vertices of the square of the smallest possible area. Find the area of triangle $A A_{1} D_{1} .(\mathbf{1 1}$ points) | 0.5 |
62,661 | 5. Ten football teams conducted two tournaments over two years. Each year, the results of the tournament were recorded in a table, listing the teams from first to tenth place. A team is considered stable if there is at least one other team that ranks lower than it in both tables. What is the minimum and maximum number of stable teams that can result from the two tournaments? Solution. According to the results of the two tournaments, there cannot be ten stable teams in the tables, as there will be no team that ranks lower than all ten teams in any table. The maximum number of stable teams is nine, and the minimum number of stable teams is zero. We will provide examples of team placements in the tables. Nine stable teams can be achieved if the same team finishes in tenth place in both tournaments. Then, for each of the remaining nine teams, the stability condition will be met.
Zero stable teams can be achieved with the following two tables for the two years:
| Place | Team Number |
| :--- | :--- |
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
| $\ldots$ | $\ldots$ |
| 9 | 9 |
| 10 | 10 |
| Place | Team Number |
| :--- | :--- |
| 1 | 10 |
| 2 | 9 |
| 3 | 8 |
| $\ldots$ | $\ldots$ |
| 9 | 2 |
| 10 | 1 |
Reversing the order of the teams results in no team having a lower-ranked team in the first year for the second year. | 09 |
62,662 | Example 11 (9th American Invitational Mathematics Examination) How many real numbers $a$ are there such that $x^{2}+a x+6 a=0$ has only integer solutions.
保留了原文的换行和格式。 | 10 |
62,663 | 29. How may pairs of integers $(x, y)$ satisfy the equation
$$
\sqrt{x}+\sqrt{y}=\sqrt{200600} ?
$$ | 11 |
62,680 | 10. Evaluate the definite integral $\int_{-1}^{+1} \frac{2 u^{332}+u^{998}+4 u^{1664} \sin u^{691}}{1+u^{666}} \mathrm{~d} u$. | \frac{2}{333}(1+\frac{\pi}{4}) |
62,701 | 1. Let $x_{1}$ and $x_{2}$ be the roots of the equation $x^{2} + p x - \frac{1}{2 p^{2}} = 0$, where $p$ is an arbitrary real parameter. For which value of the parameter $p$ does the following hold:
$$
x_{1}^{4} + x_{2}^{4} = 2 + \sqrt{2}
$$ | \\frac{1}{\sqrt[8]{2}} |
62,724 | Problem 2. In a small town, there is only one tram line. It is circular, and trams run along it in both directions. There are stops on the loop: Circus, Park, and Zoo. The journey from Park to Zoo via Circus is three times longer than not via Circus. The journey from Circus to Zoo via Park is twice as short as not via Park. Which route from Park to Circus is shorter - via Zoo or not via Zoo - and by how many times?
(A.V. Shapovalov) | 11 |
62,746 | Let $N$ denote the number of all natural numbers $n$ such that $n$ is divisible by a prime $p> \sqrt{n}$ and $p<20$. What is the value of $N$ ? | 69 |
62,749 | 12. Given the function $f(x)=a+x-b^{x}$ has a zero $x_{0} \in(n, n+1)(n \in \mathbf{Z})$, where the constants $a, b$ satisfy the conditions $2019^{a}=2020,2020^{b}=2019$. Then the value of $n$ is $\qquad$ | -1 |
62,783 | On [square](https://artofproblemsolving.com/wiki/index.php/Square) $ABCD$, point $E$ lies on side $AD$ and point $F$ lies on side $BC$, so that $BE=EF=FD=30$. Find the area of the square $ABCD$. | 810 |
62,792 | 7. For any $n$-element set $S_{n}$, if its subsets $A_{1}$, $A_{2}, \cdots, A_{k}$ satisfy $\bigcup_{i=1}^{k} A_{i}=S_{n}$, then the unordered set group $\left(A_{1}, A_{2}, \cdots, A_{k}\right)$ is called a “$k$-stage partition” of set $S_{n}$. Therefore, the number of 2-stage partitions of $S_{n}$ is $\qquad$. | \frac{1}{2}\left(3^{n}+1\right) |
62,821 | 4th Junior Balkan 2001 Problem 1 Find all positive integers a, b, c such that a 3 + b 3 + c 3 = 2001. | 10,10,1 |
62,824 | 88. Given that $x, y, z$ are three distinct non-zero natural numbers, if
$$
\overline{x y y y y}+\overline{x y y y}+\overline{x y y}+\overline{x y}+y=\overline{y y y y z} .
$$
where $\overline{x y y y y}$ and $\overline{y y y y z}$ are both five-digit numbers, when $x+y$ is maximized, the corresponding value of $x+y+z$ is | 22 |
62,825 | Problem 11.3. Let $k_{1}$ be the smallest natural number that is a root of the equation
$$
\sin k^{\circ}=\sin 334 k^{\circ}
$$
(a) (2 points) Find $k_{1}$.
(b) (2 points) Find the smallest root of this equation that is a natural number greater than $k_{1}$. | 40 |
62,837 | 10.216. Two circles, with radii of 4 and 8, intersect at a right angle. Determine the length of their common tangent. | 8 |
62,848 | 5. The altitudes of triangle $A B C$ intersect at a point $H$. Find $\angle A C B$ if it is known that $A B=C H$. Explain your reasoning. | 45 |
62,850 | 2. Solve the equation in natural numbers: $1+x+x^{2}+x^{3}=2^{y}$. | 1,2 |
62,859 | 5. Let $A, B$ be moving points on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$,
and $\overrightarrow{O A} \cdot \overrightarrow{O B}=0, O$ is the origin, then the distance from $O$ to the line $A B$ is $\qquad$ | \frac{}{\sqrt{^{2}+b^{2}}} |
62,861 | 8. A moth made a hole in the carpet in the shape of a rectangle with sides 10 cm and 4 cm. Find the smallest size of a square patch that can cover this hole (the patch covers the hole if all points of the rectangle lie inside the square or on its boundary). | 7\sqrt{2} |
62,894 | 3. An odd six-digit number is called "simply cool" if it consists of digits that are prime numbers, and no two identical digits stand next to each other. How many "simply cool" numbers exist? | 729 |
62,895 | 8. In $\triangle A B C$, $a, b, c$ are the sides opposite to angles $A, B, C$ respectively. Given $a+c=2 b, A-C=\frac{\pi}{3}, \sin B=$ $\qquad$ . | \frac{\sqrt{39}}{8} |
62,921 | ## Problem Statement
Calculate the definite integral:
$$
\int_{0}^{\pi} 2^{4} \cdot \sin ^{2}\left(\frac{x}{2}\right) \cos ^{6}\left(\frac{x}{2}\right) d x
$$ | \frac{5\pi}{8} |
62,950 | Problem 2. (Option 2)
In a box, there are 3 red, 4 gold, and 5 silver stars. Randomly, one star is taken from the box and hung on the Christmas tree. What is the probability that a red star will end up on the top of the tree, there will be no more red stars on the tree, and there will be exactly 3 gold stars, if a total of 6 stars are taken from the box? | \frac{5}{231} |
62,987 | Let $ABC$ be an equilateral triangle and $D$ and $E$ two points on $[AB]$ such that $AD = DE = EB$. Let $F$ be a point on $BC$ such that $CF = AD$. Find the value of $\widehat{CDF} + \widehat{CEF}$. | 30 |
62,996 | 908*. Does the equation
$$
x^{2}+y^{2}+z^{2}=2 x y z
$$
have a solution in non-negative integers? | (0,0,0) |
62,999 | Karcsinak has 10 identical balls, among which 5 are red, 3 are white, and 2 are green, and he also has two boxes, one of which can hold 4 balls, and the other can hold 6 balls. In how many ways can he place the balls into the two boxes? (The arrangement of the balls within the boxes is not important.) | 11 |
63,012 | Three circles each with a radius of 1 are placed such that each circle touches the other two circles, but none of the circles overlap. What is the exact value of the radius of the smallest circle that will enclose all three circles? | 1+\frac{2}{\sqrt{3}} |
63,036 | ## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow 3} \frac{\sqrt{x+13}-2 \sqrt{x+1}}{\sqrt[3]{x^{2}-9}}
$$ | 0 |
63,042 | Out of seven Easter eggs, three are red. We placed ten eggs into a larger box and six into a smaller box randomly. What is the probability that there is a red egg in both boxes? | \frac{3}{4} |
63,047 | 8. In tetrahedron $ABCD$, the dihedral angle between faces $ABC$ and $BCD$ is $30^{\circ}$, the area of $\triangle ABC$ is $120$, the area of $\triangle BCD$ is $80$, and $BC=10$. Find the volume of the tetrahedron.
(1992 American Competition Problem) | 320 |
63,056 | How many positive integers $n$ are there such that the geometric and harmonic means of $n$ and 2015 are both integers? | 5 |
63,058 | In the equation $2 b x+b=3 c x+c$, both $b$ and $c$ can take any of the values $1,2,3,4,5,6$. In how many cases will the solution of the equation be positive?
---
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 3 |
63,070 | 1. In how many different ways can $n$ women and $n$ men be arranged so that no two neighbors are of the same gender, and this is:
a) in a single row,
b) around a round table with $2n$ seats. | (n-1)!n! |
63,096 | 403*. Solve the equation:
$$
\sqrt[4]{1-x^{2}}+\sqrt[4]{1-x}+\sqrt[4]{1+x}=3
$$ | 0 |
63,124 | 4. The rod $AB$ of a thermometer hanging vertically on a wall has a length of $2r$. The eye of the observer is on a line $l$ which is normal to the plane of the wall and intersects the line $AB$ at a point whose distance from the midpoint of the segment $AB$ is equal to $h(h>r)$. At what distance from the wall should the observer's eye be located so that the angle under which the observer sees the rod is the largest? | \sqrt{^{2}-r^{2}} |
63,125 | 5.6. (Yugoslavia, 83). Find all values of $n \in \mathbf{N}$ that have the following property: if the numbers $n^{3}$ and $n^{4}$ are written next to each other (in the decimal system), then in the resulting record each of the 10 digits $0,1, \ldots, 9$ will appear exactly once. | 18 |
63,136 | 5. For any set $S$, let $|S|$ denote the number of elements in the set, and let $n(S)$ denote the number of subsets of set $S$. If $A, B, C$ are three sets that satisfy the following conditions:
(1) $n(A)+n(B)+n(C)=n(A \cup B \cup C)$;
(2) $|A|=|B|=100$.
Find the minimum value of $|A \cap B \cap C|$. | 97 |
63,139 | 4. Given $x \cdot y \in\left[-\frac{\pi}{4} \cdot \frac{\pi}{4}\right], a \in \mathbf{R}$, and $\left\{\begin{array}{l}x^{3}+\sin x-2 a=0 \\ 4 y^{3}+\frac{1}{2} \sin 2 y-a=0\end{array}\right.$, then $\cos (x+2 y)=$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 1 |
63,162 | 6.058. $\sqrt[3]{\frac{5-x}{x+3}}+\sqrt[7]{\frac{x+3}{5-x}}=2$. | 1 |
63,173 | Cut a $4 \times 4 \times 4$ cube into 64 $1 \times 1 \times 1$ small cubes, then dye 16 of the $1 \times 1 \times 1$ small cubes red, requiring that among any 4 small cubes parallel to any edge, exactly 1 small cube is dyed red. The number of different dyeing methods is $\qquad$ (dyeing methods that are the same after rotation are also considered different dyeing methods). | 576 |
63,225 | Problem 4. Point $O$ is the center of the circumcircle of triangle $ABC$ with sides $AB=8$ and $AC=5$. Find the length of side $BC$ if the length of the vector $\overrightarrow{OA}+3 \overrightarrow{OB}-4 \overrightarrow{OC}$ is 10. | 4 |
63,235 | Shapovalov A.V.
What is the maximum number of colors that can be used to color the cells of an $8 \times 8$ chessboard so that each cell shares a side with at least two cells of the same color? | 16 |
63,249 | 3. Given the point sets
$$
\begin{array}{l}
A=\left\{(x, y) \left\lvert\,(x-3)^{2}+(y-4)^{2} \leqslant\left(\frac{5}{2}\right)^{2}\right.\right\}, \\
B=\left\{(x, y) \left\lvert\,(x-4)^{2}+(y-5)^{2}>\left(\frac{5}{2}\right)^{2}\right.\right\} .
\end{array}
$$
Then the number of integer points (i.e., points with both coordinates as integers) in the point set $A \cap B$ is | 7 |
63,257 | 7. the polynomial $P(x)=x^{3}-2 x^{2}-x+1$ has the three real zeros $a>b>c$. Find the value of the expression
$$
a^{2} b+b^{2} c+c^{2} a
$$
## Solution | 4 |
63,267 | 2. If the function $f(x)=\frac{(\sqrt{1008} x+\sqrt{1009})^{2}+\sin 2018 x}{2016 x^{2}+2018}$ has a maximum value of $M$ and a minimum value of $m$, then
$M+m=$ $\qquad$ | 1 |
63,270 |
3. A finite non-empty set $S$ of integers is called 3 -good if the the sum of the elements of $S$ is divisble by 3 . Find the number of 3 -good non-empty subsets of $\{0,1,2, \ldots, 9\}$.
| 351 |
63,284 | $\left.\begin{array}{l}{[\text { Inscribed quadrilateral with perpendicular diagonals ] }} \\ {[\quad \text { Pythagorean Theorem (direct and inverse). }}\end{array}\right]$
A circle with radius 2 is circumscribed around a quadrilateral $ABCD$ with perpendicular diagonals $AC$ and $BD$. Find the side $CD$, if $AB=3$. | \sqrt{7} |
63,287 | 8 Nine consecutive positive integers are arranged in ascending order to form a sequence $a_{1}, a_{2}, \cdots, a_{9}$. If the value of $a_{1}+a_{3}+a_{5}+a_{7}+a_{9}$ is a perfect square, and the value of $a_{2}+a_{4}+a_{6}+a_{8}$ is a perfect cube, then the minimum value of the sum of these nine positive integers is $\qquad$ . | 18000 |
63,313 | 4. Determine all natural numbers $n$ for which:
$$
[\sqrt[3]{1}]+[\sqrt[3]{2}]+\ldots+[\sqrt[3]{n}]=2 n
$$ | 33 |
63,330 | 7. Given $5 \sin 2 \alpha=\sin 2^{\circ}$, then the value of $\frac{\tan \left(\alpha+1^{\circ}\right)}{\tan \left(\alpha-1^{\circ}\right)}$ is | -\frac{3}{2} |
63,360 | A sports club has 80 members and three sections: football, swimming, and chess. We know that
a) there is no athlete who plays chess but does not swim;
b) the football players who do not swim all play chess;
c) the combined number of members in the chess and football sections is the same as the number of members in the swimming section;
d) every member is part of at least two sections.
How many members are in all three sections at the same time? | 0 |
63,365 | 12. From 30 people with distinct ages, select two groups, the first group consisting of 12 people, and the second group consisting of 15 people, such that the oldest person in the first group is younger than the youngest person in the second group. The number of ways to select these groups is. $\qquad$ . | 4060 |
63,369 | 31. The following rectangle is formed by nine pieces of squares of different sizes. Suppose that each side of the square $\mathrm{E}$ is of length $7 \mathrm{~cm}$. Let the area of the rectangle be $x \mathrm{~cm}^{2}$. Find the value of $x$. | 1056 |
63,380 | Example 2 Divide a circle into $n(n \geqslant 2)$ sectors, each sector is colored with one of $r$ different colors, and it is required that adjacent sectors are colored differently. Question: How many coloring methods are there? | a_{n}=(r-1)(-1)^{n}+(r-1)^{n} |
63,399 | 4. In triangle $A B C$, side $A C$ is greater than $A B$, line $l$ is the bisector of the external angle $C$. A line passing through the midpoint of $A B$ and parallel to $l$ intersects $A C$ at point $E$. Find $C E$, if $A C=7$ and $C B=4$. (The external angle of a triangle is the angle adjacent to the internal angle at the given vertex.)
$(25$ points. $)$ | \frac{11}{2} |
63,428 | For example, there are 18 tickets to be distributed among four classes, A, B, C, and D, with the requirement that Class A gets at least 1 ticket but no more than 5 tickets, Class B gets at least 1 ticket but no more than 6 tickets, Class C gets at least 2 tickets but no more than 7 tickets, and Class D gets at least 4 tickets but no more than 10 tickets. How many different ways of distribution are there? | 140 |
63,430 | 6. Let $A, B$ be points on the graph of the function $f(x)=3-x^{2}$ on either side of the $y$-axis, then the minimum value of the area of the region enclosed by the tangent lines of $f(x)$ at points $A, B$ and the $x$-axis is $\qquad$ . | 8 |
63,443 | ## Task A-1.6.
Determine the smallest natural number $n$ such that half of $n$ is the square of some natural number, a third of $n$ is the cube of some natural number, and a fifth of $n$ is the fifth power of some natural number. | 2^{15}\cdot3^{10}\cdot5^{6} |
63,448 | Example 3 Let $S$ be a subset of $\{1,2, \cdots, 50\}$ with the following property: the sum of any two distinct elements of $S$ is not divisible by 7. What is the maximum number of elements that $S$ can have?
(43rd American High School Mathematics Examination) | 23 |
63,474 | 1. [3] Two circles centered at $O_{1}$ and $O_{2}$ have radii 2 and 3 and are externally tangent at $P$. The common external tangent of the two circles intersects the line $O_{1} O_{2}$ at $Q$. What is the length of $P Q$ ? | 12 |
63,496 | 5. In triangle $A B C$, angle $A$ is $75^{\circ}$, and angle $C$ is $60^{\circ}$. On the extension of side $A C$ beyond point $C$, segment $C D$ is laid off, equal to half of side $A C$. Find angle $B D C$. | 45 |
63,508 | A2 (1-3, Hungary) Given the quadratic equation in $\cos x$: $a \cos ^{2} x+b \cos x+c$ $=0$. Here, $a, b, c$ are known real numbers. Construct a quadratic equation whose roots are $\cos 2 x$. In the case where $a$ $=4, b=2, c=-1$, compare the given equation with the newly constructed equation. | 4\cos^{2}2x+2\cos2x-1=0 |
63,521 | 60. As shown in the figure, a transparent sealed water container consists of a cylinder and a cone, with the base diameter and height of the cylinder both being 12 cm. The container is partially filled with water. When placed upright, the water surface is 11 cm from the top, and when inverted, the water surface is 5 cm from the top. The volume of this container is $\qquad$ cubic centimeters. $(\pi$ is taken as 3.14) | 1695.6 |
63,539 | Let $[x]$ be the integer part of a number $x$, and $\{x\}=x-[x]$. Solve the equation
$$
[x] \cdot\{x\}=1991 x .
$$ | -\frac{1}{1992} |
63,540 | 247. Six examinees. Suppose that during the exam, six students sit on one bench, with aisles on both sides. They finish the exam in a random order and leave immediately. What is the probability that someone will have to disturb one of the remaining five classmates to get to the aisle? | \frac{43}{45} |
63,549 | $2 \cdot 8$ set $A=\left\{z \mid z^{18}=1\right\}, B=\left\{w \mid w^{48}=1\right\}$ are both sets of complex roots of unity, $C=\{z w \mid z \in A, w \in B\}$ is also a set of complex roots of unity, how many elements does set $C$ contain? | 144 |
63,579 | 2. Eight knights are randomly placed on a chessboard (not necessarily on distinct squares). A knight on a given square attacks all the squares that can be reached by moving either (1) two squares up or down followed by one squares left or right, or (2) two squares left or right followed by one square up or down. Find the probability that every square, occupied or not, is attacked by some knight. | 0 |
63,585 | 2. If $\alpha \neq \frac{k \pi}{2}(k \in \mathbf{Z}), T=\frac{\sin \alpha+\tan \alpha}{\cos \alpha+\cot \alpha}$, then $(\quad)$.
A. $T$ is negative
B. $T$ is positive
C. $T$ is non-negative
D. The sign of $T$ is uncertain | B.T=\frac{\sin^{2}\alpha(1+\cos\alpha)}{\cos^{2}\alpha(1+\sin\alpha)}>0 |
63,600 | 7.186. $\log _{12}\left(4^{3 x}+3 x-9\right)=3 x-x \log _{12} 27$. | 3 |
63,608 | 4. Given that $a, b, c$ are not all zero, then the maximum value of $\frac{a b+2 b c}{a^{2}+b^{2}+c^{2}}$ is | \frac{\sqrt{5}}{2} |
63,617 | Ilya Muromets meets the three-headed Zmei Gorynych. And the battle begins. Every minute Ilya cuts off one of Zmei's heads. With a probability of $1 / 4$, two new heads grow in place of the severed one, with a probability of $1 / 3$ only one new head grows, and with a probability of $5 / 12$ - no heads grow. The Zmei is considered defeated if he has no heads left. Find the probability that Ilya will eventually defeat the Zmei. | 1 |
63,636 | (2) On a certain section of railway, there are three stations $A$, $B$, and $C$ in sequence, with $A B=5 \mathrm{~km}$ and $B C=3 \mathrm{~km}$. According to the train schedule, the train is supposed to depart from station $A$ at 8:00, arrive at station $B$ at 8:07 and stop for 1 minute, and arrive at station $C$ at 8:12. In actual operation, assume the train departs from station $A$ on time, stops for 1 minute at station $B$, and travels at a constant speed of $v \mathrm{~km} / \mathrm{h}$ while moving. The absolute value of the difference between the actual arrival time of the train at a station and the scheduled time is called the operation error of the train at that station.
(1) Write down the operation errors of the train at stations $B$ and $C$;
(2) If the sum of the operation errors at stations $B$ and $C$ is required to be no more than 2 minutes, find the range of values for $v$. | [39,\frac{195}{4}] |
63,639 | Tony has an old sticky toy spider that very slowly "crawls" down a wall after being stuck to the wall. In fact, left untouched, the toy spider crawls down at a rate of one inch for every two hours it's left stuck to the wall. One morning, at around $9$ o' clock, Tony sticks the spider to the wall in the living room three feet above the floor. Over the next few mornings, Tony moves the spider up three feet from the point where he finds it. If the wall in the living room is $18$ feet high, after how many days (days after the first day Tony places the spider on the wall) will Tony run out of room to place the spider three feet higher? | 8 |
63,643 | 6・97 Let $x, y, z$ be positive numbers, and $x^{2}+y^{2}+z^{2}=1$, try to find the minimum value of the following expression
$$
S=\frac{x y}{z}+\frac{y z}{x}+\frac{z x}{y} .
$$ | \sqrt{3} |
63,647 | Two tangents are drawn from a point $A$ to a circle with center $O$, touching it at $B$ and $C$. Let $H$ be the orthocenter of triangle $A B C$, given that $\angle B A C=40^{\circ}$, find the value of the angle $\angle H C O$.
 | 40 |
63,650 | ## 4. Magic Table
In each cell of the table in the picture, one of the numbers $11, 22, 33, 44, 55$, 66, 77, 88, and 99 is written. The sums of the three numbers in the cells of each row and each column are equal. If the number 33 is written in the center cell of the table, what is the sum of the numbers written in the four colored cells?
Result: $\quad 198$ | 198 |
63,666 | 3. (3 points) As shown in the right figure, the length of rectangle $A B C D$ is 6 cm, and the width is 2 cm. A line segment $A E$ is drawn through point $A$ to divide the rectangle into two parts, one part being a right triangle, and the other part being a trapezoid. If the area of the trapezoid is 3 times the area of the right triangle, then, the difference between the perimeter of the trapezoid and the perimeter of the right triangle is $\qquad$ cm. | 6 |
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