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13,265 | One end of a bus route is at Station $A$ and the other end is at Station $B$. The bus company has the following rules:
(1) Each bus must complete a one-way trip within 50 minutes (including the stopping time at intermediate stations), and it stops for 10 minutes when reaching either end.
(2) A bus departs from both Station $A$ and Station $B$ every 6 minutes. Determine the minimum number of buses required for this bus route. | 20 |
13,266 | Triangle $ABC$ has $AB=25$ , $AC=29$ , and $BC=36$ . Additionally, $\Omega$ and $\omega$ are the circumcircle and incircle of $\triangle ABC$ . Point $D$ is situated on $\Omega$ such that $AD$ is a diameter of $\Omega$ , and line $AD$ intersects $\omega$ in two distinct points $X$ and $Y$ . Compute $XY^2$ .
*Proposed by David Altizio* | 252 |
13,286 | Task 3. (15 points) An educational center "Young Geologist" received an object for research consisting of about 300 monoliths (a container designed for 300 monoliths, which was almost completely filled). Each monolith has a specific name (sandy loam or clayey loam) and genesis (marine or lake-glacial deposits). The relative frequency (statistical probability) that a randomly selected monolith will be sandy loam is $\frac{1}{8}$. The relative frequency that a randomly selected monolith will be marine clayey loam is $\frac{22}{37}$. How many monoliths of lake-glacial genesis does the object contain, if there are no marine sandy loams among the sandy loams? | 120 |
13,300 | 3. Let $O$ be the circumcenter of acute $\triangle A B C$, with $A B=6, A C=10$. If $\overrightarrow{A O}=x \overrightarrow{A B}+y \overrightarrow{A C}$, and $2 x+10 y=5$, then $\cos \angle B A C=$ $\qquad$ . | \dfrac{1}{3} |
13,302 | Example 7 Find the range of real number $a$ for which the equation $\sqrt{a+\sqrt{a+\sin x}}=\sin x$ has real solutions.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | [-\frac{1}{4}, 0] |
13,314 | Solve the following equation:
$$
\sqrt{\frac{x-1991}{10}}+\sqrt{\frac{x-1990}{11}}=\sqrt{\frac{x-10}{1991}}+\sqrt{\frac{x-11}{1990}} .
$$ | 2001 |
13,326 | 1. Milla and Zhena came up with a number each and wrote down all the natural divisors of their numbers on the board. Milla wrote down 10 numbers, Zhena wrote down 9 numbers, and the largest number written on the board twice is 50. How many different numbers are written on the board? | 13 |
13,428 | # 6. CONDITION
Vasya has three cans of paint of different colors. In how many different ways can he paint a fence consisting of 10 planks so that any two adjacent planks are of different colors and he uses all three colors of paint? Justify your answer. | 1530 |
13,442 | What is the maximum area of a triangle if none of its side lengths exceed 2? | \sqrt{3} |
13,449 | As shown, \(U\) and \(C\) are points on the sides of triangle \(MN H\) such that \(MU = s\), \(UN = 6\), \(NC = 20\), \(CH = s\), and \(HM = 25\). If triangle \(UNC\) and quadrilateral \(MUCH\) have equal areas, what is \(s\)? | 4 |
13,450 | ## Problem Statement
Calculate the definite integral:
$$
\int_{\pi / 2}^{2 \pi} 2^{8} \cdot \cos ^{8} x d x
$$ | 105\pi |
13,460 | # 1. Option 1.
A confectionery factory received 5 rolls of ribbon, each 50 m long, for packaging cakes. How many cuts need to be made to get pieces of ribbon 2 m long? | 120 |
13,462 | A regular hexagon \( A B C D E K \) is inscribed in a circle of radius \( 3 + 2\sqrt{3} \). Find the radius of the circle inscribed in the triangle \( B C D \). | \dfrac{3}{2} |
13,504 | # Task № 3.4
## Condition:
Anton makes watches for a jewelry store on order. Each watch consists of a bracelet, a precious stone, and a clasp.
The bracelet can be silver, gold, or steel. Anton has precious stones: zircon, emerald, quartz, diamond, and agate, and clasps: classic, butterfly, and buckle. Anton is only satisfied when three watches are laid out in a row on the display according to the following rules:
- There must be steel watches with a classic clasp and a zircon stone
- Next to the watches with the classic clasp, there must be gold and silver watches;
- The three watches in a row must have different bracelets, precious stones, and clasps.
How many ways are there to make Anton happy? | 48 |
13,529 | At McDonald's restaurants, we can order Chicken McNuggets in packages of 6, 9, or 20 pieces. (For example, we can order 21 pieces because $21=6+6+9$, but there is no way to get 19 pieces.) What is the largest number of pieces that we cannot order? | 43 |
13,534 | For which values of \( n \) is the number \( P_n = 36^n + 24^n - 7^n - 5^n \) divisible by 899? | n \text{ is even} |
13,549 | Problem 100. The bisectors $A L_{1}$ and $B L_{2}$ of triangle $A B C$ intersect at point I. It is known that $A I: I L_{1}=3 ; B I: I L_{2}=2$. Find the ratio of the sides in triangle $A B C$. | 3:4:5 |
13,553 |
It is known that the tangents to the graph of the function \( y = a(x+2)^2 + 2 \) drawn from the point \( M(x_0, y_0) \) intersect at a right angle. Restore the form of the function, given that the coordinates of the point \( M(x_0, y_0) \) satisfy the following relation:
$$
\log_{x - x^2 + 3}(y - 6) = \log_{x - x^2 + 3} \left( \frac{|2x + 6| - |2x + 3|}{3x + 7.5} \sqrt{x^2 + 5x + 6.25} \right)
$$ | y = -\dfrac{1}{20}(x + 2)^2 + 2 |
13,558 | Example 9. Solve the equation
$$
y^{\prime \prime}-6 y^{\prime}+9 y=4 e^{x}-16 e^{3 x}
$$ | y = e^{x} + \left(C_1 + C_2 x - 8x^{2}\right) e^{3x} |
13,577 | In the cells of a $3 \times 3$ square, the numbers $0,1,2, \ldots, 8$ are placed. It is known that any two consecutive numbers are located in neighboring (side-adjacent) cells. What number can be in the central cell if the sum of the numbers in the corner cells is 18? | 2 |
13,579 | 8.5. On the sides $B C$ and $C D$ of the square $A B C D$, points $M$ and $K$ are marked respectively such that $\angle B A M = \angle C K M = 30^{\circ}$. Find $\angle A K D$. | 75^\circ |
13,596 | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0} \frac{e^{\sin 2 x}-e^{\sin x}}{\tan x}$ | 1 |
13,598 | 40th Putnam 1979 Problem A1 Find the set of positive integers with sum 1979 and maximum possible product. | 2 \times 3^{659} |
13,607 | ## Task 2 - 030712
In the Peace Race in 1963, an individual time trial was held between Bautzen and Dresden (57 km).
The riders started at intervals of 1 minute. Immediately before the eventual overall winner Klaus Ampler (GDR), his toughest rival Vyncke (Belgium) started. While Ampler covered an average of 42 km per hour, Vyncke achieved a "pace" of 40 km per hour.
At what time and after how many kilometers would Ampler have caught up with the Belgian rider if both had ridden at a constant speed? Justify your answer! | 14 |
13,623 | Two disks of radius 1 are drawn so that each disk's circumference passes through the center of the other disk. What is the circumference of the region in which they overlap? | \dfrac{4\pi}{3} |
13,687 | 26*. What kind of mapping is the composition of two homotheties in space? If this mapping is a homothety, what will be its coefficient? How to determine the position of its center? | k_1 k_2 |
13,690 | 8. Let $f(x)=x^{2}+b x+c$, set $A=\{x \mid f(x)=x\}, B=\{x \mid f(x-1)=$ $x+1\}$, if $A=\{2\}$, find set $B$. | \{3 + \sqrt{2}, 3 - \sqrt{2}\} |
13,706 | The real root of the equation $8x^3-3x^2-3x-1=0$ can be written in the form $\frac{\sqrt[3]{a}+\sqrt[3]{b}+1}{c}$ , where $a$ , $b$ , and $c$ are positive integers. Find $a+b+c$ .
Please give the answer directly without any intermediate steps. | 98 |
13,719 | 7. (10 points) There are 37 people standing in a line to count off. The first person reports 1, and each subsequent person reports a number that is 3 more than the previous person's number. During the counting, one person made a mistake and reported a number that was 3 less than the previous person's number. The sum of the numbers reported by these 37 people is exactly 2011. Then, the person who made the mistake is the $\qquad$th person to report a number. | 34 |
13,726 | 19. Andrey found the product of all numbers from 1 to 11 inclusive and wrote the result on the board. During the break, someone accidentally erased three digits, and the remaining number on the board is $399 * 68 * *$. Help restore the digits without recalculating the product. | 39916800 |
13,757 | 4・203 There are two coal mines, A and B. Coal from mine A releases 4 calories when burned per gram, and coal from mine B releases 6 calories when burned per gram. The price of coal at the origin is: 20 yuan per ton for mine A, and 24 yuan per ton for mine B. It is known that: the transportation cost of coal from mine A to city N is 8 yuan per ton. If coal from mine B is to be transported to city N, what should the transportation cost per ton be to make it more economical than transporting coal from mine A? | 18 |
13,767 | 1. Given that the vertices $A$ and $C$ of $\triangle A B C$ lie on the graph of the inverse proportion function $y=\frac{\sqrt{3}}{x}(x>0)$, $\angle A C B=90^{\circ}$, $\angle A B C=30^{\circ}$, $A B \perp x$-axis, point $B$ is above point $A$, and $A B=6$. Then the coordinates of point $C$ are | \left( \dfrac{\sqrt{3}}{2}, 2 \right) |
13,771 | $\underline{\text { Fomin S. }}$.
Two people toss a coin: one tossed it 10 times, the other - 11 times.
What is the probability that the second person got heads more times than the first? | \dfrac{1}{2} |
13,803 | 2. Determine all positive integers that are equal to 300 times the sum of their digits. | 2700 |
13,826 | There are four different passwords, $A$, $B$, $C$, and $D$, used by an intelligence station. Each week, one of the passwords is used, and each week it is randomly chosen with equal probability from the three passwords not used in the previous week. Given that the password used in the first week is $A$, find the probability that the password used in the seventh week is also $A$ (expressed as a simplified fraction). | \dfrac{61}{243} |
13,829 | What is the smallest four-digit positive integer that is divisible by both 5 and 9 and has only even digits? | 2880 |
13,835 | A rectangular table of size \( x \) cm \( \times 80 \) cm is covered with identical sheets of paper of size 5 cm \( \times 8 \) cm. The first sheet is placed in the bottom-left corner, and each subsequent sheet is placed 1 cm higher and 1 cm to the right of the previous one. The last sheet is adjacent to the top-right corner. What is the length \( x \) in centimeters? | 77 |
13,836 | 5. (7 points) The king decided to test his hundred sages and announced that the next day he would line them up with their eyes blindfolded and put a black or white hat on each of them. After their eyes are uncovered, each, starting from the last in line, will name the supposed color of their hat. If he fails to guess correctly, he will be executed. The sages still have time to agree on how they will act tomorrow. How many sages can definitely be saved? | 99 |
13,845 | If the ellipse \( x^{2} + 4(y-a)^{2} = 4 \) and the parabola \( x^{2} = 2y \) have a common point, find the range of the real number \( a \). | [-1, \dfrac{17}{8}] |
13,868 | 7. Four different natural numbers, one of which is an even prime number, the sum of any two is a multiple of 2, the sum of any three is a multiple of 3, and the sum of these four numbers is exactly a multiple of 4. The smallest sum of these 4 numbers is $\qquad$ _. | 44 |
13,872 | 1. Calculate $\left(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots+\frac{1}{99}-\frac{1}{100}\right) \div\left(\frac{1}{51 \times 100}+\frac{1}{52 \times 99}+\cdots+\frac{1}{75 \times 76}\right)$ | 151 |
13,880 | What is the largest integer \( n \) such that
$$
\frac{\sqrt{7}+2 \sqrt{n}}{2 \sqrt{7}-\sqrt{n}}
$$
is an integer? | 343 |
13,913 | ## Task 31/70
Determine all common solutions of the two equations
$$
\begin{array}{r}
3 x^{4}+13 x^{3}+20 x^{2}+17 x+7=0 \\
3 x^{4}+x^{3}-8 x^{2}+11 x-7=0
\end{array}
$$
without using an approximation method! | -\dfrac{7}{3} |
13,914 | On segment \(AC\) there is a point \(B\), with \(AB = 14 \, \text{cm}\) and \(BC = 28 \, \text{cm}\). On segments \(AB\), \(BC\), and \(AC\), semicircles are constructed in one half-plane relative to the boundary \(AB\). Find the radius of the circle that is tangent to all three semicircles. | 6 |
13,933 | A tetrahedron has the following properties. There is a sphere, center X, which touches each edge of the tetrahedron; there are four spheres with centers at the vertices of the tetrahedron which each touch each other externally; and there is another sphere, center X, which touches all four spheres. Prove that the tetrahedron is regular. | \text{The tetrahedron is regular.} |
14,011 | ## Task B-4.6.
Let $f_{1}(x)=\frac{1}{2-x}, f_{n}(x)=\left(f_{1} \circ f_{n-1}\right)(x), n \geqslant 2$, for all real numbers $x$ for which the given functions are defined. What is $f_{2021}(4)$? | \dfrac{6059}{6062} |
14,033 | ## Problem Statement
Calculate the definite integral:
$$
\int_{\arcsin (2 / \sqrt{5})}^{\arcsin (3 / \sqrt{10})} \frac{2 \tan x + 5}{(5 - \tan x) \sin 2x} \, dx
$$ | \ln \left( \dfrac{9}{4} \right) |
14,066 | 3. On a chessboard, $8 \times 8$, there are 63 coins of 5 denari each and one coin of 10 denari, placed such that there is exactly one coin in each square. There are enough 5, 10, and 20 denari coins available. The following exchanges of three coins on the board with other three coins are possible:
$$
\begin{array}{ll}
(5,5,5) \leftrightarrow(10,10,10), & (5,5,10) \leftrightarrow(5,10,20) \leftrightarrow(20,20,10) \\
(5,10,10) \leftrightarrow(10,10,20), & (5,20,20) \leftrightarrow(5,5,20) \leftrightarrow(20,20,20)
\end{array}
$$
where the order does not matter.
Is it possible, after a finite number of exchanges on the board, to have 60 coins of 10 denari, 3 coins of 20 denari, and 1 coin of 5 denari? | \text{No} |
14,091 | Let \( f(x) = \left\{ \begin{array}{cc} 1 & 1 \leqslant x \leqslant 2 \\ x-1 & 2 < x \leqslant 3 \end{array} \right. \). For any \( a \,(a \in \mathbb{R}) \), define \( v(a) = \max \{ f(x) - a x \mid x \in [1,3] \} - \min \{ f(x) - a x \mid x \in [1,3] \} \). Draw the graph of \( v(a) \) and find the minimum value of \( v(a) \). | \dfrac{1}{2} |
14,110 | G4.1 $x_{1}=2001$. When $n>1, x_{n}=\frac{n}{x_{n-1}}$. Given that $x_{1} x_{2} x_{3} \ldots x_{10}=a$, find the value of $a$. | 3840 |
14,139 | Two people are tossing a coin: one tossed it 10 times, and the other tossed it 11 times.
What is the probability that the second person's coin landed on heads more times than the first person's coin? | \dfrac{1}{2} |
14,183 | ## Task 2
A class estimates the length of a path on the schoolyard to be $28 \mathrm{~m}$. Two boys measure this path. They lay a measuring tape of $20 \mathrm{~m}$ length once and then measure another $12 \mathrm{~m}$.
By how many meters did the students overestimate? | 4 |
14,187 | 2B. Find $x$ for which the function $y=\left(x-x_{1}\right)^{2}+\left(x-x_{2}\right)^{2}+\ldots+\left(x-x_{1996}\right)^{2}$ takes the smallest value. | \dfrac{x_1 + x_2 + \ldots + x_{1996}}{1996} |
14,202 | Let $(a,b)=(a_n,a_{n+1}),\forall n\in\mathbb{N}$ all be positive interger solutions that satisfies $$ 1\leq a\leq b $$ and $$ \dfrac{a^2+b^2+a+b+1}{ab}\in\mathbb{N} $$ And the value of $a_n$ is **only** determined by the following recurrence relation: $ a_{n+2} = pa_{n+1} + qa_n + r$
Find $(p,q,r)$ . | (5, -1, -1) |
14,224 | 【Question 7】
Subtract 101011 from 10000000000, and the digit 9 appears $\qquad$ times in the resulting answer. | 7 |
14,241 | Let \( A, B, C, D, E, F \) be 6 points on a circle in that order. Let \( X \) be the intersection of \( AD \) and \( BE \), \( Y \) is the intersection of \( AD \) and \( CF \), and \( Z \) is the intersection of \( CF \) and \( BE \). \( X \) lies on segments \( BZ \) and \( AY \) and \( Y \) lies on segment \( CZ \). Given that \( AX = 3 \), \( BX = 2 \), \( CY = 4 \), \( DY = 10 \), \( EZ = 16 \), and \( FZ = 12 \), find the perimeter of triangle \( XYZ \). | \dfrac{77}{6} |
14,262 | In a $4 \times 4$ table, numbers are written such that the sum of the neighbors of each number is equal to 1 (neighboring cells share a common side).
Find the sum of all the numbers in the table. | 6 |
14,282 | 16. Given that the function $f(x)$ is monotonically increasing on $[2,+\infty)$, and for any real number $x$ it always holds that $f(2+x)=f(2-x)$, if $f\left(1-2 x^{2}\right)<f\left(1+2 x-x^{2}\right)$, then the range of values for $x$ is | (-2, 0) |
14,357 | Let's calculate the sum $S_{n}=1 \cdot 2^{2}+2 \cdot 3^{2}+3 \cdot 4^{2}+\ldots+n(n+1)^{2}$. | \dfrac{n(n+1)(n+2)(3n+5)}{12} |
14,358 | ## Problem Statement
Calculate the indefinite integral:
$$
\int \frac{x^{2}+\ln x^{2}}{x} d x
$$ | \frac{x^{2}}{2} + (\ln x)^2 + C |
14,363 | Variant 2.
Café "Buratino" operates 6 days a week with a day off on Mondays. Kolya made two statements: "from April 1 to April 20, the café was open for 18 days" and "from April 10 to April 30, the café was also open for 18 days." It is known that he was wrong once. How many days was the café open from April 1 to April 13? | 11 |
14,369 | Example 1. Reduce the general equations of a line to canonical form
\[
\left\{\begin{array}{l}
2 x-3 y-3 z-9=0 \\
x-2 y+z+3=0
\end{array}\right.
\] | \frac{x}{9} = \frac{y}{5} = \frac{z + 3}{1} |
14,375 | Example 8 Find the range of the function $f(x)=2 x+\sqrt{1+x-x^{2}}$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | [1 - \sqrt{5}, \dfrac{7}{2}] |
14,376 | Given two parabolas $\Gamma_{1}$ and $\Gamma_{2}$ on the Cartesian plane, both with a leading coefficient of 1, and two non-parallel lines $l_{1}$ and $l_{2}$. If the segments intercepted by $l_{1}$ on $\Gamma_{1}$ and $\Gamma_{2}$ are equal in length, and the segments intercepted by $l_{2}$ on $\Gamma_{1}$ and $\Gamma_{2}$ are also equal in length, prove that the parabolas $\Gamma_{1}$ and $\Gamma_{2}$ coincide. | \Gamma_{1} \text{ and } \Gamma_{2} \text{ coincide} |
14,379 | 3. Let $n=\left(p^{2}-1\right)\left(p^{2}-4\right)+9$. What is the smallest possible sum of the digits of the number $n$, if $p$ is a prime number? For which prime numbers $p$ is this sum achieved? | 9 |
14,426 | In a six-digit decimal number $\overline{a_{1} a_{2} a_{3} a_{4} a_{5} a_{6}}$, each digit $a_{i}(1 \leqslant i \leqslant 6)$ is an odd number, and the digit 1 is not allowed to appear consecutively (for example, 135131 and 577797 satisfy the conditions, while 311533 does not satisfy the conditions). Find the total number of such six-digit numbers. $\qquad$ . | 13056 |
14,432 | 12.219. When a circular sector rotates about one of its extreme radii, a body is formed whose spherical surface area is equal to the area of the conical surface. Find the sine of the central angle of the circular sector. | \dfrac{4}{5} |
14,462 | 60. For the National Day, hang colorful flags in the order of "4 red flags, 3 yellow flags, 2 blue flags" in a repeating sequence, a total of 50 flags are hung. Among them, there are $\qquad$ red flags. | 24 |
14,470 | 4. 1 class has 6 boys, 4 girls, and now 3 class leaders need to be selected from them, requiring that there is at least 1 girl among the class leaders, and each person has 1 role, then there are $\qquad$ ways to select. | 600 |
14,471 | 1、Calculate: $20140601=13 \times\left(1000000+13397 \times \_\right.$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
1、Calculate: $20140601=13 \times\left(1000000+13397 \times \_\right.$ | 41 |
14,488 | The odd function \( f(x) \) is increasing within its domain \((-1,1)\). Given that \( f(1-m) + f\left(m^2 - 1\right) < 0 \), find the range of the real number \( m \). | (0, 1) |
14,536 | 55. The area of triangle $D E F$ is 7, $A D=D B, B E=2 E C, C F=3 F A$, then the area of triangle $A B C$ is $\qquad$ . | 24 |
14,544 | Determine the number of ordered pairs of integers \((m, n)\) for which \(m n \geq 0\) and \(m^{3}+n^{3}+99 m n=33^{3}\). | 35 |
14,555 | Let \( T \) be the set of all positive divisors of \( 2020^{100} \). The set \( S \) satisfies:
1. \( S \) is a subset of \( T \);
2. No element in \( S \) is a multiple of another element in \( S \).
Find the maximum number of elements in \( S \). | 10201 |
14,566 | Misha and Masha had the same multi-digit whole number written in their notebooks, ending in 9876. Masha placed a plus sign between the third and fourth digits, counting from the right, while Misha did the same between the fourth and fifth digits, also counting from the right. To the students' surprise, both resulting sums turned out to be the same. What was the original number the students had written down? Provide all possible answers and prove that there are no others. | 9999876 |
14,593 | 4. Photographs are archived in the order of their numbering in identical albums, with exactly 4 photographs per page. In this case, the 81st photograph in sequence landed on the 5th page of one of the albums, and the 171st photograph landed on the 3rd page of another. How many photographs can each album hold?
# | 32 |
14,601 | A set of several numbers, none of which are the same, has the following property: the arithmetic mean of some two numbers from this set is equal to the arithmetic mean of some three numbers from the set and is equal to the arithmetic mean of some four numbers from the set. What is the smallest possible number of numbers in such a set?
# | 5 |
14,633 | 2. The integer solution to the equation $(\lg x)^{\lg (\lg x)}=10000$ is $x=$ | 10^{100} |
14,636 | Prove the following identity:
$$
\operatorname{tg} x+2 \operatorname{tg} 2 x+4 \operatorname{tg} 4 x+8 \operatorname{ctg} 8 x=\operatorname{ctg} x
$$ | \operatorname{ctg} x |
14,642 | 30. Detective Conan wrote two two-digit numbers in his notebook. He found that $\frac{3}{5}$ of one number equals $\frac{1}{3}$ of the other number. The maximum difference between these two numbers is $\qquad$ . | 44 |
14,646 | [ Angles subtended by equal arcs and equal chords]
## Law of Sines
The lengths of three sides of a cyclic quadrilateral inscribed in a circle of radius $2 \sqrt{2}$ are equal and each is 2. Find the fourth side. | 5 |
14,663 | [ Thales' Theorem and the Theorem of Proportional Segments ] [ Orthogonal (Rectangular) Projection $\quad$ ]
Given points $A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right)$ and a non-negative number $\lambda$. Find the coordinates of point $M$ on the ray $A B$, for which $A M: A B=\lambda$. | \left( (1 - \lambda)x_1 + \lambda x_2, (1 - \lambda)y_1 + \lambda y_2 \right) |
14,685 | 1. In the field of real numbers, solve the equation
$$
\sqrt{x+3}+\sqrt{x}=p
$$
with the unknown $x$ and the real parameter $p$. | \left( \dfrac{p^2 - 3}{2p} \right)^2 |
14,686 | 41. The Dozing Schoolboy. A schoolboy, waking up at the end of an algebra lesson, heard only a fragment of the teacher's phrase: "… I will only say that all the roots are real and positive." Glancing at the board, he saw there a 20th-degree equation assigned as homework, and tried to quickly write it down. He managed to write down only the first two terms $x^{20}-20 x^{19}$, before the teacher erased the board; however, he remembered that the constant term was +1. Could you help our unfortunate hero solve this equation? | 1 |
14,692 | ## Problem Statement
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty} \frac{n \sqrt[5]{n}-\sqrt[3]{27 n^{6}+n^{2}}}{(n+\sqrt[4]{n}) \sqrt{9+n^{2}}}
$$ | -3 |
14,715 | 7. (10 points) A toy store sells a type of building blocks: each starship costs 8 yuan, and each mech costs 26 yuan; one starship and one mech can be combined to form an ultimate mech, which is sold for 33 yuan per set. If the store owner sold a total of 31 starships and mechs in a week, and the total revenue was 370 yuan; then how many starships were sold individually? | 20 |
14,726 | Given a triangle \( \triangle ABC \) with interior angles \( \angle A, \angle B, \angle C \) and opposite sides \( a, b, c \) respectively, where \( \angle A - \angle C = \frac{\pi}{2} \) and \( a, b, c \) are in arithmetic progression, find the value of \( \cos B \). | \dfrac{3}{4} |
14,763 | [ Touching Circles ]
[
Find the ratio of the radii of two circles touching each other, if each of them touches the sides of an angle equal to $\alpha$.
# | \dfrac{1 - \sin(\alpha/2)}{1 + \sin(\alpha/2)} |
14,772 | ROMN is a rectangle with vertices in that order and RO = 11, OM = 5. The triangle ABC has circumcenter O and its altitudes intersect at R. M is the midpoint of BC, and AN is the altitude from A to BC. What is the length of BC? | 28 |
14,797 | 5. Given $f(x)=\sqrt{\frac{1-x}{1+x}}$. If $\alpha \in\left(\frac{\pi}{2}, \pi\right)$, then $f(\cos \alpha)+f(-\cos \alpha)$ can be simplified to | 2 \csc \alpha |
14,815 | 1. Task: Compare the numbers $\left(\frac{2}{3}\right)^{2016}$ and $\left(\frac{4}{3}\right)^{-1580}$. | \left(\frac{2}{3}\right)^{2016} < \left(\frac{4}{3}\right)^{-1580} |
14,826 | The function \( y = \cos x + \sin x + \cos x \sin x \) has a maximum value of \(\quad\). | \frac{1}{2} + \sqrt{2} |
14,850 | ## 42. Barrels
In how many ways can a 10-liter barrel be emptied using two containers with capacities of 1 liter and 2 liters? | 89 |
14,864 | ## Task 36/72
We are looking for all four-digit prime numbers with the following properties:
1. All digits in the decimal representation are different from each other.
2. If the number is split in the middle into two two-digit numbers, both numbers are prime numbers, each with a digit sum of 10.
3. The last two digits are also prime numbers, each on their own. | 1973 |
14,865 | 64. Let the four vertices of a regular tetrahedron be $A B C D$, with each edge length being 1 meter. A small insect starts from point $A$ and moves according to the following rule: at each vertex, it randomly selects one of the three edges passing through that vertex and crawls all the way to the end of that edge. What is the probability that after crawling 7 meters, it is exactly at vertex $A$? $\qquad$ | \dfrac{182}{729} |
14,875 | 9. A wire of length 1 is cut into three pieces. Then the probability that these three pieces can exactly form a triangle is | \dfrac{1}{4} |
14,911 | The sum of the digits of all counting numbers less than 13 is
$$
1+2+3+4+5+6+7+8+9+1+0+1+1+1+2=51
$$
Find the sum of the digits of all counting numbers less than 1000. | 13500 |
14,916 | The Wolf with the Three Little Pigs wrote a detective novel "Three Little Pigs-2," and then, together with Little Red Riding Hood and her grandmother, a cookbook "Little Red Riding Hood-2." The publishing house paid the royalties for both books to the pig Naf-Naf. He took his share and handed over the remaining 2100 gold coins to the Wolf. The royalties for each book are divided equally among its authors. How much money should the Wolf take for himself? | 700 |
14,922 | 6-102 Let the cube root of $m$ be a number of the form $n+r$, where $n$ is a positive integer, and $r$ is a positive real number less than $\frac{1}{1000}$. When $m$ is the smallest positive integer satisfying the above condition, find the value of $n$.
---
The translation is provided as requested, maintaining the original format and line breaks. | 19 |
14,935 | In the plane Cartesian coordinate system \(xOy\), point \(P\) is a moving point on the line \(y = -x - 2\). Two tangents to the parabola \(y = \frac{x^2}{2}\) are drawn through point \(P\), and the points of tangency are \(A\) and \(B\). Find the minimum area of the triangle \(PAB\). | 3\sqrt{3} |
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