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19,000 | A cylinder with a volume of 9 is inscribed in a cone. The plane of the top base of this cylinder cuts off a frustum from the original cone, with a volume of 63. Find the volume of the original cone. | 64 |
19,022 | Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions? | 28 |
19,044 | 13. [25] Four circles with radii $1,2,3$, and $r$ are externally tangent to one another. Compute $r$. (No proof is necessary.) | \dfrac{6}{23} |
19,075 | ## Task Condition
Find the derivative.
$y=\frac{x^{4}-8 x^{2}}{2\left(x^{2}-4\right)}$ | \dfrac{x(x^4 - 8x^2 + 32)}{(x^2 - 4)^2} |
19,076 | Squares \(ABCD\) and \(DEFG\) have side lengths 1 and \(\frac{1}{3}\), respectively, where \(E\) is on \(\overline{CD}\) and points \(A, D, G\) lie on a line in that order. Line \(CF\) meets line \(AG\) at \(X\). The length \(AX\) can be written as \(\frac{m}{n}\), where \(m, n\) are positive integers and \(\operatorname{gcd}(m, n)=1\). Find \(100m+n\). | 302 |
19,090 | 4. Let $n>1$ be a natural number. An equilateral triangle with side length $n$ is divided by lines parallel to its sides into congruent equilateral smaller triangles with side length 1. The number of smaller triangles that have at least one side on the side of the original triangle is 1 less than the number of all other smaller triangles. Determine all such $n$. | 5 |
19,098 | XX OM - II - Task 2
Find all four-digit numbers in which the thousands digit is equal to the hundreds digit, and the tens digit is equal to the units digit, and which are squares of integers. | 7744 |
19,103 | 21. In $\frac{1 \times 2}{2}, \frac{1 \times 2 \times 3}{2^{2}}, \frac{1 \times 2 \times 3 \times 4}{2^{3}}, \cdots \cdots, \frac{1 \times 2 \times 3 \times \cdots \times 100}{2^{99}}$, the number of terms that simplify to integers is $\qquad$ . | 6 |
19,114 | Given \( n = 1990 \), find the value of
\[
\frac{1}{2^{n}}\left(1 - 3 \binom{n}{2} + 3^{2} \binom{n}{4} - 3^{3} \binom{n}{6} + \cdots + 3^{994} \binom{n}{1988} - 3^{995} \binom{n}{1990}\right).
\] | -\dfrac{1}{2} |
19,117 | 8. In the figure on the right, the line $y=b-x$, where $0<b<4$, intersects the $y$-axis at $P$ and the line $x=4$ at $S$. If the ratio of the area of $\triangle Q R S$ to the area of $\triangle Q O P$ is $9: 25$, determine the value of $b$. | \dfrac{5}{2} |
19,195 | For which real numbers does the following inequality hold?
$$
\sqrt[3]{5 x+2}-\sqrt[3]{x+3} \leq 1
$$ | (-\infty, 5] |
19,206 | The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is $\tfrac{a-\sqrt{2}}{b}$ , where $a$ and $b$ are positive integers. What is $a+b$
[asy] real x=.369; draw((0,0)--(0,1)--(1,1)--(1,0)--cycle); filldraw((0,0)--(0,x)--(x,x)--(x,0)--cycle, gray); filldraw((0,1)--(0,1-x)--(x,1-x)--(x,1)--cycle, gray); filldraw((1,1)--(1,1-x)--(1-x,1-x)--(1-x,1)--cycle, gray); filldraw((1,0)--(1,x)--(1-x,x)--(1-x,0)--cycle, gray); filldraw((.5,.5-x*sqrt(2)/2)--(.5+x*sqrt(2)/2,.5)--(.5,.5+x*sqrt(2)/2)--(.5-x*sqrt(2)/2,.5)--cycle, gray); [/asy] | 11 |
19,214 | A cup is filled with a saline solution of $15\%$ concentration. There are three iron balls of different sizes: large, medium, and small, with their volumes in the ratio 10:5:3. Initially, the small ball is submerged in the cup of saline causing $10\%$ of the saline solution to overflow. Then the small ball is removed. Next, the medium ball is submerged in the cup and then removed. This is followed by submerging and removing the large ball. Finally, pure water is added to the cup until it is full again. What is the concentration of the saline solution in the cup at this point? | 10\% |
19,218 | Zhendarov R.G.
Find all such prime numbers $p$ that the number $p^{2}+11$ has exactly six distinct divisors (including one and the number itself). | 3 |
19,242 | Determine the largest multiple of 36 that has all even and different digits. | 8640 |
19,255 | Example. Find the work of the force
\[
\vec{F}=(x-y) \vec{i}+\vec{j}
\]
when moving along the curve \( L \)
\[
x^{2}+y^{2}=4 \quad(y \geq 0)
\]
from point \( M(2,0) \) to point \( N(-2,0) \). | 2\pi |
19,267 | We are installing new tires on both wheels of a motorcycle. A tire is considered completely worn out if it has run $15000 \mathrm{~km}$ on the rear wheel, or $25000 \mathrm{~km}$ on the front wheel. What is the maximum distance the motorcycle can travel before the tires are completely worn out, if we timely switch the front tire with the rear tire? | 18750 |
19,280 | 9. If $0<a<\sqrt{3} \sin \theta, \theta \in\left[\frac{\pi}{4}, \frac{5 \pi}{6}\right]$, then the minimum value of $f(a, \theta)=\sin ^{3} \theta+\frac{4}{3 a \sin ^{2} \theta-a^{3}}$ is $\qquad$ | 3 |
19,293 | In a scalene triangle, an inscribed circle is drawn, and the points of tangency with the sides are taken as the vertices of a second triangle. In this second triangle, another inscribed circle is drawn, and the points of tangency are taken as the vertices of a third triangle; this process continues indefinitely. Prove that there are no two similar triangles in the resulting sequence. | \text{No two triangles in the sequence are similar.} |
19,375 | Using the numbers $2, 4, 12, 40$ each exactly once, you can perform operations to obtain 24. | 24 |
19,380 | 330. Find $y^{\prime}$, if $y=\sqrt{x}+\cos ^{2} 3 x$. | \frac{1}{2\sqrt{x}} - 3 \sin 6x |
19,393 | Three people, A, B, and C, are taking an elevator from the 1st floor to the 3rd to 7th floors of a mall. Each floor can accommodate at most 2 people getting off the elevator. How many ways are there for them to get off the elevator? | 120 |
19,406 | 3a
Given 100 numbers $a_{1}, a_{2}$,
$a_{1}-4 a_{2}+3 a_{3} \geq 0$,
$a_{2}-4 a_{3}+3 a_{4} \geq 0$,
$a_{3}-4 a_{4}+3 a_{5} \geq 0$,
$\ldots$,
$a_{99}-4 a_{100}+3 a_{1} \geq 0$,
Given 100 numbers $a_{1}, a_{2}, a_{3}, \ldots, a_{100}$, satisfying the conditions:
$a_{100}-4 a_{1}+3 a_{2} \geq 0$
It is known that $a_{1}=1$, determine $a_{2}, a_{3}, \ldots, a_{100}$. | 1 |
19,411 | Write in the form of a fraction the number
$$
x=0,512341234123412341234123412341234 \ldots
$$ | \dfrac{51229}{99990} |
19,424 | How many pairs of positive integers $(x, y)$ are there such that
$$
\frac{x y}{x+y}=144 ?
$$ | 45 |
19,425 | 4. (8 points) There is a magical tree with 60 fruits on it. On the first day, 1 fruit will fall. Starting from the second day, the number of fruits that fall each day is 1 more than the previous day. However, if the number of fruits on the tree is less than the number that should fall on a certain day, then on that day it will start over by dropping 1 fruit, and continue according to the original rule. So, on the $\qquad$th day, all the fruits on the tree will be gone. | 14 |
19,429 | 25 chess players are participating in a tournament. Each of them has different levels of strength, and in each match, the stronger player always wins.
What is the minimum number of matches required to determine the two strongest players? | 28 |
19,436 | In an equilateral triangle \(ABC\), two points \(M\) and \(N\) are chosen inside the triangle. Can the 6 segments \(AM, BM, CM, AN, BN, CN\) always form a tetrahedron? | \text{No} |
19,467 | Suppose \( S = \{1,2, \cdots, 2005\} \). Find the minimum value of \( n \) such that every subset of \( S \) consisting of \( n \) pairwise coprime numbers contains at least one prime number. | 16 |
19,471 | G1.2 Among 50 school teams joining the HKMO, no team answered all four questions correctly in the paper of a group event. If the first question was solved by 45 teams, the second by 40 teams, the third by 35 teams and the fourth by 30 teams. How many teams solved both the third and the fourth questions? | 15 |
19,472 | 17. As shown in the figure, it is known that the three vertices of $\triangle A B C$ are on the ellipse $\frac{x^{2}}{12}+\frac{y^{2}}{4}=1$, and the coordinate origin $O$ is the centroid of $\triangle A B C$. Try to find the area of $\triangle A B C$. | 9 |
19,490 | 11. (3b,9-11) In the conditions of a chess match, the winner is declared as the one who outperforms the opponent by two wins. Draws do not count. The probabilities of winning for the opponents are equal. The number of decisive games in such a match is a random variable. Find its mathematical expectation. | 4 |
19,503 | 143. Find the general solution of the equation $y^{\prime \prime}-6 y^{\prime}+9 y=0$. | y = (C_1 + C_2 x) e^{3x} |
19,548 | 19. Given a sphere of radius $R$. At a distance equal to $2R$ from the center of the sphere, a point $S$ is taken, and from it, all lines tangent to the sphere (i.e., having exactly one common point with it) are drawn. What does the union of these tangents represent? Calculate the area of the surface formed by the segments of the tangents from point $S$ to the points of tangency. | \dfrac{3}{2} \pi R^2 |
19,581 | Two circles touch each other internally at point K. The chord \(A B\) of the larger circle is tangent to the smaller circle at point \(L\), and \(A L = 10\). Find \(B L\) if \(A K: B K = 2: 5\). | 25 |
19,582 | 5. Given the hyperbola $C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)$ with left and right foci $F_{1}, F_{2}$, and eccentricity $\frac{5}{3}$, if a line $l$ passing through $F_{1}$ is tangent to the circle $x^{2}+y^{2}=a^{2}$ at point $T$, and $l$ intersects the right branch of the hyperbola $C$ at point $P$, then $\frac{\left|\overrightarrow{F_{1} P}\right|}{\left|\overrightarrow{F_{1} T}\right|}=$ $\qquad$. | 4 |
19,615 | Task B-2.5. If $x_{1}$ and $x_{2}$ are the solutions of the equation $x^{2}+2013 x+1=0$, and $y_{1}$ and $y_{2}$ are the solutions of the equation $x^{2}+2014 x+1=0$, calculate the value of the expression $\left(x_{1}-y_{1}\right)\left(x_{2}-y_{2}\right)\left(x_{1}-y_{2}\right)\left(x_{2}-y_{1}\right)$. | 1 |
19,618 | 2. Given are sets $A, B$, and $C$ such that:
- $A \cup B \cup C=\{1,2,3, \ldots, 100\}$;
- $A$ is the set of all natural numbers not greater than 100 that are divisible by 2;
- $B$ is the set of all natural numbers not greater than 100 that are divisible by 3;
- $B \cap C$ is the set of all natural numbers not greater than 100 whose sum of digits is equal to 9;
- $(A \cap C) \backslash B$ is the set of all two-digit numbers that are not divisible by 3 and whose unit digit is equal to 4.
Determine the number of elements in the set $C$. | 49 |
19,653 | The $ABCD$ square-shaped field, measuring 19.36 hectares, has a point $E$ on the $AD$ side, 110 meters from $D$, to which a straight path leads from $B$. Antal and Béla are having a race. Antal starts from point $A$, and Béla from point $B$, both starting at the same time and running at a constant speed towards the common goal, point $D$. Béla reaches $D$ via the segments $BE + ED$. When Béla reaches $E$, Antal has a 30 m lead. Who wins the race and by how many meters? | 8 |
19,677 | 1.44 The increasing sequence $2,3,5,6,7,10,11, \cdots$, contains all positive integers that are neither perfect squares nor perfect cubes. Find the 500th term of this sequence. | 528 |
19,678 | 74. 30 students from five courses came up with 40 problems for the olympiad, with students from the same course coming up with the same number of problems, and students from different courses coming up with a different number of problems. How many students came up with exactly one problem? | 26 |
19,690 | In the country of Lakes, there are seven lakes connected by ten non-intersecting canals, and you can travel from any lake to any other. How many islands are there in this country? | 4 |
19,693 | Find the smallest positive integer \( n \) such that for any positive integer \( k \geqslant n \), in the set \( M = \{1, 2, \cdots, k\} \), for any \( x \in M \), there always exists another number \( y \in M \) (with \( y \neq x \)) such that \( x + y \) is a perfect square. | 7 |
19,708 | Example 3.2.2 Find the minimum value of $n$ that satisfies $\sin x_{1}+\sin x_{2}+\cdots+\sin x_{n}=0$, $\sin x_{1}+2 \sin x_{2}+\cdots+n \sin x_{n}=100$. | 20 |
19,731 | Let \( n, k \in \mathbf{N}^{+} \), and let \( S \) be a set of \( n \) points on a plane (no three points are collinear). For any point \( P \) in \( S \), there exist \( k \) points in \( S \) that are equidistant from \( P \). Prove: \( k \leq \frac{1}{2} + \sqrt{2n} \). | \frac{1}{2} + \sqrt{2n} |
19,735 | 14. (9) In parallelogram $A B C D$, angle $\angle B A C$ is twice the angle $\angle B D C$. Find the area of the parallelogram, given that $A B=$ $A C=2$. | 2\sqrt{3} |
19,737 | Example 2.44. Investigate the conditional and absolute convergence of the improper integral
$$
I=\int_{0}^{2}\left[2 x \sin \left(\frac{\pi}{x^{2}}\right)-\frac{2 \pi}{x} \cos \left(\frac{\pi}{x^{2}}\right)\right] d x
$$ | 2\sqrt{2} |
19,741 | Problem 4.3. Zhenya drew a square with a side of 3 cm, and then erased one of these sides. A figure in the shape of the letter "P" was obtained. The teacher asked Zhenya to place dots along this letter "P", starting from the edge, so that the next dot was 1 cm away from the previous one, as shown in the picture, and then count how many dots he got. He got 10 dots.
Then the teacher decided to complicate the task and asked to count the number of dots, but for the letter "P" obtained in the same way from a square with a side of 10 cm. How many dots will Zhenya have this time? | 31 |
19,762 | There is a box containing 3 red balls and 3 white balls, which are identical in size and shape. A fair die is rolled, and the number rolled determines the number of balls drawn from the box. What is the probability that the number of red balls drawn is greater than the number of white balls drawn? | \dfrac{19}{60} |
19,792 | 7. By the property of absolute value, replacing $x$ with $-x$ does not change this relation. This means that the figure defined by the given inequality is symmetric with respect to the OY axis. Therefore, it is sufficient to find the area of half of the figure for $x \geq 0$. In this case, we obtain the inequality $\left|x-2 y^{2}\right|+x+2 y^{2} \leq 8-4 y$. By removing the absolute value sign, we get two regions: 1: $\left\{\begin{array}{c}x \geq 2 y^{2} \\ x-2 y^{2}+x+2 y^{2} \leq 8-4 y\end{array} \Leftrightarrow\left\{\begin{array}{l}x \geq 2 y^{2} \\ y \leq 2-\frac{x}{2}\end{array}\right.\right.$
Region II: $\left\{\begin{array}{c}x \leq 2 y^{2} \\ -x+2 y^{2}+x+2 y^{2} \leq 8-4 y\end{array} \Leftrightarrow\left\{\begin{array}{c}x \leq 2 y^{2} \\ y^{2}+y-2 \leq 0\end{array} \Leftrightarrow\left\{\begin{array}{c}x \leq 2 y^{2} \\ -2 \leq y \leq 1\end{array}\right.\right.\right.$
Next, on the coordinate plane xOy, we plot the graphs of the obtained inequalities for $x \geq 0$, taking into account that $x=2 y^{2}$ is the graph of a parabola with its vertex at the origin, and the branches of this parabola are directed along the positive direction of the Ox axis. The union of these regions gives a figure which is a trapezoid $M N A B$, the area of which is equal to $S=\frac{1}{2} * 3(8+2)=15$. Then the doubled area is 30 (see Fig. 2). | 30 |
19,803 | 27. Given positive integers $a, b$ satisfy $\sqrt{2020 a}+\sqrt{2020 b}-\sqrt{2020 a b}=a \sqrt{b}+b \sqrt{a}-2020$, then the minimum value of $\sqrt{a+b}$ is $\qquad$ | 11 |
19,809 | Write in the form of a fraction (if possible) the number:
$$
x=0,5123412341234123412341234123412341234 \ldots
$$
Can you generalize this method to all real numbers with a periodic decimal expansion? And conversely | \dfrac{51229}{99990} |
19,813 | Prove that:
$$
\cos \frac{2 \pi}{2 n+1}+\cos \frac{4 \pi}{2 n+1}+\cdots+\cos \frac{2 n \pi}{2 n+1}=-\frac{1}{2}.
$$ | -\dfrac{1}{2} |
19,819 | 14. A square-based pyramid has a base side length of $\sqrt{3}$ and all the edges of the lateral faces are $\sqrt{2}$. How many degrees does the angle between two edges not belonging to the same lateral face measure? | 120 |
19,831 | Prove that
$$
-\frac{1}{2} \leq \frac{(x+y)(1-xy)}{(1+x^2)(1+y^2)} \leq \frac{1}{2}
$$
for all real numbers $x$ and $y$. | -\frac{1}{2} \leq \frac{(x+y)(1-xy)}{(1+x^2)(1+y^2)} \leq \frac{1}{2} |
19,845 | 6. If $\frac{z-1}{z+1}(z \in \mathbf{C})$ is a pure imaginary number, then the minimum value of $\left|z^{2}-z+2\right|$ is | \dfrac{\sqrt{14}}{4} |
19,885 | 4. The segments $\overline{A C}$ and $\overline{B D}$ intersect at point $O$. The perimeter of triangle $A B C$ is equal to the perimeter of triangle $A B D$, and the perimeter of triangle $A C D$ is equal to the perimeter of triangle $B C D$. Find the length of segment $\overline{A O}$, if $\overline{B O}=10$ cm. | 10 |
19,909 | Fomin C.B.
Kolya and Vasya received 20 grades each in January, and Kolya received as many fives as Vasya received fours, as many fours as Vasya received threes, as many threes as Vasya received twos, and as many twos as Vasya received fives. Moreover, their average grade for January is the same. How many twos did Kolya receive in January? | 5 |
19,914 | Let \( n \) be a natural number greater than or equal to 2. Prove that if there exists a positive integer \( b \) such that \( \frac{b^n - 1}{b - 1} \) is a power of a prime number, then \( n \) must be a prime number. | n \text{ must be a prime number} |
19,972 | In trapezoid \(ABCD\), the sides \(AB\) and \(CD\) are parallel and \(CD = 2AB\). Points \(P\) and \(Q\) are chosen on sides \(AD\) and \(BC\), respectively, such that \(DP : PA = 2\) and \(BQ : QC = 3 : 4\). Find the ratio of the areas of quadrilaterals \(ABQP\) and \(CDPQ\). | \dfrac{19}{44} |
19,974 | Given a regular $n$-sided polygon $A_{1} A_{2} \cdots A_{n}$ inscribed in the unit circle, show that for any point $P$ on the circumference, the inequality $\sum_{k=1}^{n}\left|P A_{k}\right|>n$ holds. | \sum_{k=1}^{n}\left|P A_{k}\right| > n |
20,021 | Grandma told her grandchildren: "Today I am 60 years and 50 months and 40 weeks and 30 days old."
How old was Grandma on her last birthday? | 65 |
20,022 | Let $n$ be an integer. We consider $s (n)$, the sum of the $2001$ powers of $n$ with the exponents $0$ to $2000$. So $s (n) = \sum_{k=0}^{2000}n ^k$ . What is the unit digit of $s (n)$ in the decimal system? | 1 |
20,090 | [ З адасаии на движение]
Two ferries simultaneously depart from opposite banks of a river and cross it perpendicularly to the banks. The speeds of the ferries are constant but not equal. The ferries meet at a distance of 720 m from one bank, after which they continue their journey. On the return trip, they meet 400 m from the other bank. What is the width of the river? | 1760 |
20,109 | ## Task Condition
Find the derivative of the specified order.
$$
y=\left(1-x-x^{2}\right) e^{\frac{x-1}{2}}, y^{IV}=?
$$ | -\frac{x^{2} +17x +55}{16} e^{\frac{x -1}{2}} |
20,117 | 1. Find all integers $n$ for which $0 \leqslant n \leqslant 90$ and
$$
\sin 80^{\circ}+\sin 50^{\circ}-\sin 20^{\circ}=\sqrt{2} \sin n^{\circ}
$$ | 85 |
20,129 | Prove: $\sqrt{n}=2^{n-1} \prod_{k=1}^{n-1} \sin \frac{k \pi}{2 n}(n \in \mathbf{N})$. | \sqrt{n} = 2^{n-1} \prod_{k=1}^{n-1} \sin \frac{k \pi}{2 n} |
20,202 | An infinite arithmetic progression contains a square. Prove it contains infinitely many squares. | \text{The arithmetic progression contains infinitely many squares.} |
20,212 | 17. How many ordered pairs $(x, y)$ of positive integers, where $x<y$, satisfy the equation
$$
\frac{1}{x}+\frac{1}{y}=\frac{1}{2007} .
$$ | 7 |
20,227 | 18. In the Seven-Star Country, each citizen has an ID code, which is a seven-digit number composed of the digits $1 \sim 7$ without repetition. The ID code of Citizen 1 is 1234567, the ID code of Citizen 2 is 1234576, the ID code of Citizen 3 is $1234657, \cdots \cdots$, and the citizen numbers and ID codes increase sequentially. What is the ID code of Citizen 2520? $\qquad$ | 4376521 |
20,262 | ## Task 26/76
We are looking for all quadruples $\left(p_{1} ; p_{2} ; p_{3} ; p_{4}\right)$ of prime numbers that satisfy the system of equations:
$$
p_{1}^{2}+p_{2}^{2}=p_{3} \quad(1) \quad ; \quad p_{1}^{2}-p_{2}^{2}=p_{4}
$$ | (3, 2, 13, 5) |
20,272 | 2. (7 points) Provide an example of a number $x$ for which the equality $\sin 2017 x - \operatorname{tg} 2016 x = \cos 2015 x$ holds. Justify your answer. | \dfrac{\pi}{4} |
20,296 | 4. Among the positive integers less than 20, each time three numbers are taken without repetition, so that their sum is divisible by 3. Then the number of different ways to do this is $\qquad$ . | 327 |
20,312 | ## Task 2 - 010932
Kurt is riding a tram along a long straight street. Suddenly, he sees his friend walking in the opposite direction on this street at the same level. After one minute, the tram stops. Kurt gets off and runs after his friend at twice the speed of his friend, but only at a quarter of the average speed of the tram.
How many minutes will it take for him to catch up? How did you arrive at your result? | 9 |
20,359 | 22. Suppose $A$ and $B$ are two angles such that
$$
\sin A+\sin B=1 \text { and } \cos A+\cos B=0 \text {. }
$$
Find the value of $12 \cos 2 A+4 \cos 2 B$. | 8 |
20,365 | 11. (10 points) There are 20 piles of stones, each containing 2006 stones. The rule is: taking one stone from each of any 19 piles and placing them into another pile is considered one operation. After fewer than 20 such operations, one pile has 1990 stones, and another pile has between 2080 and 2100 stones. How many stones are in this pile? | 2090 |
20,369 | # Task 1. (10 points)
How many natural numbers $n$ not exceeding 2017 exist such that the quadratic trinomial $x^{2}+x-n$ can be factored into linear factors with integer coefficients
# | 44 |
20,373 | Given that \( \cos A + \cos B + \cos C = \sin A + \sin B + \sin C = 0 \), find the value of \( \cos^4 A + \cos^4 B + \cos^4 C \). | \dfrac{9}{8} |
20,376 | Two natural numbers $\mathrm{a}, \mathrm{b}$ have a least common multiple equal to 50. How many possible values of $\mathrm{a}+\mathrm{b}$ are there? | 8 |
20,377 | Triangle A is contained within a convex polygon B. Let $\mathrm{S}(\mathrm{A})$ and $\mathrm{S}(\mathrm{B})$ be the areas, and $\mathrm{P}(\mathrm{A})$ and $\mathrm{P}(\mathrm{B})$ be the perimeters of these figures.
Prove that
$$
\frac{S(A)}{S(B)} < \frac{P(A)}{P(B)}
$$ | \frac{S(A)}{S(B)} < \frac{P(A)}{P(B)} |
20,378 | 11. The selling price of a commodity is an integer number of yuan. 100 yuan can buy at most 3 pieces. Person A and Person B each brought several hundred-yuan bills. The money A brought can buy at most 7 pieces of this commodity, and the money B brought can buy at most 14 pieces. When the two people's money is combined, they can buy 1 more piece. The price of each piece of this commodity is yuan. | 27 |
20,383 | Example 9 (1999 National High School League Question) Given $\theta=\arctan \frac{5}{12}$, then the principal value of the argument of the complex number $z=\frac{\cos 2 \theta+\mathrm{i} \sin 2 \theta}{239+\mathrm{i}}$ is | \dfrac{\pi}{4} |
20,389 | 1.97 Find the largest integer $n$ such that $\frac{(n-2)^{2}(n+1)}{2 n-1}$ is an integer.
(17th American Invitational Mathematics Examination, 1999) | 14 |
20,399 | 2. how many seven-digit numbers are there for which the product of the digits is equal to $45^{3}$?
## Solution: | 350 |
20,400 | Let
\[\mathbf{A} = \renewcommand{\arraystretch}{1.5} \begin{pmatrix} \frac{\sqrt{3}}{2} & 0 & -\frac{1}{2} \\ 0 & -1 & 0 \\ \frac{1}{2} & 0 & \frac{\sqrt{3}}{2} \end{pmatrix} \renewcommand{\arraystretch}{1}.\]Compute $\mathbf{A}^{2018}.$ | \begin{pmatrix} \frac{1}{2} & 0 & -\frac{\sqrt{3}}{2} \\ 0 & 1 & 0 \\ \frac{\sqrt{3}}{2} & 0 & \frac{1}{2} \end{pmatrix} |
20,407 | 3. If real numbers $x, y$ satisfy $4 x^{2}-4 x y+2 y^{2}=1$, then the sum of the maximum and minimum values of $3 x^{2}+x y+y^{2}$ is $\qquad$ | 3 |
20,433 | 2. In $\triangle A B C$, $A B=4, C A: C B=5: 3$. Try to find $\left(S_{\triangle A B C}\right)_{\max }$. | \dfrac{15}{2} |
20,450 | 4. Suppose $a_{1}=\frac{1}{6}$ and
$$
a_{n}=a_{n-1}-\frac{1}{n}+\frac{2}{n+1}-\frac{1}{n+2}
$$
for $n>1$. Find $a_{100}$. | \dfrac{1}{10302} |
20,452 | 7. Given positive numbers $a, b, c$ satisfying $a^{2}+b^{2}+2 c^{2}=1$. Then the maximum value of $\sqrt{2} a b+2 b c+7 a c$ is $\qquad$ | 2\sqrt{2} |
20,471 | \section*{Problem 2 - 281012}
Antje wants to determine all four-digit natural numbers \(z\) that satisfy the following conditions (1), (2), (3):
(1) The first and second digits of \(z\) are the same.
(2) The third and fourth digits of \(z\) are the same.
(3) The number \(z\) is a perfect square.
Antje wants to solve this problem without using a table of numbers, a calculator, or any other computing device. How can she proceed? | 7744 |
20,489 | 11.6. A sphere passes through all the vertices of one face of a cube and is tangent to all the edges of the opposite face of the cube. Find the ratio of the volume of the sphere to the volume of the cube. | \dfrac{41\sqrt{41}}{384}\pi |
20,499 | The Houson Association of Mathematics Educators decides to hold a grand forum on mathematics education and invites a number of politicians from the United States to participate. Around lunch time the politicians decide to play a game. In this game, players can score 19 points for pegging the coordinator of the gathering with a spit ball, 9 points for downing an entire cup of the forum's interpretation of coffee, or 8 points for quoting more than three consecutive words from the speech Senator Bobbo delivered before lunch. What is the product of the two greatest scores that a player cannot score in this game? | 1209 |
20,514 | 1. What is the maximum number of sides a convex polygon can have if all its diagonals are of equal length? | 5 |
20,520 | A circle is circumscribed around a square with side length $a$, and a regular hexagon is circumscribed around the circle. Determine the area of the hexagon. | \sqrt{3} a^2 |
20,533 | 1. (16 points) After walking one-fifth of the way from home to school, Olya realized she had forgotten her notebook. If she does not go back for it, she will arrive at school 6 minutes before the bell, but if she returns, she will be 2 minutes late. How much time (in minutes) does the journey to school take? | 20 |
20,574 | 2.077. $\frac{a^{-1}-b^{-1}}{a^{-3}+b^{-3}}: \frac{a^{2} b^{2}}{(a+b)^{2}-3 a b} \cdot\left(\frac{a^{2}-b^{2}}{a b}\right)^{-1} ; \quad a=1-\sqrt{2} ; b=1+\sqrt{2}$. | \dfrac{1}{4} |
20,584 | The TV show "The Mystery of Santa Barbara" features 20 characters. In each episode, one of the following events occurs: a certain character learns the Mystery, a certain character learns that someone knows the Mystery, or a certain character learns that someone does not know the Mystery. What is the maximum number of episodes the series can continue? | 780 |
20,593 | A circle with radius 10 is tangent to two adjacent sides $AB$ and $AD$ of the square $ABCD$. On the other two sides, the circle intersects, cutting off segments of 4 cm and 2 cm from the vertices, respectively. Find the length of the segment that the circle cuts off from vertex $B$ at the point of tangency. | 8 |
20,607 | In triangle \(ABC\) with side \(BC\) equal to 9, an incircle is drawn, touching side \(BC\) at point \(D\). It is known that \(AD = DC\) and the cosine of angle \(BCA\) equals \(\frac{2}{3}\). Find \(AC\).
Apply the Law of Cosines. | 4 |
20,658 | ## 1. Jaja
Baka Mara has four hens. The first hen lays one egg every day. The second hen lays one egg every other day. The third hen lays one egg every third day. The fourth hen lays one egg every fourth day. If on January 1, 2023, all four hens laid one egg, how many eggs in total will Baka Mara's hens lay throughout the entire year of 2023?
## Result: $\quad 762$ | 762 |
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