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54,154 | 2. In a triangle, the length of one median is half longer than the length of the side to which it is drawn. Find the angle between the other two medians. | 90 |
54,158 | 6. Solve the system
$$
\left\{\begin{array}{l}
\operatorname{tg}^{3} x+\operatorname{tg}^{3} y+\operatorname{tg}^{3} z=36 \\
\operatorname{tg}^{2} x+\operatorname{tg}^{2} y+\operatorname{tg}^{2} z=14 \\
\left(\operatorname{tg}^{2} x+\operatorname{tg} y\right)(\operatorname{tg} x+\operatorname{tg} z)(\operatorname{tg} y+\operatorname{tg} z)=60
\end{array}\right.
$$
In the answer, indicate the sum of the minimum and maximum $\operatorname{tgx}$, which are solutions to the system. | 4 |
54,161 | 1. Given the set $M=\{1,2, \cdots, 2020\}$, for any non-empty subset $A$ of $M$, $\lambda_{A}$ is the sum of the largest and smallest numbers in the set $A$. Then the arithmetic mean of all such $\lambda_{A}$ is $\qquad$ | 2021 |
54,183 | 4. $\cos \frac{\pi}{7} \cos \frac{2 \pi}{7} \cos \frac{3 \pi}{7}=$ $\qquad$ | \frac{1}{8} |
54,186 | 4. (7 points) The numbers $a, b, c, d$ belong to the interval $[-6.5,6.5]$. Find the maximum value of the expression $a+2 b+c+2 d-a b-b c-c d-d a$. | 182 |
54,243 | Task 5. (20 points) In the center of a circular field stands a geologists' cabin. From it, 6 straight roads extend, dividing the field into 6 equal sectors. Two geologists set out on a journey from their cabin at a speed of 4 km/h along a road each arbitrarily chooses. Determine the probability that the distance between them after one hour will be at least 6 km.
# | 0.5 |
54,265 | 1. In a computer game, a turtle moves across a grid on the computer screen, which contains 5 columns and 7 rows. Initially, it is located at the bottom-left corner of the screen - on the cell with coordinates $(0,0)$. If the program instructs the turtle to move off the screen, it reappears on the opposite side - for example, taking one step up from the cell $(3,6)$, the turtle will end up in the cell $(3,0)$. Where will the turtle be after executing the following program:
1) 1 step down; 2) 2 steps to the right; 3) 3 steps up; 4) 4 steps to the left; 5) 5 steps down; 6) 6 steps to the right; $\ldots$; 2016) 2016 steps to the left; 2017) 2017 steps down? | (2,6) |
54,269 | Thirty nine nonzero numbers are written in a row. The sum of any two neighbouring numbers is positive, while the sum of all the numbers is negative. Is the product of all these numbers negative or positive? (4 points)
Boris Frenkin | \text{positive} |
54,289 | 2. On the island, there live 7 natives who know mathematics and physics, 6 natives who know physics and chemistry, 3 natives who know chemistry and mathematics, and 4 natives who know physics and biology. In how many ways can a team of three people be formed who together know at least three subjects out of the four? The four subjects are: mathematics, physics, chemistry, and biology.
# | 1080 |
54,334 | 4. In the acute triangle $\triangle A B C$, if
$$
\sin A=2 \sin B \cdot \sin C,
$$
then the minimum value of $\tan A+2 \tan B \cdot \tan C+\tan A \cdot \tan B \cdot \tan C$ is $\qquad$. | 16 |
54,336 | Example 9 Let $f(x)=\left\{\begin{array}{cc}1 & 1 \leqslant x \leqslant 2 \\ x-1 & 2<x \leqslant 3\end{array}\right.$, for any $a(a \in \mathrm{R})$, denote $v(a)=\max \{f(x)-a x \mid x \in[1,3]\}-\min \{f(x)-a x \mid x \in[1,3]\}$, try to draw the graph of $v(a)$, and find the minimum value of $v(a)$. | \frac{1}{2} |
54,341 | 2. If a line segment of length 3 cm is randomly divided into three segments, the probability that these segments can form a triangle is $\qquad$ . | \frac{1}{4} |
54,342 | 4. In $\triangle A B C$, if $\sin A=2 \sin C$, and the three sides $a, b, c$ form a geometric sequence, then the value of $\cos A$ is $\qquad$ . | -\frac{\sqrt{2}}{4} |
54,345 | 1. Determine all five-digit numbers $\overline{a b c d e}$ such that $\overline{a b}, \overline{b c}, \overline{c d}$, and $\overline{d e}$ are perfect squares. Provide a detailed explanation of your answer! | 81649 |
54,358 | Given $f(x)=a x^{2}+b x+c(a, b, c$ are real numbers) and its absolute value on $[-1,1]$ is $\leqslant 1$, find the maximum value of $|a|+|b|+|c|$. | 3 |
54,371 | 3. Given complex numbers $z_{1}, z_{2}$ satisfy $\left(z_{1}-i\right)\left(z_{2}+i\right)=1$, if $\left|z_{1}\right|=\sqrt{2}$, then the range of $\left|z_{2}\right|$ is $\qquad$ | [2-\sqrt{2},2+\sqrt{2}] |
54,378 | 2. Find the area of the triangle if it is known that its medians $C M$ and $B N$ are 6 and 4.5 respectively, and $\angle B K M=45^{\circ}$, where $K$ is the point of intersection of the medians. | 9\sqrt{2} |
54,386 | 13. Given that $[x]$ represents the greatest integer not exceeding $x$, if $[x+0.1]+[x+0.2]+\ldots+[x+0.9]=104, x$'s minimum value is $\qquad$ . | 11.5 |
54,390 | 14. In the city of Meow, where mathematics is everywhere, siblings Milli and Geo face a new challenge: they need to select some different odd numbers so that their sum equals 2023. How many odd numbers can they select at most? $\qquad$ | 43 |
54,401 | 3.076. $\frac{(1-\cos 2 \alpha) \cos \left(45^{\circ}+2 \alpha\right)}{2 \sin ^{2} 2 \alpha-\sin 4 \alpha}$. | -\frac{\sqrt{2}}{4}\tan\alpha |
54,411 | 4. We will call a number $\mathrm{X}$ "25-supporting" if for any 25 real numbers $a_{1}, \ldots, a_{25}$, the sum of which is an integer, there exists at least one for which $\left|a_{i}-\frac{1}{2}\right| \geq X$.
In your answer, specify the largest 25-supporting $X$, rounded to the hundredths according to standard mathematical rules. | 0.02 |
54,454 | Let's determine those two-digit numbers which are 3 less than the sum of the cubes of their digits! | 32 |
54,476 | 7. Car A and Car B start from locations A and B simultaneously and travel towards each other. Car A travels at 40 kilometers per hour, and Car B travels at 60 kilometers per hour. After reaching B and A respectively, they immediately return. On the return trip, Car A's speed increases by half, while Car B's speed remains unchanged. It is known that the distance between the two points where the two cars meet for the second time is 50 kilometers, then the distance between A and B is $\qquad$ kilometers. | \frac{1000}{7} |
54,477 | # Problem 8. (5 points)
In an $8 \times 8$ table, some 23 cells are black, and the rest are white. In each white cell, the sum of the number of black cells on the same row and the number of black cells on the same column is written; nothing is written in the black cells. What is the maximum value that the sum of the numbers in the entire table can take? | 234 |
54,483 | 13. If the expansion of $(a+b)^{n}$ has three consecutive terms whose binomial coefficients form an arithmetic sequence, then the largest three-digit positive integer $n=$ | 959 |
54,491 | 3. A rectangle $A D E C$ is described around a right triangle $A B C$ with legs $A B=5$ and $B C=6$, as shown in the figure. What is the area of $A D E C$? | 30 |
54,494 | Determine the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x, y \in \mathbb{R}$
$$
f\left(x^{2}-y^{2}\right)=(x-y)(f(x)+f(y))
$$ | f(x)=kx |
54,512 | Problem 4. Inside a rectangular grid with a perimeter of 50 cells, a rectangular hole with a perimeter of 32 cells is cut out along the cell boundaries (the hole does not contain any boundary cells). If the figure is cut along all horizontal grid lines, 20 strips 1 cell wide will be obtained. How many strips will be obtained if, instead, it is cut along all vertical grid lines? (A $1 \times 1$ square is also a strip!)
[6 points]
(A. V. Shapovalov) | 21 |
54,515 | G4.3 Given two positive integers $x$ and $y, x y-(x+y)=\operatorname{HCF}(x, y)+\operatorname{LCM}(x, y)$, where $\operatorname{HCF}(x, y)$ and $\operatorname{LCM}(x, y)$ are respectively the greatest common divisor and the least common multiple of $x$ and $y$. If $c$ is the maximum possible value of $x+y$, find $c$. | 10 |
54,521 | 1. Given 2414 cards, on which natural numbers from 1 to 2414 are written (each card has exactly one number, and the numbers do not repeat). It is required to choose two cards such that the sum of the numbers written on them is divisible by 100. In how many ways can this be done? | 29112 |
54,525 | 3. A group of toddlers in a kindergarten has 90 teeth in total. Any two toddlers together do not have more than 9 teeth. What is the minimum number of toddlers that can be in the group? | 23 |
54,540 | If $f$ is a polynomial, and $f(-2)=3$, $f(-1)=-3=f(1)$, $f(2)=6$, and $f(3)=5$, then what is the minimum possible degree of $f$? | 4 |
54,543 | \section*{Problem 3 - 071233}
What are the last two digits of the number \(7^{7^{7^{7}}}-7^{7^{7}}\) ? | 0 |
54,552 | Task B-2.8. A path $2 \mathrm{~m}$ wide and with an area of $36 \mathrm{~m}^{2}$ has been built around a swimming pool in the shape of a regular hexagon. What is the perimeter of the pool? | 18-4\sqrt{3} |
54,576 | 3. Determine all representations of the number 2001 as a sum of 1979 squares of natural numbers. | 2001=2\cdot3^{2}+2\cdot2^{2}+1975\cdot1^{2} |
54,580 | 3. Let $a$ be the decimal part of $\sqrt{3}$, $b$ be the decimal part of $\sqrt{2}$: $\frac{a}{(a-b) b}$ has an integer part of $\qquad$ | 5 |
54,582 | 8,9 [Examples and counterexamples. Constructions]
Ten football teams each played against each other once. As a result, each team ended up with exactly $x$ points.
What is the greatest possible value of $x$? (Win - 3 points, draw - 1 point, loss - 0.)
# | 13 |
54,588 | 8. (10 points) 12 Smurfs are sitting around a round table, each Smurf hates the 2 Smurfs sitting next to him, but does not hate the other 9 Smurfs. Papa Smurf needs to send out a team of 5 Smurfs to rescue Smurfette who was captured by Gargamel, the team cannot include Smurfs who hate each other, then there are $\qquad$ ways to form the team. | 36 |
54,609 | Solve the following equation:
$$
\sqrt{0.12^{0.12 x}+5}+\sqrt{0.12^{0.12 x}+6}=5
$$ | 1.079 |
54,619 | The square of 13 is 169, which has the digit 6 in the tens place. The square of another number has the digit 7 in the tens place. What are the possible values for the digit in the units place of this square? | 6 |
54,627 | 2. Determine all natural numbers that are 33 times larger than the sum of their digits. | 594 |
54,649 | ## Zadatak A-3.4.
Dan je tetraedar kojem je jedan brid duljine 3, a svi ostali duljine 2 .
Odredi obujam tog tetraedra.
| \frac{\sqrt{3}}{2} |
54,655 | 4. (7 points) The numbers $a, b, c, d$ belong to the interval $[-7 ; 7]$. Find the maximum value of the expression $a+2 b+c+2 d-a b-b c-c d-d a$. | 210 |
54,659 | 31. In $\triangle A B C, D C=2 B D, \angle A B C=45^{\circ}$ and $\angle A D C=60^{\circ}$. Find $\angle A C B$ in degrees. | 75 |
54,701 | Task 1. (1 point)
Find the largest three-digit number ABC that is divisible by the two-digit numbers AB and BC. (Different letters do not necessarily represent different digits) | 990 |
54,702 | 17. A five-digit number is divisible by 2025, and the quotient is exactly the sum of the digits of this five-digit number. There are $\qquad$ such five-digit numbers. | 2 |
54,718 | 14. Three different numbers are chosen at random from the list $1,3,5,7,9,11,13,15,17$, 19. The probability that one of them is the mean of the other two is $p$. What is the value of $\frac{120}{p}$ ? | 720 |
54,719 | [Pythagorean Theorem (direct and inverse)] [Theorem on the sum of the squares of the diagonals] Two circles with radii $\sqrt{5}$ and $\sqrt{2}$ intersect at point $A$. The distance between the centers of the circles is 3. A line through point $A$ intersects the circles at points $B$ and $C$ such that $A B=A C$ (point $B$ does not coincide with $C$). Find $A B$. | \frac{6}{\sqrt{5}} |
54,723 | Determine the last (rightmost) three decimal digits of $n$ where:
\[
n=1 \times 3 \times 5 \times 7 \times \ldots \times 2019.
\] | 875 |
54,730 | Task 1. Fill in the squares with numbers from 1 to 5 to make the equation true (each number is used exactly once):
$$
\square+\square=\square \cdot(\square-\square)
$$
It is sufficient to provide one example. | 1+2=3\cdot(5-4) |
54,732 | Find the value of $\cos \frac{2 \pi}{5}+\cos \frac{4 \pi}{5}$. | -\frac{1}{2} |
54,738 | 11. Given $\frac{1}{\sin \theta}+\frac{1}{\cos \theta}=\frac{35}{12}, \theta \in\left(0, \frac{\pi}{2}\right)$. Find $\theta$. | \theta=\arcsin\frac{3}{5} |
54,755 | 2. In square $ABCD$ with side length $1$, there are points $P, Q$ on $AB, AD$ respectively. If the perimeter of $\triangle APQ$ is 2, find $\angle PCQ$.
| 45 |
54,767 | ## Task 5 - 010725
If you place a cube on the table, only 5 of its 6 faces are still visible. Now, three cubes with edge lengths $a_{1}=20 \mathrm{~cm}, a_{2}=10 \mathrm{~cm}$, and $a_{3}=4 \mathrm{~cm}$ are to be stacked on top of each other in size order. The largest cube stands at the bottom on the tabletop. The centers of the cubes are exactly aligned with each other.
How large is the total visible area of all three cubes? | 2464\, |
54,794 | 8. A bag contains 8 white balls and 2 red balls. Each time, one ball is randomly taken out, and then 1 white ball is put back. The probability that all the red balls are exactly taken out by the 4th draw is $\qquad$ .
| 0.0434 |
54,796 | 1. In the expansion of the polynomial $(x-1)^{3}(x+2)^{10}$, the coefficient of $x^{6}$ is $\qquad$ . | -4128 |
54,797 | ## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0}\left(e^{x}+x\right)^{\cos x^{4}}
$$ | 1 |
54,807 | 13. In parallelogram $A B C D, \angle B A D=76^{\circ}$. Side $A D$ has midpoint $P$, and $\angle P B A=52^{\circ}$. Find $\angle P C D$. | 38 |
54,838 | There are $20$ geese numbered $1-20$ standing in a line. The even numbered geese are standing at the front in the order $2,4,\dots,20,$ where $2$ is at the front of the line. Then the odd numbered geese are standing behind them in the order, $1,3,5,\dots ,19,$ where $19$ is at the end of the line. The geese want to rearrange themselves in order, so that they are ordered $1,2,\dots,20$ (1 is at the front), and they do this by successively swapping two adjacent geese. What is the minimum number of swaps required to achieve this formation?
[i]Author: Ray Li[/i] | 55 |
54,839 | In a tournament with 5 teams, there are no ties. In how many ways can the $\frac{5 \cdot 4}{2}=10$ games of the tournament occur such that there is neither a team that won all games nor a team that lost all games? | 544 |
54,915 | 10. Given real numbers $x, y$ satisfy
$$
3|x+1|+2|y-1| \leqslant 6 \text {. }
$$
Then the maximum value of $2 x-3 y$ is $\qquad$ . | 4 |
54,916 | 7. For any real numbers $a, b$, the minimum value of $\max \{|a+b|,|a-b|,|1-b|\}$ is $\qquad$ . | \frac{1}{2} |
54,929 | 10.266. Find the third side of an acute-angled triangle if two of its sides are equal to $a$ and $b$ and it is known that the medians of these sides intersect at a right angle. | \sqrt{\frac{^{2}+b^{2}}{5}} |
54,933 | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0}\left(2-e^{\sin x}\right)^{\operatorname{ctg} \pi x}$ | e^{-\frac{1}{\pi}} |
54,937 | Berlov S.L.
At the alumni meeting, 45 people attended. It turned out that any two of them who had the same number of acquaintances among those present were not acquainted with each other. What is the maximum number of pairs of acquaintances that could have been among those who attended the meeting? | 870 |
54,943 | 2. The minimum value of the function $f(x)=(\sqrt{1+x}+\sqrt{1-x}-3)\left(\sqrt{1-x^{2}}+1\right)$ is $m$, and the maximum value is $M$, then $\frac{M}{m}=$
$\qquad$ . | \frac{3-\sqrt{2}}{2} |
54,949 | For a group of children, it holds that in every trio of children from the group, there is a boy named Adam, and in every quartet, there is a girl named Beata.
How many children can be in such a group at most, and what are their names in that case?
(J. Zhouf)
Hint. If you don't know how to start, consider a specific group of children and check if the given properties hold. | 5 |
54,968 | 5. On a standard graph paper, an angle is drawn (see figure). Find its measure without using measuring instruments. Justify your answer. | 45 |
54,980 | 7. Let the ellipse $\frac{x^{2}}{m^{2}}+\frac{y^{2}}{n^{2}}=1$ pass through the fixed point $P(1,2)$. Then the minimum value of $m+n$ is . $\qquad$ | (1+2^{\frac{2}{3}})^{\frac{3}{2}} |
55,004 | (10) Let the sequence $\left\{a_{n}\right\}$ have the sum of the first $n$ terms $S_{n}$ satisfying: $S_{n}+a_{n}=\frac{n-1}{n(n+1)}, n=1$, $2, \cdots$, then the general term $a_{n}=$ $\qquad$ | a_{n}=\frac{1}{2^{n}}-\frac{1}{n(n+1)} |
55,015 | 3.193.
$$
\frac{2 \cos \left(\frac{\pi}{6}-2 \alpha\right)-\sqrt{3} \sin \left(\frac{5 \pi}{2}-2 \alpha\right)}{\cos \left(\frac{9 \pi}{2}-2 \alpha\right)+2 \cos \left(\frac{\pi}{6}+2 \alpha\right)}=\frac{\tan 2 \alpha}{\sqrt{3}}
$$ | \frac{\tan2\alpha}{\sqrt{3}} |
55,024 | 7. (IMO-10 Problem) Let $[x]$ denote the greatest integer not exceeding $x$. Find the value of $\sum_{k=0}^{\infty}\left[\frac{n+2^{k}}{2^{k+1}}\right]$, where $n$ is any natural number. | n |
55,034 | 45. Calculate the determinant
$$
D=\left|\begin{array}{rrrr}
3 & 0 & 2 & 0 \\
2 & 3 & -1 & 4 \\
0 & 4 & -2 & 3 \\
5 & 2 & 0 & 1
\end{array}\right|
$$ | -54 |
55,038 | Problem 2. On a plane, 100 points are marked. It turns out that on two different lines a and b, there are 40 marked points each. What is the maximum number of marked points that can lie on a line that does not coincide with a and b? | 23 |
55,045 | 3. On the sides $B C$ and $C D$ of the square $A B C D$, points $E$ and $F$ are chosen such that the angle $E A F$ is $45^{\circ}$. The length of the side of the square is 1. Find the perimeter of triangle $C E F$. Justify your solution. | 2 |
55,067 | 4. Given point $P(-2,5)$ lies on the circle $C: x^{2}+y^{2}-2 x-2 y+F=0$, and the line $l: 3 x+4 y+8=0$ intersects the circle at points $A, B$, then $\overrightarrow{A B} \cdot \overrightarrow{B C}=$ $\qquad$ | -32 |
55,078 | Find the smallest integer $n$ such that the "expanded" writing of $(x y-7 x-3 y+21)^{n}$ contains 2012 terms. | 44 |
55,082 | Find all ordered triples $(a,b, c)$ of positive integers which satisfy $5^a + 3^b - 2^c = 32$ | (2, 2, 1) |
55,111 | 7. Real numbers $x, y, z$ satisfy
$$
x+y+z=1 \text {, and } x^{2}+y^{2}+z^{2}=3 \text {. }
$$
Then the range of $x y z$ is $\qquad$ | [-1,\frac{5}{27}] |
55,117 | 2. Three sportsmen called Primus, Secundus and Tertius take part in a race every day. Primus wears the number ' 1 ' on his shirt, Secundus wears ' 2 ' and Tertius wears ' 3 '.
On Saturday Primus wins, Secundus is second and Tertius is third. Using their shirt numbers this result is recorded as ' 123 '.
On Sunday Primus starts the race in the lead with Secundus in second. During Sunday's race Primus and Secundus change places exactly 9 times, Secundus and Tertius change places exactly 10 times while Primus and Tertius change places exactly 11 times.
How will Sunday's result be recorded? | 231 |
55,118 | 2. Given the hyperbola $C_{1}: 2 x^{2}-y^{2}=1$, and the ellipse $C_{2}$ : $4 x^{2}+y^{2}=1$. If $M$ and $N$ are moving points on the hyperbola $C_{1}$ and the ellipse $C_{2}$ respectively, $O$ is the origin, and $O M \perp O N$, then the distance from point $O$ to the line $M N$ is $\qquad$ | \frac{\sqrt{3}}{3} |
55,138 | 6. (10 points) The calculation result of the expression $1!\times 3-2!\times 4+3!\times 5-4!\times 6+\cdots+2009!\times 2011-2010!\times 2012+2011$ ! is $\qquad$ | 1 |
55,142 | The probability of twins being born in Shvambria is $p$, and triplets are not born in Shvambria.
a) Estimate the probability that a Shvambrian met on the street is one of a pair of twins?
b) In a certain Shvambrian family, there are three children. What is the probability that there is a pair of twins among them?
c) In Shvambrian schools, twins are always enrolled in the same class. There are a total of $N$ students in Shvambria.
What is the expected number of pairs of twins among them? | \frac{Np}{p+1} |
55,156 | 12.153. A circle of radius $r$ is inscribed in a sector of radius $R$. Find the perimeter of the sector. | 2R(1+\arcsin\frac{r}{R-r}) |
55,161 | Let's determine the positive integers $a, b$ for which
$$
(\sqrt{30}-\sqrt{18})(3 \sqrt{a}+\sqrt{b})=12
$$ | =2,b=30 |
55,173 | 3. For the tetrahedron $ABCD$, the 6 edge lengths are 7, 13, 18, 27, 36, 41, and it is known that $AB=41$, then $CD=$ $\qquad$ | 13 |
55,189 | 2. In a planar quadrilateral $ABCD$, $AB=\sqrt{3}, AD=DC=CB=1$, the areas of $\triangle ABD$ and $\triangle BCD$ are $S$ and $T$ respectively, then the maximum value of $S^{2}+T^{2}$ is $\qquad$ . | \frac{7}{8} |
55,193 | 13. In a drawer, there are red and blue socks, no more than 1991 in total. If two socks are drawn without replacement, the probability that they are the same color is $\frac{1}{2}$. How many red socks can there be at most in this case? | 990 |
55,203 | Example 8 Color each vertex of a quadrilateral pyramid with one color, and make the endpoints of the same edge have different colors. If only 5 colors are available, how many different coloring methods are there? | 420 |
55,209 | (3) $\cos \frac{\pi}{11}-\cos \frac{2 \pi}{11}+\cos \frac{3 \pi}{11}-\cos \frac{4 \pi}{11}+\cos \frac{5 \pi}{11}=$ $\qquad$ (answer with a number). | \frac{1}{2} |
55,211 | 2. The numbers are $a_{1}=\log _{2}\left(3^{x}-1\right), a_{2}=\log _{4}\left(9^{x}-3^{x+1}+2\right)$ and $a_{3}=\log _{2}\left(3-3^{x}\right)$.
a) Determine all real values of $x$ for which all 3 numbers $a_{1}, a_{2}$, and $a_{3}$ are defined.
b) Determine all real values of $x$ for which $a_{1}+a_{3}=2 a_{2}$. | \log_{3}\frac{5}{2} |
55,226 | 1. A smaller square was cut out from a larger square, one of its sides lying on the side of the original square. The perimeter of the resulting octagon is $40 \%$ greater than the perimeter of the original square. By what percentage is its area less than the area of the original square? | 64 |
55,236 | Example 8. The point $z=x+i y$ describes the segment
$$
x=1, \quad-1 \leqslant y \leqslant 1
$$
What is the length of the line obtained by mapping this segment using the function $w=z^{2}$? | 2\sqrt{2}+\ln(3+2\sqrt{2}) |
55,248 | 4. A regular $n$-gon, where $n$ is an odd number greater than 1. What is the maximum number of vertices that can be painted red so that the center of the $n$-gon does not lie inside the polygon determined by the red vertices?
## 58th Mathematical Competition for High School Students in Slovenia Maribor, April 12, 2014
## Problems for 3rd Year
Solve the problems independently. You have 210 minutes for solving.
The use of notes, literature, or a pocket calculator is not allowed. | \frac{n+1}{2} |
55,255 | [The ratio of the areas of triangles with a common base or common height] [The area of a figure is equal to the sum of the areas of the figures into which it is divided
Points $M$ and $N$ are located on side $A C$ of triangle $A B C$, and points $K$ and $L$ are on side $A B$, such that $A M: M N: N C=1: 3: 1$ and $A K=K L=L B$. It is known that the area of triangle $A B C$ is 1. Find the area of quadrilateral $K L N M$. | \frac{7}{15} |
55,265 | [ Higher degree equations (other).]
Write down the equation of which the root will be the number $\alpha=\frac{1}{2}(\sqrt[3]{5 \sqrt{2}+7}-\sqrt[3]{5 \sqrt{2}-7})$. Write the number $\alpha$ without using radicals. | \alpha=1 |
55,273 | Example 2 Given $m=\frac{\sin x}{\sin (y-z)}, n=\frac{\sin y}{\sin (z-x)}, p=\frac{\sin z}{\sin (x-y)}$, find the value of $m n + n p + p m$. | -1 |
55,285 | 3.1. (13 points) Ani has blue, green, and red paints. She wants to paint a wooden cube so that after painting, the cube has two faces of each color. In how many different ways can she do this? Ways of painting that can be obtained by rotating the cube are considered the same. | 6 |
55,290 | 1.31 Find the largest natural number $k$, such that $3^{k}$ divides $2^{3^{m}}+1$, where $m$ is any natural number.
("Friendship Cup" International Mathematical Competition, 1992) | 2 |
55,300 | 9. If a sequence of numbers, except for the first and last numbers, each number is equal to the sum of the two adjacent numbers, it is called a sequence with oscillatory property, for example $2,3,1,-2,-3, \cdots \cdots$. It is known that in the following sequence, each * represents a number and satisfies the oscillatory property.
$$
1, *, *, *, *, *, *, *, *, *, *, *, *, *, *, *, *, *, *, 1,
$$
Then the sum of the 18 numbers represented by * is $\qquad$ . | 0 |
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