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6,743 | Let $x, y, z$ be positive real numbers such that $x^{3} y^{2} z=1$. What is the minimum value that $x+2 y+3 z$ can take? | 2 \sqrt[3]{9} |
6,747 | Example 14 How many different numbers must be selected from $1,2,3, \cdots, 1000$ to ensure that there are 3 different numbers among the selected ones that can form the lengths of the three sides of a triangle? | 16 |
6,765 | Booin D.A.
The whole family drank a full cup of coffee with milk, and Katya drank a quarter of all the milk and a sixth of all the coffee. How many people are in the family? | 5 |
6,769 | The sequence \(\left\{a_{n}\right\}\) satisfies: \(a_1 = 1\), and for each \(n \in \mathbf{N}^{*}\), \(a_n\) and \(a_{n+1}\) are the roots of the equation \(x^2 + 3nx + b_n = 0\). Find \(\sum_{k=1}^{20} b_k\). | 6385 |
6,773 | The cells of an $n \times n$ table are filled with the numbers $1,2,\dots,n$ for the first row, $n+1,n+2,\dots,2n$ for the second, and so on until $n^2-n,n^2-n+1,\dots,n^2$ for the $n$-th row. Peter picks $n$ numbers from this table such that no two of them lie on the same row or column. Peter then calculates the sum $S$ of the numbers he has chosen. Prove that Peter always gets the same number for $S$, no matter how he chooses his $n$ numbers. | \dfrac{n(n^2 + 1)}{2} |
6,786 | Fill the integers $1, 2, 3, 4, 5, 6, 7, 8, 9$ into a $3 \times 3$ grid, using each integer exactly once, so that the sum of the three numbers in each row and each column is odd. Determine the total number of such possible configurations. | 25920 |
6,787 | Solve the following equation in the set of real numbers:
$$
\sqrt{x_{1}-1^{2}}+2 \sqrt{x_{2}-2^{2}}+\cdots+n \sqrt{x_{n}-n^{2}}=\frac{x_{1}+x_{2}+\cdots+x_{n}}{2}
$$ | x_k = 2k^2 |
6,789 | 3 [ Completing the square. Sums of squares ]
Find all real solutions of the equation with 4 unknowns: $x^{2}+y^{2}+z^{2}+t^{2}=x(y+z+t)$. | (0, 0, 0, 0) |
6,846 | ## Problem Statement
Calculate the indefinite integral:
$$
\int \frac{x^{3}+x+2}{(x+2) x^{3}} d x
$$ | \ln|x + 2| - \frac{1}{2x^{2}} + C |
6,848 | Shapovalov A.V.
A bag of seeds was passed around the table. The first person took 1 seed, the second took 2, the third took 3, and so on: each subsequent person took one more seed than the previous one. It is known that in the second round, the total number of seeds taken was 100 more than in the first round. How many people were sitting at the table? | 10 |
6,874 | Find the maximum natural number which is divisible by 30 and has exactly 30 different positive divisors. | 11250 |
6,893 | 552. Find all four-digit numbers that are 9 times greater than their reversals. | 9801 |
6,900 | 4. In a deck of 52 cards, each person makes one cut. A cut consists of taking the top $N$ cards and placing them at the bottom of the deck, without changing their order.
- First, Andrey cut 28 cards,
- then Boris cut 31 cards,
- then Vanya cut 2 cards,
- then Gena cut several cards,
- then Dima cut 21 cards.
The last cut restored the original order. How many cards did Gena cut? | 22 |
6,901 | In a laboratory, the speed at which sound propagates along rods made of different materials is measured. In the first experiment, it turns out that the entire path, consisting of three rods connected sequentially, is traversed by sound in $a$ seconds. The path consisting of the second and third rods is traversed by sound twice as fast as the first rod alone.
In another experiment, the second rod is replaced with a new one, and then the sequential connection of the three rods is traversed by sound in $b$ seconds. The connection of the first and second rods is twice as slow as the third rod alone.
Find the speed of sound in the new rod if its length is $l$ meters. | \dfrac{3l}{2(b - a)} |
6,910 | A smaller square was cut out from a larger square in such a way that one side of the smaller square lies on a 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 the area of the octagon less than the area of the original square? | 64\% |
6,914 | ## Problem Statement
Calculate the volumes of the bodies bounded by the surfaces.
$$
\frac{x^{2}}{9}+\frac{y^{2}}{4}-z^{2}=1, z=0, z=3
$$ | 72\pi |
6,915 | The distance from the point where a diameter of a circle intersects a chord of length 18 cm to the center of the circle is 7 cm. This point divides the chord in the ratio 2:1. Find the radius.
Given:
\[ AB = 18 \, \text{cm}, \, EO = 7 \, \text{cm}, \, AE = 2 \, BE \]
Find the radius \( R \). | 11 |
6,931 | In the tetrahedron $A B C D$, the edge $C D$ is perpendicular to the plane $A B C$. Let $M$ be the midpoint of $D B$, $N$ be the midpoint of $A B$, and $K$ be a point on the edge $C D$ such that $|C K| = \frac{1}{3} |C D|$. Prove that the distance between the lines $B K$ and $C N$ is equal to the distance between the lines $A M$ and $C N$. | \text{The distances are equal.} |
6,962 | Determine the number of integers $n$ such that $1 \leqslant n \leqslant 10^{10}$, and such that for all $k=1,2, \ldots, 10$, the integer $n$ is divisible by $k$. | 3968253 |
7,015 | In the acute triangle $\triangle ABC$, prove that $\sin A + $\sin B + $\sin C + $\tan A + $\tan B + $\tan C > 2\pi$. | \sin A + \sin B + \sin C + \tan A + \tan B + \tan C > 2\pi |
7,023 | A line \( l \) passes through a point \( X \) with barycentric coordinates \( (\alpha: \beta: \gamma) \). Let \( d_{\mathrm{a}}, d_{\mathrm{b}}, d_{\mathrm{c}} \) be the signed distances from vertices \( A, B, C \) to the line \( l \) (points on opposite sides of the line \( l \) have different signs). Prove that \( d_{\mathrm{a}} \alpha + d_{\mathrm{b}} \beta + d_{\mathrm{c}} \gamma = 0 \). | d_{\mathrm{a}} \alpha + d_{\mathrm{b}} \beta + d_{\mathrm{c}} \gamma = 0 |
7,049 | It is known that the variance of each of the given independent random variables does not exceed 4. Determine the number of such variables for which the probability that the deviation of the arithmetic mean of the random variable from the arithmetic mean of their mathematical expectations by no more than 0.25 exceeds 0.99. | 6400 |
7,093 | Each unit square of a 3-by-3 unit-square grid is to be colored either blue or red. For each square, either color is equally likely to be used. The probability of obtaining a grid that does not have a 2-by-2 red square is $\frac {m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ | 929 |
7,159 | 10. If integers $a, b, c$ satisfy: $0 \leqslant a \leqslant 10,0 \leqslant b \leqslant 10,0 \leqslant c \leqslant 10,10 \leqslant a+b+c \leqslant 20$, then the number of ordered triples $(a, b, c)$ that satisfy the condition is $\qquad$ groups. | 891 |
7,180 | $C$ is a point on the extension of diameter $A B, C D$ is a tangent, angle $A D C$ is $110^{\circ}$. Find the angular measure of arc $B D$.
# | 40 |
7,189 | Inside the tetrahedron \(A B C D\), points \(X\) and \(Y\) are given. The distances from point \(X\) to the faces \(A B C\), \(A B D\), \(A C D\), \(B C D\) are \(14, 11, 29, 8\) respectively. The distances from point \(Y\) to the faces \(A B C\), \(A B D\), \(A C D\), \(B C D\) are \(15, 13, 25, 11\) respectively. Find the radius of the inscribed sphere of the tetrahedron \(A B C D\). | 17 |
7,214 | ## Problem Statement
Calculate the volumes of the bodies bounded by the surfaces.
$$
\frac{x^{2}}{16}+\frac{y^{2}}{9}+\frac{z^{2}}{100}=1, z=5, z=0
$$ | 55\pi |
7,240 | There are five distinct nonzero natural numbers; the smallest one is 7. If one of them is decreased by 20, and the other four numbers are each increased by 5, the resulting set of numbers remains the same. What is the sum of these five numbers? | 85 |
7,244 | Given that point \( P \) lies on the hyperbola \( \Gamma: \frac{x^{2}}{463^{2}} - \frac{y^{2}}{389^{2}} = 1 \). A line \( l \) passes through point \( P \) and intersects the asymptotes of hyperbola \( \Gamma \) at points \( A \) and \( B \), respectively. If \( P \) is the midpoint of segment \( A B \) and \( O \) is the origin, find the area \( S_{\triangle O A B} = \quad \). | 180107 |
7,279 | 12. On the Cartesian plane, the number of lattice points (i.e., points with both integer coordinates) on the circumference of a circle centered at $(199,0)$ with a radius of 199 is $\qquad$ . | 4 |
7,286 | A rectangular table measures \( x \) cm \(\times\) 80 cm. Identical sheets of paper measuring 5 cm \(\times\) 8 cm are placed on the table. The first sheet is placed at the bottom-left corner of the table, and each subsequent sheet is placed one centimeter higher and one centimeter to the right of the previous sheet. The last sheet is placed at the top-right corner. What is the length \( x \) in centimeters? | 77 |
7,300 | Let \( m, n \) be positive integers with \( m, n \geqslant 2 \). For any set \( A=\{a_{1}, a_{2}, \cdots, a_{n}\} \) of \( n \) integers, consider the differences \( a_{j} - a_{i} \) for each pair of distinct elements \( a_{i}, a_{j} \) with \( j > i \). Arrange these \( \binom{n}{2} \) differences in ascending order to form a sequence, called the "derived sequence" of the set \( A \), denoted as \( \bar{A} \). Let \( \bar{A}(m) \) denote the number of elements in the derived sequence \( \bar{A} \) that are divisible by \( m \).
Prove: For any positive integer \( m \geqslant 2 \), and for any set \( A=\{a_{1}, a_{2}, \cdots, a_{n}\} \) and the set \( B=\{1, 2, \cdots, n\} \), the derived sequences \( \bar{A} \) and \( \bar{B} \) satisfy the inequality \( \bar{A}(m) \geqslant \bar{B}(m) \). | \bar{A}(m) \geqslant \bar{B}(m) |
7,312 | 10,11
Find the dihedral angles of the pyramid $ABCD$, all edges of which are equal to each other.
# | \arccos\left(\frac{1}{3}\right) |
7,313 | 16. What is the largest multiple of 7 less than 10,000 which can be expressed as the sum of squares of three consecutive numbers? | 8750 |
7,314 | 11. Let $a, b, c, d$ be real numbers, find
$$
a^{2}+b^{2}+c^{2}+d^{2}+a b+a c+a d+b c+b d+c d+a+b+c+d
$$
the minimum value. | -\dfrac{2}{5} |
7,325 | Let \( x_{1}, y_{1}, x_{2}, y_{2} \) be real numbers satisfying the equations \( x_{1}^{2}+5 x_{2}^{2}=10 \), \( x_{2} y_{1}-x_{1} y_{2}=5 \) and \( x_{1} y_{1}+5 x_{2} y_{2}=\sqrt{105} \). Find the value of \( y_{1}^{2}+5 y_{2}^{2} \). | 23 |
7,333 | 19. (6 points) There is a strip of paper, on which there are three types of markings, dividing the strip into 6 equal parts, 10 equal parts, and 12 equal parts along its length. Now, if we cut the strip along all the markings with scissors, the strip will be divided into $\qquad$ parts. | 20 |
7,334 | The triangle $\triangle ABC$ is equilateral, and the point $P$ is such that $PA = 3 \, \text{cm}$, $PB = 4 \, \text{cm}$, and $PC = 5 \, \text{cm}$. Calculate the length of the sides of the triangle $\triangle ABC$. | \sqrt{25 + 12\sqrt{3}} |
7,345 | For an arbitrary triangle, prove the inequality (using the usual notations) \(\frac{bc \cos A}{b+c} + a < p < \frac{bc + a^{2}}{a}\). | \frac{bc \cos A}{b+c} + a < p < \frac{bc + a^{2}}{a} |
7,346 | In the trapezium \(ABCD\), the lines \(AB\) and \(DC\) are parallel, \(BC = AD\), \(DC = 2 \times AD\), and \(AB = 3 \times AD\). The angle bisectors of \(\angle DAB\) and \(\angle CBA\) intersect at the point \(E\). What fraction of the area of the trapezium \(ABCD\) is the area of the triangle \(ABE\)? | \dfrac{3}{5} |
7,373 | 20. Let $a_{1}, a_{2}, \ldots$ be a sequence satisfying the condition that $a_{1}=1$ and $a_{n}=10 a_{n-1}-1$ for all $n \geq 2$. Find the minimum $n$ such that $a_{n}>10^{100}$. | 102 |
7,376 | Example 18. Solve the equation
$$
x=(\sqrt{1+x}+1)(\sqrt{10+x}-4)
$$ | -1 |
7,382 | Given vectors \(\overrightarrow{O P}=\left(2 \cos \left(\frac{\pi}{2}+x\right),-1\right)\) and \(\overrightarrow{O Q}=\left(-\sin \left(\frac{\pi}{2}- x\right), \cos 2 x\right)\), and the function \(f(x)=\overrightarrow{O P} \cdot \overrightarrow{O Q}\). If \(a, b, c\) are the sides opposite angles \(A, B, C\) respectively in an acute triangle \( \triangle ABC \), and it is given that \( f(A) = 1 \), \( b + c = 5 + 3 \sqrt{2} \), and \( a = \sqrt{13} \), find the area \( S \) of \( \triangle ABC \). | \dfrac{15}{2} |
7,389 | Find all positive integers $n$, such that $n-1$ and $\frac{n(n+1)}{2}$ are both perfect numbers. | 7 |
7,398 | 1. Let the regular hexagon be $A_{1} A_{2} A_{3} A_{4} A_{5} A_{6}$. Points $P_{1}$ and $P_{2}$ are the midpoints of sides $\overline{A_{4} A_{5}}$ and $\overline{A_{3} A_{4}}$. What is the ratio of the areas of triangle $\Delta P_{1} A_{1} P_{2}$ and the regular hexagon $A_{1} A_{2} A_{3} A_{4} A_{5} A_{6}$? | \dfrac{7}{24} |
7,425 | Determine which digit is in the 1000th place after the decimal point in the decimal expansion of the number $\frac{9}{28}$.
(M. Krejčová)
Hint. What does the decimal expansion of the given number look like? | 4 |
7,470 | Using 4 different colors to paint the 4 faces of a regular tetrahedron (each face is an identical equilateral triangle) so that different faces have different colors, how many different ways are there to paint it? (Coloring methods that remain different even after any rotation of the tetrahedron are considered different.) | 2 |
7,484 | In trapezoid \(ABCD\), the side \(AD\) is perpendicular to the bases and is equal to 9. \(CD\) is 12, and the segment \(AO\), where \(O\) is the point of intersection of the diagonals of the trapezoid, is equal to 6. Find the area of triangle \(BOC\). | \dfrac{108}{5} |
7,491 | ## Task 1 - 050731
What digits does the product end with?
$$
z=345926476^{3} \cdot 125399676^{2} \cdot 2100933776^{3}
$$ | 76 |
7,506 | 3. Given the equation about $x$: $x^{2}+a|x|+a^{2}-3=0(a \in \mathbf{R})$ has a unique real solution, then $a=$ | \sqrt{3} |
7,529 | Let \( k \) be a real number and define the function \( f(x) = \frac{x^{4} + kx^{2} + 1}{x^{4} + x^{2} + 1} \). If for any three real numbers \( a, b, c \) (which can be the same), there exists a triangle with side lengths \( f(a), f(b), f(c) \), then the range of values for \( k \) is ________. | (-\frac{1}{2}, 4) |
7,548 | Given a convex quadrilateral \(ABCD\), \(X\) is the midpoint of the diagonal \(AC\). It is known that \(CD \parallel BX\). Find \(AD\), given that \(BX = 3\), \(BC = 7\), and \(CD = 6\). | 14 |
7,565 | 3. The great commander, Marshal of the Soviet Union, Georgy Konstantinovich Zhukov, was born in the village of Strelkovka, Kaluga Governorate. He lived for 78 years. In the 20th century, he lived 70 years longer than in the 19th century. In what year was G.K. Zhukov born? | 1896 |
7,571 | 83. Even or odd sum of all natural numbers from 1 to 17? | odd |
7,576 | Find the strictly increasing functions $f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*}$ such that $f(2)=2$ and if $n, m \geq 1$,
$$
f(n m)=f(n) f(m)
$$
(MEMO 2014) | f(n) = n |
7,591 | 4. Solve the following equation for positive $x$.
$$
x^{2014} + 2014^{2013} = x^{2013} + 2014^{2014}
$$ | 2014 |
7,604 | In a given trapezoid, a triangle is drawn with a base that coincides with one of the non-parallel sides $A C$ of the trapezoid, and its vertex is located at the midpoint of the side opposite to $A C$. Show that the area of this triangle is half of the area of the trapezoid. | \frac{1}{2} |
7,605 | Example. Let $m, n$ be positive integers, find the minimum value of $\left|12^{m}-5^{n}\right|$. | 7 |
7,628 | Points X, Y, and Z lie on a circle with center O such that XY = 12. Points A and B lie on segment XY such that OA = AZ = ZB = BO = 5. Compute AB. | 2\sqrt{13} |
7,648 | Three, (50 points) Find the largest positive integer $n$, such that for any integer $a$, if $(a, n)=1$, then $a^{2} \equiv 1(\bmod n)$. | 24 |
7,655 | [Theorem on the lengths of a tangent and a secant; the product of the entire secant and its external part Rectangles and squares. Properties and criteria
In a square $A B C D$ with side $a$, a circle is inscribed, touching side $C D$ at point $E$.
Find the chord connecting the points where the circle intersects the line $A E$.
# | \dfrac{2\sqrt{5}}{5}a |
7,657 | Given the sequence elements \( a_{n} \) such that \( a_{1}=1337 \) and \( a_{2n+1}=a_{2n}=n-a_{n} \) for all positive integers \( n \). Determine the value of \( a_{2004} \). | 2004 |
7,668 | In a square room, there is a bulb on each wall that can shine one of the seven colors of the rainbow. There are no bulbs in the room that shine the same color. With each move, a person can change the color of one bulb to a color that no bulb is currently shining, while also changing the colors of the two other bulbs to the remaining two unused colors. (After the move, there still won't be two bulbs with the same color in the room). What is the minimum number of moves required so that eventually, each bulb has shone each of the seven colors? | 8 |
7,680 | Example. Compute the limit
$$
\lim _{x \rightarrow 0}\left(\frac{1+x^{2} 2^{x}}{1+x^{2} 5^{x}}\right)^{1 / \sin ^{3} x}
$$ | \dfrac{2}{5} |
7,688 | \section*{Exercise 4 - 221014}
Let \(r\) be the radius of the circumcircle of a regular decagon \(P_{1} P_{2} \ldots P_{10}\) and \(s\) the length of one side of this decagon.
Calculate \(s\) in terms of \(r\)! | \dfrac{r(\sqrt{5} - 1)}{2} |
7,748 | $3-$ Area of a trapezoid $\quad$ K [ Mean proportionals in a right triangle ]
A circle inscribed in an isosceles trapezoid divides its lateral side into segments equal to 4 and 9. Find the area of the trapezoid. | 156 |
7,772 | Cosine Theorem
A circle can be circumscribed around quadrilateral $ABCD$. In addition, $AB=3, BC=4, CD=5$ and $AD=2$.
Find $AC$. | \sqrt{\dfrac{299}{11}} |
7,778 | # 5.1. Condition:
In the warehouse, there are 8 cabinets, each containing 4 boxes, each with 10 mobile phones. The warehouse, each cabinet, and each box are locked. The manager has been tasked with retrieving 52 mobile phones. What is the minimum number of keys the manager should take with them? | 9 |
7,780 | Given the Fibonacci sequence defined as follows: \( F_{1}=1, F_{2}=1, F_{n+2}=F_{n+1}+F_{n} \) (for \( n \geq 1 \)), find \( \left(F_{2017}, F_{99} F_{101}+1\right) \). | 1 |
7,785 | 3. Given a tetrahedron $S-ABC$ with the base being an isosceles right triangle with hypotenuse $AB$, and $SA=SB=SC=AB=2$. Suppose points $S, A, B, C$ all lie on a sphere with center $O$. Then the surface area of this sphere is $\qquad$ | \dfrac{16}{3}\pi |
7,786 | 187. For which integer values of $x$ will the number $\frac{5 x+2}{17}$ be an integer? | x = 17k + 3 |
7,809 | 1. Points $A$ and $B$ lie on a circle with center $O$ and radius 6, and point $C$ is equidistant from points $A, B$, and $O$. Another circle with center $Q$ and radius 8 is circumscribed around triangle $A C O$. Find $B Q$. Answer: 10 | 10 |
7,828 | 1. Determine all natural numbers $n$ such that using each of the digits $0,1,2,3, \ldots, 9$ exactly once, the numbers $n^{3}$ and $n^{4}$ can be written. | 18 |
7,850 | 5. Garland (1 b. 6-11). A string of 100 lights was hung on a Christmas tree in a row. Then the lights started switching according to the following algorithm: all lights turned on, after a second, every second light turned off, after another second, every third light switched: if it was on, it turned off and vice versa. After another second, every fourth light switched, then every fifth light, and so on. After 100 seconds, everything was over. Find the probability that a randomly chosen light is on after this (the lights do not burn out or break). | \dfrac{1}{10} |
7,854 | ## Task 2 - 030612
A pioneer group set off from the city to a holiday camp by bus at 16:00.
When they had covered nine-tenths of the distance, the pioneers had to get off the bus 2 km before the camp because the bus could no longer travel on the forest road leading to the camp. They needed half an hour for the rest of the journey and arrived at the camp at 17:00.
At what speed did the bus travel? (How many kilometers did it cover in one hour?) | 36 |
7,856 | 10.089. Find the area of the circle circumscribed around an isosceles triangle if the base of this triangle is 24 cm and the lateral side is $13 \mathrm{~cm}$. | \dfrac{28561}{100}\pi |
7,862 | Example 5 Solve the system of inequalities $\left\{\begin{array}{l}x+2<3+2 x, \\ 4 x-3<3 x-1, \\ 8+5 x \geqslant 6 x+7 .\end{array}\right.$ | (-1, 1] |
7,865 | ## 5. Middle Number
Arrange in ascending order all three-digit numbers less than 550 whose hundreds digit is equal to the product of the other two digits. Among these arranged numbers, which number is in the middle?
## Result: $\quad 331$ | 331 |
7,961 | 3. Misha calculated the products $1 \times 2, 2 \times 3$, $3 \times 4, \ldots, 2017 \times 2018$. For how many of them is the last digit zero? | 806 |
7,972 | 10.2. How many solutions in natural numbers $x, y$ does the equation $x+y+2 x y=2023$ have? | 6 |
7,976 |
For how many natural numbers \( n \) not exceeding 600 are the triples of numbers
\[ \left\lfloor \frac{n}{2} \right\rfloor, \left\lfloor \frac{n}{3} \right\rfloor, \left\lfloor \frac{n}{5} \right\rfloor \quad\text{and}\quad \left\lfloor \frac{n+1}{2} \right\rfloor, \left\lfloor \frac{n+1}{3} \right\rfloor, \left\lfloor \frac{n+1}{5} \right\rfloor \]
distinct? As always, \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \( x \). | 440 |
7,989 |
A uniform cubic die with faces numbered $1, 2, 3, 4, 5, 6$ is rolled three times independently, resulting in outcomes $a_1, a_2, a_3$. Find the probability of the event "$|a_1 - a_2| + |a_2 - a_3| + |a_3 - a_1| = 6$". | \dfrac{1}{4} |
8,009 | 8. Finding the minimum force acting on the buoy from the current, knowing the horizontal component of the tension force, is not difficult.
Consider the forces acting on the buoy along the horizontal axis: on one side, there is the force from the current, and on the other side, the horizontal component of the tension force,
$M_{5} * a_{\sigma_{-} x}=T^{*} \sin (a)-F_{\text {current_rop }}$
where $M_{5}$ is the mass of the buoy, $a_x$ is its acceleration along the horizontal axis, and $F_{\text {current_rop }}$ is the force acting on the buoy from the current in the horizontal direction.
Therefore, at the moment the anchor detaches, the equality holds
$F_{\text {current_rop }}=T^{*} \sin (a)$
which means the minimum value of the force acting on the buoy from the current is equal to its horizontal projection and is
$$
F_{\text {current }}=F_{\text {current_hor }}=400[\mathrm{H}]
$$ | 400 |
8,083 | A truck has new tires fitted on all four wheels. A tire is considered completely worn out if it has traveled $15000 \mathrm{~km}$ on the rear wheel or $25000 \mathrm{~km}$ on the front wheel. How far can the truck travel before all four tires are completely worn out if the front and rear pairs of tires are swapped at suitable intervals? | 18750 |
8,094 | ## Task 2
In a mathematics competition, 12 participants are from Varaždin, 8 from Zadar, and 2 from Osijek. They need to form a five-member team in which there will be at least one student from each of these three cities, but there cannot be three students from the same city. In how many different ways can they form the team? | 4560 |
8,097 | 5. Calculate: $128 \div(64 \div 32) \div(32 \div 16) \div(16 \div 8) \div(8 \div 4)=$ | 8 |
8,145 | Let \( a, b, \) and \( c \) be three positive real numbers whose product is 1. Prove that if the sum of these numbers is greater than the sum of their inverses, then exactly one of them is greater than 1. | \text{Exactly one of } a, b, \text{ and } c \text{ is greater than } 1 |
8,153 | ## 1. Squares
The image shows four figures composed of black and yellow squares. If we continue to draw figures in the same way, by how much will the number of all black and the number of all yellow squares differ in the first 29 figures in the sequence?
Result: $\quad 841$ | 841 |
8,156 | Given the complex numbers \( z = \cos \alpha + i \sin \alpha \) and \( u = \cos \beta + i \sin \beta \), and that \( z + u = \frac{4}{5} + \frac{3}{5} i \), find \( \tan(\alpha + \beta) \). | \dfrac{24}{7} |
8,158 | Let \( n \) be a positive integer. Find the largest integer \( k \) such that it is possible to form \( k \) subsets from a set with \( n \) elements, where the intersection of any two subsets is non-empty. | 2^{n-1} |
8,160 | Find the least positive integer $n$ such that the prime factorizations of $n$, $n + 1$, and $n + 2$ each have exactly two factors (as $4$ and $6$ do, but $12$ does not). | 33 |
8,189 | # Task 4.
In modern conditions, digitalization - the conversion of all information into digital code - is considered relevant. Each letter of the alphabet can be assigned a non-negative integer, called the code of the letter. Then, the weight of a word can be defined as the sum of the codes of all the letters in that word. Is it possible to encode the letters Е, О, С, Т, Ш, Ь with elementary codes, each consisting of a single digit from 0 to 9, so that the weight of the word "СТО" is not less than the weight of the word "ШЕСТЬСОТ"? If such encoding is possible, in how many ways can it be implemented? If such encoding is possible, does it allow for the unambiguous restoration of a word from its code? | 10 |
8,193 | There are 4 balls of different masses. How many weighings on a balance scale without weights are needed to arrange these balls in descending order of mass? | 5 |
8,214 | Find all integers $n$ such that $\frac{n^{3}-n+5}{n^{2}+1}$ is an integer.
Initial 241 | 0 |
8,230 | 24. In this question, $S_{\triangle X Y Z}$ denotes the area of $\triangle X Y Z$. In the following figure, if $D E / / B C$, $S_{\triangle A D E}=1$ and $S_{\triangle A D C}=4$, find $S_{\triangle D B C}$. | 12 |
8,271 |
For $n$ positive real numbers $a_{1}, a_{2}, \cdots, a_{n}$, let
$$S = \sum_{i=1}^{n} a_{i}.$$
Prove that:
$$\sum_{i=1}^{n-1} \sqrt{a_{i}^{2} + a_{i}a_{i+1} + a_{i+1}^{2}} \geqslant \sqrt{\left(S-a_{1}\right)^{2} + \left(S-a_{1}\right)\left(S-a_{n}\right) + \left(S-a_{n}\right)^{2}}.$$ | \sum_{i=1}^{n-1} \sqrt{a_{i}^{2} + a_{i}a_{i+1} + a_{i+1}^{2}} \geqslant \sqrt{\left(S-a_{1}\right)^{2} + \left(S-a_{1}\right)\left(S-a_{n}\right) + \left(S-a_{n}\right)^{2}} |
8,275 | Given real numbers \( x \) and \( y \) satisfy \( x+y=1 \). What is the maximum value of \(\left(x^{3}+1\right)\left(y^{3}+1\right)\)? | 4 |
8,312 | Given a regular square pyramid \( P-ABCD \) with a base side length \( AB=2 \) and height \( PO=3 \). \( O' \) is a point on the segment \( PO \). Through \( O' \), a plane parallel to the base of the pyramid is drawn, intersecting the edges \( PA, PB, PC, \) and \( PD \) at points \( A', B', C', \) and \( D' \) respectively. Find the maximum volume of the smaller pyramid \( O-A'B'C'D' \). | \dfrac{16}{27} |
8,314 | Z1) Find all integer values that the expression
$$
\frac{p q+p^{p}+q^{q}}{p+q}
$$
where $p$ and $q$ are prime numbers.
Answer: The only integer value is 3 . | 3 |
8,328 | A right triangle has acute angles of $60^{\circ}$ and $30^{\circ}$. Two congruent circles are inscribed in the triangle, touching the hypotenuse, each other, and one of the legs. How many times is the smaller leg of the triangle the radius of the circles? | 2 + \sqrt{3} |
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