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90,472 | 5. The diagonals of a cyclic quadrilateral $\mathrm{ABCD}$ are perpendicular. Perpendiculars dropped from vertices $\mathrm{B}$ and $\mathrm{C}$ to side $\mathrm{AD}$ intersect the diagonals $\mathrm{AC}$ and $\mathrm{BD}$ at points $\mathrm{M}$ and $\mathrm{N}$, respectively. Find $\mathrm{MN}$, if $\mathrm{BC}=2$. | 2 |
90,481 | 10.5. The teams participating in the quiz need to answer 50 questions. The cost (in integer points) of a correct answer to each question was determined by experts after the quiz, the cost of an incorrect answer - 0 points. The final score of the team was determined by the sum of points received for correct answers. When summarizing the results, it was found that the costs of correct answers could be assigned in such a way that the teams could take places according to any wishes of the experts. What is the maximum number of teams that could have participated in the quiz | 50 |
90,501 | 5. The square of a three-digit number ends with three identical non-zero digits. Write the largest such three-digit number. | 962 |
90,514 | 5. In the game "set," all possible four-digit numbers consisting of the digits $1,2,3$ (each digit appearing exactly once) are used. It is said that a triplet of numbers forms a set if, in each digit place, either all three numbers contain the same digit, or all three numbers contain different digits.
For example, the numbers 1232, 2213, 3221 form a set (in the first place, all three digits appear, in the second place, only the digit two appears, in the third place, all three digits appear, and in the fourth place, all three digits appear). The numbers $1123,2231,3311$ do not form a set (in the last place, two ones and a three appear).
How many sets exist in the game?
(Permuting the numbers does not create a new set: $1232,2213,3221$ and $2213,1232,3221$ are the same set.) | 1080 |
90,540 | # 4. Variant 1.
A square piece of paper is folded as follows: the four corners are folded to the center so that they meet at one point (see figure),
resulting in a square again. After performing this operation several times, a square with a side length of 3 cm and a thickness of 16 sheets of paper is obtained. Find the side length of the original square in centimeters. | 12 |
90,550 | Variant 10.6.3. A pair of natural numbers ( $a, p$ ) is called good if the number $a^{3}+p^{3}$ is divisible by $a^{2}-p^{2}$, and $a>p$.
(a) (1 point) Indicate any possible value of $a$ for which the pair $(a, 19)$ is good.
(b) (3 points) Find the number of good pairs for which $p$ is a prime number less than 24. | 27 |
90,559 | 5-6. 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 touches the bottom left corner, and each subsequent sheet is placed one centimeter higher and one centimeter to the right of the previous one. The last sheet touches the top right corner. What is the length $x$ in centimeters?
 | 77 |
90,565 | Problem 9.5. Point $M$ is the midpoint of side $B C$ of triangle $A B C$, where $A B=17$, $A C=30, B C=19$. A circle is constructed with side $A B$ as its diameter. An arbitrary point $X$ is chosen on this circle. What is the minimum value that the length of segment $M X$ can take?
 | 6.5 |
90,566 | Task 5.4. Masha drew two little people in her notebook. The area of each cell is 1.
Which of the little people has a larger area?
What is the difference? If the areas are the same, write "0" in the answer.
 | 2 |
90,576 | Problem 9.1. For a natural number $a$, the product $1 \cdot 2 \cdot 3 \cdot \ldots \cdot a$ is denoted as $a$ !.
(a) (2 points) Find the smallest natural number $m$ such that $m$ ! is divisible by $23 m$.
(b) (2 points) Find the smallest natural number $n$ such that $n$ ! is divisible by $33n$. | 12 |
90,588 | 2.1. Petya writes on the board such different three-digit natural numbers that each of them is divisible by 3, and the first two digits differ by 2. What is the maximum number of such numbers he can write if they end in 6 or 7? | 9 |
90,595 | Variant 7.8.2. The cells of a $40 \times 40$ table are colored in $n$ colors such that for any cell, the union of its row and column contains cells of all $n$ colors. Find the maximum possible number of blue cells if
(a) (1 point) $n=2$;
(b) (3 points) $n=20$. | 840 |
90,603 | 7.4 Vanya wrote the numbers $1,2,3, \ldots, 13$ in his notebook. He multiplied five of them by 3, and the rest by 7, then added all the products. Could the result have been 433? | 433 |
90,612 | 10.6. Petya and Vasya came up with ten polynomials of the fifth degree. Then Vasya, in turn, called out consecutive natural numbers (starting from some number), and Petya substituted each called number into one of the polynomials of his choice and wrote down the obtained values on the board from left to right. It turned out that the numbers written on the board formed an arithmetic progression (in this exact order). What is the maximum number of numbers Vasya could have called out? | 50 |
90,616 | 7.5. In the sum $+1+3+9+27+81+243+729$, any addends can be crossed out and the signs before some of the remaining numbers can be changed from “+” to “-”. Masha wants to use this method to first obtain an expression whose value is 1, then, starting over, obtain an expression whose value is 2, then (starting over again) obtain 3, and so on. Up to what largest integer will she be able to do this without skipping any? | 1093 |
90,720 | 5. We took ten consecutive natural numbers greater than 1, multiplied them, found all the prime divisors of the resulting number, and multiplied these prime divisors (taking each exactly once). What is the smallest number that could have resulted? Fully justify your answer.
Solution. We will prove that among ten consecutive numbers, there must be a number that has a prime divisor different from $2, 3, 5, 7$. Let's count how many of the ten consecutive numbers can have the specified four divisors. Out of ten numbers, exactly five are divisible by 2, and at least one of these five numbers is divisible by 3. (This fact can be taken for granted in the work, but let's recall one possible proof. The remainders from dividing six consecutive numbers by 6 form a set from 0 to 5, i.e., there is a number with a remainder of 0, which is divisible by both 2 and 3.) Further, the set of ten numbers contains at least three and no more than four numbers that are divisible by 3. If there are three such numbers, at least one of these three is divisible by 2 and has already been counted. If there are four such numbers, they have the form $3k, 3k+3, 3k+6, 3k+9$. Depending on the parity of $k$, either $3k, 3k+6$ or $3k+3, 3k+9$ are divisible by 2. Thus, when considering numbers divisible by 3, we can add no more than two new numbers to the five even numbers already counted. Ten consecutive numbers contain exactly two numbers that are divisible by 5, one of which is even and has already been counted. Therefore, divisibility by 5 adds no more than one number to the seven already counted. Among ten consecutive numbers, no more than two are divisible by 7, one of which is even and has already been counted. Divisibility by 7 adds no more than one number to the eight already counted. Thus, among ten consecutive numbers, no more than nine have at least one of the divisors $2, 3, 5, 7$, i.e., there is a number with a larger prime divisor and not divisible by any of the first four prime numbers. Therefore, in the product of ten numbers, there will be at least one divisor different from $2, 3, 5, 7$, so the product of all prime divisors will be greater than 210. The smallest possible product is $2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 = 2310$. | 2310 |
90,728 | 2. This year, the son and daughter are so many years old that the product of their ages is 7 times less than the father's age. And in three years, the product of their ages will already be equal to the father's age. Find the father's age. | 21 |
90,767 | 8.4. When a certain number is divided by 13 and 15 with a remainder, the same incomplete quotients are obtained. Find the largest such number. | 90 |
90,790 | 1. Cyclists Petya, Vlad, and Timur simultaneously started a warm-up race on a circular cycling track. Their speeds are 27 km/h, 30 km/h, and 32 km/h, respectively. After what shortest time will they all be at the same point on the track again? (The length of the cycling track is 400 meters.) | 24 |
90,840 | 9.1. Four non-zero numbers are written on the board, and the sum of any three of them is less than the fourth number. What is the smallest number of negative numbers that can be written on the board? Justify your answer. | 3 |
90,848 | Problem 6.6. On an island, there live knights who always tell the truth, and liars who always lie. One day, 65 inhabitants of the island gathered for a meeting. Each of them, in turn, made the statement: "Among the previously made statements, there are exactly 20 fewer true statements than false ones." How many knights were at this meeting? | 23 |
90,895 | 8.4. Sasha is a coffee lover who drinks coffee at work, so every morning before work, he prepares a thermos of his favorite drink according to a strict ritual. He brews 300 ml of the strongest aromatic coffee in a Turkish coffee pot. Then Sasha repeats the following actions 6 times: he pours 200 ml from the pot into the thermos, and adds 200 ml of water to the pot, and mixes thoroughly. After the 6th time, Sasha pours all 300 ml from the pot into the thermos. What will be the concentration of the original strongest aromatic coffee in the thermos? Initially, the thermos is empty.
The concentration is a number from 0 to 1. Provide the answer as a fraction (common or decimal), not as a percentage.
Answer, option 1. 0.2 .
Answer, option 2. 0.2 .
Answer, option 3.0 .2 .
Answer, option 4. 0.2 . | 0.2 |
90,907 | 4. When entering the shooting range, the player pays 100 rubles to the cashier. After each successful shot, the amount of money increases by $10 \%$, and after each miss - decreases by $10 \%$. Could it be that after several shots, he ends up with 80 rubles and 19 kopecks? | 80.19 |
90,913 | 1.1. On September 1, 2021, Vasya deposited 100,000 rubles in the bank. Exactly one year later, the bank accrues 10% annual interest (that is, it increases the amount by 10 percent of what was on the account at that moment, for example, on September 2, 2022, Vasya will have 110,000 rubles on his account). Find the smallest year number in which on September 2, the amount on Vasya's account will be more than 150,100 rubles. | 2026 |
90,920 | 1.1. The lines containing the bisectors of the exterior angles of a triangle with angles of 42 and 59 degrees intersected pairwise and formed a new triangle. Find the degree measure of its largest angle. | 69 |
90,921 | 10.3. Several different real numbers are written on the board. It is known that the sum of any three of them is rational, while the sum of any two of them is irrational. What is the largest number of numbers that can be written on the board? Justify your answer. | 3 |
90,922 | Problem 6.5. In the park, paths are laid out as shown in the figure. Two workers started to asphalt them, starting simultaneously from point $A$. They lay asphalt at constant speeds: the first on the section $A-B-C$, the second on the section $A-D-E-F-C$. In the end, they finished the work simultaneously, spending 9 hours on it. It is known that the second works 1.2 times faster than the first. How many minutes did the second spend laying asphalt on the section $D E$?
 | 45 |
90,928 | Problem 11.6. Inside the cube $A B C D A_{1} B_{1} C_{1} D_{1}$, there is the center $O$ of a sphere with radius 10. The sphere intersects the face $A A_{1} D_{1} D$ along a circle with radius 1, the face $A_{1} B_{1} C_{1} D_{1}$ along a circle with radius 1, and the face $C D D_{1} C_{1}$ along a circle with radius 3. Find the length of the segment $O D_{1}$.
 | 17 |
90,978 | # 2. Option 1.
Tourists Vitya and Pasha are walking from city A to city B at equal speeds, while tourists Katya and Masha are walking from city B to city A at equal speeds. Vitya met Masha at 12:00, Pasha met Masha at 15:00, and Vitya met Katya at 14:00. How many hours after noon did Pasha meet Katya? | 5 |
90,993 | 3. Three lines intersect at one point 0. Outside these lines, a point M is taken and perpendiculars are dropped from it to them. The points $\mathrm{H}_{1}, \mathrm{H}_{2}$ and $\mathrm{H}_{3}$ are the bases of these perpendiculars. Find the ratio of the length of the segment OM to the radius of the circle circumscribed around the triangle $\mathrm{H}_{1} \mathrm{H}_{2} \mathrm{H}_{3}$.
Answer: 2. | 2 |
91,019 | 5-6. On a rectangular table of size $x$ cm $\times 80$ cm, identical sheets of paper of size 5 cm $\times 8$ cm are placed. The first sheet touches the bottom left corner, and each subsequent sheet is placed one centimeter higher and one centimeter to the right of the previous one. The last sheet touches the top right corner. What is the length $x$ in centimeters?
 | 77 |
91,025 | # Task № 6.2
## Condition:
Given triangle $\mathrm{ABC}$. Median $\mathrm{BM}$ is perpendicular to bisector $\mathrm{AL}$, and $\mathrm{BL}=5$. Find LM. | 5 |
91,080 | Variant 9.2.1. On the sides $AB$ and $AD$ of rectangle $ABCD$, points $M$ and $N$ are marked, respectively. It is known that $AN=7$, $NC=39$, $AM=12$, $MB=3$.
(a) (1 point) Find the area of rectangle $ABCD$.
(b) (3 points) Find the area of triangle $MNC$.
 | 268.5 |
91,109 | 2. From one point on a straight highway, three cyclists start simultaneously (but possibly in different directions). Each of them rides at a constant speed without changing direction. An hour after the start, the distance between the first and second cyclist was 20 km, and the distance between the first and third - 5 km. At what speed is the third cyclist riding, if it is known that he is riding slower than the first, and the speed of the second is 10 km/h? List all possible options. | 25 |
91,144 | 6.5. The arithmetic mean of four numbers is 10. If one of these numbers is erased, the arithmetic mean of the remaining three increases by 1; if instead another number is erased, the arithmetic mean of the remaining numbers increases by 2; and if only the third number is erased, the arithmetic mean of the remaining increases by 3. By how much will the arithmetic mean of the remaining three numbers change if the fourth number is erased? | 6 |
91,156 | 5. Cut a $3 \times 9$ rectangle into 8 squares.
7 points are awarded for a complete solution to each problem
The maximum total score is 35 | 8 |
91,208 | # 1. CONDITION
How many integers between 100 and 10000 are there such that their representation contains exactly 3 identical digits? | 333 |
91,351 | 5. A sign engraver makes signs with letters. He engraves identical letters in the same amount of time, and different letters possibly in different times. For two signs “ДОМ МОДЫ” and “ВХОД” together, he spent 50 minutes, and one sign “В ДЫМОХОД” he made in 35 minutes. How long will it take him to make the sign “ВЫХОД”? | 20 |
91,400 | 7. (10 points) Thirty ones are written on the board. Every minute, Karlson erases two arbitrary numbers and writes their sum on the board, and then eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could have eaten in 30 minutes? | 435 |
91,473 | 6. (8 points) On the board, 28 ones are written. Every minute, Karlson erases two arbitrary numbers and writes their sum on the board, and then eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could have eaten in 28 minutes? | 378 |
91,532 | 9. (20 points) In a rectangular parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$, the lengths of the edges are known: $A B=54, A D=90, A A_{1}=60$. The midpoint of edge $A_{1} B_{1}$ is marked as point $E$, and the midpoint of edge $B_{1} C_{1}$ is marked as point $F$. Find the distance between the lines $A E$ and $B F$. | 43.20000000000000 |
91,541 | 4. (7 points) A group of 9 boys and 9 girls was randomly paired. Find the probability that at least one pair consists of two girls. Round your answer to the nearest hundredth. | 0.99 |
91,552 | 7. (10 points) Natural numbers $a, b, c$ are chosen such that $a<b<c$. Moreover, it is known that the system of equations $2 x+y=2021$ and $y=|x-a|+|x-b|+|x-c|$ has exactly one solution. Find the minimum possible value of $c$.
# | 1011 |
91,554 | 4. Four managers of responsibility shifting reported: "If they are arranged in pairs, there will be 1 left. If they are arranged in threes, there will also be 1 left. If they are arranged in fours, there will be 2 left, and if they are arranged in fives, there will also be 2 left." Should the head of the report reception department believe such a report? Determine the maximum number of true statements among these four statements (possibly all) and for each maximum set of non-contradictory statements, find the smallest number of objects being arranged, considering that there are at least a thousand. | 1042 |
91,562 | Problem 5. There are 4 numbers, not all of which are the same. If you take any two of them, the ratio of the sum of these two numbers to the sum of the other two numbers will be the same value $\mathrm{k}$. Find the value of $\mathrm{k}$. Provide at least one set of four numbers that satisfy the condition. Describe all possible sets of such numbers and determine how many there are. | -1 |
91,573 | 4. (7 points) A group of 7 boys and 7 girls was randomly paired. Find the probability that at least one pair consists of two girls. Round your answer to the nearest hundredth. | 0.96 |
91,596 | # 7. Problem 7
Thirty-nine students from seven classes came up with 60 problems, and students from the same class came up with the same number of problems (not equal to zero), while students from different classes came up with different numbers of problems. How many students came up with one problem? | 33 |
91,630 | # 5. Task 5.1
On a farm, there are pigs and horses. What is the smallest number of pigs on the farm so that they can make up from $54 \%$ to $57 \%$ of the total number of animals
# | 5 |
91,639 | # 8. Problem 8
A square of size $2018 \times 2018$ was cut into rectangles with integer side lengths. Some of these rectangles were used to form a square of size $2000 \times 2000$, and the remaining rectangles were used to form a rectangle where the length differs from the width by less than 40. Find the perimeter of this rectangle. | 1078 |
91,670 | # 9. Problem 9
Find the maximum integer $x$ for which there exists an integer ${ }^{y}$ such that the pair $(x, y)$ is a solution to the equation $x^{2}-x y-2 y^{2}=9$.
# | 3 |
91,675 | # 5. Task 5
The Wolf and Ivan the Tsarevich are 20 versts away from the source of living water, and the Wolf is carrying Ivan the Tsarevich there at a speed of 3 versts per hour. To revive Ivan the Tsarevich, one liter of water is needed, which flows from the source at a rate of half a liter per hour. At the source, there is a Crow with unlimited carrying capacity, which must collect the water, after which it will fly towards the Wolf and Ivan the Tsarevich, flying 6 versts per hour and spilling a quarter liter of water each hour. After how many hours will it be possible to revive Ivan the Tsarevich? Points for the task: 8. | 4 |
91,686 | 1. Natural numbers a and b are such that 5 LCM $(\mathrm{a}, \mathrm{b})+2$ GCD $(\mathrm{a}, \mathrm{b})=120$. Find the greatest possible value of the number a. | 20 |
91,702 | 3. (3 points) Let $x_{1}, x_{2}, \ldots, x_{60}$ be natural numbers greater than 1 (not necessarily distinct). In a $60 \times 60$ table, the numbers are arranged as follows: at the intersection of the $i$-th row and the $k$-th column, the number $\log _{x_{k}} \frac{x_{i}}{8}$ is written. Find the smallest possible value of the sum of all numbers in the table.
Answer: -7200 | -7200 |
91,709 | # 4. Task 4
At the parade, drummers stand in a neat square formation of 50 rows with 50 drummers each. The drummers are dressed either in blue or red costumes. What is the maximum number of drummers that can be dressed in blue costumes so that each drummer dressed in blue sees only drummers in red costumes? Consider the drummers to be looking in all directions (360 degrees) and to be point-like. | 625 |
91,711 | 3. From 60 right-angled triangles with legs of 2 and 3, a rectangle was formed. What is the maximum value that its perimeter can take? | 184 |
91,718 | 5. (3 points) Anya, Vanya, Danya, Manya, Sanya, and Tanya were collecting apples. It turned out that each of them collected a whole percentage of the total number of apples collected, and all these numbers are different and greater than zero. What is the minimum number of apples that could have been collected?
Answer: 25 | 25 |
91,730 | 3. It is known that the quadratic equation $\mathrm{A} x^{2}+Б x+B=0$, where $А$, Б and $В-$ are some digits, has a root -7. By what number greater than 1 is the three-digit number АБВ (i.e., the number consisting of the digits А, Б, and В in that order) definitely divisible? | 17 |
91,737 | 5. (3 points) Anya, Vanya, Danya, Sanya, and Tanya were collecting apples. It turned out that each of them collected a whole percentage of the total number of apples collected, and all these numbers are different and greater than zero. What is the minimum number of apples that could have been collected?
Answer: 20 | 20 |
91,751 | 9.2. Replace the two asterisks with two different numbers so that the identity equality is obtained: $(3 x-*)(2 x+5)-x=6 x^{2}+2(5 x-*)$. | 25 |
91,755 | 3. A year ago, Snow White was as old as the sum of the ages of the seven dwarfs. In two years, she will be as old as six of the older ones. How old is the youngest dwarf now? | 16 |
91,773 | 3. A circle with a radius of 5 touches the hypotenuse of a right triangle and the extensions of both of its legs. Find the perimeter of the triangle. | 10 |
91,896 | Problem 1. Yura has a calculator that allows multiplying a number by 3, adding 3 to a number, or (if the number is divisible by 3) dividing the number by 3. How can one use this calculator to get from the number 1 to the number 11? $\quad[3$ points] (T. I. Golenishcheva-Kutuzova) | 11 |
91,908 | 8.2. In a five-digit number, one digit was crossed out, and the resulting four-digit number was added to the original. The sum turned out to be 54321. Find the original number. | 49383 |
91,940 | 3-ча 1. In a chess tournament, students from grades IX and X participated. There were 10 times more students from grade X than from grade IX, and they scored 4.5 times more points in total than all the students from grade IX. How many points did the students from grade IX score? Find all solutions. | 10 |
91,945 | 3-ча 5. Given a segment \(O A\). From the end of the segment \(A\), 5 segments \(A B_{1}, A B_{2}, A B_{3}, A B_{4}, A B_{5}\) extend. From each point \(B_{i}\), five new segments can extend, or no new segments, and so on.
Can the number of free ends of the constructed segments be equal to \(1001\)?
By a free end of a segment, we mean a point that belongs to only one segment (except for the point \(O)\). | 1001 |
91,954 | # Task 4. (12 points)
A numerical sequence is such that $x_{n}=\frac{n+2}{n x_{n-1}}$ for all $n \geq 2$. Find the product $x_{1} x_{2} x_{3} \ldots x_{2016} x_{2017}$, if $x_{1}=1$.
# | 673 |
91,964 | 27.1. (Belgium, 80). Each of the two urns contains white and black balls, and the total number of balls in both urns is 25. One ball is randomly drawn from each urn. Knowing that the probability that both drawn balls will be white is 0.54, find the probability that both drawn balls will be black. | 0.04 |
91,973 | Example 2.25. $\int_{-\infty}^{0} e^{x} d x$.
Translate the text above into English, keeping the original text's line breaks and format, and output the translation result directly.
Example 2.25. $\int_{-\infty}^{0} e^{x} d x$. | 1 |
91,988 | 18.5. On the sides $C B$ and $C D$ of the square $A B C D$, points $M$ and $K$ are taken such that the perimeter of triangle $C M K$ is equal to twice the side of the square. Find the measure of angle $M A K$. | 45 |
92,026 | Example 5.24. Calculate $\sin 18^{\circ}$, using the first two terms of series (3), and estimate the resulting error. | 0.3091 |
92,089 | 582. $y=4x-x^{2}$. | 4 |
92,112 | 63. A large batch of car tires contains $1.5\%$ defects. What should be the volume of a random sample so that the probability of finding at least one defective car tire in it would be more than $0.92?$ | 168 |
92,149 | 307. $y^{2}=9 x, x=16, x=25$ and $y=0$ (Fig. 150). | 122 |
92,209 | Example 2. Construct a table of divided differences of various orders for the following values of $x$ and $y=f(x)$:
\[
\begin{gathered}
x_{0}=-3, x_{1}=-2, x_{2}=-1, x_{3}=1, x_{4}=2 \\
y_{0}=-9, y_{1}=-16, y_{2}=-3, y_{3}=11, y_{4}=36
\end{gathered}
\] | 1 |
92,210 | 471. A motorcyclist left A for B and at the same time a pedestrian set off from B to A. Upon meeting the pedestrian, the motorcyclist gave him a ride, brought him to A, and immediately set off again for B. As a result, the pedestrian reached A 4 times faster than he would have if he had walked the entire way. How many times faster would the motorcyclist have arrived in B if he had not had to return? | 2.75 |
92,222 | 129. The probability of an event occurring in each of the independent trials is 0.8. How many trials need to be conducted to expect with a probability of 0.9 that the event will occur at least 75 times? | 100 |
92,254 | 95. Word game. Dr. Sylvester Sharadek announced that he can always guess the word you think of if he is allowed to ask 20 questions, to which the answers should be only "yes" or "no," and if the word is in the dictionary. Do you think he is boasting? | 20 |
92,318 | Example 6. A point is thrown into the region $G$, bounded by the ellipse $x^{2}+4 y^{2}=8$. What is the probability that it will fall into the region $g$, bounded by this ellipse and the parabola $x^{2}-4 y=0$? (In Fig. 1.7, the region $g$ is shaded).
Fig. 1.7. | 0.303 |
92,325 | 132. For which $x$ and $y$ is the number $x x y y$ a square of a natural number? | 7744 |
92,339 | 205. It is required to find a number which, when multiplied by itself, added to two, then doubled, added to three again, divided by 5, and finally multiplied by 10, results in 50. | 3 |
92,345 | Problem 16. Solve the equation
$$
\lg \left(x^{2}+9\right)-3 \cdot 2^{x}+5=0
$$ | 1 |
92,346 | 382. What is the smallest number of weights and of what weight can be used to weigh any whole number of pounds from 1 to 40 on a balance scale, given that during weighing, weights can be placed on both pans of the scale. | 4 |
92,361 | 448. A cyclist set off from point A to point B, and 15 minutes later, a car set off after him. Halfway from A to B, the car caught up with the cyclist. When the car arrived at B, the cyclist still had to cover another third of the entire distance. How long will it take the cyclist to travel the distance from A to B? | 45 |
92,401 | 70. A certain amount was spent on strawberries at 2 r. 40 k. per 1 kg of one variety and the same amount on another variety at 1 r. 60 k. Find the average price of 1 kg of the strawberries purchased. | 1 |
92,408 | 471. Someone has twelve pints of wine and wants to give away half of it, but he does not have a six-pint container. He has two containers, one holds 8 pints and the other 5 pints; the question is: how can he pour six pints into the eight-pint container?
## Lebesgue's Problem. | 6 |
92,447 | 16. How many four-digit numbers can be written in total, where the digits do not repeat, using the digits $0,1,2,3$, and such that the digits 0 and 2 are not adjacent?
6 | 8 |
92,480 | 779. At least how many points of a body must be fixed in order for the entire body to be immobile? | 3 |
92,489 | 23. Given 5 numbers: $a_{1}=1, a_{2}=-1, a_{3}=-1, a_{4}=1, a_{5}=-1$. The following numbers are determined as follows: $a_{6}=a_{1} \cdot a_{2}, a_{7}=a_{2} \cdot a_{3}, a_{8}=a_{3} \cdot a_{4}$ and so on. What is $a_{1988}$? | -1 |
92,490 | 35. Is there a natural number $n$ such that $n^{n}+(n+1)^{n}$ is divisible by 1987? | 993 |
92,508 | 246. Чайная смесь. Бакалейщик купил два сорта чая: один по 32 цента за фунт, другой, лучшего качества, по 40 центов за фунт. Он решил смешать сорта и составленную смесь продать по 43 цента за фунт, чтобы получить тем самым $25 \%$ чистой прибыли.
Сколько фунтов каждого сорта пойдет на приготовление 100 фунтов смеси? | 70 |
92,519 | 105. Repeating quartet of digits. If we multiply 64253 by 365, we get 23452345, where the first four digits repeat.
What is the largest number by which 365 must be multiplied to obtain a similar product containing eight digits, the first four of which repeat? | 273863 |
92,529 | 97. The Fibonacci sequence. Consider the Fibonacci sequence $1,1,2,3,5,13, \ldots$, whose terms $F_{n}$ (Fibonacci numbers) satisfy the relation $F_{n+2}=F_{n}+F_{n+1}$, $F_{1}=F_{2}=1$. Now consider the sequence of digits in the units place of the Fibonacci numbers. Will this sequence be cyclic, that is, can it be obtained by the unlimited repetition of the same finite set of digits, as is true, for example, in the case of the sequence $055055 \ldots$. | 60 |
92,533 | 15. A runner and two cyclists and a motorcyclist are moving along a ring road, each at a constant speed, but in different directions. The runner and one of the cyclists are moving in the same direction, while the motorcyclist and the other cyclist are moving in the opposite direction. The runner meets the second cyclist every 12 minutes, the first cyclist overtakes the runner every 20 minutes, and the motorcyclist overtakes the second cyclist every 5 minutes. How often does the motorcyclist meet the first cyclist? | 3 |
92,538 | 21. When the young fishermen were asked how many fish each of them had caught, the first one answered: "I caught half the number of fish that my friend caught, plus 10 fish." The second one said: "And I caught as many as my friend, plus 20 fish." How many fish did the fishermen catch? | 100 |
92,606 | 4. There were plates on the shelf. First, from all the plates except two, $1 / 3$ part was taken, and then $1 / 2$ of the remaining plates. After this, 9 plates were left on the shelf. How many plates were on the shelf? | 26 |
92,635 | Ex. 151. In an isosceles triangle \(ABC\) (\(AB = BC\)), \(\angle ABC = 80^\circ\). Point \(M\) inside the triangle is such that \(\angle MAC = 30^\circ\) and \(\angle MCA = 10^\circ\). Find \(\angle BMC\).
 | 70 |
92,652 | 16.3. An old problem. The master hired a worker for a year and promised to give him 12 rubles and a coat. But after 7 months, the worker wanted to leave. Upon settlement, he received the coat and 5 rubles. How much did the coat cost
$$
\text { (6-8 grades) }
$$
12 | 4.8 |
92,674 | 21.3. Find all two-digit numbers that are equal to three times the product of their digits.
$$
(7-8 \text {th grades })
$$ | 1524 |
92,746 | 45. Five girls are sitting on five chairs in a row, and opposite them, on five chairs, are sitting five boys. It was decided that the boys would swap places with the girls. In how many ways can this be done? | 14400 |
92,779 | ## Problem Statement
A cylinder is filled with gas at atmospheric pressure (103.3 kPa). Assuming the gas is ideal, determine the work (in joules) done during the isothermal compression of the gas by a piston moving inward by $h$ meters (see figure).
Hint: The equation of state for the gas is
$\rho V = \text{const}$, where $\rho$ is the pressure and $V$ is the volume.
$$
H = 0.4 \text{m}, h = 0.2 \text{m}, R = 0.1 \text{m}
$$
 | 900 |
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