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51,460 | ## Task A-3.2.
A quadruple of natural numbers $(a, b, c, d)$ is called green if
$$
b=a^{2}+1, \quad c=b^{2}+1, \quad d=c^{2}+1
$$
and $D(a)+D(b)+D(c)+D(d)$ is odd, where $D(k)$ is the number of positive divisors of the natural number $k$.
How many green quadruples are there whose all members are less than 1000000? | 2 |
51,480 | 4. Find all natural numbers $n$ for which the number $n^{7}+n^{6}+n^{5}+1$ has exactly three natural divisors. (O. Nechaeva, I. Rubanov) | 1 |
51,491 | 6. In the Cartesian coordinate system $x O y$, the area of the plane region corresponding to the point set $K=\{(x, y) \mid(|x|+|3 y|-6)(|3 x|+|y|-6) \leqslant 0\}$ is $\qquad$ . | 24 |
51,495 | 203. Find the condition for the compatibility of the equations:
$$
a_{1} x+b_{1} y=c_{1}, \quad a_{2} x+b_{2} y=c_{2}, \quad a_{3} x+b_{3} y=c_{3}
$$ | a_{1}(b_{2}c_{3}-b_{3}c_{2})+a_{2}(b_{3}c_{1}-c_{3}b_{1})+a_{3}(b_{1}c_{2}-b_{2}c_{1})=0 |
51,502 | 12. An ant is at a vertex of a tetrahedron. Every minute, it randomly moves to one of the adjacent vertices. What is the probability that after one hour, it stops at the original starting point? $\qquad$ . | \frac{3^{59}+1}{4\cdot3^{59}} |
51,505 | 11. Let any real numbers $x_{0}>x_{1}>x_{2}>x_{3}>0$, to make $\log _{\frac{x_{0}}{x_{1}}} 1993+\log _{\frac{x_{1}}{x_{2}}} 1993+\log _{\frac{x_{2}}{x_{3}}} 1993 \geqslant$ $k \log _{x_{0}} 1993$ always hold, then the maximum value of $k$ is $\qquad$. | 9 |
51,507 | 13.296. Two excavator operators must complete a certain job. After the first one worked for 15 hours, the second one starts and finishes the job in 10 hours. If, working separately, the first one completed $1 / 6$ of the job, and the second one completed $1 / 4$ of the job, it would take an additional 7 hours of their combined work to finish the job. How many hours would it take each excavator operator to complete the job individually? | 20 |
51,518 | Problem 9.8. 73 children are standing in a circle. A mean Father Frost walks around the circle clockwise and distributes candies. At first, he gave one candy to the first child, then skipped 1 child, gave one candy to the next child, then skipped 2 children, gave one candy to the next child, then skipped 3 children, and so on.
After distributing 2020 candies, he left. How many children did not receive any candies? | 36 |
51,521 | What is the area of the set of points $P(x ; y)$ in the Cartesian coordinate system for which $|x+y|+|x-y| \leq 4 ?$ | 16 |
51,527 | Two sides of a regular polygon of $n$ sides when extended meet at $28$ degrees. What is smallest possible value of $n$ | 45 |
51,533 | Task 5. (20 points) At the first deposit, equipment of the highest class was used, and at the second deposit, equipment of the first class was used, with the highest class being less than the first. Initially, $40 \%$ of the equipment from the first deposit was transferred to the second. Then, $20 \%$ of the equipment that ended up on the second deposit was transferred back to the first, with half of the transferred equipment being of the first class. After this, the equipment of the highest class on the first deposit was 26 units more than on the second, and the total amount of equipment on the second deposit increased by more than $5 \%$ compared to the initial amount. Find the total amount of equipment of the first class. | 60 |
51,548 | Consider a standard ($8$-by-$8$) chessboard. Bishops are only allowed to attack pieces that are along the same diagonal as them (but cannot attack along a row or column). If a piece can attack another piece, we say that the pieces threaten each other. How many bishops can you place a chessboard without any of them threatening each other? | 14 |
51,554 | 1. If real numbers $x, y$ satisfy $x^{2}-2 x y+5 y^{2}=4$, then the range of $x^{2}+y^{2}$ is $\qquad$ . | [3-\sqrt{5},3+\sqrt{5}] |
51,572 | 15. (15 points) 100 people participate in a quick calculation test, with a total of 10 questions. The number of people who answered each question correctly is shown in the table below:
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline Question Number & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline \begin{tabular}{l}
Correct \\
Answers
\end{tabular} & 93 & 90 & 86 & 91 & 80 & 83 & 72 & 75 & 78 & 59 \\
\hline
\end{tabular}
Rule: Answering 6 or more questions correctly is considered passing. Based on the table above, calculate the minimum number of people who passed. | 62 |
51,587 | (4) Given $a>0, b>0$, if $x$ represents the smallest of the three numbers $1$, $a$, and $\frac{b}{a^{2}+b^{2}}$, when $a$ and $b$ vary, the maximum value of $x$ is $\qquad$ . | \frac{\sqrt{2}}{2} |
51,598 | For an integer $m\ge 3$, let $S(m)=1+\frac{1}{3}+…+\frac{1}{m}$ (the fraction $\frac12$ does not participate in addition and does participate in fractions $\frac{1}{k}$ for integers from $3$ until $m$). Let $n\ge 3$ and $ k\ge 3$ . Compare the numbers $S(nk)$ and $S(n)+S(k)$
. | S(nk) < S(n) + S(k) |
51,610 | 3. Let $x_{1}$ and $x_{2}$ be the roots of the equation
$$
p^{2} x^{2}+p^{3} x+1=0
$$
Determine $p$ such that the expression $x_{1}^{4}+x_{2}^{4}$ has the smallest value. | \\sqrt[8]{2} |
51,629 | 5. Find the fraction $\frac{p}{q}$ with the smallest possible natural denominator, for which $\frac{1}{2014}<\frac{p}{q}<\frac{1}{2013}$. Enter the denominator of this fraction in the provided field | 4027 |
51,630 | # Problem 6.
In the alphabet of the inhabitants of the magical planet ABV2020, there are only three letters: A, B, and V, from which all words are formed. In any word, two identical letters cannot be adjacent, and each of the three letters must be present in any word. For example, the words ABV, VABAVAB, and BVBVAB are permissible, while the words VAV, ABAAVA, and AVABBB are not. How many 20-letter words are there in the dictionary of this planet? | 1572858 |
51,653 | Example 5 There are three villages $A$, $B$, and $C$ forming a triangle (as shown in Figure 5). The ratio of the number of primary school students in villages $A$, $B$, and $C$ is $1: 2: 3$. A primary school needs to be established. Where should the school be located to minimize the total distance $S$ traveled by the students to school? | P=C |
51,671 | 1. $\log _{9 a} 8 a=\log _{3 a} 2 a$. Then $\ln a=$ | \frac{\ln2\cdot\ln3}{\ln3-2\ln2} |
51,673 | A cone is inscribed in a regular quadrilateral pyramid. Find the ratio of the area of the total surface of the cone to the area of its lateral surface, if the side of the base of the pyramid is 4, and the angle between the height of the pyramid and the plane of the lateral face is $30^{\circ}$. | 1.5 |
51,688 | 3. (10 points) $[a]$ represents the greatest integer not greater than $a$. Given that $\left(\left[\frac{1}{7}\right]+1\right) \times\left(\left[\frac{2}{7}\right]+1\right) \times\left(\left[\frac{3}{7}\right]+1\right) \times \cdots \times$ $\left(\left[\frac{k}{7}\right]+1\right)$ leaves a remainder of 7 when divided by 13, then the largest positive integer $k$ not exceeding 48 is $\qquad$ | 45 |
51,703 | 2, ** Divide a circle into $n(\geqslant 2)$ sectors $S_{1}, S_{2}, \cdots, S_{n}$. Now, color these sectors using $m(\geqslant 2)$ colors, with each sector being colored with exactly one color, and the requirement that adjacent sectors must have different colors. How many different coloring methods are there? | (-1)^{n}+(-1)^{n}(-1) |
51,733 | 1. Let $n$ be a natural number, $a, b$ be positive real numbers, and satisfy the condition $a+b=2$, then the minimum value of $\frac{1}{1+a^{n}}+\frac{1}{1+b^{n}}$ is . $\qquad$ | 1 |
51,772 | 3. calculate the sum of the digits of the number
$$
9 \times 99 \times 9999 \times \cdots \times \underbrace{99 \ldots 99}_{2^{n}}
$$
where the number of nines doubles in each factor.
## 1st solution | 9\cdot2^{n} |
51,780 | 6. Let the circumradius of $\triangle A B C$ be $R, \frac{a \cos A+b \cos B+c \cos C}{a \sin B+b \sin C+c \sin A}=\frac{a+b+c}{9 R}$. Here $a, b, c$ are the sides opposite to $\angle A, \angle B, \angle C$ respectively. Try to find the sizes of the three interior angles of $\triangle A B C$. | A=B=C=60 |
51,807 | 8. (10 points) A frog starts climbing from the bottom of a 12-meter deep well at 8:00. It climbs up 3 meters and then slides down 1 meter due to the slippery well wall. The time it takes to slide down 1 meter is one-third of the time it takes to climb up 3 meters. At 8:17, the frog reaches 3 meters below the well's mouth for the second time. The time it takes for the frog to climb from the bottom of the well to the mouth is $\qquad$ minutes. | 22 |
51,815 | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0}\left(\frac{1+\sin x \cdot \cos 2 x}{1+\sin x \cdot \cos 3 x}\right)^{\frac{1}{\sin x^{3}}}$ | e^{\frac{5}{2}} |
51,820 | 9. Given $m, n, s \in \mathbf{R}^{+}, \alpha, \beta \in\left(0 \cdot \frac{\pi}{2}\right)$, and $m \tan \alpha+n \tan \beta=s$, then the minimum value of $m \sec \alpha$ $+n \sec \beta$ is $\qquad$ | \sqrt{(+n)^{2}+^{2}} |
51,821 | 9. Given $\sin x+\sin y=0.6, \cos x+\cos y=0.8$. Then $\cos x \cdot \cos y=$ | -\frac{11}{100} |
51,839 | 57. In the figure below, there are 5 squares and 12 circles. Fill the numbers $1 \sim 12$ into the circles so that the sum of the numbers at the four corners of each square is the same. What is this sum? $\qquad$ . | 26 |
51,891 | 5. If the cube of a three-digit positive integer is an eight-digit number of the form $\overline{A B C D C D A B}$, then such a three-digit number is $\qquad$ | 303 |
51,908 | A thousand integer divisions are made: $2018$ is divided by each of the integers from $ 1$ to $1000$. Thus, a thousand integer quotients are obtained with their respective remainders. Which of these thousand remainders is the bigger? | 672 |
51,923 | There is an unlimited supply of [congruent](https://artofproblemsolving.com/wiki/index.php/Congruent) [equilateral triangles](https://artofproblemsolving.com/wiki/index.php/Equilateral_triangle) made of colored paper. Each [triangle](https://artofproblemsolving.com/wiki/index.php/Triangle) is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles. Two large triangles are considered distinguishable if it is not possible to place one on the other, using translations, rotations, and/or reflections, so that their corresponding small triangles are of the same color.
Given that there are six different colors of triangles from which to choose, how many distinguishable large equilateral triangles may be formed?
[asy] pair A,B; A=(0,0); B=(2,0); pair C=rotate(60,A)*B; pair D, E, F; D = (1,0); E=rotate(60,A)*D; F=rotate(60,C)*E; draw(C--A--B--cycle); draw(D--E--F--cycle); [/asy] | 336 |
51,936 | 2. $S$ is a subset of the set $\{1,2, \cdots, 2023\}$, satisfying that the sum of the squares of any two elements is not a multiple of 9, then the maximum value of $|S|$ is $\qquad$. (Here $|S|$ denotes the number of elements in $S$) | 1350 |
51,993 | 1. A father and son measured the length of the courtyard with their steps in winter, starting from the same place and walking in the same direction. In some places, the father's and son's footprints coincided exactly. In total, there were 61 footprints along the measurement line on the snow. What is the length of the courtyard if the father's step is $0,72 \boldsymbol{\mu}$, and the son's step is 0.54 m? | 21.6() |
51,999 | 29. 3 red marbles, 4 blue marbles and 5 green marbles are distributed to 12 students. Each student gets one and only one marble. In how many ways can the marbles be distributed so that Jamy and Jaren get the same colour and Jason gets a green marble? | 3150 |
52,000 | 3. The faces of a hexahedron and the faces of a regular octahedron are all equilateral triangles with side length $a$. The ratio of the radii of the inscribed spheres of these two polyhedra is a reduced fraction $\frac{m}{n}$. Then, the product $m \cdot n$ is $\qquad$. | 6 |
52,004 | 9. A person picks $n$ different prime numbers less than 150 and finds that they form an arithmetic sequence. What is the greatest possible value of $n$ ?
(1 mark)
某人選取了 $n$ 個不同的質數, 每個均小於 150 。他發現這些質數組成一個等差數列。求 $n$ 的最大可能值。 | 5 |
52,019 | 5. Given that $x, y, z$ are 3 real numbers greater than or equal to 1, then
$$
\left(\frac{\sqrt{x^{2}(y-1)^{2}+y^{2}}}{x y}+\frac{\sqrt{y^{2}(z-1)^{2}+z^{2}}}{y z}+\frac{\sqrt{z^{2}(x-1)^{2}+x^{2}}}{z x}\right)^{2}
$$
the sum of the numerator and denominator of the minimum value written as a simplified fraction is $\qquad$ . | 11 |
52,028 | 8. Given a sequence $\left\{a_{n}\right\}$ with 9 terms, where $a_{1}=a_{9}=1$, and for each $i \in\{1,2, \cdots, 8\}$, we have $\frac{a_{i+1}}{a_{i}} \in\left\{2,1,-\frac{1}{2}\right\}$, then the number of such sequences is $\qquad$ . | 491 |
52,029 | 【Question 7】
In a certain mathematics competition, there are a total of 6 questions, each worth 7 points (the final score for each question is an integer, with a minimum of 0 points and a maximum of 7 points). Each contestant's total score is the product of the scores of the 6 questions. If two contestants have the same score, the sum of the scores of the 6 questions is used to determine the ranking. If they are still the same, the two contestants are ranked as tied. In this competition, there are a total of $8^{6}=262144$ contestants, and there are no ties among these contestants. The score of the contestant ranked $7^{6}=117649$ is $\qquad$ points. | 1 |
52,035 | 1. Find the smallest 10-digit number, the sum of whose digits is greater than that of any smaller number. | 1999999999 |
52,052 | 11. Let $a$ and $b$ be integers for which $\frac{a}{2}+\frac{b}{1009}=\frac{1}{2018}$. Find the smallest possible value of $|a b|$. | 504 |
52,071 | * Let $n$ be a given positive integer, find the number of positive integer solutions $(x, y)$ for $\frac{1}{n}=\frac{1}{x}+\frac{1}{y}, x \neq y$. | (n^{2})-1 |
52,085 | Problem 1. Buratino, Karabas-Barabas, and Duremar are running along a path around a circular pond. They start simultaneously from the same point, with Buratino running in one direction and Karabas-Barabas and Duremar running in the opposite direction. Buratino runs three times faster than Duremar and four times faster than Karabas-Barabas. After Buratino meets Duremar, he meets Karabas-Barabas 150 meters further. What is the length of the path around the pond? | 3000 |
52,105 | # 9. Solution.
1st method. An elementary outcome in a random experiment is a triplet of places where children in red caps stand. Consider the event $A$ "all three red caps are next to each other." This event is favorable in 10 elementary outcomes. The event $B$ "two red caps are next to each other, and the third is separate" is favorable in 60 elementary outcomes (10 ways to choose two adjacent places, and the third place must be one of the 6 places not adjacent to the already chosen ones). The total number of ways to choose a triplet is $C_{10}^{3}=120$. Therefore, the required probability is $\mathrm{P}(A)+\mathrm{P}(B)=\frac{70}{120}=\frac{7}{12}$.
2nd method. Number the children in red caps. Consider the events $A_{1}$ "the second and third caps are next to each other," $A_{2}$ "the first and third caps are next to each other," and $A_{3}$ "the first and second caps are next to each other." We need to find the probability of the union:
$$
\begin{aligned}
\mathrm{P}\left(A_{1} \cup A_{2} \cup A_{3}\right) & =\mathrm{P}\left(A_{1}\right)+\mathrm{P}\left(A_{2}\right)+\mathrm{P}\left(A_{3}\right)- \\
- & \mathrm{P}\left(A_{1} \cap A_{2}\right)-\mathrm{P}\left(A_{1} \cap A_{3}\right)-\mathrm{P}\left(A_{2} \cap A_{3}\right)+\mathrm{P}\left(A_{1} \cap A_{2} \cap A_{3}\right) .
\end{aligned}
$$
The probability of event $A_{1}$ is 2/9 (if the second cap occupies some place, then for the third cap, nine places remain, but only two of them are next to the second). The probabilities of events $A_{2}$ and $A_{3}$ are the same:
$$
\mathrm{P}\left(A_{1}\right)=\mathrm{P}\left(A_{2}\right)=\mathrm{P}\left(A_{3}\right)=\frac{2}{9}
$$
The event $A_{1} \cap A_{2}$ consists in the third cap being between the first and second. The probability of this is $\frac{2}{9} \cdot \frac{1}{8}=\frac{1}{36}$. The same probability applies to the other two pairwise intersections:
$$
\mathrm{P}\left(A_{1} \cap A_{2}\right)=\mathrm{P}\left(A_{1} \cap A_{3}\right)=\mathrm{P}\left(A_{2} \cap A_{3}\right)=\frac{1}{36} .
$$
The event $A_{1} \cap A_{2} \cap A_{3}$ is impossible: $\mathrm{P}\left(A_{1} \cap A_{2} \cap A_{3}\right)=0$. Therefore,
$$
\mathrm{P}\left(A_{1} \cup A_{2} \cup A_{3}\right)=3 \cdot \frac{2}{9}-3 \cdot \frac{1}{36}+0=\frac{2}{3}-\frac{1}{12}=\frac{7}{12}
$$
The 3rd method is similar to the 2nd method from the second variant. Other methods of solution are also possible. | \frac{7}{12} |
52,111 | In a right-angled triangle, $s_{a}$ and $s_{b}$ are the medians to the legs, and $s_{c}$ is the median to the hypotenuse. Determine the maximum value of the expression $\frac{s_{a}+s_{b}}{s_{c}}$. | \sqrt{10} |
52,121 | Let $n,k$ be positive integers such that $n>k$. There is a square-shaped plot of land, which is divided into $n\times n$ grid so that each cell has the same size. The land needs to be plowed by $k$ tractors; each tractor will begin on the lower-left corner cell and keep moving to the cell sharing a common side until it reaches the upper-right corner cell. In addition, each tractor can only move in two directions: up and right. Determine the minimum possible number of unplowed cells. | (n-k)^2 |
52,134 | 1. If the function $f(x)=3 \cos \left(\omega x+\frac{\pi}{6}\right)-\sin \left(\omega x-\frac{\pi}{3}\right)(\omega>0)$ has the smallest positive period of $\pi$, then the maximum value of $f(x)$ in the interval $\left[0, \frac{\pi}{2}\right]$ is $\qquad$ . | 2\sqrt{3} |
52,152 | # 6. Variant 1.
A diagonal of a 20-gon divides it into a 14-gon and an 8-gon (see figure). How many of the remaining diagonals of the 20-gon intersect the highlighted diagonal? The vertex of the 14-gon is not considered an intersection.
 | 72 |
52,158 | 15. In an acute-angled triangle $A B C$, points $D, E$, and $F$ are the feet of the perpendiculars from $A, B$, and $C$ onto $B C, A C$ and $A B$, respectively. Suppose $\sin A=\frac{3}{5}$ and $B C=39$, find the length of $A H$, where $H$ is the intersection $A D$ with $B E$. | 52 |
52,178 | Task A-3.4. (4 points)
Determine the minimum value of the expression
$$
\sin (x+3)-\sin (x+1)-2 \cos (x+2)
$$
for $x \in \mathbb{R}$. | 2(\sin1-1) |
52,193 | In the interior of triangle $ABC$, we have chosen point $P$ such that the lines drawn through $P$ parallel to the sides of the triangle determine 3 triangles and 3 parallelograms. The areas of the resulting triangles are 4, 9, and 49 square units. What is the area of the original triangle? | 144 |
52,197 | 4. At a large conference, four people need to be selected from five volunteers, Xiao Zhang, Xiao Zhao, Xiao Li, Xiao Luo, and Xiao Wang, to undertake four different tasks: translation, tour guiding, etiquette, and driving. If Xiao Zhang and Xiao Zhao can only perform the first two tasks, and the other three can perform all four tasks, then the number of different selection schemes is. kinds. | 36 |
52,207 | The circle constructed on side $A D$ of parallelogram $A B C D$ as its diameter passes through the midpoint of diagonal $A C$ and intersects side $A B$ at point $M$. Find the ratio $A M: A B$, if $A C=3 B D$.
# | 4:5 |
52,227 | 11. Similar events. If a die is thrown, then the event $A=\{1,2,3\}$ consists of one of the faces 1, 2, or 3 appearing. Similarly, the event $B=\{1,2,4\}$ is favorable if the faces 1, 2, or 4 appear.
A die is thrown 10 times. It is known that the event $A$ occurred exactly 6 times.
a) (from 8th grade. 2 points) Find the probability that, under this condition, the event $B$ did not occur at all.
b) (from 9th grade. 3 points) Find the expected value of the random variable $X$ "The number of occurrences of event $B$". | \frac{16}{3} |
52,228 | 11. What? Where? When? Experts and Viewers play "What, Where, When" until six wins - whoever wins six rounds first is the winner. The probability of the Experts winning in one round is 0.6, and there are no ties. Currently, the Experts are losing with a score of $3: 4$. Find the probability that the Experts will still win. | 0.4752 |
52,230 | Problem 2. Given is the rectangle $ABCD$ where the length of side $AB$ is twice the length of side $BC$. On side $CD$, a point $M$ is chosen such that the angle $AMD$ is equal to the angle $AMB$.
a) Determine the angle $AMD$.
b) If $\overline{DM}=1$, what is the area of rectangle $ABCD$? | 2(7+4\sqrt{3}) |
52,231 | ## 34. Family Breakfast
Every Sunday, a married couple has breakfast with their mothers. Unfortunately, each spouse's relationship with their mother-in-law is quite strained: both know that there are two out of three chances of getting into an argument with their mother-in-law upon meeting. In the event of a conflict, the other spouse takes the side of their own mother (and thus argues with their spouse) about half the time; equally often, they defend their spouse (or husband) and argue with their mother.
Assuming that the arguments between each spouse and their mother-in-law are independent of each other, what do you think is the proportion of Sundays when the spouses manage to avoid arguing with each other? | \frac{4}{9} |
52,242 | 1. Given a set of data consisting of seven positive integers, the only mode is 6, and the median is 4. Then the minimum value of the sum of these seven positive integers is $\qquad$ . | 26 |
52,248 | 1.29. Through points $R$ and $E$, belonging to sides $A B$ and $A D$ of parallelogram $A B C D$ and such that $A R=\frac{2}{3} A B, A E=\frac{1}{3} A D$, a line is drawn. Find the ratio of the area of the parallelogram to the area of the resulting triangle. | 9:1 |
52,251 | 2-2. Find the minimum value of the function
$$
f(x)=x^{2}+(x-2)^{2}+(x-4)^{2}+\ldots+(x-102)^{2}
$$
If the result is a non-integer, round it to the nearest integer and write it as the answer. | 46852 |
52,269 | We inscribe a cone around a sphere. What is the maximum fraction of the cone's volume that the sphere can fill? | \frac{1}{2} |
52,271 | 3. In a square, the inscribed circle is drawn, in which an inscribed square is drawn with sides parallel to the sides of the original square. The area between these squares is colored dark gray. In the smaller square, the inscribed circle is drawn again with another inscribed square with parallel sides; the area between these is colored light gray. We repeat this process until there are 2023 dark gray and 2023 light gray areas. In the picture below, you can see the first steps drawn. The innermost square remains white. The total area of all dark gray areas together is exactly 1.
How large is the total area of all light gray areas together? | \frac{1}{2} |
52,277 | 5.2.4. $B L$ is the bisector of triangle $A B C$. Find its area, given that $|A L|=4$, $|B L|=2 \sqrt{15},|C L|=5$. | \frac{9\sqrt{231}}{4} |
52,290 | # Problem 6. (4 points)
Six positive numbers, not exceeding 3, satisfy the equations $a+b+c+d=6$ and $e+f=2$. What is the smallest value that the expression
$$
\left(\sqrt{a^{2}+4}+\sqrt{b^{2}+e^{2}}+\sqrt{c^{2}+f^{2}}+\sqrt{d^{2}+4}\right)^{2}
$$
can take? | 72 |
52,333 | Question 127, Find the last two digits of $\left[(2+\sqrt{3})^{2^{2020}}\right]$.
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | 53 |
52,340 | [Trigonometric Inequalities]
Find the maximum value of the expression $\sin x \sin y \sin z + \cos x \cos y \cos z$. | 1 |
52,344 | 5. Find the minimum distance from the point $(0,5 / 2)$ to the graph of $y=x^{4} / 8$. | \frac{\sqrt{17}}{2} |
52,350 | Problem 5.5. A large rectangle consists of three identical squares and three identical small rectangles. The perimeter of the square is 24, and the perimeter of the small rectangle is 16. What is the perimeter of the large rectangle?
The perimeter of a figure is the sum of the lengths of all its sides.
 | 52 |
52,365 | 4.3. A smooth sphere with a radius of 1 cm was dipped in red paint and launched between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm, respectively (this sphere ended up outside the smaller sphere but inside the larger one). Upon touching both spheres, the sphere leaves a red trail. During its movement, the sphere traveled along a closed path, resulting in a red-contoured area on the smaller sphere with an area of 47 square cm. Find the area of the region bounded by the red contour on the larger sphere. Provide the answer in square centimeters, rounding to the nearest hundredth if necessary. | 105.75 |
52,366 | 3. Maxim came up with a new way to divide numbers by a two-digit number $N$. To divide any number $A$ by the number $N$, the following steps need to be performed:
1) Divide $A$ by the sum of the digits of the number $N$;
2) Divide $A$ by the product of the digits of the number $N$;
3) Subtract the second result from the first.
For which numbers $N$ will Maxim's method give the correct result? (20 points) | 24 |
52,378 | A certain line intersects parallel lines $a$ and $b$ at points $A$ and $B$ respectively. The bisector of one of the angles formed with vertex $B$ intersects line $a$ at point $C$. Find $A C$, if $A B=1$.
# | 1 |
52,396 | 9.3 How many right-angled triangles with integer sides exist, where one of the legs is equal to 2021. | 4 |
52,404 | Problem No. 5 (15 points)
The system shown in the figure is in equilibrium. It is known that the uniform rod $AB$ and the load lying on it have the same mass $m=10$ kg. The load is located exactly in the middle of the rod. The thread, passing over the pulleys, is attached to one end of the rod and at a distance of one quarter of the rod's length from its left end. Determine the mass $m_2$ of the second load, suspended from the center of one of the pulleys. The threads and pulleys are weightless, and there is no friction in the pulley axes.
Answer: 80 kg
# | 80 |
52,407 | 7. Find the smallest positive integer $n$ such that $A_{n}=1+11+111+\ldots+1 \ldots 1$ (the last term contains $n$ ones) is divisible by 45. | 35 |
52,444 | 2. Suppose $\frac{1}{2} \leq x \leq 2$ and $\frac{4}{3} \leq y \leq \frac{3}{2}$. Determine the minimum value of
$$
\frac{x^{3} y^{3}}{x^{6}+3 x^{4} y^{2}+3 x^{3} y^{3}+3 x^{2} y^{4}+y^{6}} \text {. }
$$ | \frac{27}{1081} |
52,452 | 7. On the coordinate plane, consider a figure $M$ consisting of all points with coordinates $(x ; y)$ that satisfy the system of inequalities
$$
\left\{\begin{array}{l}
|y|+|4+y| \leqslant 4 \\
\frac{x-y^{2}-4 y-3}{2 y-x+3} \geqslant 0
\end{array}\right.
$$
Sketch the figure $M$ and find its area. | 8 |
52,470 | 6. Two acute angles $\alpha$ and $\beta$ satisfy the condition $\operatorname{Sin}^{2} \alpha+\operatorname{Sin}^{2} \beta=\operatorname{Sin}(\alpha+\beta)$. Find the sum of the angles $\alpha+\beta$ in degrees. | 90 |
52,472 | 9. $[\boldsymbol{7}]$ let $\mathcal{R}$ be the region in the plane bounded by the graphs of $y=x$ and $y=x^{2}$. Compute the volume of the region formed by revolving $\mathcal{R}$ around the line $y=x$. | \frac{\sqrt{2}\pi}{60} |
52,478 | 20. A tram ticket is called lucky in the Leningrad style if the sum of its first three digits equals the sum of the last three digits. A tram ticket is called lucky in the Moscow style if the sum of its digits in even positions equals the sum of its digits in odd positions. How many tickets are lucky both in the Leningrad style and in the Moscow style, including the ticket 000000? | 6700 |
52,515 | 2. Determine the largest even three-digit natural number whose product of digits is 24, and the digits are distinct. | 614 |
52,522 | 4. (1990 AIME Problem 8) \( n \) is the smallest positive integer that satisfies the following conditions:
(1) \( n \) is a multiple of 75.
(2) \( n \) has exactly 75 positive divisors (including 1 and itself).
Find \( \frac{n}{75} \). | 432 |
52,552 | 6. Ms. Olga Ivanovna, the class teacher of 5B, is staging a "Mathematical Ballet." She wants to arrange the boys and girls so that exactly 2 boys are 5 meters away from each girl. What is the maximum number of girls that can participate in the ballet if it is known that 5 boys are participating? | 20 |
52,557 | How many different orders can the digits $0,1,2,3,4,5,6$ form a seven-digit number divisible by four? (The number cannot start with 0.) | 1248 |
52,593 | 16. Let $P\left(x+a, y_{1}\right) 、 Q\left(x, y_{2}\right) 、 R\left(2+a, y_{3}\right)$ be three different points on the graph of the inverse function of $f(x)=2^{x}+a$. If there is exactly one real number $x$ that makes $y_{1} 、 y_{2} 、 y_{3}$ an arithmetic sequence, find the range of the real number $a$, and determine the area of $\triangle P Q R$ when the distance from the origin to point $R$ is minimized. | \frac{1}{4} |
52,595 | A convex quadrilateral's two opposite sides are each divided into 100 equal parts, and the points of the same ordinal number are connected. Among the 100 quadrilaterals thus obtained, the area of the first one is 1, and the area of the last, the 100th one, is 2 square units. What is the area of the original quadrilateral? | 150 |
52,624 | 5. (2000 World Inter-City League) A bus, starting a 100-kilometer journey at 12:20 PM, has a computer that at 1:00 PM, 2:00 PM, 3:00 PM, 4:00 PM, 5:00 PM, and 6:00 PM, says: “If the average speed from now on is the same as the average speed so far, then it will take one more hour to reach the destination.” Is this possible? If so, how far has the bus traveled by 6:00 PM? | 85 |
52,639 | 11.095. A sphere is circumscribed around a regular triangular prism, the height of which is twice the side of the base. How does its volume relate to the volume of the prism? | \frac{64\pi}{27} |
52,673 | Two dice are rolled. What is the probability that the product of the numbers obtained on the two dice is divisible by 6? | \frac{5}{12} |
52,706 | 6. For a natural number $n$, $G(n)$ denotes the number of natural numbers $m$ for which $m+n$ divides $m n$. Find $G\left(10^{\mathrm{k}}\right)$. | 2k^{2}+2k |
52,721 | 1. Among the 95 numbers $1^{2}, 2^{2}, 3^{2}, \cdots, 95^{2}$, the numbers with an odd digit in the tens place total $\qquad$.
untranslated part: $\qquad$ | 19 |
52,732 | 3. Given the function $f(x)=x^{2}-2 x+a$. If $\{x \mid f(x)=x\}=\{x \mid f(f(x))=x\}$, then the range of the real number $a$ is $\qquad$ . | [\frac{5}{4},+\infty) |
52,738 | 700*. In what number system is the number $11111_{d}$ a perfect square? | 3 |
52,745 | 18. Find the number of real roots of the equation $\log _{10}^{2} x-\left\lfloor\log _{10} x\right\rfloor-2=0$.
(Note: $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$.) | 3 |
52,747 | $ABCD$ is a square of side 1. $P$ and $Q$ are points on $AB$ and $BC$ such that $\widehat{PDQ} = 45^{\circ}$. Find the perimeter of $\Delta PBQ$. | 2 |
52,751 | 3. The isosceles trapezoid $A B C D$ has parallel sides $A B$ and $C D$. We know that $\overline{A B}=6, \overline{A D}=5$ and $\angle D A B=60^{\circ}$. A light ray is launched from $A$ that bounces off $C B$ at point $E$ and intersects $A D$ at point $F$. If $A F$ $=3$, calculate the area of triangle $A F E$. | \frac{3\sqrt{3}}{2} |
52,786 | 3. Let the function be
$$
f(x)=\sqrt{2 x^{2}+2 x+41}-\sqrt{2 x^{2}+4 x+4}(x \in \mathbf{R}) \text {. }
$$
Then the maximum value of $f(x)$ is $\qquad$ | 5 |
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