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69,008 | 2. On 8 balls, numbers are written: $2,3,4,5,6,7,8,9$. In how many ways can the balls be placed into three boxes so that no box contains a number and its divisor? | 432 |
68,814 | During a partial solar eclipse, when the apparent diameter of the Moon and the Sun was the same, at the maximum moment, the edge of the moon disk coincided with the center of the sun disk. What was the percentage of the solar eclipse? | 0.391 |
60,411 | 5. A circle is circumscribed around a right triangle $\mathrm{ABC}$ with hypotenuse $\mathrm{AB}$. On the larger leg $\mathrm{AC}$, a point $\mathrm{P}$ is marked such that $\mathrm{AP}=\mathrm{BC}$. On the arc $\mathrm{ACB}$, its midpoint $\mathrm{M}$ is marked. What can the angle $\mathrm{PMC}$ be equal to? | 90 |
59,508 | Example 2 (17th All-Soviet Union Mathematical Olympiad, 9th Grade) One side of a rectangle is $1 \mathrm{~cm}$. It is known that it is divided into four smaller rectangles by two perpendicular lines, with the area of three of them being no less than $1 \mathrm{~cm}^{2}$, and the area of the fourth one being no less than $2 \mathrm{~cm}^{2}$. How long must the other side of the original rectangle be at least? | 3+2\sqrt{2} |
51,542 | Task 5. (20 points) A three-meter gas pipe has rusted in two places. Determine the probability that all three resulting parts can be used as connections to gas stoves, if according to regulations, the stove should not be located closer than 75 cm to the main gas pipe.
# | \frac{1}{16} |
16,076 | Let \( E \) be a point moving inside the square \( ABCD \). Given that the minimum value of the sum of distances from \( E \) to the points \( A \), \( B \), and \( C \) is \( \sqrt{2} + \sqrt{6} \), find the side length of this square. | 2 |
10,123 | Let $a,b,c,d,e,f,g,h,i$ be distinct integers from $1$ to $9.$ The minimum possible positive value of \[\dfrac{a \cdot b \cdot c - d \cdot e \cdot f}{g \cdot h \cdot i}\] can be written as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | 289 |
1,667 | Let \( d \) be a positive divisor of 2015. Then the maximum value of the unit digit of \( d^{\frac{2005}{d}} \) is \(\qquad\). | 7 |
54,862 | 21311 ㅊ Let the three-digit number $n=\overline{a b c}$, if $a, b, c$ can form an isosceles (including equilateral) triangle, find the number of such three-digit numbers $n$.
| 165 |
61,710 | 2. Simplify $\sqrt[3]{10+7 \sqrt{2}}+\sqrt[3]{10-7 \sqrt{2}}$ to get the result $\qquad$ . | 2 \sqrt[3]{4} |
22,117 | Given six thin wooden sticks, the two longer ones are $\sqrt{3}a$ and $\sqrt{2}a$, while the remaining four are of length $a$. If they are used to form a triangular prism, find the cosine of the angle between the lines containing the two longer edges. | \dfrac{\sqrt{6}}{3} |
20,710 | 6. The number of triangles with different shapes that can be formed using the vertices of a regular tridecagon is $\qquad$ (Note: Congruent triangles are considered to have the same shape). | 14 |
58,895 | 3. A circle is inscribed in an isosceles trapezoid $A B C D$ and touches the base $C D$ at point $L$, and the legs $B C$ and $A D$ at points $K$ and $M$, respectively. In what ratio does the line $A L$ divide the segment $M K$? | 1:3 |
28,792 | Example 5 If real numbers $x, y, z$ satisfy $x^{2}+y^{2}+z^{2}=1$, find the maximum value of $\left(x^{2}-y z\right)\left(y^{2}-z x\right)\left(z^{2}-x y\right)$. | \frac{1}{8} |
52,093 | 4. How many natural numbers less than 10000 have exactly three equal digits? | 333 |
29,622 | Problem 11.3. (15 points) Natural numbers $a, b, c$ are such that $1 \leqslant a<b<c \leqslant 3000$. Find the greatest possible value of the quantity
$$
\text { GCD }(a, b)+\text { GCD }(b, c)+\text { GCD }(c, a)
$$ | 3000 |
54,169 | 8.1. Find ten natural numbers whose sum and product are equal to 20. | 1,1,1,1,1,1,1,1,2,10 |
34,478 | 12. (12 points) A row of 2012 balls, colored red, yellow, and blue, are arranged in a line, with a distance of 1 centimeter between each adjacent pair of balls. In every set of 4 adjacent balls, there is 1 red ball, 1 yellow ball, and 2 blue balls. The distance between the 100th red ball from the left and the 100th yellow ball from the right is 1213 centimeters. The distance between the 100th blue ball from the left and the 100th blue ball from the right is $\qquad$ centimeters. | 1615 |
19,676 | \section*{Problem 4 - 330944}
Someone finds the statement
\[
22!=11240007277 * * 607680000
\]
In this, the two digits indicated by \(*\) are illegible. He wants to determine these digits without performing the multiplications that correspond to the definition of 22!.
Conduct such a determination and justify it! It may be used that the given digits are correct.
Hint: For every positive integer \(n\), \(n\)! is defined as the product of all positive integers from 1 to \(n\). | 77 |
57,468 | 4.5.3 $\star \star$ Find all real numbers $k$ such that the inequality
$$
a^{3}+b^{3}+c^{3}+d^{3}+1 \geqslant k(a+b+c+d)
$$
holds for all $a, b, c, d \in[-1,+\infty$ ). | \frac{3}{4} |
27,567 | Let's determine the minimum of the function
$$
\sqrt{x^{2}+1}+\sqrt{y^{2}+9}+\sqrt{x^{2}+y^{2}-20 x-20 y+2 x y+104}
$$ | \sqrt{136} |
17,081 | 5. Given the function $f(x)$ as shown in Table 1. $a_{k}(0 \leqslant k \leqslant 4)$ equals the number of times $k$ appears in $a_{0}, a_{1}, \cdots, a_{4}$. Then $a_{0}+a_{1}+a_{2}+a_{3}=$ $\qquad$ . | 5 |
67,841 | Gy. 1729. Solve the system of equations,
$$
\begin{aligned}
& a_{11} x+a_{12} y+a_{13} z=0 \\
& a_{21} x+a_{22} y+a_{23} z=0 \\
& a_{31} x+a_{32} y+a_{33} z=0
\end{aligned}
$$
if the coefficients satisfy the following conditions:
a) each one is positive
b) the sum of the coefficients in each row and each column is 1,
c) $a_{11}=a_{22}=a_{33}=1 / 2$. | x=0,y=0,z=0 |
4,434 | Given that \(a \leq b < c\) are the side lengths of a right triangle, find the maximum constant \(M\) such that
$$
\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geq \frac{M}{a+b+c} .
$$ | 5 + 3\sqrt{2} |
25,006 | A farmer sold domestic rabbits. By the end of the market, he sold exactly one-tenth as many rabbits as the price per rabbit in forints. He then distributed the revenue between his two sons. Starting with the older son, the boys alternately received one-hundred forint bills, but at the end, the younger son received only a few ten-forint bills. The father then gave him his pocket knife and said that this made their shares equal in value. How much was the pocket knife worth? | 40 |
28,932 | 9-36 Find the smallest real number $c$ such that for any positive sequence $\left\{x_{n}\right\}$, if
$$
\begin{array}{l}
x_{1}+x_{2}+\cdots+x_{n} \leqslant x_{n+1}, n=1,2,3, \cdots \text { then } \\
\sqrt{x_{1}}+\sqrt{x_{2}}+\cdots+\sqrt{x_{n}} \leqslant c \sqrt{x_{1}+x_{2}+\cdots+x_{n}}, n=1,2,3, \cdots
\end{array}
$$ | \sqrt{2}+1 |
21,280 | Construct a square such that two adjacent vertices lie on a circle with a unit radius, and the side connecting the other two vertices is tangent to the circle. Calculate the sides of the square! | \dfrac{8}{5} |
55,311 | 21.3.11 If the sum of the digits of a positive integer $a$ equals 7, then $a$ is called a "lucky number". Arrange all "lucky numbers" in ascending order as $a_{1}, a_{2}, a_{3}, \cdots$, if $a_{n}-2005$, find $a_{5 n}$. | 52000 |
67,836 | \section*{Problem 3 - 021133}
In how many different ways can the number 99 be expressed as the sum of three distinct prime numbers?
(Two cases are considered the same if the same addends appear, merely in a different order.) | 21 |
3,218 | We defined an operation denoted by $*$ on the integers, which satisfies the following conditions:
1) $x * 0 = x$ for every integer $x$;
2) $0 * y = -y$ for every integer $y$;
3) $((x+1) * y) + (x * (y+1)) = 3(x * y) - x y + 2 y$ for every integer $x$ and $y$.
Determine the result of the operation $19 * 90$. | 1639 |
57,679 | B1. What is the smallest positive integer consisting of the digits 2, 4, and 8, where each of these digits appears at least twice and the number is not divisible by 4? | 244882 |
29,772 | 7. Given a quadrilateral pyramid $P-ABCD$ with the base $ABCD$ being a rhombus with a top angle of $60^{\circ}$, and each side face forms a $60^{\circ}$ angle with the base, a point $M$ inside the pyramid is equidistant from the base and each side face, with the distance being 1. Then the volume of the pyramid is $\qquad$ | 8\sqrt{3} |
4,682 | As shown in the figure, in the right triangular prism $ABC-A_{1}B_{1}C_{1}$, $\angle ACB=90^\circ$, $AC=6$, $BC=CC_{1}=\sqrt{2}$, let $P$ be a moving point on the segment $B_{1}C$. Find the minimum value of $AP + PC_{1}$. | 5\sqrt{2} |
57,667 | Example 2 As shown in Figure 2, in $\triangle A B C$, $\angle A B C$ $=46^{\circ}, D$ is a point on side $B C$, $D C=A B$,
$\angle D A B=21^{\circ}$. Try to find
the degree measure of $\angle C A D$. | 67^{\circ} |
57,056 | 2. Given a convex $n$-sided polygon $A_{1} A_{2} \cdots A_{n}(n>4)$ where all interior angles are integer multiples of $15^{\circ}$, and $\angle A_{1}+\angle A_{2}+\angle A_{3}=$ $285^{\circ}$. Then, $n=$ | 10 |
31,579 | It is known that ЖЖ + Ж = МЁД. What is the last digit of the product: В $\cdot И \cdot H \cdot H \cdot U \cdot \Pi \cdot У \cdot X$ (different letters represent different digits, the same letters represent the same digits)?
# | 0 |
17,662 | Let \( x_{i} > 0 \) (for \( i = 1, 2, \cdots, n \)), \( m \in \mathbf{R}^{+} \), \( a \geq 0 \), and \( \sum_{i=1}^{n} x_{i} = s \leq n \). Then, it holds that \( \prod_{i=1}^{n} \left( x_{i}^{m} + \frac{1}{x_{i}^{m}} + a \right) \geq \left[ \left( \frac{s}{n} \right)^{m} + \left( \frac{n}{s} \right)^{m} + a \right]^{n} \). | \prod_{i=1}^{n} \left( x_{i}^{m} + \frac{1}{x_{i}^{m}} + a \right) \geq \left[ \left( \frac{s}{n} \right)^{m} + \left( \frac{n}{s} \right)^{m} + a \right]^{n} |
32,825 | 2. Positive numbers $a, b, c$ are such that $a^{2}+b^{2}+c^{2}=3$. Find the minimum value of the expression
$$
A=\frac{a^{4}+b^{4}}{c^{2}+4 a b}+\frac{b^{4}+c^{4}}{a^{2}+4 b c}+\frac{c^{4}+a^{4}}{b^{2}+4 c a}
$$ | \frac{6}{5} |
56,773 | Bakayev E.V.
A mathematician with five children walked into a pizzeria.
Masha: I want one with tomatoes and no sausage.
Vanya: And I want one with mushrooms.
Dasha: I'll have one without tomatoes.
Nikita: I want one with tomatoes. But no mushrooms!
Igor: And I want one without mushrooms. But with sausage!
Dad: Well, with such picky eaters, one pizza definitely won't be enough...
Will the mathematician be able to order two pizzas and treat each child to the one they want, or will he have to order three pizzas? | 3 |
33,586 | In a football tournament, each team plays exactly twice against each of the others. There are no draws, a win earns two points, and a loss earns nothing. It turns out that only one team won the tournament with 26 points, and there are two teams tied for last with 20 points each. Determine the number of teams, and provide an example of a tournament where such results occur. | 12 |
59,206 | 3.259. $\frac{\operatorname{tg}\left(\frac{5}{4} \pi-4 \alpha\right) \sin ^{2}\left(\frac{5}{4} \pi+4 \alpha\right)}{1-2 \cos ^{2} 4 \alpha}$. | -\frac{1}{2} |
56,777 | 8. For positive integer $a$ and integers $b$, $c$, the quadratic equation $a x^{2}+b x+c=0$ has two roots $\alpha, \beta$. And it satisfies $0<\alpha<\beta<$ 1. Find the minimum value of $a$. | 5 |
61,663 | 6. A triangle with vertices at $(1003,0),(1004,3)$, and $(1005,1)$ in the $x y$-plane is revolved all the way around the $y$-axis. Find the volume of the solid thus obtained. | 5020\pi |
58,287 | Bakayev E.V.
In a circle, there are boys and girls (both are present), a total of 20 children. It is known that for each boy, the neighbor in the clockwise direction is a child in a blue T-shirt, and for each girl, the neighbor in the counterclockwise direction is a child in a red T-shirt. Can the number of boys in the circle be determined unambiguously? | 10 |
51,793 | A set of $8$ problems was prepared for an examination. Each student was given $3$ of them. No two students received more than one common problem. What is the largest possible number of students? | 8 |
20,430 | The sequence \(\left(a_{n}\right)\) is an arithmetic progression with a common difference of 1. It is known that \( \mathrm{S}_{2022} \) is the smallest sum among all sums \( S_{n} \) (smaller than the sum of the first \( n \) terms for any other value of \( n \) ). What values can the first term of the progression take? | (-2022, -2021) |
65,236 | 12.176. A circle is inscribed in an isosceles triangle with base $a$ and angle $\alpha$ at the base. Find the radius of the circle that is tangent to the inscribed circle and the lateral sides of the triangle. | \frac{}{2}\tan^3\frac{\alpha}{2} |
51,608 | Example 1. The roots of the equation
$$
x^{3}+a x^{2}+b x+c=0
$$
form a geometric progression. What necessary and sufficient condition must the coefficients of the equation satisfy? | ^{3}=b^{3} |
21,945 | What is the maximum number of kings, not attacking each other, that can be placed on a standard $8 \times 8$ chessboard? | 16 |
10,919 | There are 54 drivers employed in a garage. How many days off can each driver have in a month (30 days), if every day 25% of the 60 cars are in the garage for preventive maintenance? | 5 |
12,565 | In some cells of a $1 \times 2100$ strip, a chip is placed in each cell. A number equal to the absolute difference between the number of chips to the left and to the right of this cell is written in each empty cell. It is known that all recorded numbers are different and non-zero. What is the smallest number of chips that can be placed in the cells? | 1400 |
707 | In an infinite sequence of natural numbers, the product of any fifteen consecutive terms is equal to one million, and the sum of any ten consecutive terms is equal to \(S\). Find the maximum possible value of \(S\). | 208 |
26,760 | 10. $A$ is the center of a semicircle, with radius $A D$ lying on the base. $B$ lies on the base between $A$ and $D$, and $E$ is on the circular portion of the semicircle such that $E B A$ is a right angle. Extend $E A$ through $A$ to $C$, and put $F$ on line $C D$ such that $E B F$ is a line. Now $E A=1, A C=\sqrt{2}, B F=\frac{2-\sqrt{2}}{4}, C F=\frac{2 \sqrt{5}+\sqrt{10}}{4}$, and $D F=\frac{2 \sqrt{5}-\sqrt{10}}{4}$. Find $D E$. | \sqrt{2-\sqrt{2}} |
53,305 | Task 1 - 331211 Determine all natural numbers $n$ for which the following conditions are satisfied:
The number $n$ is ten-digit. For the digits of its decimal representation, denoted from left to right by $a_{0}, a_{1}$, $\ldots, a_{9}$, it holds that: $a_{0}$ matches the number of zeros, $a_{1}$ matches the number of ones, ..., $a_{9}$ matches the number of nines in the decimal representation of $n$. | 6210001000 |
56,709 | 15. (15 points) There are 8 football teams participating in a round-robin tournament. The winning team gets 1 point, the losing team gets 0 points, and in the case of a draw, both teams get 0.5 points. After the tournament, the teams are ranked based on their points from highest to lowest, and it is found that: all teams have different scores, and the second-place team's score is the same as the total score of the last four teams. Find the score of the team that finished in second place. | 6 |
68,486 | 3. In a football tournament, seven teams played: each team played once with each other. Teams that scored thirteen or more points advance to the next round. Three points are awarded for a win, one point for a draw, and zero points for a loss. What is the maximum number of teams that can advance to the next round? | 4 |
4,538 | In the convex quadrilateral $ABCD$, it is known that $\angle CBD = 58^\circ$, $\angle ABD = 44^\circ$, and $\angle ADC = 78^\circ$. Find the angle $\angle CAD$. | 58^\circ |
21,129 | A needle lies on a plane. It is allowed to rotate the needle by $45^{\circ}$ around either of its ends. Is it possible, after making several such rotations, to return the needle to its original position with its ends swapped? | \text{No} |
24,097 | Buses travel along a country road, at equal intervals in both directions and at equal speeds. A cyclist, traveling at $16 \, \mathrm{km/h}$, begins counting buses from the moment when two buses meet beside him. He counts the 31st bus approaching from the front and the 15th bus from behind, both meeting the cyclist again.
How fast were the buses traveling? | 46 |
67,859 | 9. (Adapted from the 1st "Hope Cup" Senior High School Competition) Let the function $f(n)=k$, where $n$ is a natural number, and $k$ is the digit at the $n$-th position after the decimal point of the irrational number $\pi=3.1415926535 \cdots$, with the rule that $f(0)=3$. Let $F_{n}=$ $\underbrace{f\{f\{f\{f\{f}(n)\} \cdots\}\}$, then $F[f(1990)+f(5)+f(13)]=$ $\qquad$. | 1 |
29,642 | 8. Find the minimum value of the discriminant of a quadratic trinomial, the graph of which has no common points with the regions located below the x-axis and above the graph of the function $y=\frac{1}{\sqrt{1-x^{2}}}$. | -4 |
25,862 | 7. There is a conical container with its vertex at the bottom and its base horizontal, and its axial section is an equilateral triangle with a side length of 6. The container is filled with water. Now, a square prism with a base side length of $a(a<6)$ is vertically immersed in the container. To make the water overflow from the container as much as possible, the value of $a$ should be $\qquad$ . | 2\sqrt{2} |
25,234 | What is the least possible number of cells that can be marked on an $n \times n$ board such that for each $m>\frac{n}{2}$ both diagonals of any $m \times m$ sub-board contain a marked cell?
## Answer: $n$. | n |
4,843 | Given that \( AB \) is a diameter of circle \( \odot C \) with a radius of 2, circle \( \odot D \) is internally tangent to \( \odot C \) at point \( A \), and circle \( \odot E \) is tangent to \( \odot C \), externally tangent to \( \odot D \), and tangent to \( AB \) at point \( F \). If the radius of \( \odot D \) is 3 times the radius of \( \odot E \), find the radius of \( \odot D \). | 4\sqrt{15} - 14 |
26,140 | # Problem 3.
In triangle $A B C$, the bisector $B E$ and the median $A D$ are equal and perpendicular. Find the area of triangle $A B C$, if $A B=\sqrt{13}$. | 12 |
52,006 | ## Task Condition
Calculate the areas of figures bounded by lines given in polar coordinates.
$$
r=\cos \phi-\sin \phi
$$ | \frac{\pi}{2} |
56,166 | 5. The government has decided to privatize civil aviation. For each pair of the country's 127 cities, the connecting airline is sold to one of the private airlines. Each airline must make all the purchased air routes one-way, but in such a way as to ensure the possibility of flying from any city to any other (possibly with several layovers). What is the maximum number of companies that can buy the air routes? | 63 |
55,776 | 2.2. $A L, B M, C N$ - medians of triangle $A B C$, intersecting at point $K$. It is known that quadrilateral $C L K M$ is cyclic, and $A B=2$. Find the length of median $C N$. | \sqrt{3} |
15,858 | Prove that for any positive integer \( n \), the inequality \( \sqrt{2 \sqrt{3 \sqrt{4 \cdots \sqrt{n}}}} < 3 \) holds. | \sqrt{2 \sqrt{3 \sqrt{4 \cdots \sqrt{n}}}} < 3 |
59,897 | 7. Let $M=\{1,2, \cdots, 17\}$. If there are four distinct numbers $a, b, c, d \in M$, such that $a+b \equiv c+d(\bmod 17)$, then $\{a, b\}$ and $\{c, d\}$ are called a “balanced pair” of set $M$. The number of balanced pairs in set $M$ is $\qquad$ | 476 |
62,758 | 1. The students $A, B, C, D$ competed in a race. Each of them predicted the order in which the race would finish. Student $A$: $A B D C$, student $B$: BACD, student $C$: $C B D A$, student $D$: $D C B A$. It turned out that no one correctly predicted the order of all the racers, but only one of them guessed the position of just one racer. What was the order of the finishers in the race? | CDAB |
59,235 | 19. Monochromatic pairs. In a box, there are $n$ white and $n$ black balls. Balls are drawn from the box in pairs - random pairs.
a) (from 9th grade. 2 points). Find the expected number of mixed-color pairs drawn from the box by the time it is empty.
b) (from 9th grade. 5 points). Suppose that if a pair is mixed-color, it is set aside, and if it is monochromatic, it is returned to the box. Such an operation is called an attempt. Find the expected number of attempts needed to set all the balls aside, leaving nothing in the box. | 2n-H_n |
68,981 | 4. (13 points) In a circle, there are 17 people: each of them is either a truth-teller (he always tells the truth) or a liar (he always lies). Everyone said that both of their neighbors are liars. What is the maximum number of liars that can be in this circle? | 11 |
6,295 | x₁, x₂, ..., xₙ is any sequence of positive real numbers, and yᵢ is any permutation of xᵢ. Show that ∑(xᵢ / yᵢ) ≥ n. | \sum_{i=1}^{n} \frac{x_i}{y_i} \geq n |
30,501 | 8,9 In trapezoid $A B C D$, points $K$ and $M$ are the midpoints of the bases $A B$ and $C D$, respectively. It is known that $A M$ is perpendicular to $D K$ and $C K$ is perpendicular to $B M$, and the angle $C K D$ is $60^{\circ}$. Find the area of the trapezoid if its height is 1. | \frac{4\sqrt{3}}{3} |
14,024 | In the pentagon \(ABCDE\), \(AB = BC = CD = DE\), \(\angle B = 96^\circ\) and \(\angle C = \angle D = 108^\circ\). Find angle \(\angle E\). | 102 |
30,848 | Example 3 Let the set $S=\{1,2, \cdots, 280\}$. Find the smallest positive integer $n$, such that every subset of $S$ with $n$ elements contains five pairwise coprime numbers.
(32nd IMO) | 217 |
9,055 | In a right-angled triangle, one of the angles is $30^{\circ}$. Prove that the segment of the perpendicular drawn to the hypotenuse through its midpoint, intersecting the longer leg, is three times shorter than the longer leg. | \frac{1}{3} |
27,229 | 14. $A B C D$ is a square with side length 9 . Let $P$ be a point on $A B$ such that $A P: P B=7: 2$. Using $C$ as centre and $C B$ as radius, a quarter circle is drawn inside the square. The tangent from $P$ meets the circle at $E$ and $A D$ at $Q$. The segments $C E$ and $D B$ meet at $K$, while $A K$ and $P Q$ meet at $M$. Find the length of $A M$.
$A B C D$ 是一個邊長爲 9 的正方形。設 $P$ 爲 $A B$ 上的一點, 使得 $A P: P B=7: 2 \circ$ 以 $C$ 緺圓心 、 $C B$ 爲牛徑在正方形內作一個四分一圓, 從 $P$ 點到圓的切線與圓相交於 $E$, 與 $A D$ 相交於 $Q$ 。 $C E$ 和 $D B$ 交於 $K$, 而 $A K$ 和 $P Q$ 則交於 $M$ 。求 $A M$ 的長度。 | \frac{85}{22} |
59,597 | Let $A$ be a subset of $\{1,2,3, \ldots, 2019\}$ having the property that the difference between any two of its elements is not a prime number. What is the largest possible number of elements of $A$?
# | 505 |
62,433 | $$
\begin{array}{l}
\frac{2^{2}}{1 \times 3} \times \frac{4^{2}}{3 \times 5} \times \cdots \times \frac{2016^{2}}{2015 \times 2017} \\
=\quad \text { (accurate to } 0.01 \text { ). }
\end{array}
$$ | 1.57 |
14,753 | ## Task Condition
Find the $n$-th order derivative.
$y=\log _{3}(x+5)$ | \dfrac{(-1)^{n-1} (n-1)!}{\ln 3 \cdot (x+5)^n} |
63,363 | 39. As shown in the figure, $P A=P B, \angle A P B=2 \angle A C B, A C$ intersects $P B$ at point $D$, and $P B=4, P D=3$, then $A D \cdot D C=$ . $\qquad$ | 7 |
64,266 | Question 78: Given $x, y, z > 0$, find the minimum value of $2 \sqrt{(x+y+z)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)}-\sqrt{\left(1+\frac{x}{y}\right)\left(1+\frac{y}{z}\right)}$. | 2\sqrt{2}+1 |
55,406 | $9.17 C_{x}^{x-1}+C_{x}^{x-2}+C_{x}^{x-3}+\ldots+C_{x}^{x-9}+C_{x}^{x-10}=1023$. | 10 |
61,274 | How many points does one have to place on a unit square to guarantee that two of them are strictly less than 1/2 unit apart? | 10 |
67,512 | 1767. Among the standard products of one factory, on average $15 \%$ belong to the second grade. With what probability can it be stated that the percentage of second-grade products among 1000 standard products of this factory differs from $15 \%$ by less than $2 \%$ in absolute value? | 0.92327 |
54,307 | Mary typed a six-digit number, but the two $1$s she typed didn't show. What appeared was $2002$. How many different six-digit numbers could she have typed?
$\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }15\qquad\textbf{(E) }20$ | 15 |
29,757 | Bakayev E.V.
Petya places 500 kings on the cells of a $100 \times 50$ board so that they do not attack each other. And Vasya places 500 kings on the white cells (in a chessboard coloring) of a $100 \times 100$ board so that they do not attack each other. Who has more ways to do this? | Vasya |
64,947 | $19422 *$ Find the largest integer $x$ that makes $4^{27}+4^{1000}+4^{x}$ a perfect square. | 1972 |
64,502 | 8.2. In triangle $A B C$, angle $A$ is the largest. Points $M$ and $N$ are symmetric to vertex $A$ with respect to the angle bisectors of angles $B$ and $C$ respectively. Find $\angle A$, if $\angle M A N=50^{\circ}$. | 80 |
67,531 | 1. (17 points) Find the area of the triangle cut off by the line $y=3 x+1$ from the figure defined by the inequality $|x-1|+|y-2| \leq 2$. | 2 |
51,313 | 5. In a $4 \times 4$ grid, fill each cell with 0 or 1, such that the sum of the four numbers in each $2 \times 2$ subgrid is odd. There are $\qquad$ different ways to do this. | 128 |
23,285 | Determine the range of values for \( m \) such that the line \( y = 4x + m \) has two distinct points \( A \) and \( B \) on the ellipse \( \frac{x^2}{4} + \frac{y^2}{3} = 1 \) that are symmetric with respect to the line. | \left( -\dfrac{2\sqrt{13}}{13}, \dfrac{2\sqrt{13}}{13} \right) |
50,525 | 5. A semicircle of radius 1 is drawn inside a semicircle of radius 2, as shown in the diagram, where $O A=O B=2$.
A circle is drawn so that it touches each of the semicircles and their common diameter, as shown.
What is the radius of the circle? | \frac{8}{9} |
21,166 | For \(0 \leq x \leq 1\) and positive integer \(n\), let \(f_0(x) = |1 - 2x|\) and \(f_n(x) = f_0(f_{n-1}(x))\). How many solutions are there to the equation \(f_{10}(x) = x\) in the range \(0 \leq x \leq 1\)? | 2048 |
11,637 | Given \( f(x) = \sum_{k=1}^{2017} \frac{\cos k x}{\cos^k x} \), find \( f\left(\frac{\pi}{2018}\right) \). | -1 |
6,536 | On two perpendicular axes \(OX\) and \(OY\), we choose two points, \(A\) and \(B\), such that \(OA + OB\) equals a constant \(k\). From point \(O\), we draw a line parallel to \(AB\), which intersects the circumcircle of triangle \(AOB\) at point \(M\). What is the locus of point \(M\)? | \left(x - \frac{k}{2}\right)^2 + \left(y - \frac{k}{2}\right)^2 = \frac{k^2}{2} |
29,853 | How many $(n ; k)$ number pairs are there for which $n>k$, and the difference between the interior angles of the $n$-sided and $k$-sided regular polygons is $1^{\circ}$? | 52 |
54,831 | For how many natural numbers $n$ between $1$ and $2014$ (both inclusive) is $\frac{8n}{9999-n}$ an integer?
| 1 |
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