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22,535 | An \( n \times n \) board is colored in \( n \) colors such that the main diagonal (from top-left to bottom-right) is colored in the first color; the two adjacent diagonals are colored in the second color; the two next diagonals (one from above and one from below) are colored in the third color, etc.; the two corners (top-right and bottom-left) are colored in the \( n \)-th color. It happens that it is possible to place on the board \( n \) rooks, no two attacking each other and such that no two rooks stand on cells of the same color. Prove that \( n \equiv 0 \pmod{4} \) or \( n \equiv 1 \pmod{4} \). | n \equiv 0 \pmod{4} \text{ or } n \equiv 1 \pmod{4} |
22,241 | Given the inequality \( a x < 6 \) with respect to \( x \), the solution also satisfies the inequality \( \frac{3x - 6a}{2} > \frac{a}{3} - 1 \). Find the range of values for the real number \( a \). | (-\infty, -\dfrac{3}{2}] |
1,555 | There are three segments of each length $2^{n}$ for $n=0,1, \cdots, 1009$. How many non-congruent triangles can be formed using these 3030 segments? (Answer in numerical form) | 510555 |
26,528 | 3. Given a triangle $ABC$. The angle bisector at vertex $A$ intersects side $BC$ at point $D$. Let $E$, $F$ be the centers of the circumcircles of triangles $ABD$, $ACD$. What can be the measure of angle $BAC$ if the center of the circumcircle of triangle $AEF$ lies on the line $BC$?
(Patrik Bak) | 120 |
56,564 | 1. [ $x$ ] represents the greatest integer not greater than $x$, then the real solution $x$ of the equation $\frac{1}{2} \times\left[x^{2}+x\right]=19 x+99$ is $\qquad$. | x=-\frac{181}{38} \text{ or } \frac{1587}{38} |
24,614 | Rectangle \(ABCD\) has area 2016. Point \(Z\) is inside the rectangle and point \(H\) is on \(AB\) so that \(ZH\) is perpendicular to \(AB\). If \(ZH : CB = 4 : 7\), what is the area of pentagon \(ADCZB\)? | 1440 |
34,343 | 9. $[\boldsymbol{7}] g$ is a twice differentiable function over the positive reals such that
$$
\begin{aligned}
g(x)+2 x^{3} g^{\prime}(x)+x^{4} g^{\prime \prime}(x) & =0 \quad \text { for all positive reals } x . \\
\lim _{x \rightarrow \infty} x g(x) & =1
\end{aligned}
$$
Find the real number $\alpha>1$ such that $g(\alpha)=1 / 2$. | \frac{6}{\pi} |
12,484 | 80. One day, Xiao Ben told a joke. Except for Xiao Ben himself, four-fifths of the classmates in the classroom heard it, but only three-quarters of the classmates laughed. It is known that one-sixth of the classmates who heard the joke did not laugh. Then, what fraction of the classmates who did not hear the joke laughed? | \dfrac{5}{12} |
19,104 | Given that \( m, n, k \) are positive integers, if there exists a pair of positive integers \( (a, b) \) such that
\[
(1+a) n^{2}-4(m+a) n+4 m^{2}+4 a+b(k-1)^{2}<3,
\]
then the number of possible values of \( m+n+k \) is \(\quad\) . | 4 |
2,567 | Find a positive integer \( n \) less than 2006 such that \( 2006n \) is a multiple of \( 2006 + n \). | 1475 |
24,439 |
Let \( a_{1}, a_{2}, \cdots, a_{k}\left(k \in \mathbf{Z}_{+}\right) \) be integers greater than 1, and they satisfy
\[
\left(a_{1}!\right)\left(a_{2}!\right) \cdots\left(a_{k}!\right) \mid 2017!
\]
Determine the maximum value of \( \sum_{i=1}^{k} a_{i} \) as \( k \) varies. | 5024 |
60,733 | [The Pigeonhole Principle (continued).]
In a photo studio, 20 birds flew in - 8 sparrows, 7 wagtails, and 5 woodpeckers. Each time the photographer clicks the camera shutter, one of the birds flies away (permanently). How many shots can the photographer take to be sure: he will have at least four birds of one species left, and at least three of another?
# | 7 |
64,933 | Problem 4. Three lines intersect to form 12 angles, and $n$ of them turn out to be equal. What is the maximum possible value of $n$? | 6 |
16,909 | 5. A natural number $n$ is called cubish if $n^{3}+13 n-273$ is a cube of a natural number. Find the sum of all cubish numbers. | 29 |
55,246 | 20. 11, 12, 13 are three consecutive natural numbers, the sum of their digits is $1+1+1+2+1+3=9$. There are three consecutive natural numbers, all less than 100, the sum of their digits is 18, there are $\qquad$ possible cases. | 8 |
24,373 | A circle with its center on the line \( y = b \) intersects the parabola \( y = \frac{3}{4} x^{2} \) at least at three points; one of these points is the origin, and two of the remaining points lie on the line \( y = \frac{3}{4} x + b \). Find all values of \( b \) for which the described configuration is possible. | \dfrac{25}{12} |
5,621 | For which values of the parameter \(a\) among the solutions of the inequality \(\left(x^{2}-a x+2 x-2 a\right) \sqrt{5-x} \leqslant 0\) can two solutions be found such that the difference between them is 5? | (-\infty, -7] \cup [0, \infty) |
66,288 | 26. [14] $w, x, y, z$ are real numbers such that
$$
\begin{aligned}
w+x+y+z & =5 \\
2 w+4 x+8 y+16 z & =7 \\
3 w+9 x+27 y+81 z & =11 \\
4 w+16 x+64 y+256 z & =1
\end{aligned}
$$
What is the value of $5 w+25 x+125 y+625 z$ ? | -60 |
7,258 | Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a differentiable function such that \( f(0) = 0 \), \( f(1) = 1 \), and \( |f'(x)| \leq 2 \) for all real numbers \( x \). If \( a \) and \( b \) are real numbers such that the set of possible values of \( \int_{0}^{1} f(x) \, dx \) is the open interval \( (a, b) \), determine \( b - a \). | \dfrac{3}{4} |
2,938 | In triangle \(ABC\), point \(P\) is the point on side \(AB\) closer to \(A\) such that \(P\) divides \(AB\) into a ratio of \(1:3\). Given that \( \angle CPB = \frac{1}{2} \angle CBP \), prove that:
$$
AC = BC + \frac{AB}{2}
$$ | AC = BC + \frac{AB}{2} |
23,897 | In triangle \(ABC\), point \(N\) lies on side \(AB\) such that \(AN = 3NB\); the median \(AM\) intersects \(CN\) at point \(O\). Find \(AB\) if \(AM = CN = 7\) cm and \(\angle NOM = 60^\circ\). | 4\sqrt{7} |
25,327 | Example 3 Suppose the annual interest rate is $i$, calculated on a compound interest basis, one wants to withdraw 1 yuan at the end of the first year, 4 yuan at the end of the second year, $\cdots$, and $n^{2}$ yuan at the end of the $n$-th year, and to be able to withdraw in this manner indefinitely. What is the minimum principal required?
(18th Putnam Problem) | \frac{(1+i)(2+i)}{i^{3}} |
55,760 | 9. 44 Let $k$ be a natural number. Determine for which value of $k$, $A_{k}=\frac{19^{k}+66^{k}}{k!}$ attains its maximum value. | 65 |
58,646 | A bunch of lilac consists of flowers with 4 or 5 petals. The number of flowers and the total number of petals are perfect squares. Can the number of flowers with 4 petals be divisible by the number of flowers with 5 petals? | \text{No} |
7,950 |
In triangle \(ABC\), point \(E\) is the midpoint of side \(BC\), point \(D\) lies on side \(AC\), \(AC = 1\), \(\angle BAC = 60^\circ\), \(\angle ABC = 100^\circ\), \(\angle ACB = 20^\circ\), and \(\angle DEC = 80^\circ\). What is the sum of the area of triangle \(ABC\) and twice the area of triangle \(CDE\)? | \dfrac{\sqrt{3}}{8} |
9,298 | Given that \( P \) is a moving point on the parabola \( y^2 = 2x \), and points \( B \) and \( C \) are on the \( y \)-axis, the circle \((x-1)^2 + y^2 = 1\) is the incircle of \( \triangle PBC \). Then, the minimum value of \( S_{\triangle PBC} \) is ______. | 8 |
22,330 | Prove that \(\frac{a+b-c}{2} < m_{c} < \frac{a+b}{2}\), where \(a\), \(b\), and \(c\) are the side lengths of an arbitrary triangle, and \(m_{c}\) is the median to side \(c\). | \frac{a+b-c}{2} < m_{c} < \frac{a+b}{2} |
57,859 | # 3. Clone 1
A rope was divided into 19 equal parts and arranged in a snake-like pattern. After that, a cut was made along the dotted line. The rope split into 20 pieces: the longest of them is 8 meters, and the shortest is 2 meters. What was the length of the rope before it was cut? Express your answer in meters.
 | 152 |
54,041 | 4.1. A train of length $L=600$ m, moving by inertia, enters a hill with an angle of inclination $\alpha=30^{\circ}$ and stops when exactly a quarter of the train is on the hill. What was the initial speed of the train $V$ (in km/h)? Provide the nearest whole number to the calculated speed. Neglect friction and assume the acceleration due to gravity is $g=10 \mathrm{m} /$ sec $^{2}$. | 49 |
22,341 | 1. Mrs. Pekmezac loves fruit. In her garden, she harvested $5 \frac{3}{4} \mathrm{~kg}$ of blueberries, $4 \frac{1}{2} \mathrm{~kg}$ of cherries, $2 \frac{19}{20} \mathrm{~kg}$ of raspberries, and $1 \frac{4}{5} \mathrm{~kg}$ of blackberries. She decided to store the fruit in the refrigerator, in bags of 75 dag. Fruit of different types can be stored in the same bag. How many bags does she need to prepare at a minimum? | 20 |
59,576 | 11. (12 points) In the figure, by connecting every other vertex of a regular octagon with each side measuring 12 cm, two squares can be formed. The area of the shaded region in the figure is $\qquad$ square centimeters. | 288 |
61,543 | 19. Choose three different numbers from $1,2,3,4,5,6$ to replace $a, b, c$ in the linear equation $a x+b y+c=0$. The number of different lines that can be drawn in the Cartesian coordinate system is $\qquad$. | 114 |
58,366 | Pedrinho is playing with three triangular pieces with sides $(5,8,10),(5,10,12)$ and $(5,8,12)$ as shown in the drawing below. He can join two pieces by exactly gluing the sides of the same length. For example, he can join the side 10 of the first piece with the side 10 of the second, but he cannot join the side 10 of the first piece with the side 8 of the third, as they do not have the same length. What is the largest perimeter that Pedrinho can obtain by joining the three pieces?
10
12
12
# | 49 |
31,955 | 3. Let $F(x)$ and $G(x)$ be polynomials of degree 2021. It is known that for all real $x$, $F(F(x)) = G(G(x))$ and there exists a real number $k, k \neq 0$, such that for all real $x$, $F(k F(F(x))) = G(k G(G(x)))$. Find the degree of the polynomial $F(x) - G(x)$. | 0 |
22,818 | Five cards labeled 1, 3, 5, 7, 9 are laid in a row in that order, forming the five-digit number 13579 when read from left to right. A swap consists of picking two distinct cards and then swapping them. After three swaps, the cards form a new five-digit number \( n \) when read from left to right. Compute the expected value of \( n \). | 50308 |
56,340 | 8. (5 points) The founder of a noble family received a plot of land. Each man in the family, upon dying, divided the land he inherited equally among his sons. If he had no sons, the land went to the state. No other members of the family gained or lost any land in any other way. In total, there were 150 people in the family. What is the smallest fraction of the original plot of land that any member of the family could have received? | \frac{1}{2\cdot3^{49}} |
51,225 | 4. (7 points) The numbers $a, b, c, d$ belong to the interval $[-5 ; 5]$. Find the maximum value of the expression $a+2 b+c+2 d-a b-b c-c d-d a$. | 110 |
5,062 | In how many ways can 6 people be seated at a round table, considering the ways different if at least some of the people get new neighbors? | 60 |
9,211 | The class teacher calculated the class average grades for each subject, and Kati helped her by recalculating the grades based on how many students received each consecutive grade. When comparing the results of the first subject, it turned out that Kati had used the data for consecutive fives in reverse order, taking the number of excellent grades as the number of poor grades. Feeling disheartened, Kati wanted to start the entire calculation over, but the class teacher said that this calculation would still be good for verification. Explain this statement. | 6 |
34,774 | Determine whether there exist, in the plane, 100 distinct lines having exactly 2008 distinct intersection points. | 2008 |
22,670 | 11. (20 points) Given positive real numbers $a, b, c, d$ satisfying $a b c d > 1$. Find the minimum value of $\frac{a^{2}+a b^{2}+a b c^{2}+a b c d^{2}}{a b c d-1}$. | 4 |
12,792 | If the function
$$
f(x) = |a \sin x + b \cos x - 1| + |b \sin x - a \cos x| \quad (a, b \in \mathbf{R})
$$
attains a maximum value of 11, then $a^{2} + b^{2} = \, \, \, $ . | 50 |
62,326 | 16. Given real numbers $x, y$ satisfy $2^{x+1}+2^{y+1}=4^{x}+4^{y}$, find the range of $M=8^{x}+8^{y}$. | (8,16] |
55,763 | Square $SEAN$ has side length $2$ and a quarter-circle of radius $1$ around $E$ is cut out. Find the radius of the largest circle that can be inscribed in the remaining figure. | 5 - 3\sqrt{2} |
59,589 | 3.1. The decreasing sequence $a, b, c$ is a geometric progression, and the sequence $577 a, \frac{2020 b}{7}, \frac{c}{7}$ is an arithmetic progression. Find the common ratio of the geometric progression. | 4039 |
64,213 | 15. A and B take turns tossing a fair coin, and the one who tosses heads first wins. They play several rounds, and it is stipulated that the loser of the previous round tosses first in the next round. If A tosses first in the 1st round, what is the probability that A wins in the 6th round? | \frac{364}{729} |
30,808 | Let $a, b, c, d, e, f$ be non-negative real numbers satisfying $a+b+c+d+e+f=6$. Find the maximal possible value of
$$
a b c+b c d+c d e+d e f+e f a+f a b
$$
and determine all 6-tuples $(a, b, c, d, e, f)$ for which this maximal value is achieved.
Answer: 8 . | 8 |
63,382 | 4. (7 points) The numbers $a, b, c, d$ belong to the interval $[-12.5,12.5]$. Find the maximum value of the expression $a+2 b+c+2 d-a b-b c-c d-d a$. | 650 |
56,779 | 4. In a joint-stock company, there are 2017 shareholders, and any 1500 of them hold a controlling stake (not less than $50 \%$ of the shares). What is the largest share of shares that one shareholder can have
# | 32.8 |
19,906 | 53. A store sells yogurt, each bottle of yogurt costs 4 yuan, and to recycle empty bottles, every three yogurt bottles can be exchanged for one bottle of yogurt. Xiaoliang spent a total of 96 yuan on yogurt, so the maximum number of bottles of yogurt he could have drunk is $\qquad$.
(You can borrow empty bottles, but you must return them) | 36 |
2,353 | For given \( n \) distinct numbers \( a_2, a_2, \cdots, a_n \) \(\in \mathbf{N}^{+}, n > 1\). Define \( p_{i}=\prod_{1 \leqslant j \leqslant n, j \neq i} (a_{i} - a_{j}) \) for \( i = 1, 2, \cdots, n \). Prove that for any \( k \in \mathbf{N}^{+} \), \( \sum_{i=1}^{n} \frac{p_{i}^{k}}{p_{i}} \) is an integer. | \sum_{i=1}^{n} p_{i}^{k-1} \text{ is an integer.} |
19,211 | A circle with radius 3 passes through vertex $B$, the midpoints of sides $AB$ and $BC$, and touches side $AC$ of triangle $ABC$. Given that angle $BAC$ is acute and $\sin \angle BAC = \frac{1}{3}$, find the area of triangle $ABC$. | 16\sqrt{2} |
63,578 | 8. In a regular tetrahedron $ABCD$ with edge length $\sqrt{2}$, it is known that $\overrightarrow{A P}=\frac{1}{2} \overrightarrow{A B}, \overrightarrow{A Q}=\frac{1}{3} \overrightarrow{A C}, \overrightarrow{A R}=\frac{1}{4} \overrightarrow{A D}$. If point $K$ is the centroid of $\triangle BCD$, then the volume of the tetrahedron $K P Q R$ is $\qquad$. | \frac{1}{36} |
5,164 | 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 |
53,046 | Place a circle of the largest possible radius inside a cube.
# | \frac{\sqrt{6}}{4} |
23,867 | The solid shown has a square base of side length $s$ . The upper edge is parallel to the base and has length $2s$ . All other edges have length $s$ . Given that $s=6\sqrt{2}$ , what is the volume of the solid? | 288 |
68,402 | Condition of the problem
Find the derivative.
$$
y=4 \ln \frac{x}{1+\sqrt{1-4 x^{2}}}-\frac{\sqrt{1-4 x^{2}}}{x^{2}}
$$ | \frac{2}{x^{3}\sqrt{1-4x^{2}}} |
18,428 | The edges of a tetrahedron that are opposite to each other have equal lengths and pairwise enclose the same angle. Prove that the tetrahedron is regular. | \text{The tetrahedron is regular.} |
28,879 | 27. (POL 2) Determine the maximum value of the sum
$$
\sum_{i<j} x_{i} x_{j}\left(x_{i}+x_{j}\right)
$$
over all $n$-tuples $\left(x_{1}, \ldots, x_{n}\right)$, satisfying $x_{i} \geq 0$ and $\sum_{i=1}^{n} x_{i}=1$. | \frac{1}{4} |
33,048 | 7.2. Two spheres are inscribed in a dihedral angle, touching each other. The radius of one sphere is three times that of the other, and the line connecting the centers of the spheres forms an angle of $60^{\circ}$ with the edge of the dihedral angle. Find the measure of the dihedral angle. Write the cosine of this angle in your answer, rounding it to two decimal places if necessary. | 0.33 |
34,093 | 3. The sum of positive numbers $a, b, c$ and $d$ does not exceed 4. Find the maximum value of the expression
$$
\sqrt[4]{a^{2}+3 a b}+\sqrt[4]{b^{2}+3 b c}+\sqrt[4]{c^{2}+3 c d}+\sqrt[4]{d^{2}+3 d a}
$$ | 4\sqrt{2} |
52,585 | 1.108 Let $A B C D E F$ be a regular hexagon. A frog starts at vertex $A$ and can randomly jump to one of the two adjacent vertices each time. If it reaches point $D$ within 5 jumps, it stops jumping; if it does not reach point $D$ within 5 jumps, it stops after 5 jumps. How many different ways can the frog jump from the start to the stop? | 26 |
11,479 | In the rectangular coordinate plane, find the number of integer points that satisfy the system of inequalities
\[
\left\{
\begin{array}{l}
y \leqslant 3x \\
y \geqslant \frac{x}{3} \\
x + y \leqslant 100
\end{array}
\right.
\] | 2551 |
30,006 | 6. [5 points] Two circles of the same radius 7 intersect at points $A$ and $B$. On the first circle, a point $C$ is chosen, and on the second circle, a point $D$ is chosen. It turns out that point $B$ lies on the segment $C D$, and $\angle C A D=90^{\circ}$. On the perpendicular to $C D$ passing through point $B$, a point $F$ is chosen such that $B F=B D$ (points $A$ and $F$ are on opposite sides of the line $C D$). Find the length of the segment $C F$.
Fig. 3: variant 2, problem 6 | 14 |
62,079 | 15. Let $S=10 \times \frac{2020^{2021}+2021^{2022}}{2020^{2020}+2021^{2021}}$, then the integer part of $S$ is | 20209 |
3,038 | Given a right triangle \(ABC\) with a right angle at \(A\). On the leg \(AC\), a point \(D\) is marked such that \(AD:DC = 1:3\). Circles \(\Gamma_1\) and \(\Gamma_2\) are then drawn with centers at \(A\) and \(C\) respectively, both passing through point \(D\). \(\Gamma_2\) intersects the hypotenuse at point \(E\). Another circle \(\Gamma_3\) with center at \(B\) and radius \(BE\) intersects \(\Gamma_1\) inside the triangle at a point \(F\) such that \(\angle AFB\) is a right angle. Find \(BC\), given that \(AB = 5\). | 13 |
31,921 | 6. In trapezoid $P Q R S$, it is known that $\angle P Q R=90^{\circ}, \angle Q R S<90^{\circ}$, diagonal $S Q$ is 24 and is the bisector of angle $S$, and the distance from vertex $R$ to line $Q S$ is 16. Find the area of trapezoid PQRS.
Answer. $\frac{8256}{25}$. | \frac{8256}{25} |
68,228 | ## 51. Congratulations from Methuselah
Every New Year, starting from the first year of our era, Methuselah, who is still alive to this day, sends a greeting to his best friend, who, naturally, has changed many times over the centuries and decades. However, the formula for the greeting, on the contrary, has remained unchanged for almost two millennia. It is very simple: "Happy New Year 1", "Happy New Year 2", "Happy New Year 3", and so on, "Happy New Year 1978" and finally, "Happy New Year 1979".
Which digit has Methuselah used the least so far? | 0 |
28,715 | Ayaev A.V.
On a plane, a regular hexagon is drawn, the length of its side is 1. Using only a ruler, construct a segment whose length is $\sqrt{7}$.
# | \sqrt{7} |
53,909 | 9-6. Petya wants to place 99 coins in the cells of a $2 \times 100$ board so that no two coins are in cells that share a side, and no more than one coin is in any cell. How many ways are there to place the coins? | 396 |
5,605 | On a chessboard of size $8 \times 8$, $k$ rooks and $k$ knights are placed so that no piece is attacking any other piece. What is the maximum possible value of $k$ for which this is possible? | 5 |
60,527 | 12. Let $f(x)=x^{2}+6 x+c$ for all real numbers $x$, where $c$ is some real number. For what values of $c$ does $f(f(x))$ have exactly 3 distinct real roots? | \frac{11-\sqrt{13}}{2} |
63,101 | 7. Let the dihedral angles between the three lateral faces and the base of a tetrahedron all be $60^{\circ}$. The side lengths of the base are $7, 8, 9$. Then the lateral surface area of the pyramid is | 24\sqrt{5} |
51,897 | Example 2 (2005 Daoxi College Entrance Examination Question) Given the vectors $\boldsymbol{a}=\left(2 \cos \frac{x}{2}, \tan \left(\frac{x}{2}+\frac{\pi}{4}\right)\right), \boldsymbol{b}=$ $\left(\sqrt{2} \sin \left(\frac{x}{2}+\frac{\pi}{4}\right), \tan \left(\frac{x}{2}-\frac{\pi}{4}\right)\right)$, let $f(x)=\boldsymbol{a} \cdot \boldsymbol{b}$. Find the maximum value of the function $f(x)$, its smallest positive period, and the monotonic intervals of $f(x)$ on $\left[0, \frac{\pi}{2}\right]$. | \sqrt{2} |
69,140 | 1. Let $i_{1}, i_{2}, \cdots, i_{10}$ be a permutation of $1,2, \cdots, 10$. Define $S=\left|i_{1}-i_{2}\right|+\left|i_{3}-i_{4}\right|+\cdots+\left|i_{9}-i_{10}\right|$. Find all possible values of $S$. ${ }^{[2]}$
(2012, Zhejiang Province High School Mathematics Competition) | 5,7,9,11,13,15,17,19,21,23,25 |
58,817 | 4. Find the minimum value of the function $f(x)=\sqrt{4 x^{2}-12 x+8}+\sqrt{4+3 x-x^{2}}$. | \sqrt{6} |
23,784 | Let \( n \geq 2 \) be a fixed integer. Find the least constant \( C \) such that the inequality
\[ \sum_{i<j} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leq C\left(\sum_{i} x_{i}\right)^{4} \]
holds for every \( x_{1}, \ldots, x_{n} \geq 0 \) (the sum on the left consists of \(\binom{n}{2}\) summands). For this constant \( \bar{C} \), characterize the instances of equality. | \dfrac{1}{8} |
51,928 | 24. $[\mathbf{1 4}]$ In how many ways may thirteen beads be placed on a circular necklace if each bead is either blue or yellow and no two yellow beads may be placed in adjacent positions? (Beads of the same color are considered to be identical, and two arrangements are considered to be the same if and only if each can be obtained from the other by rotation). | 41 |
10,708 | Let \( a_{1}, a_{2}, \cdots, a_{21} \) be a permutation of \( 1, 2, \cdots, 21 \) that satisfies
\[ \left|a_{20} - a_{21}\right| \geq \left|a_{19} - a_{21}\right| \geq \left|a_{18} - a_{21}\right| \geq \cdots \geq \left|a_{1} - a_{21}\right|. \]
Determine the number of such permutations. | 3070 |
63,091 | Problem 6.2. A square with a side of 1 m is cut into three rectangles with equal perimeters. What can these perimeters be? List all possible options and explain why there are no others. | \frac{8}{3} |
63,329 | 8. A country has 21 cities, and some airlines can implement air transportation between these cities. Each airline connects pairs of cities with non-stop flights (and several airlines can operate flights between the same two cities at the same time). Every two cities are connected by at least one non-stop flight. How many airlines are needed at least to meet the above requirements? | 21 |
63,983 | Pov L. A.
Do there exist three pairwise distinct natural numbers $a, b$, and $c$ such that the numbers $a+b+c$ and $a \cdot b \cdot c$ are squares of some natural numbers? | =1,b=3,=12 |
29,383 | Three, (50 points) Given ten points in space, where no four points lie on the same plane. Connect some of the points with line segments. If the resulting figure contains no triangles and no spatial quadrilaterals, determine the maximum number of line segments that can be drawn. | 15 |
53,851 | The 61st question: Find the maximum value of the positive integer $n$, such that for any simple graph of order $n$ with vertices $v_{1}, v_{2}, \ldots, v_{n}$, there always exist $n$ subsets $A_{1}, A_{2}, \ldots, A_{n}$ of the set $\{1,2, \ldots, 2020\}$, satisfying: $A_{i} \cap A_{j} \neq \varnothing$ if and only if $v_{i}$ is adjacent to $v_{j}$. | 89 |
23,674 | From $A$ to $B$ it is 999 km. Along the road, there are kilometer markers with distances written to $A$ and to $B$:
$0|999,1|998, \ldots, 999|0$.
How many of these markers have only two different digits? | 40 |
2,643 | Let $k$ be a positive integer. Lexi has a dictionary $\mathbb{D}$ consisting of some $k$ -letter strings containing only the letters $A$ and $B$ . Lexi would like to write either the letter $A$ or the letter $B$ in each cell of a $k \times k$ grid so that each column contains a string from $\mathbb{D}$ when read from top-to-bottom and each row contains a string from $\mathbb{D}$ when read from left-to-right.
What is the smallest integer $m$ such that if $\mathbb{D}$ contains at least $m$ different strings, then Lexi can fill her grid in this manner, no matter what strings are in $\mathbb{D}$ ? | 2^{k-1} |
60,281 | Task B-3.4. In trapezoid $A B C D$ with perpendicular diagonals, the lengths of the bases $|A B|=a=4,|D C|=c=3$ are known. If the leg $\overline{B C}$ with base $a$ forms an angle of $60^{\circ}$, what is its length? | \frac{1+\sqrt{193}}{4} |
64,869 | 3. Solve the inequality $\log _{\frac{3 x-6}{3 x+5}}(3 x-9)^{10} \geq-10 \log _{\frac{3 x+5}{3 x-6}}(3 x+6)$.
Answer. $x \in\left(-2 ;-\frac{5}{3}\right) \cup(2 ; 3) \cup(3 ;+\infty)$. | x\in(-2;-\frac{5}{3})\cup(2;3)\cup(3;+\infty) |
58,203 | ## 3. Hussar
The captain of a hussar ship divides gold coins among a trio of his hussars in the following way:
first division: one for me, one for each of you
second division: two for me, one for each of you
third division: three for me, one for each of you.
In each subsequent division, the captain takes one more gold coin than in the previous division, and gives each of the other hussars one gold coin.
How many more gold coins does the captain have in total compared to each of the other hussars after 44 divisions?
Result: $\quad 858$ | 858 |
15,982 | Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$ -axis. What is the least positive
integer $n$ such that performing the sequence of transformations $T_1, T_2, T_3, \cdots, T_n$ returns the point $(1,0)$ back to itself? | 359 |
31,168 | 7. Given real numbers $a, b, c$ satisfy $\left|a x^{2}+b x+c\right|$ has a maximum value of 1 on $x \in[-1,1]$. Then the maximum possible value of $\left|c x^{2}+b x+a\right|$ on $x \in[-1,1]$ is $\qquad$ | 2 |
28,339 | 7. Given the hyperbola $C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)$ with eccentricity $\frac{\sqrt{17}}{3}, F$ is the right focus, points $A, B$ are on the right branch, and $D$ is the point symmetric to $A$ with respect to the origin $O$, and $D F \perp A B$. If $\overrightarrow{A F}=\lambda \overrightarrow{F B}$, then $\lambda=$ $\qquad$ . | \frac{1}{2} |
54,746 | Example 1.11 A and B are playing table tennis, and the final score is $21: 17$. Find the number of scoring scenarios in which A has always been leading B during the game. | \frac{4\times37!}{21!17!} |
4,927 | Given \( A = \{ x \mid x^2 - 4x + 3 < 0, x \in \mathbf{R} \} \) and \( B = \{ x \mid 2^{1-x} + a \leq 0 \text{ and } x^2 - 2(a+7)x + 5 \leq 0, x \in \mathbf{R} \} \), if \( A \subseteq B \), then the range of \( a \) is ______. | [-4, -1] |
11,696 | Given that the radius of the incircle of an equilateral triangle \(ABC\) is 2 and the center of this circle is \(I\). If point \(P\) satisfies \(P I = 1\), what is the maximum value of the ratio of the areas of \(\triangle APB\) to \(\triangle APC\)? | \dfrac{3 + \sqrt{5}}{2} |
29,773 | 13. $[8]$ Let $a, b$, and $c$ be the side lengths of a triangle, and assume that $a \leq b$ and $a \leq c$. Let $x=\frac{b+c-a}{2}$. If $r$ and $R$ denote the inradius and circumradius, respectively, find the minimum value of $\frac{a x}{r R}$. | 3 |
19,335 | Example 2 Given $a_{1}=1, a_{n}=\frac{2}{3} a_{n-1}+n^{2}-15(n \geqslant 2)$, find $a_{n}$. | a_n = 25 \left( \dfrac{2}{3} \right)^{n-1} + 3n^2 - 12n - 15 |
52,602 | 10.014. A chord of a circle is equal to $10 \mathrm{~cm}$. Through one end of the chord, a tangent to the circle is drawn, and through the other end, a secant parallel to the tangent is drawn. Determine the radius of the circle if the inner segment of the secant is $12 \mathrm{~cm}$. | 6.25 |
5,228 | 8. Let $A B C$ be a triangle with $A B=A C$ and $\angle B A C=20^{\circ}$. Let $D$ be the point on the side $A B$ such that $\angle B C D=70^{\circ}$. Let $E$ be the point on the side $A C$ such that $\angle C B E=60^{\circ}$. Determine the value of the angle $\angle C D E$.
Answer: $\angle C D E=20^{\circ}$. | 20^\circ |
24,013 | A square is inscribed in an equilateral triangle such that each vertex of the square touches the perimeter of the triangle. One side of the square intersects and forms a smaller equilateral triangle within which we inscribe another square in the same manner, and this process continues infinitely. What fraction of the equilateral triangle's area is covered by the infinite series of squares? | \dfrac{3 - \sqrt{3}}{2} |
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