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|---|---|---|
How many trials must we conduct at a minimum so that the probability of:
a) Rolling a sum of 15 exactly once with 3 dice,
b) Rolling a sum of at least 15 exactly once with 3 dice,
is greater than $1/2$?
|
8
| 0.125 |
Let \( a \) be a prime number and \( b \) be a positive integer. Given the equation \( 9(2a + b)^2 = 509(4a + 511b) \), find the values of \( a \) and \( b \).
|
a = 251, \, b = 7
| 0.625 |
A number $a$ is randomly selected from $1, 2, 3, \cdots, 10$, and a number $b$ is randomly selected from $-1, -2, -3, \cdots, -10$. What is the probability that $a^{2} + b$ is divisible by 3?
|
\frac{37}{100}
| 0.75 |
If $n$ is a natural number, we denote by $n!$ the product of all integers from 1 to $n$. For example: $5! = 1 \times 2 \times 3 \times 4 \times 5$ and $13! = 1 \times 2 \times 3 \times 4 \times 5 \times \ldots \times 12 \times 13$. By convention, $0! = 1$. Find three different integers $a, b$, and $c$ between 0 and 9 such that the three-digit number $abc$ is equal to $a! + b! + c!$.
|
145
| 0.125 |
Each New Year, starting from the first year AD, Methuselah, who is still alive today, sends a greeting to his best friend. The greeting formula remains unchanged for almost two millennia: "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 frequently so far?
|
0
| 0.125 |
Two lines enclose an angle of $36^{\circ}$. A grasshopper jumps back and forth from one line to the other such that the length of each of its jumps is always the same. Show that it can reach at most ten different points.
|
10
| 0.875 |
Let \( Q \) be the set of \( n \)-term sequences, each element of which is 0 or 1. Let \( A \) be a subset of \( Q \) with \( 2^{n-1} \) elements. Show that there are at least \( 2^{n-1} \) pairs \( (a, b) \), where \( a \in A \), \( b \in Q \backslash A \), and the sequences \( a \) and \( b \) differ in exactly one position.
|
|A| = 2^{n-1}
| 0.375 |
If a certain number of cats ate a total of 999,919 mice, and all cats ate the same number of mice, how many cats were there in total? Additionally, each cat ate more mice than there were cats.
|
991
| 0.5 |
In a pocket, there are six small balls that are identical in size and shape, each labeled with one of the numbers $-5, -4, -3, -2, 2, 1$. We randomly draw one ball from the pocket and denote the number on it as $a$. Determine the probability that the vertex of the parabola given by \( y = x^2 + 2x + a + 2 \) falls in the third quadrant, and the equation
\[
\frac{ax}{x-2} = \frac{3x+2}{2-x} + 2
\]
has an integer solution.
|
\frac{1}{3}
| 0.75 |
Given the real number \( x \) satisfies the equation \(\sin \left(x + 20^\circ \right) = \cos \left( x + 10^\circ \right) + \cos \left( x - 10^\circ \right)\), find the value of \(\tan x\).
|
\sqrt{3}
| 0.875 |
Two real numbers \(x\) and \(y\) are such that \(8y^4 + 4x^2y^2 + 4xy^2 + 2x^3 + 2y^2 + 2x = x^2 + 1\). Find all possible values of \(x + 2y^2\).
|
\frac{1}{2}
| 0.875 |
Given point \( A(4,0) \) and \( B(2,2) \), while \( M \) is a moving point on the ellipse \(\frac{x^{2}}{25} + \frac{y^{2}}{9} = 1\), the maximum value of \( |MA| + |MB| \) is ______.
|
10 + 2 \sqrt{10}
| 0.75 |
On one side of a street, six neighboring houses will be built. The houses can be made of brick or wood, but as a fire safety measure, two wooden houses cannot be adjacent to each other. In how many ways can the construction of these houses be planned?
|
21
| 0.875 |
Formulate the equation of a line passing through the intersection point of the lines \( x + y - 2 = 0 \) and \( 3x + 2y - 5 = 0 \), and perpendicular to the line \( 3x + 4y - 12 = 0 \).
|
4x - 3y - 1 = 0
| 0.875 |
Find the largest \( \mathrm{C} \) such that for all \( \mathrm{y} \geq 4 \mathrm{x}>0 \), the inequality \( x^{2}+y^{2} \geq \mathrm{C} x y \) holds.
|
\frac{17}{4}
| 0.75 |
The solution to the equation \(\arcsin x + \arcsin 2x = \arccos x + \arccos 2x\) is
|
\frac{\sqrt{5}}{5}
| 0.625 |
What is the ratio of the areas of a regular hexagon inscribed in a circle to a regular hexagon circumscribed about the same circle?
|
\frac{3}{4}
| 0.875 |
In the cells of a \(75 \times 75\) table, pairwise distinct natural numbers are placed. Each of them has no more than three different prime divisors. It is known that for any number \(a\) in the table, there exists a number \(b\) in the same row or column such that \(a\) and \(b\) are not coprime. What is the maximum number of prime numbers that can be in the table?
|
4218
| 0.625 |
Determine all distinct positive integers \( x \) and \( y \) such that
\[ \frac{1}{x} + \frac{1}{y} = \frac{2}{7} \]
|
(4, 28)
| 0.75 |
Svetlana takes a triplet of numbers and transforms it by the rule: at each step, each number is replaced by the sum of the other two. What is the difference between the largest and smallest numbers in the triplet on the 1580th step of applying this rule, if the initial triplet of numbers was $\{80, 71, 20\}$? If the problem allows for multiple answers, list them all as a set.
|
60
| 0.875 |
On the lateral side \(CD\) of trapezoid \(ABCD (AD \parallel BC)\), point \(M\) is marked. A perpendicular \(AH\) is dropped from vertex \(A\) to segment \(BM\). It turns out that \(AD = HD\). Find the length of segment \(AD\), given that \(BC = 16\), \(CM = 8\), and \(MD = 9\).
|
18
| 0.125 |
On a rectangular sheet of graph paper of size \( m \times n \) cells, several squares are placed such that their sides align with the vertical and horizontal lines of the paper. It is known that no two squares coincide, and no square contains another square within itself. What is the maximum number of such squares?
|
m \times n
| 0.125 |
If \( R = 1 \times 2 + 2 \times 2^2 + 3 \times 2^3 + \ldots + 10 \times 2^{10} \), find the value of \( R \).
|
18434
| 0.125 |
What is the minimum value of the function \( y = \sin^4 x + \cos^4 x + \sec^4 x + \csc^4 x \)?
|
8.5
| 0.875 |
The hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ has a right focus at $F$ and an eccentricity of $e$. A line passing through point $F$ with a slope angle of $\frac{\pi}{3}$ intersects this hyperbola at points $A$ and $B$. If the midpoint of $A B$ is $M$ and the distance $|FM|$ equals the semi-focal distance, find $e$.
|
\sqrt{2}
| 0.875 |
Find the sum of the integers that belong to the set of values of the function \(f(x) = \log_{2}(5 \cos 2x + 11)\) for \(x \in \left[1.25 \left(\arctan\left(\frac{1}{3}\right)\right) \cos\left(\pi + \arcsin(-0.6)\right), \arctan 2\right]\).
|
7
| 0.875 |
All angles of an $n$-gon $A_{1} A_{2} A_{3} \ldots A_{n}$ are equal; and
$$
A_{1} A_{2} \leqslant A_{2} A_{3} \leqslant A_{3} A_{4} \leqslant \ldots \leqslant A_{n-1} A_{n} \leqslant A_{n} A_{1} .
$$
What values can the ratio $\frac{A_{1} A_{n}}{A_{1} A_{2}}$ take?
|
1
| 0.375 |
Given that \( x \) and \( y \) are greater than 0, and \( x^{2}+y \geq x^{3}+y^{2} \), find the maximum value of \( x^{2}+y^{2} \).
|
2
| 0.625 |
As shown, \(U\) and \(C\) are points on the sides of triangle \(MN H\) such that \(MU = s\), \(UN = 6\), \(NC = 20\), \(CH = s\), and \(HM = 25\). If triangle \(UNC\) and quadrilateral \(MUCH\) have equal areas, what is \(s\)?
|
s = 4
| 0.5 |
Given that \( n \in \mathbf{N}^{+} \), and subsets \( A_{1}, A_{2}, \cdots, A_{2n+1} \) of a set \( B \) satisfy:
1. \( \left|A_{i}\right| = 2n \) for \( i = 1, 2, \cdots, 2n+1 \);
2. \( \left|A_{i} \cap A_{j}\right| = 1 \) for \( 1 \leqslant i < j \leqslant 2n+1 \);
3. Each element in \( B \) belongs to at least two of the \( A_{i} \).
For which values of \( n \) is it possible to label the elements of \( B \) with 0 or 1 such that each \( A_{i} \) contains exactly \( n \) elements labeled 0?
|
n \text{ is even}
| 0.25 |
Find the two-digit number whose digits are distinct and whose square is equal to the cube of the sum of its digits.
|
27
| 0.625 |
Several schoolchildren went mushroom picking. The schoolchild who gathered the most mushrooms collected \( \frac{1}{5} \) of the total amount of mushrooms, while the one who gathered the least collected \( \frac{1}{7} \) of the total amount. How many schoolchildren were there?
|
6
| 0.75 |
There are 11 members in total from two units, A and B, with unit A having 7 members and unit B having 4 members. A group of 5 people is to be formed from these members.
1. How many ways can the group be formed if it must include 2 people from unit B?
2. How many ways can the group be formed if it must include at least 2 people from unit B?
3. How many ways can the group be formed if a specific person from unit B and a specific person from unit A cannot be in the group simultaneously?
|
378
| 0.875 |
Let \( k \) be a given positive integer, and let \( P \) be a point on plane \( \alpha \). In plane \( \alpha \), if it is possible to draw \( n \) lines that do not pass through point \( P \) such that any ray starting from \( P \) and lying in this plane intersects with at least \( k \) of these \( n \) lines, find the minimum value of \( n \).
|
2k + 1
| 0.375 |
1. How many different ways are there to choose 4 letters from $a, b, b, c, c, c, d, d, d, d$?
2. How many different four-digit numbers can be formed by choosing 4 digits from $1, 2, 2, 3, 3, 3, 4, 4, 4, 4$?
|
175
| 0.25 |
Given an isosceles triangle \(ABC\) with base \(AC\). A circle with radius \(R\) and center at point \(O\) passes through points \(A\) and \(B\) and intersects the line \(BC\) at point \(M\), distinct from \(B\) and \(C\). Find the distance from point \(O\) to the center of the circumcircle of triangle \(ACM\).
|
R
| 0.75 |
For any positive integer \( n \), connect the origin \( O \) and the point \( A_{n}(n, n+3) \). Let \( f(n) \) denote the number of lattice points on the line segment \( O A_{n} \) excluding the endpoints. Find \( f(1) + f(2) + \cdots + f(2006) \).
|
1336
| 0.875 |
Find all pairs of integers \( x \) and \( y \) such that
$$
(x+2)^{4} - x^{4} = y^{3}
$$
|
(-1, 0)
| 0.625 |
Calculate \(\cos (\alpha + \beta)\) if \(\cos \alpha - \cos \beta = -\frac{3}{5}\) and \(\sin \alpha + \sin \beta = \frac{7}{4}\).
|
-\frac{569}{800}
| 0.75 |
Mark has a cursed six-sided die that never rolls the same number twice in a row, and all other outcomes are equally likely. Compute the expected number of rolls it takes for Mark to roll every number at least once.
|
\frac{149}{12}
| 0.5 |
Given a regular hexagon \( A B C D E F \) with side length \( 10 \sqrt[4]{27} \), find the area of the union of triangles \( ACE \) and \( BDF \).
|
900
| 0.375 |
Find all values of \( n \in \mathbf{N} \) possessing the following property: if you write the numbers \( n^3 \) and \( n^4 \) next to each other (in decimal system), then in the resulting sequence, each of the 10 digits \( 0, 1, \ldots, 9 \) appears exactly once.
|
18
| 0.375 |
Let \( a, b, \) and \( c \) be strictly positive real numbers such that \( a^2 + b^2 + c^2 = \frac{1}{2} \). Show that
\[
\frac{1 - a^2 + c^2}{c(a + 2b)} + \frac{1 - b^2 + a^2}{a(b + 2c)} + \frac{1 - c^2 + b^2}{b(c + 2a)} \geq 6
\]
|
6
| 0.875 |
There are 200 different cards on the table with numbers $201, 203, 205, \ldots, 597, 599$ (each card has exactly one number, and each number appears exactly once). In how many ways can you choose 3 cards such that the sum of the numbers on the chosen cards is divisible by 3?
|
437844
| 0.125 |
From the first 2005 natural numbers, \( k \) of them are arbitrarily chosen. What is the least value of \( k \) to ensure that there is at least one pair of numbers such that one of them is divisible by the other?
|
1004
| 0.75 |
In the Martian calendar, a year consists of 5882 days, with each month having either 100 or 77 days. How many months are in the Martian calendar?
|
74
| 0.375 |
Given positive integers \( n \) and \( m \), let \( A = \{1, 2, \cdots, n\} \) and define \( B_{n}^{m} = \left\{\left(a_{1}, a_{2}, \cdots, a_{m}\right) \mid a_{i} \in A, i=1,2, \cdots, m\} \right. \) satisfying:
1. \( \left|a_{i} - a_{i+1}\right| \neq n-1 \), for \( i = 1, 2, \cdots, m-1 \);
2. Among \( a_{1}, a_{2}, \cdots, a_{m} \) (with \( m \geqslant 3 \)), at least three of them are distinct.
Find the number of elements in \( B_{n}^{m} \) and in \( B_{6}^{3} \).
|
104
| 0.25 |
The sequence \( 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, \ldots \) is formed as follows: write down infinitely many '1's, insert a '2' between the first and the second '1's, insert two '2's between the second and the third '1's, insert three '2's between the third and the fourth '1's, and so on. If \( a_{n} \) denotes the \( n \)-th term of the sequence, find the value of \( a_{1} a_{2} + a_{2} a_{3} + \cdots + a_{2013} a_{2014} \).
|
7806
| 0.5 |
A circle with radius \( r \) is inscribed in a circle with radius \( R \). Point \( A \) is the point of tangency. A line perpendicular to the line connecting the centers intersects the smaller circle at point \( B \) and the larger circle at point \( C \). Find the radius of the circumcircle of triangle \( A B C \).
|
\sqrt{rR}
| 0.125 |
Given for any \( x_{1}, x_{2}, \cdots, x_{2020} \in [0, 4] \),
the equation
$$
\left|x - x_{1}\right| + \left|x - x_{2}\right| + \cdots + \left|x - x_{2020}\right| = 2020a
$$
has at least one root in the interval \([0, 4]\), find the value of \( a \).
|
2
| 0.75 |
Arrange all powers of 3 and any finite sums of distinct powers of 3 into an increasing sequence:
$$
1, 3, 4, 9, 10, 12, 13, \cdots
$$
Find the 100th term of this sequence.
|
981
| 0.25 |
Determinants of nine digits. Nine positive digits can be arranged in the form of a 3rd-order determinant in 9! ways. Find the sum of all such determinants.
|
0
| 0.875 |
Find the number of first kind circular permutations formed by selecting 6 elements (with repetition allowed) from the 3-element set \( A = \{a, b, c\} \).
|
130
| 0.5 |
Points \( A_1 \), \( B_1 \), and \( C_1 \) are located on the sides \( BC \), \( AC \), and \( AB \) of triangle \( ABC \) respectively, such that \( BA_1 : A_1C = CB_1 : B_1A = AC_1 : C_1B = 1 : 3 \). Find the area of the triangle formed by the intersections of the lines \( AA_1 \), \( BB_1 \), and \( CC_1 \), given that the area of triangle \( ABC \) is 1.
|
\frac{4}{13}
| 0.375 |
Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for all real numbers \( x \) and \( y \),
\[ f\left(x^{2}-y^{2}\right)=x f(x)-y f(y) \]
|
f(x) = kx
| 0.5 |
In the trapezoid \(ABCD (BC \parallel AD)\), the diagonals intersect at point \(M\). Given \(BC = b\) and \(AD = a\), find the ratio of the area of triangle \(ABM\) to the area of trapezoid \(ABCD\).
|
\frac{ab}{(a+b)^2}
| 0.5 |
How many solutions does the equation
\[ x^{2}+y^{2}+2xy-1988x-1988y=1989 \]
have in the set of positive integers?
|
1988
| 0.75 |
Find the maximum value of the natural number \( a \) such that the inequality \(\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{3n+1}>2a-5\) holds for all natural numbers \( n \).
|
3
| 0.5 |
Given \( n \) real numbers, satisfying \( \left|x_{i}\right| < 1 \) for \( i = 1, \ldots, n \), and \( \left| x_1 \right| + \left| x_2 \right| + \cdots + \left| x_n \right| \geq 19 + \left| x_1 + \cdots + x_n \right| \), find the smallest possible value of \( n \).
|
20
| 0.875 |
A regular 2017-gon \( A_1 A_2 \cdots A_{2017} \) is inscribed in a unit circle \( O \). If two different vertices \( A_i \) and \( A_j \) are chosen randomly, what is the probability that \( \overrightarrow{O A_i} \cdot \overrightarrow{O A_j} > \frac{1}{2} \)?
|
\frac{1}{3}
| 0.5 |
Given points \( A(1,1) \), \( B\left(\frac{1}{2},0\right) \), and \( C\left(\frac{3}{2},0\right) \), the plane region \( G \) is enclosed by the lines passing through points \( A \) and \( B \), through points \( A \) and \( C \), and the line \( y=a \) where \( 0 < a < 1 \). If the probability of any point in the rectangular region \(\{(x, y) \mid 0<x<2, 0<y<1\}\) falling into the region \( G \) is \(\frac{1}{16}\), find the value of \( a \).
|
\frac{1}{2}
| 0.75 |
Teacher Li plans to buy 25 souvenirs for students from a store that has four types of souvenirs: bookmarks, postcards, notebooks, and pens, with 10 pieces available for each type (souvenirs of the same type are identical). Teacher Li intends to buy at least one piece of each type. How many different purchasing plans are possible? (Answer in numeric form.).
|
592
| 0.375 |
Given a sequence of integers $\left\{a_{n}\right\}$ satisfying $a_{n}=a_{n-1}-a_{n-2}$ for $n \geq 3$, if the sum of the first 1492 terms is 1985 and the sum of the first 1985 terms is 1492, find the sum of the first 2001 terms.
|
986
| 0.125 |
Let \( I \) be the universal set, and let non-empty sets \( P \) and \( Q \) satisfy \( P \subsetneq Q \subsetneq I \). If there exists a set operation expression involving \( P \) and \( Q \) such that the result of the operation is the empty set \( \varnothing \), then one such expression could be _____ (just write one expression).
|
P \cap Q^c
| 0.125 |
Given that $\sin ^{10} x+\cos ^{10} x=\frac{11}{36}$, find the value of $\sin ^{14} x+\cos ^{14} x$.
|
\frac{41}{216}
| 0.625 |
It is sufficient to check that $DF_{1}=BC$. Similarly, $DF_{2}=BC$ and the problem is solved.
|
D F_{1} = B C
| 0.625 |
Daniel and Scott are playing a game where a player wins as soon as he has two points more than his opponent. Both players start at zero points, and points are earned one at a time. If Daniel has a 60% chance of winning each point, what is the probability that Daniel will win the game?
|
\frac{9}{13}
| 0.875 |
Two snowplows are working to clear snow. The first can clear the entire street in 1 hour, while the second can do it in 75% of that time. They started working together and after 20 minutes, the first machine stopped working. How much more time is needed for the second machine to finish the job?
|
10 \text{ minutes}
| 0.875 |
Find the number of all Young diagrams with weight $s$, if
a) $s=4$; b) $s=5$;
c) $s=6$; d) $s=7$.
Refer to the definition of Young diagrams in the handbook.
|
15
| 0.125 |
Two circles are constructed on a plane in such a way that each passes through the center of the other. Points $P$ and $Q$ are the points of their intersection. A line passing through point $P$ intersects the first circle at point $A$ and the second circle at point $B$ such that $P$ is between $A$ and $B$. This line makes an angle of $15^{\circ}$ with the line connecting the centers. Find the area of triangle $ABQ$ if it is known that $PQ = 2\sqrt{3}$.
|
S=3+2 \sqrt{3}
| 0.125 |
Given a pyramid \(S-ABC\) with height \(SO = 3\) and a base whose side length is 6, a perpendicular is drawn from point A to its opposite face \(SBC\) with the foot of the perpendicular being \(O'\). On the line segment \(AO'\), a point \(P\) is selected such that \(\frac{AP}{PO'} = 8\). Find the area of the cross-section through point \(P\) and parallel to the base.
|
\sqrt{3}
| 0.375 |
How many four-digit numbers are there that are divisible by 17 and end in 17?
|
5
| 0.75 |
Given a regular 200-sided polygon \( A_{1} A_{2} \cdots A_{200} \), connect the diagonals \( A_{i} A_{i+9} \) for \( i=1, 2, \cdots, 200 \), where \( A_{i+200}=A_{i} \) for \( i=1,2, \cdots, 9 \). How many different intersection points are there inside the regular 200-sided polygon formed by these 200 diagonals?
|
800
| 0.375 |
If the inequality $(x - 2) e^x < a x + 1$ has exactly three distinct integer solutions, find the minimum value of the integer $a$.
|
3
| 0.625 |
Isosceles trapezoid \(ABCD\) with bases \(AB\) and \(CD\) has a point \(P\) on \(AB\) with \(AP = 11\), \(BP = 27\), \(CD = 34\), and \(\angle CPD = 90^\circ\). Compute the height of isosceles trapezoid \(ABCD\).
|
15
| 0.875 |
Does \(xy = z\), \(yz = x\), \(zx = y\) have any solutions in positive reals apart from \(x = y = z = 1\)?
|
x = y = z = 1
| 0.25 |
Find the solutions of the system of equations
$$
\begin{aligned}
a^{3}+3 a b^{2}+3 a c^{2}-6 a b c & =1 \\
b^{3}+3 b a^{2}+3 b c^{2}-6 a b c & =1 \\
c^{3}+3 c a^{2}+3 c b^{2}-6 a b c & =1
\end{aligned}
$$
|
(1, 1, 1)
| 0.375 |
A positive integer is written on each corner of a square such that numbers on opposite vertices are relatively prime while numbers on adjacent vertices are not relatively prime. What is the smallest possible value of the sum of these 4 numbers?
|
60
| 0.125 |
We form a number from three consecutive digits. If we write these digits in reverse order and subtract the resulting number from the original one, the difference is always 198.
|
198
| 0.75 |
Maruska wrote one of ten consecutive natural numbers on each of ten cards. However, she lost one card. The sum of the numbers on the remaining nine cards was 2012. What number was written on the lost card?
|
223
| 0.75 |
Given that \(\alpha\) and \(\beta\) are real numbers, for any real numbers \(x\), \(y\), and \(z\), it holds that
$$
\alpha(x y + y z + z x) \leqslant M \leqslant \beta\left(x^{2} + y^{2} + z^{2}\right),
$$
where \(M = \sum \sqrt{x^{2} + x y + y^{2}} \sqrt{y^{2} + y z + z^{2}}\,\) and \(\sum\) represents cyclic sums. Find the maximum value of \(\alpha\) and the minimum value of \(\beta\).
|
3
| 0.875 |
Find the maximum length of a horizontal segment with endpoints on the graph of the function \( y = x^3 - x \).
|
2
| 0.875 |
The total length of the highway from Lishan Town to the provincial capital is 189 kilometers, passing through the county town. The county town is 54 kilometers away from Lishan Town. In the morning at 8:30, a bus departs from Lishan Town to the county town and arrives at 9:15. After a 15-minute stop, it heads to the provincial capital, arriving by 11:00 AM. Another bus departs from the provincial capital directly to Lishan Town at 9:00 AM on the same day, traveling at 60 kilometers per hour. When the two buses meet, the one traveling from the provincial capital to Lishan Town has been traveling for how many minutes?
|
72
| 0.75 |
Five people play several games of dominoes (2 vs 2) in such a way that each player partners with each other player exactly once, and opposes each other player exactly twice. Find the number of games played and all possible ways to arrange the players.
|
5 \text{ games}
| 0.375 |
An archipelago consists of $N \geqslant 7$ islands. Any two islands are connected by at most one bridge. It is known that from each island no more than 5 bridges lead out, and among any 7 islands there must be two connected by a bridge. What is the maximum value that $N$ can take?
|
36
| 0.5 |
The height \( AH \) of triangle \( ABC \) is equal to its median \( BM \). On the extension of side \( AB \) beyond point \( B \), point \( D \) is marked such that \( BD = AB \). Find the angle \( BCD \).
|
30^\circ
| 0.625 |
Pile up 2019 stones into one pile. First, person A splits this pile into two piles and writes the product of the number of stones in each pile on the blackboard. Then, person A selects one pile from the two and splits it into two more piles, again writing the product of the number of stones in each pile on the blackboard. Person A continues this process until all piles have exactly 1 stone. At this point, what is the total sum of the numbers on the blackboard?
|
2037171
| 0.25 |
In the right triangle \(ABC\), the hypotenuse \(AB\) is 10. \(AD\) is the angle bisector of \(\angle A\). Segment \(DC\) is 3. Find \(DB\).
|
5
| 0.875 |
Find all positive integers \( n \) such that \( n \) divides \( 2^{n} - 1 \).
|
1
| 0.5 |
The sequence \(\left(x_{n}\right)\) is defined recursively by \(x_{0}=1, x_{1}=1\), and:
\[ x_{n+2}=\frac{1+x_{n+1}}{x_{n}} \]
for all \(n \geq 0\). Calculate \(x_{2007}\).
|
2
| 0.75 |
Arrange all positive odd numbers in ascending order. Take the first number as \(a_1\), take the sum of the subsequent two numbers as \(a_2\), then take the sum of the subsequent three numbers as \(a_3\), and so on. This generates the sequence \(\{a_n\}\), i.e.
\[ a_1 = 1, \quad a_2 = 3 + 5, \quad a_3 = 7 + 9 + 11, \quad \cdots \cdots \]
Find the value of \(a_1 + a_2 + \cdots + a_{20}\).
|
44100
| 0.875 |
Calculate the integrals:
1) $\int_{0}^{1} x e^{-x} \, dx$
2) $\int_{1}^{2} x \log_{2} x \, dx$
3) $\int_{1}^{e} \ln^{2} x \, dx$
|
e - 2
| 0.625 |
Can a finite set of points in space, not lying in the same plane, have the following property: for any two points \(A\) and \(B\) from this set, there exist two other points \(C\) and \(D\) from this set such that \(AB \parallel CD\) and these lines do not coincide?
|
\text{Yes}
| 0.875 |
In quadrilateral \(ABCD\), the diagonals \(AC\) and \(BD\) intersect at point \(O\). The areas of \(\triangle AOB\), \(\triangle AOD\), and \(\triangle BOC\) are \(12\), \(16\), and \(18\), respectively. What is the area of \(\triangle COD\)?
|
24
| 0.75 |
In the city where the Absent-Minded Scientist lives, phone numbers consist of 7 digits. The Scientist easily remembers a phone number if it is a palindrome, meaning it reads the same forwards and backwards. For example, the number 4435344 is easy for the Scientist to remember because it is a palindrome. However, the number 3723627 is not a palindrome, making it difficult for the Scientist to remember. Find the probability that the phone number of a new random acquaintance is easy for the Scientist to remember.
|
0.001
| 0.625 |
Given that \(b\) and \(c\) are both integers and \(c < 2000\). If the quadratic equation \(x^{2} - bx + c = 0\) has roots whose real parts are both greater than 1, how many pairs \((b, c)\) satisfy this condition?
|
1995003
| 0.25 |
Find the smallest positive integer \( b \) for which \( 7 + 7b + 7b^2 \) is a fourth power.
|
18
| 0.5 |
In a parallelogram with sides 2 and 4, a diagonal equal to 3 is drawn. A circle is inscribed in each of the resulting triangles. Find the distance between the centers of the circles.
|
\frac{\sqrt{51}}{3}
| 0.5 |
The line \( K M_{1} \) intersects the extension of \( A B \) at point \( N \). Find the measure of angle \( DNA \).
|
90^\circ
| 0.25 |
Does there exist a natural number that, when divided by the sum of its digits, gives 2014 as both the quotient and the remainder? If there is more than one such number, write their sum as the answer. If no such number exists, write 0 as the answer.
|
0
| 0.5 |
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