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There are 300 black and white pieces in total. The black crow divides the black and white pieces into 100 piles, each containing 3 pieces. There are 27 piles with exactly $l$ white pieces, 42 piles with 2 or 3 black pieces, and the number of piles with 3 white pieces is equal to the number of piles with 3 black pieces. How many white pieces are there in total?
|
158
| 0.625 |
A natural number \( 1 \leq n \leq 221 \) is called lucky if, when dividing 221 by \( n \), the remainder is wholly divisible by the incomplete quotient (the remainder can be equal to 0). How many lucky numbers are there?
|
115
| 0.75 |
A $4 \times 8$ paper rectangle is folded along the diagonal as shown in the figure. What is the area of the triangle that is covered twice?
|
10
| 0.125 |
The different ways to obtain the number of combinations of dice, as discussed in Example 4-15 of Section 4.6, can also be understood using the generating function form of Pólya’s enumeration theorem as follows:
$$
\begin{aligned}
P= & \frac{1}{24} \times\left[\left(x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}\right)^{6}\right. \\
& +6\left(x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}\right)^{2}\left(x_{1}^{4}+x_{2}^{4}+x_{3}^{4}+x_{4}^{4}+x_{5}^{4}+x_{6}^{4}\right) \\
& +3\left(x_{1}+x+x_{3}+x_{4}+x_{5}+x_{6}\right)^{2}\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}\right)^{2} \\
& \left.+6\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}\right)^{3}+8\left(x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+x_{4}^{3}+x_{5}^{3}+x_{6}^{3}\right)^{2}\right],
\end{aligned}
$$
where $x_{i}$ represents the $i^{th}$ color for $i=1,2,\cdots,6$.
|
30
| 0.5 |
The last two digits of the decimal representation of the square of a natural number are the same and are not zero. What are these digits? Find all solutions.
|
44
| 0.5 |
A function \( f(x, y, z) \) is linear in \( x \), \( y \), and \( z \) such that \( f(x, y, z) = \frac{1}{xyz} \) for \( x, y, z \in \{3, 4\} \). What is \( f(5, 5, 5) \)?
|
\frac{1}{216}
| 0.125 |
Find \(\log_{30} 8\), given that \(\lg 5 = a\) and \(\lg 3 = b\).
|
\frac{3(1-a)}{b+1}
| 0.125 |
The largest prime factor of 101101101101 is a four-digit number \(N\). Compute \(N\).
|
9901
| 0.75 |
There are 23 socks in a drawer: 8 white and 15 black. Every minute, Marina goes to the drawer and pulls out a sock. If at any moment Marina has pulled out more black socks than white ones, she exclaims, "Finally!" and stops the process.
What is the maximum number of socks Marina can pull out before she exclaims, "Finally!"? The last sock Marina pulled out is included in the count.
|
17
| 0.625 |
The quadrilateral $A B C D$ is inscribed in a circle with center $O$. The diagonals $A C$ and $B D$ are perpendicular. Show that the distance from $O$ to the line $(A D)$ is equal to half the length of the segment $[B C]$.
|
\frac{BC}{2}
| 0.375 |
Let \( a \), \( b \), and \( c \) be the three roots of \( x^3 - x + 1 = 0 \). Find \( \frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1} \).
|
-2
| 0.875 |
The RSA Factoring Challenge, which ended in 2007, involved factoring extremely large numbers that were the product of two prime numbers. The largest number successfully factored in this challenge was RSA-640, which has 193 decimal digits and carried a prize of $20,000. The next challenge number carried a prize of $30,000, and contains $N$ decimal digits. Your task is to submit a guess for $N`. Only the team(s) that have the closest guess(es) receives points. If $k$ teams all have the closest guesses, then each of them receives $\left\lceil\frac{20}{k}\right\rceil$ points.
|
212
| 0.375 |
In which cases does a month have the largest number of Saturdays? What is this number?
|
5
| 0.875 |
Solve the equation \(\frac{x}{2+\frac{x}{2+\frac{x}{2+\ddots \frac{x}{2+\frac{x}{1+\sqrt{1+x}}}}}}=1\)
(where there are 1985 terms of 2 in the expression).
|
3
| 0.625 |
The polynomial $a x^{2} + b x + c$ has integer coefficients and two distinct roots that are greater than 0 but less than 1. How small can $|a|$ be?
|
5
| 0.375 |
Given \( x = \frac{\pi}{12} \), find the value of the function \( f(x) = \sqrt{\sin^4{x} + 4\cos^2{x}} - \sqrt{\cos^4{x} + 4\sin^2{x}} \).
|
\frac{\sqrt{3}}{2}
| 0.875 |
Let complex numbers \(z_{1}\) and \(z_{2}\) satisfy \(\operatorname{Re}(z_{1}) > 0\) and \(\operatorname{Re}(z_{2}) > 0\) and \(\operatorname{Re}(z_{1}^2) = \operatorname{Re}(z_{2}^2) = 2\) (where \(\operatorname{Re}(z)\) denotes the real part of the complex number \(z\)).
(1) Find the minimum value of \(\operatorname{Re}(z_{1} z_{2})\).
(2) Find the minimum value of \(\left|z_{1} + 2\right| + \left|\overline{z_{2}} + 2\right| - \left|\overline{z_{1}} - z_{2}\right|\).
|
4\sqrt{2}
| 0.25 |
Find the intersection point of a line and a plane.
$$
\begin{aligned}
& \frac{x-1}{2}=\frac{y-2}{0}=\frac{z-4}{1} \\
& x-2 y+4 z-19=0
\end{aligned}
$$
|
(3, 2, 5)
| 0.875 |
The numbers from 1 to 10 were divided into two groups such that the product of the numbers in the first group is divisible by the product of the numbers in the second group.
What is the smallest possible value of the quotient obtained by dividing the product of the first group by the product of the second group?
|
7
| 0.5 |
Let \(a\), \(b\), and \(c\) be the lengths of the three sides of a triangle. Suppose \(a\) and \(b\) are the roots of the equation
\[
x^2 + 4(c + 2) = (c + 4)x,
\]
and the largest angle of the triangle is \(x^\circ\). Find the value of \(x\).
|
90
| 0.875 |
Let \( p_{1}, p_{2}, \ldots, p_{97} \) be prime numbers (not necessarily distinct). What is the greatest integer value that the expression
\[ \sum_{i=1}^{97} \frac{p_{i}}{p_{i}^{2}+1}=\frac{p_{1}}{p_{1}^{2}+1}+\frac{p_{2}}{p_{2}^{2}+1}+\ldots+\frac{p_{97}}{p_{97}^{2}+1} \]
can take?
|
38
| 0.5 |
The Dorokhov family plans to purchase a vacation package to Crimea. The vacation will include the mother, father, and their eldest daughter Polina, who is 5 years old. They have chosen to stay at the "Bristol" hotel and have reached out to two travel agencies, "Globus" and "Around the World," to calculate the cost of the tour from July 10 to July 18, 2021.
The deals from each agency are as follows:
- At "Globus":
- 11,200 rubles per person for those under 5 years old.
- 25,400 rubles per person for those above 5 years old.
- A 2% discount on the total cost as regular customers.
- At "Around the World":
- 11,400 rubles per person for those under 6 years old.
- 23,500 rubles per person for those above 6 years old.
- A 1% commission fee is applied to the total cost.
Determine which travel agency offers the best deal for the Dorokhov family and identify the minimum cost for their vacation in Crimea. Provide only the number in your answer, without units of measurement.
|
58984
| 0.75 |
Let the sequences \(\left\{x_{n}\right\}\) and \(\left\{y_{n}\right\}\) satisfy \( x_{n} + i y_{n} = \left(\frac{\sqrt{3}i - 1}{2}\right)^{n} \). Find the sum of the first 1994 terms of the sequence \(\left\{x_{n}\right\}\), denoted by \(S_{1994}\).
|
-1
| 0.5 |
A cylinder has a height that is 3 times its base radius. If it is divided into a large and a small cylinder, the surface area of the large cylinder is 3 times the surface area of the small cylinder. Find the ratio of the volume of the large cylinder to the volume of the small cylinder.
|
11
| 0.75 |
For what value of \( k \) do we have the identity (true for any \( a, b, c \)):
$$
(a+b)(b+c)(c+a) = (a+b+c)(ab + bc + ca) + k \cdot abc?
$$
|
-1
| 0.625 |
A clock chimes 2 times at 2 o'clock, taking 2 seconds to complete. How many seconds does it take to chime 12 times at 12 o'clock?
|
22
| 0.875 |
Find the sum of all possible distinct two-digit numbers, all of whose digits are odd.
|
1375
| 0.75 |
During the draw before the math marathon, the team captains were asked to name the smallest possible sum of the digits in the decimal representation of the number \( n+1 \), given that the sum of the digits of the number \( n \) is 2017. What answer did the captain of the winning team give?
|
2
| 0.5 |
In the expression \( S = \sqrt{x_{1} - x_{2} + x_{3} - x_{4}} \), \( x_{1}, x_{2}, x_{3}, x_{4} \) are a permutation of 1, 2, 3, and 4. Determine the number of distinct permutations that make \( S \) a real number.
|
16
| 0.375 |
Sum the following series:
a) \(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots\)
b) \(\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\ldots\)
c) \(\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\ldots\)
|
\frac{1}{3}
| 0.25 |
How many times should two dice be rolled so that the probability of getting two sixes at least once is greater than $1/2$?
|
25
| 0.75 |
Given natural numbers \( k \) and \( n \) such that \( 1 < k \leq n \). What is the smallest \( m \) for which the following statement holds: for any placement of \( m \) rooks on an \( n \times n \) board, it is always possible to select \( k \) rooks from these \( m \) such that no two of the selected rooks can attack each other?
|
n(k-1) + 1
| 0.125 |
On a rectangular table of size $x$ cm $\times 80$ cm, identical sheets of paper of size 5 cm $\times 8$ cm are placed. The first sheet is placed in the bottom left corner, and each subsequent sheet is placed one centimeter above and one centimeter to the right of the previous one. The last sheet touches the top right corner. What is the length of $x$ in centimeters?
|
77 \text{ cm}
| 0.875 |
How many pairs of integers \(x, y\) between 1 and 1000 are there such that \(x^{2} + y^{2}\) is divisible by 7?
|
20164
| 0.5 |
How many pairs of natural numbers \(a\) and \(b\) exist such that \(a \geq b\) and the equation
\[ \frac{1}{a} + \frac{1}{b} = \frac{1}{6} \]
is satisfied?
|
5
| 0.75 |
Calculate: \(3 \times 995 + 4 \times 996 + 5 \times 997 + 6 \times 998 + 7 \times 999 - 4985 \times 3\)
|
9980
| 0.25 |
Given the sequence \(\{a_n\}\):
\[ a_n = 2^n + 3^n + 6^n + 1 \text{ for } n \in \mathbb{Z}_+ \]
Is there an integer \( k \geq 2 \) such that \( k \) is coprime with all numbers in the sequence \(\{a_n\}\)? If it exists, find the smallest integer \( k \); if it does not exist, explain why.
|
23
| 0.5 |
Let point \( C \) be a moving point on the parabola \( y^2 = 2x \). From \( C \), two tangent lines \( CA \) and \( CB \) are drawn to the circle \((x-1)^2 + y^2 = 1\), intersecting the negative half of the \( y \)-axis at \( A \) and the positive half of the \( y \)-axis at \( B \). Find the minimum area of triangle \( \triangle ABC \).
|
8
| 0.125 |
If \( f(x) = 5x - 3 \) for \( x \in \{0, 2, 3, 4, 5\} \) and \( g(x) = x^2 - 2x - 3 \) for \( x \in \{-2, -1, 1, 4, 5\} \), then what is the domain of the function \( F(x) = \log f(x) + \frac{1}{\sqrt{g(x}} \)?
|
\{4, 5\}
| 0.875 |
Find the number of natural numbers that do not exceed 2016 and are coprime with it. Recall that two integers are called coprime if they have no common natural divisors other than one.
|
576
| 0.75 |
Two girls knit at constant, but different speeds. The first girl takes a tea break every 5 minutes, and the second girl every 7 minutes. Each tea break lasts exactly 1 minute. When the girls went for a tea break together, it turned out that they had knitted the same amount. By what percentage is the first girl's productivity higher if they started knitting at the same time?
|
5\%
| 0.125 |
On the side \( BC \) of an equilateral triangle \( ABC \), points \( K \) and \( L \) are marked such that \( BK = KL = LC \). On the side \( AC \), point \( M \) is marked such that \( AM = \frac{1}{3} AC \). Find the sum of the angles \( \angle AKM \) and \( \angle ALM \).
|
30^\circ
| 0.75 |
Find the length of the segment of the line that is parallel to the bases of a trapezoid and passes through the intersection point of the diagonals, if the bases of the trapezoid are equal to \( a \) and \( b \).
|
\frac{2ab}{a+b}
| 0.875 |
Given a monotonically increasing sequence of positive integers $\left\{a_{n}\right\}$ that satisfies the recurrence relation $a_{n+2}=3 a_{n+1}-a_{n}$, with $a_{6}=280$, find the value of $a_{7}$.
|
733
| 0.875 |
Find all positive integers \( n \) for which we can find one or more integers \( m_1, m_2, ..., m_k \), each at least 4, such that:
1. \( n = m_1 m_2 ... m_k \)
2. \( n = 2^M - 1 \), where \( M = \frac{(m_1 - 1)(m_2 - 1) ... (m_k - 1)}{2^k} \)
|
7
| 0.75 |
Can a rectangle of size $55 \times 39$ be divided into rectangles of size $5 \times 11$?
|
\text{No}
| 0.75 |
Among all proper fractions where both the numerator and the denominator are two-digit numbers, find the smallest fraction that is greater than $\frac{5}{6}$. Provide the numerator of this fraction in your answer.
|
81
| 0.5 |
There is a type of cell that dies at a rate of 2 cells per hour, and each remaining cell divides into 2. If the number of cells is 1284 after 8 hours, how many cells were there initially?
|
9
| 0.5 |
Given the numbers $\log _{\sqrt{29-x}}\left(\frac{x}{7}+7\right), \log _{(x+1)^{2}}(29-x), \log _{\sqrt{\frac{x}{7}+7}}(-x-1)$. For what values of $x$ are two of these numbers equal, and the third is greater than them by 1?
|
x=-7
| 0.875 |
The radius of the circle circumscribed around the triangle $K L M$ is $R$. A line is drawn through vertex $L$, perpendicular to the side $K M$. This line intersects at points $A$ and $B$ the perpendicular bisectors of the sides $K L$ and $L M$. It is known that $A L = a$. Find $B L$.
|
\frac{R^2}{a}
| 0.125 |
Given \( x = \cos \frac{2}{5} \pi + i \sin \frac{2}{5} \pi \), evaluate \( 1 + x^4 + x^8 + x^{12} + x^{16} \).
|
0
| 0.625 |
The plane vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ have an included angle of $\frac{\pi}{3}$. If $|\boldsymbol{a}|$, $|\boldsymbol{b}|$, and $|\boldsymbol{a}+\boldsymbol{b}|$ form an arithmetic sequence, find the ratio $|\boldsymbol{a}| : |\boldsymbol{b}| : |\boldsymbol{a} + \boldsymbol{b}|$.
|
3 : 5 : 7
| 0.875 |
How many triples of non-negative integers \((x, y, z)\) satisfy the equation
\[ x y z + x y + y z + z x + x + y + z = 2012 \]
|
27
| 0.625 |
In \( \triangle ABC \), \( AB = 4 \), \( BC = 7 \), \( CA = 5 \). Let \(\angle BAC = \alpha\). Find the value of \( \sin^6 \frac{\alpha}{2} + \cos^6 \frac{\alpha}{2} \).
|
\frac{7}{25}
| 0.625 |
An ant starts at vertex \( A \) of a cube \( ABCD-EFGH \). Each time, it crawls with equal probability to one of the three neighboring vertices. What is the probability that the ant is back at vertex \( A \) after six moves?
|
\frac{61}{243}
| 0.25 |
A circular cone has vertex \( I \), a base with radius 1, and a slant height of 4. Point \( A \) is on the circumference of the base and point \( R \) is on the line segment \( IA \) with \( IR = 3 \). Shahid draws the shortest possible path starting at \( R \), traveling once around the cone, and ending at \( A \). If \( P \) is the point on this path that is closest to \( I \), what is the length \( IP \)?
|
\frac{12}{5}
| 0.75 |
An arbitrary finite sequence composed of 0s and 1s is called a word. The word \(A A A\) obtained by repeating word \(A\) three times is called the triplet of \(A\). For example, if \(A = 101\), its triplet is 101101101. For a word, the following two operations are allowed:
(1) Adding the triplet of any word at any position (including the beginning and the end) of the word;
(2) Removing the triplet of any word from it.
For example, from the word 0001, we can get 0111001 or 1, and so on. Is it possible, through a number of operations, to transform word 10 into word 01?
|
\text{No}
| 0.875 |
Joe knows that to convert from pounds to kilograms, you need to divide the mass in pounds by 2 and then decrease the resulting number by 10%. From this, Joe concluded that to convert from kilograms to pounds, you need to multiply the mass in kilograms by 2 and then increase the resulting number by 10%. By what percentage does he err from the correct mass in pounds?
|
1\% \text{ less}
| 0.375 |
In the sequence of positive integers, starting with \(2018, 121, 16, \ldots\), each term is the square of the sum of digits of the previous term. What is the \(2018^{\text{th}}\) term of the sequence?
|
256
| 0.875 |
Given complex numbers \( z_{1}, z_{2}, z_{3} \) such that \( \frac{z_{1}}{z_{2}} \) is a purely imaginary number, \( \left|z_{1}\right|=\left|z_{2}\right|=1 \), and \( \left|z_{1}+z_{2}+z_{3}\right|=1 \), find the minimum value of \( \left|z_{3}\right| \).
|
\sqrt{2} - 1
| 0.875 |
Damao, Ermao, and Sanmao are three brothers. Damao tells Sanmao: “When Dad was 36 years old, my age was 4 times yours, and Ermao’s age was 3 times yours.” Ermao adds: “Yes, at that time, the sum of our three ages was exactly half of Dad’s current age.” Sanmao says: “Now, the total age of all four of us adds up to 108 years.” How old is Sanmao this year?
|
15
| 0.75 |
Let's call a natural number semi-prime if it is greater than 25 and is the sum of two distinct prime numbers. What is the largest number of consecutive natural numbers that can be semi-prime? Justify your answer.
|
5
| 0.125 |
Let \( A B C D E F \) be a regular hexagon. Let \( G \) be a point on \( E D \) such that \( E G = 3 G D \). If the area of \( A G E F \) is 100, find the area of the hexagon \( A B C D E F \).
|
240
| 0.25 |
A Vandal and a Moderator are editing a Wikipedia article. The article originally is error-free. Each day, the Vandal introduces one new error into the Wikipedia article. At the end of the day, the moderator checks the article and has a \( \frac{2}{3} \) chance of catching each individual error still in the article. After 3 days, what is the probability that the article is error-free?
|
\frac{416}{729}
| 0.875 |
\(a, b, m, n\) satisfy: \(a m^{2001} + b n^{2001} = 3\); \(a m^{2002} + b n^{2002} = 7\); \(a m^{2003} + b n^{2003} = 24\); \(a m^{2004} + b m^{2004} = 102\). Find the value of \(m^{2}(n-1)\).
|
6
| 0.5 |
Let \( x_{1}, x_{2}, \ldots, x_{n} \) be strictly positive real numbers. Show that
\[ \frac{x_{1}}{x_{n}} + \frac{x_{2}}{x_{n-1}} + \ldots + \frac{x_{n}}{x_{1}} \geq n. \]
|
n
| 0.625 |
Consider a table with $m$ rows and $n$ columns. In how many ways can this table be filled with all zeros and ones so that there is an even number of ones in every row and every column?
|
2^{(m-1)(n-1)}
| 0.5 |
Given four points \( A, B, C, D \) in space, with at most one of the segments \( AB, AC, AD, BC, BD, CD \) having a length greater than 1, find the maximum sum of the lengths of these six segments.
|
5 + \sqrt{3}
| 0.375 |
Let \([x]\) be the largest integer not greater than \(x\). If \(B = [10 + \sqrt{10 + \sqrt{10 + \sqrt{10 + \cdots}}}]\), find the value of \(B\).
|
13
| 0.875 |
Given positive integer \( n = abc < 10000 \), where \( a \), \( b \), and \( c \) are prime numbers, and the equations \( 2a + 3b = c \) and \( 4a + c + 1 = 4b \) hold, find the value of \( n \).
|
1118
| 0.625 |
Let $F_{1}$ and $F_{2}$ be the two foci of an ellipse $C$. Let $AB$ be a chord of the ellipse passing through the point $F_{2}$. In the triangle $\triangle F_{1}AB$, the lengths are given as follows:
$$
|F_{1}A| = 3, \; |AB| = 4, \; |BF_{1}| = 5.
$$
Find $\tan \angle F_{2}F_{1}B$.
|
\frac{1}{7}
| 0.625 |
Let \( A \) and \( B \) be two sets, and \((A, B)\) be called a "pair". If \( A \neq B \), then \((A, B)\) and \((B, A)\) are considered different "pairs". Find the number of different pairs \((A, B)\) that satisfy the condition \( A \cup B = \{1,2,3,4\} \).
|
81
| 0.625 |
Given \( x, y \in \mathbf{R} \) and \( 2 x^{2} + 3 y^{2} \leq 12 \), find the maximum value of \( |x + 2y| \).
|
\sqrt{22}
| 0.5 |
Find the smallest natural number \( N \) such that \( N+2 \) is divisible by 2, \( N+3 \) by 3, ..., \( N+10 \) by 10.
|
2520
| 0.75 |
A group of 8 boys and 8 girls was paired up randomly. Find the probability that there is at least one pair with two girls. Round your answer to the nearest hundredth.
|
0.98
| 0.5 |
In the captain's log, we found the formula $d = p \sqrt{h}$ for determining the distance to the horizon, but the number in place of $p$ is not legible. In the formula, $d$ represents the distance to the horizon in kilometers, and $h$ is the observer's height above sea level in meters. Determine the value of $p$ so that we obtain a usable formula. (Consider the Earth's radius to be 6370 km.)
|
3.57
| 0.625 |
Two granaries, A and B, originally each stored whole bags of grain. If 90 bags are transferred from granary A to granary B, the number of bags in granary B will be twice the number of bags in granary A. If a certain number of bags are transferred from granary B to granary A, then the number of bags in granary A will be six times the number of bags in granary B. What is the minimum number of bags originally stored in granary A?
|
153
| 0.75 |
Let \( ABC \) be a triangle such that \( AB = 2 \), \( CA = 3 \), and \( BC = 4 \). A semicircle with its diameter on \(\overline{BC}\) is tangent to \(\overline{AB}\) and \(\overline{AC}\). Compute the area of the semicircle.
|
\frac{27 \pi}{40}
| 0.25 |
Inside a convex 13-sided polygon, there are 200 points such that no three of these 213 points (including the vertices of the polygon) lie on the same line. The polygon is divided into triangles, each vertex of which is any three of the given 213 points. What is the maximum number of triangles that could result?
|
411
| 0.75 |
Let \( P \) be a polynomial with \( P(1) = P(2) = \cdots = P(2007) = 0 \) and \( P(0) = 2009! \). \( P(x) \) has leading coefficient 1 and degree 2008. Find the largest root of \( P(x) \).
|
4034072
| 0.875 |
Find the possible value of \(x+y\) if it is known that \(x^{3}-6x^{2}+15x=12\) and \(y^{3}-6y^{2}+15y=16\).
|
4
| 0.875 |
Calculate the sum
$$
a^{2000}+\frac{1}{a^{2000}}
$$
if $a^{2}-a+1=0$.
|
-1
| 0.625 |
Find the sum of the coefficients of the polynomial \( P(x) = x^4 - 29x^3 + ax^2 + bx + c \), given that \( P(5) = 11 \), \( P(11) = 17 \), and \( P(17) = 23 \).
|
-3193
| 0.5 |
Let the function \( f_{0}(x) = |x| \), \( f_{1}(x) = \left| f_{0}(x) - 1 \right| \), \( f_{2}(x) = \left| f_{1}(x) - 2 \right| \). Determine the area of the enclosed region formed by the graph of \( f_{2}(x) \) and the x-axis.
|
7
| 0.5 |
It is easy to see for $n=1,2,3$ that $n$ circles divide the plane into at most $2^n$ parts. Into how many parts can the plane be divided by drawing four circles, by suitably choosing the sizes and mutual positions of the circles?
|
14
| 0.75 |
There are 1987 sets, each with 45 elements. The union of any two sets has 89 elements. How many elements are there in the union of all 1987 sets?
|
87429
| 0.125 |
There are 2011 street lamps numbered \(1, 2, 3, \ldots, 2011\). For the sake of saving electricity, it is required to turn off 300 of these lamps. However, the conditions are that no two adjacent lamps can be turned off simultaneously, and the lamps at both ends cannot be turned off. How many ways are there to turn off the lamps under these conditions? (Express your answer in terms of binomial coefficients).
|
\binom{1710}{300}
| 0.5 |
In an isosceles triangle \( ABC \) where \( AB = BC \), side \( AC = 10 \). A circle with a diameter of 15 is inscribed in the angle \( ABC \) such that it touches the side \( AC \) at its midpoint. Find the radius of the circle inscribed in the triangle \( ABC \).
|
\frac{10}{3}
| 0.5 |
Given an acute-angled triangle \(ABC\). Point \(M\) is the intersection point of its altitudes. Find angle \(A\) if it is known that \(AM = BC\).
|
45^\circ
| 0.875 |
Let \( A B C D E F \) be a regular hexagon of area 1. Let \( M \) be the midpoint of \( D E \). Let \( X \) be the intersection of \( A C \) and \( B M \), let \( Y \) be the intersection of \( B F \) and \( A M \), and let \( Z \) be the intersection of \( A C \) and \( B F \). If \([P]\) denotes the area of polygon \( P \) for any polygon \( P \) in the plane, evaluate \([B X C] + [A Y F] + [A B Z] - [M X Z Y]\).
|
0
| 0.375 |
In a pasture where the grass grows evenly every day, the pasture can feed 10 sheep for 20 days, or 14 sheep for 12 days. How many days of grass growth each day is enough to feed 2 sheep?
|
2 \text{ days}
| 0.625 |
Which is greater: \(1234567 \cdot 1234569\) or \(1234568^{2}\)?
|
1234568^2
| 0.875 |
In a certain meeting, there are 30 participants. Each person knows at most 5 others among the remaining participants, and in any group of 5 people, there are always two who do not know each other. Find the largest positive integer \( k \) such that there exists a subset of \( k \) people from these 30 people who do not know each other.
|
6
| 0.75 |
We will call a natural number interesting if all its digits are different, and the sum of any two adjacent digits is the square of a natural number. Find the largest interesting number.
|
6310972
| 0.375 |
How many ordered pairs of positive integers \((x, y)\) satisfy the equation
\[
x \sqrt{y} + y \sqrt{x} + \sqrt{2006 x y} - \sqrt{2006 x} - \sqrt{2006 y} - 2006 = 0 ?
\]
|
8
| 0.5 |
Let \(ABCDEF\) be a regular hexagon. A frog starts at vertex \(A\). Each time, it can jump to one of the two adjacent vertices. If the frog reaches point \(D\) within 5 jumps, it stops jumping; if it does not reach point \(D\) within 5 jumps, it stops after completing 5 jumps. How many different ways can the frog jump from the start until it stops?
|
26
| 0.125 |
Dima needed to arrive at the station at 18:00, where his father was supposed to pick him up by car. However, Dima caught an earlier train and arrived at the station at 17:05. Instead of waiting for his father, he started walking towards him. They met on the way, Dima got into the car, and they arrived home 10 minutes earlier than the calculated time. What was Dima's walking speed before he met his father, if the car's speed was $60 \ \text{km/h}$?
|
6 \text{ km/h}
| 0.375 |
There are 8 lily pads in a pond numbered \(1, 2, \ldots, 8\). A frog starts on lily pad 1. During the \(i\)-th second, the frog jumps from lily pad \(i\) to \(i+1\), falling into the water with probability \(\frac{1}{i+1}\). The probability that the frog lands safely on lily pad 8 without having fallen into the water at any point can be written as \(\frac{m}{n}\), where \(m, n\) are positive integers and \(\operatorname{gcd}(m, n) = 1\). Find \(100m + n\).
|
108
| 0.875 |
Find the volume of a regular quadrilateral prism if its diagonal makes an angle of \(30^{\circ}\) with the plane of the lateral face, and the side of the base is equal to \(a\).
|
a^3 \sqrt{2}
| 0.5 |
In the equation
$$
\frac{x^{2}+p}{x}=-\frac{1}{4},
$$
with roots \(x_{1}\) and \(x_{2}\), determine \(p\) such that:
a) \(\frac{x_{1}}{x_{2}}+\frac{x_{2}}{x_{1}}=-\frac{9}{4}\),
b) one root is 1 less than the square of the other root.
|
-\frac{15}{8}
| 0.125 |
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