problem
stringlengths 18
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0.92
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|---|---|---|
Show that for all \(a, b, c \geq 0\) such that \(abc = 1\), \(\left(a^2 + 1\right)\left(b^3 + 2\right)\left(c^6 + 5\right) \geq 36\).
|
36
| 0.875 |
A rectangular table with dimensions $x$ cm $\times 80$ cm is covered with identical sheets of paper measuring 5 cm $\times 8$ cm. The first sheet is placed in the bottom-left corner, and each subsequent sheet is placed one centimeter higher and one centimeter to the right of the previous sheet. The last sheet is positioned in the top-right corner. What is the length $x$ in centimeters?
|
77
| 0.375 |
The sequence $\{a_{n}\}$ satisfies: $a_{1}=\frac{1}{4}$, $a_{2}=\frac{1}{5}$, and $a_{1}a_{2} + a_{2}a_{3} + \cdots + a_{n}a_{n+1} = n a_{1}a_{n+1}$ for any positive integer $n$. Then, find the value of $\frac{1}{a_{1}} + \frac{1}{a_{2}} + \cdots + \frac{1}{a_{97}}$.
|
5044
| 0.5 |
In how many ways can two disjoint subsets be selected from a set with $n$ elements?
|
3^n
| 0.875 |
Two circles touch each other internally at point K. Chord \( AB \) of the larger circle touches the smaller circle at point L, with \( A L = 10 \). Find \( B L \) given that \( A K : B K = 2 : 5 \).
|
25
| 0.625 |
Define a sequence of polynomials as follows: let \( a_{1}=3 x^{2}-x \), let \( a_{2}=3 x^{2}-7 x+3 \), and for \( n \geq 1 \), let \( a_{n+2}=\frac{5}{2} a_{n+1}-a_{n} \). As \( n \) tends to infinity, what is the limit of the sum of the roots of \( a_{n} \)?
|
\frac{13}{3}
| 0.625 |
Find the maximum value of the expression \((\sqrt{36-4 \sqrt{5}} \sin x-\sqrt{2(1+\cos 2 x)}-2) \cdot (3+2 \sqrt{10-\sqrt{5}} \cos y-\cos 2 y)\). If the answer is not an integer, round it to the nearest whole number.
|
27
| 0.5 |
\(\frac{\cos 67^{\circ} \cos 7^{\circ} - \cos 83^{\circ} \cos 23^{\circ}}{\cos 128^{\circ} \cos 68^{\circ} - \cos 38^{\circ} \cos 22^{\circ}} - \tan 164^{\circ}\).
|
0
| 0.875 |
How many integers \( b \) exist for which the equation \( x^2 + b x - 9600 = 0 \) has an integer solution that is divisible by both 10 and 12? Indicate the largest possible \( b \).
|
9599
| 0.375 |
A 10,000-meter race involves three participants: A, B, and C. A starts last. During the race, A exchanges positions with B and C a total of 9 times. Determine the final position of A in the race.
|
2
| 0.5 |
Given that for any \( x \geqslant 1 \), the following inequality holds:
\[
\ln x - a\left(1 - \frac{1}{x}\right) \geqslant 0.
\]
Find the range of the real number \( a \).
|
(-\infty, 1]
| 0.5 |
On an infinite tape, numbers are written in a row. The first number is one, and each subsequent number is obtained by adding the smallest non-zero digit of its decimal representation to the previous number. How many digits are in the decimal representation of the number that is in the $9 \cdot 1000^{1000}$-th place in this sequence?
|
3001
| 0.375 |
Given \(a\) and \(b\) are real numbers, satisfying:
\[
\sqrt[3]{a} - \sqrt[3]{b} = 12, \quad ab = \left( \frac{a + b + 8}{6} \right)^3.
\]
Find \(a - b\).
|
468
| 0.875 |
In a drawer, there are 5 distinct pairs of socks. Four socks are drawn at random. The probability of drawing two pairs is one in $n$. Determine the value of $n$.
|
21
| 0.875 |
Given the function \( f(x) = x + \frac{4}{x} - 1 \), if there exist \( x_{1}, x_{2}, \cdots, x_{n} \in \left[ \frac{1}{4}, 4 \right] \) such that
$$
\sum_{i=1}^{n-1} f(x_{i}) = f(x_{n}),
$$
then the maximum value of the positive integer \( n \) is...
|
6
| 0.75 |
a) In how many different ways can we paint the circles in the letter "O," not necessarily using all 3 colors?
b) In how many different ways can we paint the circles in the letter "E," necessarily using all 3 colors?
c) In how many different ways can we paint the letter "B" such that circles connected by a segment must have different colors? (The three circles aligned vertically have TWO segments separating them).
|
12
| 0.5 |
Let \( f: \mathbf{N}^{*} \rightarrow \mathbf{N}^{\top} \), and for all \( m, n \in \mathbf{N}^{\top} \), \( f(f(m) + f(n)) = m + n \). Find \( f(2005) \).
|
2005
| 0.875 |
How many solutions does the equation \( x_1 + x_2 + x_3 = 1000 \) have
a) in natural numbers?
b) in non-negative integers?
|
\binom{1002}{2}
| 0.25 |
In an integer triangle, two sides are equal to 10. Find the third side, given that the radius of the inscribed circle is an integer.
|
12
| 0.5 |
A cuckoo clock is on the wall. At the beginning of every hour, the cuckoo makes a number of "cuckoo" sounds equal to the hour displayed by the hour hand (for example, at 19:00 the cuckoo makes 7 sounds). One morning, Maxim approached the clock when it showed 9:05. He started turning the minute hand until he moved the time forward by 7 hours. How many times did the cuckoo make a sound during this time?
|
43
| 0.625 |
The area of the base of a right triangular prism is 4, and the areas of the lateral faces are 9, 10, and 17. Find the volume of the prism.
|
12
| 0.5 |
Find all functions \( f: \mathbf{Z}^{*} \rightarrow \mathbf{R} \) (where \(\mathbf{Z}^{*}\) is the set of positive integers) that satisfy the equation \( f(n+m)+f(n-m)=f(3n) \) for \( m, n \in \mathbf{Z}^{*} \) and \( n \geq m \).
|
f(n) = 0
| 0.875 |
The sum of the lengths of all of the edges of rectangular prism \( ABCDEFGH \) is 24. If the total surface area of the prism is 11, determine the length of the diagonal \( AH \).
|
5
| 0.875 |
Find all values of \( a \) for which the equation \( a^{2}(x-2) + a(39-20x) + 20 = 0 \) has at least two distinct roots.
|
20
| 0.625 |
Given that a parabola \( P \) has the center of the ellipse \( E \) as its focus, \( P \) passes through the two foci of \( E \), and \( P \) intersects \( E \) at exactly three points, determine the eccentricity of the ellipse.
|
\frac{2\sqrt{5}}{5}
| 0.125 |
Show that the fractional part of \( (2 + \sqrt{3})^n \) tends to 1 as \( n \) tends to infinity.
|
1
| 0.75 |
Find all pairs of integers \((x, y)\) that are solutions to the equation
$$
7xy - 13x + 15y - 37 = 0.
$$
Indicate the sum of all found values of \(x\).
|
4
| 0.625 |
A $12 \times 12$ grid is colored such that each cell is either black or white. Ensure that within any $3 \times 4$ or $4 \times 3$ subgrid, there is at least one black cell. Find the minimum number of black cells needed.
|
12
| 0.625 |
Let $A, B, C,$ and $D$ be four cocyclic points (in this order) on a circle with center $O$. Suppose that $(AC) \perp (BD)$. Let $I$ be the foot of the perpendicular from $O$ in triangle $AOD$. Show that $OI = \frac{BC}{2}$.
|
\frac{B C}{2}
| 0.875 |
If \( a \) is the smallest cubic number divisible by 810, find the value of \( a \).
|
729000
| 0.75 |
Find a curve passing through the point $(0, -2)$ such that the tangent of the angle of inclination of the tangent at any of its points equals the ordinate of that point increased by three units.
|
y = e^x - 3
| 0.75 |
Given the complex numbers \( z = \cos \alpha + i \sin \alpha \) and \( u = \cos \beta + i \sin \beta \), and that \( z + u = \frac{4}{5} + \frac{3}{5} i \), find \( \tan(\alpha + \beta) \).
|
\frac{24}{7}
| 0.875 |
布尔图 rode a motorcycle from home to pick peaches and observe monkeys at Huaguo Mountain. He rode slowly and enjoyed the scenery along the way. Upon reaching the top, he found no peaches and was attacked by monkeys. He immediately rode back home at a speed of 42 kilometers per hour. If the average speed for the entire journey up and down the mountain was 21 kilometers per hour, what was his speed in kilometers per hour when going up the mountain?
|
14
| 0.875 |
Point \( K \) lies on edge \( AB \) of pyramid \( ABCD \). Construct the cross-section of the pyramid with a plane passing through point \( K \) parallel to lines \( BC \) and \( AD \).
|
K L M N
| 0.125 |
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 |
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 |
The robot vacuum cleaner is programmed to move on the floor according to the law:
$$\left\{\begin{array}{l} x = t(t-6)^2 \\ y = 0, \quad 0 \leq t \leq 7 \\ y = (t-7)^2, \quad t \geq 7 \end{array}\right.$$
where the axes are chosen parallel to the walls and the movement starts from the origin. Time $t$ is measured in minutes, and coordinates are measured in meters. Find the distance traveled by the robot in the first 7 minutes and the absolute change in the velocity vector during the eighth minute.
|
\sqrt{445}
| 0.5 |
Find the maximum real number \( k \) such that for any positive numbers \( a \) and \( b \), the following inequality holds:
$$
(a+b)(ab+1)(b+1) \geqslant k \, ab^2.
$$
|
\frac{27}{4}
| 0.625 |
Calculate the limit of the function:
$$
\lim_{{x \rightarrow 0}} \frac{\sqrt{1+x \sin x}-1}{e^{x^{2}}-1}
$$
|
\frac{1}{2}
| 0.875 |
Given $\overrightarrow{OA}=(4,3)$, the graph of the function $y=x^{2}+bx+c$ is translated by the vector $\overrightarrow{OA}$ to obtain a new graph that is tangent to the line $4x + y - 8 = 0$ at the point $T(1, 4)$. Find the analytical expression of the original function.
|
y = x^2 + 2x - 2
| 0.875 |
What is the largest \( x \) such that \( x^2 \) divides \( 24 \cdot 35 \cdot 46 \cdot 57 \)?
|
12
| 0.625 |
There are fewer than 100 students performing a group dance, and there are two possible formations: one formation is with a group of 5 students in the middle, with the remaining students in groups of 8 around the outer circle; the other formation is with a group of 8 students in the middle, with the remaining students in groups of 5 around the outer circle. What is the maximum number of students?
|
93
| 0.875 |
Given that \( x + y + z = xy + yz + zx \), find the minimum value of \( \frac{x}{x^2 + 1} + \frac{y}{y^2 + 1} + \frac{z}{z^2 + 1} \).
|
-\frac{1}{2}
| 0.625 |
Determine the measure of angle $\hat{A}$ of triangle $ABC$ if it is known that the angle bisector of this angle is perpendicular to the line passing through the orthocenter and the circumcenter of the triangle.
|
60^\circ
| 0.25 |
Sarah and Hagar play a game of darts. Let \(O_0\) be a circle of radius 1. On the \(n\)th turn, the player whose turn it is throws a dart and hits a point \(p_n\) randomly selected from the points of \(O_{n-1}\). The player then draws the largest circle that is centered at \(p_n\) and contained in \(O_{n-1}\), and calls this circle \(O_n\). The player then colors every point that is inside \(O_{n-1}\) but not inside \(O_n\) her color. Sarah goes first, and the two players alternate turns. Play continues indefinitely. If Sarah's color is red, and Hagar's color is blue, what is the expected value of the area of the set of points colored red?
|
\frac{6 \pi}{7}
| 0.75 |
A person's age in 1962 was one more than the sum of the digits of the year in which they were born. How old are they?
|
23
| 0.625 |
In quadrilateral \( \square ABCD \), \(\angle B < 90^\circ\) and \(AB < BC\). Tangents are drawn from point \( D \) to the circumcircle \( \Gamma \) of triangle \( \triangle ABC \), touching the circle at points \( E \) and \( F \). Given that \(\angle EDA = \angle FDC\), find \(\angle ABC\).
|
60^\circ
| 0.875 |
In a $15 \times 15$ grid, some centers of adjacent squares are connected by segments to form a closed line that does not intersect itself and is symmetric with respect to one of the main diagonals. Show that the length of the line is less than or equal to 200.
|
200
| 0.75 |
The sequence \(\left\{a_{n}\right\}\) satisfies \(a_{1}=2\) and \(a_{n+1}=\frac{2(n+2)}{n+1} a_{n}\) for \(n \in \mathbf{N}^{*}\). What is \(\frac{a_{2014}}{a_{1}+a_{2}+\cdots+a_{2013}}\)?
|
\frac{2015}{2013}
| 0.25 |
There are knights who always tell the truth and liars who always lie living on an island. The island's population is 1000 people, spread across 10 villages (each village has at least two people). One day, each islander claimed that all other villagers in their village are liars. How many liars live on the island? (Two villagers are considered to be from the same village if they live in the same village.)
|
990
| 0.75 |
The sequence $1001, 1004, 1009$ has a general term $a_{n} = n^{2} + 1000$, where $n \in \mathbf{N}_{+}$. For each $n$, let $d_{n}$ denote the greatest common divisor (GCD) of $a_{n}$ and $a_{n+1}$. Determine the maximum value of $d_{n}$ as $n$ ranges over all positive integers.
|
4001
| 0.75 |
In a $6 \times 6$ grid of numbers:
(i) All the numbers in the top row and the leftmost column are the same;
(ii) Each other number is the sum of the number above it and the number to the left of it;
(iii) The number in the bottom right corner is 2016.
What are the possible numbers in the top left corner?
|
8
| 0.125 |
In triangle \(K L M\), side \(K L\) is equal to 24, angle bisector \(L N\) is equal to 24, and segment \(M N\) is equal to 9. Find the perimeter of triangle \(L M N\).
|
60
| 0.5 |
Find \(\operatorname{ctg} 2 \alpha\) if it is known that \(\sin \left(\alpha-90^{\circ}\right)=-\frac{2}{3}\) and \(270^{\circ}<\alpha<360^{\circ}\).
|
\frac{\sqrt{5}}{20}
| 0.875 |
Consider a circle of radius \( R \) centered at the origin on the Cartesian plane. Specify at least one value of \( R \) for which there are exactly 32 integer points (a point is called an integer point if both its x-coordinate and y-coordinate are integers) lying on this circle.
|
\sqrt{1105}
| 0.125 |
In the quadrilateral \( ABCD \), angle \( B \) is \( 150^{\circ} \), angle \( C \) is a right angle, and the sides \( AB \) and \( CD \) are equal.
Find the angle between side \( BC \) and the line passing through the midpoints of sides \( BC \) and \( AD \).
|
60^\circ
| 0.75 |
What is the third smallest number that is both a triangular number and a square number? Obviously, the first two numbers that have this property are 1 and 36. What is the next number?
|
1225
| 0.625 |
Given the sequence $\left\{a_{n}\right\}$ where $a_{1}=1$, $a_{2}=2$, and $a_{n}a_{n+1}a_{n+2}=a_{n}+a_{n+1}+a_{n+2}$ for all $n$, and $a_{n+1}a_{n+2} \neq 1$, find $S_{1999} = \sum_{n=1}^{1999} a_{n} = \quad$.
|
3997
| 0.5 |
A pedestrian walked 5.5 kilometers in 1 hour but did not reach point \( B \) (short by \(2 \pi - 5.5\) km). Therefore, the third option is longer than the first and can be excluded.
In the first case, they need to cover a distance of 5.5 km along the alley. If they move towards each other, the required time is \(\frac{5.5}{20 + 5.5}\) hours.
In the second case, moving towards each other, after \(\frac{2 \pi - 5.5}{5.5}\) hours the pedestrian will reach point \( B \), while the cyclist will still be riding on the highway (since \(\frac{4}{15} > \frac{2 \pi - 5.5}{5.5}\)). Thus, the cyclist will always be on the highway and the closing speed of the pedestrian and the cyclist will be \(15 + 5.5 = 20.5\) km/h. They will meet in \(\frac{2 \pi - 1.5}{20.5}\) hours.
Compare the numbers obtained in cases 1 and 2:
$$
\frac{5.5}{25.5} = \frac{11}{51} < 0.22 < 0.23 < \frac{4 \cdot 3.14 - 3}{41} < \frac{2 \pi - 1.5}{20.5}
$$
Therefore, the answer is given by the first case.
**Note**: Some participants interpreted the condition differently, considering that the pedestrian and the cyclist met after an hour, and at that time, the pedestrian asked the cyclist to bring him the keys. Since the condition does not specify where exactly the neighbor was, and with a different interpretation of the condition, a similar problem arises (although with slightly more cumbersome calculations), the jury decided to accept both interpretations of the condition. The plan for solving the second interpretation is provided.
|
\frac{11}{51}
| 0.25 |
The terms of a certain sequence are sums of the corresponding terms of two geometric progressions, given that the first two terms are equal to 0. Determine the sequence.
|
0
| 0.125 |
Find all positive \( x \) and \( y \) for which the expression
\[
\frac{x y}{2}+\frac{18}{x y}
\]
takes the minimum value and among all such \( x \) and \( y \), the expression
\[
\frac{y}{2}+\frac{x}{3}
\]
is also minimized.
|
x=3, y=2
| 0.5 |
Solve the equation
$$
\cos ^{2} 8 x+\cos ^{2} x=2 \cos ^{2} x \cdot \cos ^{2} 8 x
$$
In the answer, indicate the number equal to the sum of the roots of the equation that belong to the interval $[3 \pi ; 6 \pi]$, rounding this number to two decimal places if necessary.
|
56.55
| 0.875 |
Find the probability that, when five different numbers are randomly chosen from the set $\{1, 2, \ldots, 20\}$, at least two of them are consecutive.
|
\frac{232}{323}
| 0.5 |
By how much is the sum of the squares of the first one hundred even numbers greater than the sum of the squares of the first one hundred odd numbers?
|
20100
| 0.375 |
Write the first 10 prime numbers in a row. How can you remove 6 digits to get the largest possible number?
|
7317192329
| 0.25 |
Compute the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty} n\left(\sqrt{n^{4}+3}-\sqrt{n^{4}-2}\right)
$$
|
0
| 0.625 |
Given the sequence \(\left\{a_{n}\right\}\), which satisfies
\[
a_{1}=0,\left|a_{n+1}\right|=\left|a_{n}-2\right|
\]
Let \(S\) be the sum of the first 2016 terms of the sequence \(\left\{a_{n}\right\}\). Determine the maximum value of \(S\).
|
2016
| 0.625 |
A container is already filled with water. There are three lead balls: large, medium, and small. The first time, the small ball is submerged in the water; the second time, the small ball is removed, and the medium ball is submerged in the water; the third time, the medium ball is removed, and the large ball is submerged in the water. It is known that the water spilled the first time is 3 times the water spilled the second time, and the water spilled the third time is 3 times the water spilled the first time. Find the ratio of the volumes of the three balls.
|
3 : 4 : 13
| 0.25 |
Let \( p \) be a given odd prime number. If there exists a positive integer \( k \) such that \( \sqrt{k^2 - pk} \) is also a positive integer, find the value of \( k \).
|
\frac{(p + 1)^2}{4}
| 0.625 |
A Tim number is a five-digit positive integer with the following properties:
1. It is a multiple of 15.
2. Its hundreds digit is 3.
3. Its tens digit is equal to the sum of its first three (leftmost) digits.
How many Tim numbers are there?
|
16
| 0.375 |
Calculate the following expression:
$$
\lim _{n \rightarrow \infty} \sum_{k=0}^{n} \frac{k^{2}+3k+1}{(k+2)!}
$$
|
2
| 0.625 |
Given that $f(x)$ is a linear function, $f[f(1)] = -1$, and the graph of $f(x)$ is symmetric about the line $x - y = 0$ is $C$. If the point $\left(n, \frac{a_{n+1}}{a_{n}}\right)$ $(n \in \mathbf{N}^{\cdot})$ lies on the curve $C$, and given $a_{1} = 1$, $\frac{a_{n+1}}{a_{n}} - \frac{a_{n}}{a_{n-1}} = 1 (n \geqslant 2)$:
1. Find the analytic expression of $f(x)$ and the equation of the curve $C$.
2. Let $S_{n} = \frac{a_{1}}{3!} + \frac{a_{2}}{4!} + \frac{a_{3}}{5!} + \cdots + \frac{a_{n}}{(n+2)!}$. For all $n \in \mathbf{N}^{\cdot}$, if $S_{n} > m$ holds, find the maximum value of the natural number $m$.
|
0
| 0.75 |
Find all functions \( f: \mathbf{Z} \rightarrow \mathbf{Z} \) such that for all \( n \in \mathbf{Z} \), \( f[f(n)]+f(n)=2n+3 \), and \( f(0)=1 \).
|
f(n) = n + 1
| 0.75 |
Let \( x \) and \( y \) be positive real numbers, and \( x + y = 1 \). Find the minimum value of \( \frac{x^2}{x+2} + \frac{y^2}{y+1} \).
|
\frac{1}{4}
| 0.75 |
Let \((a_{1}, a_{2}, \ldots, a_{8})\) be a permutation of \((1,2, \ldots, 8)\). Find, with proof, the maximum possible number of elements of the set
\[ \left\{a_{1}, a_{1}+a_{2}, \ldots, a_{1}+a_{2}+\cdots+a_{8}\right\} \]
that can be perfect squares.
|
5
| 0.375 |
Determine the largest integer \( n \) such that \( 7^{2048} - 1 \) is divisible by \( 2^{n} \).
|
14
| 0.375 |
In how many ways can the number \( n \) be represented as the sum of positive odd addends? (Representations that differ only in the order of the addends are considered different.)
|
F_n
| 0.125 |
For each positive integer \( n \geq 1 \), we define the recursive relation given by
\[ a_{n+1} = \frac{1}{1 + a_{n}}. \]
Suppose that \( a_{1} = a_{2012} \). Find the sum of the squares of all possible values of \( a_{1} \).
|
3
| 0.625 |
In an isosceles triangle \(ABC\) with base \(AB\), the angle bisectors \(CL\) and \(AK\) are drawn. Find \(\angle ACB\) of triangle \(ABC\), given that \(AK = 2CL\).
|
108^\circ
| 0.25 |
Given a five-digit palindromic number that equals 45 times a four-digit palindromic number (i.e., $\overline{\mathrm{abcba}}=45 \times \overline{\text{deed}}$), what is the largest possible value of the five-digit palindromic number?
|
59895
| 0.375 |
A function \( f \) defined on ordered pairs of positive integers satisfies the following properties:
1. \( f(x, x) = x \);
2. \( f(x, y) = f(y, x) \);
3. \( (x + y) f(x, y) = y f(x, x + y) \).
Calculate \( f(14,52) \).
|
364
| 0.875 |
Consider triangle \(ABC\) with \(\angle A = 2 \angle B\). The angle bisectors from \(A\) and \(C\) intersect at \(D\), and the angle bisector from \(C\) intersects \(\overline{AB}\) at \(E\). If \(\frac{DE}{DC} = \frac{1}{3}\), compute \(\frac{AB}{AC}\).
|
\frac{7}{9}
| 0.125 |
Find a natural number \( n \), knowing that it has two prime divisors and satisfies the conditions \(\tau(n) = 6\) and \(\sigma(n) = 28\).
|
12
| 0.625 |
Two cars leave points A and B simultaneously and meet at 12 PM. If the speed of the first car is doubled while keeping the speed of the second car the same, the meeting will occur 56 minutes earlier. If the speed of the second car is doubled while keeping the speed of the first car the same, they will meet 65 minutes earlier. Determine the meeting time if the speeds of both cars were doubled.
|
10:29 \text{ AM}
| 0.625 |
It is known that the polynomial \( f(x) = 8 + 32x - 12x^2 - 4x^3 + x^4 \) has 4 distinct real roots \(\{x_{1}, x_{2}, x_{3}, x_{4}\}\). The polynomial of the form \( g(x) = b_{0} + b_{1} x + b_{2} x^2 + b_{3} x^3 + x^4 \) has roots \(\{x_{1}^2, x_{2}^2, x_{3}^2, x_{4}^2\}\). Find the coefficient \( b_{1} \) of the polynomial \( g(x) \).
|
-1216
| 0.5 |
On the board, 26 ones are written. Every minute, Karlson erases two arbitrary numbers and writes their sum on the board, and then eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could eat in 26 minutes?
|
325
| 0.25 |
What is the greatest possible number of rays in space emanating from a single point and forming obtuse angles pairwise?
|
4
| 0.5 |
Two pedestrians set out simultaneously from $A$ to $B$ and from $B$ to $A$, respectively. When the first pedestrian has traveled halfway, the second pedestrian has 24 kilometers left to go. When the second pedestrian has traveled halfway, the first pedestrian has 15 kilometers left to go. How many kilometers will the second pedestrian have left to walk after the first pedestrian finishes their journey?
|
8 \text{ km}
| 0.75 |
The warehouse has nails in boxes weighing 24, 23, 17, and 16 kg. Can the storekeeper release 100 kg of nails from the warehouse without opening the boxes?
|
Yes
| 0.625 |
In the right triangle \(ABC\), the altitude \(BH\) is drawn to the hypotenuse \(AC\). Points \(X\) and \(Y\) are the centers of the circles inscribed in triangles \(ABH\) and \(CBH\) respectively. The line \(XY\) intersects the legs \(AB\) and \(BC\) at points \(P\) and \(Q\). Find the area of triangle \(BPQ\), given that \(BH = h\).
|
\frac{h^2}{2}
| 0.75 |
In triangle \(ABC\) with sides \(BC=7\), \(AC=5\), and \(AB=3\), an angle bisector \(AD\) is drawn. A circle is circumscribed around triangle \(ABD\), and a circle is inscribed in triangle \(ACD\). Find the product of their radii.
|
\frac{35}{32}
| 0.625 |
A number \( \mathrm{X} \) is called "25-supporting" if for any 25 real numbers \( a_{1}, \ldots, a_{25} \) whose sum is an integer, there is at least one for which \( \left|a_{i} - \frac{1}{2}\right| \geq X \).
Provide the largest 25-supporting \( X \), rounded to the nearest hundredth according to standard mathematical rules.
|
0.02
| 0.625 |
At the namesake festival, 45 Alexanders, 122 Borises, 27 Vasily, and several Gennady attended. At the beginning of the festival, all of them lined up so that no two people with the same name stood next to each other. What is the minimum number of Gennadys that could have attended the festival?
|
49
| 0.5 |
Rearrange the four digits of 2016 to form a four-digit perfect square. What is this four-digit perfect square? $\qquad$ .
|
2601
| 0.5 |
The line $c$ is defined by the equation $y = 2x$. Points $A$ and $B$ have coordinates $A(2, 2)$ and $B(6, 2)$. On line $c$, find the point $C$ from which the segment $AB$ is seen at the largest angle.
|
(2, 4)
| 0.875 |
Find a polynomial with integer coefficients for which $\cos 18^{\circ}$ is a root.
|
16x^4 - 20x^2 + 5
| 0.5 |
If \( A \) is a prime number and \( A-4 \), \( A-6 \), \( A-12 \), \( A-18 \) are also prime numbers, then \( A = \) ?
|
23
| 0.875 |
Gavrila found that the front tires of a car last for 21,000 km, and the rear tires last for 28,000 km. He decided to switch them at some point to maximize the possible distance the car can travel. Find this maximum distance (in km).
|
24000 \text{ km}
| 0.875 |
Compute the sum \( S = \sum_{i=0}^{101} \frac{x_{i}^{3}}{1 - 3x_{i} + 3x_{i}^{2}} \) for \( x_{i} = \frac{i}{101} \).
|
51
| 0.75 |
Let the random variables $\xi$ and $\eta$ denote the lifetimes of the blue and red light bulbs, respectively. The lifetime of the flashlight is equal to the minimum of these two values. Clearly, $\min (\xi, \eta) \leq \xi$. Let's move to the expected values: $\operatorname{E} \min (\xi, \eta) \leq \mathrm{E} \xi=2$. Therefore, the expected lifetime of the flashlight is no more than 2 years.
|
2
| 0.875 |
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