problem
stringlengths 18
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0.92
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|---|---|---|
In the diagram, \(ABCD\) is a parallelogram. \(E\) is on side \(AB\), and \(F\) is on side \(DC\). \(G\) is the intersection point of \(AF\) and \(DE\), and \(H\) is the intersection point of \(CE\) and \(BF\). Given that the area of parallelogram \(ABCD\) is 1, \(\frac{\mathrm{AE}}{\mathrm{EB}}=\frac{1}{4}\), and the area of triangle \(BHC\) is \(\frac{1}{8}\), find the area of triangle \(ADG\).
|
\frac{7}{92}
| 0.625 |
Let $A$ be a finite set of integers and $n \geq 1$. Let $N$ be the number of $n$-tuples $\left(x_{1}, \ldots, x_{n}\right)$ in $A^{n}$ satisfying $x_{1}+\cdots+x_{n}=-1$. Let $M$ be the number of $n$-tuples in $A^{n}$ satisfying $x_{1}+\cdots+x_{n}=-1$ and for all integers $i$ satisfying $1 \leq i \leq n-1$, $x_{1}+\cdots+x_{i} \geq 0$.
Show that $M = \frac{N}{n}$.
|
M = \frac{N}{n}
| 0.875 |
The vertices of a regular hexagon are labeled \(\cos (\theta), \cos (2 \theta), \ldots, \cos (6 \theta)\). For every pair of vertices, Bob draws a blue line through the vertices if one of these functions can be expressed as a polynomial function of the other (that holds for all real \(\theta\)), and otherwise Roberta draws a red line through the vertices. In the resulting graph, how many triangles whose vertices lie on the hexagon have at least one red and at least one blue edge?
|
14
| 0.375 |
In a certain class, there are 28 boys and 22 girls. If 5 students are to be elected to different class committee positions, and it's desired that both boys and girls are represented among the 5 students, how many different election outcomes are possible?
|
239297520
| 0.5 |
A pair of natural numbers \(a > b\) is called good if the LCM of these numbers is divisible by their difference. Among all natural divisors of the number \( n \), exactly one good pair was found. What can \( n \) be equal to?
|
n = 2
| 0.875 |
Given the function
\[ f(x)=\begin{cases}
2^{2-x}, & x<2, \\
\log_{3}(x+1), & x \geq 2,
\end{cases} \]
if the equation \( f(x)=m \) has two distinct real roots, then the range of real number \( m \) is (express in interval notation).
|
(1,+\infty)
| 0.5 |
Let \( f(x) \) be a function defined on \(\mathbf{R}\) such that \( f(1) = 1 \) and for any \( x \in \mathbf{R} \), it holds that \( f(x+5) \geq f(x) + 5 \) and \( f(x+1) \leq f(x) + 1 \). If \( g(x) = f(x) + 1 - x \), find \( g(2002) \).
(Note: This problem is from the 2002 National High School Mathematics Competition in China.)
|
1
| 0.5 |
Solve the system $\left\{\begin{array}{l}2 x + y + 8 \leq 0, \\ x^{4} + 2 x^{2} y^{2} + y^{4} + 9 - 10 x^{2} - 10 y^{2} = 8 x y.\end{array}\right.$
|
(-3, -2)
| 0.625 |
We have an infinite grid, and we need to write a positive integer in each cell with the following conditions. The number $n$ should appear exactly $n$ times (for example, the number 1 appears once, the number 2 appears twice, etc.), and the difference between numbers in any two adjacent cells (cells sharing a common side) should be less than a given $k$. What is the smallest integer $k$ for which this arrangement is possible?
|
3
| 0.375 |
a) What is the maximum number of edges in a 30-vertex graph that contains no triangles?
b) What is the maximum number of edges in a 30-vertex graph that contains no complete subgraph of four vertices?
|
300
| 0.75 |
Let \( S = \{ 1, 2, 3, \ldots, 280 \} \). Find the smallest integer \( n \) such that each \( n \)-element subset of \( S \) contains five numbers which are pairwise relatively prime.
|
217
| 0.25 |
Compute the limit of the function:
$$
\lim _{x \rightarrow 0}\left(\frac{1+\operatorname{tg} x \cdot \cos 2 x}{1+\operatorname{tg} x \cdot \cos 5 x}\right)^{\frac{1}{x^{3}}}
$$
|
e^{\frac{21}{2}}
| 0.5 |
Clara leaves home by bike at 1:00 p.m. for a meeting scheduled with Quinn later that afternoon. If Clara travels at an average of $20 \mathrm{~km} / \mathrm{h}$, she would arrive half an hour before their scheduled meeting time. If Clara travels at an average of $12 \mathrm{~km} / \mathrm{h}$, she would arrive half an hour after their scheduled meeting time. At what average speed, in $\mathrm{km} / \mathrm{h}$, should Clara travel to meet Quinn at the scheduled time?
|
15
| 0.875 |
Bernie has 2020 marbles and 2020 bags labeled \( B_{1}, \ldots, B_{2020} \) in which he randomly distributes the marbles (each marble is placed in a random bag independently). If \( E \) is the expected number of integers \( 1 \leq i \leq 2020 \) such that \( B_{i} \) has at least \( i \) marbles, compute the closest integer to \( 1000 E \).
|
1000
| 0.375 |
Given \(0 < a < b\), lines \(l\) and \(m\) pass through two fixed points \(A(a, 0)\) and \(B(b, 0)\) respectively, intersecting the parabola \(y^{2} = x\) at four distinct points. When these four points lie on a common circle, find the locus of the intersection point \(P\) of lines \(l\) and \(m\).
|
x = \frac{a+b}{2}
| 0.375 |
All cells of an \( n \times n \) square grid are numbered in some order with numbers from 1 to \( n^{2} \). Petya makes moves according to the following rules. On the first move, he places a rook on any cell. On each subsequent move, Petya can either place a new rook on some cell or move the rook from a cell numbered \( a \) horizontally or vertically to a cell with a number greater than \( a \). Each time a rook lands on a cell, that cell is immediately colored; placing a rook on a colored cell is prohibited. What is the minimum number of rooks Petya will need to color all cells of the grid regardless of the initial numbering?
|
n
| 0.75 |
Two classes are planting trees. Each student in the first class plants 3 trees, and each student in the second class plants 5 trees. A total of 115 trees are planted. What is the maximum possible total number of students in both classes combined?
|
37
| 0.875 |
For which values of \(a\) does the equation \(|x-3| = a x - 1\) have two solutions? Enter the midpoint of the interval of parameter \(a\) in the provided field. Round the answer to three significant digits according to rounding rules and enter it in the provided field.
|
0.667
| 0.75 |
The equation \( x^{3} - 9x^{2} + 8x + 2 = 0 \) has three real roots \( p, q, r \). Find \( \frac{1}{p^{2}} + \frac{1}{q^{2}} + \frac{1}{r^{2}} \).
|
25
| 0.75 |
Given a quadratic polynomial \( f(x) \) such that the equation \( (f(x))^3 - f(x) = 0 \) has exactly three solutions. Find the ordinate of the vertex of the polynomial \( f(x) \).
|
0
| 0.75 |
When young fishermen were asked how many fish each of them caught, the first one replied, "I caught half the number of fish that my friend caught, plus 10 fish." The second one said, "And I caught as many as my friend, plus 20 fish." How many fish did the fishermen catch?
|
100
| 0.125 |
What is the maximum number of closed intervals that can be given on the number line such that among any three intervals, there are two that intersect, but the intersection of any four intervals is empty?
|
6
| 0.125 |
For each real number \( x \), \( f(x) \) is defined to be the minimum of the values of \( 2x + 3 \), \( 3x - 2 \), and \( 25 - x \). What is the maximum value of \( f(x) \)?
|
\frac{53}{3}
| 0.875 |
The sum of two sides of a rectangle is 7 cm, and the sum of three of its sides is 9.5 cm. Find the perimeter of the rectangle.
|
14 \text{ cm}
| 0.875 |
Sasha chose a natural number \( N > 1 \) and wrote down all of its natural divisors in ascending order: \( d_{1}<\ldots<d_{s} \) (such that \( d_{1}=1 \) and \( d_{s}=N \)). Then for each pair of adjacent numbers, he calculated their greatest common divisor; the sum of the \( s-1 \) resulting numbers turned out to be equal to \( N-2 \). What values could \( N \) take?
|
N = 3
| 0.875 |
A palindrome is a string that does not change when its characters are written in reverse order. Let \( S \) be a 40-digit string consisting only of 0s and 1s, chosen uniformly at random out of all such strings. Let \( E \) be the expected number of nonempty contiguous substrings of \( S \) which are palindromes. Compute the value of \( \lfloor E \rfloor \).
|
113
| 0.625 |
Find the maximum value of \( S = \sin^2 \theta_1 + \sin^2 \theta_2 + \cdots + \sin^2 \theta_n \), where \( 0 \leq \theta_i \leq \pi \) and \( \theta_1 + \theta_2 + \cdots + \theta_n = \pi \).
|
\frac{9}{4}
| 0.625 |
Charlie folds a \(\frac{17}{2}\)-inch by 11-inch piece of paper in half twice, each time along a straight line parallel to one of the paper's edges. What is the smallest possible perimeter of the piece after two such folds?
|
\frac{39}{2}
| 0.375 |
\( AB \) is a diameter; \( BC \) and \( AC \) are chords, with \( \angle B C = 60^\circ \). \( D \) is the point where the extension of the diameter \( AB \) meets the tangent \( CD \). Find the ratio of the areas of triangles \( DCB \) and \( DCA \).
|
\frac{1}{3}
| 0.75 |
Each hotel room can accommodate no more than 3 people. The hotel manager knows that a group of 100 football fans, who support three different teams, will soon arrive. A room can only house either men or women; and fans of different teams cannot be housed together. How many rooms need to be booked to accommodate all the fans?
|
37
| 0.125 |
Calculate \(\int_{0}^{1} e^{-x^{2}} \, dx\) to an accuracy of 0.001.
|
0.747
| 0.5 |
The three-digit number \(\overline{abc}\) consists of three non-zero digits. The sum of the other five three-digit numbers formed by rearranging \(a, b, c\) is 2017. Find \(\overline{abc}\).
|
425
| 0.625 |
A hotel has 5 distinct rooms, each with single beds for up to 2 people. The hotel has no other guests, and 5 friends want to stay there for the night. In how many ways can the 5 friends choose their rooms?
|
2220
| 0.25 |
Neznayka, Doctor Pilyulkin, Knopochka, Vintik, and Znayka participated in a math contest. Exactly four out of them solved each problem. Znayka solved more problems than each of the others - 10 problems, while Neznayka solved fewer problems than each of the others - 6 problems. How many problems in total were there in the math contest?
|
10
| 0.625 |
Let \( S=\{1,2, \cdots, 10\} \). \( A_{1}, A_{2}, \cdots, A_{k} \) are subsets of \( S \) that satisfy the following conditions:
1. \( \left|A_{i}\right|=5 \) for \( i=1,2, \cdots, k \).
2. \( \left|A_{i} \cap A_{j}\right| \leqslant 2 \) for \( 1 \leqslant i < j \leqslant k \).
Find the maximum value of \( k \).
|
6
| 0.625 |
As shown in the figure, \( M \) and \( N \) are points on the diagonals \( AC \) and \( CE \) of the regular hexagon \( ABCDEF \), respectively. The points divide the diagonals such that \( AM:AC = CN:CE = r \). If the points \( B \), \( M \), and \( N \) are collinear, find the value of \( r \).
|
\frac{\sqrt{3}}{3}
| 0.875 |
Find the minimum value of the algebraic expression $|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x|$ where $x$ is a real number.
|
2\sqrt{2} - 1
| 0.5 |
The arithmetic mean of three two-digit natural numbers \( x, y, z \) is 60. What is the maximum value that the expression \( \frac{x + y}{z} \) can take?
|
17
| 0.875 |
Given \( a_i \in \mathbf{R} \) (for \( i = 1, 2, \cdots, 10 \)) and \( \sum_{i=1}^{10} \frac{a_i^2}{a_i^2 + 1} = 1 \), find the range of values for \( \sum_{i=1}^{10} \frac{a_i}{a_i^2 + 1} \).
|
[-3, 3]
| 0.875 |
There are eight cards, each with a natural number from 1 to 8. If three cards are drawn so that the sum of their numbers is 9, how many different ways can this be done?
|
3
| 0.75 |
How many common terms (terms with the same value) are there between the arithmetic sequences \(2, 5, 8, \cdots, 2015\) and \(4, 9, 14, \cdots, 2014\)?
|
134
| 0.75 |
The sequence $\left\{x_{n}\right\}: 1,3,3,3,5,5,5,5,5, \cdots$ is formed by arranging all positive odd numbers in increasing order, and each odd number $k$ appears consecutively $k$ times, where $k=1,3,5, \cdots$. If the general term formula of this sequence is given by $x_{n}=a[\sqrt{b n+c}]+d$, find the value of $a+b+c+d$.
|
3
| 0.5 |
Determine the value of the following expression with as little computation as possible:
$$
\frac{49 \cdot 91^{3}+338 \cdot 343^{2}}{66^{3}-176 \cdot 121}: \frac{39^{3} \cdot 7^{5}}{1331000}
$$
|
\frac{5}{13}
| 0.75 |
In a football championship with 16 teams, each team played with every other team exactly once. A win was awarded 3 points, a draw 1 point, and a loss 0 points. A team is considered successful if it scored at least half of the maximum possible points. What is the maximum number of successful teams that could have participated in the tournament?
|
15
| 0.25 |
Let \( P(x) \) be a polynomial of degree \( n \) such that for \( k = 1, 2, 4, \ldots, 2^{n} \), we have \( P(k) = \frac{1}{k} \). Find \( P(0) \).
|
2 - \frac{1}{2^n}
| 0.75 |
Let \( x \) and \( y \) be complex numbers such that \( x + y = \sqrt{20} \) and \( x^2 + y^2 = 15 \). Compute \( |x - y| \).
|
\sqrt{10}
| 0.875 |
Two of the lightest kittens weigh a total of 80 grams. Four of the heaviest kittens weigh 200 grams. The total weight of all the kittens is \(X\) grams. What is the smallest value of \(X\) for which it is not possible to uniquely determine the number of kittens born?
|
480
| 0.125 |
You need to find out a five-digit phone number by asking questions that can be answered with "yes" or "no". What is the minimum number of questions required to guarantee finding the number (assuming all answers are correct)?
|
17
| 0.875 |
Given that \( \boldsymbol{m} \) is a non-zero vector, \( \boldsymbol{n} \) is a unit vector, \( \boldsymbol{m} \neq \boldsymbol{n} \), and the angle between \( \boldsymbol{m} \) and \( \boldsymbol{m} - \boldsymbol{n} \) is \( 60^{\circ} \), with \( |\boldsymbol{m}| \in (0, a] \), find the minimum value of \( a \).
|
\frac{2\sqrt{3}}{3}
| 0.75 |
As shown in the diagram, there are two cubes of different sizes. The edge length of the larger cube is 6 times the edge length of the smaller cube. All 6 faces of the larger cube are painted red, and all 6 faces of the smaller cube are painted yellow. Then, the two cubes are glued together. What is the ratio of the red surface area to the yellow surface area on the resulting combined figure?
|
43
| 0.75 |
Optimus Prime departs in robot form from point $A$ to point $B$ and arrives at $B$ on time. If he transforms into a car from the beginning, his speed increases by $\frac{1}{4}$, allowing him to arrive at $B$ 1 hour earlier. If he travels 150 km in robot form first, then transforms into a car, and his speed increases by $\frac{1}{5}$, he can arrive 40 minutes earlier. How far apart are points $A$ and $B$ in kilometers?
|
750
| 0.625 |
Exactly at noon, a truck left the village heading towards the city, and at the same time, a car left the city heading towards the village. If the truck had left 45 minutes earlier, they would have met 18 kilometers closer to the city. If the car had left 20 minutes earlier, they would have met $k$ kilometers closer to the village. Find $k$.
|
8
| 0.25 |
A train has 18 identical cars. In some of the cars, half of the seats are free, in others, one third of the seats are free, and in the remaining cars, all the seats are occupied. In the entire train, exactly one ninth of all seats are free. How many cars have all seats occupied?
|
13
| 0.75 |
Find the sum of the coefficients of the even-degree terms in the polynomial obtained from the expression \( f(x) = \left( x^3 - x + 1 \right)^{100} \) after expanding and combining like terms.
|
1
| 0.875 |
In a regular 1000-gon, all diagonals are drawn. What is the maximum number of diagonals that can be selected such that among any three of the chosen diagonals, at least two have the same length?
|
2000
| 0.375 |
If the four-digit number \(5ab4\) is a perfect square, then \(a + b =\)
|
9
| 0.75 |
The entire area is divided into squares, denoted by two integer indices \( M \) and \( N \) such that, for example, a point with coordinates \( x=12.25, y=20.9 \) is in the square numbered \([12; 20]\), and a point with coordinates \( x=-12.34, y=0.1239 \) is in the square numbered \([-13; 0]\), and so on. A mysterious object moves in the plane \( Oxy \) along the trajectory \( y=\left(\left(\left(x^{5}-2013\right)^{5}-2013\right)^{5}-2013\right)^{5} \), and a radar beam on the ground is directed along the line \( y=x+2013 \). Specify the numbers of all squares in which the radar will detect the appearance of the mysterious object.
|
[4; 2017]
| 0.25 |
Observe the following operation:
If \(\overline{\mathrm{abc}}\) is a three-digit number, then since \(\overline{\mathrm{abc}} = 100a + 10b + c = 99a + 9b + (a + b + c)\),
Therefore, if \(a + b + c\) is divisible by 9, \(\overline{\mathrm{abc}}\) is divisible by 9.
This conclusion can be generalized to any number of digits.
Using this conclusion, solve the following questions:
1. \(N\) is a 2011-digit number, and each digit is 2. Find the remainder when \(N\) is divided by 9.
2. \(N\) is an \(n\)-digit number, and each digit is 7. \(n\) leaves a remainder of 3 when divided by 9. Find the remainder when \(N\) is divided by 9.
|
3
| 0.625 |
The side of a triangle is $\sqrt{2}$, and the angles adjacent to it are $75^{\circ}$ and $60^{\circ}$. Find the segment connecting the bases of the altitudes dropped from the vertices of these angles.
|
1
| 0.5 |
Let \(a, b, c\) be integers. Define \(f(x) = ax^2 + bx + c\). Suppose there exist pairwise distinct integers \(u, v, w\) such that \(f(u) = 0\), \(f(v) = 0\), and \(f(w) = 2\). Find the maximum possible value of the discriminant \(b^2 - 4ac\) of \(f\).
|
16
| 0.5 |
Person A and Person B start from point A to point B at the same time. If both travel at a constant speed, Person A takes 4 hours to complete the journey, and Person B takes 6 hours. When the remaining distance for Person B is 4 times the remaining distance for Person A, how many hours have they been traveling?
|
3.6
| 0.875 |
Dima's mother told him he needed to eat 13 spoons of porridge. Dima told his friend that he ate 26 spoons of porridge. Each subsequent child, when talking about Dima's feat, increased the number of spoons by 2 or 3 times. Eventually, one of the children told Dima's mother that Dima ate 33,696 spoons of porridge. How many times in total, including Dima, did the children talk about Dima's feat?
|
9
| 0.75 |
The judging panel for the Teen Singer Grand Prix consists of several judges. Each judge can award a maximum score of 10 points to a singer. After the first singer's performance, the following scoring situation was observed: The average score given by all judges was 9.64 points; if the highest score was removed, the average score given by the remaining judges was 9.60 points; if the lowest score was removed, the average score given by the remaining judges was 9.68 points. Therefore, the minimum possible lowest score given by the judges is $\qquad$ points, and the total number of judges in this Grand Prix is $\qquad$.
|
10
| 0.125 |
What is the smallest square number that, when divided by a cube number, results in a fraction in its simplest form where the numerator is a cube number (other than 1) and the denominator is a square number (other than 1)?
|
64
| 0.125 |
Determine the smallest positive value of \( a \) for which the equation
$$
\frac{\frac{x-a}{2}+\frac{x-2a}{3}}{\frac{x+4a}{5}-\frac{x+3a}{4}}=\frac{\frac{x-3a}{4}+\frac{x-4a}{5}}{\frac{x+2a}{3}-\frac{x+a}{2}}
$$
has an integer root.
|
\frac{419}{421}
| 0.125 |
An empty swimming pool was filled with water by two taps \(A\) and \(B\), both with constant flow rates. For 4 hours, both taps were open and filled 50% of the pool. Then, tap \(B\) was closed and for the next 2 hours, tap \(A\) filled 15% of the pool's volume. After this period, tap \(A\) was closed and tap \(B\) was opened. How long did tap \(B\) need to remain open to fill the pool by itself?
|
7
| 0.875 |
I place \( n \) pairs of socks (thus \( 2n \) socks) in a line such that the left sock is to the right of the right sock for each pair. How many different ways can I place my socks in this manner?
|
\frac{(2n)!}{2^n}
| 0.75 |
Find the general solution of the equation \( y'' - 5y' + 6y = 0 \) and identify the particular solution that satisfies the initial conditions \( y = 1 \) and \( y' = 2 \) at \( x = 0 \).
|
e^{2x}
| 0.875 |
Find the sum of all four-digit numbers in which the digits $0, 4, 5, 9$ are absent.
|
6479352
| 0.875 |
\(\frac{\cos 70^{\circ} \cos 10^{\circ}+\cos 80^{\circ} \cos 20^{\circ}}{\cos 69^{\circ} \cos 9^{\circ}+\cos 81^{\circ} \cos 21^{\circ}}\).
|
1
| 0.5 |
Compute the lengths of the arcs of the curves given by the equations in the rectangular coordinate system.
$$
y=\arccos \sqrt{x}-\sqrt{x-x^{2}}+4, \quad 0 \leq x \leq \frac{1}{2}
$$
|
\sqrt{2}
| 0.875 |
Given the set \( I = \{1, 2, \ldots, 2020\} \):
$$
\begin{array}{l}
W = \{w(a, b) = (a + b) + ab \mid a, b \in I\} \cap I, \\
Y = \{y(a, b) = (a + b) \cdot ab \mid a, b \in I\} \cap I, \\
X = W \cap Y,
\end{array}
$$
where \( W \) is the 'Wu' set, \( Y \) is the 'Yue' set, and \( X \) is the 'Xizi' set. Find the sum of the maximum and minimum numbers in the set \( X \).
|
2020
| 0.125 |
If \( 0 < x < \frac{\pi}{2} \) and \( \sin x - \cos x = \frac{\pi}{4} \) and \( \tan x + \frac{1}{\tan x} = \frac{a}{b - \pi^c} \), where \( a, b \), and \( c \) are positive integers, find the value of \( a + b + c \).
|
50
| 0.875 |
Calculate the result of the expression:
\[
\frac{\frac{1}{1}-\frac{1}{3}}{\frac{1}{1} \times \frac{1}{2} \times \frac{1}{3}} + \frac{\frac{1}{2}-\frac{1}{4}}{\frac{1}{2} \times \frac{1}{3} \times \frac{1}{4}} + \frac{\frac{1}{3}-\frac{1}{5}}{\frac{1}{3} \times \frac{1}{4} \times \frac{1}{5}} + \cdots + \frac{\frac{1}{9}-\frac{1}{11}}{\frac{1}{9} \times \frac{1}{10} \times \frac{1}{11}}
\]
|
108
| 0.75 |
Let \( S \) be a set of 2017 distinct points in the plane. Let \( R \) be the radius of the smallest circle containing all points in \( S \) on either the interior or boundary. Also, let \( D \) be the longest distance between two of the points in \( S \). Let \( a, b \) be real numbers such that \( a \leq \frac{D}{R} \leq b \) for all possible sets \( S \), where \( a \) is as large as possible and \( b \) is as small as possible. Find the pair \( (a, b) \).
|
(\sqrt{3}, 2)
| 0.125 |
Assume integers \( u \) and \( v \) satisfy \( 0 < v < u \), and let \( A \) be \((u, v)\). Points are defined as follows: \( B \) is the reflection of \( A \) over the line \( y = x \), \( C \) is the reflection of \( B \) over the \( y \)-axis, \( D \) is the reflection of \( C \) over the \( x \)-axis, and \( E \) is the reflection of \( D \) over the \( y \)-axis. The area of pentagon \( ABCDE \) is 451. Find \( u+v \).
|
21
| 0.5 |
How many ordered pairs of integers \((x, y)\) satisfy \(x^{2} \leq y \leq x+6\)?
|
26
| 0.875 |
It is known that for real numbers \(a\) and \(b\), the following equalities hold:
\[
a^3 - 3ab^2 = 11, \quad b^3 - 3a^2b = 2
\]
What values can the expression \(a^2 + b^2\) take?
|
5
| 0.875 |
Find the minimum value of \(\sum_{k=1}^{100} |n - k|\), where \(n\) ranges over all positive integers.
|
2500
| 0.25 |
First-grader Petya was arranging his available chips into the outline of an equilateral triangle such that each side, including the vertices, contains the same number of chips. Then, using the same chips, he managed to arrange them into the outline of a square in the same manner. How many chips does Petya have if the side of the square contains 2 fewer chips than the side of the triangle?
|
24
| 0.875 |
Given the equation
\[ x^{2} + p x + q = 0 \]
with roots \( x_{1} \) and \( x_{2} \), determine the coefficients \( r \) and \( s \) as functions of \( p \) and \( q \) such that the equation
\[ y^{2} + r y + s = 0 \]
has roots \( y_{1} \) and \( y_{2} \) which are related to the roots \( x_{1} \) and \( x_{2} \) by
\[ y_{1} = \frac{x_{1}}{x_{1}-1} \quad \text{and} \quad y_{2} = \frac{x_{2}}{x_{2}-1} \]
What condition must hold between \( p \) and \( q \) for the equations (1) and (2) to be identical?
|
p + q = 0
| 0.75 |
The lateral edges of a triangular pyramid are pairwise perpendicular and have lengths $a, b,$ and $c$. Find the volume of the pyramid.
|
\frac{a b c}{6}
| 0.375 |
There are $N$ ($N \geqslant 9$) distinct non-negative real numbers less than 1 written on the blackboard. It is known that for any eight numbers on the blackboard, there exists another number on the blackboard such that the sum of these nine numbers is an integer. Find all possible values of $N$.
|
N=9
| 0.5 |
Carlson and Baby have several jars of jam, each weighing an integer number of pounds.
The total weight of all Carlson's jars of jam is 13 times the total weight of all Baby's jars. Carlson gave Baby the jar with the smallest weight (among those he had), after which the total weight of his jars turned out to be 8 times the total weight of Baby's jars.
What is the maximum possible number of jars Carlson could have initially had?
|
23
| 0.75 |
Sarah is deciding whether to visit Russia or Washington, DC for the holidays. She makes her decision by rolling a regular 6-sided die. If she gets a 1 or 2, she goes to DC. If she rolls a 3, 4, or 5, she goes to Russia. If she rolls a 6, she rolls again. What is the probability that she goes to DC?
|
\frac{2}{5}
| 0.875 |
A surveillance service will be installed in a park in the form of a network of stations. The stations must be connected by telephone lines, so that any one of the stations can communicate with all the others, either through a direct connection or through at most one other station.
Each station can be directly connected by a cable to at most three other stations. What is the largest number of stations that can be connected in this way?
|
10
| 0.625 |
In the plane Cartesian coordinate system \( xOy \), the set \( S=\left\{(x, y) \mid (|x|+|y|-1)\left(x^{2}+y^{2}-1\right) \leqslant 0\right\} \) represents a plane region. Find the area of this region.
|
\pi - 2
| 0.875 |
Inside an equilateral triangle, a point \( M \) is taken, which is at distances \( b, c, \) and \( d \) from its sides. Find the height of the triangle.
|
b + c + d
| 0.875 |
The parallelogram \(ABCD\) is such that \(\angle B < 90^\circ\) and \(AB < BC\). Points \(E\) and \(F\) are chosen on the circumcircle \(\omega\) of triangle \(ABC\) such that the tangents to \(\omega\) at these points pass through \(D\). It turns out that \(\angle EDA = \angle FDC\). Find the angle \(\angle ABC\).
(A. Yakubov)
|
60^\circ
| 0.75 |
Out of the four inequalities \(2x > 70\), \(x < 100\), \(4x > 25\), and \(x > 5\), two are true and two are false. Find the value of \(x\), given that it is an integer.
|
x = 6
| 0.875 |
What is the largest integer \( k \) whose square \( k^2 \) is a factor of \( 10! \)?
|
720
| 0.625 |
A cube with a side length of 20 is divided into 8000 unit cubes, and a number is written in each small cube. It is known that in every column of 20 cubes parallel to the edge of the cube, the sum of the numbers is 1 (columns in all three directions are considered). In a certain unit cube, the number 10 is written. Through this cube, there are three $1 \times 20 \times 20$ layers parallel to the faces of the large cube. Find the sum of all numbers outside these layers.
|
333
| 0.625 |
Calculate the limit of the numerical sequence:
\[
\lim _{n \rightarrow \infty} \frac{\sqrt{n-1}-\sqrt{n^{2}+1}}{\sqrt[3]{3 n^{3}+3}+\sqrt[3]{n^{5}+1}}
\]
|
0
| 0.75 |
Given the convex quadrilateral \(ABCD\) with an area of 1 unit. Reflect vertex \(A\) over \(B\), \(B\) over \(C\), \(C\) over \(D\), and \(D\) over \(A\). What is the area of the quadrilateral formed by the reflected points?
|
5
| 0.75 |
The vertices of a rhombus are located on the sides of a parallelogram, and the sides of the rhombus are parallel to the diagonals of the parallelogram. Find the ratio of the areas of the rhombus and the parallelogram if the ratio of the diagonals of the parallelogram is \( k \).
|
\frac{2k}{(1+k)^2}
| 0.125 |
A two-digit number ' $a b$ ' is multiplied by its reverse ' $b a$ '. The ones (units) and tens digits of the four-digit answer are both 0.
What is the value of the smallest such two-digit number ' $a b$ '?
|
25
| 0.625 |
Calculate $\lim _{n \rightarrow \infty}\left(\sqrt[3^{2}]{3} \cdot \sqrt[3^{3}]{3^{2}} \cdot \sqrt[3^{4}]{3^{3}} \ldots \sqrt[3^{n}]{3^{n-1}}\right)$.
|
\sqrt[4]{3}
| 0.75 |
Given that \( |\overrightarrow{AB}| = 10 \), if a point \( P \) on the plane satisfies \( |\overrightarrow{AP} - t \overrightarrow{AB}| \geq 3 \) for any \( t \in \mathbf{R} \), find the minimum value of \( \overrightarrow{PA} \cdot \overrightarrow{PB} \) and the corresponding value of \( |\overrightarrow{PA} + \overrightarrow{PB}| \).
|
6
| 0.25 |
Find the least positive integral value of \( n \) for which the equation
\[ x_{1}^{3} + x_{2}^{3} + \cdots + x_{n}^{3} = 2002^{2002} \]
has integer solutions \(\left(x_{1}, x_{2}, x_{3}, \cdots, x_{n}\right)\).
|
4
| 0.875 |
Suppose \(x\) and \(y\) are integers such that
\[
(x-2004)(x-2006)=2^{y}.
\]
Find the largest possible value of \(x+y\).
|
2011
| 0.875 |
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