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stringlengths 18
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|---|---|---|
Find the relationship between the coefficients \(a, b, c\) such that the system of equations
$$
\left\{\begin{array}{l}
a x^{2}+b x+c=0 \\
b x^{2}+c x+a=0 \\
c x^{2}+a x+b=0
\end{array}\right.
$$
has real solutions.
|
a + b + c = 0
| 0.875 |
Calculate the volumes of solids formed by rotating the region bounded by the function graphs about the \( O y \) (y-axis).
$$
y = x^{2} + 1, \quad y = x, \quad x = 0, \quad x = 1
$$
|
\frac{5\pi}{6}
| 0.875 |
How many numbers from 1 to 1000 (inclusive) cannot be represented as the difference of two squares of integers?
|
250
| 0.875 |
Let \( a, b, c \) be three pairwise distinct positive integers. If \( \{ a+b, b+c, c+a \} = \{ n^2, (n+1)^2, (n+2)^2 \} \), then what is the minimum value of \( a^2 + b^2 + c^2 \)?
|
1297
| 0.5 |
Find the limits:
1) $\lim _{x \rightarrow 0} \frac{x}{\sin x}$
2) $\lim _{x \rightarrow 0} \frac{\sin 4 x}{\sin 5 x}$
3) $\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}$
|
\frac{1}{2}
| 0.5 |
It takes 60 grams of paint to paint a cube on all sides. How much paint is needed to paint a "snake" composed of 2016 such cubes? The beginning and end of the snake are shown in the diagram, and the rest of the cubes are indicated by ellipses.
|
80660
| 0.75 |
Two players, Blake and Ruby, play the following game on an infinite grid of unit squares, all initially colored white. The players take turns starting with Blake. On Blake's turn, Blake selects one white unit square and colors it blue. On Ruby's turn, Ruby selects two white unit squares and colors them red. The players alternate until Blake decides to end the game. At this point, Blake gets a score, given by the number of unit squares in the largest (in terms of area) simple polygon containing only blue unit squares. What is the largest score Blake can guarantee?
|
4
| 0.25 |
Let's call a non-empty (finite or infinite) set $A$ consisting of real numbers complete if for any real numbers $a$ and $b$ (not necessarily distinct and not necessarily in $A$) such that $a+b$ lies in $A$, the number $ab$ also lies in $A$. Find all complete sets of real numbers.
|
\mathbb{R}
| 0.125 |
How many 12-element subsets \( T = \{a_1, a_2, \cdots, a_{12}\} \) from the set \( S = \{1, 2, 3, \cdots, 2009\} \) exist such that the absolute difference between any two elements is not 1?
|
\binom{1998}{12}
| 0.125 |
Calculate the definite integral:
$$
\int_{-\pi / 2}^{0} 2^{8} \cdot \cos ^{8} x \, dx
$$
|
35 \pi
| 0.875 |
Given that \( a \) is a constant and \( x \in \mathbf{R} \), \( f(x)=\frac{f(x-a)-1}{f(x-a)+1} \). The period of \( f(x) \) is \_\_\_\_.
|
4a
| 0.875 |
A positive integer with \( n+3 \) digits, \( 144\cdots430 \) (with \( n \) number of 4s), is a multiple of 2015. What is the smallest value of \( n \)?
|
14
| 0.375 |
If a square is divided into acute-angled triangles, what is the minimum number of parts that can be created?
|
8
| 0.375 |
For any positive integer \( n \), if \( f(n) \) is defined as:
\[ f(n) = \begin{cases}
\log_8 n & \text{if } \log_8 n \text{ is a rational number;} \\
0 & \text{if } \log_8 n \text{ is not a rational number.}
\end{cases} \]
Then evaluate the sum:
\[ \sum_{n=1}^{197} f(n) \]
Options:
A. \( \log_8 2047 \)
B. 6
C. \( \frac{55}{3} \)
D. \( \frac{58}{3} \)
|
\frac{55}{3}
| 0.875 |
A cleaner was mopping the stairs in a skyscraper. To make the work more enjoyable, she counted the cleaned steps. When she had cleaned exactly half of the steps, she took a break. After a while, she started working again and wanted to continue counting the steps. However, when trying to recall the number of steps she had already cleaned, she mistakenly read the correct three-digit number backward, creating a smaller number, and counted from there. After mopping all the stairs, she ended at the number 746. How many steps could she have actually cleaned?
|
1142
| 0.125 |
Given an odd prime number \( p \). If there exists a positive integer \( k \) such that \( \sqrt{k^{2} - p k} \) is also a positive integer, then \( k = \) ______.
|
\frac{(p+1)^2}{4}
| 0.625 |
For what value of \(a\) does the inequality \(\log \frac{1}{\div}\left(\sqrt{x^{2}+a x+5}+1\right) \cdot \log _{5}\left(x^{2}+a x+6\right)+\log _{a} 3 \geqslant 0\) have exactly one solution?
|
a = 2
| 0.5 |
Suppose that \((a_{1}, b_{1}), (a_{2}, b_{2}), \ldots, (a_{100}, b_{100})\) are distinct ordered pairs of nonnegative integers. Let \(N\) denote the number of pairs of integers \((i, j)\) satisfying \(1 \leq i < j \leq 100\) and \(|a_{i} b_{j} - a_{j} b_{i}| = 1\). Determine the largest possible value of \(N\) over all possible choices of the 100 ordered pairs.
|
197
| 0.625 |
What are the values of \(p\) such that \(p\), \(p+2\), and \(p+4\) are all prime numbers?
|
3
| 0.875 |
Does there exist an angle \(\alpha \in (0, \pi / 2)\) such that \(\sin \alpha\), \(\cos \alpha\), \(\tan \alpha\), and \(\cot \alpha\), taken in some order, are consecutive terms of an arithmetic progression?
|
\text{No}
| 0.5 |
In a unit cube \( ABCD-A_{1}B_{1}C_{1}D_{1} \), points \( E, F, G \) are the midpoints of edges \( AA_{1}, C_{1}D_{1}, D_{1}A_{1} \) respectively. Find the distance from point \( B_{1} \) to the plane containing points \( E, F, G \).
|
\frac{\sqrt{3}}{2}
| 0.625 |
Shuai Shuai memorized more than one hundred words in seven days. The number of words memorized in the first three days is $20\%$ less than the number of words memorized in the last four days, and the number of words memorized in the first four days is $20\%$ more than the number of words memorized in the last three days. How many words did Shuai Shuai memorize in total over the seven days?
|
198
| 0.75 |
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 |
Find the continuous functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for all \( x \) and \( y \),
\[ f\left(\frac{x+y}{2}\right) = \frac{f(x) + f(y)}{2} \]
|
f(x) = ax + b
| 0.5 |
Consider the two hands of an analog clock, each of which moves with constant angular velocity. Certain positions of these hands are possible (e.g. the hour hand halfway between the 5 and 6 and the minute hand exactly at the 6), while others are impossible (e.g. the hour hand exactly at the 5 and the minute hand exactly at the 6). How many different positions are there that would remain possible if the hour and minute hands were switched?
|
143
| 0.125 |
Inside the square \(ABCD\), point \(M\) is chosen such that \(MA = 1\), \(MB = 2\), and \(MC = 3\). Find \(MD\).
|
\sqrt{6}
| 0.375 |
Under what condition on \((x, y) \in \mathbb{N}^{2}\) is the integer \((x+y)^{2} + 3x + y + 1\) a perfect square?
|
x = y
| 0.5 |
We move on a grid by taking only one-unit steps to the right or upward. What is the number of ways to go from point \((0,0)\) to point \((m, n)\)?
|
\binom{m+n}{m}
| 0.875 |
Each chocolate costs 1 dollar, each licorice stick costs 50 cents, and each lolly costs 40 cents. How many different combinations of these three items cost a total of 10 dollars?
|
36
| 0.25 |
Given two linear functions \( f(x) \) and \( g(x) \) such that the graphs \( y=f(x) \) and \( y=g(x) \) are parallel lines not parallel to the coordinate axes. Find the minimum value of the function \( (g(x))^{2} + f(x) \) if the minimum value of the function \( (f(x))^{2} + g(x) \) is 4.
|
- \frac{9}{2}
| 0.5 |
Sergey Stanislavovich is 36 years, 36 months, 36 weeks, 36 days, and 36 hours old.
How many full years old is Sergey Stanislavovich?
|
39
| 0.625 |
Is a triangle necessarily isosceles if the center of its inscribed circle is equidistant from the midpoints of two of its sides?
|
\text{No}
| 0.375 |
Does the equation \( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 = 0 \) have any negative roots?
Find the common fraction with a denominator of 21 that lies between the fractions \(\frac{5}{14}\) and \(\frac{5}{12}\).
|
\frac{8}{21}
| 0.75 |
Kevin has four red marbles and eight blue marbles. He arranges these twelve marbles randomly in a ring. Determine the probability that no two red marbles are adjacent.
|
\frac{7}{33}
| 0.125 |
Find all real solutions to the equation with 4 unknowns: \( x^{2} + y^{2} + z^{2} + t^{2} = x(y + z + t) \).
|
(0, 0, 0, 0)
| 0.875 |
There are 2013 cards with the digit 1 and 2013 cards with the digit 2. Vasya arranges these cards to form a 4026-digit number. On each move, Petya can swap any two cards and pay Vasya 1 ruble. The process ends when Petya forms a number divisible by 11. What is the maximum amount of money Vasya can earn if Petya aims to pay as little as possible?
|
5
| 0.375 |
If the system of equations involving \( x \) and \( y \)
\[
\left\{
\begin{array}{l}
\sin x = m \sin^3 y, \\
\cos x = m \cos^3 y
\end{array}
\right.
\]
has real solutions, then what is the range of positive values for \( m \)?
|
[1, 2]
| 0.75 |
2.305. Verify by transforming the left part that:
a) $\sqrt[3]{7+5 \sqrt{2}}-\sqrt[3]{5 \sqrt{2}-7}=2$
b) $\sqrt{3+\sqrt{3}+\sqrt[3]{10+6 \sqrt{3}}}=\sqrt{3}+1$.
|
\sqrt{3} + 1
| 0.125 |
Let \( a_0, a_1, \ldots \) be a sequence such that \( a_0 = 3 \), \( a_1 = 2 \), and \( a_{n+2} = a_{n+1} + a_n \) for all \( n \geq 0 \). Find
\[
\sum_{n=0}^{8} \frac{a_n}{a_{n+1} a_{n+2}}.
\]
|
\frac{105}{212}
| 0.625 |
Given that $\alpha^{2005}+\beta^{2005}$ can be expressed as a two-variable polynomial in terms of $\alpha+\beta$ and $\alpha \beta$, find the sum of the coefficients of this polynomial.
|
1
| 0.375 |
Misha painted all the integers in several colors such that any two numbers whose difference is a prime number are painted in different colors. What is the minimum number of colors that could have been used by Misha? Justify your answer.
|
4
| 0.625 |
A random variable $X$ is distributed according to the normal law. The expected value $a=0$ and the standard deviation of this variable is $\sigma=0.5$. Find the probability that the deviation of the random variable $X$ in absolute value will be less than one.
|
0.9544
| 0.875 |
Let \( S_{n} = 1 + 2 + \cdots + n \). How many of \( S_{1}, S_{2}, \cdots, S_{2015} \) are multiples of 2015?
|
8
| 0.375 |
In the sequence $\left\{a_{n}\right\}$, if $a_{n}^{2}-a_{n-1}^{2}=p$ (where $n \geq 2, n \in \mathbf{N^{*}}$, and $p$ is a constant), then $\left\{a_{n}\right\}$ is called an "equal variance sequence." Below are the judgments for the "equal variance sequence":
1. The sequence $\left\{(-1)^{n}\right\}$ is an equal variance sequence;
2. If $\left\{a_{n}\right\}$ is an equal variance sequence, then $ \left\{a_{n}^{2}\right\}$ is an arithmetic sequence;
3. If $\left\{a_{n}\right\}$ is an equal variance sequence, then $\left\{a_{k n}\right\}\left(k \in \mathbf{N}^{*}, k\right.$ being a constant) is also an equal variance sequence;
4. If $\left\{a_{n}\right\} $ is both an equal variance sequence and an arithmetic sequence, then this sequence is a constant sequence.
Among them, the correct proposition numbers are $\qquad$ (Fill in all the correct proposition numbers on the line).
|
1234
| 0.875 |
Calculate the number of trailing zeroes in 2019!.
|
502
| 0.875 |
What is the smallest number of distinct integers needed so that among them one can select both a geometric progression and an arithmetic progression of length 5?
|
6
| 0.125 |
Find the maximum real number $\lambda$ such that for the real coefficient polynomial $f(x)=x^{3}+a x^{2}+b x+c$ having all non-negative real roots, it holds that $f(x) \geqslant \lambda(x-a)^{3}$ for all $x \geqslant 0$. Additionally, determine when equality holds in the given expression.
|
-\frac{1}{27}
| 0.375 |
Given 100 numbers. Each number is increased by 2. The sum of the squares of the numbers remains unchanged. Each resulting number is then increased by 2 again. How has the sum of the squares changed now?
|
800
| 0.75 |
In how many ways can you select two letters from the word "УЧЕБНИК" such that one of the letters is a consonant and the other is a vowel?
|
12
| 0.75 |
Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that
\[ \forall x, y \in \mathbb{R}, f(x^3) - f(y^3) = (x^2 + xy + y^2)(f(x) - f(y)) \]
|
f(x) = ax + b
| 0.25 |
In Anchuria, a checkers championship takes place over several rounds. The days and cities for hosting the rounds are determined by a draw. According to the championship rules, no two rounds can be held in the same city and no two rounds can be held on the same day. A lottery is held among the fans: the grand prize goes to the person who correctly guesses the cities and days for all the rounds before the championship starts. If no one guesses correctly, the grand prize is awarded to the championship organizing committee. There are eight cities in Anchuria, and the championship is scheduled to take place over eight days. How many rounds should there be in the championship to maximize the probability that the organizing committee wins the grand prize?
|
6
| 0.125 |
Find the smallest prime number that can be expressed as the sum of five different prime numbers.
|
43
| 0.75 |
The distance from platform $A$ to platform $B$ was covered by an electric train in $X$ minutes ($0<X<60$). Find $X$ if it is known that at both the moment of departure from $A$ and the moment of arrival at $B$, the angle between the hour and minute hands of the clock was $X$ degrees.
|
48
| 0.875 |
In a store, there are four types of nuts: hazelnuts, almonds, cashews, and pistachios. Stepan wants to buy 1 kilogram of nuts of one type and 1 kilogram of nuts of another type. He has calculated the cost of such a purchase depending on which two types of nuts he chooses. Five of Stepan's six possible purchases would cost 1900, 2070, 2110, 2330, and 2500 rubles. How many rubles is the cost of the sixth possible purchase?
|
2290
| 0.375 |
A sphere is inscribed in a dihedral angle.
Point \( O \) is located in the section \( A A^{\prime} C^{\prime} C \) of a rectangular parallelepiped \( A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime} \) with dimensions \( 2 \times 6 \times 9 \) such that \( \angle O A B + \angle O A D + \angle O A A^{\prime} = 180^{\circ} \). The sphere with center at point \( O \) touches the planes \( A^{\prime} B^{\prime} C^{\prime} \) and \( A A^{\prime} B \) and does not intersect the plane \( A A^{\prime} D \). Find the distance from point \( O \) to this plane.
|
3
| 0.875 |
A three-digit number \(abc\) satisfies \(abc = a + b^2 + c^3\). How many such three-digit numbers \(\overline{abc}\) satisfy this condition?
|
4
| 0.5 |
In quadrilateral \(ABCD\), \(\angle ABD = 70^\circ\), \(\angle CAD = 20^\circ\), \(\angle BAC = 48^\circ\), \(\angle CBD = 40^\circ\). Find \(\angle ACD\).
|
22^\circ
| 0.125 |
Express \( x \) using the real parameter \( a \) if
$$
\sqrt{\log _{a} (a x) + \log _{x} (a x)} + \sqrt{\log _{a} \left(\frac{x}{a}\right) + \log _{x} \left(\frac{a}{x}\right)} = 2.
$$
|
x = a
| 0.875 |
At lunch, Abby, Bart, Carl, Dana, and Evan share a pizza divided radially into 16 slices. Each one takes one slice of pizza uniformly at random, leaving 11 slices. The remaining slices of pizza form "sectors" broken up by the taken slices; for example, if they take five consecutive slices then there is one sector, but if none of them take adjacent slices then there will be five sectors. What is the expected number of sectors formed?
|
\frac{11}{3}
| 0.875 |
If \( n \) is a positive integer such that \( n^{3} + 2n^{2} + 9n + 8 \) is the cube of an integer, find \( n \).
|
7
| 0.5 |
Two people travel toward each other from points \(A\) and \(B\) with speeds \(v_{1} = 6 \, \mathrm{m/s}\) and \(v_{2} = 4 \, \mathrm{m/s}\). At the moment they meet, one of them turns around and walks in the opposite direction, while the other does not change direction. The person who did not change direction reached their final destination \(t_{2} = 10\) minutes earlier than the person who turned around. Determine the time \(t_{1}\) that passed from the start of the journey until the meeting. (15 points)
|
30 \text{ minutes}
| 0.125 |
Two runners, starting simultaneously and running at constant speeds, run in opposite directions on a circular track. One completes the circuit in 5 minutes, and the other in 8 minutes. Find the number of distinct meeting points of the runners on the track if they run for at least one hour.
|
13
| 0.625 |
How many integers at minimum must be selected from the set $\{1, 2, \ldots, 20\}$ to ensure that this selection includes two integers \(a\) and \(b\) such that \(a - b = 2\)?
|
11
| 0.625 |
Let \(\mathbb{N}\) be the set of positive integers, i.e., \(\mathbb{N}=\{1,2, \ldots\}\). Find all functions \(f: \mathbb{N} \rightarrow \mathbb{N}\) such that
$$
f(f(m)+f(n)) = m+n \text{ for all } m, n \in \mathbb{N}.
$$
|
f(n) = n
| 0.75 |
Determine the functions \( f \) from \( \mathbb{Q}_{+}^{*} \) to \( \mathbb{Q}_{+}^{*} \) that satisfy \( f(x+1)=f(x)+1 \) and \( f\left(x^{3}\right)=f(x)^{3} \) for all \( x \in \mathbb{Q}_{+}^{*} \).
|
f(x) = x
| 0.875 |
Given \( 0 < x < 1 \) and \( a, b \) are both positive constants, the minimum value of \( \frac{a^{2}}{x}+\frac{b^{2}}{1-x} \) is ______.
|
(a + b)^2
| 0.875 |
Find the smallest natural number greater than 1 that is at least 600 times greater than each of its prime divisors.
|
1944
| 0.5 |
Three hedgehogs are dividing three pieces of cheese with masses of 5 g, 8 g, and 11 g. A fox comes to help them. She can simultaneously cut 1 g of cheese from any two pieces and eat it. Can the fox leave the hedgehogs with equal pieces of cheese?
|
Yes
| 0.25 |
In a closed bag, there are apples. Three friends tried to lift the bag and guess how many fruits were inside. The first friend guessed that there were 16 apples, the second guessed 19, and the third guessed 25. When they opened the bag, it turned out that one of them was off by 2, another by 4, and the last one by 5. How many apples were in the bag? Find all possible answers.
|
21
| 0.875 |
For a homework assignment, Tanya was asked to come up with 20 examples of the form \( * + * = * \), where different natural numbers need to be inserted in place of \( * \) (i.e., a total of 60 different numbers should be used). Tanya loves prime numbers very much, so she decided to use as many of them as possible while still getting correct examples. What is the maximum number of prime numbers Tanya can use?
|
41
| 0.625 |
From eleven school students and three teachers, a school committee needs to be formed consisting of eight people. At least one teacher must be included in the committee. In how many ways can the committee be formed?
|
2838
| 0.875 |
A perpendicular dropped from the vertex of a rectangle to the diagonal divides the right angle into two parts in the ratio $1: 3$.
Find the angle between this perpendicular and the other diagonal.
|
45^\circ
| 0.875 |
The fare in Moscow with the "Troika" card in 2016 is 32 rubles for one trip on the metro and 31 rubles for one trip on ground transportation. What is the minimum total number of trips that can be made at these rates, spending exactly 5000 rubles?
|
157
| 0.625 |
You have a \(2 \times 3\) grid filled with integers between 1 and 9. The numbers in each row and column are distinct. The first row sums to 23, and the columns sum to 14, 16, and 17 respectively.
Given the following grid:
\[
\begin{array}{c|c|c|c|}
& 14 & 16 & 17 \\
\hline
23 & a & b & c \\
\hline
& x & y & z \\
\hline
\end{array}
\]
What is \(x + 2y + 3z\)?
|
49
| 0.75 |
What integer can be the sum of the lengths of the diagonals of a rhombus with side length 2?
|
5
| 0.625 |
Three candles were lit simultaneously. When the first candle burned out, $\frac{2}{5}$ of the second candle and $\frac{3}{7}$ of the third candle remained. What fraction of the third candle will remain when the second candle burns out?
|
\frac{1}{21}
| 0.875 |
Find the smallest real number $x$ such that $45\%$ of $x$ and $24\%$ of $x$ are natural numbers.
|
\frac{100}{3}
| 0.875 |
What is the probability that a randomly chosen positive integer is relatively prime to 6? What is the probability that at least one of two randomly chosen integers is relatively prime to 6?
|
\frac{5}{9}
| 0.875 |
Given that \(7,999,999,999\) has at most two prime factors, find its largest prime factor.
|
4,002,001
| 0.625 |
On a digital clock, the date is always displayed as an eight-digit number, such as January 1, 2011, which is displayed as 20110101. What is the last date in 2011 that is divisible by 101? This date is represented as $\overline{2011 \mathrm{ABCD}}$. What is $\overline{\mathrm{ABCD}}$?
|
1221
| 0.375 |
From the natural numbers $1, 2, \ldots, 101$, select a group of numbers such that the greatest common divisor of any two numbers in the group is greater than two. What is the maximum number of such numbers in this group?
|
33
| 0.375 |
There are coins of denominations 50 kopecks, 1 ruble, 2 rubles, 5 rubles, and 10 rubles. In a wallet, there are several coins. It is known that no matter which 20 coins are taken out of the wallet, among them there will be at least one 1-ruble coin, at least one 2-ruble coin, and at least one 5-ruble coin. What is the maximum number of coins possible in the wallet such that this condition holds?
|
28
| 0.625 |
Let a positive integer \( k \) be called interesting if the product of the first \( k \) prime numbers is divisible by \( k \) (for example, the product of the first two prime numbers is \(2 \cdot 3 = 6\), and 2 is an interesting number).
What is the largest number of consecutive interesting numbers that can occur?
|
3
| 0.5 |
A ship travels downstream from point $A$ to point $B$ and takes 1 hour. On the return trip, the ship doubles its speed and also takes 1 hour. How many minutes will it take to travel from point $A$ to point $B$ if the ship initially uses double its speed?
|
36 \text{ minutes}
| 0.875 |
In the plane Cartesian coordinate system $xOy$, given two points $M(-1, 2)$ and $N(1, 4)$, point $P$ moves along the $x$-axis. When the angle $\angle MPN$ reaches its maximum value, find the x-coordinate of point $P$.
|
1
| 0.625 |
Inside the cube \( ABCD A_1B_1C_1D_1 \), there is a center \( O \) of a sphere with a radius of 10. The sphere intersects the face \( AA_1D_1D \) along a circle of radius 1, the face \( A_1B_1C_1D_1 \) along a circle of radius 1, and the face \( CDD_1C_1 \) along a circle of radius 3. Find the length of the segment \( OD_1 \).
|
17
| 0.75 |
Josh takes a walk on a rectangular grid of \( n \) rows and 3 columns, starting from the bottom left corner. At each step, he can either move one square to the right or simultaneously move one square to the left and one square up. In how many ways can he reach the center square of the topmost row?
|
2^{n-1}
| 0.5 |
Determine the digits $A, B, C, D$ if
$$
A^{\overline{A B}}=\overline{C C B B D D C A}
$$
|
A=2, B=5, C=3, D=4
| 0.625 |
Define a "position" as a point $(x, y)$ on the Cartesian plane, where $x$ and $y$ are positive integers not exceeding 20. Initially, all 400 positions are empty. Two players, A and B, take turns placing stones, starting with A. Each time it is A's turn, he places a new red stone on an empty position such that any two red stones are not at a distance of $\sqrt{5}$. Each time it is B's turn, he places a new blue stone on any empty position (the distance between the position of a blue stone and other stones can be any value). The game continues until one player can no longer place a stone. Determine the largest integer $K$ such that no matter how B places the blue stones, A can always guarantee placing at least $K$ red stones.
|
100
| 0.875 |
Given that the difference between two 2-digit numbers is 58 and the last two digits of the squares of these two numbers are the same, find the smaller number.
|
21
| 0.625 |
What is the smallest square of an integer that ends with the longest sequence of the same digits?
For example, if the longest sequence of the same digits were five, then a suitable number would be 24677777 (of course, if it were the smallest square, but it is not). Zero is not considered an acceptable digit.
|
38^2 = 1444
| 0.375 |
If \( S = \frac{1}{1 + 1^{2} + 1^{4}} + \frac{2}{1 + 2^{2} + 2^{4}} + \frac{3}{1 + 3^{2} + 3^{4}} + \ldots + \frac{200}{1 + 200^{2} + 200^{4}} \), find the value of \( 80402 \times S \).
|
40200
| 0.5 |
The center of ellipse \( \Gamma \) is at the origin \( O \). The foci of the ellipse lie on the x-axis, and the eccentricity \( e = \sqrt{\frac{2}{3}} \). A line \( l \) intersects the ellipse \( \Gamma \) at two points \( A \) and \( B \) such that \( \overrightarrow{C A} = 2 \overrightarrow{B C} \), where \( C \) is a fixed point \((-1, 0)\). When the area of the triangle \( \triangle O A B \) is maximized, find the equation of the ellipse \( \Gamma \).
|
x^2 + 3y^2 = 5
| 0.5 |
Given the function \( f(x)=\left(x^{2}-4\right)^{-\frac{1}{2}}(x>2), f^{-1}(x) \) is the inverse function of \( f(x) \).
(1) If the sequence \(\{a_{n}\}\) is defined by \(a_{1}=1\) and \(\frac{1}{a_{n+1}}=f^{-1}\left(a_{n}\right) \left(n \in \mathbf{N}_{+}\right)\), find \(a_{n}\).
(2) Let \(S_{n}=a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}\) and \(b_{n}=S_{2n+1}-S_{n}\). Does there exist a maximum positive integer \(p\) such that \(b_{n}<\frac{1}{p}\) holds for all \(n \in \mathbf{N}_{+}\)? Provide an explanation or proof.
|
p = 3
| 0.5 |
How many times does 24 divide into 100 factorial (100!)?
|
32
| 0.875 |
In a trapezoid, the lengths of the diagonals are 6 and 8, and the length of the midline is 5. Find the height of the trapezoid.
|
4.8
| 0.75 |
Leo the fox has a 5 by 5 checkerboard grid with alternating red and black squares. He fills in the grid with the numbers \(1, 2, 3, \ldots, 25\) such that any two consecutive numbers are in adjacent squares (sharing a side) and each number is used exactly once. He then computes the sum of the numbers in the 13 squares that are the same color as the center square. Compute the maximum possible sum Leo can obtain.
|
169
| 0.25 |
In the arithmetic sequence \(\left(a_{n}\right)\) where \(a_{1}=1\) and \(d=4\),
Calculate
\[
A=\frac{1}{\sqrt{a_{1}}+\sqrt{a_{2}}}+\frac{1}{\sqrt{a_{2}}+\sqrt{a_{3}}}+\ldots+\frac{1}{\sqrt{a_{1579}}+\sqrt{a_{1580}}}
\]
Report the smallest integer greater than \(A\).
|
20
| 0.5 |
On the island of Unfortune, there live knights who always tell the truth and liars who always lie. One day, 2022 natives gathered at a round table, and each of them made the statement:
"Next to me sit a knight and a liar!"
It is known that three knights made a mistake (i.e., unintentionally lied). What is the maximum number of knights that could have been at the table?
|
1349
| 0.25 |
Find the number of real solutions of the equation
\[ x^{2}+\frac{1}{x^{2}}=2006+\frac{1}{2006}. \]
|
4
| 0.75 |
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