task_type stringclasses 1
value | problem stringlengths 23 3.94k | answer stringlengths 1 231 | problem_tokens int64 8 1.39k | answer_tokens int64 1 200 |
|---|---|---|---|---|
math | In a sequence of numbers, every (intermediate) term is half as large as the arithmetic mean of its neighboring terms. What relationship holds between an arbitrary term, the term with an index two less, and the term with an index two more? - Let the first term of the sequence be 1, and the 9th term be 40545. What is the $25-\mathrm{th}$ term? | 57424611447841 | 89 | 14 |
math | 6. The number of pairs of positive integer solutions $(n, m)$ for the equation $1!+2!+3!+\cdots+n!=m$ ! is $\qquad$ pairs. | 1 | 42 | 1 |
math | 9. (16 points) There are six piles of apples, and their numbers form an arithmetic sequence. Junjun picks one of the piles and takes out 150 apples, distributing them to the other 5 piles, giving 10, 20, 30, 40, and 50 apples to each pile in sequence. After the distribution, Junjun finds that the number of apples in these 5 piles is exactly 2 times, 3 times, 4 times, 5 times, and 6 times the number of apples in the pile he picked. How many apples are there in total in the six piles? $\qquad$ | 735 | 140 | 3 |
math | Nick is taking a $10$ question test where each answer is either true or false with equal probability. Nick forgot to study, so he guesses randomly on each of the $10$ problems. What is the probability that Nick answers exactly half of the questions correctly? | \frac{63}{256} | 57 | 10 |
math | 1. (10 points) The creative competition at the institute consisted of four tasks. In total, there were 70 applicants. The first test was successfully passed by 35, the second by 48, the third by 64, and the fourth by 63 people, with no one failing all 4 tasks. Those who passed both the third and fourth tests were admitted to the institute. How many were admitted? | 57 | 91 | 2 |
math | 15. Given that $f$ is a real-valued function on the set of all real numbers such that for any real numbers $a$ and $b$,
$$
\mathrm{f}(a \mathrm{f}(b))=a b
$$
Find the value of $\mathrm{f}(2011)$. | 2011 | 71 | 4 |
math | 81. A farmer has 15 cows, 5 of which produce more than 4500 liters of milk per year. Three cows are randomly selected from those owned by the farmer. Find the distribution law of the random variable $X$ representing the number of cows with the specified high milk yield among the selected ones. | X:\begin{pmatrix}x_{i}&0&1&2&3\\p_{i}&24/91&45/91&20/91&2/91\end{pmatrix} | 68 | 51 |
math | Michelle has a word with $2^n$ letters, where a word can consist of letters from any alphabet. Michelle performs a swicheroo on the word as follows: for each $k = 0, 1, \ldots, n-1$, she switches the first $2^k$ letters of the word with the next $2^k$ letters of the word. For example, for $n = 3$, Michelle changes \[
ABCDEFGH \to BACDEFGH \to CDBAEFGH \to EFGHCDBA
\] in one switcheroo.
In terms of $n$, what is the minimum positive integer $m$ such that after Michelle performs the switcheroo operation $m$ times on any word of length $2^n$, she will receive her original word?
| 2^n | 171 | 2 |
math | Let $m$ be the number of five-element subsets that can be chosen from the set of the first $14$ natural numbers so that at least two of the five numbers are consecutive. Find the remainder when $m$ is divided by $1000$. | 750 | 55 | 3 |
math | Triangle $\vartriangle ABC$ has circumcenter $O$ and orthocenter $H$. Let $D$ be the foot of the altitude from $A$ to $BC$, and suppose $AD = 12$. If $BD = \frac14 BC$ and $OH \parallel BC$, compute $AB^2$.
. | 160 | 75 | 3 |
math | 6. In $\triangle A B C$, $a, b, c$ are the sides opposite to $\angle A, \angle B, \angle C$ respectively. If $\frac{a}{b}+\frac{b}{a}=4 \cos C$, and $\cos (A-B)=\frac{1}{6}$, then $\cos C=$ | \frac{2}{3} | 75 | 7 |
math | 1166. Find the integrals:
1) $\int\left(5 x^{3}-4 x^{2}+2\right) d x$
2) $\int \frac{d x}{x-2}$
3) $\int \frac{5 d x}{(x+3)^{7}}$
4) $\int \frac{2 x-5}{x^{2}+4 x+8} d x$ | \begin{aligned}1)&\quad\frac{5}{4}x^{4}-\frac{4}{3}x^{3}+2x+C\\2)&\quad\ln|x-2|+C\\3)&\quad-\frac{5}{6(x+3)^{6}}+C\\4)&\quad\ln(x^{2}+4x+8)-\frac | 95 | 87 |
math | 10. Find the smallest real number $M$, such that for all real numbers $a, b, c$, we have
$$
\left|a b\left(a^{2}-b^{2}\right)+b c\left(b^{2}-c^{2}\right)+c a\left(c^{2}-a^{2}\right)\right| \leqslant M\left(a^{2}+b^{2}+c^{2}\right)^{2} .
$$ | \frac{9\sqrt{2}}{32} | 105 | 13 |
math | 10. To pack books when moving a school library, you can buy small boxes that hold 12 books or large ones that hold 25 books. If all the books are packed in small boxes, 7 books will remain, and if all the books are packed in large boxes, there will be room for 5 more books. The library's collection contains between 500 and 650 books. How many books are in the library? | 595 | 95 | 3 |
math | Example. Find the indefinite integral
$$
\int \operatorname{ctg} x \ln \sin x d x
$$ | \frac{\ln^{2}\sinx}{2}+C | 28 | 14 |
math | 11-29 Let $1990=2^{\alpha_{1}}+2^{\alpha_{2}}+\cdots+2^{\alpha_{n}}$, where $\alpha_{1}, \alpha_{2}, \cdots, \alpha_{n}$ are distinct non-negative integers. Find $\alpha_{1}+\alpha_{2}+\cdots+\alpha_{n}$.
(China Beijing High School Grade 1 Mathematics Competition, 1990) | 43 | 102 | 2 |
math | 3. If the function $f(x)=x^{2}+a|x-1|$ is monotonically increasing on $[0,+\infty)$, then the range of the real number $a$ is | [-2,0] | 45 | 5 |
math | Find how many committees with a chairman can be chosen from a set of n persons. Hence or otherwise prove that
$${n \choose 1} + 2{n \choose 2} + 3{n \choose 3} + ...... + n{n \choose n} = n2^{n-1}$$ | n \cdot 2^{n-1} | 67 | 11 |
math | 3. Let $A=\{1,2,3,4,5\}$. Then the number of mappings $f: A \rightarrow A$ that satisfy the condition $f(f(x))$ $=f(x)$ is $\qquad$ (answer with a number) | 196 | 58 | 3 |
math | Mr. Krbec with his cat Kokeš were selling tickets at Kulikov Castle. On Saturday, they sold 210 children's tickets for 25 groschen each and also some adult tickets for 50 groschen each. In total, they earned 5950 groschen that day.
How many adult tickets did they sell?
(M. Krejčová)
Hint. How much did they earn from the adult tickets? | 14 | 95 | 2 |
math | Let's determine all pairs of positive integers $(n, p)$ such that $p$ is prime, $n \leq 2p$, and $(p-1)^{n}+1$ is divisible by $n^{p-1}$. | p=2,n=2p=3,n=3 | 53 | 12 |
math | Let A be the sum of the digits of 2012 ${ }^{2012}$. We define B as the sum of the digits of A, and $\mathrm{C}$ as the sum of the digits of $B$. Determine $C$. | 7 | 55 | 1 |
math | 9,10
Solve the system of equations:
$x^{2}+4 \sin ^{2} y-4=0$,
$\cos x-2 \cos ^{2} y-1=0$. | 0,\pi/2+k\pi | 48 | 8 |
math | ## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0}\left(\frac{\operatorname{arctg} 3 x}{x}\right)^{x+2}
$$ | 9 | 48 | 1 |
math | (3) If the function $f(x)=\log _{a}\left(4 x+\frac{a}{x}\right)$ is increasing on the interval $[1$,
2], then the range of values for $a$ is $\qquad$ | (1,4] | 55 | 5 |
math | 7.021. $3 \log _{5} 2+2-x=\log _{5}\left(3^{x}-5^{2-x}\right)$.
7.021. $3 \log _{5} 2+2-x=\log _{5}\left(3^{x}-5^{2-x}\right)$. | 2 | 78 | 1 |
math | 7.4. Arseny sat down at the computer between 4 and 5 PM, when the hour and minute hands were pointing in opposite directions, and got up from it on the same day between 10 and 11 PM, when the hands coincided. How long did Arseny sit at the computer? | 6 | 68 | 1 |
math | [ Sequences ]
Continue the sequence of numbers: 1, 11, 21, 1112, 3112, 211213, 312213, 212223, 114213...
# | 31121314 | 66 | 8 |
math | 7.218. $\left(\frac{3}{5}\right)^{2 \log _{9}(x+1)} \cdot\left(\frac{125}{27}\right)^{\log _{27}(x-1)}=\frac{\log _{5} 27}{\log _{5} 243}$. | 2 | 80 | 1 |
math | 9. Let $x, y$ be positive real numbers, and $x+y=1$. Then the minimum value of $\frac{x^{2}}{x+2}+\frac{y^{2}}{y+1}$ is $\qquad$ . | \frac{1}{4} | 54 | 7 |
math | 19(1260). Find the smallest natural number that, when multiplied by 2, becomes a square, and when multiplied by 3, becomes a cube of a natural number. | 72 | 40 | 2 |
math | Someone leaves 46800 frts to their children in such a way that this amount would be divided equally among them. However, before the inheritance is divided, two children die, which results in each of the remaining children receiving 1950 frts more than they would have otherwise. Question: How many children were there? | 8 | 70 | 1 |
math | 3. Inside a right triangle $ABC$ with hypotenuse $AC$, a point $M$ is taken such that the areas of triangles $ABM$ and $BCM$ are one-third and one-fourth of the area of triangle $ABC$, respectively. Find $BM$, if $AM=60$ and $CM=70$. If the answer is not an integer, round it to the nearest integer. | 38 | 87 | 2 |
math | 7. Given $x=\frac{1}{\sqrt{3}+\sqrt{2}}, y=\frac{1}{\sqrt{3}-\sqrt{2}}$. Then, $x^{2}+y^{2}$ is
untranslated part: 轩隹
Note: The term "轩隹" does not have a clear meaning in this context and has been left untranslated. If you can provide more context or clarify the term, I can attempt to translate it accurately. | 10 | 101 | 2 |
math | 5・145 Let the polynomial $R(x)$ have a degree less than 4, and there exists a polynomial $P(x)$ such that
$$
\begin{array}{c}
7 \sin ^{31} t+8 \sin ^{13} t-5 \sin ^{5} t \cos ^{4} t-10 \sin ^{7} t+5 \sin ^{5} t-2 \\
\equiv P(\sin t)\left[\sin ^{4} t-(1+\sin t)\left(\cos ^{2} t-2\right)\right]+R(\sin t),
\end{array}
$$
where $t \in \mathbb{R}$. Try to find all such $R(x)$. | 13x^{3}+5x^{2}+12x+3 | 168 | 18 |
math | Find the number of ways to choose 2005 red, green, and yellow balls such that the number of red balls is even or the number of green balls is odd. | \binom{2007}{2}-\binom{1004}{2} | 37 | 22 |
math | In an isosceles trapezoid, the parallel bases have lengths $\log 3$ and $\log 192$, and the altitude to these bases has length $\log 16$. The perimeter of the trapezoid can be written in the form $\log 2^p 3^q$, where $p$ and $q$ are positive integers. Find $p + q$. | 18 | 86 | 2 |
math | ## Task Condition
Find the derivative.
$$
y=\frac{1}{2 \sin \frac{\alpha}{2}} \cdot \operatorname{arctg} \frac{2 x \sin \frac{\alpha}{2}}{1-x^{2}}
$$ | \frac{1+x^{2}}{(1-x^{2})^{2}+4x^{2}\cdot\sin^{2}\frac{\alpha}{2}} | 57 | 35 |
math | Example 2 Find the monotonic intervals of $f(x)=\log _{\frac{1}{2}}\left(3-2 x-x^{2}\right)$. | f(x)isdecreasingon(-3,-1)increasingon(-1,1) | 37 | 20 |
math | Example 6 Suppose $0<\theta<\pi$, then the maximum value of $\sin \frac{\theta}{2}(1+\cos \theta)$ is $\qquad$
(1994, National High School Mathematics Competition) | \frac{4 \sqrt{3}}{9} | 50 | 12 |
math | 11th Putnam 1951 Problem A3 Find ∑ 0 ∞ (-1) n /(3n + 1). | \frac{1}{3}\ln2+\frac{\pi}{3\sqrt{3}} | 31 | 20 |
math | 2. Let planar vectors $\boldsymbol{a} 、 \boldsymbol{b}$ satisfy $|\boldsymbol{a}+\boldsymbol{b}|=3$. Then the maximum value of $a \cdot b$ is $\qquad$ | \frac{9}{4} | 53 | 7 |
math | Determine all real numbers $x$ such that for all positive integers $n$ the inequality $(1+x)^n \leq 1+(2^n -1)x$ is true. | x \in [0, 1] | 39 | 10 |
math | 7.082. $x^{\frac{\lg x+5}{3}}=10^{5+\lg x}$. | 10^{-5};10^{3} | 29 | 10 |
math | ## problem statement
Find the cosine of the angle between vectors $\overrightarrow{A B}$ and $\overrightarrow{A C}$.
$A(-4 ; 3 ; 0), B(0 ; 1 ; 3), C(-2 ; 4 ;-2)$ | 0 | 58 | 1 |
math | 12. After removing one element from the set $\{1!, 2!, \cdots, 24!\}$, the product of the remaining elements is exactly a perfect square. | 12! | 39 | 3 |
math | Example 6 Given $x, y, z \in \mathbf{R}_{+}$, and $x+y+z=1$. Find
$$\frac{\sqrt{x}}{4 x+1}+\frac{\sqrt{y}}{4 y+1}+\frac{\sqrt{z}}{4 z+1}$$
the maximum value. | \frac{3 \sqrt{3}}{7} | 76 | 12 |
math | 4.1. The sea includes a bay with more saline water. The salinity of the water in the sea is 120 per mille, in the bay 240 per mille, in the part of the sea not including the bay - 110 per mille. How many times is the volume of water in the sea larger than the volume of water in the bay? The volume of water is considered, including the volume of salt. Per mille - one-thousandth of a number; salinity is determined as the ratio of the volume of salt to the total volume of the mixture. | 13 | 128 | 2 |
math | Four rays spread out from point $O$ in a $3$-dimensional space in a way that the angle between every two rays is $a$. Find $\cos a$. | -\frac{1}{3} | 37 | 7 |
math | ## Task B-2.1.
Determine for which value of the variable $x$ the expression $(3 a x+2015)^{2}+(2 a x+2015)^{2}, a \in \mathbb{R}$, $a \neq 0$ has the smallest value and what is the value of that minimum. | 312325 | 78 | 6 |
math | 1. (5 points) Find the value of $n$ for which the following equality holds:
$$
\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\ldots+\frac{1}{\sqrt{n}+\sqrt{n+1}}=2011
$$ | 4048143 | 88 | 7 |
math | Consider non-negative real numbers $a, b, c$ satisfying the condition $a^2 + b^2 + c^2 = 2$ . Find the maximum value of the following expression $$P=\frac{\sqrt{b^2+c^2}}{3-a}+\frac{\sqrt{c^2+a^2}}{3-b}+a+b-2022c$$ | 3 | 84 | 1 |
math |
2. Determine the smallest prime that does not divide any five-digit number whose digits are in a strictly increasing order.
| 11 | 24 | 2 |
math | ## 96. Math Puzzle 5/73
A copper wire that is $l_{1}=50 \mathrm{~m}$ long and $d_{1}=1 \mathrm{~mm}$ thick is drawn to a length of $l_{2}=1800 \mathrm{~m}$. What is the new diameter $d_{2}$? | 0.17 | 79 | 4 |
math | 1. How many times is number $A$ greater or smaller than number $B$, if
$$
\begin{gathered}
A=\underbrace{1+\ldots+1}_{2022 \text { times }}+\underbrace{2+\ldots+2}_{2021 \text { times }}+\ldots+2021+2021+2022, \\
B=\underbrace{2023+\ldots+2023}_{2022 \text { times }}+\underbrace{2022+\ldots+2022}_{2021 \text { times }}+\ldots+3+3+2
\end{gathered}
$$ | \frac{1}{2}B | 159 | 8 |
math | 1A. Given are two quadratic equations $x^{2}+a x+1=0$ and $x^{2}+x+a=0$. Determine all values of the parameter $a$ for which the two equations have at least one common solution. | =1=-2 | 54 | 4 |
math | Anetka's uncle has his birthday on the same day of the year as Anetka's aunt. The uncle is older than the aunt, but not by more than ten years, and both are adults. At the last celebration of their birthdays, Anetka realized that if she multiplied their celebrated ages and then multiplied the resulting product by the number of dogs that gathered at the celebration, she would get the number 2024.
How many dogs could have been at this celebration? Determine all possibilities.
(M. Petrová) | 1or4 | 110 | 3 |
math | 12. The sequence $\left\{a_{n}\right\}$ is defined as follows: $a_{0}=1, a_{1}=2, a_{n+2}=a_{n}+a_{n+1}^{2}$, then $a_{2006} \equiv$ $\qquad$ $(\bmod 7)$ | 6 | 78 | 1 |
math | 10. An integer $n$ allows the polynomial $f(x)=3 x^{3}-n x-n-2$ to be expressed as the product of two non-constant polynomials with integer coefficients, then the sum of all possible values of $n$ is $\qquad$ . | 192 | 60 | 3 |
math | 1. Ana, her mother, and her grandmother are celebrating their birthday on June 23rd. Today, June 23rd, 2021, the sum of their ages is 127. When Ana was 4 years old, her grandmother was twice as old as her mother. Two years ago, her mother was three times as old as Ana. In which year was Ana's mother born? How old will Ana be when she and her mother together have the same number of years as her grandmother? In which year will that be? | 2036 | 115 | 4 |
math | 176 Given the parabola $y^{2}=x, A B$ is a chord passing through the focus $F$, $A$ and $B$ are in the $\mathrm{I}$ and $\mathrm{N}$ quadrants respectively, if $A F-F B=1$, then the slope of $A B$ $=$ $\qquad$ | \frac{1+\sqrt{5}}{2} | 76 | 12 |
math | 4. 12 baskets of apples and 14 baskets of pears weigh 6 c 92 kg. Moreover, the weight of one basket of pears is 10 kg less than the weight of one basket of apples. How much does one basket of pears and one basket of apples weigh separately? | 22 | 66 | 2 |
math | Find the greatest value $M$ that the expression $7 x+10 y+z$ can take when $x, y, z$ are real numbers satisfying $x^{2}+2 x+\frac{1}{5} y^{2}+7 z^{2}=6$. In which cases is this value achieved? Go to the puzzle at the cell corresponding to $M$.
Go to the puzzle at the cell corresponding to $M$. | 55 | 93 | 2 |
math | 6. 62 Given that if $y=\sin x,-\frac{\pi}{2}<x<\frac{\pi}{2}$, then $x=\sin ^{-1} y$, now if $y=\sin x,\left(1992+\frac{1}{2}\right) \pi \leqslant x \leqslant\left(1993+\frac{1}{2}\right) \pi$, try to express $x$ in terms of $y$. | 1993\pi-\sin^{-1}y | 109 | 12 |
math | 12. (20 points) Given the ellipse $\frac{x^{2}}{4}+y^{2}=1$, and $P$ is any point on the circle $x^{2}+y^{2}=16$. Tangents $PA$ and $PB$ are drawn from $P$ to the ellipse, touching the ellipse at points $A$ and $B$ respectively. Find the maximum and minimum values of $\overrightarrow{P A} \cdot \overrightarrow{P B}$. | \frac{33}{4} \text{ and } \frac{165}{16} | 107 | 23 |
math | In the expansion of $(x+y)^{n}$ using the binomial theorem, the second term is 240, the third term is 720, and the fourth term is 1080. Find $x, y$, and $n$.
# | 2,3,n=5 | 58 | 6 |
math | 5. 10 Let $A$ and $B$ be the sums of the odd and even terms, respectively, in the expansion of $(x+a)^{n}$. Find $A^{2}-B^{2}$. | \left(x^{2}-a^{2}\right)^{n} | 48 | 15 |
math | 7.273. $\log _{9}\left(x^{3}+y^{3}\right)=\log _{3}\left(x^{2}-y^{2}\right)=\log _{3}(x+y)$. | (1;0),(2;1) | 51 | 9 |
math | On the board we write a series of $n$ numbers, where $n \geq 40$, and each one of them is equal to either $1$ or $-1$, such that the following conditions both hold:
(i) The sum of every $40$ consecutive numbers is equal to $0$.
(ii) The sum of every $42$ consecutive numbers is not equal to $0$.
We denote by $S_n$ the sum of the $n$ numbers of the board. Find the maximum possible value of $S_n$ for all possible values of $n$. | 20 | 124 | 2 |
math | ## Problem Statement
Write the equation of the plane passing through point $A$ and perpendicular to vector $\overrightarrow{B C}$.
$A(4, -2, 0)$
$B(1, -1, -5)$
$C(-2, 1, -3)$ | -3x+2y+2z+16=0 | 63 | 14 |
math | 7.7. Petya told Misha that in his class exactly two thirds of all the girls are blondes, exactly one seventh of the boys are blonds, and in total, a third of the class has light hair. Misha said: "You once told me that there are no more than 40 people in your class. 0 ! I know how many girls are in your class!" How many? | 12 | 87 | 2 |
math | 4. If $x$ is a four-digit integer in decimal notation, let the product of the digits of $x$ be $T(x)$, the sum of the digits be $S(x)$, $p$ be a prime number, and $T(x)=p^{k}, S(x)=p^{p}-5$, then the smallest $x$ is $\qquad$ . | 1399 | 80 | 4 |
math | Task 2. The sets $A$ and $B$ are subsets of the positive integers. The sum of any two different elements from $A$ is an element of $B$. The quotient of any two different elements of $B$ (where we divide the larger by the smaller) is an element of $A$. Determine the maximum number of elements in $A \cup B$. | 5 | 79 | 1 |
math | 4. (8 points) In the movie "The Monkey King's Return", there is a scene where Monkey King battles mountain demons. Some of the demons are knocked down, and the number of those knocked down is one third more than those standing; After a while, 2 more demons are knocked down, but then 10 demons stand up again. At this point, the number of standing demons is one quarter more than those knocked down. How many demons are standing now?
| 35 | 97 | 2 |
math | 2. From one point on a straight highway, three cyclists start simultaneously (but possibly in different directions). Each of them rides at a constant speed without changing direction. An hour after the start, the distance between the first and second cyclist was 20 km, and the distance between the first and third - 5 km. At what speed is the third cyclist riding, if it is known that he is riding slower than the first, and the speed of the second is 10 km/h? List all possible options. | 25 | 106 | 2 |
math | 12. Find the real solution to the inequality $\log _{6}(1+\sqrt{x})>\log _{25} x$, | 0<x<25 | 30 | 5 |
math | 5. (10 points) A small railway wagon with a jet engine is standing on the tracks. The tracks are laid in the form of a circle with a radius of $R=4$ m. The wagon starts from rest, with the jet force having a constant value. What is the maximum speed the wagon will reach after one full circle, if its acceleration over this period should not exceed $a=2 \mathrm{M} / \mathrm{c}^{2}$? | 2.8\mathrm{~}/\mathrm{} | 99 | 11 |
math | \section*{Problem 6 - 081236}
All real numbers \(a\) are to be determined for which the equation
\[
\sin ^{6} x+\cos ^{6} x=a\left(\sin ^{4} x+\cos ^{4} x\right)
\]
has at least one real solution. Furthermore, all solutions for \(a=\frac{5}{6}\) are to be specified. | (2k+1)\frac{\pi}{8}\quad(k\in\mathbb{Z}) | 96 | 22 |
math | 10.3. Vasya has $n$ candies of several types, where $n \geqslant 145$. It is known that if any group of at least 145 candies is chosen from these $n$ candies (in particular, the group can consist of all $n$ candies), then there exists a type of candy such that the chosen group contains exactly 10 candies of this type. Find the largest possible value of $n$.
(A. Antropov) | 160 | 106 | 3 |
math | Let $A$ be a positive real number. What are the possible values of $\displaystyle\sum_{j=0}^{\infty} x_j^2, $ given that $x_0, x_1, \cdots$ are positive numbers for which $\displaystyle\sum_{j=0}^{\infty} x_j = A$? | (0, A^2) | 78 | 8 |
math | 1. Given the equations
$$
x^{2}-a x+b-4=0 \text { and } y^{2}-b y+a-\frac{1}{4}=0, a, b \in \mathbb{R}
$$
For which values of $a$ and $b$, the roots of the second equation are the reciprocal values of the roots of the first equation? | a_{1}=0,a_{2}=-\frac{7}{4},a_{3}=\frac{9}{4};b_{1}=0,b_{2}=\frac{7}{2},b_{3}=\frac{9}{2} | 82 | 55 |
math | 8. The set $S=\{1,2, \ldots, 2022\}$ is to be partitioned into $n$ disjoint subsets $S_{1}, S_{2}, \ldots, S_{n}$ such that for each $i \in\{1,2, \ldots, n\}$, exactly one of the following statements is true:
(a) For all $x, y \in S_{i}$ with $x \neq y, \operatorname{gcd}(x, y)>1$.
(b) For all $x, y \in S_{i}$ with $x \neq y, \operatorname{gcd}(x, y)=1$.
Find the smallest value of $n$ for which this is possible. | 15 | 166 | 2 |
math | If $\frac{1}{\sqrt{2011+\sqrt{2011^2-1}}}=\sqrt{m}-\sqrt{n}$, where $m$ and $n$ are positive integers, what is the value of $m+n$? | 2011 | 58 | 4 |
math | 1. Given an $n$-degree polynomial $p(x)$ satisfying $p(k)=\frac{1}{C_{n+1}^{k}}$, where $k=0,1,2, \cdots, n$. Find $p(n+1)$.
(IMO - 22 Preliminary Problem) | p(n+1)={\begin{pmatrix}1,\text{when}n\text{iseven,}\\0,\text{when}n\text{isodd.}\end{pmatrix}.} | 68 | 46 |
math | 4. Find a factor of 1464101210001 , such that it lies between 1210000 and 1220000 .求 1464101210001 在 1210000 和 1220000 之間的一個因數。 | 1211101 | 84 | 7 |
math | 38. In a tournament where each pair of teams played each other twice, 4 teams participated. For each win, two points were awarded, for a draw - one, and for a loss - 0. The team that finished in last place scored 5 points. How many points did the team that finished in first place score? | 7 | 69 | 1 |
math | 6. Let $A B C$ be an equilateral triangle of side length 1 . For a real number $0<x<0.5$, let $A_{1}$ and $A_{2}$ be the points on side $B C$ such that $A_{1} B=A_{2} C=x$, and let $T_{A}=\triangle A A_{1} A_{2}$. Construct triangles $T_{B}=\triangle B B_{1} B_{2}$ and $T_{C}=\triangle C C_{1} C_{2}$ similarly.
There exist positive rational numbers $b, c$ such that the region of points inside all three triangles $T_{A}, T_{B}, T_{C}$ is a hexagon with area
$$
\frac{8 x^{2}-b x+c}{(2-x)(x+1)} \cdot \frac{\sqrt{3}}{4} \text {. }
$$
Find $(b, c)$. | (8,2) | 210 | 5 |
math | 143*. Find the condition under which the difference of two irreducible fractions is equal to their product. | \frac{}{b},\frac{}{+b} | 22 | 13 |
math | Find the smallest $\lambda \in \mathbb{R}$ such that for all $n \in \mathbb{N}_+$, there exists $x_1, x_2, \ldots, x_n$ satisfying $n = x_1 x_2 \ldots x_{2023}$, where $x_i$ is either a prime or a positive integer not exceeding $n^\lambda$ for all $i \in \left\{ 1,2, \ldots, 2023 \right\}$. | \frac{2}{2024} | 117 | 10 |
math | 12. Given the equation $x^{5}-x^{2}+5=0$ with its five roots being $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, f(x)=x^{2}+1$, then $\prod_{k=1}^{5} f\left(x_{k}\right)=$ | 37 | 77 | 2 |
math | Task 1. Three girls, Ana, Maja, and Aleksandra, collected 770 strawberries in the forest and decided to divide them proportionally according to their ages. Whenever Maja took 4 strawberries, Ana took 3 strawberries, and for every 6 strawberries Maja took, Aleksandra took 7 strawberries. How old is each of the girls if it is known that together they are 35 years old? How many strawberries did each of them get? | Ana:198,Maia:264,Aleksar:308 | 98 | 20 |
math | Example 1 Find the sum of the $n^{2}$ numbers:
$$
\begin{array}{l}
1,2,3, \cdots, n \\
2,3,4, \cdots, n+1 \\
3,4,5, \cdots, n+2 \\
\cdots \cdots \\
n, n+1, n+2, \cdots, 2 n-1
\end{array}
$$ | n^3 | 97 | 3 |
math | The sum of two numbers is 581; the quotient of their least common multiple and greatest common divisor is 240. Which are these numbers? | A=560,B=21 | 33 | 9 |
math | 7. If the quadratic function $f(x)=a x^{2}-2 x-a$ satisfies
$$
f(2)<f(1)<f(3)<f(0) \text {, }
$$
then the range of the real number $a$ is $\qquad$. | \left(\frac{1}{2}, \frac{2}{3}\right) | 61 | 18 |
math | Find all $ 3$-digit numbers such that placing to the right side of the number its successor we get a $ 6$-digit number which is a perfect square. | n \in \{183, 328, 528, 715\} | 37 | 25 |
math | 6. (5 points) The sum of all digits in the product of $2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17$ is $\qquad$ | 12 | 50 | 2 |
math | 3. Calculate the value of the product $\frac{2^{3}-1}{2^{3}+1} \cdot \frac{3^{3}-1}{3^{3}+1} \cdot \frac{4^{3}-1}{4^{3}+1} \cdot \ldots \cdot \frac{200^{3}-1}{200^{3}+1}$. | \frac{40201}{60300} | 88 | 15 |
math | 14. Given $O(0,0), A(1,0), B(0,2)$. Let $P_{1}, P_{2}, P_{3}$ be the midpoints of $AB, OB, OP_{1}$ respectively. For any positive integer $n$, $P_{n+3}$ is the midpoint of the segment $P_{n} P_{n+1}$. Let $P_{n}(x_{n}, y_{n})$, and set
$$
\begin{array}{l}
a_{n}=\frac{x_{n}}{2}+x_{n+1}+x_{n+2}, \\
b_{n}=a_{n} y_{n}+y_{n+1}+y_{n+2} .
\end{array}
$$
(1) Find $a_{1}, a_{2}, a_{3}$;
(2) Prove: $y_{n+4}=1-\frac{y_{n}}{4}$;
(3) Let $c_{n}=1-\frac{5}{4} y_{4 n}$, prove that $\{c_{n}\}$ is a geometric sequence, and find $c_{n}$. | c_{n}=(-1)^{n} \frac{1}{4^{n}} | 263 | 19 |
math | Alice and Bob are in a hardware store. The store sells coloured sleeves that fit over keys to distinguish them. The following conversation takes place:
[color=#0000FF]Alice:[/color] Are you going to cover your keys?
[color=#FF0000]Bob:[/color] I would like to, but there are only $7$ colours and I have $8$ keys.
[color=#0000FF]Alice:[/color] Yes, but you could always distinguish a key by noticing that the red key next to the green key was different from the red key next to the blue key.
[color=#FF0000]Bob:[/color] You must be careful what you mean by "[i]next to[/i]" or "[i]three keys over from[/i]" since you can turn the key ring over and the keys are arranged in a circle.
[color=#0000FF]Alice:[/color] Even so, you don't need $8$ colours.
[b]Problem:[/b] What is the smallest number of colours needed to distinguish $n$ keys if all the keys are to be covered. | 2 | 240 | 1 |
math | 5 Find all real numbers $a$ such that any positive integer solution of the inequality $x^{2}+y^{2}+z^{2} \leqslant a(x y+y z+z x)$ are the lengths of the sides of some triangle. | 1 \leqslant a < \frac{6}{5} | 55 | 15 |
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