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 | Points $A$, $B$, $C$, $D$, and $E$ are on the same plane such that $A,E,C$ lie on a line in that order, $B,E,D$ lie on a line in that order, $AE = 1$, $BE = 4$, $CE = 3$, $DE = 2$, and $\angle AEB = 60^\circ$. Let $AB$ and $CD$ intersect at $P$. The square of the area of quadrilateral $PAED$ can be expressed as $\frac{... | 967 | 162 | 3 |
math | ## Task Condition
Find the derivative.
$y=\frac{3}{2} \cdot \ln \left(\tanh \frac{x}{2}\right)+\cosh x-\frac{\cosh x}{2 \sinh^{2} x}$ | \frac{\operatorname{ch}^{4}x}{\operatorname{sh}^{3}x} | 54 | 24 |
math | The school store is running out of supplies, but it still has five items: one pencil (costing $\$1$), one pen (costing $\$1$), one folder (costing $\$2$), one pack of paper (costing $\$3$), and one binder (costing $\$4$). If you have $\$10$, in how many ways can you spend your money? (You don't have to spend all of you... | 31 | 104 | 2 |
math | For some positive integer $n$, the sum of all odd positive integers between $n^2-n$ and $n^2+n$ is a number between $9000$ and $10000$, inclusive. Compute $n$.
[i]2020 CCA Math Bonanza Lightning Round #3.1[/i] | 21 | 73 | 2 |
math | 29. [18] Compute the remainder when
$$
\sum_{k=1}^{30303} k^{k}
$$
is divided by 101 . | 29 | 42 | 2 |
math | 8. Spring has arrived, and the school has organized a spring outing for the students. However, due to certain reasons, the spring outing is divided into indoor and outdoor activities. The number of people participating in outdoor activities is 480 more than those participating in indoor activities. Now, if 50 people fr... | 870 | 116 | 3 |
math | Example 11 Let $a b c \neq 0$, if $(a+2 b+3 c)^{2}=14\left(a^{2}+\right.$ $b^{2}+c^{2}$ ), then the value of $\frac{b+c}{a}+\frac{a+c}{b}+\frac{a+b}{c}$ is $\qquad$. | 8 | 83 | 1 |
math | 10. Given a sequence $\left\{a_{n}\right\}$ with 100 terms, satisfying $a_{1}=0, a_{100}=475$,
and $\left|a_{k+1}-a_{k}\right|=5(k=1,2, \cdots, 99)$.
Then the number of different sequences that meet the conditions is $\qquad$. | 4851 | 90 | 4 |
math | 1*. Into how many regions do $n$ planes divide space if every three of them have exactly one common point, and no four of them have a common point? | F_{3}(n)=C_{n}^{3}+C_{n}^{2}+C_{n}^{1}+C_{n}^{0} | 34 | 37 |
math | \section*{Problem 1 - 261031}
For a natural number \(x\), the following conditions must be satisfied:
(1) When written in the binary system, \(x\) has exactly seven digits.
(2) When \(x\) is written in the ternary system, no digit appears more than twice.
(3) When written in the quinary system, \(x\) has exactly fo... | 126 | 112 | 3 |
math | ## Task 1 - 120731
At a secondary school with exactly 500 students, there are mathematical-scientific, artistic, and sports working groups. The following is known about the participation of students in these working groups:
(1) Exactly 250 students are members of at least one sports working group.
(2) Exactly 125 st... | 20 | 229 | 2 |
math | Shapovalov A.V.
For which $N$ can the numbers from 1 to $N$ be rearranged in such a way that the arithmetic mean of any group of two or more consecutive numbers is not an integer? | 2m | 47 | 2 |
math | Example 3. A pocket contains 7 white balls and 3 black balls of the same size. If two balls are drawn at random, what is the probability of getting one white ball and one black ball? (Exercise 6, Question 6) | \frac{7}{15} | 52 | 8 |
math | ## T-1 A
Determine all triples $(a, b, c)$ of real numbers satisfying the system of equations
$$
\begin{aligned}
& a^{2}+a b+c=0 \\
& b^{2}+b c+a=0 \\
& c^{2}+c a+b=0
\end{aligned}
$$
Answer. The solutions are
$$
(a, b, c) \in\left\{(0,0,0),\left(-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right)\right\}
$$
| (,b,)\in{(0,0,0),(-\frac{1}{2},-\frac{1}{2},-\frac{1}{2})} | 133 | 36 |
math | 3. Find the numerical value of the expression
$$
\frac{1}{x^{2}+1}+\frac{1}{y^{2}+1}+\frac{2}{x y+1}
$$
if it is known that $x$ is not equal to $y$ and the sum of the first two terms is equal to the third. | 2 | 77 | 1 |
math | 6. If a real-coefficient quartic polynomial with the leading coefficient of 1 has four imaginary roots, and the product of two of them is $32+\mathrm{i}$, while the sum of the other two is $7+\mathrm{i}$, then the coefficient of the quadratic term is $\qquad$
Translate the above text into English, please retain the or... | 114 | 90 | 3 |
math | 5. (8 points) Through the sides of a regular $2 n$-gon, lines are drawn. Into how many parts do these lines divide the plane | 2n^ | 33 | 3 |
math | Two balls, the larger one has a mass of $M$, the smaller one has a mass of $m$. Let $A$ be a point within the segment defined by the centers of the two balls, and $B$ be a point outside this segment, chosen such that both balls exert equal gravitational force on a unit mass placed at this point, according to Newton's l... | 6.56\mathrm{~} | 146 | 9 |
math | ## 46. Math Puzzle $3 / 69$
Write down a three-digit number and subtract the number with the reversed digit order. Divide the result by the difference between the 1st and 3rd digit of the original number, then divide by 11 again. If you now take the square root, the result is always 3. This is a neat trick to amaze yo... | 3 | 95 | 1 |
math | Problem 7.7. The numbers from 1 to 200 were arranged in a random order on a circle such that the distances between adjacent numbers on the circle are the same.
For any number, the following is true: if you consider 99 numbers standing clockwise from it and 99 numbers standing counterclockwise from it, there will be an... | 114 | 100 | 3 |
math | Example 7 Let $M$ be a moving point on the unit circle $x^{2}+y^{2}=1$, and $N$ be a vertex of an equilateral triangle with fixed point $A(3,0)$ and $M$. Assume that $M \rightarrow N \rightarrow A \rightarrow M$ forms a counterclockwise direction. Find the equation of the trajectory of point $N$ as point $M$ moves. | (x-\frac{3}{2})^{2}+(y+\frac{3\sqrt{3}}{2})^{2}=1 | 93 | 29 |
math | Example $1^{\bullet}$. Consider the sequence $\left\{a_{n}\right\}$ defined inductively by $a_{1}=1, a_{n+1}=\frac{1}{2} a_{n}+1$ $(n=1,2,3, \cdots)$. Try to find the general term of the sequence $\left\{a_{n}\right\}$. | a_{n}=2-\left(\frac{1}{2}\right)^{n-1} | 89 | 21 |
math | 5. Given a circle and two points $P$ and $Q$ inside it, inscribe a right triangle such that its legs pass through $P$ and $Q$. For which positions of $P$ and $Q$ does the problem have no solution?
SOLUTION: | OM+\frac{PQ}{2}<r | 57 | 10 |
math | 8 Utilize the discriminant
Example 10. Try to determine all real solutions of the system of equations
$$
\left\{\begin{array}{l}
x+y+z=3, \\
x^{2}+y^{2}+z^{2}=3, \\
x^{5}+y^{5}+z^{5}=3
\end{array}\right.
$$ | x=1, y=1, z=1 | 84 | 11 |
math | 6. Black and white balls are arranged in a circle, with black balls being twice as many as white ones. It is known that among pairs of adjacent balls, there are three times as many monochromatic pairs as polychromatic ones. What is the smallest number of balls that could have been arranged? (B. Trushin) | 24 | 69 | 2 |
math | For three unknown natural numbers, it holds that:
- the greatest common divisor of the first and second is 8,
- the greatest common divisor of the second and third is 2,
- the greatest common divisor of the first and third is 6,
- the least common multiple of all three numbers is 1680,
- the largest of the numbers is ... | 120,16,42or168,16,30 | 122 | 19 |
math | Find all solutions $(x_1, x_2, . . . , x_n)$ of the equation
\[1 +\frac{1}{x_1} + \frac{x_1+1}{x{}_1x{}_2}+\frac{(x_1+1)(x_2+1)}{x{}_1{}_2x{}_3} +\cdots + \frac{(x_1+1)(x_2+1) \cdots (x_{n-1}+1)}{x{}_1x{}_2\cdots x_n} =0\] | E = \{(x_1, x_2, \ldots, x_n) \mid x_i \neq 0 \text{ for all } i \text{ and at least one } x_j = -1\} | 126 | 51 |
math | 17. Among the positive integers less than $10^{4}$, how many positive integers $n$ are there such that $2^{n}-n^{2}$ is divisible by 7? | 2857 | 42 | 4 |
math | 8. $[\mathbf{6}]$ Let $A:=\mathbb{Q} \backslash\{0,1\}$ denote the set of all rationals other than 0 and 1. A function $f: A \rightarrow \mathbb{R}$ has the property that for all $x \in A$,
$$
f(x)+f\left(1-\frac{1}{x}\right)=\log |x| .
$$
Compute the value of $f(2007)$. | \log(2007/2006) | 112 | 13 |
math | 3. It is known that the polynomial $f(x)=8+32 x-12 x^{2}-4 x^{3}+x^{4}$ has 4 distinct real roots $\left\{x_{1}, x_{2}, x_{3}, x_{4}\right\}$. The polynomial of the form $g(x)=b_{0}+b_{1} x+b_{2} x^{2}+b_{3} x^{3}+x^{4}$ has roots $\left\{x_{1}^{2}, x_{2}^{2}, x_{3}^{2}, x_{4}^{2}\right\}$. Find the coefficient $b_{1}$... | -1216 | 159 | 5 |
math | 2.3.5 * In $\triangle A B C$, let $B C=a, C A=b, A B=c$. If $9 a^{2}+9 b^{2}-19 c^{2}=0$, then $\frac{\cot C}{\cot A+\cot B}=$ $\qquad$ . | \frac{5}{9} | 68 | 7 |
math | Compute the area of the trapezoid $ABCD$ with right angles $BAD$ and $ADC$ and side lengths of $AB=3$, $BC=5$, and $CD=7$. | 15 | 43 | 2 |
math | Three. (20 points) Given the ellipse $\frac{y^{2}}{a^{2}}+\frac{x^{2}}{b^{2}}=1(a>b>0)$, draw two perpendicular chords $A C$ and $B D$ through the center $O$. Let the eccentric angles of points $A$ and $B$ be $\theta_{1}$ and $\theta_{2}$, respectively. Find the range of $\left|\cos \left(\theta_{1}-\theta_{2}\right)\r... | 0 \leqslant\left|\cos \left(\theta_{1}-\theta_{2}\right)\right| \leqslant \frac{a^{2}-b^{2}}{a^{2}+b^{2}} | 113 | 52 |
math | 8. Let $A B C$ be an equilateral triangle with side length 8 . Let $X$ be on side $A B$ so that $A X=5$ and $Y$ be on side $A C$ so that $A Y=3$. Let $Z$ be on side $B C$ so that $A Z, B Y, C X$ are concurrent. Let $Z X, Z Y$ intersect the circumcircle of $A X Y$ again at $P, Q$ respectively. Let $X Q$ and $Y P$ inters... | 304 | 134 | 3 |
math | 10. For any real numbers $x, y$, define the operation $x * y$ as $x * y=a x+b y+c x y$, where $a, b, c$ are constants, and the operations on the right side of the equation are the usual real number addition and multiplication. It is known that $1 * 2=3, 2 * 3=4$, and there is a non-zero real number $d$, such that for a... | 4 | 119 | 1 |
math | 3. Given the equation $x e^{-2 x}+k=0$ has exactly two real roots in the interval $(-2,2)$, then the range of values for $k$ is | (-\frac{1}{2e},-\frac{2}{e^{4}}) | 42 | 19 |
math | Consider a round-robin tournament with $2 n+1$ teams, where each team plays each other team exactly once. We say that three teams $X, Y$ and $Z$, form a cycle triplet if $X$ beats $Y, Y$ beats $Z$, and $Z$ beats $X$. There are no ties.
(a) Determine the minimum number of cycle triplets possible.
(b) Determine the max... | \frac{n(n+1)(2n+1)}{6} | 94 | 15 |
math | Task 1. Represent in the form of an irreducible fraction
$$
6 \frac{3}{2015} \times 8 \frac{11}{2016}-11 \frac{2012}{2015} \times 3 \frac{2005}{2016}-12 \times \frac{3}{2015}
$$ | \frac{11}{112} | 89 | 10 |
math | ## 137. Math Puzzle 10/76
A card is drawn blindly from a Skat deck. After it is shuffled back in, the "draw" is repeated.
What is the probability that the first card is an Ace and the second card is a King? | 0.0156 | 59 | 6 |
math | Example 6. Solve the integral equation
$$
\frac{1}{\sqrt{\pi x}} \int_{0}^{\infty} e^{-t^{2} /(4 x)} \varphi(t) d t=1
$$ | \varphi(x)\equiv1 | 53 | 7 |
math | If $a, b, x$ and $y$ are real numbers such that $ax + by = 3,$ $ax^2+by^2=7,$ $ax^3+bx^3=16$, and $ax^4+by^4=42,$ find $ax^5+by^5$. | 20 | 71 | 2 |
math | Let's calculate the expression
$$
\frac{x_{1}}{x_{2}}+\frac{x_{2}}{x_{1}}
$$
if $x_{1}$ and $x_{2}$ are the roots of the equation
$$
x^{2}+p x+q=0
$$ | \frac{p^{2}-2q}{q} | 66 | 12 |
math | 7. If a small ball with a radius of 1 can move freely in all directions inside a regular tetrahedron container with an edge length of $6 \sqrt{6}$, then the area of the container's inner wall that the ball can never touch is $\qquad$ . | 120\sqrt{3} | 60 | 8 |
math | 15. (16 points) Let $x_{1}, x_{2}, x_{3} \in [0,12]$,
$$
x_{1} x_{2} x_{3}=\left(\left(12-x_{1}\right)\left(12-x_{2}\right)\left(12-x_{3}\right)\right)^{2} \text {. }
$$
Find the maximum value of $f=x_{1} x_{2} x_{3}$. | 729 | 109 | 3 |
math | The surface of the Earth consists of $70\%$ water and $30\%$ land. Two fifths of the land are deserts or covered by ice and one third of the land is pasture, forest, or mountain; the rest of the land is cultivated. What is the percentage of the total surface of the Earth that is cultivated? | 8 | 73 | 1 |
math | 1. Determine the integer solutions of the equation $p(x+y)=xy$, where $p$ is a given prime number. | (0,0),(2p,2p),(p(p+1),p+1),(p(1-p),p-1),(p+1,p(p+1)),(p-1,p(1-p)) | 26 | 46 |
math | 4. For a nonzero integer $a$, denote by $v_{2}(a)$ the largest nonnegative integer $k$ such that $2^{k} \mid a$. Given $n \in \mathbb{N}$, determine the largest possible cardinality of a subset $A$ of set $\left\{1,2,3, \ldots, 2^{n}\right\}$ with the following property:
$$
\text { for all } x, y \in A \text { with } ... | 2^{[\frac{n+1}{2}]} | 139 | 11 |
math | 18. (3 points) When Xiaoming goes from home to school, he walks the first half of the distance and takes a ride for the second half; when he returns from school to home, he rides for the first $\frac{1}{3}$ of the time and walks for the last $\frac{2}{3}$ of the time. As a result, the time it takes to go to school is 2... | 150 | 145 | 3 |
math | Example 3 Let the logarithmic equation be $\lg (a x)=2 \lg (x-1)$, discuss the range of values for $a$ such that the equation has a solution, and find its solution. | \frac{1}{2}(2++\sqrt{^{2}+4}) | 46 | 18 |
math | 3. Find all positive real numbers $t$ such that there exists an infinite set $X$ of real numbers, for which, for any $x, y, z \in X$ (where $x, y, z$ can be the same), and any real number $a$ and positive real number $d$, we have
$$
\max \{|x-(a-d)|,|y-a|,|z-(a+d)|\}>t d
$$ | 0<t<\frac{1}{2} | 99 | 10 |
math | ## Task A-2.4.
Determine all real numbers $a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{2020} \geqslant 0$ for which
$$
a_{1}+a_{2}+\cdots+a_{2020}=1 \quad \text { and } \quad a_{1}^{2}+a_{2}^{2}+\cdots+a_{2020}^{2}=a_{1}
$$ | a_{1}=\cdots=a_{n}=\frac{1}{n}\quad\text{}\quada_{n+1}=\cdots=a_{2020}=0 | 122 | 40 |
math | Given a sequence of numbers:
$$
\begin{array}{l}
1, \frac{1}{2}, \frac{2}{2}, \frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \frac{3}{3}, \frac{2}{3}, \frac{1}{3}, \\
\frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \frac{4}{4}, \frac{3}{4}, \frac{2}{4}, \frac{1}{4}, \cdots
\end{array}
$$
Question: (1) What is the position of $\frac{5}... | 54 \text{ or } 60, \frac{4}{15} | 171 | 19 |
math | Find two such common fractions - one with a denominator of 8, the other with a denominator of 13 - so that they are not equal, but the difference between the larger and the smaller of them is as small as possible.
# | \frac{3}{8},\frac{5}{13} | 49 | 15 |
math | Consider a chessboard that is infinite in all directions. Alex the T-rex wishes to place a positive integer in each square in such a way that:
[list]
[*] No two numbers are equal.
[*] If a number $m$ is placed on square $C$, then at least $k$ of the squares orthogonally adjacent to $C$ have a multiple of $m$ written on... | k = 2 | 103 | 5 |
math | Example 1. If $x=2-\sqrt{3}$, find the value of $\frac{x^{4}-4 x^{3}-x^{2}+9 x-4}{x^{2}-4 x+5}$.
(1989, Shanghai Junior High School Mathematics Competition) | -\frac{\sqrt{3}}{4} | 63 | 10 |
math | 2. Let the edge length of the regular tetrahedron $ABCD$ be 1 meter. A small insect starts from point $A$ and moves according to the following rule: at each vertex, it randomly chooses one of the three edges connected to this vertex and crawls along this edge to another vertex. What is the probability that it returns t... | \frac{7}{27} | 87 | 8 |
math | Example 5 Given
$$
A=\left\{z \mid z^{18}=1\right\} \text { and } B=\left\{\omega \mid \omega^{48}=1\right\}
$$
are sets of complex roots of unity,
$$
C=\{z w \mid z \in A, w \in B\}
$$
is also a set of complex roots of unity. How many distinct elements are there in the set $C$? ${ }^{[3]}$ | 144 | 111 | 3 |
math | 8. A uniform cube die has the numbers $1,2, \cdots, 6$ marked on its faces. Each time two identical dice are rolled, the sum of the numbers on the top faces is defined as the point number for that roll. Then, the probability that the product of the 3 point numbers obtained from 3 rolls can be divided by 14 is $\qquad$.... | \frac{1}{3} | 92 | 7 |
math | 18. (6 points) Given the four-digit number $\overline{\mathrm{ABCD}}$, the conclusions of three people, Jia, Yi, and Bing, are as follows:
Jia: “The unit digit is half of the hundred digit”;
Yi: “The tens digit is 1.5 times the hundred digit”;
Bing: “The average of the four digits is 4”.
According to the above informa... | 4462 | 108 | 4 |
math | 4. For natural numbers $m$ and $n$, it is known that $3 n^{3}=5 m^{2}$. Find the smallest possible value of $m+n$. | 60 | 38 | 2 |
math | 1. Given that $x$ and $y$ are nonzero real numbers such that $x+\frac{1}{y}=10$ and $y+\frac{1}{x}=\frac{5}{12}$, find all possible values of $x$. | 4,6 | 56 | 3 |
math | 8. In tetrahedron $ABCD$, $\triangle ABD$ is an equilateral triangle, $\angle BCD=90^{\circ}$, $BC=CD=1$, $AC=\sqrt{3}$, $E, F$ are the midpoints of $BD, AC$ respectively, then the cosine value of the angle formed by line $AE$ and $BF$ is $\qquad$ | \frac{\sqrt{2}}{3} | 88 | 10 |
math | 135. Find $\lim _{x \rightarrow \infty} \frac{\sqrt{x^{2}+4}}{x}$. | 1 | 31 | 1 |
math | 9 Given $0<a_{1}<a_{2}<\cdots<a_{n}$, for any permutation $b_{1}, b_{2}, \cdots$, $b_{n}$ of $a_{1}, a_{2}, \cdots, a_{n}$. Let $M=\prod_{i=1}^{n}\left(a_{i}+\frac{1}{b_{i}}\right)$, find the permutation $b_{1}, b_{2}, \cdots, b_{n}$ that maximizes $M$. | b_{1}=a_{1}, b_{2}=a_{2}, \cdots, b_{n}=a_{n} | 117 | 28 |
math | 9. (16 points) Let the real-coefficient polynomial
$$
P_{i}(x)=x^{2}+b_{i} x+c_{i}\left(b_{i}, c_{i} \in \mathbf{R}, i=1,2, \cdots, n\right)
$$
be distinct, and for any $1 \leqslant i<j \leqslant n, P_{i}(x)+$ $P_{j}(x)$ has exactly one real root. Find the maximum value of $n$. | 3 | 119 | 1 |
math | 9. (3 points) The product of two numbers, A and B, is 1.6. If A is multiplied by 5 and B is also multiplied by 5, then the product of A and B is $\qquad$ . | 40 | 51 | 2 |
math | ## Task B-4.1.
Solve the equation
$$
\binom{n}{n-2}+2\binom{n-1}{n-3}=\binom{n+1}{n-1}+\binom{n-2}{n-3}
$$ | 4 | 60 | 1 |
math | 11. Find all values of $b$ for which the equation
$$
a^{2-2 x^{2}}+(b+4) a^{1-x^{2}}+3 b+4=0
$$
has no solutions for any $a>1$. | [-\frac{4}{3};+\infty) | 58 | 12 |
math | 83. Given two concentric circles with radii $r$ and $R (r < R)$. Through a point $P$ on the smaller circle, a line is drawn intersecting the larger circle at points $B$ and $C$. The perpendicular to $BC$ at point $P$ intersects the smaller circle at point $A$. Find $|PA|^2 + |PB|^2 + |PC|^2$. | 2(R^2+r^2) | 89 | 8 |
math | Let $n$ be a positive integer, with prime factorization $$n = p_1^{e_1}p_2^{e_2} \cdots p_r^{e_r}$$ for distinct primes $p_1, \ldots, p_r$ and $e_i$ positive integers. Define $$rad(n) = p_1p_2\cdots p_r,$$ the product of all distinct prime factors of $n$.
Find all polynomials $P(x)$ with rational coefficients such th... | P(x) = \frac{x}{b} | 128 | 11 |
math | Example 6 Find three prime numbers such that their product is 5 times their sum.
The above text has been translated into English, retaining the original text's line breaks and format. | 2,5,7 | 38 | 5 |
math | 8. Let the general term formula of the sequence $\left\{a_{n}\right\}$ be $a_{n}=n^{3}-n$ $\left(n \in \mathbf{Z}_{+}\right)$, and the terms in this sequence whose unit digit is 0, arranged in ascending order, form the sequence $\left\{b_{n}\right\}$. Then the remainder when $b_{2} 018$ is divided by 7 is $\qquad$ . | 4 | 107 | 1 |
math | Let $b$ be a real number randomly sepected from the interval $[-17,17]$. Then, $m$ and $n$ are two relatively prime positive integers such that $m/n$ is the probability that the equation \[x^4+25b^2=(4b^2-10b)x^2\] has $\textit{at least}$ two distinct real solutions. Find the value of $m+n$. | 63 | 98 | 2 |
math | Find the greatest value of $M$ $\in \mathbb{R}$ such that the following inequality is true $\forall$ $x, y, z$ $\in \mathbb{R}$
$x^4+y^4+z^4+xyz(x+y+z)\geq M(xy+yz+zx)^2$. | \frac{2}{3} | 70 | 7 |
math | 7. Given in $\triangle A B C$, $a=2 b, \cos B=\frac{2 \sqrt{2}}{3}$, then $\sin \frac{A-B}{2}+\sin \frac{C}{2}=$ $\qquad$ | \frac{\sqrt{10}}{3} | 57 | 11 |
math | Task 1. In an urn, there are 8 balls, 5 of which are white. A ball is drawn from the urn 8 times, and after recording its color, it is returned to the urn. Find the probability that the white color was recorded 3 times. | 0.101 | 58 | 5 |
math | Example 6 Let $a_{1}, a_{2}, \cdots, a_{n}(n \geqslant 2)$ be $n$ distinct real numbers, $a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}=S$. Try to find $\max \min _{1 \leqslant i<j \leqslant n}\left(a_{i}-a_{j}\right)^{2}$. | \frac{12 S}{n\left(n^{2}-1\right)} | 105 | 18 |
math | 2. Find all integer solutions of the equation
$$
x+\frac{1}{y+\frac{1}{z}}=\frac{7}{3}
$$ | (2;2;1);(2;4;-1);(3;-1;-2);(3;-2;2);(1;1;-4) | 34 | 35 |
math | 3.15. Find all positive solutions ($x_{1}>0, x_{2}>0$, $x_{3}>0, x_{4}>0, x_{5}>0$) of the system of equations
$$
\left\{\begin{array}{l}
x_{1}+x_{2}=x_{3}^{2} \\
x_{2}+x_{3}=x_{4}^{2} \\
x_{3}+x_{4}=x_{5}^{2} \\
x_{4}+x_{5}=x_{1}^{2} \\
x_{5}+x_{1}=x_{2}^{2}
\end{array}\right.
$$
## 3.4. The number of solutions of t... | x_{\}=x_{\max}=2 | 168 | 10 |
math | 7. The function
$$
f(x)=\frac{\sin x-1}{\sqrt{3-2 \cos x-2 \sin x}}(0 \leqslant x \leqslant 2 \pi)
$$
has the range . $\qquad$ | [-1,0] | 61 | 5 |
math | 7. Given real numbers $a, b, c$ satisfy $\left|a x^{2}+b x+c\right|$ has a maximum value of 1 on $x \in[-1,1]$. Then the maximum possible value of $\left|c x^{2}+b x+a\right|$ on $x \in[-1,1]$ is $\qquad$ | 2 | 82 | 1 |
math | 9. Let $m$ be a positive integer, and let $T$ denote the set of all subsets of $\{1,2, \ldots, m\}$. Call a subset $S$ of $T \delta$-good if for all $s_{1}, s_{2} \in S, s_{1} \neq s_{2},\left|\Delta\left(s_{1}, s_{2}\right)\right| \geq \delta m$, where $\Delta$ denotes symmetric difference (the symmetric difference of... | 2048 | 172 | 4 |
math | IMO 1960 Problem A1 Determine all 3 digit numbers N which are divisible by 11 and where N/11 is equal to the sum of the squares of the digits of N. | 550,803 | 43 | 7 |
math | 2. Find a natural number $N$, such that it is divisible by 5 and 49, and including 1 and $N$, it has a total of 10 divisors. | 5 \times 7^{4} | 41 | 8 |
math | ## Task Condition
Write the equation of the plane passing through point $A$ and perpendicular to vector $\overrightarrow{B C}$.
$A(1; -1; 8)$
$B(-4; -3; 10)$
$C(-1; -1; 7)$ | 3x+2y-3z+23=0 | 64 | 13 |
math | 13. (10 points) A class of 30 students goes to the movies, and their student IDs are $1, 2, \cdots, 30$; the movie tickets they hold are exactly the 1st, 2nd, ..., 30th seats in a row. Now, the movie tickets are to be distributed to these students according to the following requirements: for any two students, A and B, ... | 48 | 140 | 2 |
math | 3. Solve the system of equations:
$$
\left\{\begin{array}{l}
3 x y-5 y z-x z=3 y \\
x y+y z=-y \\
-5 x y+4 y z+x z=-4 y
\end{array}\right.
$$ | (2,-\frac{1}{3},-3),(x,0,0),(0,0,z) | 62 | 24 |
math | 1. Andrei, Boris, and Valentin participated in a 1 km race. (We assume that each of them ran at a constant speed). Andrei was 100 m ahead of Boris at the finish line. And Boris was 60 m ahead of Valentin at the finish line. What was the distance between Andrei and Valentin at the moment Andrei finished? | 154\mathrm{} | 80 | 6 |
math | Determine the sum of the real numbers $x$ for which $\frac{2 x}{x^{2}+5 x+3}+\frac{3 x}{x^{2}+x+3}=1$. | -4 | 46 | 2 |
math | 14.5. 14 ** A storybook contains 30 stories, with lengths of $1, 2, \cdots, 30$ pages respectively. Starting from the first page of the book, stories are published, and each subsequent story begins on a new page. Question: What is the maximum number of stories that can start on an odd-numbered page? | 23 | 80 | 2 |
math | On a clean board, we wrote a three-digit natural number with yellow chalk, composed of mutually different non-zero digits. Then we wrote on the board with white chalk all other three-digit numbers that can be obtained by changing the order of the digits of the yellow number. The arithmetic mean of all the numbers on th... | 361 | 111 | 3 |
math | 4. Let the foci of an ellipse be $F_{1}(-1,0)$ and $F_{2}(1,0)$ with eccentricity $e$, and let the parabola with vertex at $F_{1}$ and focus at $F_{2}$ intersect the ellipse at point $P$. If $\frac{\left|P F_{1}\right|}{\left|P F_{2}\right|}=e$, then the value of $e$ is . $\qquad$ | \frac{\sqrt{3}}{3} | 106 | 10 |
math | 2. Solve the equation $\left(\frac{x}{400}\right)^{\log _{5}\left(\frac{x}{8}\right)}=\frac{1024}{x^{3}}$. | \frac{8}{5},16 | 45 | 9 |
math | $\triangle ABC$, $\angle A=23^{\circ},\angle B=46^{\circ}$. Let $\Gamma$ be a circle with center $C$, radius $AC$. Let the external angle bisector of $\angle B$ intersects $\Gamma$ at $M,N$. Find $\angle MAN$. | 60^\circ | 66 | 4 |
math | Dudeney, Amusements in Mathematics Problem 21 I paid a man a shilling for some apples, but they were so small that I made him throw in two extra apples. I find that made them cost just a penny a dozen less than the first price he asked. How many apples did I get for my shilling? | 18 | 69 | 2 |
math | Example 5.11. Estimate the error made when replacing the sum of the series
$$
1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots+(-1)^{n-1} \frac{1}{n}+\ldots
$$
with the sum of its first four terms. | 0.2 | 77 | 3 |
math | 6.5 Arrange 5 different red beads and 3 different blue beads around a circular plate. How many ways are there to arrange them? If the blue beads are not adjacent, how many ways are there? What if the blue beads are together? | 7!,1440,5!3! | 51 | 11 |
math | 1. Let $a, b$ be two positive integers, their least common multiple is 232848, then the number of such ordered pairs of positive integers $(a, b)$ is $\qquad$
$\qquad$ groups. | 945 | 52 | 3 |
math | Example 1 Let $f: \mathbf{R} \rightarrow \mathbf{R}$ satisfy
$$
f\left(x^{3}\right)+f\left(y^{3}\right)=(x+y) f\left(x^{2}\right)+f\left(y^{2}\right)-f(x y) \text {. }
$$
Find the analytical expression of the function $f(x)$. | f(x)=0 | 86 | 4 |
math | On a board, the numbers from 1 to 2009 are written. A couple of them are erased and instead of them, on the board is written the remainder of the sum of the erased numbers divided by 13. After a couple of repetition of this erasing, only 3 numbers are left, of which two are 9 and 999. Find the third number. | 8 | 83 | 1 |
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