task_type stringclasses 4
values | problem stringlengths 21 5.23k | answer stringlengths 1 8.29k | problem_tokens int64 11 1.16k | answer_tokens int64 1 2.04k |
|---|---|---|---|---|
math | 2.066. $\frac{\sqrt{x^{3}}+\sqrt{x y^{2}}-\sqrt{x^{2} y}-\sqrt{y^{3}}}{\sqrt[4]{y^{5}}+\sqrt[4]{x^{4} y}-\sqrt[4]{x y^{4}}-\sqrt[4]{x^{5}}}$.
2.066. $\frac{\sqrt{x^{3}}+\sqrt{x y^{2}}-\sqrt{x^{2} y}-\sqrt{y^{3}}}{\sqrt[4]{y^{5}}+\sqrt[4]{x^{4} y}-\sqrt[4]{x y^{4}}-\sqrt[4]{x^{5}}}$.
The translation is the same as the... | -(\sqrt[4]{x}+\sqrt[4]{y}) | 351 | 15 |
math | 4. What is the size of the largest rectangle that can be drawn inside of a 3-4-5 right triangle with one of the rectangle's sides along one of the legs of the triangle? | 3 | 41 | 1 |
math | Example 3. Five people stand in a row, requiring that A does not stand at the head, B does not stand at the end, and C and D do not stand together. How many ways are there to arrange them? | 50 | 47 | 2 |
math | 8・33 The sequence of real numbers $a_{1}, a_{2}, \cdots, a_{n}, \cdots$ is defined by the following equation:
$$a_{n+1}=2^{n}-3 a_{n}, n=0,1,2, \cdots$$
(1) Find the expression for $a_{n}$ in terms of $a_{0}$ and $n$.
(2) Find $a_{0}$ such that for any positive integer $n, a_{n+1}>a_{n}$. | a_{0}=\frac{1}{5} | 120 | 11 |
math | 3. Determine for which real number $p$ the equations
$$
\begin{array}{r}
x^{3}+x^{2}-36 x-p=0 \\
x^{3}-2 x^{2}-p x+2 p=0
\end{array}
$$
have a common root.
The school-based - closed part of the first round in category A takes place
## on Tuesday, December 3, 2002
so that it starts in the morning and the contestant... | 0,-60,36 | 162 | 7 |
math | Find the largest real $ T$ such that for each non-negative real numbers $ a,b,c,d,e$ such that $ a\plus{}b\equal{}c\plus{}d\plus{}e$: \[ \sqrt{a^{2}\plus{}b^{2}\plus{}c^{2}\plus{}d^{2}\plus{}e^{2}}\geq T(\sqrt a\plus{}\sqrt b\plus{}\sqrt c\plus{}\sqrt d\plus{}\sqrt e)^{2}\] | \frac{\sqrt{30}}{30 + 12\sqrt{6}} | 108 | 20 |
math |
Problem 8'.3. Find all natural numbers $n$ such that there exists an integer number $x$ for which $499\left(1997^{n}+1\right)=x^{2}+x$.
| 1 | 53 | 1 |
math | Given a digits {$0,1,2,...,9$} . Find the number of numbers of 6 digits which cantain $7$ or $7$'s digit and they is permulated(For example 137456 and 314756 is one numbers). | 2002 | 64 | 4 |
math | Which is larger: $1997^{1999}$ or $1999^{1997}$? | 1997^{1999}>1999^{1997} | 28 | 20 |
math | Given $N$ positive integers such that the greatest common divisors of all nonempty subsets of them are pairwise distinct. What is the smallest number of prime factors of the product of all $N$ numbers?
[i] Proposed by Aleksandr Golovanov [/i] | N | 54 | 2 |
math | 2. Given numbers $x, y, z \in [0, \pi]$. Find the maximum value of the expression
$$
A=\sin (x-y)+\sin (y-z)+\sin (z-x)
$$ | 2 | 49 | 1 |
math | 1003. A mathematician was asked: "How old are you?" He answered in a complicated way: "The number of my years has an odd number of distinct divisors, and the number of these divisors also has an odd number of distinct divisors." So how old is he? | 36or100 | 62 | 6 |
math | 232. Power series. Expand $\frac{1}{(1+x)\left(1+x^{2}\right)\left(1+x^{4}\right)\left(1+x^{8}\right)}$ into a power series. | 1-x+x^{16}-x^{17}+x^{32}-x^{33}+\ldots | 50 | 26 |
math | 12. (12 points) Person A and Person B start from points A and B respectively at the same time, moving towards each other at a constant speed. When A and B meet at point C, Person C starts from B, moving at a constant speed towards A. When A and C meet at point D, A immediately turns around and reduces their speed to 80... | 5265 | 136 | 4 |
math | Find the smallest positive real number $k$ such that the following inequality holds $$\left|z_{1}+\ldots+z_{n}\right| \geqslant \frac{1}{k}\big(\left|z_{1}\right|+\ldots+\left|z_{n}\right|\big) .$$ for every positive integer $n \geqslant 2$ and every choice $z_{1}, \ldots, z_{n}$ of complex numbers with non-negative re... | \sqrt{2} | 150 | 6 |
math | 15. (15 points) There is a road between places $A$ and $B$. Two cars, Car A and Car B, start from $A$ and $B$ respectively at the same time. Car A's speed is 50 kilometers/hour. After 1 hour, the two cars meet for the first time. Then the two cars continue to drive, each reaching $B$ and $A$ respectively, and immediate... | 110 | 137 | 3 |
math | Find the greatest possible value of $ sin(cos x) \plus{} cos(sin x)$ and determine all real
numbers x, for which this value is achieved. | \sin(1) + 1 | 33 | 9 |
math | David has a collection of 40 rocks, 30 stones, 20 minerals and 10 gemstones. An operation consists of removing three objects, no two of the same type. What is the maximum number of operations he can possibly perform?
[i]Ray Li[/i] | 30 | 60 | 2 |
math | 11.5 Find the smallest positive integer that can be expressed as the sum of 2002 positive integers with equal sums of their digits, and also as the sum of 2003 positive integers with equal sums of their digits. | 10010 | 51 | 5 |
math | 3. Given that $x, y$ are integers, and
$$
y=\frac{4012}{\sqrt{x+2005}-\sqrt{x-2007}} \text {. }
$$
Then the maximum value of $y$ is $\qquad$ . | 2006 | 63 | 4 |
math | 7. In $\triangle A B C$, the sides opposite to $\angle A 、 \angle B 、 \angle C$ are $a 、 b 、 c$ respectively. If $\frac{\cos B}{\cos C}=-\frac{b}{2 a+c}$. Then $\angle B=$ | \frac{2\pi}{3} | 66 | 9 |
math | 2. Solve in the set of real numbers the equation $[x+2015]-\left[\frac{5 x-2015}{2}\right]=\frac{x}{2}+2015$, where $[a]$ represents the integer part of the real number a. | 504 | 64 | 3 |
math | 10.2. Find all values of parameters $a, b, c$, for which the system of equations $\left\{\begin{array}{l}a x+b y=c \\ b x+c y=a \\ c x+a y=b,\end{array}\right\}$
has at least one negative solution (when $x, y<0$). | +b+=0 | 75 | 3 |
math | A candy company makes $5$ colors of jellybeans, which come in equal proportions. If I grab a random sample of $5$ jellybeans, what is the probability that I get exactly $2$ distinct colors? | \frac{12}{125} | 45 | 10 |
math | 1. If for all $x$ such that $|x| \leqslant 1$, $t+1>(t^2-4)x$ always holds, then the range of values for $t$ is $\qquad$ | (\frac{\sqrt{13}-1}{2},\frac{\sqrt{21}+1}{2}) | 51 | 25 |
math | 9.182. $\log _{2 x}\left(x^{2}-5 x+6\right)<1$. | x\in(0;\frac{1}{2})\cup(1;2)\cup(3;6) | 27 | 25 |
math | Example 1. Two students took four tests, their average scores were different, but both were below 90 and were integers. They then took a fifth test, after which their average scores both increased to 90. What were the scores of the two students on the fifth test? | 98 \text{ and } 94 | 59 | 10 |
math | 13. Solve the equation $\frac{1}{\cos x \cos 2 x}+\frac{1}{\cos 2 x \cos 3 x}+\cdots+\frac{1}{\cos 100 x \cos 101 x}=0$. | \frac{}{25}\pi,\in{Z}, | 61 | 13 |
math | 6. Given $\sin (x+\sin x)=\cos (x-\cos x)$, where $x \in[0, \pi]$. Then $x=$ | \frac{\pi}{4} | 36 | 7 |
math | 1. Given that $a$ and $b$ are distinct real numbers, and
$$
(\sqrt[3]{a}+\sqrt[3]{b})^{3}=a^{2} b^{2} \text {. }
$$
then $(3 a+1)(3 b+1)-3 a^{2} b^{2}=$ | 1 | 73 | 1 |
math | 21 In a regular tetrahedron wooden block $ABCD$ with edge length 2, there is a point $P (AP<1)$ on the edge $AB$. A cross-section perpendicular to the edge $AB$ is to be sawed through point $P$. When the sawing stops at a certain position, it is measured that the saw cut $PM=1$ on the face $ABD$. The saw cut $PN=\frac{... | 1 | 109 | 1 |
math | 10. If $x, y$ are real numbers, and $x^{2}+x y+y^{2}=3$, then the maximum and minimum values of $x^{2}-x y+y^{2}$ are $\qquad$. | 9,1 | 51 | 3 |
math | 6. If $3 \sin ^{3} x+\cos ^{3} x=3$, then the value of $\sin ^{2018} x+\cos ^{2018} x$ is $\qquad$ | 1 | 52 | 1 |
math | 2.135. $\frac{x^{3}-a^{-2 / 3} \cdot b^{-1}\left(a^{2}+b^{2}\right) x+b^{1 / 2}}{b^{3 / 2} \cdot x^{2}} ; x=a^{2 / 3} b^{-1 / 2}$. | 0 | 76 | 1 |
math | ## Task 2 - 320522
In a cabinet, there are 11 checkered, 7 lined, and 12 unlined notepads, and no others. It is too dark to distinguish the notepads, and they are unordered. Someone wants to take out a number of notepads and only then determine how many notepads of each type they have taken.
a) What is the smallest n... | 13 | 183 | 2 |
math | 2. (3 points) There is a division equation, the sum of the dividend and divisor is 136, and the quotient is 7, then the divisor is | 17 | 36 | 2 |
math | 4. Each bag contains 5 small balls that are identical in shape, size, and material, numbered $1 \sim 5$. Without seeing the balls, one ball is randomly drawn from the bag, the number is noted, and the ball is then returned. Another ball is then drawn from the bag. The probability that the number on the ball drawn the s... | \frac{2}{5} | 92 | 7 |
math | Let $\{a_{n}\}$ be a sequence which satisfy
$a_{1}=5$ and $a_{n=}\sqrt[n]{a_{n-1}^{n-1}+2^{n-1}+2.3^{n-1}} \qquad \forall n\geq2$
[b](a)[/b] Find the general fomular for $a_{n}$
[b](b)[/b] Prove that $\{a_{n}\}$ is decreasing sequences | a_n = \sqrt[n]{2^n + 3^n} | 108 | 14 |
math | 2. Solve the inequality $\log _{x}\left(36-60 x+25 x^{2}\right)<0$. | x\in(0;1)\cup(1;6/5)\cup(6/5;7/5) | 30 | 26 |
math | 26.2. Find the smallest natural number divisible by 11, which after being increased by 1, would be divisible by 13.
$$
\text { (7-8 grades) }
$$ | 77 | 45 | 2 |
math | Find all strictly increasing functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that $f(2)=2$ and for all $m, n$ that are coprime, $f(m n)=f(m) f(n)$. | f(k)=k | 57 | 4 |
math | Solve in positive integers the following equation $$\left [\sqrt{1}\right]+\left [\sqrt{2}\right]+\left [\sqrt{3}\right]+\ldots+\left [\sqrt{x^2-2}\right]+\left [\sqrt{x^2-1}\right]=125,$$ where $[a]$ is the integer part of the real number $a$. | 6 | 80 | 1 |
math | Example 1. Find the equation of the tangent line at point $P\left(x_{1}, y_{1}\right)$ on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$. | \frac{x_{1} x}{a}+\frac{y_{1} y}{b}=1 | 56 | 22 |
math | 10.1. Find at least one four-digit number that has the following property: if the sum of all its digits is multiplied by the product of all its digits, the result is 3990. (I. Rubanov) | 1567 | 51 | 4 |
math | 1. If the function
$$
f(x)=3 \cos \left(\omega x+\frac{\pi}{6}\right)-\sin \left(\omega x-\frac{\pi}{3}\right)(\omega>0)
$$
has the smallest positive period of $\pi$, then the maximum value of $f(x)$ in the interval $\left[0, \frac{\pi}{2}\right]$ is $\qquad$ | 2\sqrt{3} | 91 | 6 |
math | 14. How many natural numbers less than 100 are there that: a) are divisible by 2 but not by 3; b) are divisible by 2 or by 3; c) are not divisible by 2 or by 3? | 33 | 55 | 2 |
math | ## Task 4 - 261244
Determine the smallest positive integer $a$ for which $(a+1)^{5}-a^{5}-1$ is divisible by 18305. | 60 | 48 | 2 |
math | Given a prime number $p > 2$. Find the least $n\in Z_+$, for which every set of $n$ perfect squares not divisible by $p$ contains nonempty subset with product of all it's elements equal to $1\ (\text{mod}\ p)$ | \frac{p-1}{2} | 60 | 9 |
math | 3. Let $x$ and $y$ be two distinct non-negative integers, and satisfy $x y + 2x + y = 13$. Then the minimum value of $x + y$ is $\qquad$ | 5 | 48 | 1 |
math | Find all non empty subset $ S$ of $ \mathbb{N}: \equal{} \{0,1,2,\ldots\}$ such that $ 0 \in S$ and exist two function $ h(\cdot): S \times S \to S$ and $ k(\cdot): S \to S$ which respect the following rules:
i) $ k(x) \equal{} h(0,x)$ for all $ x \in S$
ii) $ k(0) \equal{} 0$
iii) $ h(k(x_1),x_2) \equal{} x_1$ for ... | \{0\} | 157 | 6 |
math | 23. Form the equation of the line passing through two points $A\left(x_{1} ; y_{1}\right)$ and $B\left(x_{2} ; y_{2}\right)$ | (x-x_{1})(y_{2}-y_{1})=(y-y_{1})(x_{2}-x_{1}) | 44 | 27 |
math | In an exam every question is solved by exactly four students, every pair of questions is solved by exactly one student, and none of the students solved all of the questions. Find the maximum possible number of questions in this exam. | 13 | 45 | 2 |
math | 1. Determine all prime numbers of the form $\frac{11 \ldots 1}{11}$, where $n$ is a natural number. | 101 | 33 | 3 |
math | Example 2.4.4 Divide a circle into 10 sectors, and color each sector with one of the four colors: red, yellow, blue, and green. Each sector is to be colored with one color, and adjacent sectors must be colored with different colors. How many ways are there to color the sectors? If all four colors must be used, how many... | 54960 | 84 | 5 |
math | Let $a$, $b$, $c$, $d$, $e$, $f$ and $g$ be seven distinct positive integers not bigger than $7$. Find all primes which can be expressed as $abcd+efg$ | 179 | 48 | 3 |
math | How many five-digit numbers are there that contain at least one digit 3 and are multiples of 3?
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | 12504 | 47 | 5 |
math | 10. Let $f$ be a function from non-negative real numbers to non-negative real numbers, satisfying
$$
f\left(a^{3}\right)+f\left(b^{3}\right)+f\left(c^{3}\right)=3 f(a) f(b) f(c) \text {, (1) }
$$
and $f(1) \neq 1$, where $a, b, c$ are non-negative real numbers. Then $f(2019)=$ $\qquad$ | 0 | 112 | 1 |
math | 2. The function $f(x)=x+\frac{1}{(x+1)^{3}}+1(x>0)$. Then, the value of $x$ when the function reaches its minimum is
保留了源文本的换行和格式。 | \sqrt[4]{3}-1 | 55 | 8 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 1} \frac{3-\sqrt{10-x}}{\sin 3 \pi x}$ | -\frac{1}{18\pi} | 40 | 10 |
math | The third question: An exam consists of $m$ questions, with $n$ students participating, where $m, n \geqslant 2$ are given integers. The scoring rule for each question is: if exactly $x$ students fail to answer the question correctly, then each student who answers the question correctly gets $x$ points, and those who f... | m(n-1) | 164 | 5 |
math | ## Task A-3.1.
If $2 \sin x - 3 \cos y = a$ and $2 \cos x + 3 \sin y = b$, what is $\sin (x - y)$? | \sin(x-y)=\frac{13-^{2}-b^{2}}{12} | 48 | 22 |
math | Let $n$ be a positive integer. For each $4n$-tuple of nonnegative real numbers $a_1,\ldots,a_{2n}$, $b_1,\ldots,b_{2n}$ that satisfy $\sum_{i=1}^{2n}a_i=\sum_{j=1}^{2n}b_j=n$, define the sets
\[A:=\left\{\sum_{j=1}^{2n}\frac{a_ib_j}{a_ib_j+1}:i\in\{1,\ldots,2n\} \textup{ s.t. }\sum_{j=1}^{2n}\frac{a_ib_j}{a_ib_j+1}\neq... | \frac{n}{2} | 318 | 6 |
math | 7. Solve the equation $\sqrt{x}\left(x^{2}-3 x-4\right)^{3}+\sqrt{4-x}\left(x^{2}-8 x\right)^{5}=0$. | 0;4 | 45 | 3 |
math | Boats are sailing on the sea at a speed of $v$. Exactly opposite them, an airplane is flying at a height of $h$ at a speed of $V$. At what distance from the boat should the pilot drop the aid package so that it lands precisely in the boat? What would be the distance in question if the airplane is flying behind the boat... | =(V+v)\sqrt{\frac{2}{}},\=(V-v)\sqrt{\frac{2}{}},\=V\sqrt{\frac{2}{}} | 85 | 36 |
math | 1. A thread is strung with 75 blue, 75 red, and 75 green beads. We will call a sequence of five consecutive beads good if it contains exactly 3 green beads and one each of red and blue. What is the maximum number of good quintets that can be on this thread? | 123 | 67 | 3 |
math | 6.2. There are 7 safes and 7 codes for them, but it is unknown which code belongs to which safe. What is the minimum number of attempts required to guarantee matching the codes to the safes? | 21 | 46 | 2 |
math | 5 Mathematical Induction
Mathematical induction can be used to solve function problems on the set of natural numbers.
Example 6 Find all functions $f: Z_{+} \rightarrow Z_{+}$, such that for any positive integer $n$, we have
$$
f(f(f(n)))+f(f(n))+f(n)=3 n \text {. }
$$
(2008, Dutch National Team Selection Exam) | f(n)=n | 90 | 4 |
math | Example 11 Given $0<a<1$, and
$$
\left[a+\frac{1}{30}\right]+\left[a+\frac{2}{30}\right]+\cdots+\left[a+\frac{29}{30}\right]=18 \text {. }
$$
Then $[10 a$ ] equals $\qquad$
$(2009$, Beijing Mathematical Competition (Grade 8)) | 6 | 91 | 1 |
math | ## Task 31/70
Determine all common solutions of the two equations
$$
\begin{array}{r}
3 x^{4}+13 x^{3}+20 x^{2}+17 x+7=0 \\
3 x^{4}+x^{3}-8 x^{2}+11 x-7=0
\end{array}
$$
without using an approximation method! | -\frac{7}{3} | 93 | 7 |
math | Three. (This question is worth 16 points) Given the set $\{1,2,3,4,5, 6,7,8,9,10\}$. Find the number of subsets of this set that have the following property: each subset contains at least 2 elements, and the absolute difference between any two elements in each subset is greater than 1.
| 133 | 81 | 3 |
math | 1. Let $x, y$ and $a$ be real numbers such that $x+y=a-1$ and $x y=a^{2}-7 a+12$. For which $a$ does the expression $x^{2}+y^{2}$ attain its maximum possible value? What are $x$ and $y$ then? | 6,2,3or3,2 | 73 | 9 |
math | 40. Find all positive integers $n$ and non-negative integers $x_{1}, x_{2}, \cdots, x_{n}$, such that $\sum_{i=1}^{n} x_{i}^{2}=1+\frac{4}{4 n+1}\left(\sum_{i=1}^{n} x_{i}\right)^{2}$.
(2002, Taiwan Mathematical Olympiad) | (1) n=6, (x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6})=(0,1,1,1,1,1), (1,0,1,1,1,1), (1,1,0,1,1,1), (1,1,1,0,1,1), (1,1,1,1,0,1), ( | 94 | 98 |
math | Two players play a game on a pile of $n$ beans. On each player's turn, they may take exactly $1$, $4$, or $7$ beans from the pile. One player goes first, and then the players alternate until somebody wins. A player wins when they take the last bean from the pile. For how many $n$ between $2014$ and $2050$ (inclusive) ... | 14 | 97 | 2 |
math | 4. 144 If $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$ satisfy the following system of equations
$$
\left\{\begin{array}{l}
2 x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=6, \\
x_{1}+2 x_{2}+x_{3}+x_{4}+x_{5}=12, \\
x_{1}+x_{2}+2 x_{3}+x_{4}+x_{5}=24, \\
x_{1}+x_{2}+x_{3}+2 x_{4}+x_{5}=48, \\
x_{1}+x_{2}+x_{3}+x_{4}+2 x_{5}=96 .
\end{array}\r... | 181 | 216 | 3 |
math | Example 1.15 Let $N$ denote the number of different ways to distribute $r$ identical items to $n$ people, find $N$.
| \binom{n+r-1}{r} | 34 | 10 |
math | $4 \cdot 261$ Find all integers $k$, such that there exists an integer $x$, satisfying the equation
$$
\sqrt{39-6 \sqrt{12}}+\sqrt{k x(k x+\sqrt{12})+3}=2 k .
$$ | k=3ork=6 | 62 | 6 |
math | 2. (5 points) The product of $1 \times 2 \times 3 \times 4 \times \cdots \times 2014$ ends with $\qquad$ zeros. | 501 | 44 | 3 |
math | 4・241 Given $2 \lg (x-2 y)=\lg x+\lg y$, try to find $x: y$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | \frac{x}{y}=4 | 57 | 7 |
math | Find all functions $ f : Z\rightarrow Z$ for which we have $ f (0) \equal{} 1$ and $ f ( f (n)) \equal{} f ( f (n\plus{}2)\plus{}2) \equal{} n$, for every natural number $ n$. | f(n) = 1 - n | 63 | 9 |
math | Find all surjective functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $m,n\in \mathbb{N}$: \[m \vert n \Longleftrightarrow f(m) \vert f(n).\] | f(n) = g(p_1)^{e_1} g(p_2)^{e_2} \cdots g(p_k)^{e_k} | 55 | 36 |
math | 4. Find the sum: $S_{n}=2^{3}+5^{3}+8^{3}+\cdots+(3 n-1)^{3}$. | S_{n}=\frac{n}{4}(27n^{3}+18n^{2}-9n-4) | 38 | 28 |
math | Find all triples of positive integers $(x, y, z)$ satisfying
$$
2(x+y+z+2 x y z)^{2}=(2 x y+2 y z+2 z x+1)^{2}+2023 .
$$ | (3,3,2) | 55 | 7 |
math | 15. Given that $\sqrt[3]{17-\frac{27}{4} \sqrt{6}}$ and $\sqrt[3]{17+\frac{27}{4} \sqrt{6}}$ are the roots of the equation
$$
x^{2}-a x+b=0 \text {, }
$$
find the value of $a b$. | 10 | 81 | 2 |
math | 5. For which natural numbers $n \geq 3$ can a set of $n$ identical numbers be obtained from the set of numbers $1,2, \ldots, n$ in a finite number of steps, if in one step you can choose two arbitrary numbers and increase each by the same arbitrary natural number? | Foralln\neq4k+2 | 68 | 10 |
math | Example 2 Find four consecutive integers, each of which is divisible by $2^{2}, 3^{2}, 5^{2}$, and $7^{2}$ respectively. | 29348,29349,29350,29351 | 38 | 23 |
math | Let $M, \alpha, \beta \in \mathbb{R} $ with $M > 0$ and $\alpha, \beta \in (0,1)$. If $R>1$ is a real number, we say that a sequence of positive real numbers $\{ C_n \}_{n\geq 0}$ is $R$-[i]inoceronte[/i] if $ \sum_{i=1}^n R^{n-i}C_i \leq R^n \cdot M$ for all $n \geq 1$. Determine the smallest real $R>1$ for which exis... | R = \beta^{-\frac{1}{\alpha}} | 191 | 14 |
math | What is the difference between the largest possible three-digit positive integer with no repeated digits and the smallest possible three-digit positive integer with no repeated digits? | 885 | 29 | 3 |
math | Example 5 Let the three sides of a right-angled triangle be $a$, $b$, and $c$, all positive integers, and the hypotenuse $c$ satisfies $87 \leqslant c \leqslant 91$. Find the lengths of the three sides of such a right-angled triangle. | (63,60,87),(39,80,89),(54,72,90),(35,84,91) | 70 | 37 |
math | Find all function $ f: \mathbb{R}^{+} \to \mathbb{R}^{+}$ such that for every three real positive number $x,y,z$ :
$$ x+f(y) , f(f(y)) + z , f(f(z))+f(x) $$
are length of three sides of a triangle and for every postive number $p$ , there is a triangle with these sides and perimeter $p$.
[i]Proposed by Amirhossein Zo... | f(x) = x | 108 | 6 |
math | 10. In $\triangle A B C, A B=9, B C=8$ and $A C=7$. The bisector of $\angle A$ meets $B C$ at $D$. The circle passing through $A$ and touching $B C$ at $D$ cuts $A B$ and $A C$ at $M$ and $N$ respectively. Find $M N$.
在 $\triangle A B C$ 中, $A B=9, B C=8$ 及 $A C=7 \circ \angle A$ 的角平分線交 $B C$ 於 $D$ 。穿過 $A$且與 $B C$ 相切於 ... | 6 | 186 | 1 |
math | 8-(22nd $I M O$ Problem) Let $P$ be any point inside triangle $A B C$, and let the distances from $P$ to the three sides $B C, C A, A B$ be $d_{1}, d_{2}, d_{3}$ respectively, with $B C=a, C A=b, A B=c$. Find the minimum value of $u=\frac{a}{d_{1}}+\frac{b}{d_{2}}+\frac{c}{d_{3}}$. | \frac{(a+b+c)^{2}}{2 S} | 113 | 14 |
math | 13.031. A group of students went on a hike through the Moscow region during their vacation. The first 30 km they walked, $20 \%$ of the remaining part of the route they traveled by raft along the river, and then they walked again, covering a distance 1.5 times greater than the distance they traveled by raft. The remain... | 150 | 123 | 3 |
math | Let $n$ be a given positive integer. Find the smallest positive integer $u_{n}$, satisfying: for every positive integer $d$, in any $u_{n}$ consecutive positive odd numbers, the number of numbers divisible by $d$ is not less than the number of numbers in $1,3,5, \cdots, 2 n-1$ that are divisible by $d$. | 2n-1 | 84 | 4 |
math | 6.005. $\frac{1}{x(x+2)}-\frac{1}{(x+1)^{2}}=\frac{1}{12}$. | x_{1,2}\in\varnothing,x_{3}=-3,x_{4}=1 | 38 | 21 |
math | 5. In the expansion of $(a+b)^{n}$, there are $n+1$ different terms. Then, the expansion of $(a+b+c+d)^{21}$ has different terms. | 2024 | 43 | 4 |
math | 12. Suppose that $a, b$ and $c$ are real numbers greater than 1 . Find the value of
$$
\frac{1}{1+\log _{a^{2} b}\left(\frac{c}{a}\right)}+\frac{1}{1+\log _{b^{2} c}\left(\frac{a}{b}\right)}+\frac{1}{1+\log _{c^{2} a}\left(\frac{b}{c}\right)} .
$$ | 3 | 109 | 1 |
math | Exercise 12. For all integers $n \geqslant 1$, determine all $n$-tuples of real numbers $\left(x_{1}, \ldots, x_{n}\right)$ such that
$$
\sqrt{x_{1}-1^{2}}+2 \sqrt{x_{2}-2^{2}}+\ldots+n \sqrt{x_{n}-n^{2}}=\frac{1}{2}\left(x_{1}+\ldots+x_{n}\right)
$$ | x_{i}=2i^{2} | 108 | 9 |
math | 6. If the lengths of all edges of the regular quadrilateral pyramid $P-ABCD$ are equal, and $M$ is the midpoint of edge $AB$, then the cosine value of the angle formed by the skew lines $BP$ and $CM$ is $\qquad$ | \frac{\sqrt{5}}{10} | 59 | 11 |
math | G8.2 From 1 to 121 , there are $b$ numbers which are not divisible by 5 nor 7 . Find $b$. | 83 | 34 | 2 |
math | Suppose the function $f(x)-f(2x)$ has derivative $5$ at $x=1$ and derivative $7$ at $x=2$. Find the derivative of $f(x)-f(4x)$ at $x=1$. | 19 | 54 | 2 |
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