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 | ## Task 18/75
Determine all triples of real solutions $(x ; y ; z)$ of the equation
$$
x^{2}+y^{4}+z^{6}+14=2 x+4 y^{2}+6 z^{3}
$$ | (1;\sqrt{2};\sqrt[3]{3})(1;-\sqrt{2};\sqrt[3]{3}) | 62 | 28 |
math | 4. Let $\tau(k)$ denote the number of all positive divisors of a natural number $k$, and suppose the number $n$ is a solution to the equation $\tau(1.6 n)=1.6 \tau(n)$. Determine the value of the ratio $\tau(0.16 n): \tau(n)$. | 1 | 71 | 1 |
math | 2.34 Let the natural number $n$ have the following property: from $1,2, \cdots, n$, any 50 different numbers chosen will have at least two numbers whose difference is 7. Find the maximum value of such an $n$.
(China Junior High School Mathematics League, 1987) | 98 | 72 | 2 |
math | 21. Six numbers are randomly selected from the integers 1 to 45 inclusive. Let $p$ be the probability that at least three of the numbers are consecutive. Find the value of $\lfloor 1000 p\rfloor$. (Note: $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$ ). | 56 | 77 | 2 |
math | 11. Suppose that $\log _{2}\left[\log _{3}\left(\log _{4} a\right)\right]-\log _{3}\left[\log _{4}\left(\log _{2} b\right)\right]=\log _{4}\left[\log _{2}\left(\log _{3} c\right)\right]=0$. Find the value of $a+b+c$. | 89 | 94 | 2 |
math | Example 7 The parabola $C_{1}: y=x^{2}+2 a x+b$ intersects the $x$-axis at $A, B$. If the vertex of $C_{1}$ is always inside the circle $C_{2}$ with segment $A B$ as its diameter. Try to find the relationship that $a, b$ should satisfy. | ^{2}-1<b<^{2} | 79 | 9 |
math | Find all real numbers $x, y, z$ such that
$$
x+y+z=3, \quad x^{2}+y^{2}+z^{2}=3, \quad x^{3}+y^{3}+z^{3}=3
$$
## - Polynomials with integer coefficients -
We now present some specific properties of polynomials with integer coefficients:
* We have already seen that if $P, Q \in \mathbb{Z}[X]$ and $\operatorname{deg}... | 1 | 746 | 1 |
math | Find whether among all quadrilaterals, whose interiors lie inside a semi-circle of radius $r$, there exist one (or more) with maximum area. If so, determine their shape and area. | \frac{3\sqrt{3}}{4} r^2 | 40 | 16 |
math | We have $2021$ colors and $2021$ chips of each color. We place the $2021^2$ chips in a row. We say that a chip $F$ is [i]bad[/i] if there is an odd number of chips that have a different color to $F$ both to the left and to the right of $F$.
(a) Determine the minimum possible number of bad chips.
(b) If we impose the ... | 1010 | 126 | 6 |
math | 1. Determine the natural number $\overline{a b c d}$ such that for its digits the following conditions hold:
$$
a \cdot d + c \cdot d = 72, \quad a \cdot c + c \cdot d = 56 \text{ and } a \cdot b \cdot c \cdot d = 0
$$ | 8046 | 76 | 4 |
math | 19.6.7 ** Let $S=\{1,2, \cdots, 2005\}$, if any $n$ pairwise coprime numbers in $S$ contain at least one prime number, find the minimum value of $n$.
| 16 | 58 | 2 |
math | 13. There are several red and white balls in a bag. If each time 2 red balls and 3 white balls are taken out, when there are no white balls left in the bag, there are still 18 red balls left; if each time 5 red balls and 3 white balls are taken out, when there are no red balls left in the bag, there are still 18 white ... | 50 | 102 | 2 |
math | 13.410 Along the sides of a right angle, towards the vertex, two spheres with radii of 2 and 3 cm are moving, with the centers of these spheres moving along the sides of the angle at unequal but constant speeds. At a certain moment, the center of the smaller sphere is 6 cm from the vertex, and the center of the larger ... | 1 | 123 | 1 |
math | 7. A die (a uniform cube with faces numbered $1,2,3,4$, 5,6) is rolled three times. What is the probability that the three numbers on the top faces can form the side lengths of a triangle whose perimeter is divisible by 3? $\qquad$ | \frac{11}{72} | 62 | 9 |
math | # Problem 1. (2 points)
Let $x, y, z$ be pairwise coprime three-digit natural numbers. What is the greatest value that the GCD $(x+y+z, x y z)$ can take? | 2994 | 48 | 4 |
math | Toner Drum and Celery Hilton are both running for president. A total of $2015$ people cast their vote, giving $60\%$ to Toner Drum. Let $N$ be the number of "representative'' sets of the $2015$ voters that could have been polled to correctly predict the winner of the election (i.e. more people in the set voted for Drum... | 605 | 116 | 3 |
math | Problem 3. Mario distributed the numbers 1, 2, ..., 8 at the vertices of a cube. When he calculated the sums obtained from each face, he noticed that all the sums were equal.
a) What is the value of each such sum?
b) Find one arrangement of the numbers 1, 2, ..., 8 at the vertices of the cube that has the desired pro... | 18 | 83 | 2 |
math | 3 Given a positive integer $n$, find the smallest positive number $\lambda$, such that for any $\theta_{i} \in$ $\left(0, \frac{\pi}{2}\right)(i=1,2, \cdots, n)$, if $\tan \theta_{1} \cdot \tan \theta_{2} \cdots \cdot \tan \theta_{n}=2^{\frac{n}{2}}$, then $\cos \theta_{1}+\cos \theta_{2}+\cdots+\cos \theta_{n}$ is not... | n-1 | 125 | 3 |
math | Solve the system of equation
$$x+y+z=2;$$$$(x+y)(y+z)+(y+z)(z+x)+(z+x)(x+y)=1;$$$$x^2(y+z)+y^2(z+x)+z^2(x+y)=-6.$$ | \{(0, 3, -1), (0, -1, 3), (3, 0, -1), (3, -1, 0), (-1, 0, 3), (-1, 3, 0)\} | 58 | 56 |
math | A non-equilateral triangle $\triangle ABC$ of perimeter $12$ is inscribed in circle $\omega$ .Points $P$ and $Q$ are arc midpoints of arcs $ABC$ and $ACB$ , respectively. Tangent to $\omega$ at $A$ intersects line $PQ$ at $R$.
It turns out that the midpoint of segment $AR$ lies on line $BC$ . Find the length of the se... | 4 | 112 | 1 |
math | 4. Compute $\sum_{k=1}^{\infty} \frac{k^{4}}{k!}$. | 15e | 26 | 3 |
math | 8. The King's Path (from 7th grade, 2 points). A chess king is on the a1 square of a chessboard and wants to move to the h8 square, moving right, up, or diagonally up-right. In how many ways can he do this? | 48639 | 60 | 5 |
math | 15.12. For which real numbers $x$ does the inequality
$$
2 \log _{x}\left(\frac{a+b}{2}\right) \leq \log _{x} a+\log _{x} b
$$
hold for all positive numbers $a$ and $b$ ? | 0<x<1 | 70 | 4 |
math | Example 2 Solve the equation $x^{3}+(1+\sqrt{2}) x^{2}-2=0$.
| -\sqrt{2},\frac{-1\\sqrt{1+4\sqrt{2}}}{2} | 27 | 23 |
math | 13 Find the maximum value of the product $x^{2} y^{2} z^{2} u$ under the conditions $x, y, z, u \geqslant 0$ and $2 x+x y+z+y z u=1$. | \frac{1}{512} | 56 | 9 |
math | IMO 1986 Problem B2 Find all functions f defined on the non-negative reals and taking non-negative real values such that: f(2) = 0, f(x) ≠ 0 for 0 ≤ x < 2, and f(xf(y)) f(y) = f(x + y) for all x, y. | f(x)=0forx\ge2,\frac{2}{2-x}for0\lex<2 | 73 | 23 |
math | ## Task A-4.1.
Determine all natural numbers $b$ for which the equality $11 \cdot 22 \cdot 33=13310$ holds in the number system with base $b$. | 6 | 50 | 1 |
math | Problem 9.3. Natural numbers $a$ and $b$ are such that $a$ is divisible by $b+1$ and 43 is divisible by $a+b$.
(a) (1 point) Indicate any possible value of $a$.
(b) (3 points) What can $b$ be? Indicate all possible options. | =22,33,40,42;\,b=1,3,10,21 | 76 | 25 |
math | 13.351. The road from post office $A$ to village $B$ goes uphill for 2 km, then on flat ground for 4 km, and finally downhill for 3 km. The postman takes 2 hours and 16 minutes to travel from $A$ to $B$, and 2 hours and 24 minutes to return. If the final destination of his route were located on the same road but twice ... | )3\mathrm{}/\mathrm{};b)4\mathrm{}/\mathrm{};)5\mathrm{}/\mathrm{} | 148 | 29 |
math | Example 2. The probability density of a random variable $X$ is given by the function
$$
p(x)= \begin{cases}0 & \text { if } x \leq 0 \\ x / 2 & \text { if } 0 < x \leq 2\end{cases}
$$
Find the probability that in the experiment, the variable $X$ will take a value from the interval $(1,2)$. | 0.75 | 96 | 4 |
math | 3. For a children's party, pastries were prepared: 10 eclairs, 20 mini tarts, 30 chocolate brownies, 40 cream puffs. What is the maximum number of children who can each take three different pastries? | 30 | 56 | 2 |
math | 3.261. $\frac{\sin \left(4 \alpha+\frac{5 \pi}{2}\right)}{1+\cos \left(4 \alpha-\frac{3 \pi}{2}\right)}$.
3.261. $\frac{\sin \left(4 \alpha+\frac{5 \pi}{2}\right)}{1+\cos \left(4 \alpha-\frac{3 \pi}{2}\right)}$.
(Note: The mathematical expression is already in a universal format and does not require translation.) | \tan(\frac{\pi}{4}+2\alpha) | 117 | 14 |
math | A student marked the product of an integer and 467 in their homework as 1925 817. The teacher underlined the 9 and the 7 as errors. What are the correct digits and what was the multiplicand? | 2839 | 53 | 4 |
math | Consider $P(x)=\sum_{k=0}^{n} a_{k} x^{k} \in \mathbb{R}[X]$ of degree $n$. What is $P^{(j)}(0)$ for $j \in \mathbb{N}$? | P^{(j)}(0)=\begin{cases}0&\text{if}j>n\\j!a_{j}&\text{if}j\leqslantn\end{cases} | 61 | 46 |
math | ## Task A-4.1.
Let $a_{0}, a_{1}, \ldots, a_{n}$ be real numbers such that
$$
a_{0}+a_{1} x+\ldots+a_{n} x^{n}=(x+1)^{3}(x+2)^{3} \cdots(x+672)^{3}
$$
Determine the sum
$$
a_{2}+a_{4}+a_{6}+\ldots+a_{2016}
$$ | \frac{1}{2}(673!)^{3}-(672!)^{3} | 116 | 22 |
math | 1. Given circles $\odot O_{1}$ and $\odot O_{2}$ intersect at points $A$ and $B$, with radii $5$ and $7$ respectively, and $O_{1} O_{2}=6$. A line through point $A$ intersects $\odot O_{1}$ and $\odot O_{2}$ at points $C$ and $D$, respectively. Points $P$ and $O$ are the midpoints of segments $CD$ and $O_{1} O_{2}$, re... | 2\sqrt{7} | 124 | 6 |
math | Let $0 \leq a, b, c, d \leq 2005$ be integers. What is the probability that $a b+c d$ is an even number? | 0.625 | 41 | 5 |
math | 1. [3 points] Find the number of eight-digit numbers, the product of the digits of each of which is equal to 64827. The answer should be presented as an integer. | 1120 | 42 | 4 |
math | $$
\begin{array}{l}
\text { 4. If } x_{1}>x_{2}>x_{3}>x_{4}>0, \text { and the inequality } \\
\log _{\frac{x_{1}}{x_{2}}} 2014+\log _{\frac{x_{2}}{x_{3}}} 2014+\log _{\frac{x_{3}}{x_{4}}} 2014 \\
\geqslant k \log _{\frac{x_{1}}{}} 2014
\end{array}
$$
always holds, then the maximum value of the real number $k$ is $\qq... | 9 | 150 | 1 |
math | 3. In $\triangle A B C$, $D$ is a point on side $B C$. It is known that $A B=13, A D=12, A C=15, B D=5$, find $D C$. | 9 | 54 | 1 |
math | For which positive integers $a, b, c$ is it true that $2^{a}-1$ is divisible by $b$, $2^{b}-1$ is divisible by $c$, and $2^{c}-1$ is divisible by $a$? | =1,b=1,=1 | 56 | 8 |
math | 6、Among the first 10000 positive integers $1,2, \ldots, 10000$, the number of integers whose digits contain at least two of the digits $2, 0, 1, 7$ (if the same digit appears multiple times: for example, 2222, it is counted as containing 1 digit) is $\qquad$
---
Note: The translation preserves the original format an... | 3862 | 100 | 4 |
math | For each positive integer $n$, the expression $1+2+3+\cdots+(n-1)+n$ represents the sum of all of the integers from 1 to $n$ inclusive. What integer is equal to
$$
(1+2+3+\cdots+2020+2021)-(1+2+3+\cdots+2018+2019) ?
$$ | 4041 | 91 | 4 |
math | We know that there exists a positive integer with $7$ distinct digits which is multiple of each of them. What are its digits?
(Paolo Leonetti) | \{1, 2, 3, 6, 7, 8, 9\} | 33 | 24 |
math | If $a, b, c \in [-1, 1]$ satisfy $a + b + c + abc = 0$, prove that $a^2 + b^2 + c^2 \ge 3(a + b + c)$ .
When does the equality hold?
| a^2 + b^2 + c^2 \ge 3(a + b + c) | 61 | 22 |
math | In America, temperature is measured in Fahrenheit. This is a uniform scale where the freezing point of water is $32^{\circ} \mathrm{F}$ and the boiling point is $212^{\circ} \mathrm{F}$.
Someone gives the temperature rounded to the nearest whole Fahrenheit degree, which we then convert to Celsius and round to the near... | \frac{13}{18} | 97 | 9 |
math | 10.4. Find the maximum value of the expression $a+b+c+d-ab-bc-cd-da$, if each of the numbers $a, b, c$ and $d$ belongs to the interval $[0 ; 1]$. | 2 | 52 | 1 |
math | 9. Uncle Zhang and Uncle Li's combined age is 56 years old. When Uncle Zhang was half of Uncle Li's current age, Uncle Li's age at that time was Uncle Zhang's current age. So, Uncle Zhang is $\qquad$ years old now. | 24 | 56 | 2 |
math | 6. In $\triangle A B C$, $\cos A, \sin A, \tan B$ form a geometric sequence with a common ratio of $\frac{3}{4}$, then $\cot C=$ $\qquad$ | -\frac{53}{96} | 47 | 9 |
math | 9. Find such four-digit numbers, each of which is divisible by 11, and the sum of the digits of each of them is 11. | 2090,3080,4070,5060,6050,7040,8030,9020 | 33 | 39 |
math | [Algorithm Theory (Miscellaneous).]
A journalist has come to a company of $\mathrm{N}$ people. He knows that in this company there is a person $\mathrm{Z}$ who knows all the other members of the company, but no one knows him. The journalist can ask each member of the company the question: "Do you know so-and-so?" Find... | N-1 | 103 | 3 |
math | 4.5.7 ** Let $\lambda>0$, find the largest constant $C=C(\lambda)$, such that for all non-negative real numbers $x, y$, we have $x^{2}+y^{2}+\lambda x y \geqslant C(x+y)^{2}$. | C(\lambda)={\begin{pmatrix}1,&\text{when}\lambda\geqslant2\text{,}\\\frac{2+\lambda}{4},&\text{when}0<\lambda<2\text{0}\end{pmatrix}.} | 65 | 62 |
math | 6. Let $1 \leqslant r \leqslant n$. Then the arithmetic mean of the smallest numbers in all $r$-element subsets of the set $M=\{1,2, \cdots, n\}$ is $\qquad$ | \frac{n+1}{r+1} | 57 | 10 |
math | For any positive integer $m\ge2$ define $A_m=\{m+1, 3m+2, 5m+3, 7m+4, \ldots, (2k-1)m + k, \ldots\}$.
(1) For every $m\ge2$, prove that there exists a positive integer $a$ that satisfies $1\le a<m$ and $2^a\in A_m$ or $2^a+1\in A_m$.
(2) For a certain $m\ge2$, let $a, b$ be positive integers that satisfy $2^a\in A_m... | a_0 = 2b_0 + 1 | 190 | 13 |
math | Bogganov I.I.
Given an infinite supply of white, blue, and red cubes. Any \$N\$ of them are arranged in a circle. A robot, starting at any point on the circle, moves clockwise and, until only one cube remains, repeatedly performs the following operation: it destroys the two nearest cubes in front of it and places a ne... | 2^k | 166 | 3 |
math | 7. The minor axis of the ellipse $\rho=\frac{1}{2-\cos \theta}$ is equal to
保留了源文本的换行和格式。 | \frac{2\sqrt{3}}{3} | 35 | 12 |
math | 1. Let set $A=\{(x, y) \mid x+y=1\}, B=\left\{(x, y) \mid x^{2}+y^{2}=2\right\}, C=A \cap B$, then the number of subsets of set $C$ is $\qquad$ . | 4 | 67 | 1 |
math | Example 3 If $\left(1+x+x^{2}+x^{3}\right)^{5}\left(1-x+x^{2}-\right.$ $\left.x^{3}\right)^{5}=a_{30}+a_{29} x+\cdots+a_{1} x^{29}+a_{0} x^{30}$, find $a_{15}$. | 0 | 88 | 1 |
math | Given are the positive real numbers $a$ and $b$ and the natural number $n$.
Determine, in dependence on $a, b$, and $n$, the largest of the $n+1$ terms in the expansion of $(a+b)^{n}$. | \text{If } i=\frac{nb+b}{a+b} \text{ is not an integer, then } G(i) \text{ is the largest term, otherwise } G(i) \text{ and } G(i+1) \text{ are maximal.} | 58 | 58 |
math | Find minimum of $x+y+z$ where $x$, $y$ and $z$ are real numbers such that $x \geq 4$, $y \geq 5$, $z \geq 6$ and $x^2+y^2+z^2 \geq 90$ | 16 | 66 | 2 |
math | 2. Let $x_{1}$ and $x_{2}$ be the largest roots of the polynomials
$$
\begin{gathered}
f(x)=1-x-4 x^{2}+x^{4} \\
\text { and } \\
g(x)=16-8 x-16 x^{2}+x^{4}
\end{gathered}
$$
respectively. Find $\frac{x_{2}}{x_{1}}$. | 2 | 98 | 1 |
math | 7. Let $z=x+y \mathrm{i} (x, y \in \mathbf{R}, \mathrm{i}$ be the imaginary unit), the imaginary part of $z$ and the real part of $\frac{z-\mathrm{i}}{1-z}$ are both non-negative. Then the area of the region formed by the points $(x, y)$ on the complex plane that satisfy the conditions is $\qquad$ . | \frac{3 \pi+2}{8} | 90 | 11 |
math | Let $ P(x) \in \mathbb{Z}[x]$ be a polynomial of degree $ \text{deg} P \equal{} n > 1$. Determine the largest number of consecutive integers to be found in $ P(\mathbb{Z})$.
[i]B. Berceanu[/i] | n | 66 | 2 |
math | 10. Let $a, b \in [0,1]$, find the maximum and minimum values of $S=\frac{a}{1+b}+\frac{b}{1+a}+(1-a)(1-b)$. | 1 | 49 | 1 |
math | 8. (10 points) In the expression $(x+y+z)^{2034}+(x-y-z)^{2034}$, the brackets were expanded and like terms were combined. How many monomials $x^{a} y^{b} z^{c}$ with a non-zero coefficient were obtained? | 1036324 | 69 | 7 |
math | 12.224. In a regular triangular pyramid, the dihedral angle at the base is equal to $\alpha$, the lateral surface area is $S$. Find the distance from the center of the base to the lateral face. | \frac{\sin\alpha}{3}\sqrt{S\sqrt{3}\cos\alpha} | 49 | 21 |
math | Example 16. Solve the equation
$$
\log _{4}(x+3)+\log _{4}(x-1)=2-\log _{4} 8
$$ | \sqrt{6}-1 | 42 | 6 |
math | 6. (48th Slovenian Mathematical Olympiad) Find all prime numbers $p$ such that $p+28$ and $p+56$ are also prime. | 3 | 38 | 1 |
math | Let $(x_n)$ be a sequence of positive integers defined as follows: $x_1$ is a fixed six-digit number and for any $n \geq 1$, $x_{n+1}$ is a prime divisor of $x_n + 1$. Find $x_{19} + x_{20}$. | 5 | 70 | 1 |
math | 12. In 2019, the Mathematics Olympiad underwent a trial reform: A city held 5 joint competitions in the second year of high school. A student who ranks in the top 20 of the city in 2 out of these 5 competitions can enter the provincial team training and does not need to participate in the remaining competitions. Each s... | \frac{269}{64} | 219 | 10 |
math | Question 31: Let the function $\mathrm{f}(\mathrm{x})=\ln \mathrm{x}$ have the domain $(\mathrm{m},+\infty)$, and $M>0$. If for any $a, b, c \in (M,+\infty)$, $a, b, c$ are the three sides of a right triangle, then $f(a), f(b), f(c)$ can also be the three sides of a triangle. Find the minimum value of M. | \sqrt{2} | 106 | 5 |
math | 14. Given 10 positive integers, the sum of any 9 of them takes only the following 9 different values: $86, 87, 88, 89$, $90, 91, 93, 94, 95$, then among these 10 positive integers, when arranged in descending order, find the sum of the 3rd and the 7th numbers. | 22 | 93 | 2 |
math | Find the probability that heads will appear an even number of times in an experiment where:
a) a fair coin is tossed $n$ times;
b) a coin, for which the probability of heads on a single toss is $p(0<p<1)$, is tossed $n$ times. | )0.5;b)\frac{1+(1-2p)^{n}}{2} | 61 | 21 |
math | Example 5 The license plates of motor vehicles in a city are consecutively numbered from "10000" to "99999". Among these 90000 license plates, the number of plates that contain at least one digit 9 and have the sum of their digits as a multiple of 9 is $\qquad$.
| 4168 | 75 | 4 |
math | Suppose you are given that for some positive integer $n$, $1! + 2! + \ldots + n!$ is a perfect square. Find the sum of all possible values of $n$. | 4 | 44 | 1 |
math | 3. Find the smallest distance from the point with coordinates $(10 ; 5 ; 10)$ to a point with positive coordinates that satisfy the inequality $(x+y+z)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right) \geq$ $9 \sqrt{1-(2 x+y)^{2}}$.
In your answer, write the square of the found distance.
points) | 115.2 | 95 | 5 |
math | 12.049. The plane angle at the vertex of a regular hexagonal pyramid is equal to the angle between a lateral edge and the plane of the base. Find this angle. | 2\arcsin\frac{\sqrt{3}-1}{2} | 39 | 16 |
math | Let $d(n)$ be the number of all positive divisors of a natural number $n \ge 2$.
Determine all natural numbers $n \ge 3$ such that $d(n -1) + d(n) + d(n + 1) \le 8$.
[i]Proposed by Richard Henner[/i] | n = 3, 4, 6 | 75 | 10 |
math | 19. Determine the number of sets of solutions $(x, y, z)$, where $x, y$ and $z$ are integers, of the the equation $x^{2}+y^{2}+z^{2}=x^{2} y^{2}$. | 1 | 59 | 1 |
math | 8. Let the quartic polynomial with integer coefficients $f(x)$ satisfy:
$$
f(1+\sqrt[3]{3})=1+\sqrt[3]{3}, f(1+\sqrt{3})=7+\sqrt{3} \text {. }
$$
Then $f(x)=$ $\qquad$ | x^{4}-3x^{3}+3x^{2}-3x | 68 | 17 |
math | 4. The Petrovs family has decided to renovate their apartment. They can hire a company for a "turnkey renovation" for 50,000 or buy materials for 20,000 and do the renovation themselves, but for that, they will have to take unpaid leave. The husband earns 2000 per day, and the wife earns 1500. How many working days can... | 8 | 108 | 1 |
math | 6. The smallest natural number $n$ that satisfies $n \sin 1 > 1 + 5 \cos 1$ is $\qquad$ . | 5 | 34 | 1 |
math | 1. Use $1,2,3,4,5$ to form a five-digit number, such that the difference between any two adjacent digits is at least 2. Then the number of such five-digit numbers is $\qquad$ . | 14 | 50 | 2 |
math | 1. Three athletes start from the same point on a closed running track that is 400 meters long and run in the same direction. The first runs at a speed of 155 m/min, the second at 200 m/min, and the third at 275 m/min. After what least amount of time will they all be at the same point again? How many overtakes will occu... | \frac{80}{3} | 99 | 8 |
math | 19. Suppose $x, y, z$ and $\lambda$ are positive real numbers such that
$$
\begin{aligned}
y z & =6 \lambda x \\
x z & =6 \lambda y \\
x y & =6 \lambda z \\
x^{2}+y^{2}+z^{2} & =1
\end{aligned}
$$
Find the value of $(x y z \lambda)^{-1}$. | 54 | 96 | 2 |
math | At a certain school, there are 6 subjects offered, and a student can take any combination of them. It is noticed that for any two subjects, there are fewer than 5 students taking both of them and fewer than 5 students taking neither. Determine the maximum possible number of students at the school. | 20 | 62 | 2 |
math | There is a cube of size $10 \times 10 \times 10$, consisting of small unit cubes. In the center $O$ of one of the corner cubes sits a grasshopper. The grasshopper can jump to the center of a cube that shares a face with the one it is currently in, and in such a way that the distance to point $O$ increases. In how many ... | \frac{27!}{(9!)^{3}} | 101 | 13 |
math | 17. 5555 children, numbered 1 to 5555 , sit around a circle in order. Each child has an integer in hand. The child numbered 1 has the integer 1 ; the child numbered 12 has the integer 21; the child numbered 123 has the integer 321 and the child numbered 1234 has the integer 4321. It is known that the sum of the integer... | -4659 | 296 | 5 |
math | 11.037. The base of the prism is a square with a side equal to $a$. One of the lateral faces is also a square, the other is a rhombus with an angle of $60^{\circ}$. Determine the total surface area of the prism. | ^{2}(4+\sqrt{3}) | 61 | 9 |
math | 5. If $a>b>c>d$, and $\frac{1}{a-b}+\frac{1}{b-c}+\frac{1}{c-d} \geq \frac{n}{a-d}$, then the maximum value of the integer $n$ is $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 9 | 85 | 1 |
math | 3. Given that $P$ is any point on the arc $\overparen{A D}$ of the circumcircle of square $A B C D$, find the value of $\frac{P A+P C}{P B}$.
untranslated text:
已知 $P$ 是正方形 $A B C D$ 的外接圆 $\overparen{A D}$ 上任一点, 求 $\frac{P A+P C}{P B}$ 的值. | \sqrt{2} | 99 | 5 |
math | ## Task 4 - 341234
Determine the smallest natural number $n$ with $n \geq 2$ and the following property ( $\left.{ }^{*}\right)$ :
$(*)$ In every set of $n$ natural numbers, there are (at least) two numbers whose sum or whose difference is divisible by 7. | 5 | 78 | 1 |
math | [ Arithmetic progression ]
[ Arithmetic. Mental calculation, etc. ]
What is the sum of the digits of all numbers from one to a billion? | 40500000001 | 29 | 11 |
math | Example 4 Solve the inequalities:
(1) $7 x+5+\frac{2}{x-4}>6 x+\frac{2}{x-4}+3$;
(2) $\frac{x^{2}(x-1)(x-2)^{2}}{(x-2)(x-1)^{2}} \leqslant 0$. | {0}\cup(1,2) | 79 | 9 |
math | One. (20 points) Given $x=\sqrt{1+\sqrt{1+\sqrt{1+x}}}$. Find the integer part of $x^{6}+x^{5}+2 x^{4}-4 x^{3}+3 x^{2}+4 x-4$.
| 36 | 64 | 2 |
math | 1. Positive integers $x_{1}, x_{2}, \cdots, x_{n}$ satisfy $x_{1}+x_{2}+\cdots+x_{n}=$ $m$ (constant). Try to find the maximum value of $f=x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}$. | (n-1)+(m-n+1)^2 | 81 | 10 |
math | Let $x_{0}+\sqrt{2003} y_{0}$ be the fundamental solution of the Pell equation
$$
x^{2}-2003 y^{2}-1
$$
Find the solution $(x, y)$ of (1), where $x, y$ are positive numbers and all prime factors of $x$ divide $x_{0}$. | x_{0},y_{0} | 81 | 8 |
math | ## Task 2 - 180712
Calculate
$$
\begin{aligned}
& a=1.25: \frac{13}{12} \cdot \frac{91}{60} \\
& b=2.225-\frac{5}{9}-\frac{5}{6} \\
& c=\frac{32}{15}: \frac{14}{15}+6+\left(\frac{45}{56}-0.375\right) \\
& d=c-\frac{b}{a}
\end{aligned}
$$
without using approximate values! | \frac{5189}{630} | 138 | 12 |
math | 1. Calculate: $\left(0.25+1 \frac{3}{5}-\frac{3}{5} \cdot 2 \frac{11}{12}\right): 10=$ | \frac{1}{100} | 46 | 9 |
math | 1. Find all integer solutions of the equation
$$
3^{x}-5^{y}=z^{2}
$$ | (x,y,z)=(2,1,2) | 25 | 10 |
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