task_type stringclasses 1
value | problem stringlengths 46 985 | answer stringlengths 1 146 | problem_tokens int64 20 349 | answer_tokens int64 1 92 |
|---|---|---|---|---|
math | ## Task 1 - 330711
On a chicken farm, 2500 chickens were kept. At the beginning of a month, there was enough feed for exactly 30 days. After exactly 14 days, 500 chickens were slaughtered. By how many days was the time for which the feed was sufficient extended? | 4 | 75 | 1 |
math | 13.078. A tourist traveled $5 / 8$ of the total distance by car and the remaining part by boat. The boat's speed is 20 km/h less than the car's speed. The tourist traveled by car for 15 minutes longer than by boat. What are the speeds of the car and the boat if the total distance of the tourist's journey is 160 km? | 100 | 87 | 3 |
math | 12.3 If $a$ and $b$ are distinct prime numbers and $a^{2}-a Q+R=0$ and $b^{2}-b Q+R=0$, find the value of $R$. | 6 | 49 | 1 |
math | Compute the number of nonempty subsets $S$ of $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ such that $\frac{\max \,\, S + \min \,\,S}{2}$ is an element of $S$. | 234 | 71 | 3 |
math | The difference of fractions $\frac{2024}{2023} - \frac{2023}{2024}$ was represented as an irreducible fraction $\frac{p}{q}$. Find the value of $p$. | 4047 | 53 | 4 |
math | 2. To fight against the mice, the cat Leopold must drink animalin daily. The cat has a bottle of animalin solution in water (a colorless transparent liquid) with a volume of $V=0.5$ l and a volumetric concentration of animalin $C_{0}=40 \%$. Every day, Leopold drinks $v=50$ ml of the solution, and to avoid being noticed by the mice, he adds the same volume of water to the bottle. Find the volumetric concentration of animalin in the bottle after the cat has drunk $n=5$ times, each time $v=50$ ml of liquid, and after each time, added water to the bottle. | 23.6 | 150 | 4 |
math | 8. Solve the equation $2 \sqrt{x^{2}-121}+11 \sqrt{x^{2}-4}=7 \sqrt{3} x$. | 14 | 36 | 2 |
math | $\left.\begin{array}{l}\text { [Inclusion-Exclusion Principle]} \\ {[\quad \text { Word Problems (Miscellaneous). }}\end{array}\right]$
In the garden, Anya and Vitya had 2006 rose bushes. Vitya watered half of all the bushes, and Anya watered half of all the bushes. It turned out that exactly three bushes, the most beautiful ones, were watered by both Anya and Vitya. How many rose bushes were left unwatered? | 3 | 117 | 1 |
math | a) Is it possible to distribute the numbers $1,2,3, \ldots, 10$ into five pairs such that the five sums of the pairs give five different prime numbers?
b) Is it possible to distribute the numbers $1,2,3, \ldots, 20$ into ten pairs such that the ten sums of the pairs give ten different prime numbers? | Yes:1+4=5,3+8=11,5+2=7,7+6=13,9+10=19 | 82 | 35 |
math | Task B-3.2. The axial section of a right circular cylinder is a square. Determine the length of the radius of the base so that its surface area and volume have the same numerical value. | 3 | 41 | 1 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0} \frac{e^{2 x}-e^{x}}{x+\tan x^{2}}$ | 1 | 42 | 1 |
math | At what smallest $n$ is there a convex $n$-gon for which the sines of all angles are equal and the lengths of all sides are different? | 5 | 34 | 1 |
math | $\textbf{Problem 4.}$ The number of perfect inhabitants of a city was a perfect square, in other words, a whole number squared. with $100$ people plus the new number of inhabitants turned out to be a perfect square plus one. Now, with another increase of $100$ people, the number of inhabitants is again a perfect square. What was the number of inhabitants original city? | 49^2 | 86 | 4 |
math | For what values of parameters $a$ and $b$ does the equation
$$
\frac{x-a}{x-2}+\frac{x-b}{x-3}=2
$$
have a solution? For what values of $a$ and $b$ is the solution unique? | +b\neq5,\quad\neq2,\quadb\neq3 | 60 | 18 |
math | 7. In triangle $A B C$, side $B C$ is equal to 4, and angle $A C B$ is equal to $\frac{2 \pi}{3}$. Circle $\Gamma$ with radius $2 \sqrt{3}$ touches sides $B C$ and $A C$ of triangle $A B C$ at points $K$ and $L$ respectively, and intersects side $A B$ at points $M$ and $N$ ( $M$ lies between $A$ and $N$ ) such that segment $M K$ is parallel to $A C$. Find the lengths of segments $C K, M K, A B$ and the area of triangle $C M N$. | CK=2,MK=6,AB=4\sqrt{13},S_{\triangleCMN}=\frac{72\sqrt{3}}{13} | 151 | 38 |
math | 9. Let $a, b$ be real numbers, for any real number $x$ satisfying $0 \leqslant x \leqslant 1$ we have $|a x+b| \leqslant 1$. Then the maximum value of $|20 a+14 b|+|20 a-14 b|$ is $\qquad$. | 80 | 82 | 2 |
math | 6. (15 points) A pedestrian is moving in a straight line towards a crosswalk at a constant speed of 3.6 km/h. At the initial moment, the distance from the pedestrian to the crosswalk is 40 m. The length of the crosswalk is 6 m. At what distance from the crosswalk will the pedestrian be after two minutes? | 74\mathrm{~} | 77 | 7 |
math | Three, (17 points) The quadratic trinomial $x^{2}-x-2 n$ can be factored into the product of two linear factors with integer coefficients.
(1) If $1 \leqslant n \leqslant 30$, and $n$ is an integer, how many such $n$ are there?
(2) When $n \leqslant 2005$, find the largest integer $n$.
| 1953 | 100 | 4 |
math | 9.75 A rectangular prism is constructed from small cubes of the same size. Three faces that share a common vertex are painted. As a result, exactly half of the small cubes have at least one face painted. How many small cubes have at least one face painted?
The rectangular prism is constructed from small cubes of the same size. Three faces that share a common vertex are painted. As a result, exactly half of the small cubes have at least one face painted. How many small cubes have at least one face painted? | 60,72,84,90,120 | 106 | 15 |
math | Can the position of a pawn on a chessboard be determined by the following equation, where $x, y$ represent the row and column numbers of the square:
$$
x^{2}+x y-2 y^{2}=13
$$ | (5,4) | 52 | 5 |
math | Ha $9^{-1 \frac{1}{6}}=0.0770401$, what is $9^{-\frac{2}{3}}$? | 0.2311203 | 37 | 9 |
math | 4. Calculate $\sqrt{3+\sqrt{5}}-\sqrt{3-\sqrt{5}}$. | \sqrt{2} | 22 | 5 |
math | # Task 8.2
Find the greatest value that the product of natural numbers can take, the sum of which is 2020.
## Number of points 7
# | 2^{2}\cdot3^{672} | 39 | 11 |
math | 3. The solution to the equation $\sqrt{13-\sqrt{13+x}}$ $=x$ is $\qquad$ | 3 | 29 | 1 |
math | 1. In a recipe for soup, it says that for 5 servings of this soup, 30 dag of carrots are needed. If the cook wants to cook soup for 60 people and prepare 2 servings of soup per person, how many carrots should be put into this soup? | 720 | 60 | 3 |
math | 1. Find all real numbers $a$ and $b$
$$
\left\{\begin{array}{l}
\left|\frac{x^{y}-1}{x^{y}+1}\right|=a \\
x^{2}+y^{2}=b
\end{array}\right.
$$
given that $x>0$, there is a unique solution. | 00<b\leq1 | 79 | 7 |
math | 2. A team consisting of boys and girls from the Rostov region went to a CS:GO esports tournament. The average number of points scored by the girls turned out to be 22, by the boys - 47, and the average number of points for the entire team - 41. What is the percentage of girls in this team? | 24 | 74 | 2 |
math | Given $f(x)$ is a function defined on the set of real numbers with a period of 2, and it is an even function. It is known that when $x \in [2,3]$, $f(x) = x$. Find the analytical expression of $f(x)$ when $x \in [-2,0]$. | f(x)=3-|x+1|(x\in[-2,0]) | 71 | 18 |
math | 9. (i) (Grade 11) Given $\sin \theta+\cos \theta=\frac{\sqrt{2}}{5}$ $\left(\frac{\pi}{2}<\theta<\pi\right)$. Then $\tan \theta-\cot \theta=$ $\qquad$ .
(ii) (Grade 12) The function $f(x)=\frac{1}{\sin ^{2} x}+\frac{2}{\cos ^{2} x}$ $\left(0<x<\frac{\pi}{2}\right)$ has a minimum value of $\qquad$ . | 3 + 2 \sqrt{2} | 128 | 9 |
math | ## 65. Matheknobelei 10/70
Die Seiten eines Rechtecks werden durch ganze Zahlen dargestellt. Wie lang müssen die Seiten sein, damit der Umfang des Rechtecks zahlenmäßig gleich dem Flächeninhalt ist?
| 3und6 | 60 | 3 |
math | 6. Petya formed all possible natural numbers that can be formed from the digits $2,0,1$, 8 (each digit can be used no more than once). Find their sum. Answer: 78331 | 78331 | 49 | 5 |
math | 10.11. What condition must the coefficients of the polynomial $x^{3}+a x^{2}+b x+c$ satisfy so that its three roots form an arithmetic progression? | \frac{2}{27}^{3}-\frac{}{3}+=0 | 41 | 19 |
math | Example 1 Let $X$ be the set of irreducible proper fractions with a denominator of 800, and $Y$ be the set of irreducible proper fractions with a denominator of 900, and let $A=\{x+y \mid x \in X, y \in Y\}$. Find the smallest denominator of the irreducible fractions in $A$. | 288 | 80 | 3 |
math | ```
Let $x_{i} \in \mathbf{R}_{+}(i=1,2, \cdots, 5)$. Find
\[
\begin{array}{l}
f= \frac{x_{1}+x_{3}}{x_{5}+2 x_{2}+3 x_{4}}+\frac{x_{2}+x_{4}}{x_{1}+2 x_{3}+3 x_{5}}+ \\
\frac{x_{3}+x_{5}}{x_{2}+2 x_{4}+3 x_{1}}+\frac{x_{4}+x_{1}}{x_{3}+2 x_{5}+3 x_{2}}+ \\
\frac{x_{5}+x_{2}}{x_{4}+2 x_{1}+3 x_{3}}
\end{array}
\]
the minimum value.
``` | \frac{5}{3} | 202 | 7 |
math | 1. In triangle $ABC$, the median $AD$ is drawn. $\widehat{D A C} + \widehat{A B C} = 90^{\circ}$. Find $\widehat{B A C}$, given that $|A B| = |A C|$. | 90 | 64 | 2 |
math | Twelve toddlers went out to the yard to play in the sandbox. Each one who brought a bucket also brought a shovel. Nine toddlers forgot their bucket at home, and two forgot their shovel. By how many fewer toddlers brought a bucket than those who brought a shovel but forgot their bucket? | 4 | 58 | 1 |
math | Let $p$ be a function associated with a permutation. We call the order of this permutation the smallest integer $k$ such that $p^{(k)}=I d$.
What is the largest order for a permutation of size 11? | 30 | 52 | 2 |
math | 5. Let $x, y \geqslant 0$, and $x+y \leqslant \sqrt{\frac{2}{3}}$. Then
$$
\sqrt{2-3 x^{2}}+\sqrt{2-3 y^{2}}
$$
the minimum value is $\qquad$ | \sqrt{2} | 68 | 5 |
math | 3. (GDR 1) Knowing that the system
$$
\begin{aligned}
x+y+z & =3, \\
x^{3}+y^{3}+z^{3} & =15, \\
x^{4}+y^{4}+z^{4} & =35,
\end{aligned}
$$
has a real solution $x, y, z$ for which $x^{2}+y^{2}+z^{2}<10$, find the value of $x^{5}+y^{5}+z^{5}$ for that solution. | 83 | 128 | 2 |
math | 5. Take a total of ten coins of one cent, two cents, and five cents, to pay eighteen cents. How many different ways are there to do this?
保留源文本的换行和格式，直接输出翻译结果如下：
5. Take a total of ten coins of one cent, two cents, and five cents, to pay eighteen cents. How many different ways are there to do this? | 3 | 83 | 1 |
math | 5. Let $N=2019^{2}-1$. How many positive factors of $N^{2}$ do not divide $N$ ? | 157 | 32 | 3 |
math | Of $450$ students assembled for a concert, $40$ percent were boys. After a bus containing an equal number of boys and girls brought more students to the concert, $41$ percent of the students at the concert were boys. Find the number of students on the bus. | 50 | 61 | 2 |
math | 7. Line segments $A B, C D$ are between two parallel planes $\alpha$ and $\beta$, $A C \subset \alpha, B D \subset \beta, A B \perp \alpha, A C=B D=5$, $A B=12, C D=13, E, F$ divide $A B, C D$ in the ratio $1: 2$. Find the length of line segment $E F$. | \frac{5}{3}\sqrt{7} | 97 | 11 |
math | Let $ X_n\equal{}\{1,2,...,n\}$,where $ n \geq 3$.
We define the measure $ m(X)$ of $ X\subset X_n$ as the sum of its elements.(If $ |X|\equal{}0$,then $ m(X)\equal{}0$).
A set $ X \subset X_n$ is said to be even(resp. odd) if $ m(X)$ is even(resp. odd).
(a)Show that the number of even sets equals the number of odd sets.
(b)Show that the sum of the measures of the even sets equals the sum of the measures of the odd sets.
(c)Compute the sum of the measures of the odd sets. | \binom{n+1}{2} 2^{n-2} | 151 | 16 |
math | 7. Let the function $f(x)$ defined on $(0,+\infty)$ satisfy that for any $x \in(0,+\infty)$, we have $f(x)>-\frac{3}{x}$, and $f\left(f(x)+\frac{3}{x}\right)=2$. Then $f(5)=$ $\qquad$ . | \frac{12}{5} | 79 | 8 |
math | Example 12. Expand into a Laurent series in the annulus $0<|z-1|<2$ the function
$$
f(z)=\frac{1}{\left(z^{2}-1\right)^{2}}
$$ | \frac{1}{(z^{2}-1)^{2}}=\frac{1}{4}\frac{1}{(z-1)^{2}}-\frac{1}{4}\frac{1}{z-1}+\frac{3}{16}-\frac{1}{8}(z-1)+\frac{5}{64}(z-1)^{2}-\frac{3}{64}(z | 52 | 92 |
math | Markus has 9 candies and Katharina has 5 candies. Sanjiv gives away a total of 10 candies to Markus and Katharina so that Markus and Katharina each end up with the same total number of candies. How many candies does Markus have now? | 12 | 59 | 2 |
math | 10.
The answer to the task should be some integer or a number written as a finite decimal. If the answer contains a fractional number, use a comma when writing it. Enter all symbols (comma, digits) without spaces.
Find the range of the function given below. In your answer, specify the value of the difference between the largest and smallest values of the function.
$y=x \cdot|3-x|-(x-3) \cdot|x|$ | 4,5 | 95 | 3 |
math | Triangle $ABC$ satisfies $\tan A \cdot \tan B = 3$ and $AB = 5$. Let $G$ and $O$ be the centroid and circumcenter of $ABC$ respectively. The maximum possible area of triangle $CGO$ can be written as $\frac{a\sqrt{b}}{c}$ for positive integers $a$, $b$, and $c$ with $a$ and $c$ relatively prime and $b$ not divisible by the square of any prime. Find $a + b + c$. | 100 | 114 | 3 |
math | 1. Does there exist a positive integer divisible by 2020, in whose representation the digits $0,1, \cdots, 9$ appear the same number of times? | 12123434565679798080 | 40 | 20 |
math | 4. Find the minimum value of the sum
$$
\left|x-1^{2}\right|+\left|x-2^{2}\right|+\left|x-3^{2}\right|+\ldots+\left|x-10^{2}\right|
$$ | 275 | 57 | 3 |
math | 272. Compute the approximate value: 1) $\sqrt[4]{17} ; 2$ ) $\operatorname{arc} \operatorname{tg} 0.98$ ; 3) $\sin 29^{\circ}$. | \sqrt[4]{17}\approx2.031,\operatorname{arctg}0.98\approx0.7754,\sin29\approx0.4848 | 57 | 46 |
math | 4. The wolf saw a roe deer several meters away from him and chased after her along a straight forest path. The wolf's jump is $22\%$ shorter than the roe deer's jump. Both animals jump at a constant speed. All the roe deer's jumps are of the same length, and the wolf's jumps are also equal to each other. There is a period of time during which both the wolf and the roe deer make a certain whole number of jumps. Each time, it turns out that the wolf has made $t\%$ more jumps than the roe deer. Find the greatest integer value of $\mathrm{t}$, for which the wolf will not be able to catch the roe deer. | 28 | 152 | 2 |
math | $[$ Arithmetic of residues (miscellaneous) $]$ $[$ Equations in integers $]$
Solve the equation $2^{x}-1=5^{y}$ in integers. | 1,0 | 38 | 3 |
math | There are $N{}$ mess-loving clerks in the office. Each of them has some rubbish on the desk. The mess-loving clerks leave the office for lunch one at a time (after return of the preceding one). At that moment all those remaining put half of rubbish from their desks on the desk of the one who left. Can it so happen that after all of them have had lunch the amount of rubbish at the desk of each one will be the same as before lunch if a) $N = 2{}$ and b) $N = 10$?
[i]Alexey Zaslavsky[/i] | 2a_1 = a_2 | 129 | 10 |
math | Find all natural numbers $n$, such that $\min_{k\in \mathbb{N}}(k^2+[n/k^2])=1991$. ($[n/k^2]$ denotes the integer part of $n/k^2$.) | n \in \mathbb{N}, 990208 \leq n \leq 991231 | 56 | 31 |
math | 15. Given the parabola $y^{2}=2 p x$ passes through the fixed point $C(1,2)$, take any point $A$ on the parabola different from point $C$, the line $A C$ intersects the line $y=x+3$ at point $P$, and through point $P$ a line parallel to the $x$-axis intersects the parabola at point $B$.
(1) Prove that the line $A B$ passes through a fixed point;
(2) Find the minimum value of the area of $\triangle A B C$. | 4\sqrt{2} | 129 | 6 |
math | ## Task 4 - 130614
Jörg and Claudia are arguing about whether there are more natural numbers from 0 to 1000 whose decimal representation contains (at least) one 5, or whether there are more that do not contain a 5.
Determine the correct answer to this question! | 271 | 70 | 3 |
math | 9. (16 points) Let the constant $a \in \mathbf{R}$, and the function
$$
f(x)=(a-x)|x|
$$
has an inverse function $f^{-1}(x)$. If the inequality
$$
f^{-1}\left(x^{2}+m\right)<f(x)
$$
holds for all $x \in[-2,2]$, find the range of the real number $m$. | \in(12,+\infty) | 97 | 10 |
math | Problem 1. A wooden rectangular prism with edge lengths as natural numbers has a volume of $250 \mathrm{~cm}^{2}$. With one cut, it is divided into two equal cubes. What is the surface area of the rectangular prism? | 250\mathrm{~}^{2} | 54 | 11 |
math | Find all integers $a$ and $b$ such that $7^{a}-3 \times 2^{b}=1$. | (1,1)(2,4) | 27 | 9 |
math | 11. Let real numbers $x_{1}, x_{2}, \cdots, x_{2014}$ satisfy
$$
\left|x_{1}\right|=99,\left|x_{n}\right|=\left|x_{n-1}+1\right| \text {, }
$$
where, $n=2,3, \cdots, 2014$. Find the minimum value of $x_{1}+x_{2}+\cdots+x_{2014}$. | -5907 | 113 | 5 |
math | Subject 4. Let $a, b, c, d \geq 2$ be natural numbers such that
$$
\log _{a} b=\frac{3}{2}, \quad \log _{c} d=\frac{5}{4}
$$
and $a-c=9$. Calculate $b-d$.
Mathematical Gazette 2013
## GRADING SCALE
10th GRADE | 93 | 92 | 2 |
math | 1. Let $|a|>1$, simplify
$$
\left(a+\sqrt{a^{2}-1}\right)^{4}+2\left(1-2 a^{2}\right)\left(a+\sqrt{a^{2}-1}\right)^{2}+3
$$
the result is $\qquad$ . | 2 | 74 | 1 |
math | Let $ p$ be a prime number. Solve in $ \mathbb{N}_0\times\mathbb{N}_0$ the equation $ x^3\plus{}y^3\minus{}3xy\equal{}p\minus{}1$. | (1, 0, 2), (0, 1, 2), (2, 2, 5) | 55 | 28 |
math | 2. Fifteen numbers are arranged in a circle. The sum of any six consecutive numbers is 50. Petya covered one of the numbers with a card. The two numbers adjacent to the card are 7 and 10. What number is under the card? | 8 | 57 | 1 |
math | Example. Find the indefinite integral
$$
\int \sin ^{4} 3 x \cos ^{4} 3 x d x
$$ | \frac{3}{2^{7}}x-\frac{1}{3\cdot2^{7}}\sin12x+\frac{1}{3\cdot2^{10}}\sin24x+C | 33 | 46 |
math | Example 2.1.1 How many odd numbers with different digits are there between 1000 and 9999? | 2240 | 29 | 4 |
math | 3. Find the numerical value of the expression
$$
\frac{1}{a^{2}+1}+\frac{1}{b^{2}+1}+\frac{2}{a b+1}
$$
if it is known that $a$ is not equal to $b$ and the sum of the first two terms is equal to the third. | 2 | 77 | 1 |
math | 39. From point $K$, located outside a circle with center $O$, two tangents $K M$ and $K N$ are drawn to this circle ($M$ and $N$ are the points of tangency). A point $C(|M C|<|C N|)$ is taken on the chord $M N$. A line perpendicular to segment $O C$ is drawn through point $C$, intersecting segment $N K$ at point $B$. It is known that the radius of the circle is $R, \widehat{M K N}=\alpha,|M C|=b$. Find $|C B|$. | \frac{\sqrt{R^{2}+b^{2}-2Rb\cos\frac{\alpha}{2}}}{\sin\frac{\alpha}{2}} | 136 | 36 |
math | 4. Given that $\overline{73 a b c 6}$ is divisible by 56 $(b<4)$, and $a$ leaves the same remainder when divided by 40, 61, and 810, find all six-digit numbers that satisfy the requirements. | 731136,737016,737296 | 63 | 20 |
math | $[$ Theorem on the lengths of a tangent and a secant; the product of the entire secant and its external part
From point $A$, two rays intersect a given circle: one - at points $B$ and $C$, the other - at points $D$ and $E$. It is known that $A B=7, B C=7, A D=10$. Find $D E$.
# | 0.2 | 90 | 3 |
math | Suppose that a sequence $a_0, a_1, \ldots$ of real numbers is defined by $a_0=1$ and \[a_n=\begin{cases}a_{n-1}a_0+a_{n-3}a_2+\cdots+a_0a_{n-1} & \text{if }n\text{ odd}\\a_{n-1}a_1+a_{n-3}a_3+\cdots+a_1a_{n-1} & \text{if }n\text{ even}\end{cases}\] for $n\geq1$. There is a positive real number $r$ such that \[a_0+a_1r+a_2r^2+a_3r^3+\cdots=\frac{5}{4}.\] If $r$ can be written in the form $\frac{a\sqrt{b}-c}{d}$ for positive integers $a,b,c,d$ such that $b$ is not divisible by the square of any prime and $\gcd (a,c,d)=1,$ then compute $a+b+c+d$.
[i]Proposed by Tristan Shin[/i] | 1923 | 257 | 4 |
math | 1. (3 points) The value of the expression $10-10.5 \div[5.2 \times 14.6-(9.2 \times 5.2+5.4 \times 3.7-4.6 \times 1.5)]$ is | 9.3 | 65 | 3 |
math | Determine, with proof, the smallest positive integer $n$ with the following property: For every choice of $n$ integers, there exist at least two whose sum or difference is divisible by $2009$. | n = 1006 | 45 | 8 |
math | 56. Up the Hill. Willie-Lazybones climbed up the hill at a speed of $1 \frac{1}{2}$ km/h, and descended at a speed of $4 \frac{1}{2}$ km/h, so the entire journey took him exactly 6 hours. How many kilometers is it from the base to the top of the hill? | 6\frac{3}{4} | 75 | 8 |
math | 135. Solve the equation
$$
x(x+1)(x+2)(x+3)=5040
$$ | x_{1}=-10,x_{2}=7 | 29 | 12 |
math | 13.213. Usually, two mechanisms are involved in performing a certain task simultaneously. The productivity of these mechanisms is not the same, and when working together, they complete the task in 30 hours. However, this time the joint operation of the two mechanisms lasted only 6 hours, after which the first mechanism was stopped and the second mechanism completed the remaining part of the task in 40 hours. How long would it take for each mechanism to complete the same task, working separately at its inherent productivity? | 75 | 107 | 2 |
math | Example 3 Find the equation of the curve $E^{\prime}$ symmetric to the curve $E: 2 x^{2}+4 x y+5 y^{2}-22=0$ with respect to the line $l: x-y+1=0$. | 5x^{2}+4xy+2y^{2}+6x-19=0 | 58 | 22 |
math | How many three-digit natural numbers are there such that the sum of these is equal to 24?
---
The text has been translated while preserving the original line breaks and format. | 10 | 36 | 2 |
math | 9. Let the sequence $\left\{a_{n}\right\}$ have the sum of the first $n$ terms denoted by $S_{n}$. The sequence of these sums satisfies
$S_{n}+S_{n+1}+S_{n+2}=6 n^{2}+9 n+7(n \geq 1)$. Given that $a_{1}=1, a_{2}=5$, find the general term formula for the sequence $\left\{a_{n}\right\}$. | a_{n}=4n-3 | 114 | 8 |
math | 3. Let $x$ be a real number. Then the maximum value of the function $y=\sqrt{8 x-x^{2}}-$ $\sqrt{14 x-x^{2}-48}$ is | 2 \sqrt{3} | 44 | 6 |
math | 11.142. In a regular triangular prism, a plane is drawn through the side of the lower base and the opposite vertex of the upper base, forming an angle of $45^{\circ}$ with the plane of the lower base. The area of the section is $S$. Find the volume of the prism. | \frac{S\sqrt{S}\cdot\sqrt[4]{6}}{2} | 68 | 20 |
math | 36. To multiply a two-digit number by 99, you need to decrease this number by one and append the complement of the number to 100. | 100(10A+B-1)+[100-(10A+B)] | 35 | 21 |
math | Example 9 Given a natural number $n \geqslant 2$, find the smallest positive number $\lambda$, such that for any positive numbers $a_{1}, a_{2}, \cdots, a_{n}$, and any $n$ numbers $b_{1}, b_{2}, \cdots, b_{n}$ in $\left[0, \frac{1}{2}\right]$, if $a_{1}+a_{2}+\cdots+a_{n}=b_{1}+b_{2}+\cdots+b_{n}=1$, then $a_{1} a_{2} \cdots a_{n} \leqslant \lambda\left(a_{1} b_{1}+a_{2} b_{2}+\cdots+a_{n} b_{n}\right)$. | \frac{1}{2}(\frac{1}{n-1})^{n-1} | 181 | 21 |
math | ## Task 1 - 220731
The consumer cooperative refunds 1.6% of the amount for which consumer vouchers were redeemed each year. For four families $A, B, C$, and $D$, it is known from one year:
$A$ had redeemed for twice the amount as $B$ or, which is the same, for three times the amount as $C$ and for four times the amount as $D$; the four families $A, B, C, D$ together received $336 \mathrm{DM}$ in refunds.
For each of the four families $A, B, C, D$, the following should be determined from this information:
a) For what amount did this family redeem consumer vouchers in this year?
b) What amount did this family therefore receive in refunds? | =10080M,b=5040M,=3360M,=2520M,FamilyA:161.28M,FamilyB:80.64M,FamilyC:53.76M,FamilyD:40.32M | 172 | 69 |
math | $4 \cdot 200$ A dance party has 42 participants. Lady $A_{1}$ danced with 7 male partners, Lady $A_{2}$ danced with 8 male partners, ..., Lady $A_{n}$ danced with all male partners. How many ladies and male partners are there at the dance party? | 18 | 70 | 2 |
math | Find the number of partitions of the set $\{1,2,3,\cdots ,11,12\}$ into three nonempty subsets such that no subset has two elements which differ by $1$.
[i]Proposed by Nathan Ramesh | 1023 | 54 | 4 |
math | Let $ABC$ be an isosceles triangle with $AB=AC$ and incentre $I$. If $AI=3$ and the distance from $I$ to $BC$ is $2$, what is the square of length on $BC$? | 80 | 55 | 2 |
math | 3. 20 Let $[x]$ denote the greatest integer not exceeding $x$. Try to compute the sum $\sum_{k=0}^{\infty}\left[\frac{n+2^{k}}{2^{k+1}}\right]$ for any positive integer $n$. | n | 62 | 1 |
math | Example 27 (2005 Western China Mathematical Olympiad) Let $S=\{1,2, \cdots, 2005\}$, if any set of $n$ pairwise coprime numbers in $S$ contains at least one prime number, find the minimum value of $n$.
---
The translation maintains the original format and line breaks as requested. | 16 | 81 | 2 |
math | A positive integer is said to be "nefelibata" if, upon taking its last digit and placing it as the first digit, keeping the order of all the remaining digits intact (for example, 312 -> 231), the resulting number is exactly double the original number. Find the smallest possible nefelibata number. | 105263157894736842 | 70 | 18 |
math | 2. $f(-x)=3(-x)^{3}-(-x)=-3 x^{3}+x=-\left(3 x^{3}-x\right)=-f(x)$ $g(-x)=f^{3}(-x)+f\left(\frac{1}{-x}\right)-8(-x)^{3}-\frac{2}{-x}=-f^{3}(x)-f\left(\frac{1}{x}\right)+8 x^{3}+\frac{2}{x}=-g(x)$
Therefore, $g$ is an odd function $\Rightarrow$ if $x_{0}$ is a root of the original equation, then $-x_{0}$ is also a root of the equation $\Rightarrow$ the sum of the roots is zero, if the roots exist
Check $\quad x=1: \quad f(1)=3 \cdot 1^{3}-1=2, \quad f\left(\frac{1}{1}\right)=f(1)=2, \quad$ substitute $\quad$ into the original equation:10=10 - correct $\Rightarrow x=1$ is a root $\Rightarrow$ the roots exist | 0 | 253 | 1 |
math | Folknor
Given two two-digit numbers $-X$ and $Y$. It is known that $X$ is twice as large as $Y$, one digit of the number $Y$ is equal to the sum, and the other digit is equal to the difference of the digits of the number $X$.
Find these numbers. | 3417 | 70 | 4 |
math | 434. The well-known expression $\boldsymbol{F}(n)$, where $n$ denotes a variable positive integer, should be prefixed with a + sign when $n$ is even, and with a - sign when $n$ is odd. How can this be expressed concisely algebraically, without adding any verbal explanation to the formula? | (-1)^{n}F(n) | 73 | 9 |
math | 4. Let three different lines be:
$$
\begin{array}{l}
l_{1}: a x+2 b y+3(a+b+1)=0, \\
l_{2}: b x+2(a+b+1) y+3 a=0, \\
l_{3}:(a+b+1) x+2 a y+3 b=0,
\end{array}
$$
Then the necessary and sufficient condition for them to intersect at one point is $\qquad$ | -\frac{1}{2} | 104 | 7 |
math | 1. Container $A$ contains 6 liters of saltwater with a concentration of $a \%$, and container $B$ contains 4 liters of saltwater with a concentration of $b \%$. After pouring 1 liter of solution from $A$ into $B$ and mixing, then pouring 1 liter of solution from $B$ back into $A$, this process is repeated $k$ times (pouring from $A$ to $B$, then from $B$ to $A$ counts as one repetition). After $k$ repetitions, the concentrations of the saltwater in $A$ and $B$ are $a_{k} \%$ and $b_{k} \%$ respectively.
（1）Find the sequence formed by $b_{k}-a_{k}$;
（2）Find $a_{k}$ and $b_{k}$. | a_{k}=\frac{3+2b}{5}+(-b)\frac{2}{5}(\frac{2}{3})^{k},\quadb_{k}=\frac{3+2b}{5}-(-b)\cdot\frac{3}{5}(\frac{2}{3})^{k} | 180 | 71 |
math | Find all functions $f:\mathbb{N}_0\to\mathbb{N}_0$ for which $f(0)=0$ and
\[f(x^2-y^2)=f(x)f(y) \]
for all $x,y\in\mathbb{N}_0$ with $x>y$. | f(x) = 0 | 69 | 7 |
math | 1. A seagull is being fed from a moving boat. A piece of bread is thrown down, the seagull takes 3 seconds to pick up the piece from the sea surface, and then it takes 12 seconds to catch up with the boat. Upon entering the bay, the boat reduces its speed by half. How much time will it now take the seagull to catch up with the boat after picking up the piece of bread?
Om vem: 2 seconds. | 2 | 100 | 1 |
math | 1. Given $t \in \mathbf{R}_{+}$. Then the minimum value of $\frac{1}{\sqrt{1+3 t^{4}}}+\frac{t^{3}}{\sqrt[4]{12}}$ is $\qquad$ . | \frac{2 \sqrt{2}}{3} | 58 | 12 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.