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 | Determine all natural numbers n for which the number $A = n^4 + 4n^3 +5n^2 + 6n$ is a perfect square of a natural number. | n = 1 | 41 | 5 |
math | Example 7 Find all positive integer pairs $(m, n)$ that satisfy the equation
$$3^{m}=2 n^{2}+1$$ | (m, n)=(1,1),(2,2),(5,11) | 31 | 17 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0}\left(\frac{x^{3}+8}{3 x^{2}+10}\right)^{x+2}$ | 0.64 | 47 | 4 |
math | 4. How many four-digit natural numbers greater than 7777 have the sum of their digits equal to 32? Write them down! | 31 | 31 | 2 |
math | 28th CanMO 1996 Problem 5 Let x 1 , x 2 , ... , x m be positive rationals with sum 1. What is the maximum and minimum value of n - [n x 1 ] - [n x 2 ] - ... - [n x m ] for positive integers n? | 0-1 | 71 | 3 |
math | 10.150. Determine the area of the circular ring enclosed between two concentric circles, the lengths of which are $C_{1}$ and $C_{2}\left(C_{1}>C_{2}\right)$. | \frac{C_{1}^{2}-C_{2}^{2}}{4\pi} | 49 | 22 |
math | Let $n>k>1$ be positive integers. Determine all positive real numbers $a_1, a_2, \ldots, a_n$ which satisfy
$$
\sum_{i=1}^n \sqrt{\frac{k a_i^k}{(k-1) a_i^k+1}}=\sum_{i=1}^n a_i=n .
$$ | a_i = 1 | 82 | 6 |
math | 28. Two brothers sold a flock of sheep that belonged to both of them, taking as many rubles for each sheep as there were sheep in the flock. The money received was divided as follows: first, the elder brother took ten rubles from the total amount, then the younger brother took ten rubles, after that the elder brother t... | 2 | 136 | 1 |
math | 10. [20] A positive real number $x$ is such that
$$
\sqrt[3]{1-x^{3}}+\sqrt[3]{1+x^{3}}=1
$$
Find $x^{2}$. | x^{2}=\frac{\sqrt[3]{28}}{3} | 52 | 17 |
math | 7.291. $\left\{\begin{array}{l}\lg (x+y)-\lg 5=\lg x+\lg y-\lg 6, \\ \frac{\lg x}{\lg (y+6)-(\lg y+\lg 6)}=-1 .\end{array}\right.$ | (2;3) | 68 | 5 |
math | \section*{Problem 2 - 320922}
In the plane, there are four pairwise distinct lines given.
a) What is the maximum possible number of points that are the intersection of (at least) two of the given lines?
b) Determine which of the numbers 0, 1, 2, 3, 4, 5, 6 are possible as the number of such intersection points and w... | 0,1,3,4,5,6 | 94 | 11 |
math | 1. Given $\triangle A B C$ with the three interior angles $\angle A, \angle B, \angle C$ opposite to the sides $a, b, c$ respectively, and satisfying $a \sin A \sin B + b \cos ^{2} A = \sqrt{2} a$. Then the value of $\frac{b}{a}$ is $\qquad$. | \sqrt{2} | 82 | 5 |
math | Example 2.25. $\int_{-\infty}^{0} e^{x} d x$.
Translate the text above into English, keeping the original text's line breaks and format, and output the translation result directly.
Example 2.25. $\int_{-\infty}^{0} e^{x} d x$. | 1 | 74 | 1 |
math | # Task 9.2
Factorize $x^{4}+2021 x^{2}+2020 x+2021$.
## Number of points 7
# | (x^{2}+x+1)(x^{2}-x+2021) | 44 | 20 |
math | 18. $A 、 n$ are natural numbers, and
$$
A=n^{2}+15 n+26
$$
is a perfect square. Then $n$ equals $\qquad$ | 23 | 46 | 2 |
math | Problem 6. Calculate
$$
\operatorname{tg} \frac{\pi}{47} \cdot \operatorname{tg} \frac{2 \pi}{47}+\operatorname{tg} \frac{2 \pi}{47} \cdot \operatorname{tg} \frac{3 \pi}{47}+\ldots+\operatorname{tg} \frac{k \pi}{47} \cdot \operatorname{tg} \frac{(k+1) \pi}{47}+\ldots+\operatorname{tg} \frac{2021 \pi}{47} \cdot \operat... | -2021 | 157 | 5 |
math | ## Task 1 - 100611
A LPG had planted potatoes on two fields. A total of 810 t was harvested from the first field, and a total of 640 t from the second field. The average yield on the first field was 180 dt per ha, and on the second field, it was 200 dt per ha.
Which of the two fields has the larger area? By how many ... | 1300 | 105 | 4 |
math | ## Task 3 - 080923
Give all pairs $(x, y)$ of natural numbers for which $x^{3}-y^{3}=999$! | (12,9)(10,1) | 40 | 11 |
math | Question 28, Given $k \in[-500,500]$, and $k \in Z$, if the equation $\lg (k x)=2 \lg (x+2)$ has exactly one real root, then the number of $k$ that satisfies the condition is $\qquad$ —. | 501 | 68 | 3 |
math | Task B-1.3. Determine the four-digit number which is 594 greater than the number obtained by swapping the two-digit beginning and the two-digit end (moving the first two digits to the end). The difference of the squares of the two-digit beginning and the two-digit end of the given number is 204. | 2014 | 69 | 4 |
math | Cla
Two circles with centers $O$ and $Q$, intersecting each other at points $A$ and $B$, intersect the bisector of angle $O A Q$ at points $C$ and $D$ respectively. Segments $A D$ and $O Q$ intersect at point $E$, and the areas of triangles $O A E$ and $Q A E$ are 18 and 42 respectively. Find the area of quadrilateral... | 200;3:7 | 113 | 7 |
math | ## Task B-2.5.
In how many ways can three numbers be selected from the set $\{1,2,3, \ldots, 12\}$ such that their sum is divisible by 3? | 76 | 47 | 2 |
math | 7. If $\tan 4 x=$
$$
\frac{\sqrt{3}}{3} \text {, then } \frac{\sin 4 x}{\cos 8 x \cos 4 x}+\frac{\sin 2 x}{\cos 4 x \cos 2 x}+\frac{\sin x}{\cos 2 x \cos x}+\frac{\sin x}{\cos x}=
$$
$\qquad$ | \sqrt{3} | 98 | 5 |
math | 5. Given that vectors $\boldsymbol{\alpha}, \boldsymbol{\beta}$ are two unit vectors in a plane with an angle of $60^{\circ}$ between them, and $(2 \boldsymbol{\alpha}-\boldsymbol{\gamma}) \cdot(\boldsymbol{\beta}-\boldsymbol{\gamma})=0$, then the maximum value of $|\gamma|$ is $\qquad$ . | \frac{\sqrt{7}+\sqrt{3}}{2} | 85 | 15 |
math | Does there exist natural numbers $a,b,c$ all greater than $10^{10}$ such that their product is divisible by each of these numbers increased by $2012$? | a = n(\lambda^2 - 1), b = n\lambda, c = n\lambda | 40 | 23 |
math | . A cylindrical hole $6 \mathrm{~cm}$ long is drilled through a sphere, the axis of the cylinder passing through the center of the sphere. What is the remaining volume?
(It is recalled that the volume of a spherical cap is $\pi h^{2}(R-h / 3)$, where $R$ is the radius of the sphere and $h$ is the height of the cap.) | 36\pi\mathrm{}^{3} | 83 | 10 |
math | 238 Test: Can 2007 be expressed in the form
$$
a_{1}^{x_{1}}+a_{2}^{x_{2}}+\cdots+a_{m}^{x_{n}}-b_{1}^{y_{1}}-b_{2}^{y_{2}}-\cdots-b_{n}^{y_{n}}
$$
where $m, n$ are both positive integers greater than 130 and less than 140 (allowing $m$ to equal $n$), $a_{1}, a_{2}, \cdots, a_{m}, b_{1}, b_{2}, \cdots, b_{n}$ are all ... | 2007 | 361 | 4 |
math | 6. For $x>0$, let $f(x)=x^{x}$. Find all values of $x$ for which $f(x)=f^{\prime}(x)$. | 1 | 39 | 1 |
math | 12. Find the unit digit of $17^{17} \times 19^{19} \times 23^{23}$. | 1 | 34 | 1 |
math | ## Task A-2.1.
If $x, y, z$ and $w$ are real numbers such that
$$
x^{2}+y^{2}+z^{2}+w^{2}+x+3 y+5 z+7 w=4
$$
determine the maximum possible value of the expression $x+y+z+w$. | 2 | 78 | 1 |
math | 6. If in the real number range there is
$$
x^{3}+p x+q=(x-a)(x-b)(x-c) \text {, }
$$
and $q \neq 0$, then $\frac{a^{3}+b^{3}+c^{3}}{a b c}=$ $\qquad$ . | 3 | 77 | 1 |
math | 3.322. $\frac{2 \cos ^{2}\left(\frac{9}{4} \pi-\alpha\right)}{1+\cos \left(\frac{\pi}{2}+2 \alpha\right)}-\frac{\sin \left(\alpha+\frac{7}{4} \pi\right)}{\sin \left(\alpha+\frac{\pi}{4}\right)} \cdot \cot\left(\frac{3}{4} \pi-\alpha\right)$. | \frac{4\sin2\alpha}{\cos^{2}2\alpha} | 107 | 19 |
math | Example 7 (Problem from the 28th Russian Mathematical Olympiad): There is a red card box and $k$ blue card boxes ($k>1$), and a deck of cards, totaling $2n$ cards, numbered from 1 to $2n$. Initially, these cards are stacked in the red card box in any order. From any card box, the top card can be taken out, placed in an... | k-1 | 128 | 3 |
math | ## 7. Centipede
Centipede Milica has exactly a hundred legs, 50 left and 50 right. Every morning she puts on 50 pairs of shoes, first all the left ones, and then all the right ones. It takes her one second to put on each left shoe. But then she gets tired, and it takes her more time for the right shoes. It takes her t... | 260 | 183 | 3 |
math | 32nd Putnam 1971 Problem A5 A player scores either A or B at each turn, where A and B are unequal positive integers. He notices that his cumulative score can take any positive integer value except for those in a finite set S, where |S| =35, and 58 ∈ S. Find A and B. Solution | A=11,B=8 | 75 | 7 |
math | $A B$ draw a right-angled triangle $A B C$ over the leg $A B$ such that the sum of the hypotenuse $B C$ and the leg $C A$ is equal to twice the length of $A B$.
Let's denote the length of $A B$ as $x$. Therefore, we have:
\[ BC + CA = 2x \]
Given that $A B C$ is a right-angled triangle with the right angle at $A$... | AC=\frac{3}{4},\quadCB=\frac{5}{4} | 508 | 18 |
math | 11. We define: $a @ b=a \times(a+1) \times \ldots \times(a+b-1)$. Given that $x @ y @ 2=420$, then $y @ x=(\quad)$ | 20or120 | 53 | 6 |
math | 7. (10 points) 30 tigers and 30 foxes are divided into 20 groups, with 3 animals in each group. Tigers always tell the truth, while foxes always lie. When asked if there are any foxes in the group, 39 out of the 60 animals answered “no”. How many groups consist of 3 tigers?
Groups. | 3 | 85 | 1 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0} \frac{e^{3 x}-e^{2 x}}{\sin 3 x-\tan 2 x}$ | 1 | 45 | 1 |
math | 1. Given the sum of 12 distinct positive integers is 2010. Then the maximum value of the greatest common divisor of these positive integers is . $\qquad$ | 15 | 38 | 2 |
math | 16. Check Digit (8th grade. 3 points). The Simple-Dairy Telephone Company “Simple-Telecom” uses three-digit telephone numbers. The equipment is old, so during connection, errors in individual digits of the transmitted subscriber number are possible - each digit, independently of the others, has a probability of $p=0.02... | 0.000131 | 181 | 8 |
math | 8. (SPA 2) In a test, $3 n$ students participate, who are located in three rows of $n$ students in each. The students leave the test room one by one. If $N_{1}(t), N_{2}(t), N_{3}(t)$ denote the numbers of students in the first, second, and third row respectively at time $t$, find the probability that for each $t$ duri... | \frac{(3!)^{n}(n!)^{3}}{(3n)!} | 140 | 18 |
math | 5. [6] Suppose $x$ and $y$ are positive real numbers such that
$$
x+\frac{1}{y}=y+\frac{2}{x}=3 \text {. }
$$
Compute the maximum possible value of $x y$. | 3+\sqrt{7} | 55 | 6 |
math | 8. Let two strictly increasing sequences of positive integers $\left\{a_{n}\right\},\left\{b_{n}\right\}$ satisfy: $a_{10}=b_{10}<2017$, for any positive integer $n$, there is $a_{n+2}=a_{n+1}+a_{n}, b_{n+1}=2 b_{n}$, then all possible values of $a_{1}+b_{1}$ are $\qquad$ . | 13,20 | 110 | 5 |
math | 10. Let $a_{1}=1, a_{n+1}=\sqrt{a_{n}^{2}-2 a_{n}+2}+b,\left(n \in N^{*}\right)$. (1) If $b=1$, find $a_{2}, a_{3}$ and the general term formula of the sequence $\left\{a_{n}\right\}$; (2) If $b=-1$, does there exist a real number $c$ such that $a_{2 n}<c<a_{2 n+1}$, for all $n \in N^{*}$? | \frac{1}{4} | 134 | 7 |
math | Folkolo
Find the maximum value of the expression $ab + bc + ac + abc$, if $a + b + c = 12$ (where $a, b$, and $c$ are non-negative numbers). | 112 | 48 | 3 |
math | 2. Find all four-digit numbers A that satisfy the following three conditions:
a) the first digit of A is twice as small as the last;
b) the second and third digits of A are the same;
c) if the number A is decreased by 2, the result is divisible by 143. | 2004 | 65 | 4 |
math | 2. Given tetrahedron $ABCD$ with volume $V, E$ is a point on edge $AD$, extend $AB$ to $F$ such that $BF=AB$, let the plane through points $C, E, F$ intersect $BD$ at $G$. Then the volume of tetrahedron $CDGE$ is $\qquad$ . | \frac{1}{3} V | 79 | 8 |
math | 4. In the division equation $26 \div$ $\square$
$\square$ $\square$.. .2, both the divisor and the quotient are single-digit numbers. Please write down all the division equations that meet the requirements: $\qquad$ . | 26\div3=8\ldots\ldots\ldots0.2;26\div4=6\ldots\ldots00.2;26\div6=4\ldots\ldots\ldots0.2;26\div8=3\ldots\ldots\ldots0.2 | 53 | 77 |
math | 4・203 There are two coal mines, A and B. Coal from mine A releases 4 calories when burned per gram, and coal from mine B releases 6 calories when burned per gram. The price of coal at the origin is: 20 yuan per ton for mine A, and 24 yuan per ton for mine B. It is known that: the transportation cost of coal from mine A... | 18 | 128 | 2 |
math | Problem 3. The equations $x^{2}+2019 a x+b=0$ and $x^{2}+2019 b x+a=0$ have one common root. What can this root be, given that $a \neq b$? | \frac{1}{2019} | 60 | 10 |
math | Find the smallest prime $p>100$ for which there exists an integer $a>1$ such that $p$ divides $\frac{a^{89}-1}{a-1}$. | 179 | 43 | 3 |
math | Solve the following system of equations:
$$
\begin{aligned}
x+y+z & =2 \\
x^{3}+y^{3}+z^{3} & =20 \\
x^{7}+y^{7}+z^{7} & =2060
\end{aligned}
$$ | 3,1,-2 | 68 | 5 |
math | 5. For which $n$ can a square grid $n \times n$ be divided into one $2 \times 2$ square and some number of strips of five cells, such that the square is adjacent to the side of the board? | 5k+2 | 51 | 4 |
math | 5.12. a) Represent as a sum of squares
\[
\left(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}\right)\left(b_{1}^{2}+b_{2}^{2}+b_{3}^{2}\right)-\left(a_{1} b_{1}+a_{2} b_{2}+a_{3} b_{3}\right)^{2}
\]
b) Represent as a sum of squares
\[
\left(a_{1}^{2}+\ldots+a_{n}^{2}\right)\left(b_{1}^{2}+\ldots+b_{n}^{2}\right)-\left(a_{1} b_{1}+\... | \sum(a_{i}b_{j}-a_{j}b_{i})^{2} | 198 | 21 |
math | Let $ABCD$ and $ABEF$ be two squares situated in two perpendicular planes and let $O$ be the intersection of the lines $AE$ and $BF$. If $AB=4$ compute:
a) the distance from $B$ to the line of intersection between the planes $(DOC)$ and $(DAF)$;
b) the distance between the lines $AC$ and $BF$. | \frac{4\sqrt{3}}{3} | 83 | 13 |
math | 10. Given that $f(x)$ is a function defined on $\mathbf{R}$, $f(1)=1$ and for any $x \in \mathbf{R}$, $f(x+5) \geqslant f(x)+5, f(x+1) \leqslant f(x)+1$. If $g(x)=f(x)+1-x$, then $g(2002)=$ $\qquad$ . | 1 | 99 | 1 |
math | ## Task B-1.1.
Calculate
$$
\frac{20182019^{2}-20182018^{2}}{20182018 \cdot 20182020-20182017 \cdot 20182019}
$$ | 1 | 79 | 1 |
math | 2. Dima, Sasha, Kolya, and Gleb participated in an olympiad and took the first four places. A year later, their classmates managed to recall only three facts: "Dima took first place or Gleb took third," "Gleb took second place or Kolya took first," "Sasha took third place or Dima took second." Who performed better - Di... | Dima | 113 | 2 |
math | # 6. Variant 1
A doll maker makes one doll in 1 hour 45 minutes. After every three dolls made, the master has to rest for half an hour. Ten dolls are needed for gifts. At what time the next day (specify hours and minutes) will the order be completed if the master started making dolls at 10:00 AM and worked through the... | 5:00 | 84 | 4 |
math | 1. (3 points) Calculate: $100-99+98-97+96-95+94-93+93-92+91=$ | 96 | 43 | 2 |
math | At an Antarctic station, there are $n$ polar explorers, all of different ages. With probability $p$, any two polar explorers establish friendly relations, independently of other sympathies or antipathies. When the wintering period ends and it's time to return home, in each pair of friends, the older one gives a friendl... | \frac{1}{p}(1-(1-p)^{n}) | 90 | 15 |
math | 30.9. Find the largest three-digit number such that the number minus the sum of its digits is a perfect square. | 919 | 26 | 3 |
math | Consider the integer $$N = 9 + 99 + 999 + 9999 + \cdots + \underbrace{99\ldots 99}_\text{321 digits}.$$ Find the sum of the digits of $N$. | 342 | 60 | 3 |
math | 333. Find the remainder of the division of the number $\left(85^{70}+19^{32}\right)^{16}$ by 21. | 16 | 40 | 2 |
math | The right vertex is $A, P$ is any point on the ellipse $C_{1}$, and the maximum value of $\overrightarrow{P F_{1}} \cdot \overrightarrow{P F_{2}}$ is in the range $\left[c^{2}, 3 c^{2}\right]$, where $c=\sqrt{a^{2}-b^{2}}$.
(1) The range of the eccentricity $e$ of the ellipse $C_{1}$;
(2) Suppose the hyperbola $C_{2}$ ... | \lambda=2 | 233 | 4 |
math | 37 Let $a$, $b$, $c$ all be positive integers greater than 1. Find the minimum possible value of the algebraic expression $\frac{a+b+c}{2}-\frac{[a, b]+[b, c]+[c, a]}{a+b+c}$. | \frac{3}{2} | 64 | 7 |
math | 1. Pasha, Masha, Tolya, and Olya ate 88 candies, and each of them ate at least one candy. Masha and Tolya ate 57 candies, but Pasha ate the most candies. How many candies did Olya eat? | 1 | 59 | 1 |
math |
Problem 8.3 Find all pairs of prime numbers $p$ and $q$, such that $p^{2}+3 p q+q^{2}$ is:
a) a perfect square;
b) a power of 5 .
| p=3,q=7 | 51 | 6 |
math | 25. In how many ways can the number $n$ be represented as the sum of three positive integer addends, if representations differing in the order of the addends are considered different? | \frac{(n-1)(n-2)}{2} | 39 | 14 |
math | 1. Find the last three digits of $9^{100}-1$. | 0 | 17 | 1 |
math | What is the remainder in the following division:
$$
\left(x^{1001}-1\right):\left(x^{4}+x^{3}+2 x^{2}+x+1\right)
$$
and also when the divisor is the following polynomial:
$$
x^{8}+x^{6}+2 x^{4}+x^{2}+1
$$
(the remainder being the polynomial that results after determining the last non-negative power term of the quot... | -2x^{7}-x^{5}-2x^{3}-1 | 105 | 16 |
math | Let $k \in \mathbb{N},$ $S_k = \{(a, b) | a, b = 1, 2, \ldots, k \}.$ Any two elements $(a, b)$, $(c, d)$ $\in S_k$ are called "undistinguishing" in $S_k$ if $a - c \equiv 0$ or $\pm 1 \pmod{k}$ and $b - d \equiv 0$ or $\pm 1 \pmod{k}$; otherwise, we call them "distinguishing". For example, $(1, 1)$ and $(2, 5)$ are un... | r_5 = 5 | 227 | 7 |
math | Initially given $31$ tuplets
$$(1,0,0,\dots,0),(0,1,0,\dots,0),\dots, (0,0,0,\dots,1)$$
were written on the blackboard. At every move we choose two written $31$ tuplets as $(a_1,a_2,a_3,\dots, a_{31})$ and $(b_1,b_2,b_3,\dots,b_{31})$, then write the $31$ tuplet $(a_1+b_1,a_2+b_2,a_3+b_3,\dots, a_{31}+b_{31})$ to the... | 87 | 223 | 2 |
math | 11. Let real numbers $x, y, z, w$ satisfy $x+y+z+w=1$. Then the maximum value of $M=x w+2 y w+3 x y+3 z w+4 x z+5 y z$ is $\qquad$ . | \frac{3}{2} | 60 | 7 |
math | Example 13 (1993 Shanghai Competition Question) Given real numbers $x_{1}, x_{2}, x_{3}$ satisfy the equations $x_{1}+\frac{1}{2} x_{2}+\frac{1}{3} x_{3}=$ 1 and $x_{1}^{2}+\frac{1}{2} x_{2}^{2}+\frac{1}{3} x_{3}^{2}=3$, then what is the minimum value of $x_{3}$? | -\frac{21}{11} | 114 | 9 |
math | 8. Given that $\triangle A B C$ is an inscribed triangle in the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$, and $A B$ passes through the point $P(1,0)$. Then the maximum area of $\triangle A B C$ is $\qquad$ . | \frac{16\sqrt{2}}{3} | 73 | 13 |
math | ## Task B-2.7.
Determine the value of the real parameter $m$ so that the solutions of the equation
$$
(m x-1) \cdot x=m x-2
$$
represent the lengths of the legs of a right triangle with a hypotenuse of length $\frac{5}{6}$. | 6 | 69 | 1 |
math | \section*{Problem 2 - 131032}
Determine all pairs \((x ; y)\) of integers \(x, y\) that satisfy the equation \(2 x^{3}+x y-7=0\)! | (-7,-99),(-1,-9),(1,5),(7,-97) | 53 | 20 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0} \frac{\sin ^{2} x-\tan^{2} x}{x^{4}}$ | -1 | 42 | 2 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0}\left(2-3^{\sin ^{2} x}\right)^{\frac{1}{\ln (\cos x)}}$ | 9 | 49 | 1 |
math | Let $m,n$ be natural numbers and let $d = gcd(m,n)$. Let $x = 2^{m} -1$ and $y= 2^n +1$
(a) If $\frac{m}{d}$ is odd, prove that $gcd(x,y) = 1$
(b) If $\frac{m}{d}$ is even, Find $gcd(x,y)$ | 2^d + 1 | 83 | 6 |
math | 1. Let $a_{m+n}=A$ and $a_{m-n}=B$ be terms of an arithmetic progression. Express the terms $a_{n}$ and $a_{m}$ in terms of $A$ and $B$. | a_{}=\frac{A+B}{2} | 51 | 11 |
math | Example 2. Factorize in the set of real numbers $R$:
$$
a^{4}-14 a^{2}+25
$$ | (a+1+\sqrt{6})(a+1-\sqrt{6})(a-1+\sqrt{6})(a-1-\sqrt{6}) | 33 | 32 |
math | 11.8. In an alphabet of $n>1$ letters; a word is any finite sequence of letters in which any two adjacent letters are different. A word is called good if it is impossible to erase all letters from it, except for four, so that the remaining sequence is of the form $a a b b$, where $a$ and $b$ are different letters. Find... | 2n+1 | 99 | 4 |
math | ## 243. Math Puzzle $8 / 85$
In a section of a steelworks, there is a small and a large Siemens-Martin furnace. The larger one produces three times as much as the smaller one. Together, they melt a certain amount of steel in one week.
In how many days can each of the furnaces produce this amount of steel on its own? | 28 | 80 | 2 |
math | Example 22 (2003 China National Training Team Test Question) A positive integer cannot be divisible by $2$ or $3$, and there do not exist non-negative integers $a, b$, such that $\left|2^{a}-3^{b}\right|=n$, find the minimum value of $n$.
untranslated text remains in its original format and line breaks are preserved. | 35 | 82 | 2 |
math | 3. Find all functions $f: \mathbb{Z}^{+} \cup\{0\} \rightarrow \mathbb{R}$ such that
$$
f(n+m)+f(n-m)=f(3 n), \quad \forall n, m \in \mathbb{Z}^{+} \cup\{0\}, n \geq m
$$ | f(n)=0 | 82 | 4 |
math | 9. Let $f, g: \mathbf{N} \rightarrow \mathbf{N}, f(n)=2 n+1, g(1)=3$ and
$$
g(n) \geqslant f[g(n-1)], \forall n \geqslant 2 .
$$
Find $g(n)$. | (n)=3\times2^{n-1}+2^{n-1}-1 | 73 | 19 |
math | Let $N$ be a positive integer; a divisor of $N$ is called [i]common[/i] if it's great than $1$ and different of $N$. A positive integer is called [i]special[/i] if it has, at least, two common divisors and it is multiple of all possible differences between any two of their common divisors.
Find all special integers. | 6, 8, 12 | 84 | 8 |
math | H5. A two-digit number is divided by the sum of its digits. The result is a number between 2.6 and 2.7 .
Find all of the possible values of the original two-digit number. | 29 | 45 | 2 |
math | $5$ airway companies operate in a country consisting of $36$ cities. Between any pair of cities exactly one company operates two way flights. If some air company operates between cities $A, B$ and $B, C$ we say that the triple $A, B, C$ is [i]properly-connected[/i]. Determine the largest possible value of $k$ such that... | k = 3780 | 100 | 8 |
math | A given radius sphere is to be enclosed by a frustum of a cone, whose volume is twice that of the sphere. The radii of the base and top of the frustum, as well as the radius of the circle along which the mantle of the frustum touches the sphere, are to be determined. | \zeta=\frac{2R\sqrt{5}}{5} | 63 | 16 |
math | 1. (8 points) Calculate $(235-2 \times 3 \times 5) \times 7 \div 5=$ | 287 | 31 | 3 |
math | 1. Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{0}=3, a_{1}=9, a_{n}=4 a_{n-1}-3 a_{n-2}-4 n+2(n \geqslant 2)$. Try to find all non-negative integers $n$, such that $a_{n}$ is divisible by 9.
(Wu Weizhao) | 1,9k+7,9k+8(k=0,1,\cdots) | 92 | 20 |
math | Find a positive integrer number $n$ such that, if yor put a number $2$ on the left and a number $1$ on the right, the new number is equal to $33n$. | 87 | 45 | 2 |
math | $12.55 \int_{0}^{\pi / 4}(\sin 2 t-\cos 2 t)^{2} d t$. | 0.25(\pi-2) | 35 | 9 |
math | G3.1 Let $a=\sqrt{1997 \times 1998 \times 1999 \times 2000+1}$, find the value of $a$. | 3994001 | 46 | 7 |
math | 3-2. Two identical polygons were cut out of cardboard, aligned, and pierced with a pin at some point. When one of the polygons is rotated around this "axis" by $25^{\circ} 30^{\prime}$, it aligns again with the second polygon. What is the smallest possible number of sides of such polygons? | 240 | 73 | 3 |
math | Problem 6.8. In class 6 "A", there are several boys and girls. It is known that in 6 "A"
- girl Tanya is friends with 12 boys;
- girl Dasha is friends with 12 boys;
- girl Katya is friends with 13 boys;
- any girl will have a friend among any three boys.
How many boys can there be in 6 "A"? List all possible options. | 13,14 | 95 | 5 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.