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 | 1. Add the same integer $a(a>0)$ to the numerator and denominator of $\frac{2008}{3}$, making the fraction an integer. Then the integer $a$ added has $\qquad$ solutions. | 3 | 49 | 1 |
math | Let $k \geq 1$ and $N>1$ be two integers. On a circle are placed $2N+1$ coins all showing heads. Calvin and Hobbes play the following game. Calvin starts and on his move can turn any coin from heads to tails. Hobbes on his move can turn at most one coin that is next to the coin that Calvin turned just now from tails to... | k \in \{1, 2, \dots, N+1\} | 140 | 19 |
math | Aurumban the value of worked gold items is proportional to the square of their mass. Thieves steal a gold item worth 100 petaks, and from this they make identical medals whose total value is 10 petaks. A jeweler buys the medals and from them (not necessarily of the same mass) makes necklaces in such a way that an integ... | 1,9,36 | 109 | 6 |
math | Example 5 Let $n(\geqslant 3)$ be a positive integer, and $M$ be an $n$-element set. Find the maximum positive integer $k$ such that:
there exists a family $\psi$ of $k$ three-element subsets of $M$ such that the intersection of any two elements of $\psi$ (note: elements of $\psi$ are three-element sets) is non-empty. | C_{n-1}^{2} | 90 | 9 |
math | 13.389 From two pieces of alloy of the same mass but with different percentage content of copper, pieces of equal mass were cut off. Each of the cut pieces was melted with the remainder of the other piece, after which the percentage content of copper in both pieces became the same. How many times smaller is the cut pie... | 2 | 72 | 1 |
math | 4 $[\quad$ Equifacetal Tetrahedron $\quad]$
In the triangular pyramid $A B C D$, the sums of the three dihedral angles at each of the vertices $B$ and $C$ are $180^{\circ}$ and $A D=B C$. Find the volume of the pyramid if the area of the face $B C D$ is 100, and the distance from the center of the circumscribed sphere... | 400 | 114 | 3 |
math | 3. If the orthocenter of $\triangle O A B$ is exactly the focus of the parabola $y^{2}=4 x$, where $O$ is the origin, and points $A$ and $B$ are on the parabola, then the area $S$ of $\triangle O A B$ is $\qquad$ . | 10\sqrt{5} | 74 | 7 |
math | 175. On the curve $y=x^{2}-3 x+5$, find the point where the ordinate $y$ increases 5 times faster than the abscissa $x$. | (4,9) | 41 | 5 |
math | Let's determine those three-digit prime numbers in which the product of the digits is 189. | 379,397,739,937 | 21 | 15 |
math | Find all positive integers $n$ for which both $837 + n$ and $837 - n$ are cubes of positive integers. | 494 | 31 | 3 |
math | (13) Given the sequence $\left\{a_{n}\right\}$ satisfies: $a_{1}=1, a_{n+1}=1+a_{n}+\sqrt{1+4 a_{n}}(n \in$ $\mathbf{N}^{*}$ ), then the general term of the sequence $a_{n}=$ $\qquad$ . | 1+(n-1)(n+\sqrt{5}-1) | 81 | 14 |
math | Problem 2. The lighthouse keeper of Finisterre has received communication that there will be a power outage and he must operate the lighthouse using a diesel generator. This generator consumes 6 liters of diesel per hour and an additional half liter each time it needs to be started (initially, it is off). During the 10... | 47.5 | 136 | 4 |
math | A derangement is a permutation of size $n$ such that for all $1 \leq i \leq n$ we have $p(i) \neq i$.
How many derangements of size $n$ are there? | n!(\frac{1}{2!}-\frac{1}{3!}+\cdots+\frac{(-1)^{n}}{n!}) | 51 | 34 |
math | Example 2 Given that the sum of 10 natural numbers is 1001. What is the maximum possible value of their greatest common divisor?
untranslated text:
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
Note: The last part of the text is a note about the translation instruction and is not part of the content to be translated. Here is ... | 91 | 125 | 2 |
math | ## 26.
So, let's start with a well-known problem. Three inhabitants of the island (A, B, and C) were talking to each other in the garden. A passerby asked A: "Are you a knight or a liar?" A answered, but so indistinctly that the passerby could not understand anything. Then the passerby asked B: "What did A say?" "A sa... | B | 130 | 1 |
math | 2. Find the product of all roots of the equation $x^{4}+4 x^{3}-2015 x^{2}-4038 x+2018=0$. | 2018 | 43 | 4 |
math | 2. (10 points) The little rabbit and the little turtle start from location $A$ to the forest amusement park at the same time. The little rabbit jumps forward 36 meters per minute, and after every 3 minutes of jumping, it plays on the spot. The first time it plays for 0.5 minutes, the second time for 1 minute, the third... | 12 | 180 | 2 |
math | Example 2. Solve the system of equations:
$$
\left\{\begin{array}{l}
x+y+z=6 \\
x^{2}+y^{2}+z^{2}=14 \\
x^{3}+y^{3}+z^{3}=36
\end{array}\right.
$$ | (1,2,3),(1,3,2),(2,1,3),(2,3,1),(3,1,2),(3,2,1) | 70 | 37 |
math | 1. Find all polynomials $P(x)$ such that $(x+100) P(x)-x P(x+1)=1$ for every real number $x$. | P(x)=(x+1)(x+2)\cdots(x+99)+\frac{1}{100} | 37 | 27 |
math | 2. In $\square A B C D$, $\angle B<90^{\circ}, A B<B C$. From point $D$ draw tangents to the circumcircle $\Gamma$ of $\triangle A B C$, the points of tangency are $E$ and $F$. It is known that $\angle E D A=\angle F D C$. Find $\angle A B C$ | 60^{\circ} | 82 | 6 |
math | Let $a,b$ and $c$ be real numbers such that $| (a-b) (b-c) (c-a) | = 1$. Find the smallest value of the expression $| a | + | b | + | c |$. (K.Satylhanov ) | \sqrt[3]{4} | 61 | 7 |
math | 13. (3 points) A math competition has 10 questions, with a rule that each correct answer earns 5 points, and each wrong answer or no answer deducts 2 points. $A$ and $B$ each answer the questions, and their total score is 58 points, with $A$ scoring 14 points more than $B$. Then $A$ answered $\qquad$ questions correctl... | 8 | 89 | 1 |
math | ## Task A-4.7.
Determine all ordered triples $(x, y, p)$ where $p$ is prime, and $x$ and $y$ are natural numbers such that
$$
p^{x}-1=y^{3}
$$ | (1,1,2)(2,2,3) | 53 | 13 |
math | 4. The length of the escalator is 200 steps. When Petya walks down the escalator, he manages to count 50 steps. How many steps will he count if he runs twice as fast? | 80 | 47 | 2 |
math | If a sequence $\left\{a_{1}, a_{2}, \ldots, a_{n}\right\}$ of positive integers (where $n$ is a positive integer) has the property that the last digit of $a_{k}$ is the same as the first digit of $a_{k+1}$ (here $k=1,2, \ldots, n$ and we define $\left.a_{n+1}=a_{1}\right)$, then the sequence is said to be a 'dragon seq... | 46 | 371 | 2 |
math | ## Task 4 - 090834
Let $K_{1}, K_{2}, K_{3}, K_{4}$ be four concentric circles, for whose radii $r_{1}, r_{2}, r_{3}$, and $r_{4}$
$$
r_{4}-r_{3}=r_{3}-r_{2}=r_{2}-r_{1}=r_{1} \quad \text { holds. }
$$
Determine the ratio of the area of $K_{1}$ to the areas of the three annuli formed by $K_{1}$ and $K_{2}$, $K_{2}$ ... | 1:3:5:7 | 162 | 7 |
math | Jonah recently harvested a large number of lychees and wants to split them into groups. Unfortunately, for all $n$ where $3\leq n\leq8$, when the lychees are distributed evenly into $n$ groups, $n-1$ lychees remain. What is the smallest possible number of lychees that Jonah could have? | 839 | 77 | 3 |
math | 163. Reduced Share. A father gave his children 6 dollars for entertainment, which was to be divided equally. But two young cousins joined the company. The money was divided equally among all the children, so that each child received 25 cents less than originally intended. How many children were there in total? | 8 | 65 | 1 |
math | Example 2 The equation $x^{10}+(13 x-1)^{10}=0$ has 10 complex roots $r_{1}, \overline{r_{1}}, \overline{r_{2}}, \overline{r_{2}}, \overline{r_{3}}, \overline{r_{3}}, \overline{r_{4}}, \overline{r_{4}}, \overline{r_{5}}, \overline{r_{5}}$, where $\overline{r_{i}}$ is the complex conjugate of $r_{i}$ $(i=1,2,3,4,5)$, fi... | 850 | 224 | 3 |
math | 9. (14 points) In $\triangle ABC$, it is known that the sides opposite to $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, and $c$ respectively, the circumradius of $\triangle ABC$ is $R=\sqrt{3}$, and it satisfies
$\tan B + \tan C = \frac{2 \sin A}{\cos C}$.
Find: (1) $\angle B$, $b$;
(2) the maximum area of $\tri... | \frac{9 \sqrt{3}}{4} | 112 | 12 |
math | 8. $A B$ is the common perpendicular segment of skew lines $a, b$, $A$ is on line $a$, $B$ is on line $b$, $A B=2$, the skew lines $a, b$ form a $30^{\circ}$ angle, and on line $a$ take $A P=4$, then the distance from point $P$ to line $b$ is | 2\sqrt{2} | 89 | 6 |
math | Example 2 Let $a \in \mathbf{N}_{+}$. Find the function $f: \mathbf{N}_{+} \rightarrow \mathbf{R}$, such that for any $x, y \in \mathbf{N}_{+}, x y>a$, we have
$$
f(x+y)=f(x y-a) .
$$ | f(x)=f(1) | 78 | 7 |
math | Determine all prime numbers $p$ such that $p^2 - 6$ and $p^2 + 6$ are both prime numbers. | p = 5 | 32 | 5 |
math | Example 1 (2002 National College Entrance Examination) Given $\sin ^{2} 2 \alpha+\sin 2 \alpha \cos \alpha-\cos 2 \alpha=1, \alpha \in\left(0, \frac{\pi}{2}\right)$, find the values of $\sin \alpha$ and $\tan \alpha$. | \sin\alpha=\frac{1}{2},\tan\alpha=\frac{\sqrt{3}}{3} | 77 | 25 |
math | Problem 2. A group of adventurers is showing off their loot. It is known that exactly 4 adventurers have rubies; exactly 10 have emeralds; exactly 6 have sapphires; exactly 14 have diamonds. Moreover, it is known that
- if an adventurer has rubies, then they have either emeralds or diamonds (but not both at the same t... | 18 | 128 | 2 |
math | 4. For any $a, b \in \mathbf{R}$,
$$
\max \{|a+b|,|a-b|,|1-b|\}
$$
the minimum value is $\qquad$. | \frac{1}{2} | 47 | 7 |
math | Determine all pairs of natural numbers $a$ and $b$ such that $\frac{a+1}{b}$ and $\frac{b+1}{a}$ they are natural numbers. | (1, 1), (1, 2), (2, 3), (2, 1), (3, 2) | 42 | 30 |
math | Find all polynomials with real coefficients such that $\forall a, b, c \in \mathbb{R}$,
$$
P(a+b-2 c)+P(b+c-2 a)+P(c+a-2 b)=3 P(a-b)+3 P(b-c)+3 P(c-a)
$$ | P(X)=aX^{2}+bX | 64 | 11 |
math | Kristýna chose a certain odd natural number divisible by three. Jakub and David then examined triangles whose perimeter in millimeters was equal to the number chosen by Kristýna and whose sides had lengths in millimeters expressed by different whole numbers.
Jakub found such a triangle in which the longest side had th... | 2019 | 196 | 4 |
math | 1. Determine all irreducible fractions $\frac{a}{b}, a, b \in \mathbb{N}$, such that
$$
\frac{a}{b}-\frac{b}{a}=2 \frac{71}{80}
$$ | \frac{16}{5} | 56 | 8 |
math | 3B. Determine all complex numbers $z$ for which
$$
z \bar{z}+1=-i(z-\bar{z})
$$ | i | 32 | 1 |
math | 13.011. Due to the reconstruction of equipment, the labor productivity of a worker increased twice during the year by the same percentage. By what percentage did the labor productivity increase each time, if during the same time a worker used to produce goods worth 2500 rubles, and now produces goods worth 2809 rubles? | 6 | 74 | 1 |
math | G4.4 If $W=2006^{2}-2005^{2}+2004^{2}-2003^{2}+\ldots+4^{2}-3^{2}+2^{2}-1^{2}$, find the value of $W$. | 2013021 | 66 | 7 |
math | The numbers $\frac{1}{1}, \frac{1}{2}, \cdots , \frac{1}{2012}$ are written on the blackboard. Aïcha chooses any two numbers from the blackboard, say $x$ and $y$, erases them and she writes instead the number $x + y + xy$. She continues to do this until only one number is left on the board. What are the possible values... | 2012 | 98 | 4 |
math | 5. The cells of a $9 \times 9$ board are painted in black and white in a checkerboard pattern. How many ways are there to place 9 rooks on the same-colored cells of the board so that they do not attack each other? (A rook attacks any cell that is in the same row or column as itself.) | 2880 | 72 | 4 |
math | 7. In the expansion of $\left(x+\frac{4}{x}-4\right)^{5}$, the coefficient of $x^{3}$ is $\qquad$ .(answer with a specific number) | 180 | 45 | 3 |
math | 7. The real numbers $x, y, z, w$ satisfy
$$
\begin{array}{l}
2 x+y+z+w=1 \\
x+3 y+z+w=2 \\
x+y+4 z+w=3 \\
x+y+z+5 w=25 .
\end{array}
$$
Find the value of $w$. | \frac{11}{2} | 75 | 8 |
math | There were seven boxes. In some of them, seven more boxes (not nested within each other) were placed, and so on. In the end, there were 10 non-empty boxes.
How many boxes are there in total? | 77 | 48 | 2 |
math | Example 3 Solve the equation
$$
\sqrt{x}+\sqrt{y-1}+\sqrt{z-2}=\frac{1}{2}(x+y+z) .
$$ | x=1, y=2, z=3 | 40 | 11 |
math | Yesterday, Alex, Beth, and Carl raked their lawn. First, Alex and Beth raked half of the lawn together in $30$ minutes. While they took a break, Carl raked a third of the remaining lawn in $60$ minutes. Finally, Beth joined Carl and together they finished raking the lawn in $24$ minutes. If they each rake at a constant... | 3 | 99 | 1 |
math | Example 3 Let real numbers $x_{1}, x_{2}, \cdots, x_{1997}$ satisfy the following two conditions:
(1) $-\frac{1}{\sqrt{3}} \leqslant x_{i} \leqslant \sqrt{3}(i=1,2, \cdots, 1997)$;
(2) $x_{1}+x_{2}+\cdots+x_{1997}=-318 \sqrt{3}$.
Try to find the maximum value of $x_{1}^{12}+x_{2}^{12}+\cdots+x_{197}^{12}$, and explain... | 189548 | 159 | 6 |
math | [ Combinatorial geometry (other).] [ Symmetric strategy ]
Kolya and Vitya are playing the following game on an infinite grid paper. Starting with Kolya, they take turns marking the nodes of the grid paper - the points of intersection of vertical and horizontal lines. Each of them, on their turn, must mark such a node ... | Vitya | 116 | 3 |
math | How many six-digit numbers can be formed with six different digits? Among these numbers, how many have four odd digits? | 33600 | 24 | 5 |
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 | 3. 17 (1) Simplify $\frac{1-a^{2}}{(1+a x)^{2}-(a+x)^{2}}$;
(2) When $x=0.44$, find the value of $\sqrt{1-x-x^{2}+x^{3}}$. | 0.672 | 66 | 5 |
math | 2. Natural numbers $x$ and $y$ are such that the following equality holds:
$$
x^{2}-3 x=25 y^{2}-15 y
$$
How many times greater is the number $x$ than the number $y$? | 5 | 57 | 1 |
math | 198. Calculate the sum:
$$
S=(1+1^2)+(2+2^2)+(3+3^2)+\cdots+(n+n^2)
$$ | \frac{1}{3}n(n+1)(n+2) | 40 | 16 |
math | 10.5. We will call a natural number semi-prime if it is greater than 25 and is the sum of two distinct prime numbers. What is the maximum number of consecutive natural numbers that can be semi-prime? Justify your answer. | 5 | 53 | 1 |
math | 11. If $\left\{\begin{array}{l}A=\frac{1}{1 \times 2}+\frac{1}{3 \times 4}+\cdots+\frac{1}{2003 \times 2004} \\ B=\frac{1}{1003 \times 2004}+\frac{1}{1004 \times 2003}+\cdots+\frac{1}{2004 \times 1003}\end{array}\right.$, find $\frac{A}{B}$.
(1 mark)
若 $\left\{\begin{array}{l}A=\frac{1}{1 \times 2}+\frac{1}{3 \times 4}... | \frac{3007}{2} | 261 | 10 |
math | Let $n$ be a positive integer. Determine, in terms of $n,$ the largest integer $m$ with the following property: There exist real numbers $x_1,\ldots, x_{2n}$ with $-1<x_1<x_2<\ldots<x_{2n}<1$ such that the sum of the lengths of the $n$ intervals $$[x_1^{2k-1},x_2^{2k-1}], [x_3^{2k-1},x_4^{2k-1}], \ldots, [x_{2n-1}^{2k-... | n | 171 | 1 |
math | 11. (This question is worth 20 points) Let the ellipse $C_{1}: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ have left and right foci $F_{1}$ and $F_{2}$, respectively, and the right vertex be $A$. Let $P$ be any point on the ellipse $C_{1}$, and the maximum value of $\overrightarrow{P F_{1}} \cdot \overrightarrow{... | 2 | 308 | 1 |
math | Let $A$ and $B$ be digits between $0$ and $9$, and suppose that the product of the two-digit numbers $\overline{AB}$ and $\overline{BA}$ is equal to $k$. Given that $k+1$ is a multiple of $101$, find $k$.
[i]Proposed by Andrew Wu[/i] | 403 | 78 | 3 |
math | 3. The equation $x^{2}+a x+3=0$ has two distinct roots $x_{1}$ and $x_{2}$; in this case,
$$
x_{1}^{3}-\frac{99}{2 x_{2}^{2}}=x_{2}^{3}-\frac{99}{2 x_{1}^{2}}
$$
Find all possible values of $a$. | -6 | 92 | 2 |
math | 4. Let $x, y, z$ be non-negative real numbers, and satisfy the equation
$$
4^{\sqrt{5 x+9 y+4 z}}-68 \times 2^{\sqrt{5 x+9 y+4 x}}+256=0 \text {, }
$$
Then the product of the maximum and minimum values of $x+y+z$ is $\qquad$ . | 4 | 91 | 1 |
math | Determine all functions $f$ from the set of non-negative integers to itself such that $f(a + b) = f(a) + f(b) + f(c) + f(d)$, whenever $a, b, c, d$, are non-negative integers satisfying $2ab = c^2 + d^2$. | f(n) = n^2 f(1) | 68 | 12 |
math | 336. Using Euler's theorem, find such values of $x$ for which $a x-b$ is divisible by $c$, if $(a, c)=1$. | ^{\varphi()-1}+ | 37 | 8 |
math | Example 4.2.3. Let $x_{1}, x_{2}, \ldots, x_{2005}$ be real numbers belonging to $[-1,1]$. Find the minimum value for the following expression
$$P=x_{1} x_{2}+x_{2} x_{3}+\ldots+x_{2004} x_{2005}+x_{2005} x_{1}$$ | -2003 | 99 | 5 |
math | One, (20 points) Given that $a$ is an integer, the system of equations about $x, y$
$$
\left\{\begin{array}{l}
x+y=(a+2) x, \\
x y=\left(a^{2}+1\right) x-2 a^{3}+2
\end{array}\right.
$$
all solutions $(x, y)$ are integers. Try to find the value of $a$. | -1,0,1 | 99 | 6 |
math | 10. All positive integer solutions to the equation $5(x y+y z+z x)=4 x y z$, with $x \leqslant y \leqslant z$ are
$\qquad$ . | (2,5,10),(2,4,20) | 47 | 15 |
math | 6. Let $a, b, c, d$ be odd numbers, $0<a<b<c<d$, and $a d=b c, a+d=2^{k}, b+c=2^{m}, k, m$ be integers, find the value of $a$.
| 1 | 59 | 1 |
math | ## Condition of the problem
Find the derivative.
$$
y=\frac{1+8 \operatorname{ch}^{2} x \cdot \ln (\operatorname{ch} x)}{2 \operatorname{ch}^{2} x}
$$ | \frac{\sinhx\cdot(4\cosh^{2}x-1)}{\cosh^{3}x} | 55 | 27 |
math | 371. Calculate: a) $\sin 110^{\circ}$; b) $\operatorname{tg} 945^{\circ}$; c) $\cos \frac{25 \pi}{4}$. | \frac{1}{2},1,\frac{\sqrt{2}}{2} | 51 | 18 |
math | For a real number $x$ let $\lfloor x\rfloor$ be the greatest integer less than or equal to $x$, and define $\{x\} = x - \lfloor x \rfloor$ to be the fractional part of $x$. For example, $\{3\} = 0$ and $\{4.56\} = 0.56$. Define $f(x)=x\{x\}$, and let $N$ be the number of real-valued solutions to the equation $f(f(f(x))... | 10 | 153 | 2 |
math | 33. Find the number of even digits in the product of the two 10 -digit numbers
$$
2222222222 \times 9999999999 .
$$ | 10 | 49 | 2 |
math | 7.242. $\left(16 \cdot 5^{2 x-1}-2 \cdot 5^{x-1}-0.048\right) \lg \left(x^{3}+2 x+1\right)=0$. | 0 | 57 | 1 |
math | 4.004. Find the first three terms $a_{1}, a_{2}, a_{3}$ of an arithmetic progression, given that $a_{1}+a_{3}+a_{5}=-12$ and $a_{1} a_{3} a_{5}=80$. | 2,-1,-4;-10,-7,-4 | 67 | 12 |
math | Let $k$ be a positive integer. Find the smallest positive integer $n$ for which there exists $k$ nonzero vectors $v_1,v_2,…,v_k$ in $\mathbb{R}^n$ such that for every pair $i,j$ of indices with $|i-j|>1$ the vectors $v_i$ and $v_j$ are orthogonal.
[i]Proposed by Alexey Balitskiy, Moscow Institute of Physics and Techno... | \left\lceil \frac{k}{2} \right\rceil | 108 | 15 |
math | 5. [7] A triangle has altitudes of length 15,21 , and 35 . Find its area. | 245\sqrt{3} | 28 | 8 |
math | 2. Having walked $4 / 9$ of the length of the bridge, the traveler noticed that a car was catching up to him, but it had not yet entered the bridge. Then he turned back and met the car at the beginning of the bridge. If he had continued his movement, the car would have caught up with him at the end of the bridge. Find ... | 9 | 90 | 1 |
math | 11.16 For what value of $m$ does the system of equations
$$
\left\{\begin{array}{l}
2 x+(m-1) y=3 \\
(m+1) x+4 y=-3
\end{array}\right.
$$
have an infinite number of solutions? No solutions? | -3 | 71 | 2 |
math | 3.16. Reduce the equation of the circle $x^{2}-2 x+y^{2}+$ $+6 y=6$ to its canonical form. Determine the coordinates of the center and the radius. | (x-1)^{2}+(y+3)^{2}=4^{2},centerO(1,-3),radiusR=4 | 45 | 30 |
math | Task 4. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy
$$
f(x y-1)+f(x) f(y)=2 x y-1
$$
for all $x, y \in \mathbb{R}$. | f(x)=x | 64 | 4 |
math | 8. Given that $p$ and $q$ are both prime numbers, and $7p+q$, $2q+11$ are also prime numbers. Then $p^{q}+q^{p}=$ $\qquad$ . | 17 | 52 | 2 |
math | Example 3.10. Find the integral $\int \frac{d x}{\sqrt{2 x+3}+\sqrt[3]{(2 x+3)^{2}}}$. | \frac{3}{2}\sqrt[3]{2x+3}-3\sqrt[6]{2x+3}+3\ln|\sqrt[6]{2x+3}+1|+C | 41 | 46 |
math | Task B-3.3. (20 points) Calculate the sum
$$
\frac{\sin 1}{\cos 0 \cdot \cos 1}+\frac{\sin 1}{\cos 1 \cdot \cos 2}+\frac{\sin 1}{\cos 2 \cdot \cos 3}+\ldots+\frac{\sin 1}{\cos 2008 \cdot \cos 2009}
$$ | \tan2009 | 100 | 6 |
math | \section*{Problem 4 - 061234}
Determine all and only those real numbers \(x\) that satisfy the equation
\[
\left[\frac{5+6 x}{8}\right]=\frac{15 x-7}{5}
\]
where \([a]\) denotes the greatest integer not greater than \(a\); for example, \(\left[\frac{13}{2}\right]=6,[-6.5]=-7\) and \([6]=6\). | \frac{7}{15} | 111 | 8 |
math | $1 \cdot 111$ An increasing integer sequence, if its 1st term is odd, the 2nd term is even, the 3rd term is odd, the 4th term is even, and so on, is called an alternating sequence. The empty set is also considered an alternating sequence. The number of all alternating sequences with each term taken from the set $\{1,2,... | 17711 | 129 | 5 |
math | In a triangle $ABC$, the incircle touches the sides $BC, CA, AB$ at $D, E, F$ respectively. If the radius if the incircle is $4$ units and if $BD, CE , AF$ are consecutive integers, find the sides of the triangle $ABC$. | 13, 14, 15 | 63 | 10 |
math | The initial terms of a sequence are: $c_{0}=2, c_{1}=3$, and subsequent terms for $k=2,3,4, \ldots$ can be calculated using the relation $c_{k}=3 c_{k-1}-2 c_{k-2}$. Write $c_{k}$ as a function of $k$ alone. What is the sum $S_{n}$ of the first $n$ terms of this sequence? Express $S_{n}$ in terms of $S_{n-1}$ and $S_{n... | S_{n}=3S_{n-1}-2S_{n-2}-1 | 124 | 19 |
math | 7. Variant 1.
103 natural numbers are written in a circle. It is known that among any 5 consecutive numbers, there are at least two even numbers. What is the minimum number of even numbers that can be in the entire circle? | 42 | 52 | 2 |
math | BMO 1966 Problem 4 A 1 , A 2 , A 3 , A 4 are consecutive vertices of a regular n-gon. 1/A 1 A 2 = 1/A 1 A 3 + 1/A 1 A 4 . What are the possible values of n? Solution | 7 | 71 | 1 |
math | \section*{Problem 1 - 111031}
Determine all ordered pairs \((a ; b)\) of real numbers \(a, b\) with \(a \neq 0, b \neq 0\), for which the following holds:
(1) The sum of the two numbers is 6.
(2) The sum of the reciprocals of both numbers is also 6. | (3+2\sqrt{2};3-2\sqrt{2})(3-2\sqrt{2};3+2\sqrt{2}) | 90 | 33 |
math | Problem condition
Find the derivative.
$y=\sqrt{9 x^{2}-12 x+5} \cdot \operatorname{arctg}(3 x-2)-\ln \left(3 x-2+\sqrt{9 x^{2}-12 x+5}\right)$ | \frac{(9x-6)\cdot\operatorname{arctg}(3x-2)}{\sqrt{9x^{2}-12x+5}} | 63 | 36 |
math | 70. A certain amount was spent on strawberries at 2 r. 40 k. per 1 kg of one variety and the same amount on another variety at 1 r. 60 k. Find the average price of 1 kg of the strawberries purchased. | 1 | 57 | 1 |
math | 5. (10 points) The distance from home to work is $s=6 \kappa$ km. At the moment Ivan left work, his favorite dog ran out of the house and ran towards his owner. They met at a distance of one third of the entire path from work. The dog instantly turned around and ran back home. Upon reaching home, he instantly turned ar... | 12\kappan | 114 | 6 |
math | 67. A material point moves in such a way that its speed is proportional to the distance traveled. At the initial moment, the point was 1 m away from the origin, and after 2 s - at a distance of $e$ m. Find the law of motion of the material point (see problem 21). | e^{/2} | 68 | 5 |
math | 253. Find the dimensions of a rectangular parallelepiped if they are expressed as integers, and the total surface area and volume are numerically equal. | (6,6,6),(5,5,10),(4,8,8),(3,12,12),(3,7,42),(3,8,24),(3,9,18),(3,10,15),(4,5,20),(4,6,12) | 32 | 71 |
math | Problem 9.6. Find all pairs $(x, y)$ of natural numbers that satisfy the equation
$$
x^{2}-6 x y+8 y^{2}+5 y-5=0
$$ | {(2,1),(4,1),(11,4),(13,4)} | 46 | 19 |
math | 4. 240 Find three natural numbers $x, y, z$ that satisfy the equation $x^{3}+y^{4}=z^{5}$, and determine whether the solution set of this equation in the set of natural numbers is finite or infinite. | x=256, y=64, z=32 | 56 | 15 |
math | 7. Find all prime triples $(p, q, r)$ such that
$$p \mid \left(q^{r}+1\right), q \mid \left(r^{p}+1\right), r \mid \left(p^{q}+1\right) .$$ | (5, 3, 2), (3, 2, 5), (2, 5, 3) | 61 | 27 |
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