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 | MIC test question, there is a unique small question: If $\frac{a+b}{a-b}=\frac{b+c}{b-c}=\frac{c+a}{c-a}$, then
$$
\begin{aligned}
a+b+c & =\ldots, a^{2}+b^{2}+c^{2}= \\
\frac{1}{a}+\frac{1}{b}+\frac{1}{c} & =\ldots
\end{aligned}
$$ | 0 | 106 | 1 |
math | 1. If from one page of a book three letters are removed from each line and then two such lines are removed, the number of all letters will decrease by 145. If, on the other hand, we add four letters to each line and write three such lines, then the number
of all letters will increase by 224. How many lines are there on... | 29,32 | 90 | 5 |
math | ## 1. Solve the system of equations in the set of real numbers
$$
\begin{aligned}
|x-5|+|y-9| & =6 \\
\left|x^{2}-9\right|+\left|y^{2}-5\right| & =52
\end{aligned}
$$ | (1;7),(4\sqrt{2}+1;4\sqrt{2}-1) | 69 | 22 |
math | A $350 \mathrm{~cm}^{3}$ volume brass cone, whose height is equal to the radius of its base, floats in mercury with its tip downward. How deep does this cone sink into the mercury, given that the specific gravity of the former is 7.2 and that of the latter is $13,6 \mathrm{gr}$. | 5.6141 | 77 | 6 |
math | Let $N = 100^2 + 99^2 - 98^2 - 97^2 + 96^2 + \cdots + 4^2 + 3^2 - 2^2 - 1^2$, where the additions and subtractions alternate in pairs. Find the remainder when $N$ is divided by $1000$. | 100 | 86 | 3 |
math | 2. If the real number $x$ satisfies $\log _{2} \log _{2} x=\log _{4} \log _{4} x$, then $x=$ $\qquad$ . | \sqrt{2} | 46 | 5 |
math | Find all triples $(k, m, n)$ of positive integers such that $m$ is a prime and:
(1) $kn$ is a perfect square;
(2) $\frac{k(k-1)}{2}+n$ is a fourth power of a prime;
(3) $k-m^2=p$ where $p$ is a prime;
(4) $\frac{n+2}{m^2}=p^4$. | (k, m, n) = (28, 5, 2023) | 94 | 22 |
math | 10. In $\triangle A B C$, $A B=A C, \angle A=80^{\circ}, D$ is a point inside the triangle, and $\angle D A B=\angle D B A=10^{\circ}$, find the degree measure of $\angle A C D$.
(Problem 432 from "Mathematics Teaching") | 30 | 78 | 2 |
math | 7.3. A natural number is called a palindrome if it remains unchanged when its digits are written in reverse order (for example, the numbers 4, 55, 626 are palindromes, while 20, 201, 2016 are not). Represent the number 2016 as a product of three palindromes greater than 1 (find all possible options and explain why ther... | 2\cdot4\cdot252 | 98 | 9 |
math | 2. Find all solutions to the system of 3 equations with 3 unknowns $x_{1}, x_{2}, x_{3}$ :
$$
\left\{\begin{array}{l}
x_{1}\left|x_{1}\right|-\left(x_{1}-a\right)\left|x_{1}-a\right|=x_{2}\left|x_{2}\right| \\
x_{2}\left|x_{2}\right|-\left(x_{2}-a\right)\left|x_{2}-a\right|=x_{3}\left|x_{3}\right| \\
x_{3}\left|x_{3}\... | x_{1}=x_{2}=x_{3}= | 187 | 12 |
math | 6. Define the sequence $\left\{a_{n}\right\}: a_{n}$ is the last digit of $1+2+\cdots+n$, and $S_{n}$ is the sum of the first $n$ terms of the sequence $\left\{a_{n}\right\}$. Then $S_{2016}=$ $\qquad$ | 7066 | 79 | 4 |
math | Let $x$ be in the interval $\left(0, \frac{\pi}{2}\right)$ such that $\sin x - \cos x = \frac12$ . Then $\sin^3 x + \cos^3 x = \frac{m\sqrt{p}}{n}$ , where $m, n$, and $p$ are relatively prime positive integers, and $p$ is not divisible by the square of any prime. Find $m + n + p$. | 28 | 102 | 2 |
math | The 83 interns from the Animath internship each choose an activity for the free afternoon from 5 proposed activities. We know that:
$\triangleright$ Shopping has been at least as popular as Laser Game;
$\Delta$ Movie tickets are sold in lots of 6;
$\Delta$ At most 5 students go to the beach;
$\triangleright$ At mos... | 3570 | 119 | 4 |
math | # 6. Option 1.
Nезнайка named four numbers, and Ponchik wrote down all their pairwise sums on six cards. Then he lost one card, and the numbers left on the remaining cards were $270, 360, 390, 500, 620$. What number did Ponchik write on the lost card? | 530 | 82 | 3 |
math |
1. Find all real solutions of the system
$$
\begin{aligned}
& \sqrt{x-y^{2}}=z-1, \\
& \sqrt{y-z^{2}}=x-1, \\
& \sqrt{z-x^{2}}=y-1 .
\end{aligned}
$$
| 1 | 68 | 1 |
math | 11. If the solution set of the inequality $\frac{x^{2}+\left(2 a^{2}+2\right) x-a^{2}+4 a-7}{x^{2}+\left(a^{2}+4 a-5\right) x-a^{2}+4 a-7}<0$ with respect to $x$ is the union of some intervals, and the sum of the lengths of these intervals is not less than 4, find the range of real number $a$. | \leqslant1or\geqslant3 | 109 | 13 |
math | 1. Find all natural numbers such that if you add their smallest divisor greater than one to them, the result is 30. | 25,27,28 | 27 | 8 |
math | 10. In the tetrahedron $P-ABC$, the three edges $PA$, $PB$, and $PC$ are pairwise perpendicular, and $PA=1$, $PB=PC=2$. If $Q$ is any point on the surface of the circumscribed sphere of the tetrahedron $P-ABC$, then the maximum distance from $Q$ to the plane $ABC$ is | \frac{3}{2}+\frac{\sqrt{6}}{6} | 86 | 17 |
math | 13. How many ways are there to arrange four 3 s and two 5 s into a six-digit number divisible by 11 ? | 9 | 30 | 1 |
math | In the equation $2 b x+b=3 c x+c$, both $b$ and $c$ can take any of the values $1,2,3,4,5,6$. In how many cases will the solution of the equation be positive?
---
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 3 | 80 | 1 |
math | 3. The sum of positive numbers $a, b, c$ and $d$ does not exceed 4. Find the maximum value of the expression
$$
\sqrt[4]{a(b+2 c)}+\sqrt[4]{b(c+2 d)}+\sqrt[4]{c(d+2 a)}+\sqrt[4]{d(a+2 b)}
$$ | 4\sqrt[4]{3} | 79 | 8 |
math | 9. (16 points) Let the complex number $z=\cos \frac{2 \pi}{13}+\mathrm{i} \sin \frac{2 \pi}{13}$. Find the value of $\left(z^{-12}+z^{-11}+z^{-10}\right)\left(z^{3}+1\right)\left(z^{6}+1\right)$. | -1 | 88 | 2 |
math | Example 3 If numbers $1,2, \cdots, 14$ are taken in ascending order as $a_{1}, a_{2}, a_{3}$, such that $a_{2}-a_{1} \geqslant 3$, and $a_{3}-a_{2} \geqslant 3$, find the number of different ways to choose them. | 120 | 85 | 3 |
math | 10. Evaluate the definite integral $\int_{-1}^{+1} \frac{2 u^{332}+u^{998}+4 u^{1664} \sin u^{691}}{1+u^{666}} \mathrm{~d} u$. | \frac{2}{333}(1+\frac{\pi}{4}) | 68 | 17 |
math | 2.037. $\frac{1-x^{-2}}{x^{1/2}-x^{-1/2}}-\frac{2}{x^{3/2}}+\frac{x^{-2}-x}{x^{1/2}-x^{-1/2}}$. | -\sqrt{x}(1+\frac{2}{x^{2}}) | 59 | 15 |
math | G7.3 The equation of the line through $(4,3)$ and $(12,-3)$ is $\frac{x}{a}+\frac{y}{b}=1$. Find $a$. | 8 | 42 | 1 |
math | A7. Two 2-digit multiples of 7 have a product of 7007 . What is their sum? | 168 | 26 | 3 |
math | ## Task 4 - 330524
Rita calculates the three numbers
$$
1+9-9+3=a, \quad 1 \cdot 9+9-3=b, \quad 1 \cdot 9 \cdot 9 \cdot 3=c
$$
She considers further possibilities for filling in the boxes in the row
$$
1 \square 9 \square 9 \square 3=
$$
with signs that are either + or - or $\cdot$. She is looking for all those in... | 1993 | 204 | 4 |
math | Two unit squares are selected at random without replacement from an $n \times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected unit squares are horizontally or vertically adjacent is less than $\frac{1}{2015}$. | 90 | 59 | 2 |
math | [ Decimal number system ] $[$ Equations in integers $]$
Find a four-digit number that is a perfect square and such that the first two digits are the same as each other and the last two digits are also the same. | 7744 | 47 | 4 |
math | Determine all pairs $ (n,p)$ of positive integers, where $ p$ is prime, such that $ 3^p\minus{}np\equal{}n\plus{}p$. | (n, p) = (6, 3) | 40 | 13 |
math | 9. Determine all integers $n > 1$ such that: for all integers $a, b \in \mathbf{Z}$ that are coprime with $n$, $a \equiv b(\bmod n) \Leftrightarrow a b \equiv 1(\bmod n)$. | 2,3,4,6,8,12,24 | 65 | 15 |
math | 13. Let $P$ be a moving point on the circle $x^{2}+y^{2}=36$, and point $A(20,0)$. When $P$ moves on the circle, the equation of the trajectory of the midpoint $M$ of line segment $P A$ is $\qquad$. | (x-10)^{2}+y^{2}=9 | 70 | 14 |
math | 1. [5 points] The altitudes $C F$ and $A E$ of an acute triangle $A B C$ intersect at point $H$. Points $M$ and $N$ are the midpoints of segments $A H$ and $C H$ respectively. It is known that $F M=1, E N=7$, and $F M \| E N$. Find $\angle A B C$, the area of triangle $A B C$, and the radius of the circumscribed circle... | \angleABC=60,S_{\triangleABC}=45\sqrt{3},R=2\sqrt{19} | 148 | 28 |
math | 12. (20 points) Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{0}=\frac{1}{k}, a_{n}=a_{n-1}+\frac{1}{n^{2}} a_{n-1}^{2}\left(k \in \mathbf{Z}_{+}\right) \text {. }
$$
If for all $n \in \mathbf{Z}_{+}$, we have $a_{n}<1$, find the range of values for $k$. | k\geqslant3 | 119 | 7 |
math | 5. For all positive real numbers $a, b, c, d$,
$$
\left(\frac{a^{3}}{a^{3}+15 b c d}\right)^{\frac{1}{2}} \geqslant \frac{a^{x}}{a^{x}+b^{x}+c^{x}+d^{x}}
$$
the real number $x=$ | \frac{15}{8} | 89 | 8 |
math | 1. Let the function $f(x)=x \sin x(x \in \mathbf{R})$ attain an extremum at $x=x_{0}$, then $\left(1+x_{0}^{2}\right)\left(1+\cos 2 x_{0}\right)=$
$\qquad$ . | 2 | 69 | 1 |
math | Five, try to find two different natural numbers, whose arithmetic mean $A$ and geometric mean $G$ are both two-digit numbers. Among $A, G$, one can be obtained by swapping the units and tens digits of the other. | 98 \text{ and } 32 | 49 | 10 |
math | ## Task 2.
Let $N$ be a natural number. We call a staircase a part of a square plate of dimensions $N \times N$ that consists of the first $K$ fields in the $K$-th row for $K=1,2, \ldots, N$. In how many ways can the staircase be cut into rectangles of different areas that consist of the fields of the given plate? | 2^{N-1} | 87 | 6 |
math | 3.1. Solve the system of equations:
$$
\left\{\begin{array}{l}
2^{x+2 y}+2^{x}=3 \cdot 2^{y} \\
2^{2 x+y}+2 \cdot 2^{y}=4 \cdot 2^{x}
\end{array}\right.
$$ | (\frac{1}{2},\frac{1}{2}) | 74 | 14 |
math |
3. Let $X=\{0,1,2,3,4,5,6,7,8,9\}$. Let $S \subseteq X$ be such that any nonnegative integer $n$ can be written as $p+q$ where the nonnegative integers $p, q$ have all their digits in $S$. Find the smallest possible number of elements in $S$.
| 5 | 86 | 1 |
math | 7. The solution to the equation $\sqrt{\frac{2 x-6}{x-11}}=\frac{3 x-7}{x+6}$ is | x_{1}=19, x_{2}=\frac{13+5 \sqrt{2}}{7} | 35 | 26 |
math | 2nd Centromerican 2000 Problem A1 Find all three digit numbers abc (with a ≠ 0) such that a 2 + b 2 + c 2 divides 26. | 100,110,101,302,320,230,203,431,413,314,341,134,143,510,501,150,105 | 45 | 67 |
math | 6. From $m$ boys and $n$ girls $(10 \geqslant m>n \geqslant 4)$, 2 people are randomly selected to be class leaders. Let event $A$ represent the selection of 2 people of the same gender, and event $B$ represent the selection of 2 people of different genders. If the probability of $A$ is the same as the probability of $... | (10,6) | 108 | 6 |
math | 11. Determine all functions \( f: \mathbf{N} \rightarrow \mathbf{N} \), such that
\[
f(a+b)=f(a)+f(b)+f(c)+f(d)
\]
for all non-negative integers \( a, b, c, d \) satisfying \( 2 a b=c^{2}+d^{2} \). | f(n)=kn^{2}(k\in{N}) | 80 | 13 |
math | (given to Juliette Fournier). Find all strictly positive integers $a$ and $b$ such that $a b^{2}+b+7$ divides $a^{2} b+a+b$.
| (11,1),(49,1),(7k^2,7k) | 45 | 19 |
math | 4. Given vectors $\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}$ satisfy
$$
|\boldsymbol{a}|:|\boldsymbol{b}|:|\boldsymbol{c}|=1: k: 3\left(k \in \mathbf{Z}_{+}\right) \text {, }
$$
and $\boldsymbol{b}-\boldsymbol{a}=2(\boldsymbol{c}-\boldsymbol{b})$.
If $\alpha$ is the angle between $\boldsymbol{a}$ and $\boldsymbol... | -\frac{1}{12} | 135 | 8 |
math | Let $P$ and $Q$ be the midpoints of sides $AB$ and $BC,$ respectively, of $\triangle ABC.$ Suppose $\angle A = 30^{\circ}$ and $\angle PQC = 110^{\circ}.$ Find $\angle B$ in degrees. | 80^\circ | 63 | 4 |
math | 335. Find the last two digits of the number $3^{5^{17}}$. | 43 | 21 | 2 |
math | 11. (CZS 5) Let $n$ be a positive integer. Find the maximal number of noncongruent triangles whose side lengths are integers less than or equal to $n$. | p_{n}= \begin{cases}n(n+2)(2 n+5) / 24, & \text { for } 2 \mid n, \\ (n+1)(n+3)(2 n+1) / 24, & \text { for } 2 \nmid n .\end{cases} | 42 | 74 |
math |
6. In the domain of real numbers solve the following system of equations
$$
\begin{aligned}
& x^{4}+y^{2}+4=5 y z \\
& y^{4}+z^{2}+4=5 z x \\
& z^{4}+x^{2}+4=5 x y .
\end{aligned}
$$
(Jaroslav Švrček)
| (\sqrt{2},\sqrt{2},\sqrt{2})(-\sqrt{2},-\sqrt{2},-\sqrt{2}) | 91 | 30 |
math | 26. Multiply the month number of a student's birthday by 31 and the day number by 12, then add the two products together, the sum is 376. What is the student's birthday? | April\21 | 47 | 4 |
math | For how many different values of the parameter $p$ does the system of equations
$$
x^{2}-y^{2}=0 \quad x y+p x-p y=p^{2}
$$
have exactly one solution? | 1 | 47 | 1 |
math | 1. N1 (UZB) What is the smallest positive integer \( t \) such that there exist integers \( x_{1}, x_{2}, \ldots, x_{t} \) with
\[ x_{1}^{3} + x_{2}^{3} + \cdots + x_{t}^{3} = 2002^{2002} ? \] | 4 | 88 | 1 |
math | One. (25 points) Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{1}=2, a_{n}=2^{2 n} a_{n-1}+2^{n^{2}} n \quad (n=2,3, \cdots) \text {. }
$$
Find the general term $a_{n} (n=1,2, \cdots)$.
(Wu Shuxun, problem contributor) | a_{n}=2^{n^{2}}\left(2^{n+1}-n-2\right) | 103 | 25 |
math | 12.033. A rectangular trapezoid with an acute angle $\alpha$ is circumscribed around a circle. Find the height of the trapezoid if its perimeter is $P$. | \frac{P\sin\alpha}{4\cdot\cos^{2}(\frac{\pi}{4}-\frac{\alpha}{2})} | 44 | 32 |
math | 6. The endpoints of a line segment $AB$ of fixed length 3 move on the parabola $y^{2}=x$, and $M$ is the midpoint of segment $AB$. Find the shortest distance from $M$ to the $y$-axis.
(1987 National College Entrance Examination) | \frac{5}{4} | 67 | 7 |
math | G2.4 Given that $x$ and $y$ are positive integers and $x+y+x y=54$. If $t=x+y$, find the value of $t$. | 14 | 39 | 2 |
math | Example 1. The probability of hitting the target with one shot for a given shooter is 0.7 and does not depend on the shot number. Find the probability that exactly 2 hits will occur out of 5 shots. | 0.1323 | 47 | 6 |
math | In $10\times 10$ square we choose $n$ cells. In every chosen cell we draw one arrow from the angle to opposite angle. It is known, that for any two arrows, or the end of one of them coincides with the beginning of the other, or
the distance between their ends is at least 2. What is the maximum possible value of $n$? | 50 | 84 | 2 |
math | 8.2. Find the largest natural number with all distinct digits such that the sum of any two of its digits is a prime number. | 520 | 28 | 3 |
math | 10. (3 points) Xiaoming and Xiaohong took out the same amount of money to buy exercise books together. In the end, Xiaoming took 8 books, and Xiaohong took 12 books. As a result, Xiaohong gave Xiaoming 1.1 yuan. The unit price of each exercise book is $\qquad$ yuan. | 0.55 | 77 | 4 |
math | A die is symmetrical but unusual: two of its faces have two points each, while the other four have one point each. Sergey rolled the die several times, and as a result, the sum of all the points that came up was 3. Find the probability that at some roll, a face with 2 points came up.
# | 0.6 | 69 | 3 |
math | 1. Let $[x]$ denote the greatest integer not exceeding the real number $x$. Given a sequence of positive integers $\left\{a_{n}\right\}$ satisfying $a_{1}=a$, and for any positive integer $n$,
$$
a_{n+1}=a_{n}+2\left[\sqrt{a_{n}}\right] \text {. }
$$
(1) If $a=8$, find the smallest positive integer $n$ such that $a_{n}... | 5 | 157 | 1 |
math | Find all nonnegative integer solutions $(a, b, c, d)$ to the equation
$$
2^{a} 3^{b}-5^{c} 7^{d}=1 \text {. }
$$ | (1,0,0,0),(3,0,0,1),(1,1,1,0),(2,2,1,1) | 45 | 33 |
math | Example 3 Find the range of $y=\frac{x^{2}+7 x+10}{x+1}(x \neq-1)$ | 9 | 33 | 1 |
math | Example 4 Given that $a, b, c$ satisfy $a+b+c=0, abc=8$. Then the range of values for $c$ is $\qquad$ | c<0 \text{ or } c \geqslant 2 \sqrt[3]{4} | 38 | 23 |
math | 7. A. For any real numbers $x, y$, define the operation
$$
x \oplus y=x+2 y+3 \text {. }
$$
It is known that real numbers $a, b$ satisfy
$$
\left(a^{3} \oplus a^{2}\right) \oplus a=a^{3} \oplus\left(a^{2} \oplus a\right)=b \text {. }
$$
Then $a+b=$ $\qquad$ | \frac{21}{8} | 106 | 8 |
math | Which natural numbers $m$ and $n$ satisfy the equation $2^{n}+1=m^{2}$? | =n=3 | 25 | 3 |
math | Define a \emph{crossword puzzle} to be a $15 \times 15$ grid of squares, each of which is either black or white. In a crossword puzzle, define a \emph{word} to be a sequence of one or more consecutive white squares in a row or column such that the squares immediately before and after the sequence both are either black ... | 4900 | 170 | 4 |
math | 2. Given a positive integer $n \geq 2$, let the polynomial $f(x)=x^{n}+b_{n-1} x^{n-1}+\ldots+b_{0}$, where $b_{i} \in \mathbb{R}, i=0,1,2, \ldots, n-1$. If there exists a non-constant infinite positive integer geometric sequence $\left\{a_{k}\right\}$ such that for any positive integer $k$, there exists a positive int... | f(x)=(x+)^{n}, | 143 | 9 |
math | 10. A hotel has 100 standard rooms, with a room rate of 400 yuan/day, but the occupancy rate is only $50 \%$. If the room rate is reduced by 20 yuan, it can increase the number of occupied rooms by 5. Find the appropriate room rate to maximize the hotel's room revenue,
| 22500 | 73 | 5 |
math | 5. The first four members of a sequence are: $2,0,1,8$. Each subsequent member of the sequence is the unit digit of the sum of the previous four members. (e.g., the fifth member is 1). Is the 2018th member of the sequence an even or odd number? Justify your answer! | odd | 73 | 1 |
math | 10.311. The lengths of the diagonals of a rhombus are in the ratio $3: 4$. How many times larger is the area of the rhombus compared to the area of the circle inscribed in it? | \frac{25}{6\pi} | 52 | 10 |
math | 5. Find all positive integers $x, y, z, w$ such that:
(1) $x, y, z, w$ are four consecutive terms of an arithmetic sequence,
(2) $x^{3}+y^{3}+z^{3}=w^{3}$. | x=3d, y=4d, z=5d, w=6d | 62 | 19 |
math | 1. $\log _{9 a} 8 a=\log _{3 a} 2 a$. Then $\ln a=$ | \frac{\ln2\cdot\ln3}{\ln3-2\ln2} | 28 | 20 |
math | Given a circle, point $C$ on it and point $A$ outside the circle. The equilateral triangle $ACP$ is constructed on the segment $AC$. Point $C$ moves along the circle. What trajectory will the point $P$ describe? | \text{The trajectory of point } P \text{ is a circle with center } \left( \frac{a - \sqrt{3} b}{2}, \frac{\sqrt{3} a + b}{2} \right) \text{ and radius } r. | 53 | 59 |
math | \section*{Problem 2 - 141022}
Give all (ordered) triples \((x, y, z)\) that satisfy the following conditions!
(1) \(x-y=96\),
(2) \(y-z=96\),
(3) \(x, y\) and \(z\) are squares of natural numbers. | (x,y,z)=(196,100,4) | 77 | 14 |
math | Proizvoov V.v.
Ten consecutive natural numbers were written on the board. When one of them was erased, the sum of the nine remaining numbers turned out to be 2002.
What numbers remained on the board? | 218,219,220,221,222,224,225,226,227 | 48 | 35 |
math | 12. (18 points) Let $n (n \geqslant 11)$ be a positive integer. The set $A$ consists of the sums of 10 consecutive positive integers not greater than $n$, and the set $B$ consists of the sums of 11 consecutive positive integers not greater than $n$. If the number of elements in $A \cap B$ is 181, find the maximum and m... | 2011 \text{ and } 2001 | 100 | 14 |
math | 2. (8 points) A shepherd was herding a flock of sheep to graze. After one ram ran out, he counted the number of sheep and found that the ratio of rams to ewes among the remaining sheep was 7:5. After a while, the ram that ran out returned to the flock, but a ewe ran out instead. The shepherd counted the sheep again and... | 25 | 107 | 2 |
math | 4. A national football association stipulates: In the league, a team gets $a$ points for a win, $b$ points for a draw, and 0 points for a loss, where real numbers $a>b>0$. If a team has exactly 2015 possible total scores after $n$ matches, find the minimum value of $n$. | 62 | 76 | 2 |
math | N4. Alice is given a rational number $r>1$ and a line with two points $B \not R$, where point $R$ contains a red bead and point $B$ contains a blue bead. Alice plays a solitaire game by performing a sequence of moves. In every move, she chooses a (not necessarily positive) integer $k$, and a bead to move. If that bead ... | All\r=(b+1)/b\with\b=1,\ldots,1010 | 179 | 21 |
math | Example 4.15. Find the particular solution of the equation
$$
y^{\prime \prime}=\sin x-1,
$$
satisfying the given initial conditions $y(0)=-1$, $y^{\prime}(0)=1$. | -\sinx-\frac{x^{2}}{2}+2x-1 | 57 | 17 |
math | $14 \cdot 41$ Try for any positive integer $n$, to calculate the sum $\sum_{k=0}^{\infty}\left[\frac{n+2^{k}}{2^{k+1}}\right]$.
(10th International Mathematical Olympiad, 1968) | n | 68 | 1 |
math | 4. Draw four chords $P_{1} Q_{1}, P_{2} Q_{2}, P_{3} Q_{3}, P_{4} Q_{4}$ of the parabola $y^{2}=x$ through the point $M(2,-1)$, and the y-coordinates of points $P_{1}, P_{2}, P_{3}, P_{4}$ form an arithmetic sequence. Try to compare $\frac{P_{1} M}{M Q_{1}}-\frac{P_{2} M}{M Q_{2}}$ with $\frac{P_{3} M}{M Q_{3}}-\frac{P... | 4d^2 | 152 | 4 |
math | # Problem 6. (10 points)
Vasily is planning to graduate from college in a year. Only 270 out of 300 third-year students successfully pass their exams and complete their bachelor's degree. If Vasily ends up among the 30 expelled students, he will have to work with a monthly salary of 25,000 rubles. It is also known tha... | 45025 | 293 | 5 |
math | Let $ H$ be the piont of midpoint of the cord $ PQ$ that is on the circle centered the origin $ O$ with radius $ 1.$
Suppose the length of the cord $ PQ$ is $ 2\sin \frac {t}{2}$ for the angle $ t\ (0\leq t\leq \pi)$ that is formed by half-ray $ OH$ and the positive direction of the $ x$ axis. Answer the following que... | S = \frac{\pi + 2}{8} \text{ square units} | 222 | 18 |
math | I1.1 Let $[x]$ represents the integral part of the decimal number $x$. Given that $[3.126]+\left[3.126+\frac{1}{8}\right]+\left[3.126+\frac{2}{8}\right]+\ldots+\left[3.126+\frac{7}{8}\right]=P$, find the value of $P$. | 25 | 90 | 2 |
math | 6.1. How many numbers from 1 to 1000 (inclusive) cannot be represented as the difference of two squares of integers | 250 | 30 | 3 |
math | Problem 5. Nikola bought 4 different pencils. All pencils, except the first one, together cost 42 denars, all pencils, except the second one, together cost 40 denars, all pencils, except the third one, together cost 38 denars, and all pencils, except the fourth one, together cost 36 denars. How much does each pencil co... | =10,b=12,=14,=16 | 84 | 15 |
math | 15. (25 points) Find all non-negative integer solutions to the equation
$$
x^{3}+y^{3}-x^{2} y^{2}-(x+y)^{2} z=0
$$ | (2,2,0),(0,0, m),(0, m, m),(m, 0, m) | 49 | 26 |
math | ## Problem Statement
Write the decomposition of vector $x$ in terms of vectors $p, q, r$:
$x=\{8 ; 1 ; 12\}$
$p=\{1 ; 2 ;-1\}$
$q=\{3 ; 0 ; 2\}$
$r=\{-1 ; 1 ; 1\}$ | -p+4q+3r | 75 | 7 |
math | 3. A coin-flipping game, starting from the number $n$, if the coin lands heads up, subtract 1, if the coin lands tails up, subtract 2. Let $E_{n}$ be the expected number of coin flips before the number becomes zero or negative. If $\lim _{n \rightarrow \infty}\left(E_{n}-a n-b\right)=0$, then the pair $(a, b)=$ $\qquad... | \left(\frac{2}{3}, \frac{2}{9}\right) | 96 | 18 |
math | 8. Find all integers $x$ such that $2 x^{2}+x-6$ is a positive integral power of a prime positive integer. | -3,2,5 | 32 | 6 |
math | 12. Following the order of King Pea, General Mushtralkin tried to arrange all the soldiers in rows first by 2, and then by $3,4,5,6,7,8,9,10$, but to his surprise, each time the last row was incomplete, as there were respectively
$$
1,2,3,4,5,6,7,8,9
$$
soldiers left. What is the smallest number of soldiers there cou... | 2519 | 106 | 4 |
math | Task 1. Gjorgja's phone number consists of two three-digit numbers written side by side. Each of them is divisible by 45, and the middle digit is 8. Determine the phone number, if the three-digit number written from left in the phone number is smaller than the three-digit number from the right. | 180585 | 68 | 6 |
math | 1. Does there exist a natural number for which the sum of the digits of its square is:
a) 80;
b) 81? | 111111111 | 32 | 9 |
math | 1. Among the natural numbers from $1 \sim 10000$, the integers that are neither perfect squares nor perfect cubes are $\qquad$ in number. | 9883 | 36 | 4 |
math | 6. Given the function $f(x)=[x[x]]$, where $[x]$ denotes the greatest integer not exceeding $x$. If $x \in [0, n] (n \in \mathbf{N}_{+})$, the range of $f(x)$ is $A$, and let $a_{n}=\operatorname{card}(A)$, then $a_{n}=$ $\qquad$ | \frac{1}{2}\left(n^{2}-n+4\right) | 89 | 18 |
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