task_type stringclasses 1
value | problem stringlengths 23 3.94k | answer stringlengths 1 231 | problem_tokens int64 10 1.39k | answer_tokens int64 1 98 |
|---|---|---|---|---|
math | Let $S$ be the set of positive integer divisors of $20^9.$ Three numbers are chosen independently and at random with replacement from the set $S$ and labeled $a_1,a_2,$ and $a_3$ in the order they are chosen. The probability that both $a_1$ divides $a_2$ and $a_2$ divides $a_3$ is $\tfrac{m}{n},$ where $m$ and $n$ are ... | 77 | 113 | 2 |
math | 7. (15 points) Insert 2 " $\div$ " and 2 "+" between the 9 "1"s below to make the calculation result an integer. The smallest integer is $\qquad$
\begin{tabular}{|lllllllllll|}
\hline 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & $=$ & $?$ \\
\hline
\end{tabular} | 3 | 103 | 1 |
math | Let $ABC$ be an isosceles triangle with $AB=4$, $BC=CA=6$. On the segment $AB$ consecutively lie points $X_{1},X_{2},X_{3},\ldots$ such that the lengths of the segments $AX_{1},X_{1}X_{2},X_{2}X_{3},\ldots$ form an infinite geometric progression with starting value $3$ and common ratio $\frac{1}{4}$. On the segment $CB... | (a, b, c) = (1, 2c, c) \text{ or } (2c, 1, c) | 307 | 32 |
math | Robin is playing notes on an 88-key piano. He starts by playing middle C, which is actually the 40th lowest note on the piano (i.e. there are 39 notes lower than middle C). After playing a note, Robin plays with probability $\tfrac12$ the lowest note that is higher than the note he just played, and with probability $\t... | \frac{13}{29} | 116 | 9 |
math | Find all polynomials $p(x)$ with real coe ffcients such that
\[p(a + b - 2c) + p(b + c - 2a) + p(c + a - 2b) = 3p(a - b) + 3p(b - c) + 3p(c - a)\]
for all $a, b, c\in\mathbb{R}$.
[i](2nd Benelux Mathematical Olympiad 2010, Problem 2)[/i] | p(x) = \lambda x^2 + \mu x | 113 | 14 |
math | Marta and Carmem have won many chocolates each. They mixed all the chocolates and now they no longer know how many chocolates each of them won. Let's help them find out these numbers? It is known that:
(a) together, they won 200 chocolates;
(b) Marta remembers that she won fewer than 100 chocolates, but more than $4 ... | Martawon96chocolatesCarmem104 | 106 | 14 |
math | 3. Given $\triangle A B C, \tan A, \tan B, \tan C$ are all integers, and $\angle A>\angle B>\angle C$. Then $\tan B=$ $\qquad$ | 2 | 45 | 1 |
math | Problem 4. Joseph solved 6 problems correctly, Darko solved 5 problems incorrectly, and Petre had the same number of correct and incorrect solutions. How many problems did all three solve correctly together, if Joseph solved twice as many problems correctly as Darko? | 13 | 54 | 2 |
math | 11. Given $x+y=1$, for what values of real numbers $x, y$ does $\left(x^{3}+1\right)\left(y^{3}+1\right)$ achieve its maximum value. | 4 | 48 | 1 |
math | Example 36 (1978 Kyiv Mathematical Olympiad) Find the smallest natural numbers $a$ and $b (b>1)$, such that
$$
\sqrt{a \sqrt{a \sqrt{a}}}=b .
$$ | =256,b=128 | 54 | 9 |
math | Example 12 Let $x, y, z$ be real numbers greater than -1. Find the minimum value of
$$\frac{1+x^{2}}{1+y+z^{2}}+\frac{1+y^{2}}{1+z+x^{2}}+\frac{1+z^{2}}{1+x+y^{2}}$$ | 2 | 73 | 1 |
math | [ Formulas for abbreviated multiplication (other).] Case analysis
Solve the equation $\left(x^{2}-y^{2}\right)^{2}=16 y+1$ in integers.
# | (\1,0),(\4,3),(\4,5) | 42 | 15 |
math | 8. In tetrahedron $ABCD$, $\triangle ABD$ is an equilateral triangle, $\angle BCD=90^{\circ}$, $BC=CD=1$, $AC=\sqrt{3}$, $E, F$ are the midpoints of $BD, AC$ respectively, then the cosine value of the angle formed by line $AE$ and $BF$ is $\qquad$ | \frac{\sqrt{2}}{3} | 88 | 10 |
math | Example 2 Given $a+b+c=1$,
$$
b^{2}+c^{2}-4 a c+6 c+1=0 \text{. }
$$
Find the value of $a b c$. | 0 | 49 | 1 |
math | 1. Solve the equation
$$
\sqrt{x^{2}+x}+\sqrt{1+\frac{1}{x^{2}}}=\sqrt{x+3}
$$ | -1 | 37 | 2 |
math | Let $G{}$ be an arbitrary finite group, and let $t_n(G)$ be the number of functions of the form \[f:G^n\to G,\quad f(x_1,x_2,\ldots,x_n)=a_0x_1a_1\cdots x_na_n\quad(a_0,\ldots,a_n\in G).\]Determine the limit of $t_n(G)^{1/n}$ as $n{}$ tends to infinity. | \frac{|G|}{|Z(G)|} | 103 | 11 |
math | 18. (3 points) When Xiaoming goes from home to school, he walks the first half of the distance and takes a ride for the second half; when he returns from school to home, he rides for the first $\frac{1}{3}$ of the time and walks for the last $\frac{2}{3}$ of the time. As a result, the time it takes to go to school is 2... | 150 | 145 | 3 |
math | Consider coins with positive real denominations not exceeding 1. Find the smallest $C>0$ such that the following holds: if we have any $100$ such coins with total value $50$, then we can always split them into two stacks of $50$ coins each such that the absolute difference between the total values of the two stacks is ... | \frac{50}{51} | 90 | 9 |
math | Example 5. Find the sum of the series $f(x)=\sum_{n=0}^{\infty}\left(n^{2}+3 n-5\right) \cdot x^{n}$. | f(x)=\frac{7x^{2}-14x+5}{(x-1)^{3}} | 46 | 25 |
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 | For what values of the parameter $a$ is the sum of the squares of the roots of the equation $x^{2}+2 a x+2 a^{2}+4 a+3=0$ the greatest? What is this sum? (The roots are considered with their multiplicity.)
# | 18 | 63 | 2 |
math | Solve the inequality
$$
\frac{1}{x}+\frac{1}{x^{3}}+\frac{1}{x^{5}}+\frac{1}{x^{7}}+\frac{1}{x^{9}}+\frac{1}{x^{11}}+\frac{1}{x^{13}} \leq \frac{7}{x^{7}}
$$ | 1orx<0 | 83 | 5 |
math | ## Task A-4.3.
A die is rolled three times in a row. Determine the probability that each subsequent roll (after the first) results in a number that is not less than the previous one. | \frac{7}{27} | 43 | 8 |
math | Find the number of positive integers $n$ such that a regular polygon with $n$ sides has internal angles with measures equal to an integer number of degrees. | 22 | 32 | 2 |
math | Putnam 1993 Problem B1 What is the smallest integer n > 0 such that for any integer m in the range 1, 2, 3, ... , 1992 we can always find an integral multiple of 1/n in the open interval (m/1993, (m + 1)/1994)? Solution | 3987 | 79 | 4 |
math | Determine the integers $m \geqslant 2, n \geqslant 2$ and $k \geqslant 3$ having the following property: $m$ and $n$ each have $k$ positive divisors and, if we denote $d_{1}<\ldots<d_{k}$ the positive divisors of $m$ (with $d_{1}=1$ and $d_{k}=m$) and $d_{1}^{\prime}<\ldots<d_{k}^{\prime}$ the positive divisors of $n$ ... | (4,9,3)(8,15,4) | 194 | 14 |
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 | The class received the following task. Determine the smallest natural number $H$ such that its half is equal to a perfect cube, its third is equal to a perfect fifth power, and its fifth is equal to a perfect square.
Pista and Sanyi were absent that day, and one of their friends told them the problem from memory, so P... | P | 209 | 1 |
math | 57. There are two triangles with respectively parallel sides and areas $S_{1}$ and $S_{2}$, where one of them is inscribed in triangle $A B C$, and the other
is circumscribed around it. Find the area of triangle $A B C$. | S_{ABC}=\sqrt{S_{1}S_{2}} | 60 | 15 |
math | Solve the following equation if $x$ and $y$ are real numbers:
$$
\left(16 x^{2}+1\right)\left(y^{2}+1\right)=16 x y .
$$ | (\frac{1}{4},1)(-\frac{1}{4},-1) | 49 | 19 |
math | 6. Let's call the distance between numbers the absolute value of their difference. It is known that the sum of the distances from eleven consecutive natural numbers to some number $a$ is 902, and the sum of the distances from these same eleven numbers to some number $b$ is 374. Find all possible values of $a$, given th... | =107,=-9,=25 | 82 | 11 |
math | Given a positive integer $n$ greater than 2004, fill the numbers $1, 2, \cdots, n^2$ into an $n \times n$ chessboard (consisting of $n$ rows and $n$ columns of squares) such that each square contains exactly one number. If a number in a square is greater than the numbers in at least 2004 other squares in its row and at... | n(n-2004) | 138 | 8 |
math | 6.24 In a geometric sequence with a common ratio greater than 1, what is the maximum number of terms that are integers between 100 and 1000.
(4th Canadian Mathematics Competition, 1972) | 6 | 52 | 1 |
math | 5. From the smallest six-digit number divisible by 2 and 3, subtract the smallest five-digit number divisible by 5 and 9. What is the resulting difference? Round that difference to the nearest hundred. What number did you get?
## Tasks worth 10 points: | 90000 | 58 | 5 |
math | Adam, Bendeguz, Cathy, and Dennis all see a positive integer $n$. Adam says, "$n$ leaves a remainder of $2$ when divided by $3$." Bendeguz says, "For some $k$, $n$ is the sum of the first $k$ positive integers." Cathy says, "Let $s$ be the largest perfect square that is less than $2n$. Then $2n - s = 20$." Dennis says,... | 210 | 141 | 3 |
math | Let $ x,y$ are real numbers such that $x^2+2cosy=1$. Find the ranges of $x-cosy$. | [-1, 1 + \sqrt{3}] | 32 | 12 |
math | 10. (10 points) When the fraction $\frac{1}{2016}$ is converted to a repeating decimal, the repeating block has exactly $\qquad$ digits. | 6 | 39 | 1 |
math | Find all polynomials $P \in \mathbb{R}[X]$ such that $16 P\left(X^{2}\right)=P(2 X)^{2}$
Hint: use the previous question | P(x)=16(\frac{x}{4})^{i} | 45 | 14 |
math | (Netherlands). Find all triples $(a, b, c)$ of real numbers such that $a b+b c+$ $c a=1$ and
$$
a^{2} b+c=b^{2} c+a=c^{2} a+b
$$ | (0,1,1), (0,-1,-1), (1,0,1), (-1,0,-1), (1,1,0), (-1,-1,0), \left(\frac{1}{3} \sqrt{3}, \frac{1}{3} \sqrt{3}, \frac{1}{3} \sqrt{3}\right), \left(-\frac{1}{3} \sqrt{3}, - | 54 | 98 |
math | 6. The sum of $n$ terms of the geometric progression $\left\{b_{n}\right\}$ is 6 times less than the sum of their reciprocals. Find the product $b_{1} b_{n}$. | \frac{1}{6} | 51 | 7 |
math | Seven boys and three girls are playing basketball. I how many different ways can they make two teams of five players so that both teams have at least one girl? | 105 | 32 | 3 |
math | 4. Let's fix a point $O$ on the plane. Any other point $A$ is uniquely determined by its radius vector $\overrightarrow{O A}$, which we will simply denote as $\vec{A}$. We will use this concept and notation repeatedly in the following. The fixed point $O$ is conventionally called the initial point. We will also introdu... | \vec{C}=\frac{k\vec{B}-\vec{A}}{k-1} | 528 | 23 |
math | 2. Let the real-coefficient quadratic equation $x^{2}+a x+2 b-$ $2=0$ have two distinct real roots, one of which lies in the interval $(0,1)$, and the other in the interval $(1,2)$. Then the range of $\frac{b-4}{a-1}$ is . $\qquad$ | (\frac{1}{2},\frac{3}{2}) | 78 | 14 |
math | 6. Given $a=\frac{11 \times 66+12 \times 67+13 \times 68+14 \times 69+15 \times 70}{11 \times 65+12 \times 66+13 \times 67+14 \times 68+15 \times 69} \times 100$, find: the integer part of $a$ is
what? | 101 | 108 | 3 |
math | 8-6 Let the sequence $a_{0}, a_{1}, a_{2}, \cdots$ satisfy
$$
a_{0}=a_{1}=11, a_{m+n}=\frac{1}{2}\left(a_{2 m}+a_{2 n}\right)-(m-n)^{2}, m, n \geqslant 0 .
$$
Find $a_{45}$. | 1991 | 91 | 4 |
math | 3. An Indian engineer thought of a natural number, listed all its proper natural divisors, and then increased each of the listed numbers by 1. It turned out that the new numbers are all the proper divisors of some other natural number. What number could the engineer have thought of? Provide all options and prove that t... | n=4n=8 | 95 | 6 |
math | Aurick throws $2$ fair $6$-sided dice labeled with the integers from $1$ through $6$. What is the probability that the sum of the rolls is a multiple of $3$? | \frac{1}{3} | 45 | 7 |
math | Determine all positive integers $n$ such that all positive integers less than or equal to $n$ and relatively prime to $n$ are pairwise coprime. | \{1, 2, 3, 4, 6, 8, 12, 18, 24, 30\} | 34 | 36 |
math | ## Task 1 - 200621
An airplane flying from A to B at a constant speed was still $2100 \mathrm{~km}$ away from B at 10:05 AM, and only $600 \mathrm{~km}$ away at 11:20 AM.
At what time will it arrive in B if it continues to fly at the same speed? | 11:50 | 89 | 5 |
math | 1. Let $i_{1}, i_{2}, \cdots, i_{10}$ be a permutation of $1,2, \cdots, 10$. Define $S=\left|i_{1}-i_{2}\right|+\left|i_{3}-i_{4}\right|+\cdots+\left|i_{9}-i_{10}\right|$. Find all possible values of $S$.
[2] | 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25 | 94 | 39 |
math | ## Task 3 - 170513
Fritz wants to write down a subtraction problem where the difference between two natural numbers is to be formed.
The result should be a three-digit number whose three digits are all the same. The minuend should be a number that ends in zero. If this zero is removed, the subtrahend should result.
... | \begin{pmatrix}370\\-37\\\hline333\end{pmatrix}\quad\begin{pmatrix}740\\-\quad74\\\hline666\end{pmatrix}\quad\begin{pmatrix}1110\\-111\\\hline999\end{pmatrix} | 87 | 80 |
math | 33. Two players $A$ and $B$ play rock-paper-scissors continuously until player $A$ wins 2 consecutive games. Suppose each player is equally likely to use each hand-sign in every game. What is the expected number of games they will play? | 12 | 55 | 2 |
math | 1. In $\triangle A B C$, it is known that: $a=2, b=2 \sqrt{2}$. Find the range of values for $\angle A$. | A\in(0,\frac{\pi}{4}] | 38 | 12 |
math | Example 14. In a batch of 12 parts, 8 are standard. Find the probability that among 5 randomly selected parts, 3 will be standard. | \frac{14}{33} | 36 | 9 |
math | PROBLEM 4. - Determine all pairs of integers $(x, y)$ that satisfy the equation $x^{2}-y^{4}=2009$.
## Solution: | (x,y)=(\45,\2) | 38 | 9 |
math | The quartic (4th-degree) polynomial P(x) satisfies $P(1)=0$ and attains its maximum value of $3$ at both $x=2$ and $x=3$. Compute $P(5)$. | -24 | 53 | 3 |
math | Several gnomes, loading their luggage onto a pony, set off on a long journey. They were spotted by trolls, who counted 36 legs and 15 heads in the caravan. How many gnomes and how many ponies were there?
# | 12 | 54 | 2 |
math | 4. 122 Solve the system of equations $\left\{\begin{array}{l}\sqrt{x-1}+\sqrt{y-3}=\sqrt{x+y}, \\ \lg (x-10)+\lg (y-6)=1 .\end{array}\right.$ | no solution | 63 | 2 |
math | The twelve students in an olympiad class went out to play soccer every day after their math class, forming two teams of six players each and playing against each other. Each day they formed two different teams from those formed in previous days. By the end of the year, they found that every group of five students had p... | 132 | 83 | 3 |
math | Let $ABCD$ be a quadrilateral with an inscribed circle $\omega$ and let $P$ be the intersection of its diagonals $AC$ and $BD$. Let $R_1$, $R_2$, $R_3$, $R_4$ be the circumradii of triangles $APB$, $BPC$, $CPD$, $DPA$ respectively. If $R_1=31$ and $R_2=24$ and $R_3=12$, find $R_4$. | 19 | 115 | 2 |
math | Example 4 Given that $\alpha$ is an acute angle, $\beta$ is an obtuse angle, and $\sec (\alpha-2 \beta) 、 \sec \alpha 、 \sec (\alpha+2 \beta)$ form an arithmetic sequence, find the value of $\frac{\cos \alpha}{\cos \beta}$. | -\sqrt{2} | 71 | 5 |
math | [Help me] Determine the smallest value of the sum M =xy-yz-zx where x; y; z are real numbers satisfying the following condition $x^2+2y^2+5z^2 = 22$. | \frac{-55 - 11\sqrt{5}}{10} | 50 | 18 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \frac{(2 n+1)^{3}+(3 n+2)^{3}}{(2 n+3)^{3}-(n-7)^{3}}$ | 5 | 61 | 1 |
math | 7.130. $\left\{\begin{array}{l}\lg \left(x^{2}+y^{2}\right)=2-\lg 5, \\ \lg (x+y)+\lg (x-y)=\lg 1.2+1 .\end{array}\right.$ | (4;2)(4;-2) | 65 | 9 |
math | ## Task 24/83
Let $p_{1}$ and $p_{2}$ be two consecutive prime numbers with $p_{1}<p_{2}$ and $f(x)$ a polynomial of degree $n$ in $x$ with integer coefficients $a_{i}(i=0 ; 1 ; 2 ; \ldots ; n)$. Determine $p_{1}$ and $p_{2}$ from $f\left(p_{1}\right)=1234$ and $f\left(p_{2}\right)=4321$. | p_{1}=2,p_{2}=3 | 121 | 10 |
math | 7. The school organizes 1511 people to go on an outing, renting 42-seat coaches and 25-seat minibuses. If it is required that there is exactly one seat per person and one person per seat, then there are $\qquad$ rental options. | 2 | 60 | 1 |
math | 7. Without rearranging the digits on the left side of the equation, insert two "+" signs between them so that the equation $8789924=1010$ becomes true. | 87+899+24=1010 | 42 | 14 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \frac{\sqrt{n^{3}+1}-\sqrt{n-1}}{\sqrt[3]{n^{3}+1}-\sqrt{n-1}}$ | \infty | 59 | 3 |
math | ## Problem Statement
Based on the definition of the derivative, find $f^{\prime}(0)$:
$$
f(x)=\left\{\begin{array}{c}
\sqrt{1+\ln \left(1+3 x^{2} \cos \frac{2}{x}\right)}-1, x \neq 0 \\
0, x=0
\end{array}\right.
$$ | 0 | 88 | 1 |
math | 8,9
On the plane, points $A(1 ; 2), B(2 ; 1), C(3 ;-3), D(0 ; 0)$ are given. They are the vertices of a convex quadrilateral $A B C D$. In what ratio does the point of intersection of its diagonals divide the diagonal $A C$? | 1:3 | 75 | 3 |
math | 18.2. (Jury, Sweden, 79). Find the maximum value of the product $x^{2} y^{2} z^{2} u$ given that $x, y, z, u \geqslant 0$ and
$$
2 x+x y+z+y z u=1
$$ | \frac{1}{512} | 71 | 9 |
math | In convex quadrilateral $ABCD$, $\angle BAD = \angle BCD = 90^o$, and $BC = CD$. Let $E$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}$. Given that $\angle AED = 123^o$, find the degree measure of $\angle ABD$. | 78^\circ | 77 | 4 |
math | \section*{Task 1 - 131021}
Determine all (in the decimal number system) three-digit prime numbers with the following properties!
(1) If each digit of the three-digit prime number is written separately, each one represents a prime number.
(2) The first two and the last two digits of the three-digit prime number repre... | 373 | 88 | 3 |
math | 32nd Putnam 1971 Problem A2 Find all possible polynomials f(x) such that f(0) = 0 and f(x 2 + 1) = f(x) 2 + 1. Solution | f(x)=x | 50 | 4 |
math | 9.5. The cells of a chessboard are painted in 3 colors - white, gray, and black - in such a way that adjacent cells, sharing a side, must differ in color, but a sharp change in color (i.e., a white cell adjacent to a black cell) is prohibited. Find the number of such colorings of the chessboard (colorings that coincide... | 2^{33} | 98 | 5 |
math | The teacher said to Xu Jun: "Two years ago, my age was three times your age." Xu Jun said to the teacher: "In eight years, your age will be twice mine." Xu Jun is $\qquad$ years old this year. | 12 | 51 | 2 |
math | The Garfield Super Winners play $100$ games of foosball, in which teams score a non-negative integer number of points and the team with more points after ten minutes wins (if both teams have the same number of points, it is a draw). Suppose that the Garfield Super Winners score an average of $7$ points per game but all... | 81 | 134 | 2 |
math | The number sequence $1,4,5,7,12,15,16,18,23, \ldots$ was written based on a certain rule. Continue the number sequence until it exceeds the number 50.
Continue the number sequence until it exceeds the number 50. | 51 | 65 | 2 |
math | 11. Given a plane that makes an angle of $\alpha$ with all 12 edges of a cube, then $\sin \alpha=$ $\qquad$ | \frac{\sqrt{3}}{3} | 34 | 10 |
math | 3. If the expansion of $(a+2 b)^{n}$ has three consecutive terms whose binomial coefficients form an arithmetic sequence, then the largest three-digit positive integer $n$ is $\qquad$ . | 959 | 44 | 3 |
math | 52. The median line of a trapezoid divides it into two quadrilaterals. The difference in the perimeters of these two quadrilaterals is 24, and the ratio of their areas is $\frac{20}{17}$. If the height of the trapezoid is 2, then the area of this trapezoid is $\qquad$ _. | 148 | 82 | 3 |
math | 7. In the Magic Land, there is a magic stone that grows uniformly upwards. To prevent it from piercing the sky, the Elders of the Immortal World decided to send plant warriors to consume the magic stone and suppress its growth. Each plant warrior consumes the same amount every day. If 14 plant warriors are dispatched, ... | 17 | 124 | 2 |
math | In a new school $40$ percent of the students are freshmen, $30$ percent are sophomores, $20$ percent are juniors, and $10$ percent are seniors. All freshmen are required to take Latin, and $80$ percent of the sophomores, $50$ percent of the juniors, and $20$ percent of the seniors elect to take Latin. The probability t... | 25 | 126 | 2 |
math | (4) Given that the coordinates of points $M$ and $N$ satisfy the system of inequalities $\left\{\begin{array}{l}x \geqslant 0, \\ y \geqslant 0, \\ x+2 y \leqslant 6, \\ 3 x+y \leqslant 12,\end{array} \quad \vec{a}=\right.$ $(1,-1)$, then the range of $\overrightarrow{M N} \cdot \vec{a}$ is . $\qquad$ | [-7,7] | 121 | 5 |
math | ## Task Condition
Find the $n$-th order derivative.
$y=\lg (2 x+7)$ | y^{(n)}=(-1)^{n-1}\cdot\frac{2^{n}\cdot(n-1)!}{\ln10}\cdot(2x+7)^{-n} | 24 | 42 |
math | 8. (10 points) The number of pages in the 5 books, The Book of Songs, The Book of Documents, The Book of Rites, The Book of Changes, and The Spring and Autumn Annals, are all different: The Book of Songs and The Book of Documents differ by 24 pages. The Book of Documents and The Book of Rites differ by 17 pages. The Bo... | 34 | 173 | 2 |
math | Kazitsyna T.v.
Four mice: White, Gray, Fat, and Thin were dividing a piece of cheese. They cut it into 4 visually identical slices. Some slices had more holes, so Thin's slice weighed 20 grams less than Fat's, and White's slice weighed 8 grams less than Gray's. However, White was not upset because his slice weighed ex... | 14 | 139 | 2 |
math | Example 6 If $x^{2}+x+2 m$ is a perfect square, then $m=$ $\qquad$
$(1989$, Wu Yang Cup Junior High School Mathematics Competition) | \frac{1}{8} | 43 | 7 |
math | 11. (12 points) 1 kilogram of soybeans can be made into 3 kilograms of tofu, and 1 kilogram of soybean oil requires 6 kilograms of soybeans. Tofu is 3 yuan per kilogram, and soybean oil is 15 yuan per kilogram. A batch of soybeans totals 460 kilograms, and after being made into tofu or soybean oil and sold, 1800 yuan i... | 360 | 119 | 3 |
math | 9・53 Let $x, y, z$ be positive numbers and $x^{2}+y^{2}+z^{2}=1$, find
$$\frac{x}{1-x^{2}}+\frac{y}{1-y^{2}}+\frac{z}{1-z^{2}}$$
the minimum value. | \frac{3 \sqrt{3}}{2} | 71 | 12 |
math | An $m \times n$ chessboard where $m \le n$ has several black squares such that no two rows have the same pattern. Determine the largest integer $k$ such that we can always color $k$ columns red while still no two rows have the same pattern. | n - m + 1 | 58 | 7 |
math | 39. Find a two-digit number that has the following properties:
a) if the desired number is multiplied by 2 and 1 is added to the product, the result is a perfect square;
b) if the desired number is multiplied by 3 and 1 is added to the product, the result is a perfect square. | 40 | 68 | 2 |
math | ## Task 4 - 320734
Determine the number of all six-digit natural numbers that are divisible by 5 and whose cross sum is divisible by 9! | 20000 | 39 | 5 |
math | Problem 7.3. (15 points) Several boxes are stored in a warehouse. It is known that there are no more than 60 boxes, and each of them contains either 59 apples or 60 oranges. After a box with a certain number of oranges was brought to the warehouse, the number of fruits in the warehouse became equal. What is the smalles... | 30 | 90 | 2 |
math | $7 \cdot 8$ If a positive divisor is randomly selected from $10^{99}$, the probability that it is exactly a multiple of $10^{88}$ is $\frac{m}{n}$, where $m$ and $n$ are coprime, find $m+n$.
The prime factorization of $10^{99}$ is $2^{99} \cdot 5^{99}$. The total number of positive divisors of $10^{99}$ is $(99+1)(99+... | 634 | 330 | 3 |
math | How many ways to fill the board $ 4\times 4$ by nonnegative integers, such that sum of the numbers of each row and each column is 3? | 2008 | 36 | 4 |
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 | Can a regular triangle be divided into
a) 2007,
b) 2008 smaller regular triangles? | 2007 | 27 | 4 |
math | The bisector of $\angle BAC$ in $\triangle ABC$ intersects $BC$ in point $L$. The external bisector of $\angle ACB$ intersects $\overrightarrow{BA}$ in point $K$. If the length of $AK$ is equal to the perimeter of $\triangle ACL$, $LB=1$, and $\angle ABC=36^\circ$, find the length of $AC$. | 1 | 84 | 1 |
math | Solve the following equation:
$$
x-1=\left[\frac{x}{2}\right]+\left[\frac{x}{3}\right]+\left[\frac{x}{6}\right]
$$ | 6k+1,6k+2,6k+3,6k+4 | 40 | 19 |
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