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 | 15. Let $p \geqslant 3$, try to calculate the value of the following expression:
$$\left(\frac{1 \cdot 2}{p}\right)+\left(\frac{2 \cdot 3}{p}\right)+\cdots+\left(\frac{(p-2)(p-1)}{p}\right)$$ | -1 | 77 | 2 |
math | 12. Cat and Mouse. If $n$ cats eat $n$ mice in $n$ hours, then how many mice will $p$ cats eat in $p$ hours? | \frac{p^2}{n} | 39 | 9 |
math | 12. Distribute $n$ identical (i.e., indistinguishable) footballs to $r$ football teams, so that each team gets at least $s_{1}$ footballs, but no more than $s_{2}\left(r s_{1} \leqslant n \leqslant r s_{2}\right)$ footballs. How many different ways are there to do this? | \sum_{k=0}^{r}(-1)^{k}\binom{r}{k}\binom{n+r-s_{1}r-(s_{2}-s_{1}+1)k-1}{r-1} | 87 | 52 |
math | 3. (6 points) Dad is 24 years older than his son. This year, Dad's age is five times that of his son. $\qquad$ years later, Dad's age will be
three times that of his son. | 6 | 51 | 1 |
math | Example 3 Let $n$ be a fixed integer, $n \geqslant 2$.
(1) - Determine the smallest constant $c$ such that the inequality $\sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leqslant c\left(\sum_{1 \leqslant i \leqslant n} x_{i}\right)^{4}$ holds for all non-negative numbers;
(2) - For this... | \frac{1}{8} | 148 | 7 |
math | 4. After adding another tug to the one pushing the barge, they started pushing the barge with double the force. How will the power spent on movement change if the water resistance is proportional to the first power of the barge's speed? | increased\\4\times | 50 | 6 |
math | 5. Let $[|x|]$ be the integer part of $x$ and $\{x\}=x-[|x|]$, the decimal part of $x$. Solve $2[|x|]=x+2\{x\}$. | 0,\frac{4}{3},\frac{8}{3} | 54 | 15 |
math | Consider a regular $n$-gon $A_1A_2\ldots A_n$ with area $S$. Let us draw the lines $l_1,l_2,\ldots,l_n$ perpendicular to the plane of the $n$-gon at $A_1,A_2,\ldots,A_n$ respectively. Points $B_1,B_2,\ldots,B_n$ are selected on lines $l_1,l_2,\ldots,l_n$ respectively so that:
(i) $B_1,B_2,\ldots,B_n$ are all on the sa... | S \cdot \frac{\sum_{i=1}^n h_i}{n} | 239 | 19 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow \frac{1}{2}} \frac{\sqrt[3]{\frac{x}{4}}-\frac{1}{2}}{\sqrt{\frac{1}{2}+x}-\sqrt{2 x}}$ | -\frac{2}{3} | 64 | 7 |
math | 6. Find all values of the parameter $a$ for which there exists a value of the parameter $b$ such that the system
$$
\left\{\begin{array}{l}
\arcsin \left(\frac{a+y}{2}\right)=\arcsin \left(\frac{x+3}{3}\right) \\
x^{2}+y^{2}+6 x+6 y=b
\end{array}\right.
$$
has exactly two solutions. | \in(-\frac{7}{2};\frac{19}{2}) | 104 | 18 |
math | 946. Find the divergence of the vector field:
1) $\bar{r}=x i+y \bar{j}+z k$
2) $\bar{p}=\frac{\bar{i}+\bar{i}+\bar{k}}{\sqrt[3]{(x+y+z)^{2}}}$
3) $\vec{q}=e^{x y}(y \bar{j}-x \bar{i}+x y \bar{k})$. | 3,-2(x+y+z)^{-} | 95 | 9 |
math | 12*. In how many ways can milk be transferred from a 12-liter barrel, filled with milk, to another empty barrel of the same volume using two empty cans of 1 liter and 2 liters? Transferring milk from one can to another is not allowed.
Note that the question in this problem is different from the previous problems. | 233 | 71 | 3 |
math | 22. Find the value of the series
$$
\sum_{k=0}^{\infty}\left\lfloor\frac{20121+2^{k}}{2^{k+1}}\right\rfloor
$$ | 20121 | 55 | 5 |
math | 11. Let $a$ and $b$ be real numbers such that $a>b, 2^{a}+2^{b}=75$ and $2^{-a}+2^{-b}=12^{-1}$. Find the value of $2^{a-b}$. | 4 | 61 | 1 |
math | There are $11$ members in the competetion committee. The problem set is kept in a safe having several locks.
The committee members have been provided with keys in such a way that every six members can open the safe, but no fi ve members can do that.
What is the smallest possible number of locks, and how many keys are ... | 2772 | 77 | 4 |
math | Problem 5. Solve the system of equations in real numbers:
$$
\left\{\begin{array}{l}
a+c=-4 \\
a c+b+d=6 \\
a d+b c=-5 \\
b d=2
\end{array}\right.
$$ | (-3,2,-1,1)(-1,1,-3,2) | 57 | 18 |
math | ## Problem 6.
Find all real solutions of the equation
$$
3^{x^{2}-x-y}+3^{y^{2}-y-z}+3^{z^{2}-z-x}=1
$$ | (x,y,z)=(1,1,1) | 47 | 10 |
math | 14. The roots of the equation $x^{2}-2 x-a^{2}-a=0$ are $\left(\alpha_{a}\right.$,
$$
\begin{array}{r}
\left.\beta_{a}\right)(a=1,2, \cdots, 2011) . \\
\text { Find } \sum_{a=1}^{2011}\left(\frac{1}{\alpha_{a}}+\frac{1}{\beta_{a}}\right) \text { . }
\end{array}
$$ | -\frac{2011}{1006} | 123 | 13 |
math | ## Task 5/73
Determine all prime numbers of the form $p=x^{4}+4 y^{4}$, where x and y are natural numbers. | 5 | 37 | 1 |
math | Suppose that $20^{21} = 2^a5^b = 4^c5^d = 8^e5^f$ for positive integers $a,b,c,d,e,$ and $f$. Find $\frac{100bdf}{ace}$.
[i]Proposed by Andrew Wu[/i] | 75 | 72 | 2 |
math | In a circus, there are $n$ clowns who dress and paint themselves up using a selection of 12 distinct colours. Each clown is required to use at least five different colours. One day, the ringmaster of the circus orders that no two clowns have exactly the same set
of colours and no more than 20 clowns may use any one par... | 48 | 99 | 2 |
math | Let $\mathcal{S}$ be the set of all perfect squares whose rightmost three digits in base $10$ are $256$. Let $\mathcal{T}$ be the set of all numbers of the form $\frac{x-256}{1000}$, where $x$ is in $\mathcal{S}$. In other words, $\mathcal{T}$ is the set of numbers that result when the last three digits of each number ... | 170 | 130 | 3 |
math | There are $15$ people, including Petruk, Gareng, and Bagong, which will be partitioned into $6$ groups, randomly, that consists of $3, 3, 3, 2, 2$, and $2$ people (orders are ignored). Determine the probability that Petruk, Gareng, and Bagong are in a group. | \frac{3}{455} | 80 | 9 |
math | 36. Multiply your shoe size by 2, add 39 to the product, multiply the resulting sum by 50, add 40 to the product, and subtract your birth year from the sum. You will get a four-digit number, the first two digits of which are your shoe size, and the last two digits are your age at the end of the calendar year 1990. Expl... | 100\overline{}+15 | 96 | 10 |
math | 34. (1984 American Invitational Mathematics Examination) A gardener is to plant three maple trees, four oaks, and five birch trees in a row. He randomly determines the order of these trees, with all different arrangements being equally likely. Express the probability that no two birch trees are adjacent as $\frac{m}{n}... | 106 | 85 | 3 |
math | 25. Three integers are selected from the set $S=1,2,3, \ldots, 19,20$. Find the number of selections where the sum of the three integers is divisible by 3 . | 384 | 48 | 3 |
math | XIII OM - III - Task 4
In how many ways can a set of $ n $ items be divided into two sets? | 2^{n-1}-1 | 28 | 7 |
math | Example 3. Evaluate the integral $I=\int_{-\pi / 2}^{\pi / 2} \frac{2 d x}{7-3 \cos 2 x}$. | \frac{\pi}{5}<I<\frac{\pi}{2} | 42 | 16 |
math | 2. Variant 1.
Athletes started in groups of 3 people with a delay between groups of several seconds. Petya, Vasya, and Kolya started simultaneously, and they were in the seventh trio from the beginning and the fifth trio from the end. How many athletes participated in the race? | 33 | 65 | 2 |
math | Example 1. Find $\int \frac{7 x^{3}-4 x^{2}-32 x-37}{(x+2)(2 x-1)\left(x^{2}+2 x+3\right)} d x$. | 3\ln|x+2|-\frac{5}{2}\ln|2x-1|+\frac{3}{2}\ln(x^{2}+2x+3)-\frac{4}{\sqrt{2}}\operatorname{arctg}\frac{x+1}{\sqrt{2}}+C | 53 | 69 |
math | 4. Variant 1.
A rectangle was cut into three rectangles, two of which have dimensions 9 m $\times$ 12 m and 10 m $\times$ 15 m. What is the maximum area the original rectangle could have had? Express your answer in square meters. | 330 | 61 | 3 |
math | 1. Two cars started from the same place but in opposite directions. At the moment when their distance was $18 \mathrm{~km}$, one car had traveled $3 \mathrm{~km}$ less than twice the distance of the other. Determine the distance each car had traveled by that moment. | 11,7 | 63 | 4 |
math | Solve the following equation in the set of integers:
$$
\underbrace{\sqrt{x+\sqrt{x+\sqrt{x+\cdots+\sqrt{x}}}}}_{1992 \text { square root signs }}=y
$$ | (0,0) | 49 | 5 |
math | \section*{Problem 1 - 151221}
a) Investigate whether there exist natural numbers \(n\) such that in the expansion formed according to the binomial theorem
\[
(a+b)^{n}=c_{0} a^{n}+c_{1} a^{n-1} \cdot b+c_{2} a^{n-2} \cdot b^{2}+\ldots+c_{n} b^{n}
\]
the coefficients \(c_{0}, c_{1}, c_{2}\) have the sum \(c_{0}+c_{1}... | 12 | 321 | 2 |
math | (12) The minimum distance from the origin to a point on the curve $x^{2}-2 x y-3 y^{2}=1$ is $\qquad$ . | \frac{\sqrt{\sqrt{5}+1}}{2} | 38 | 15 |
math | 10. A dealer bought two cars. He sold the first one for $40 \%$ more than he paid for it and the second one for $60 \%$ more than he paid for it. The total sum he received for the two cars was $54 \%$ more that the total sum he paid for them. When written in its lowest terms, the ratio of the prices the dealer paid for... | 124 | 111 | 3 |
math | Let $n(n \geq 1)$ be a positive integer and $U=\{1, \ldots, n\}$. Let $S$ be a nonempty subset of $U$ and let $d(d \neq 1)$ be the smallest common divisor of all elements of the set $S$. Find the smallest positive integer $k$ such that for any subset $T$ of $U$, consisting of $k$ elements, with $S \subset T$, the great... | 1+\left[\frac{n}{d}\right] | 118 | 11 |
math | Example 3. Find $\lim _{\substack{x \rightarrow 0 \\ y \rightarrow 0}} \frac{x^{2} y}{x^{2}+y^{2}}$. | 0 | 41 | 1 |
math | ## Zadatak B-2.5.
Ako je $x=2017 \cdot 2018 \cdot 2021 \cdot 2022+4$, izračunajte $\sqrt{x}-2020^{2}$.
| -2024 | 62 | 5 |
math | 2. Let
$$
\sqrt{49-a^{2}}-\sqrt{25-a^{2}}=3
$$
Calculate the value of the expression
$$
\sqrt{49-a^{2}}+\sqrt{25-a^{2}} .
$$ | 8 | 58 | 1 |
math | 1. The distance between cities $A$ and $B$ is 435 km. A train departed from $A$ at a speed of 45 km/h. After 40 minutes, another train departed from city $B$ towards it at a speed of 55 km/h. How far apart will they be one hour before they meet? | 100 | 75 | 3 |
math | 3. The value of the complex number $\left(\frac{1}{2}+\frac{\sqrt{3}}{2} \mathrm{i}\right)^{6 n}\left(n \in \mathbf{Z}_{+}\right)$ is | 1 | 52 | 1 |
math | The set $X$ of $N$ four-digit numbers formed from the digits $1,2,3,4,5,6,7,8$ satisfies the following condition:
[i]for any two different digits from $1,2,3,4,,6,7,8$ there exists a number in $X$ which contains both of them. [/i]\\
Determine the smallest possible value of $N$. | 6 | 89 | 1 |
math | Lirfisches. $\underline{\text {... }}$.
Yura laid out 2001 coins of 1, 2, and 3 kopecks in a row. It turned out that between any two 1-kopeck coins there is at least one coin, between any two 2-kopeck coins there are at least two coins, and between any two 3-kopeck coins there are at least three coins. How many 3-kope... | 500or501 | 106 | 7 |
math | 17. (10 points) In $\triangle A B C$, let the sides opposite to $\angle A$, $\angle B$, and $\angle C$ be $a$, $b$, and $c$, respectively, satisfying $a \cos C=(2 b-c) \cos A$.
(1) Find the size of $\angle A$;
(2) If $a=\sqrt{3}$, and $D$ is the midpoint of side $B C$, find the range of $A D$. | (\frac{\sqrt{3}}{2},\frac{3}{2}] | 107 | 17 |
math | Example 12 The sports meet lasted for $n$ days $(n>1)$, and a total of $m$ medals were awarded. On the first day, 1 medal was awarded, and then $\frac{1}{7}$ of the remaining $m-1$ medals. On the second day, 2 medals were awarded, and then $\frac{1}{7}$ of the remaining medals, and so on, until on the $n$-th day exactl... | 6 | 135 | 1 |
math | 4th ASU 1964 Problem 3 Reduce each of the first billion natural numbers (billion = 10 9 ) to a single digit by taking its digit sum repeatedly. Do we get more 1s than 2s? | 1 | 53 | 1 |
math | Find all primes $ p$ for that there is an integer $ n$ such that there are no integers $ x,y$ with $ x^3 \plus{} y^3 \equiv n \mod p$ (so not all residues are the sum of two cubes).
E.g. for $ p \equal{} 7$, one could set $ n \equal{} \pm 3$ since $ x^3,y^3 \equiv 0 , \pm 1 \mod 7$, thus $ x^3 \plus{} y^3 \equiv 0 ,... | 7 | 135 | 1 |
math | 12.5 The sequence of natural numbers $\left\{x_{n}\right\}$ is constructed according to the following rule:
$$
x_{1}=a, \quad x_{2}=b, \quad x_{n+2}=x_{n}+x_{n+1}, \quad n \geqslant 1 .
$$
It is known that one of the terms in the sequence is 1000. What is the smallest possible value of the sum $a+b$?
(Recommended by t... | 10 | 123 | 2 |
math | Exercise 2. We place the integers from 1 to 9 in each of the cells of a $3 \times 3$ grid. For $i=1$, 2 and 3, we denote $\ell_{i}$ as the largest integer present in the $i$-th row and $c_{i}$ as the smallest integer present in the $i^{\text{th}}$ column.
How many grids exist such that $\min \left\{\ell_{1}, \ell_{2},... | 25920 | 148 | 5 |
math | 5. Three Thieves, Bingo, Bunko, and Balko, robbed a bank and carried away 22 bags of banknotes. They placed them in a row so that the first bag contained the least amount of money, and each subsequent bag contained one more stack of banknotes than the previous one. Chief Bingo divided the stolen bags of money according... | 500000 | 330 | 6 |
math | Find all second degree polynomial $d(x)=x^{2}+ax+b$ with integer coefficients, so that there exists an integer coefficient polynomial $p(x)$ and a non-zero integer coefficient polynomial $q(x)$ that satisfy: \[\left( p(x) \right)^{2}-d(x) \left( q(x) \right)^{2}=1, \quad \forall x \in \mathbb R.\] | d(x) = x^2 + ax + b | 90 | 12 |
math | Aerith bakes some cookes. On the first day, she gives away 1 cookie and then $1 / 8$ of the remaining cookies; on the second day, she gives away 2 cookies and then $1 / 8$ of the remaining cookies, and so on. On the 7th day, she gives away 7 cookies and then there are none left. How many cookies did she bake? | 49 | 88 | 2 |
math | 6.4. Petya and Masha made apple juice. In total, they got 10 liters. They poured it into two jugs. But it turned out that it was hard for Masha to carry her jug, so she poured some of the juice into Petya's jug. As a result, Petya ended up with three times more, and Masha ended up with three times less. How many liters... | 7.5 | 97 | 3 |
math | 13. $[\mathbf{9}]$ Find the smallest positive integer $n$ for which
$$
1!2!\cdots(n-1)!>n!^{2} .
$$ | 8 | 42 | 1 |
math | 6.131. Without solving the equation $x^{2}-(2 a+1) x+a^{2}+2=0$, find the value of $a$ for which one of the roots is twice the other. | 4 | 49 | 1 |
math | 7. Given that the parabola $P$ has the center of the ellipse $E$ as its focus, $P$ passes through the two foci of $E$, and $P$ intersects $E$ at exactly three points, then the eccentricity of $E$ is equal to | \frac{2\sqrt{5}}{5} | 61 | 12 |
math | 8. (2002 Hunan Province Competition Question) Given $a_{1}=1, a_{2}=3, a_{n+2}=(n+3) a_{n+1}-(n+2) a_{n}$. If for $m \geqslant n$, the value of $a_{m}$ can always be divided by 9, find the minimum value of $n$.
Translate the above text into English, please retain the original text's line breaks and format, and output ... | 5 | 115 | 1 |
math | 22. Alice and Bob are playing a game with dice. They each roll a die six times, and take the sums of the outcomes of their own rolls. The player with the higher sum wins. If both players have the same sum, then nobody wins. Alice's first three rolls are 6,5 , and 6 , while Bob's first three rolls are 2,1 , and 3 . The ... | 3895 | 111 | 4 |
math | 9. Let the line $l: y=x+b\left(0<b<\frac{1}{2}\right)$ intersect the parabola $y^{2}=2 x$ at points $A$ and $B$, and let $O$ be the origin. Then, when the area of $\triangle A O B$ is maximized, the equation of the line $l$ is $\qquad$ | y=x+\frac{1}{3} | 86 | 9 |
math | In how many different ways one can place 3 rooks on the cells of $6 \times 2006$ chessboard such that they don't attack each other? | 20\cdot2006\cdot2005\cdot2004 | 37 | 20 |
math | The number $N$ is the product of two primes. The sum of the positive divisors of $N$ that are less than $N$ is $2014$. Find $N$. | 4022 | 41 | 4 |
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 | 3. (10 points) Divide students into 35 groups, each with 3 people. Among them, there are 10 groups with only 1 boy, 19 groups with no less than 2 boys, and the number of groups with 3 boys is twice the number of groups with 3 girls. Then the number of boys is $\qquad$ people. | 60 | 80 | 2 |
math | Example 5 Find all positive integers $x, y$ such that
$$y^{x}=x^{50}$$ | (x, y) = (1,1), \left(2,2^{25}\right), \left(2^{2}, 2^{25}\right), \left(5,5^{10}\right), \left(5^{2}, 5^{4}\right), \left(10,10^{5}\right), (50,50), (100,10) | 26 | 91 |
math | My favorite numbers are those which, when multiplied by their own digit sum, become ten times larger. The product of three of my favorite numbers is 71668. Which are these numbers?
(Birdie) | 19,46,82 | 45 | 8 |
math | 2.42. The areas of the bases of a truncated pyramid are $S_{1}$ and $S_{2}$ $\left(S_{1}<S_{2}\right)$, and its volume is $V$. Determine the volume of the complete pyramid. | \frac{VS_{2}\sqrt{S_{2}}}{S_{2}\sqrt{S_{2}}-S_{1}\sqrt{S_{1}}} | 54 | 35 |
math | B1. Let the sequence $a_{n}$ satisfy the recursive relation $a_{n+1}=\frac{1+a_{n}}{1-a_{n}}$ for all natural numbers $n$. Determine all possible values of the first term $a_{1}$, for which the sequence $a_{n}$ will contain a term $\mathrm{z}$ with the value 2022. | {2022,\frac{2021}{2023},-\frac{1}{2022},-\frac{2023}{2021}} | 85 | 41 |
math | Find all polynomials $p(x)\in\mathbb{R}[x]$ such that for all $x\in \mathbb{R}$:
$p(5x)^2-3=p(5x^2+1)$ such that:
$a) p(0)\neq 0$
$b) p(0)=0$ | p(x) = \frac{1 + \sqrt{13}}{2} | 71 | 19 |
math | 8.2. The product of two natural numbers $a$ and $b$ is a three-digit number, which is the cube of some natural number $k$. The quotient of the numbers $a$ and $b$ is the square of this same number $k$. Find $a, b$, and $k$. | =243,b=3,k=9 | 66 | 10 |
math | 2. Let the set
$$
A=\left\{a_{1}, a_{2}, \cdots, a_{5}\right\}\left(a_{i} \in \mathbf{R}_{+}, i=1,2, \cdots, 5\right) .
$$
If the set of the products of the four elements in all four-element subsets of set $A$ is $B=\{2,3,4,6,9\}$, then the sum of the elements in set $A$ is $\qquad$ | \frac{49}{6} | 118 | 8 |
math | 4.14. Find the coordinates of the point $A^{\prime}$, symmetric to the point $A(2,0,2)$ with respect to the plane $4 x+6 y+4 z-50=0$. | (6,6,6) | 51 | 7 |
math | 21. For a positive integer $n$, define $s(n)$ as the smallest positive integer $t$ such that $n$ is a factor of $t$ !. Compute the number of positive integers $n$ for which $s(n)=13$. | 792 | 55 | 3 |
math | 18. Find the sum of the coefficients of the polynomial
$$
\left(4 x^{2}-4 x+3\right)^{4}\left(4+3 x-3 x^{2}\right)^{2} \text {. }
$$ | 1296 | 55 | 4 |
math | Let $a_{n}$ denote the integer closest to $\sqrt{n}$. What is the value of
$$
\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{1994}}
$$ | \frac{3974}{45} | 58 | 11 |
math | In the class, there are fewer than 30 people. The probability that a randomly chosen girl is an excellent student is $3 / 13$, and the probability that a randomly chosen boy is an excellent student is $4 / 11$. How many excellent students are there in the class? | 7 | 62 | 1 |
math | 【Question 23】
A positive integer, if from the first digit (the highest place) to a certain digit in the middle the digits are increasingly larger, and from this digit to the last digit (the unit place) the digits are increasingly smaller, is called a "convex number" (for example, 1357642 is a "convex number", while 753... | 254 | 122 | 3 |
math | One plant is now $44$ centimeters tall and will grow at a rate of $3$ centimeters every $2$ years. A second plant is now $80$ centimeters tall and will grow at a rate of $5$ centimeters every $6$ years. In how many years will the plants be the same height? | 54 | 71 | 2 |
math | 2. a) (4p) Let $x \neq-1, y \neq-2, z \neq-3$ be rational numbers such that $\frac{2015}{x+1}+\frac{2015}{y+2}+\frac{2015}{z+3}=2014$. Calculate $\frac{x-1}{x+1}+\frac{y}{y+2}+\frac{z+1}{z+3}$.
Mathematical Gazette No. 10/2014
b) (3p) Let $a=\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\ldots+\frac{1}{2014}-\frac{... | \frac{2017}{2015} | 192 | 13 |
math | Exercise 3. Let $x, y$ and $z$ be three real numbers such that $x^{2}+y^{2}+z^{2}=1$. Find the minimum and maximum possible values of the real number $x y+y z-z x$. | -1 | 56 | 2 |
math | $ABCD$ is a rectangle. $I$ is the midpoint of $CD$. $BI$ meets $AC$ at $M$. Show that the line $DM$ passes through the midpoint of $BC$. $E$ is a point outside the rectangle such that $AE = BE$ and $\angle AEB = 90^o$. If $BE = BC = x$, show that $EM$ bisects $\angle AMB$. Find the area of $AEBM$ in terms of $x$. | \frac{x^2 (3 + 2\sqrt{2})}{6} | 108 | 19 |
math | \section*{Exercise 1 - 141221}
Let \(x_{n}(n=0,1,2,3, \ldots)\) be the sequence of numbers for which \(x_{0}=1\) and
\[
x_{n}=\frac{x_{n-1}}{x_{n-1}+1}
\]
\((n=1,2,3, \ldots)\) holds.
Determine the terms \(x_{1}, x_{2}\) and \(x_{3}\) of this sequence. Provide a term \(f(n)\) with the property \(f(n)=x_{n}(n=1,2,3,... | x_{n}=\frac{1}{n+1} | 153 | 13 |
math | 2. Given numbers $x, y \in\left(0, \frac{\pi}{2}\right)$. Find the maximum value of the expression
$$
A=\frac{\sqrt{\cos x \cos y}}{\sqrt{\operatorname{ctg} x}+\sqrt{\operatorname{ctg} y}}
$$ | \frac{\sqrt{2}}{4} | 70 | 10 |
math | Let's determine the sum of the following series:
$$
\binom{n}{0}+\binom{n+1}{1}+\binom{n+2}{2}+\ldots+\binom{n+k}{k}
$$
where $n$ and $k$ are natural numbers. (The symbol $\binom{n}{k}$ represents the number of ways to choose $k$ elements from a set of $n$ elements, disregarding the order of the elements. As is known... | \binom{n+k+1}{k} | 186 | 10 |
math | G2.2 Let $f_{1}=9$ and $f_{n}=\left\{\begin{array}{ll}f_{n-1}+3 & \text { if } n \text { is a multiple of } 3 \\ f_{n-1}-1 & \text { if } n \text { is not a multiple of } 3\end{array}\right.$.
If $B$ is the number of possible values of $k$ such that $f_{k}<11$, determine the value of $B$. | 5 | 118 | 1 |
math | 1. Let $x^{2}+y^{2} \leqslant 2$. Then the maximum value of $\left|x^{2}-2 x y-y^{2}\right|$ is $\qquad$ . | 2 \sqrt{2} | 47 | 6 |
math | Example 3 Given $x \in \mathbf{R}$, let $M=\max \{1-x, 2 x-$ $3, x\}$. Find $M_{\min }$. | \frac{1}{2} | 44 | 7 |
math | # Problem T-1
Given a pair $\left(a_{0}, b_{0}\right)$ of real numbers, we define two sequences $a_{0}, a_{1}, a_{2}, \ldots$ and $b_{0}, b_{1}, b_{2}, \ldots$ of real numbers by
$$
a_{n+1}=a_{n}+b_{n} \quad \text { and } \quad b_{n+1}=a_{n} \cdot b_{n}
$$
for all $n=0,1,2, \ldots$ Find all pairs $\left(a_{0}, b_{0... | b_{0}=0 | 207 | 5 |
math | Problem 3. Several (more than one) consecutive natural numbers are written on the board, the sum of which is 2016. What can the largest of these numbers be? | 63,106,228,291,673 | 39 | 18 |
math | Example 3 Let $A, B, C, D$ be four points in space, and connect $AB, AC, AD, BC, BD, CD$ where at most one of these has a length greater than 1. Try to find the maximum value of the sum of the lengths of the six segments.
Connect $AB, AC, AD, BC, BD, CD$ where at most one of these has a length greater than 1, try to f... | 5 + \sqrt{3} | 110 | 7 |
math | 4. For any positive real numbers $x, y$,
$$
\frac{(x y+x+y)(x+y+1)}{(x+y)(x+1)(y+1)}
$$
the range of values is | (1,\frac{9}{8}] | 47 | 9 |
math | 80*. Find the condition under which the polynomial $x^{3}+p x^{2}+q x+n$ is a perfect cube. | q=\frac{p^{2}}{3};n=\frac{p^{3}}{27} | 31 | 23 |
math | 2.2.45 ** In $\triangle A B C$, if $\frac{\cos A}{\sin B}+\frac{\cos B}{\sin A}=2$, and the perimeter of $\triangle A B C$ is 12. Find the maximum possible value of its area. | 36(3-2\sqrt{2}) | 61 | 11 |
math | Problem 1. The lengths of the sides of triangle $ABC$ are $54 \mathrm{~mm}$, $39 \mathrm{~mm}$, and $47 \mathrm{~mm}$, and the lengths of the sides of triangle $KLM$ are $8 \mathrm{~cm}$, $4 \mathrm{~cm}$, and $5 \mathrm{~cm}$. Which one has the larger perimeter? | 17 | 94 | 2 |
math | 5. Given two points $A(0,1), B(6,9)$. If there is an integer point $C$ (Note: A point with both coordinates as integers is called an integer point), such that the area of $\triangle A B C$ is minimized. Then the minimum value of the area of $\triangle A B C$ is $\qquad$ | 1 | 77 | 1 |
math | $4.77 \operatorname{tg} 9^{\circ}+\operatorname{tg} 15^{\circ}-\operatorname{tg} 27^{\circ}-\operatorname{ctg} 27^{\circ}+\operatorname{ctg} 9^{\circ}+\operatorname{ctg} 15^{\circ}=8$. | 8 | 86 | 1 |
math | In the coordinate plane, a set of $2000$ points $\{(x_1, y_1), (x_2, y_2), . . . , (x_{2000}, y_{2000})\}$ is called [i]good[/i] if $0\leq x_i \leq 83$, $0\leq y_i \leq 83$ for $i = 1, 2, \dots, 2000$ and $x_i \not= x_j$ when $i\not=j$. Find the largest positive integer $n$ such that, for any good set, the interior and... | 25 | 171 | 2 |
math | 12. Suppose that $a, b$ and $c$ are real numbers greater than 1 . Find the value of
$$
\frac{1}{1+\log _{a^{2} b}\left(\frac{c}{a}\right)}+\frac{1}{1+\log _{b^{2} c}\left(\frac{a}{b}\right)}+\frac{1}{1+\log _{c^{2} a}\left(\frac{b}{c}\right)} .
$$ | 3 | 109 | 1 |
math | 3. The number of lattice points (points with integer coordinates) inside the region (excluding the boundary) bounded by the right branch of the hyperbola $x^{2}-y^{2}=1$ and the line $x=100$ is $\qquad$ . | 9800 | 58 | 4 |
math | Let $ a$, $ b$, $ c$, $ x$, $ y$, and $ z$ be real numbers that satisfy the three equations
\begin{align*}
13x + by + cz &= 0 \\
ax + 23y + cz &= 0 \\
ax + by + 42z &= 0.
\end{align*}Suppose that $ a \ne 13$ and $ x \ne 0$. What is the value of
\[ \frac{13}{a - 13} + \frac{23}{b - 23} + \frac{42}{c -... | -2 | 156 | 2 |
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