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 | ### 2.142.
$$
\frac{\frac{1}{\sqrt{3+x} \cdot \sqrt{x+2}}+\frac{1}{\sqrt{3-x} \cdot \sqrt{x-2}}}{\frac{1}{\sqrt{3+x} \cdot \sqrt{x+2}}-\frac{1}{\sqrt{3-x} \cdot \sqrt{x-2}}} ; \quad x=\sqrt{6} \text {. }
$$ | -\frac{\sqrt{6}}{2} | 103 | 10 |
math | 1. In the addition example: $\square+\triangle+\square=\square \square$, fill in the same digit in each square and a different digit in the triangle so that the example becomes correct. | 1+9+1=11 | 40 | 8 |
math | 3. The difference of two natural numbers is 5 times less than their sum and 24 times less than their product. Find these numbers. | 12,8 | 30 | 4 |
math | 6.7. (New York, 78). Find all pairs of natural numbers $A \neq B$ for which the system
$$
\left\{\begin{array}{l}
\cos A x+\cos B x=0 \\
A \sin A x+B \sin B x=0
\end{array}\right.
$$
has a solution. | A=2^p,B=2^q | 79 | 10 |
math | 6. Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{n+1}+(-1)^{n} a_{n}=2 n-1 \text {, }
$$
and the sum of the first 2019 terms of the sequence $\left\{a_{n}-n\right\}$ is 2019. Then the value of $a_{2020}$ is $\qquad$ .
6.1.
$$
\begin{array}{l}
\text { From } a_{n+1}+(-1)^{n} a_{n}=2 n-1 \\
\Rightarrow\left... | 1 | 378 | 1 |
math | 2. Let $n$ be the smallest positive integer satisfying the following conditions:
(1) $n$ is a multiple of 75;
(2) $n$ has exactly 75 positive divisors (including 1 and itself).
Find $\frac{n}{75}$.
(Eighth American Mathematical Invitational) | 432 | 67 | 3 |
math | 3. Let $A=\{1,2,3,4,5\}$. Then the number of mappings $f: A \rightarrow A$ that satisfy the condition $f(f(x))$ $=f(x)$ is $\qquad$ (answer with a number) | 196 | 58 | 3 |
math | Problem 3. Marco was born between 1300 and 1400, and Darko between 1400 and 1500. Both were born on April 6th, in years that are perfect squares of natural numbers. Both lived for 110 years. In which year, while they were alive, were their ages on April 7th perfect squares of natural numbers. | 1469 | 87 | 4 |
math | Example 1 Find the value of $\xi_{0}=\langle-1,1,4, \overline{3,1,1,1,3,7}\rangle$.
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | \frac{3-\sqrt{57}}{24} | 65 | 14 |
math | Ask 31 The sequence $\left\{a_{n}\right\}$ is defined as follows:
$$
\begin{array}{r}
a_{1}=0, a_{2}=1, \cdots, a_{n}=\frac{1}{2} n a_{n-1}+\frac{1}{2} n(n-1) a_{n-2}+(-1)^{n}\left(1-\frac{n}{2}\right) \\
(n \geqslant 3) .
\end{array}
$$
Try to find the simplest expression for $f_{n}=a_{n}+2 \mathrm{C}_{n}^{1} a_{n-1}... | 2n!-(n+1) | 214 | 8 |
math | A sequence $\{a_n\}_{n\geq0}$ obeys the recurrence $a_n=1+a_{n-1}+\alpha a_{n-2}$ for all $n\geq2$ and for some $\alpha>0$. Given that $a_0=1$ and $a_1=2$, compute the value of $\alpha$ for which
$$\sum_{n=0}^{\infty}\frac{a_n}{2^n}=10$$ | \frac{6}{5} | 105 | 7 |
math | 6. Given a constant $a \in(0,1)$, $|x|+|y| \leqslant 1$, the maximum value of the function $f(x, y)=a x+y$ is $\qquad$ . | 1 | 53 | 1 |
math | ## Task 1 - 201221
For each $n=1,2,3, \ldots$ let
$$
a_{n}=\frac{1}{n^{2}} \sum_{k=1}^{n} k
$$
Furthermore, let $I_{1}, I_{2}, I_{3}$, and $I_{4}$ be the closed intervals
$$
I_{1}=[1 ; 2], \quad I_{2}=[0.53 ; 0.531], \quad I_{3}=[0.509 ; 0.51], \quad I_{4}=[0.4 ; 0.5]
$$
Investigate for each of these intervals whe... | a_{n}\inI_{1}exactlyforn=1,\,a_{n}\inI_{2}fornon,\,a_{n}\inI_{3}exactlyforn=50,51,52,53,54,55,\,a_{n}\inI_{4}fornon | 209 | 76 |
math | 3. Determine all ordered triples of numbers $(x, y, z)$ for which: $x-y=y-z=$ 96, where $x, y, z$ are squares of natural numbers. | 196,100,4 | 42 | 9 |
math | 6、Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=1, a_{n+1}=\frac{(n+1) a_{n}}{2 n+a_{n}}\left(n \in N_{+}\right)$. Then $\sum_{k=1}^{2017} \frac{k}{a_{k}}=$ | 2^{2018}-2019 | 83 | 11 |
math | For what $n$ can the following system of inequalities be solved?
$$
1<x<2 ; \quad 2<x^{2}<3 ; \quad \ldots, \quad n<x^{n}<n+1
$$ | 1,2,3,4 | 49 | 7 |
math | # Problem 2. (3 points)
Natural numbers $a$ and $b$ are such that $2a + 3b = \operatorname{LCM}(a, b)$. What values can the number $\frac{\operatorname{LCM}(a, b)}{a}$ take? List all possible options in ascending or descending order, separated by commas. If there are no solutions, write the number 0. | 0 | 90 | 1 |
math | Sonkin $M$.
Solve the equation $\left(x^{2}-y^{2}\right)^{2}=1+16 y$ in integers. | (\1,0),(\4,3),(\4,5) | 34 | 15 |
math | $A$ says to $B$: "Take any number. Write its digits in reverse order and subtract the smaller number from the larger one. Multiply this difference by any number. In the resulting product, cross out any non-zero digit of your choice, and tell me the truncated number." $B$'s response: 35407, to which $A$ says that the cr... | 8 | 95 | 1 |
math | ## Task 2
Divide the sum of the numbers 24 and 16 by 5. | 8 | 23 | 1 |
math | 6. Given that the hyperbola has asymptotes $2 x \pm y=0$, and it passes through the intersection point of the lines $x+y-3=0$ and $2 x-y+3 t=0$, where $-2 \leqslant t \leqslant 5$. Then the maximum possible value of the real axis length of the hyperbola is $\qquad$ . | 4 \sqrt{3} | 88 | 6 |
math | 7. In the Cartesian coordinate system, if the circle with center $(r+1,0)$ and radius $r$ has a point $(a, b)$ satisfying $b^{2} \geq 4 a$, then the minimum value of $r$ is $\qquad$ . | 4 | 60 | 1 |
math | Let $\triangle P_1P_2P_3$ be an equilateral triangle. For each $n\ge 4$, [i]Mingmingsan[/i] can set $P_n$ as the circumcenter or orthocenter of $\triangle P_{n-3}P_{n-2}P_{n-1}$. Find all positive integer $n$ such that [i]Mingmingsan[/i] has a strategy to make $P_n$ equals to the circumcenter of $\triangle P_1P_2P_3$. ... | n \equiv 0 \pmod{4} | 140 | 12 |
math | 3. It is known that the quadratic trinomial $x^{2}+b x+c$ has two distinct roots. If we add the coefficients $b$ and $c$ and the two roots (four numbers), we get the number -3, and if we multiply these same four numbers, we get the number 36. Find all such quadratic trinomials. | x^{2}+4x-3 | 79 | 9 |
math | ## Task Condition
Find the derivative.
$y=x^{e^{\cos x}}$ | x^{e^{\cosx}}\cdote^{\cosx}\cdot(\frac{1}{x}-\sinx\cdot\lnx) | 19 | 33 |
math | 4.052. Given two infinite geometric progressions with a common ratio $|q|<1$, differing only in the sign of their common ratios. Their sums are respectively equal to $S_{1}$ and $S_{2}$. Find the sum of the infinite geometric progression formed by the squares of the terms of any of the given progressions. | S_{1}\cdotS_{2} | 74 | 9 |
math | (7) $\cos \left(\sqrt{1-\sqrt{x^{2}+5 x+7}}+\sqrt{x^{2}+5 x+6}\right)=$ | 1 | 38 | 1 |
math | 1. What is the smallest number of digits that can be erased from the number 20162016 so that the result is divisible by 2016 (it is not allowed to erase nothing)? Please note that you need to not only provide an example but also explain why it is impossible to do with fewer digits. | 3 | 70 | 1 |
math | ## [ Systems of nonlinear algebraic equations ]
Solve the system
$$
\begin{aligned}
& x^{2}+y^{2}=1 \\
& 4 x y\left(2 y^{2}-1\right)=1 .
\end{aligned}
$$ | (\\frac{\sqrt{2-\sqrt{2}}}{2},\\frac{\sqrt{2+\sqrt{2}}}{2}),(\\frac{\sqrt{2+\sqrt{2}}}{2},\\frac{\sqrt{2-\sqrt{2}}}{2}) | 58 | 58 |
math | 33. It is known that there is only one pair of positive integers $a$ and $b$ such that $a \leq b$ and $a^{2}+b^{2}+8 a b=2010$. Find the value of $a+b$. | 42 | 60 | 2 |
math | Find the smallest positive real number $r$ with the following property: For every choice of 2023 unit vectors $\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{2023} \in \mathbb{R}^{2}$, a point $\mathbf{p}$ can be found in the plane such that for each subset $S$ of $\{1,2, \ldots, 2023\}$, the sum
$$
\sum_{i \in S}... | \frac{2023}{2} | 177 | 10 |
math | In a triangle, two angles $\beta$ and $\gamma$ and the radius $R$ of the circumscribed circle are given. Find the radius of the inscribed circle.
# | 4R\sin\beta/2\sin\gamma/2\cos(\beta+\gamma/2) | 38 | 23 |
math | 3A. Given the sequence $a_{1}, a_{2}, \ldots, a_{n} \ldots$, defined by:
$$
a_{1}=2 \text { and } a_{n+1}=1+a_{1} a_{2} \ldots a_{n}
$$
for every natural number, $n \geq 2$. Calculate $S_{2009}+P_{2009}$, where
$$
S_{k}=\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{a_{n}} \text { and } P_{k}=\frac{1}{a_{1} a_{2} \... | 1 | 176 | 1 |
math | Let $x$ be a real number such that $$4^{2x}+2^{-x}+1=(129+8\sqrt2)(4^{x}+2^{-x}-2^{x}).$$ Find $10x$. | 35 | 53 | 2 |
math | 8. Given that the ellipse $C_{1}$ and the hyperbola $C_{2}$ share the foci $F_{1}(3,0), F_{2}(-3,0)$, and that the minor axis and the imaginary axis coincide. Then the number of lattice points inside the region enclosed by the trajectory of the intersection points of $C_{1}$ and $C_{2}$ is $\qquad$ . | 25 | 90 | 2 |
math | Five. (20 points) The increasing sequence $1,3,4,9,10,12$, $13, \cdots$ consists of some positive integers, which are either powers of 3 or the sum of several different powers of 3. Find the value of the 2014th term. | 88329 | 70 | 5 |
math | Let $ S$ be the set of natural numbers $ n$ satisfying the following conditions:
$ (i)$ $ n$ has $ 1000$ digits,
$ (ii)$ all the digits of $ n$ are odd, and
$ (iii)$ any two adjacent digits of $ n$ differ by $ 2$.
Determine the number of elements of $ S$. | 8 \cdot 3^{499} | 80 | 10 |
math | 4. From 9 classmates, select 5 to form the class committee, requiring that A and B either both be selected or both not be selected, and C and D not be selected at the same time. The number of selection methods that meet the requirements is $\qquad$ (answer with a number).
Translate the above text into English, please ... | 41 | 88 | 2 |
math | 8.35 How many natural numbers $n$ make $n^{2}-19 n+91$ a perfect square?
(China Beijing Junior High School Grade 2 Mathematics Competition, 1991) | 2 | 46 | 1 |
math | 4. Given the dihedral angle $\alpha-l-\beta$ is $60^{\circ}$, moving points $P, Q$ are in planes $\alpha, \beta$ respectively, the distance from $P$ to $\beta$ is $\sqrt{3}$, and the distance from $Q$ to $\alpha$ is $2 \sqrt{3}$, then the minimum distance between points $P, Q$ is $\qquad$ . | 2\sqrt{3} | 95 | 6 |
math | In one of three boxes there is a prize, the other two boxes are empty. You do not know which box contains the prize, but the host does. You must point to one of the boxes, which you think contains the prize. After this, the host opens one of the two remaining boxes. Since he does not want to give away the prize immedia... | \frac{2}{3} | 108 | 7 |
math | 4. Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{1}=0, a_{n+1}=a_{n}+4 \sqrt{a_{n}+1}+4(n \geqslant 1) \text {. }
$$
Then $a_{n}=$ $\qquad$ | 4n^{2}-4n | 76 | 7 |
math | 4. Determine the polynomial $P(x)$ if it is of the fourth degree, $P(0)=0$ and $P(x)-P(x-1)=x^{3}$, for any real number $x$. Using the obtained result, calculate the sum $1^{3}+2^{3}+\cdots+n^{3}$. | 1^{3}+2^{3}+\cdots+n^{3}=\frac{1}{4}n^{2}(n+1)^{2} | 72 | 34 |
math | 9.5. To a natural number $N$, the largest divisor of $N$ less than $N$ was added, and the result was a power of ten. Find all such $N$. (N. Agakhanov) | 75 | 49 | 2 |
math | Example 2. Find the integral curve of the equation $y^{\prime \prime}=x+1$, passing through the point $M_{0}(1,1)$ and tangent to the line $y=\frac{1}{2} x+\frac{1}{2}$ at this point. | \frac{x^{3}}{6}+\frac{x^{2}}{2}-x+\frac{4}{3} | 62 | 26 |
math | SUBIECTUL IV
Consider the sequence:
$\left(x_{n}\right)_{n \geq 1}$ of real numbers defined by $x_{1}=\sqrt{\frac{1}{2}}, \quad x_{n+1}=\sqrt{\frac{1+x_{n}}{2}}, \quad n \geq 1$. Calculate $\lim _{n \rightarrow \infty} x_{1} \cdot x_{2} \cdot \ldots . . x_{n}$
## BAREM CLASA XI | \frac{2}{\pi} | 116 | 8 |
math | 6.014. $\frac{4}{x^{2}+4}+\frac{5}{x^{2}+5}=2$. | 0 | 32 | 1 |
math | 2. The range of the function $y=\sqrt{3 x+6}+\sqrt{8-x}$ is $\qquad$ | [\sqrt{10},2\sqrt{10}] | 28 | 13 |
math | (Kazakstan 2003). Let $\left(a_{n}\right)$ and $\left(b_{n}\right)$ be the sequences defined by $a_{0}=b_{0}=0$ and $a_{n}=a_{n-1}^{2}+3$ and $b_{n}=b_{n-1}^{2}+2^{n}$.
Compare the numbers $a_{2003}$ and $b_{2003}$. | b_{2003}<a_{2003} | 104 | 14 |
math | Example 11 Let $p(x)$ be a polynomial of degree $3n$, such that $P(0)=P(3) \cdots=P(3n)=2, P(1)=$ $P(4)=\cdots=P(3n-2)=1, P(2)=P(5)=\cdots=P(3n-1)=0, P(3n+1)=730$. Determine $n$.
(13th US Olympiad Problem) | 4 | 106 | 1 |
math | ## Problem Statement
Calculate the area of the parallelogram constructed on vectors $a$ and $b$.
$a=3 p-2 q$
$b=p+5 q$
$|p|=4$
$|q|=\frac{1}{2}$
$(\widehat{p, q})=\frac{5 \pi}{6}$ | 17 | 73 | 2 |
math | 3. Calculate the maximum value of the expression $\sin \alpha+\sin \beta+\sin \gamma$, if $\alpha, \beta$ and $\gamma$ are angles in a triangle. | \frac{3\sqrt{3}}{2} | 39 | 12 |
math | 40. For what values of $n$ is the expression $2^{n}+1$ a non-trivial power of a natural number? | 3 | 31 | 1 |
math | 8. [7] Triangle $A B C$ has side lengths $A B=231, B C=160$, and $A C=281$. Point $D$ is constructed on the opposite side of line $A C$ as point $B$ such that $A D=178$ and $C D=153$. Compute the distance from $B$ to the midpoint of segment $A D$. | 208 | 94 | 3 |
math | Find all functions $f: \mathbb{Q} \rightarrow \mathbb{Q}$ satisfying
$$
\forall x, y \in \mathbb{Q}, \quad f(x+f(y))=f(x)+y
$$ | f(x)=xf(x)=-x | 51 | 8 |
math | The ninth question: Given a positive integer $n$ greater than 1, let $x_{1}, x_{2}, \ldots, x_{n}$ be real numbers, and $\sum_{i=1}^{n} x_{i}^{2}=1$. Try to find the maximum value of $\sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{kx}_{\mathrm{k}}{ }^{2}+\sum_{1 \leq \mathrm{i}<\mathrm{j} \leq \mathrm{n}}(\mathrm{i}+\mathrm{... | \frac{n(n+1)}{4}+\frac{n}{2}\sqrt{\frac{(n+1)(2n+1)}{6}} | 140 | 32 |
math | ## Task 2 - 310732
A person answers the question about their birthday:
"In the year 1989, I was $a$ years old. I was born on the $t$-th day of the $m$-th month of the year $(1900+j)$. The numbers $a, j, m, t$ are natural numbers; for them, $a \cdot j \cdot m \cdot t=105792.$"
Determine whether the numbers $a, j, m, ... | 57,i=32,=2,=29 | 138 | 13 |
math | 18. (3 points) Li Shuang rides a bike at a speed of 320 meters per minute from location $A$ to location $B$. On the way, due to a bicycle malfunction, he pushes the bike and walks for 5 minutes to a place 1800 meters from $B$ to repair the bike. After 15 minutes, he continues towards $B$ at 1.5 times his original ridin... | 72 | 124 | 2 |
math | 8. Let the integer pair $(m, n)$ satisfy $\frac{m^{2}+m n+n^{2}}{m+2 n}=\frac{13}{3}$. Then $m+2 n=$ $\qquad$ . | 9 | 53 | 1 |
math | A number $x_n$ of the form 10101...1 has $n$ ones. Find all $n$ such that $x_n$ is prime. | n = 2 | 37 | 5 |
math | \section*{Problem 4 - 161044}
Determine all integer pairs \((x ; y)\) that satisfy the following equation!
\[
x y + 3 x - 2 y - 3 = 0
\] | (-1,-2),(1,0),(3,-6),(5,-4) | 55 | 17 |
math | ## Task Condition
Find the derivative.
$$
y=\frac{2\left(3 x^{3}+4 x^{2}-x-2\right)}{15 \sqrt{1+x}}
$$ | x\sqrt{1+x} | 45 | 7 |
math | Let $n \ge 2$ be a positive integer, and write in a digit form \[\frac{1}{n}=0.a_1a_2\dots.\] Suppose that $n = a_1 + a_2 + \cdots$. Determine all possible values of $n$. | 8 | 63 | 1 |
math | One, (20 points) If positive numbers $a, b$ satisfy $ab=1$, find
$$
M=\frac{1}{1+a}+\frac{1}{1+2b}
$$
the minimum value. | 2\sqrt{2}-2 | 50 | 7 |
math | Solve the following equation:
$$
\sqrt[3]{x}+\sqrt[3]{1-x}=\frac{3}{2}
$$ | x_{1}=\frac{1}{2}+\frac{31\sqrt{5}}{216},\quadx_{2}=\frac{1}{2}-\frac{31\sqrt{5}}{216} | 31 | 54 |
math | 【4】The smallest natural number that leaves a remainder of 2 when divided by 3, a remainder of 4 when divided by 5, and a remainder of 4 when divided by 7 is ( ). | 74 | 45 | 2 |
math | 11. Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=1, a_{2}=2, \frac{a_{n+2}}{a_{n}}=\frac{a_{n+1}^{2}+1}{a_{n}^{2}+1}(n=1,2, \cdots)$. Try to find $\left[a_{2017}\right]([a]$ represents the greatest integer not exceeding the real number $a$) | 63 | 111 | 2 |
math | Calculate: $10 \times\left(\frac{1}{1 \times 2}+\frac{5}{2 \times 3}+\frac{11}{3 \times 4}+\cdots+\frac{89}{9 \times 10}\right)$. | 81 | 62 | 2 |
math | 15. Given the function $f(x)=a x^{2}+8 x+3(a<0)$, for a given negative number $a$ there is a maximum positive number $l(a)$ such that the inequality $|f(x)| \leqslant 5$ holds for all $x$ in the interval $[0, l(a)]$. Find the value of $a$ for which $l(a)$ is maximized, and determine the maximum $l(a)$. | \frac{\sqrt{5}+1}{2} | 103 | 12 |
math | [b]p1.[/b] At a conference a mathematician and a chemist were talking. They were amazed to find that they graduated from the same high school. One of them, the chemist, mentioned that he had three sons and asked the other to calculate the ages of his sons given the following facts:
(a) their ages are integers,
(b) the ... | 2, 2, 9 | 504 | 7 |
math | 8. (10 points) In the expression $(x+y+z)^{2026}+(x-y-z)^{2026}$, the brackets were expanded and like terms were combined. How many monomials $x^{a} y^{b} z^{c}$ with a non-zero coefficient were obtained? | 1028196 | 69 | 7 |
math | ## 31. Math Puzzle 12/67
From a statistic, it is evident that only about $15 \%$ of the agriculturally used land area of our Earth is regularly irrigated, and this area produces $25 \%$ of the world's harvest.
How many times greater is the average yield per hectare of the irrigated area compared to that of the non-ir... | 1.9 | 87 | 3 |
math | [b]O[/b]n a blackboard a stranger writes the values of $s_7(n)^2$ for $n=0,1,...,7^{20}-1$, where $s_7(n)$ denotes the sum of digits of $n$ in base $7$. Compute the average value of all the numbers on the board. | 3680 | 73 | 4 |
math | 2. Color the positive integers $1,2, \cdots, 15$ in blue or red. Find the number of coloring methods that satisfy the following conditions:
(1) The positive integer 15 is red;
(2) If the positive integers $x$ and $y$ are of different colors, and $x+y \leqslant 15$, then $x+y$ is colored blue;
(3) If the positive intege... | 4 | 129 | 1 |
math | Determine the maximal value of $ k $, such that for positive reals $ a,b $ and $ c $ from inequality $ kabc >a^3+b^3+c^3 $ it follows that $ a,b $ and $ c $ are sides of a triangle. | k = 5 | 57 | 5 |
math | 1. Solve the equation $x^{\log _{3}\left(27 x^{2}\right)}=\frac{x^{9}}{81}$. | 3,9 | 35 | 3 |
math | I3.1 Given that $\frac{1-\sqrt{3}}{2}$ satisfies the equation $x^{2}+p x+q=0$, where $p$ and $q$ are rational numbers. If $A=|p|+2|q|$, find the value of $A$. | 2 | 66 | 1 |
math | To be calculated $\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+2 x+1}+\sqrt{x^{2}+4 x+1}-\sqrt{4 x^{2}+1}\right)$. | 3 | 53 | 1 |
math | ## Task Condition
Find the derivative.
$y=x+\frac{1}{1+e^{x}}-\ln \left(1+e^{x}\right)$ | \frac{1}{(1+e^{x})^{2}} | 35 | 15 |
math | ## Problem 1
Consider the sum $S=x_{1} x_{2}+x_{3} x_{4}+\ldots+x_{2015} x_{2016}$, where $x_{1}, x_{2}, \ldots, x_{2016} \in\{\sqrt{3}-\sqrt{2}, \sqrt{3}+\sqrt{2}\}$. Is it possible to have $S=2016$?
Cristian Lazăr | =756,b==126 | 110 | 9 |
math | Example 7 Given an integer $n \geqslant 2$, for any coprime positive integers $a_{1}, a_{2}, \cdots, a_{n}$, let
$$
A=a_{1}+a_{2}+\cdots+a_{n} .
$$
Let the greatest common divisor of $A$ and $a_{i}(i=1,2, \cdots, n)$ be $d_{i}$; the greatest common divisor of the remaining $n-1$ numbers after removing $a_{i}$ from $a_... | (n-1)^{n} | 181 | 7 |
math | 15. (12 points) Chen Chen, a little kid, found that he has a total of 20 coins of 1 jiao and 5 jiao, and the total amount of money is 8 yuan. How many 1 jiao coins does he have? | 5 | 59 | 1 |
math | 12.390. A sphere is inscribed in a spherical sector of radius $R$ (Fig. 12.239). Find the radius of the circle of contact between the surfaces of the sphere and the sector, if the central angle in the axial section of the spherical sector is $\alpha$. | \frac{R\sin\alpha}{4\cos^{2}(\frac{\pi}{4}-\frac{\alpha}{4})} | 66 | 30 |
math | 28*. Variables \(x\) and \(y\) are positive, \(x+y=6\). Find the minimum value of the sum \(\frac{1}{x}+\frac{1}{y}\). | \frac{2}{3} | 44 | 7 |
math | 11. In the expansion of $\left(1+x+x^{2}+\cdots+x^{100}\right)^{3}$, after combining like terms, the coefficient of $x^{150}$ is $\qquad$ (answer with a number). | 7651 | 57 | 4 |
math | 1. Let the function $f(x)$ have a domain and range both equal to $R$, and for any $a, b \in R$ there is $f[a f(b)]=a b$. Then the value of $|f(1995)|$ is $\qquad$ | 1995 | 62 | 4 |
math | Example 3. Find the mass of the body $\Omega$ with density $\mu=20 z$, bounded by the surfaces
$$
z=\sqrt{1-x^{2}-y^{2}}, \quad z=\sqrt{\frac{x^{2}+y^{2}}{4}}
$$ | 4\pi | 62 | 3 |
math | Find the polynomial $P$ of degree 2 in $\mathbb{R}[X]$ such that $P(0)=1, P(1)=2$ and $P(2)=5$. | X^{2}+1 | 42 | 6 |
math | Example 12 Given $x, y, z \in(-1,1)$, and $x y z=\frac{1}{36}$, try to find the minimum value of the function $u=\frac{1}{1-x^{2}}+\frac{4}{4-y^{2}}+\frac{9}{9-z^{2}}$. | \frac{108}{35} | 75 | 10 |
math | ## Zadatak B-1.2.
Za koje je realne brojeve $x$ vrijednost izraza
$$
\frac{\left(x+\frac{1}{x}\right)^{3}-\left(x^{3}+\frac{1}{x^{3}}\right)-3}{\left(x+\frac{1}{x}\right)^{2}-\left(x^{2}+\frac{1}{x^{2}}\right)+1}
$$
veća od 1 ?
| \langle0,\infty\rangle\backslash{1} | 112 | 14 |
math | + Find all positive integers $n$ such that there exist $k \in \mathbf{N}^{*}, k \geqslant 2$ and positive rational numbers $a_{1}, a_{2}, \cdots, a_{k}$, satisfying
$$
a_{1}+a_{2}+\cdots+a_{k}=a_{1} \cdots a_{k}=n .
$$ | 4orn\geqslant6 | 90 | 8 |
math | Four, $n^{2}(n \geqslant 4)$ positive numbers are arranged in $n$ rows and $n$ columns,
$$
\begin{array}{llllll}
a_{11} & a_{12} & a_{13} & a_{14} & \cdots & a_{1 n} \\
a_{21} & a_{22} & a_{23} & a_{24} & \cdots & a_{2 n} \\
a_{31} & a_{32} & a_{33} & a_{34} & \cdots & a_{3 n} \\
a_{41} & a_{42} & a_{43} & a_{44} & \cd... | 2-\frac{1}{2^{n-1}}-\frac{n}{2^{n}} | 340 | 20 |
math | 10. Real numbers $x, y$ satisfy $x^{2}+y^{2}=20$, then the maximum value of $x y+8 x+y$ is
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 42 | 64 | 2 |
math | 9. Buratino the Statistician (from 7th grade, 2 points). Every month, Buratino plays in the "6 out of 45" lottery organized by Karabas-Barabas. In the lottery, there are 45 numbered balls, and in each draw, 6 random winning balls are drawn.
Buratino noticed that in each subsequent draw, there are no balls that appeare... | 7 | 180 | 1 |
math | 3. Three-digit number $\overline{a b c}=a^{2}+1+(\overline{b c})^{2}$. Then $\overline{a b c}=$ | 726 | 40 | 3 |
math | Let $ABCD$ be a rhombus of sides $AB = BC = CD= DA = 13$. On the side $AB$ construct the rhombus $BAFC$ outside $ABCD$ and such that the side $AF$ is parallel to the diagonal $BD$ of $ABCD$. If the area of $BAFE$ is equal to $65$, calculate the area of $ABCD$. | 120 | 91 | 3 |
math | 5. In a certain school, a teacher prepared a certain number of thank-you cards for the participants of the Math Night. The number of students who registered to participate was $\frac{2}{13}$ more than the number of thank-you cards he prepared, but on the actual Math Night, $\frac{1}{15}$ of the registered students did ... | 280 | 108 | 3 |
math | 4. If the complex coefficient equation with respect to $x$
$$
(1+2 \mathrm{i}) x^{2}+m x+1-2 \mathrm{i}=0
$$
has real roots, then the minimum value of the modulus of the complex number $m$ is $\qquad$ | 2 | 65 | 1 |
math | 12. Let $F_{1}$ and $F_{2}$ be the two foci of the ellipse $C$, and $AB$ be a chord of the ellipse passing through point $F_{2}$. In $\triangle F_{1} A B$,
$$
\left|F_{1} A\right|=3,|A B|=4,\left|B F_{1}\right|=5 \text {. }
$$
Then $\tan \angle F_{2} F_{1} B=$ $\qquad$ | \frac{1}{7} | 112 | 7 |
math | [Law of Cosines]
In triangle $ABC$, angle $A$ is $60^{\circ}$, $AB=1$, $BC=a$. Find $AC$.
# | if\<\frac{\sqrt{3}}{2},\no\solution;\\if\=\frac{\sqrt{3}}{2},\AC=\frac{1}{2};\\if\\frac{\sqrt{3}}{2}<<1,\AC=\frac{1\\sqrt{4a^2-3}}{2};\\if\\geqslant1,\AC=\frac{1+\sqrt} | 39 | 89 |
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