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 | 3. The sum of two numbers is 135, and $35\%$ of one number is equal to $28\%$ of the other number. Determine these numbers. | 60,75 | 41 | 5 |
math | ## Task 4
A rectangle is $4 \mathrm{~cm} 8 \mathrm{~mm}$ wide and twice as long. Calculate the sum of all side lengths of the rectangle! | 28 | 41 | 2 |
math | Find the number of addition problems in which a two digit number is added to a second two digit number to give a two digit answer, such as in the three examples: \[\begin{tabular}{@{\hspace{3pt}}c@{\hspace{3pt}}}23\\42\\\hline 65\end{tabular}\,,\qquad\begin{tabular}{@{\hspace{3pt}}c@{\hspace{3pt}}}36\\36\\\hline 72\end... | 3240 | 166 | 4 |
math | Example 3. Solve the equation
$$
\sqrt{1+3 x}=x+1
$$ | x_{1}=0,x_{2}=1 | 23 | 10 |
math | 6.164. $\sqrt{x+8+2 \sqrt{x+7}}+\sqrt{x+1-\sqrt{x+7}}=4$. | 2 | 33 | 1 |
math | 25. How many even four-digit numbers can be formed from the digits $0,1,2,3,4,5$? | 540 | 29 | 3 |
math | 2. The sequence $(\left.a_{n}\right)$ is defined by:
(i) $a_{1}=1$
(ii) $\quad(\forall n \in N) a_{n+1}=a_{n}+\frac{1}{\left[a_{n}\right]}$
For which values of $n$ does $a_{n}>20$ hold? ( $[x]$ is the greatest integer less than or equal to $x$.) | n>191 | 97 | 5 |
math | # 1. Option 1.
Vasya strikes the strings of a 6-string guitar from 1 to 6 and back. Each subsequent strike hits the adjacent string. On which string number will the 2000th strike fall? (The order of striking the strings: $1-2-3-4-5-6-5-4-3-2-1-2-\ldots$) | 2 | 89 | 1 |
math | [ Combinatorics (miscellaneous) $]$ $[$ Estimation + example ]
In a pond, 30 pikes were released, which gradually eat each other. A pike is considered full if it has eaten at least three pikes (full or hungry). What is the maximum number of pikes that can become full? | 9 | 70 | 1 |
math | 2. Given unit vectors $\boldsymbol{a} 、 \boldsymbol{b}$ satisfy $a \perp b$, vector $\boldsymbol{c}$
satisfies
$$
|c-a|+|c-2 b|=\sqrt{6} .
$$
Then the range of $|\boldsymbol{c}-\boldsymbol{a}|$ is $\qquad$ . | \left[\frac{\sqrt{6}-\sqrt{5}}{2}, \frac{\sqrt{6}+\sqrt{5}}{2}\right] | 84 | 34 |
math | 6.1. How many triangles with integer sides have a perimeter equal to 27? (Triangles that differ only in the order of the sides - for example, $7,10,10$ and $10,10,7$ - are considered the same triangle.) | 19 | 60 | 2 |
math | 11.2. Is the number $4^{2019}+6^{2020}+3^{4040}$ prime? | No | 34 | 1 |
math | 3.1. Philatelist Andrey decided to distribute all his stamps equally into 2 envelopes, but it turned out that one stamp was left over. When he distributed them equally into 3 envelopes, one stamp was again left over; when he distributed them equally into 5 envelopes, 3 stamps were left over; finally, when he tried to d... | 223 | 128 | 3 |
math | 14. Given that the weights of $A$, $B$, $C$, and $D$ are all integers in kilograms, where $A$ is the lightest, followed by $B$, $C$, and $D$, the weights of each pair of them are as follows (unit: kg):
$45,49,54,55,60,64$.
Then the weight of $D$ is $\qquad$ kg. | 35 | 97 | 2 |
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 | $5 \cdot 85$ If all the coefficients of a polynomial are equal to $0, 1, 2$, or 3, the polynomial is called compatible. For a given natural number $n$, find the number of all compatible polynomials that satisfy the condition $p(2)=n$.
| \left[\frac{n}{2}\right]+1 | 65 | 11 |
math | We define the weight $W$ of a positive integer as follows: $W(1) = 0$, $W(2) = 1$, $W(p) = 1 + W(p + 1)$ for every odd prime $p$, $W(c) = 1 + W(d)$ for every composite $c$, where $d$ is the greatest proper factor of $c$. Compute the greatest possible weight of a positive integer less than 100. | 12 | 100 | 2 |
math | The quadrilateral $ABCD$ has $AB$ parallel to $CD$. $P$ is on the side $AB$ and $Q$ on the side $CD$ such that $\frac{AP}{PB}= \frac{DQ}{CQ}$. M is the intersection of $AQ$ and $DP$, and $N$ is the intersection of $PC$ and $QB$. Find $MN$ in terms of $AB$ and $CD$. | \frac{AB \cdot CD}{AB + CD} | 96 | 12 |
math | 14. In $\triangle A B C$, $\angle B A C=80^{\circ}, \angle A B C=60^{\circ}, D$ is a point inside the triangle, and $\angle D A B=10^{\circ}, \angle D B A=$ $20^{\circ}$, find the degree measure of $\angle A C D$.
(Mathematical Bulletin Problem 1142) | 30 | 93 | 2 |
math | 16. (15 points) Let the function
$$
f(x)=x^{2}-\left(k^{2}-5 a k+3\right) x+7(a, k \in \mathbf{R}) \text {. }
$$
For any $k \in[0,2]$, if $x_{1}, x_{2}$ satisfy
$$
x_{1} \in[k, k+a], x_{2} \in[k+2 a, k+4 a] \text {, }
$$
then $f\left(x_{1}\right) \geqslant f\left(x_{2}\right)$. Find the maximum value of the positive ... | \frac{2\sqrt{6}-4}{5} | 153 | 13 |
math | Four, it is known that the obtuse triangle $\triangle A B C$ satisfies the following conditions:
(1) The lengths of $A B, B C, C A$ are all positive integers;
(2) The lengths of $A B, B C, C A$ are all no more than 50;
(3) The lengths of $A B, B C, C A$ form an arithmetic sequence with a positive common difference.
Fi... | 157 | 128 | 3 |
math | 31st IMO 1990 shortlist Problem 26 Find all positive integers n such that every positive integer with n digits, one of which is 7 and the others 1, is prime. Solution | 1,2 | 45 | 3 |
math | 1. If $a, b$ are both integers, the equation
$$
a x^{2}+b x-2008=0
$$
has two distinct roots that are prime numbers, then $3 a+b=$ $\qquad$ (2008, Taiyuan Junior High School Mathematics Competition) | 1000 | 69 | 4 |
math | Example 3 (1990 National Training Team Training Question) Find all positive integers $M$ less than 10 such that 5 divides
$$
1989^{M}+M^{1989} .
$$ | 1or4 | 51 | 3 |
math | 32nd IMO 1991 shortlist Problem 23 f(n) is an integer-valued function defined on the integers which satisfies f(m + f( f(n) ) ) = - f( f(m+1)) - n for all m, n. The polynomial g(n) has integer coefficients and g(n) = g( f(n) ) for all n. Find f(1991) and the most general form for g. | f(1991)=-1992,\(x) | 96 | 16 |
math | Determine all integer pairs $(m, n)$ such that $\frac{n^{3}+1}{m n-1}$ is an integer. | (1,2),(1,3),(2,1),(3,1),(2,5),(3,5),(5,2),(5,3),(2,2) | 30 | 37 |
math | Which is larger: $1985^{1986}$ or $1986^{1985}$? | 1985^{1986}>1986^{1985} | 28 | 20 |
math | ## Task 35/69
We are looking for a natural number $n$ with a four-digit decimal representation, which has the following properties:
1. Its cross sum is an odd square number.
2. It is the product of exactly two different prime numbers.
3. The sum of the two prime numbers is ten times the number obtained by removing th... | 1969 | 121 | 4 |
math | 2. Given that $a$ is a natural number, there exists a linear polynomial with integer coefficients and $a$ as the leading coefficient, which has two distinct positive roots less than 1. Then, the minimum value of $a$ is $\qquad$ . | 5 | 55 | 1 |
math | ## Task B-3.2.
Determine all even natural numbers $n$ for which
$$
\left|\log _{0.5} 20^{\cos \pi}+\log _{0.5} 30^{\cos 2 \pi}+\log _{0.5} 42^{\cos 3 \pi}+\cdots+\log _{0.5}\left(n^{2}+7 n+12\right)^{\cos n \pi}\right|=1
$$ | 4 | 115 | 1 |
math | ## Task B-2.3.
There are 8 steps to the top of the staircase. In how many ways can we reach the top if we can climb by one step or by two steps? | 34 | 41 | 2 |
math | II. (16 points) Find all three-digit numbers such that the last three digits of their square are the original two-digit number.
The above text has been translated into English, retaining the original text's line breaks and format. | 625 \text{ or } 376 | 47 | 12 |
math | Fomin D:
The numbers $1, 1/2, 1/3, \ldots, 1/100$ are written on the board. We choose two arbitrary numbers $a$ and $b$ from those written on the board, erase them, and write the number
$a + b + ab$. We perform this operation 99 times until only one number remains. What is this number? Find it and prove that it does ... | 100 | 104 | 3 |
math | 24. $\triangle A B C$ is an equilateral triangle of side length 30 . Fold the triangle so that $A$ touches a point $X$ on $B C$. If $B X=6$, find the value of $k$, where $\sqrt{k}$ is the length of the crease obtained from folding. | 343 | 70 | 3 |
math | How many distinct values can $x^{9}$ take modulo 999? | 15 | 17 | 2 |
math | 11.4. We will call a natural number interesting if it is the product of exactly two (distinct or equal) prime numbers. What is the greatest number of consecutive numbers, all of which are interesting | 3 | 42 | 1 |
math | Richard likes to solve problems from the IMO Shortlist. In 2013, Richard solves $5$ problems each Saturday and $7$ problems each Sunday. He has school on weekdays, so he ``only'' solves $2$, $1$, $2$, $1$, $2$ problems on each Monday, Tuesday, Wednesday, Thursday, and Friday, respectively -- with the exception of Decem... | 1099 | 129 | 4 |
math | 7. $n$ is a positive integer, $f(n)=\sin \frac{n \pi}{2}$. Then
$$
f(1991)+f(1992)+\cdots+f(2003)=
$$
$\qquad$ | -1 | 60 | 2 |
math | 4.38. Find the highest power of two that divides the number $(n+1)(n+2) \cdot \ldots \cdot 2 n$.
$$
* * *
$$ | 2^n | 42 | 2 |
math | If $f(x)$ is a linear function with $f(k)=4, f(f(k))=7$, and $f(f(f(k)))=19$, what is the value of $k$ ? | \frac{13}{4} | 43 | 8 |
math | Determine the maximal value of $k$ such that the inequality
$$\left(k +\frac{a}{b}\right) \left(k + \frac{b}{c}\right)\left(k + \frac{c}{a}\right)
\le \left( \frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\right) \left( \frac{b}{a}+ \frac{c}{b}+ \frac{a}{c}\right)$$
holds for all positive reals $a, b, c$. | \sqrt[3]{9} - 1 | 127 | 11 |
math | 6. Variant 1 Sasha wrote the number 765476547654 on the board. He wants to erase several digits so that the remaining digits form the largest possible number divisible by 9. What is this number? | 7654765464 | 52 | 10 |
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 | 18. School A and School B each send out 5 students to participate in a long-distance running competition. The rule is: The $\mathrm{K}$-th student to reach the finish line gets $K$ points (no students arrive at the finish line simultaneously). The team with fewer total points wins. Therefore, the winning team's total s... | 13 | 77 | 2 |
math | ## Task A-4.1.
Determine all prime numbers $p$ and $q$ such that $p^{q}+1$ is also prime. | p=2,q=2 | 34 | 6 |
math | $2 \cdot 107$ Let the natural number $n \geqslant 5, n$ different natural numbers $a_{1}, a_{2}, \cdots, a_{n}$ have the following property: for any two different non-empty subsets $A$ and $B$ of the set
$$
S=\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}
$$
the sum of all numbers in $A$ and the sum of all numbers in $B$ ... | 2-\frac{1}{2^{n-1}} | 165 | 12 |
math | 5. Solve the inequality $\sqrt{x+2-|x+1|} \leq x+5-|2 x+3|$.
# | x\in[-1.5;1] | 32 | 10 |
math | What is the largest integer $n$ for which
$$
\frac{\sqrt{7}+2 \sqrt{n}}{2 \sqrt{7}-\sqrt{n}}
$$
is an integer? | 343 | 43 | 3 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty} \frac{\sqrt{\left(n^{2}+5\right)\left(n^{4}+2\right)}-\sqrt{n^{6}-3 n^{3}+5}}{n}
$$ | \frac{5}{2} | 68 | 7 |
math | \section*{Task 1 - 041231}
In a mathematical circle, six participants agree on a real number \(a\), which the seventh participant, who had left the room beforehand, is supposed to determine. Upon his return, he receives the following information:
1. \(a\) is a rational number.
2. \(a\) is an integer that is divisible ... | 7 | 214 | 1 |
math | I3.1 If the product of numbers in the sequence $10^{\frac{1}{11}}, 10^{\frac{2}{11}}, 10^{\frac{3}{11}}, \ldots, 10^{\frac{\alpha}{11}}$ is 1000000 , determine the value of the positive integer $\alpha$. | 11 | 85 | 2 |
math | 3. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that
$$
n+f(m) \text { divides } f(n)+n f(m)
$$
for all $m, n \in \mathbb{N}$. (We denote the set of natural numbers by $\mathbb{N}$.)
(Albania) | f(x)\equivx^{2}f(x)\equiv1 | 81 | 13 |
math | Find the law that forms the sequence and give its next 2 terms:
$425,470,535,594,716,802, \ldots$ | 870983 | 44 | 6 |
math | Find $A^2$, where $A$ is the sum of the absolute values of all roots of the following equation: \begin{eqnarray*}x &=& \sqrt{19} + \frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{x}}}}}}}}}\end{eqnarray*} | 383 | 126 | 3 |
math | 11. (20 points) Let the sequence $\left\{a_{n}\right\}$ satisfy $a_{1}=a_{2}=1$, and $a_{n+1} a_{n-1}=a_{n}^{2}+n a_{n} a_{n-1}(n=2,3, \cdots)$.
(1) Find the general term formula for $\left\{a_{n}\right\}$;
(2) Find $\sum_{k=2}^{n} \frac{a_{k}}{(k-2) \text { ! }}$. | \frac{(n+1)!}{2^{n-1}}-2 | 131 | 16 |
math | 3. It is known that for positive numbers $a, b$, and $c$, each of the three equations $a x^{2} +$ param $1 b x + c = 0$, $b x^{2} +$ param $1 c x + a = 0$, $c x^{2} +$ param $1 a x + b = 0$ has at least one real root.
What is the smallest possible value for the product of the roots of the second equation if the produc... | 0.24 | 379 | 4 |
math | A $5\times5$ grid of squares is filled with integers. Call a rectangle [i]corner-odd[/i] if its sides are grid lines and the sum of the integers in its four corners is an odd number. What is the maximum possible number of corner-odd rectangles within the grid?
Note: A rectangles must have four distinct corners to be c... | 60 | 113 | 2 |
math | ## 1. Find all primes $p, q$ such that the equation $x^{2}+p x+q=0$ has at least one integer root. (Patrik Bak)
| (p,q)=(3,2) | 43 | 7 |
math | 12. For a positive integer $n \leqslant 500$, it has the property: when an element $m$ is randomly selected from the set $\{1,2, \cdots, 500\}$, the probability that $m \mid n$ is $\frac{1}{100}$. Then the maximum value of $n$ is $\qquad$ | 81 | 86 | 2 |
math | In triangle $ABC$, $AD$ is angle bisector ($D$ is on $BC$) if $AB+AD=CD$ and $AC+AD=BC$, what are the angles of $ABC$? | \angle A = 120^\circ | 46 | 11 |
math | 13.361. A red pencil costs 27 kopecks, a blue one - 23 kopecks. No more than 9 rubles 40 kopecks can be spent on purchasing pencils. It is necessary to purchase the maximum possible total number of red and blue pencils. At the same time, the number of red pencils should be as few as possible, but the number of blue pen... | 14 | 120 | 2 |
math | 7. Let $F_{1}$ and $F_{2}$ be the left and right foci of the hyperbola $C: \frac{x^{2}}{4}-\frac{y^{2}}{12}=1$, respectively. Point $P$ lies on the hyperbola $C$, and $G$ and $I$ are the centroid and incenter of $\triangle F_{1} P F_{2}$, respectively. If $G I / / x$-axis, then the circumradius $R$ of $\triangle F_{1} ... | 5 | 130 | 1 |
math | We have $x_i >i$ for all $1 \le i \le n$.
Find the minimum value of $\frac{(\sum_{i=1}^n x_i)^2}{\sum_{i=1}^n \sqrt{x^2_i - i^2}}$ | n(n+1) | 63 | 5 |
math | 13. Let $x_{1}, x_{2}, x_{3}$ be the roots of the equation $x^{3}-17 x-18=0$, with $-4<x_{1}<-3$, and $4<x_{3}<5$.
(1) Find the integer part of $x_{2}$;
(2) Find the value of $\arctan x_{1}+\arctan x_{2}+\arctan x_{3}$. | -\frac{\pi}{4} | 104 | 7 |
math | 3. Given $0 \leq a_{k} \leq 1(k=1,2, \ldots, 2020)$, let $a_{2021}=a_{1}, a_{2022}=a_{2}$, then the maximum value of $\sum_{k=1}^{2020}\left(a_{k}-\right.$ $\left.a_{k+1} a_{k+2}\right)$ is $\qquad$ . | 1010 | 107 | 4 |
math | 1. (2004 National High School Mathematics Competition Sichuan Preliminary) Given the inequality $m^{2}+\left(\cos ^{2} \theta-5\right) m+4 \sin ^{2} \theta \geqslant 0$ always holds, find the range of the real number $m$. | \geqslant4or\leqslant0 | 73 | 13 |
math | 4. The integer $n$ satisfies the inequality $n+(n+1)+(n+2)+\cdots+(n+20)>2019$. What is the minimum possible value of $n$ ? | 87 | 46 | 2 |
math | Task 2. Determine the largest real number $M$ such that for every infinite sequence $x_{0}, x_{1}, x_{2}, \ldots$ of real numbers that satisfies
a) $x_{0}=1$ and $x_{1}=3$,
b) $x_{0}+x_{1}+\cdots+x_{n-1} \geq 3 x_{n}-x_{n+1}$,
it holds that
$$
\frac{x_{n+1}}{x_{n}}>M
$$
for all $n \geq 0$. | 2 | 129 | 1 |
math | Example 4. Find $\lim _{x \rightarrow \infty}\left(\frac{x+2}{x-3}\right)^{x}$.
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly.
However, since the text is already in English, there's no need for translation. If you meant to tr... | e^5 | 149 | 3 |
math | Example 11 The set of all positive integers can be divided into two disjoint subsets of positive integers $\{f(1), f(2), \cdots, f(n), \cdots\},\{g(1), g(2), \cdots, g(n), \cdots\}$, where $f(1)<f(2)<\cdots<f(n)<\cdots g(1)<$ $g(2)<\cdots<g(n)<\cdots$ and $g(n)=f[f(n)]+1(n \geqslant 1)$. Find $f(240)$.
(IMO - 20 Proble... | 388 | 142 | 3 |
math | For nine non-negative real numbers $a_{1}, a_{2}, \cdots, a_{9}$ whose sum is 1, let
$$
\begin{array}{l}
S=\min \left\{a_{1}, a_{2}\right\}+2 \min \left\{a_{2}, a_{3}\right\}+\cdots+8 \min \left\{a_{8}, a_{9}\right\}+9 \min \left\{a_{9}, a_{1}\right\}, \\
T=\max \left\{a_{1}, a_{2}\right\}+2 \max \left\{a_{2}, a_{3}\ri... | [\frac{36}{5},\frac{31}{4}] | 274 | 16 |
math | 2. Find the minimum value of the function $f(x)=\sqrt{x^{2}+2 x+10}+\sqrt{x^{2}-2 x+10}$. | 2\sqrt{10} | 39 | 7 |
math | Let $ABCD$ be a square of side length $1$, and let $P$ be a variable point on $\overline{CD}$. Denote by $Q$ the intersection point of the angle bisector of $\angle APB$ with $\overline{AB}$. The set of possible locations for $Q$ as $P$ varies along $\overline{CD}$ is a line segment; what is the length of this segmen... | 3 - 2\sqrt{2} | 95 | 9 |
math | ## Task B-1.6.
Determine the value of the real parameter $a$ for which each of the equations
$$
2 a-1=\frac{3-3 a}{x-1} \quad \text { and } \quad a^{2}(2 x-4)-1=a(4-5 x)-2 x
$$
has a unique solution and their solutions are equal. | -1 | 85 | 2 |
math | ## Task 4 - 110734
Fritz tells:
"In our class, there are exactly twice as many girls as boys. If there were 5 fewer boys and 5 fewer girls, then we would have exactly three times as many girls as boys."
Determine the number of all girls and all boys in this class! | 20 | 71 | 2 |
math | 516. Find approximately $\sin 31^{\circ}$. | 0.515 | 16 | 5 |
math | The trinomial $f(x)$ is such that $(f(x))^3-f(x)=0$ has three real roots. Find the y-coordinate of the vertex of $f(x)$. | 0 | 39 | 1 |
math | The road from city $A$ to city $B$ has an uphill, a flat, and a downhill section. A pedestrian walked from $A$ to $B$ in 2.3 hours and returned in 2.6 hours. On the uphill, they walked at 3 km/h, on the flat section at 4 km/h, and on the downhill at 6 km/h. What can we determine about the length of the $A B$ road secti... | 9.8 | 142 | 3 |
math | 2. $y=\arcsin ^{2} x-2 \arcsin x-2, y_{\max }=$ $\qquad$ | \frac{\pi^{2}}{4}+\pi-2 | 33 | 14 |
math | How many subsets of $\{1, 2, ... , 2n\}$ do not contain two numbers with sum $2n+1$? | 3^n | 32 | 2 |
math | 17 Positive integer $n$ cannot be divided by $2$ or $3$, and there do not exist non-negative integers $a$, $b$ such that $\left|2^{a}-3^{b}\right|=n$. Find the minimum value of $n$. | 35 | 57 | 2 |
math | 6・80 Let two complex numbers $x, y$, the sum of their squares is 7, and the sum of their cubes is 10. What is the largest possible real value of $x+y$? | 4 | 46 | 1 |
math | Spivak $A . B$.
Over two years, the factory reduced the volume of its production by $51 \%$. Each year, the volume of production decreased by the same percentage. By how much? | 30 | 44 | 2 |
math | 10. let $m$ be any natural number. As a function of $m$, determine the smallest natural number $k$ for which the following applies: If $\{m, m+1, \ldots, k\}=A \cup B$ is any decomposition into two sets $A$ and $B$, then $A$ or $B$ contains three elements $a, b, c$ (which do not necessarily have to be different) with $... | ^{^{+2}} | 106 | 5 |
math | Consider the sum
\[
S_n = \sum_{k = 1}^n \frac{1}{\sqrt{2k-1}} \, .
\]
Determine $\lfloor S_{4901} \rfloor$. Recall that if $x$ is a real number, then $\lfloor x \rfloor$ (the [i]floor[/i] of $x$) is the greatest integer that is less than or equal to $x$.
| 98 | 104 | 2 |
math | 2) Let $2 n$ real numbers $a_{1}, a_{2}, \cdots, a_{2 n}$ satisfy the condition $\sum_{i=1}^{2 n-1}\left(a_{i+1}-a_{i}\right)^{2}=$
1. Find the maximum value of $\left(a_{n-1}+a_{n-2}+\cdots+a_{2 n}\right)-\left(a_{1}+a_{2}+\cdots+a_{n}\right)$. (Leng Gangsong provided the problem) | \sqrt{\frac{n(2n^{2}+1)}{3}} | 123 | 17 |
math | Three, (25 points) The integers $x_{0}, x_{1}, \cdots, x_{2000}$ satisfy the conditions: $x_{0}=0,\left|x_{1}\right|=\left|x_{0}+1\right|,\left|x_{2}\right|=\left|x_{1}+1\right|$, $\cdots,\left|x_{2004}\right|=\left|x_{2003}+1\right|$. Find the minimum value of $\mid x_{1}+x_{2}+\cdots+$ $x_{2004} \mid$. | 10 | 138 | 2 |
math | 5. Find the number of pairs of natural numbers $(x, y), 1 \leqslant x, y \leqslant 1000$, such that $x^{2}+y^{2}$ is divisible by 5. | 360000 | 54 | 6 |
math | ## Task $1 / 61$
Determine all three-digit numbers that, when divided by 11, yield a number equal to the sum of the squares of the digits of the original number! | 550,803 | 42 | 7 |
math | The neighbors Elza, Sueli, Patrícia, Heloísa, and Cláudia arrive together from work and start climbing the stairs of the five-story building where they live. Each one lives on a different floor. Heloísa arrives at her floor after Elza, but before Cláudia. When Sueli reaches her floor, Heloísa still has two floors to cl... | \begin{pmatrix}\hline5^{ | 124 | 10 |
math | 11. (6 points) It snowed heavily at night, and in the morning, Xiaolong and his father measured the length of a circular path in the garden together, starting from the same point and walking in the same direction. Xiaolong's step length is 54 cm, and his father's step length is 72 cm. After each of them walked one comp... | 21.6 | 115 | 4 |
math | Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is called [i]$k$-good[/i] if $\gcd(f(m) + n, f(n) + m) \le k$ for all $m \neq n$. Find all $k$ such that there exists a $k$-good function.
[i]Proposed by James Rickards, Canada[/... | k \ge 2 | 119 | 6 |
math | 11. Let $a$ and $b$ be integers for which $\frac{a}{2}+\frac{b}{1009}=\frac{1}{2018}$. Find the smallest possible value of $|a b|$. | 504 | 55 | 3 |
math | Example 3-37 A drawer contains 20 shirts, of which 4 are blue, 7 are gray, and 9 are red. How many should be randomly taken to ensure that 4 are of the same color?
How many more should be drawn to ensure that there are 5, 6, 7, 8, 9 shirts of the same color? | 10,16,15,17,19,20 | 80 | 17 |
math | ## Task 3 - 010933
Find the fraction that equals 0.4, and whose numerator and denominator sum up to a two-digit square number!
How did you find the solution? | \frac{14}{35} | 44 | 9 |
math | Solve the equation $p q r = 7(p + q + r)$ in prime numbers. | {3,5,7} | 21 | 7 |
math | 8.268. $\operatorname{tg} \frac{3 x}{2}-\operatorname{tg} \frac{x}{2}=2 \sin x$.
8.268. $\tan \frac{3 x}{2}-\tan \frac{x}{2}=2 \sin x$. | x_{1}=2\pik,x_{2}=\\arccos\frac{-1+\sqrt{17}}{4}+2\pin,k,n\inZ | 67 | 38 |
math | 7. Petya is playing a computer game called "Pile of Stones." Initially, there are 16 stones in the pile. Players take turns taking 1, 2, 3, or 4 stones from the pile. The player who takes the last stone wins. Petya is playing for the first time and therefore takes a random number of stones each time, without violating ... | \frac{1}{256} | 133 | 9 |
math | G4.3 It is given that $m$ and $n$ are two natural numbers and both are not greater than 10 . If $c$ is the number of pairs of $m$ and $n$ satisfying the equation $m x=n$, where $\frac{1}{4}<x<\frac{1}{3}$, find $c$. | 2 | 76 | 1 |
math | 1. Calculate $\frac{20042003^{2}+1}{20042002^{2}+20042004^{2}}=$ | \frac{1}{2} | 44 | 7 |
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