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 | ## 5. Cards
There were seven cards in a box with numbers from 3 to 9 written on them (each card had one number). Mirko randomly took three cards from the box, and Slavko took two cards, while two cards remained in the box. Mirko looked at his cards and said to Slavko: "I know that the sum of the numbers on your cards ... | 192 | 105 | 3 |
math | 4. Let $x, y, z$ be positive numbers, and $x^{2}+y^{2}+z^{2}=1$, find the minimum value of the following expression:
$$
S=\frac{x y}{z}+\frac{y z}{x}+\frac{z x}{y} .
$$
(22nd All-Soviet Union Olympiad) | \sqrt{3} | 82 | 5 |
math | Example 4.16. Find the general solution of the equation
$$
y^{\prime \prime}-5 y^{\prime}-6 y=0
$$ | C_{1}e^{-x}+C_{2}e^{6x} | 36 | 18 |
math | 5. If for any real numbers $x \neq y$, we have
$$
\frac{\mathrm{e}^{x}-\mathrm{e}^{y}}{x-y}+k(x+y)>1,
$$
then the range of the real number $k$ is | -\frac{1}{2} | 61 | 7 |
math | 27. Solve the equation $(a+b) c!=(a!+b!) c$ in natural numbers | =1,=1,b=2;=1,=2,b=1;=1,=2,b=2;==b | 22 | 29 |
math | 1. Let $A D, B E, C F$ be the three altitudes of an acute triangle $A B C$, with the coordinates of $D, E, F$ being $(4,0),\left(\frac{80}{17}, \frac{20}{17}\right),\left(\frac{5}{2}, \frac{5}{2}\right)$, respectively. Find the coordinates of $A, B, C$. | A(4,4) | 98 | 6 |
math | Example 1. Derive the equations of the tangent and normal lines to the curve $y=x \ln x$, drawn at the point with abscissa $x=e$. | ():2x-e,(n):-\frac{1}{2}x+\frac{3}{2}e | 36 | 23 |
math | A3. What is the value of $\left(\frac{4}{5}\right)^{3}$ as a decimal? | 0.512 | 26 | 5 |
math | Determine all triples $(a, b, c)$ of positive integers for which $a b-c, b c-a$, and $c a-b$ are powers of 2. Explanation: A power of 2 is an integer of the form $2^{n}$, where $n$ denotes some nonnegative integer. (Serbia) Answer. There are sixteen such triples, namely $(2,2,2)$, the three permutations of $(2,2,3)$, a... | (2,2,2), (2,2,3), (2,6,11), (3,5,7) | 123 | 29 |
math | $$
\text { II. (40 points) Let } a_{i} \in(0,1](1 \leqslant i \leqslant 45) \text {. }
$$
Find the maximum value of $\lambda$ such that the inequality
$$
\sqrt{\frac{45}{\sum_{i=1}^{45} a_{i}}} \geqslant 1+\lambda \prod_{i=1}^{45}\left(1-a_{i}\right)
$$
holds. | 8(\frac{81}{80})^{45} | 118 | 14 |
math | Example 12 If $\alpha, \beta, \gamma$ are the roots of the equation $x^{3}-x-1=0$, find the value of $\frac{1+\alpha}{1-\alpha}+\frac{1+\beta}{1-\beta}+\frac{1+\gamma}{1-\gamma}$. | -7 | 69 | 2 |
math | 9. The three-digit number $\overline{a b c}$ consists of three non-zero digits. The sum of the other five three-digit numbers formed by rearranging $a, b, c$ is 2017. Find $\overline{a b c}$.
δΈδ½ζΈ $\overline{a b c}$ η±δΈειιΆζΈεη΅ζγθ₯ζ $a γ b γ c$ ιζ°ζε, εε
Άι€δΊεε―η΅ζηδΈδ½ζΈδΉεζ― 2017 γζ± $\overline{a b c}$ γ | 425 | 125 | 3 |
math | [
The Perimeter of a Triangle
The bisector drawn from vertex $N$ of triangle $M N P$ divides side $M P$ into segments of 28 and 12.
Find the perimeter of triangle $M N P$, given that $M N - N P = 18$.
# | 85 | 67 | 2 |
math | 12. How many positive integers are there whose digits do not include 0 , and whose digits have sum 6 ? | 32 | 25 | 2 |
math | 6. Solve the equation $\sqrt{\frac{x+3}{3 x-5}}+1=2 \sqrt{\frac{3 x-5}{x+3}}$. | 4 | 37 | 1 |
math |
4. Determine all integers $n \geq 2$ for which there exist integers $x_{1}, x_{2}, \ldots, x_{n-1}$ satisfying the condition that if $0<i<n, 0<j<n, i \neq j$ and $n$ divides $2 i+j$, then $x_{i}<x_{j}$.
Proposed by Merlijn Staps, NLD
The answer is that $n=2^{k}$ with $k \geq 1$ or $n=3 \cdot 2^{k}$ where $k \geq 0$.... | n=2^{k}withk\geq1orn=3\cdot2^{k}wherek\geq0 | 134 | 27 |
math | 5. Let any real numbers $x_{0}>x_{1}>x_{2}>x_{3}>0$, to make $\log _{\frac{x_{0}}{x_{1}}} 1993+$ $\log \frac{x_{1}}{x_{2}} 1993+\log _{\frac{x_{2}}{x_{3}}} 1993>k \log _{\frac{x_{0}}{x_{3}}} 1993$ always hold, then the maximum value of $k$ is | 9 | 117 | 1 |
math | 1. A three-digit number has its middle digit three times smaller than the sum of the other two, and the sum of the last two digits is half of the first digit. If the digits in the tens and units places are swapped, the resulting number is 18 less than the given number. What is that number? | 831 | 66 | 3 |
math | # 7.1. Condition:
On the Christmas tree, there is a garland of 100 bulbs. It is known that the first and third bulbs are yellow. Moreover, among any five consecutive bulbs, exactly two are yellow and exactly three are blue. Father Frost can see only part of the garland from behind the tree. Help him find out the color... | 97;99;100-Bluecolor,98-Yellowcolor | 128 | 18 |
math | In the village, there are 100 houses. What is the maximum number of closed, non-intersecting fences that can be built so that each fence encloses at least one house and no two fences enclose the same set of houses
# | 199 | 52 | 3 |
math | 4. In $\triangle A B C$, if $\qquad$
$$
\frac{\overrightarrow{A B} \cdot \overrightarrow{B C}}{3}=\frac{\overrightarrow{B C} \cdot \overrightarrow{C A}}{2}=\frac{\overrightarrow{C A} \cdot \overrightarrow{A B}}{1}
$$
then $\tan A=$ $\qquad$ | \sqrt{11} | 90 | 6 |
math | Integers a, b, c, d, and e satisfy the following three properties:
(i) $2 \le a < b <c <d <e <100$
(ii)$ \gcd (a,e) = 1 $
(iii) a, b, c, d, e form a geometric sequence.
What is the value of c? | 36 | 77 | 2 |
math | Consider the set $E$ of all positive integers $n$ such that when divided by $9,10,11$ respectively, the remainders(in that order) are all $>1$ and form a non constant geometric progression. If $N$ is the largest element of $E$, find the sum of digits of $E$
| 13 | 71 | 2 |
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 | 43. (FIN 1) Evaluate
$$
S=\sum_{k=1}^{n} k(k+1) \cdots(k+p),
$$
where $n$ and $p$ are positive integers. | \frac{n(n+1)\cdots(n+p+1)}{p+2} | 48 | 19 |
math | Ana has $22$ coins. She can take from her friends either $6$ coins or $18$ coins, or she can give $12$ coins to her friends. She can do these operations many times she wants. Find the least number of coins Ana can have. | 4 | 59 | 1 |
math | 12. If the set $A=\{(x, y)|y=-| x |-2\}, B=\left\{(x, y) \mid(x-a)^{2}+y^{2}=a^{2}\right\}$ satisfies $A \cap B$ $=\varnothing$, then the range of the real number $a$ is $\qquad$ . | -2\sqrt{2}-2<<2\sqrt{2}+2 | 79 | 17 |
math | 1. Solve the equation in integers
$$
2025^{x}-100 x y+3-y^{2}=0
$$ | (0;2),(0;-2) | 31 | 9 |
math | 1. It is known that $x+\frac{1}{x} \leqslant 4$. Find the range of the function
$$
f(x)=x^{3}+\frac{1}{x^{3}}
$$ | f(x)\in[-\infty;-2]\cup[2;52] | 49 | 18 |
math | An arithmetic sequence consists of $ 200$ numbers that are each at least $ 10$ and at most $ 100$. The sum of the numbers is $ 10{,}000$. Let $ L$ be the [i]least[/i] possible value of the $ 50$th term and let $ G$ be the [i]greatest[/i] possible value of the $ 50$th term. What is the value of $ G \minus{} L$? | \frac{8080}{199} | 115 | 12 |
math | $14.4 .23^{\star \star}$ Find all prime numbers $p$ such that the sum of all divisors of $p^{4}$ is a perfect square. | 3 | 40 | 1 |
math | Let $n{}$ be a positive integer. What is the smallest sum of digits that $5^n + 6^n + 2022^n$ can take? | 8 | 36 | 1 |
math | 13.335. If a two-digit number is divided by the product of its digits, the quotient is 3 and the remainder is 8. If the number, formed by the same digits but in reverse order, is divided by the product of the digits, the quotient is 2 and the remainder is 5. Find this number. | 53 | 72 | 2 |
math | 12. For any positive integer $n$, define the function $\mu(n)$:
$$
\mu(1)=1 \text {, }
$$
and when $n=p_{1}^{\alpha_{1}} p_{2}^{\alpha_{2}} \cdots p_{t}^{\alpha_{4}} \geqslant 2$,
$$
\mu(n)=\left\{\begin{array}{ll}
(-1)^{t}, & \alpha_{1}=\alpha_{2}=\cdots=\alpha_{t}=1 ; \\
0, & \text { otherwise, }
\end{array}\right.
... | 0 | 241 | 1 |
math | 2. Given 50 numbers. It is known that among their pairwise products, exactly 500 are negative. Determine the number of zeros among these numbers. | 5 | 34 | 1 |
math | 1. (16 points) Given a complex number $z$ satisfying $|z|=1$. Find
$$
u=\left|z^{3}-3 z+2\right|
$$
the maximum value. | 3 \sqrt{3} | 46 | 6 |
math | 364. Once I decided to take a ride on a chairlift. At some point, I noticed that the chair coming towards me had the number 95, and the next one had the number 0, followed by 1, 2, and so on. I looked at the number on my chair; it turned out to be 66. Have I passed the halfway point? At which chair will I pass the half... | 18 | 92 | 2 |
math | Let $a$ and $b$ be real numbers. Define $f_{a,b}\colon R^2\to R^2$ by $f_{a,b}(x;y)=(a-by-x^2;x)$. If $P=(x;y)\in R^2$, define $f^0_{a,b}(P) = P$ and $f^{k+1}_{a,b}(P)=f_{a,b}(f_{a,b}^k(P))$ for all nonnegative integers $k$.
The set $per(a;b)$ of the [i]periodic points[/i] of $f_{a,b}$ is the set of points $P\in R^2... | \frac{-(b+1)^2}{4} | 210 | 12 |
math | 5. Each pair of numbers $x$ and $y$ is assigned a number $x * y$. Find $1993 * 1935$, given that for any three numbers $x, y$ and $z$ the identities $x * x=0$ and $x *(y * z)=(x * y)+z$ are satisfied. | 58 | 76 | 2 |
math | 2. (6 points) It is known that no digit of a three-digit number is zero and the sum of all possible two-digit numbers formed from the digits of this number is equal to this number. Find the largest such three-digit number. | 396 | 49 | 3 |
math | Example 4 Given the numbers $1,2,2^{2}, \cdots, 2^{n-1}$. For any permutation $\sigma=\left(x_{1}, x_{2}, \cdots, x_{n}\right)$ of them, define $S_{1}(\sigma)=$ $x_{1}, S_{2}(\sigma)=x_{1}+x_{2}, \cdots, S_{n}(\sigma)=x_{1}+x_{2}+\cdots+$ $x_{n}$. Also let $Q(\sigma)=S_{1}(\sigma) S_{2}(\sigma) \cdots S_{n}(\sigma)$, f... | 2^{-\frac{n(n-1)}{2}} | 180 | 12 |
math | 2. Find the maximum value of the quantity $x^{2}+y^{2}+z^{2}$, given that
$$
x^{2}+y^{2}+z^{2}=3 x+8 y+z
$$ | 74 | 52 | 2 |
math | 1. Given that for every pair of real numbers $x, y$, the function $f$ satisfies $f(x)+f(y)=f(x+y)-x y-1$. If $f(1)=1$, then the number of integers $n$ that satisfy $f(n)=n$ is $\qquad$ . | 2 | 67 | 1 |
math | SI. 4 The solution of the inequality $x^{2}+5 x-2 c \leq 0$ is $d \leq x \leq 1$. Find $d$. | -6 | 43 | 2 |
math | 11.022. Determine the volume of a regular quadrilateral prism if its diagonal forms an angle of $30^{\circ}$ with the plane of a lateral face, and the side of the base is $a$. | ^{3}\sqrt{2} | 48 | 7 |
math | 1A. Dimitar arranged the digits $1,2,3,4,5,6,7,8,9$ in a circle in some way. Any three consecutive digits, in the clockwise direction, form a three-digit number. Dimitar added all such numbers. What sum did Dimitar get? | 4995 | 64 | 4 |
math | 9. The minimum distance from a point on the curve $C: \sqrt{x}+\sqrt{y}=1$ to the origin is $\qquad$ | \frac{\sqrt{2}}{4} | 33 | 10 |
math | Let $a, b, c, d, e, f, g$ be non-negative numbers whose sum is 1. Select the largest of the sums $a+b+c, b+c+d, c+d+e, d+e+f$, $e+f+g$. What is the minimum value of the thus obtained value? | \frac{1}{3} | 68 | 7 |
math | Example 1. Find $\int\left(x^{2}-x+1\right) \cos 2 x d x$. | \frac{2x^{2}-2x+1}{4}\sin2x+\frac{2x-1}{4}\cos2x+C | 27 | 32 |
math | Let $\mathcal{L}$ be a finite collection of lines in the plane in general position (no two lines in $\mathcal{L}$ are parallel and no three are concurrent). Consider the open circular discs inscribed in the triangles enclosed by each triple of lines in $\mathcal{L}$. Determine the number of such discs intersected by ... | \frac{(|\mathcal{L}|-1)(|\mathcal{L}|-2)}{2} | 108 | 25 |
math | 101. A special number. What number is formed from five consecutive digits (not necessarily in order) such that the number formed by the first two digits, multiplied by the middle digit, gives the number formed by the last two digits. (For example, if we take the number 12896, then 12, multiplied by 8, gives 96. However... | 13452 | 109 | 5 |
math | Let $n$ be a positive integer, $[x]$ be the greatest integer not exceeding the real number $x$, and $\{x\}=x-[x]$.
(1) Find all positive integers $n$ that satisfy $\sum_{k=1}^{2013}\left[\frac{k n}{2013}\right]=2013+n$;
(2) Find all positive integers $n$ that maximize $\sum_{k=1}^{2013}\left\{\frac{k n}{2013}\right\}$,... | 3 | 128 | 1 |
math | An 8-by-8 square is divided into 64 unit squares in the usual way. Each unit square is colored black or white. The number of black unit squares is even. We can take two adjacent unit squares (forming a 1-by-2 or 2-by-1 rectangle), and flip their colors: black becomes white and white becomes black. We call this oper... | 32 | 134 | 2 |
math | Exercise 7. Let $k$ be a strictly positive integer. For any real number $x$, the real number $|x|$ is the absolute value of $x$, which is $x$ if $x$ is positive, $-x$ if $x$ is negative. Find the number of triplets $(x, y, z)$ where $x, y, z$ are integers such that $x+y+z=0$ and $|x|+|y|+|z|=2k$. | 6k | 107 | 2 |
math | 5. The maximum value of the function $y=|\sin x|+|\sin 2 x|$ is $\qquad$ . | \frac{\sqrt{414+66 \sqrt{33}}}{16} | 28 | 21 |
math | 19. [9] Calculate $\sum_{n=1}^{2001} n^{3}$. | 4012013006001 | 25 | 13 |
math | 13. (2004 National College Entrance Examination - Jiangsu Paper) Given: $0<\alpha<\frac{\pi}{2}, \tan \frac{\alpha}{2}+\cot \frac{\alpha}{2}=\frac{5}{2}$. Find the value of $\sin \left(\alpha-\frac{\pi}{3}\right)$. | \frac{1}{10}(4-3\sqrt{3}) | 78 | 16 |
math | The sequence $(a_{n})$ is defined by $a_1=1$ and $a_n=n(a_1+a_2+\cdots+a_{n-1})$ , $\forall n>1$.
[b](a)[/b] Prove that for every even $n$, $a_{n}$ is divisible by $n!$.
[b](b)[/b] Find all odd numbers $n$ for the which $a_{n}$ is divisible by $n!$. | n = 1 | 106 | 5 |
math | 6. For which prime numbers $p$ and $q$ does the equation $p^{2 x}+q^{2 y}=z^{2}$ have a solution in natural numbers $x, y$ and $z ?$ | p=2,q=3orp=3,q=2 | 48 | 12 |
math | 7.253. $\log _{a} \sqrt{4+x}+3 \log _{a^{2}}(4-x)-\log _{a^{4}}\left(16-x^{2}\right)^{2}=2$. For which values of $a$ does the equation have a solution? | 4-^{2},where\in(0;1)\cup(1;2\sqrt{2}) | 70 | 23 |
math | Example 8. The probability of producing a non-standard item with a certain technological process is 0.06. The inspector takes an item from the batch and immediately checks its quality. If it turns out to be non-standard, further tests are stopped, and the batch is held. If the item is standard, the inspector takes the ... | \begin{pmatrix}X&1&2&3&4&5\\\hlineP&0.06&0.056&0.053&0.050&0.781\\\end{pmatrix} | 103 | 56 |
math | 15. Let $a=\frac{1+\sqrt{2009}}{2}$. Find the value of $\left(a^{3}-503 a-500\right)^{10}$. | 1024 | 48 | 4 |
math | 5. To make the equation
$$
x^{4}+(m-4) x^{2}+2(1-m)=0
$$
have exactly one real root that is not less than 2, the range of values for $m$ is $\qquad$. | m \leqslant -1 | 58 | 8 |
math | Among $100$ points in the plane, no three collinear, exactly $4026$ pairs are connected by line segments. Each point is then randomly assigned an integer from $1$ to $100$ inclusive, each equally likely, such that no integer appears more than once. Find the expected value of the number of segments which join two points... | 1037 | 97 | 4 |
math | A1. The average of the numbers $2,5, x, 14,15$ is $x$. Determine the value of $x$. | 9 | 33 | 1 |
math | A5. Dani wrote the integers from 1 to $N$. She used the digit 1 fifteen times. She used the digit 2 fourteen times.
What is $N$ ? | 41 | 38 | 2 |
math | 2.128. $\frac{\sqrt[3]{\sqrt{3}+\sqrt{6}} \cdot \sqrt[6]{9-6 \sqrt{2}}-\sqrt[6]{18}}{\sqrt[6]{2}-1}=-\sqrt[3]{3}$. | -\sqrt[3]{3} | 64 | 7 |
math | ## Task Condition
Find the derivative.
$y=\left(x^{8}+1\right)^{\text{th } x}$ | (x^{8}+1)^{\operatorname{}x}\cdot(\frac{\ln(x^{8}+1)}{\operatorname{ch}^{2}x}+\frac{8x^{7}\cdot\operatorname{}x}{x^{8}+1}) | 28 | 58 |
math | ## Task 1 - 210931
Four pairs of statements are made about a natural number $x$:
Pair A: (1) $x$ is a two-digit number.
(2) $x$ is less than 1000.
Pair B: (1) The second digit of the number $x$ is 0.
(2) The sum of the digits of the number $x$ is 11.
Pair C: (1) $x$ is written with exactly three digits, and all t... | 703,740 | 233 | 7 |
math | Let $A=2012^{2012}$, B the sum of the digits of $A$, C the sum of the digits of $B$, and D the sum of the digits of $C$.
What is the value of $\mathrm{D}$? | 7 | 58 | 1 |
math | 10. Solve the equation $\sqrt{x^{2}+6 x+11}+\sqrt{x^{2}-6 x+11}=10$. | \\frac{5}{4}\sqrt{14} | 34 | 12 |
math | $\triangle ABC$ has area $240$. Points $X, Y, Z$ lie on sides $AB$, $BC$, and $CA$, respectively. Given that $\frac{AX}{BX} = 3$, $\frac{BY}{CY} = 4$, and $\frac{CZ}{AZ} = 5$, find the area of $\triangle XYZ$.
[asy]
size(175);
defaultpen(linewidth(0.8));
pair A=(0,15),B=(0,-5),C=(25,0.5),X=origin,Y=(4C+B)/5,Z=(5A+C)/6... | 122 | 203 | 3 |
math | 2. Find the value of the expression $\frac{p}{q}+\frac{q}{p}$, where $p$ and $q$ are the largest and smallest roots of the equation $x^{3}+6 x^{2}+6 x=-1$, respectively. | 23 | 59 | 2 |
math | 2. Suppose $A B C$ is a scalene right triangle, and $P$ is the point on hypotenuse $\overline{A C}$ such that $\angle A B P=45^{\circ}$. Given that $A P=1$ and $C P=2$, compute the area of $A B C$. | \frac{9}{5} | 72 | 7 |
math | 124*. A checker can move in one direction along a strip divided into cells, moving either to the adjacent cell or skipping one cell in a single move. In how many ways can it move 10 cells? | 89 | 45 | 2 |
math | [ Counting in two ways ] $[\quad$ Trees $]$
A travel agency ran a promotion: "Buy a trip to Egypt, bring four friends who also buy a trip, and get the cost of your trip back." During the promotion, 13 buyers came on their own, and the rest were brought by friends. Some of them brought exactly four new customers, while... | 29 | 103 | 2 |
math | 13. (15 points) Zhang Qiang rides a bike from bus station $A$, along the bus route, traveling 250 meters per minute. After a period of time, a bus also departs from station $A$, traveling 450 meters per minute, and needs to stop at a station for 1 minute after every 6 minutes of travel. If the bus catches up with Zhang... | 2100 | 109 | 4 |
math | Task B-2.1. Determine all values of the real parameter $a$ for which the sum of the squares of the solutions of the quadratic equation $x^{2}+2 a x+a-3=0$ is greater than 6. | \in\langle-\infty,0\rangle\cup\langle\frac{1}{2},\infty\rangle | 52 | 27 |
math | 6.200. $\left\{\begin{array}{l}(x+y)^{2}+2 x=35-2 y, \\ (x-y)^{2}-2 y=3-2 x\end{array}\right.$ | (-5,-2),(3,2),(-3,-4),(1,4) | 54 | 18 |
math | 371. Calculate: a) $\sin 110^{\circ}$; b) $\operatorname{tg} 945^{\circ}$; c) $\cos \frac{25 \pi}{4}$. | \frac{1}{2},1,\frac{\sqrt{2}}{2} | 51 | 18 |
math | 9. (16 points) Given the function
$$
\begin{array}{l}
f(x)=a x^{3}+b x^{2}+c x+d(a \neq 0), \\
\text { when } 0 \leqslant x \leqslant 1 \text {, }|f^{\prime}(x)| \leqslant 1 .
\end{array}
$$
Try to find the maximum value of $a$. | \frac{8}{3} | 104 | 7 |
math | $14 \cdot 16$ Solve the equation $x^{2}-2 x-3=12 \cdot\left[\frac{x-1}{2}\right]$.
(China Sichuan Province Junior High School Mathematics League, 1990) | 1+2\sqrt{7} | 57 | 8 |
math | On the lateral sides $A B$ and $C D$ of trapezoid $A B C D$, points $M$ and $N$ are taken such that segment $M N$ is parallel to the bases and divides the area of the trapezoid in half. Find the length of $M N$, if $B C=a$ and $A D=b$. | x^2=\frac{^2+b^2}{2} | 79 | 14 |
math | A positive integer is called [i]sabroso [/i]if when it is added to the number obtained when its digits are interchanged from one side of its written form to the other, the result is a perfect square. For example, $143$ is sabroso, since $143 + 341 =484 = 22^2$. Find all two-digit sabroso numbers. | 29, 38, 47, 56, 65, 74, 83, 92 | 91 | 30 |
math | 3. Let the general term formula of the sequence $\left\{a_{n}\right\}$ be $a_{n}=n^{2}+$ $\lambda n(n \in \mathbb{N})$. If $\left\{a_{n}\right\}$ is a monotonically increasing sequence, then the range of the real number $\lambda$ is | \lambda > -3 | 76 | 5 |
math | Given real numbers $a, b, c$ satisfy $a+b+c=ab+bc+ca$, try to find
$$
\frac{a}{a^{2}+1}+\frac{b}{b^{2}+1}+\frac{c}{c^{2}+1}
$$
the minimum value. | -\frac{1}{2} | 70 | 7 |
math | 3. In a certain football league, a double round-robin system (i.e., two teams play each other twice) is used, with $m$ teams participating. At the end of the competition, a total of $9 n^{2}+6 n+32$ matches were played, where $n$ is an integer. Then $m=$ $\qquad$ | 8or32 | 78 | 4 |
math | 7. The minimum value of the function $y=\sqrt{x^{2}+2 x+2}+\sqrt{x^{2}-2 x+2}$ is $\qquad$ . | 2 \sqrt{2} | 39 | 6 |
math | 4. Given the sequence $\left\{a_{n}\right\}$ with the sum of the first $n$ terms as $S_{n}$, and
$$
a_{1}=3, S_{n}=2 a_{n}+\frac{3}{2}\left((-1)^{n}-1\right) \text {. }
$$
If $\left\{a_{n}\right\}$ contains three terms $a_{1} γ a_{p} γ a_{q}(p γ q \in$ $\left.\mathbf{Z}_{+}, 1<p<q\right)$ that form an arithmetic seque... | 1 | 144 | 1 |
math | 1. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f(2 x y)+f(f(x+y))=x f(y)+y f(x)+f(x+y)
$$
for all real numbers $x$ and $y$. | f(x)=0,f(x)=x,f(x)=2-x | 64 | 13 |
math | Test $\mathbf{A}$ Given that $a, b, c, d, e$ are real numbers satisfying
$$
\begin{array}{c}
a+b+c+d+e=8, \\
a^{2}+b^{2}+c^{2}+d^{2}+e^{2}=16
\end{array}
$$
determine the maximum value of $e$. | \frac{16}{5} | 87 | 8 |
math | Example 3. Find $\lim _{n \rightarrow \infty}\left(1+\frac{1}{n}+\frac{1}{n^{2}}\right)^{n}$. | e | 42 | 1 |
math | In the Cartesian plane is given a set of points with integer coordinate \[ T=\{ (x;y)\mid x,y\in\mathbb{Z} ; \ |x|,|y|\leq 20 ; \ (x;y)\ne (0;0)\} \] We colour some points of $ T $ such that for each point $ (x;y)\in T $ then either $ (x;y) $ or $ (-x;-y) $ is coloured. Denote $ N $ to be the number of couples $ {(x_1;... | 420 | 202 | 3 |
math | There are real numbers $a, b, c,$ and $d$ such that $-20$ is a root of $x^3 + ax + b$ and $-21$ is a root of $x^3 + cx^2 + d.$ These two polynomials share a complex root $m + \sqrt{n} \cdot i,$ where $m$ and $n$ are positive integers and $i = \sqrt{-1}.$ Find $m+n.$ | 330 | 101 | 3 |
math | Example 8 Define the sequence $\left\{a_{n}\right\}$:
$$
\begin{array}{l}
a_{1}=0, \\
a_{n}=a_{\left[\frac{n}{2}\right]}+(-1)^{\frac{n(n+1)}{2}}(n=2,3, \cdots) .
\end{array}
$$
For each non-negative integer $k$, find the number of indices $n$ that satisfy $2^{k} \leqslant n<2^{k+1}$ and $a_{n}=0$. | \begin{cases}0&ifkisodd\\\mathrm{C}_{k}^{\frac{k}{2}}& | 127 | 26 |
math | 7. Let positive real numbers $x, y$ satisfy
$$
x^{2}+y^{2}+\frac{1}{x}+\frac{1}{y}=\frac{27}{4} \text {. }
$$
Then the minimum value of $P=\frac{15}{x}-\frac{3}{4 y}$ is | 6 | 76 | 1 |
math | Find out how many positive integers $n$ not larger than $2009$ exist such that the last digit of $n^{20}$ is $1$. | 804 | 35 | 3 |
math | 8. (10 points) From 1 to 1000, the maximum number of numbers that can be selected such that the difference between any two of these numbers does not divide their sum is $\qquad$. | 334 | 46 | 3 |
math | Let $ABC$ be a right-angled triangle with $\angle ABC=90^\circ$, and let $D$ be on $AB$ such that $AD=2DB$. What is the maximum possible value of $\angle ACD$? | 30^\circ | 51 | 4 |
math | ## Task 6/82
Given the inequality $|x|+|y| \leq n$ with $n \in N, x ; y \in G$ (where $G$ denotes the set of integers). Determine the number of ordered solution pairs $(x ; y)$. | n^{2}+(n+1)^{2} | 62 | 12 |
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