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 | Example 3 Find the smallest positive number $c$, such that for all infinite positive sequences $\left\{x_{n}\right\}$ satisfying $x_{1}+x_{2}+\cdots+x_{n} \leqslant x_{n+1}$, the inequality
$$
\begin{array}{l}
\sqrt{x_{1}}+\sqrt{x_{2}}+\cdots+\sqrt{x_{n}} \\
\leqslant c \sqrt{x_{1}+x_{2}+\cdots+x_{n}}(n=1,2, \cdots) .
... | \sqrt{2}+1 | 138 | 7 |
math | 4. It is known that construction teams A and B each have several workers. If team A lends 90 workers to team B, then the total number of workers in team B will be twice that of team A; if team B lends several workers to team A, then the total number of workers in team A will be six times that of team B. The original mi... | 153 | 83 | 3 |
math | Given a real number $t\neq -1$. Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
\[(t+1)f(1+xy)-f(x+y)=f(x+1)f(y+1)\]
for all $x,y\in\mathbb{R}$. | f(x) = 0 \text{ or } f(x) = t \text{ or } f(x) = (t+1)x - (t+2) | 72 | 36 |
math | Find a formula for the sum of the squares of the numbers in the $n$th row of Pascal's triangle (i.e. the numbers $\left(\begin{array}{l}n \\ 0\end{array}\right),\left(\begin{array}{l}n \\ 1\end{array}\right), \ldots,\left(\begin{array}{l}n \\ n\end{array}\right)$ ). | (\begin{pmatrix}2n\\n\end{pmatrix}) | 92 | 16 |
math | Write in the form of a fraction the number
$$
x=0,512341234123412341234123412341234 \ldots
$$ | \frac{51229}{99990} | 54 | 15 |
math | (3) Let there be a non-empty set $A \subseteq\{1,2,3,4,5,6,7\}$, and when $a \in A$, it must also be that $8-a \in A$. The number of such sets $A$ is $\qquad$. | 15 | 65 | 2 |
math | Two distinct points $A$ and $B$ are chosen at random from 15 points equally spaced around a circle centered at $O$ such that each pair of points $A$ and $B$ has the same probability of being chosen. The probability that the perpendicular bisectors of $OA$ and $OB$ intersect strictly inside the circle can be expressed i... | 11 | 109 | 4 |
math | Every member, starting from the third one, of two sequences $\{a_n\}$ and $\{b_n\}$ equals to the sum of two preceding ones. First members are: $a_1 = 1, a_2 = 2, b_1 = 2, b_2 = 1$. How many natural numbers are encountered in both sequences (may be on the different places)? | 3 | 84 | 1 |
math | 13 Color 8 small squares on a $4 \times 4$ chessboard black, such that each row and each column has exactly two black squares, then there are $\qquad$ different ways to do so. (Answer with a number) | 90 | 52 | 2 |
math | Example 5 Find a polynomial $p(x)$ such that $\left(x^{2}+1\right) \mid p(x), \left(x^{3}+x^{2}+1\right) \mid p(x)+1$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | (x^{2}+1)(x^{2}+x-1) | 77 | 16 |
math | Determine the number of real number $a$, such that for every $a$, equation $x^3=ax+a+1$ has a root $x_0$ satisfying following conditions:
(a) $x_0$ is an even integer;
(b) $|x_0|<1000$. | 999 | 65 | 3 |
math | 1. (2003 Anhui Province Competition Question) Among three-digit numbers, if the digit in the tens place is smaller than the digits in the hundreds and units places, the number is called a concave number, such as 504, 746, etc., which are all concave numbers. Therefore, among three-digit numbers with no repeated digits,... | 240 | 92 | 3 |
math | 1. There are fewer than 500 children on the playground. The percentage of boys among all children equals the number of all girls. How many boys and how many girls are on the playground? Find all possible solutions. | (320,80),(225,75),(90,60),(50,50),(5,20) | 46 | 32 |
math | Inside triangle $ABC$ there are three circles $k_1, k_2, k_3$ each of which is tangent to two sides of the triangle and to its incircle $k$. The radii of $k_1, k_2, k_3$ are $1, 4$, and $9$. Determine the radius of $k.$ | 11 | 76 | 2 |
math | XXXIX OM - I - Problem 1
For each positive number $ a $, determine the number of roots of the polynomial $ x^3+(a+2)x^2-x-3a $. | 3 | 42 | 1 |
math | Two bikers, Bill and Sal, simultaneously set off from one end of a straight road. Neither biker moves at a constant rate, but each continues biking until he reaches one end of the road, at which he instantaneously turns around. When they meet at the opposite end from where they started, Bill has traveled the length of ... | 8 | 92 | 1 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \frac{\sqrt{n\left(n^{5}+9\right)}-\sqrt{\left(n^{4}-1\right)\left(n^{2}+5\right)}}{n}$ | -\frac{5}{2} | 64 | 7 |
math | 22. Let circles $\Gamma$ and $\Gamma'$ be two concentric circles with center $O$, with radii $20$ and $17$ respectively. An equilateral triangle $\triangle ABC$ is inside circle $\Gamma$ and outside circle $\Gamma'$, with point $A$ on circle $\Gamma$, and line $BC$ tangent to circle $\Gamma'$ at point $S$. Line $AO$ in... | 130 | 210 | 3 |
math | 5. There are 211 students and four different types of chocolate, with each type having more than 633 pieces. It is stipulated that each student can take at most three chocolates, or they can choose not to take any. If the students are grouped according to the types and quantities of chocolates they take, then the large... | 7 | 81 | 1 |
math | 31. [15] Find the sum of all primes $p$ for which there exists a prime $q$ such that $p^{2}+p q+q^{2}$ is a square. | 8 | 44 | 1 |
math | 6. 148 Find all functions \( f(x) \) and \( g(x) \) that satisfy
\[
\sin x + \cos y \equiv f(x) + f(y) + g(x) - g(y), \quad x, y \in \mathbb{R}.
\] | f(x)=\frac{\sinx+\cosx}{2},(x)=\frac{\sinx-\cosx}{2}+ | 66 | 29 |
math | ерешин D.A.
Once, Rabbit was in a hurry to meet with Donkey Eeyore, but Winnie-the-Pooh and Piglet unexpectedly came to him. Being well-mannered, Rabbit offered his guests some refreshments. Pooh tied a napkin around Piglet's mouth and single-handedly ate 10 pots of honey and 22 cans of condensed milk, with each pot o... | 30 | 213 | 2 |
math | If we take the square of a number not divisible by 5 and add 1 to it, or subtract 4 from it, we get a number divisible by 5. | 37 | 36 | 2 |
math | 8. The function $J(x)$ is defined by:
$$
J(x)=\left\{\begin{array}{ll}
4+x & \text { for } x \leq-2, \\
-x & \text { for }-20 .
\end{array}\right.
$$
How many distinct real solutions has the equation $J(J(J(x)))=0$ ? | 4 | 81 | 1 |
math | Task 2. Determine all quadruples $(a, b, c, d)$ of positive real numbers that satisfy $a+b+c+d=1$ and
$$
\max \left(\frac{a^{2}}{b}, \frac{b^{2}}{a}\right) \cdot \max \left(\frac{c^{2}}{d}, \frac{d^{2}}{c}\right)=(\min (a+b, c+d))^{4}
$$ | (\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4}) | 102 | 28 |
math | 2.129. $\frac{25 \cdot \sqrt[4]{2}+2 \sqrt{5}}{\sqrt{250}+5 \sqrt[4]{8}}-\sqrt{\frac{\sqrt{2}}{5}+\frac{5}{\sqrt{2}}+2}=-1$. | -1 | 71 | 2 |
math | Let $x_0=1$, and let $\delta$ be some constant satisfying $0<\delta<1$. Iteratively, for $n=0,1,2,\dots$, a point $x_{n+1}$ is chosen uniformly form the interval $[0,x_n]$. Let $Z$ be the smallest value of $n$ for which $x_n<\delta$. Find the expected value of $Z$, as a function of $\delta$. | 1 - \ln \delta | 98 | 6 |
math | 1. Given $f(x)=x^{2}-2 x$, set $A=\{x \mid f(f(x))=0\}$, then the sum of all elements in set $A$ is $\qquad$ . | 4 | 48 | 1 |
math | Consider the function $f(x)=5x^4-12x^3+30x^2-12x+5$. Let $f(x_1)=p$, wher $x_1$ and $p$ are non-negative integers, and $p$ is prime. Find with proof the largest possible value of $p$.
[i]Proposed by tkhalid[/i] | 5 | 85 | 1 |
math | The length of the diagonal of a square is 10 . What is the area of this square? | 50 | 21 | 2 |
math | 11. Find all prime numbers \( p, q, r \) that satisfy the equation \( p(p+1) + q(q+1) = r(r+1) \). | (2,2,3) | 39 | 7 |
math | 1. Let $f(x)$ be a function defined on $\mathbf{R}$, for any real number $x$ we have $f(x+3) f(x-4)=-1$. When $0 \leqslant x<7$, $f(x)=\log _{2}(9-x)$. Then the value of $f(-100)$ is $\qquad$ | -\frac{1}{2} | 84 | 7 |
math | .
An arithmetic progression is a set of the form $\{a, a+d, \ldots, a+k d\}$, where $a, d, k$ are positive integers and $k \geqslant 2$. Thus an arithmetic progression has at least three elements and the successive elements have difference $d$, called the common difference of the arithmetic progression.
Let $n$ be a ... | n^2 | 166 | 3 |
math | 5. The solution set of the equation $\frac{\sqrt{x}+2^{x}}{\sqrt{x}+2^{x+1}}+\frac{\sqrt{x}+3^{x}}{\sqrt{x}+3^{x+1}}+\frac{\sqrt{x}+6^{x}}{\sqrt{x}+6^{x+1}}=1$ is | {0} | 78 | 3 |
math | 2. In the region $\left\{\begin{array}{l}0 \leqslant x \leqslant 2 \pi, \\ 0 \leqslant y \leqslant 3\end{array}\right.$, a point $P(a, b)$ is randomly taken, the probability that $b>\left(\sin \frac{a}{2}+\cos \frac{a}{2}\right)^{2}$ is $ـ$. $\qquad$ | \frac{2}{3} | 106 | 7 |
math | 17. On a roulette wheel, any number from 0 to 2007 can come up with equal probability. The roulette wheel is spun repeatedly. Let $P_{k}$ be the probability that at some point the sum of the numbers that have come up in all the spins equals $k$. Which number is greater: $P_{2007}$ or $P_{2008} ?$ | P_{2007} | 87 | 7 |
math | Example 15. Let $a, b \in N$, when $a^{2}+b^{2}$ is divided by $a+b$, the quotient is $q$, and the remainder is $r$. Find all pairs of numbers $(a, b)$ such that $q^{2}+r=1977$. (19th 1MO problem) | (a, b)=(50,37),(50,7),(37,50),(7,50) | 79 | 26 |
math | 2. A total of $2^{n}$ coins are distributed among several children. If during the distribution of the coins, one of the children has at least half of the coins, then that child gives to each of the remaining children as many coins as they already have. In the case where all the coins are with one child, there is no red... | n | 202 | 1 |
math | 1. Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$, such that for any real numbers $x, y$, we have
$$
f\left(x^{2}\right) f\left(y^{2}\right)+|x| f\left(-x y^{2}\right)=3|y| f\left(x^{2} y\right) \text {. }
$$ | f(x)=0\text{or}f(x)=2|x| | 91 | 15 |
math | Let $a$, $b$, and $c$ be positive real numbers such that $a^2+b^2+c^2=989$ and $(a+b)^2+(b+c)^2+(c+a)^2=2013$. Find $a+b+c$. | 32 | 60 | 2 |
math | 19 In a carriage, any $m(m \geqslant 3)$ passengers have a unique common friend (if A is a friend of B, then B is also a friend of A, and no one is a friend of themselves), how many friends does the person with the most friends have in this carriage?
| m | 66 | 1 |
math | 53rd Putnam 1992 Problem A3 Find all positive integers a, b, m, n with m relatively prime to n such that (a 2 + b 2 ) m = (ab) n . Solution | ,b,,n=2^r,2^r,2r,2r+1foranypositiveintegerr | 49 | 25 |
math | 9. Let $[x]$ be the greatest integer not exceeding the real number $x$. Given the sequence $\left\{a_{n}\right\}$ satisfies: $a_{1}=\frac{1}{2}, a_{n+1}=a_{n}^{2}+3 a_{n}+1, \quad n \in N^{*}$, find $\left[\sum_{k=1}^{2017} \frac{a_{k}}{a_{k}+2}\right]$.
| 2015 | 113 | 4 |
math | Tomongho A.K.
Find all increasing arithmetic progressions with a finite number of terms, the sum of which is 1, and each term has the form $1 / k$, where $k$ is a natural number. | 1/6,1/3,1/2 | 47 | 11 |
math | ## Problem Statement
Find the point of intersection of the line and the plane.
$$
\begin{aligned}
& \frac{x-2}{-1}=\frac{y-3}{-1}=\frac{z+1}{4} \\
& x+2 y+3 z-14=0
\end{aligned}
$$ | (1;2;3) | 73 | 7 |
math | 11. (5 points) In a basketball game, a shot made from outside the three-point line scores 3 points, a shot made from inside the three-point line scores 2 points, and a free throw scores 1 point. A team scored a total of 65 points by making 32 shots in one game. It is known that the number of two-point shots is 3 more t... | 4 | 110 | 1 |
math | 5.19 It is known that 1982 people attended a gathering, and among any 4 people, at least 1 person knows the other 3. How many people at least know all the attendees at this gathering? | 1979 | 49 | 4 |
math | 8 The line $\sqrt{3} x-y-\sqrt{3}=0$ intersects the parabola $y^{2}=4 x$ at points $A$ and $B$ ($A$ is above the x-axis), and intersects the x-axis at point $F$. If $\overrightarrow{O F}=\lambda \overrightarrow{O A}+\mu \overrightarrow{O B}$, then $\mu^{2}-$ $\lambda^{2}=$ $\qquad$ . | \frac{1}{2} | 103 | 7 |
math | 63rd Putnam 2002 Problem B1 An event is a hit or a miss. The first event is a hit, the second is a miss. Thereafter the probability of a hit equals the proportion of hits in the previous trials (so, for example, the probability of a hit in the third trial is 1/2). What is the probability of exactly 50 hits in the first... | \frac{1}{99} | 92 | 8 |
math | 7. In $\triangle A B C$, if $a+c=2 b$, then $\tan \frac{A}{2} \cdot \tan \frac{C}{2}=$ | \frac{1}{3} | 39 | 7 |
math | 1. Calculate: $1^{2}-2^{2}+3^{2}-4^{2}+\ldots+2017^{2}$. | 2035153 | 34 | 7 |
math | 368. Which of the three functions
\[
\begin{aligned}
& y=100 x \\
& y=1000+100 x \\
& y=10000+99 x
\end{aligned}
\]
increases faster than the others? | 100 | 65 | 3 |
math | ## Task 4 - 280834
For a school sports festival, class $8 c$ wants to select a team of four students from the seven best students in the 100-m sprint to participate in the $4 \times 100$-m relay race.
a) How many different teams could be selected from the seven students?
b) How many different teams could be selected... | 35,10,4,24 | 150 | 10 |
math | ## Task A-2.2.
Determine all triples of prime numbers whose sum of squares decreased by 1 is equal to the square of some natural number. | (3,2,2),(2,3,2),(2,2,3) | 33 | 19 |
math | 1. (2 points) Boy Vasya wrote down the non-zero coefficients of the polynomial $P(x)$ of the eighth degree in his notebook. Then he calculated the derivative of the resulting polynomial and wrote down its non-zero coefficients, and so on, until he obtained a constant, which he also wrote down.
What is the smallest num... | 8 | 114 | 1 |
math | Let $ABC$ be a triangle with $AB = 26$, $BC = 51$, and $CA = 73$, and let $O$ be an arbitrary point in the interior of $\vartriangle ABC$. Lines $\ell_1$, $\ell_2$, and $\ell_3$ pass through $O$ and are parallel to $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$, respectively. The intersections of $\ell_1$, $\ell... | 280 | 144 | 3 |
math | 6. 156 Find all continuous functions \( f:(1,+\infty) \rightarrow \mathbb{R} \) that satisfy
$$f(x y) \equiv x f(y)+y f(x), \quad x, y>1$$ | f(x) = c x \ln x | 55 | 9 |
math | Example 3-12 In the permutations of 4 $x$s, 3 $y$s, and 2 $z$s, find the number of permutations that do not contain the patterns $x x x x$, $y y y$, and $z z$. Let $A_{1}$ be the class of permutations of 9 characters where the pattern $x x x x$ appears, $A_{2}$ be the class where the pattern $y y y$ appears; and $A_{3}... | 871 | 118 | 3 |
math | 9.179. $|x-6|>\left|x^{2}-5 x+9\right|$. | x\in(1;3) | 26 | 8 |
math | 12. Given in $\triangle A B C$, $\boldsymbol{A} \boldsymbol{B}=\boldsymbol{a}, \boldsymbol{A} \boldsymbol{C}=\boldsymbol{b}$. Try to express $S_{\triangle A B C}=$ $\qquad$ using vector operations of $\boldsymbol{a}$ and $\boldsymbol{b}$. | \frac{1}{2} \sqrt{(|\boldsymbol{a}| \cdot|\boldsymbol{b}|)^{2}-(\boldsymbol{a} \cdot \boldsymbol{b})^{2}} | 83 | 47 |
math | Example 12. Solve the equation
$$
\sqrt{2 x+5}+\sqrt{x-1}=8
$$ | 10 | 28 | 2 |
math | Example 4 (29th Russian Olympiad) Find all angles $\alpha$ such that the set $\{\sin \alpha, \sin 2 \alpha, \sin 3 \alpha\}=\{\cos \alpha, \cos 2 \alpha, \cos 3 \alpha\}$. | \alpha=\frac{k\pi}{2}+\frac{\pi}{8}\quad(k\in{Z}) | 64 | 24 |
math | 10. (5 points) There are several apples and pears. If each bag contains 5 apples and 3 pears, then when the pears are exactly finished, there are still 4 apples left; if each bag contains 7 apples and 3 pears, then when the apples are exactly finished, there are still 12 pears left. How many apples and pears are there ... | 132 | 91 | 3 |
math | 13. (16 points) In an acute triangle $\triangle ABC$, $\sin A=\frac{4 \sqrt{5}}{9}$, and $a$, $b$, $c$ are the sides opposite to $\angle A$, $\angle B$, $\angle C$ respectively.
(1) Find the value of $\sin 2(B+C)+\sin ^{2} \frac{B+C}{2}$;
(2) If $a=4$, try to find the value of the area $S$ of $\triangle ABC$ when $\ove... | \frac{45-8\sqrt{5}}{81} | 136 | 16 |
math | 10.35 For what real values of $x$ and $y$ are the complex numbers $z_{1}=4 i-2 x y-x y i$ and $z_{2}=y^{2} i-x^{2}+3$ equal? | x_{1}=3,y_{1}=1;\quadx_{2}=-3,y_{2}=-1;\quadx_{3}=\frac{\sqrt{3}}{3},y_{3}=-\frac{4\sqrt{3}}{3};\quadx_{4}=-\frac{\sqrt{3}}{3},y_{4}=\frac{4\sqrt{3}}{3} | 56 | 90 |
math | 2. Given the quadratic trinomial $2 x^{2}-x-36$. Find all integer $x$ for which the values of this trinomial are equal to the square of a prime number. | 5or13 | 44 | 4 |
math | 10.231. A circle of radius $r$ is inscribed in a rectangular trapezoid. Find the sides of the trapezoid if its smaller base is equal to $4 r / 3$. | 4r,\frac{10r}{3},2r | 48 | 13 |
math |
Problem 3. Let $A$ be a set with 8 elements. Find the maximal number of 3 -element subsets of $A$, such that the intersection of any two of them is not a 2-element set.
| 8 | 48 | 1 |
math | 3.172. Find $\operatorname{ctg} 2 \alpha$, if it is known that $\sin \left(\alpha-90^{\circ}\right)=-\frac{2}{3}$ and $270^{\circ}<\alpha<360^{\circ}$. | \frac{\sqrt{5}}{20} | 67 | 11 |
math | 11. The sequences $\left\{a_{n}\right\},\left\{b_{n}\right\}$ satisfy $a_{1}=2, b_{1}=1, a_{n+1}=5 a_{n}+3 b_{n}+7, b_{n+1}=3 a_{n}+5 b_{n}$ (where $n \in \mathbf{N}_{+}$), then the general term formula of $\left\{a_{n}\right\}$ is $a_{n}=$ $\qquad$ . | a_{n}=2^{3n-2}+2^{n+1}-4(n\in{N}_{+}) | 122 | 27 |
math | Example 4 Let $x_{i}>0(i=1,2, \cdots, n), \mu>0$, $\lambda-\mu>0$, and $\sum_{i=1}^{n} x_{i}=1$. For a fixed $n\left(n \in \mathbf{N}_{+}\right)$, find the minimum value of $f\left(x_{1}, x_{2}, \cdots, x_{n}\right)=\sum_{i=1}^{n} \frac{x_{i}}{\lambda-\mu x_{i}}$. | \frac{n}{n \lambda-\mu} | 124 | 10 |
math | Question 87: Let unit vectors $\vec{a} 、 \vec{b}$ have an acute angle between them. If for any
$$
(x, y) \in\{(x, y)|| x \vec{a}+y \vec{b} \mid=1, x y \geq 0\}
$$
it holds that $|x+2 y| \leq \frac{8}{\sqrt{15}}$, find the minimum possible value of $\vec{a} \cdot \vec{b}$. | \frac{1}{4} | 118 | 7 |
math | 1. If the function $f(x)=\log _{a}\left(x+\sqrt{b x^{2}+2 a^{2}}\right) \quad(a>0$ and $a \neq-1)$ is an odd function, then the real number pair $(a, b)=$ $\qquad$ . | (\frac{\sqrt{2}}{2},1) | 70 | 12 |
math | 6. The number 1 is written on the board. If the number a is written on the board, it can be replaced by any number of the form a+d, where $d$ is coprime with a and $10 \leq d \leq 20$. Is it possible to obtain the number $18!=1 \cdot 2 \cdot 3 \cdot \ldots \cdot 18$ on the board after several such operations? (I. Ruban... | 18! | 106 | 3 |
math | 4. Calculate $\sec \frac{2 \pi}{9}+\sec \frac{4 \pi}{9}+\sec \frac{6 \pi}{9}+\sec \frac{8 \pi}{9}$. | 4 | 48 | 1 |
math | C3 (6-5, Romania) On the plane, given 5 points, the lines connecting these points pairwise are neither parallel, nor perpendicular, nor coincident. From each point, draw perpendiculars to the lines formed by connecting the other four points pairwise. How many intersection points (excluding the given 5 points) can these... | 310 | 74 | 3 |
math | 217. Solve the equation: a) $x^{2}-6 x+13=0$; b) $9 x^{2}+12 x+$ $+29=0$ | x_1=3-2i,x_2=3+2i | 44 | 16 |
math | Given a positive integer $n$, there are $n$ boxes $B_1,...,B_n$. The following procedure can be used to add balls.
$$\text{(Procedure) Chosen two positive integers }n\geq i\geq j\geq 1\text{, we add one ball each to the boxes }B_k\text{ that }i\geq k\geq j.$$
For positive integers $x_1,...,x_n$ let $f(x_1,...,x_n)$ be ... | 2m + 1 \text{ for } n = 3m, \text{ and } 2m + 2 \text{ for } n = 3m + 1 \text{ or } 3m + 2 | 202 | 51 |
math | 6. Equation
$$
2^{m} \times 3^{n}-3^{n+1}+2^{m}=13
$$
The non-negative integer solution $(m, n)=$ $\qquad$ | (3,0),(2,2) | 48 | 9 |
math | 8. If one element is deleted from the set $\{1!, 2!, \cdots, 100!\}$, and the product of the remaining elements is exactly a perfect square, then the deleted element is $\qquad$ . | 50! | 51 | 3 |
math | Let $\alpha\geq 1$ be a real number. Define the set
$$A(\alpha)=\{\lfloor \alpha\rfloor,\lfloor 2\alpha\rfloor, \lfloor 3\alpha\rfloor,\dots\}$$
Suppose that all the positive integers that [b]does not belong[/b] to the $A(\alpha)$ are exactly the positive integers that have the same remainder $r$ in the division by $20... | \frac{2021}{2020} | 128 | 13 |
math | Given a string of length $2n$, we perform the following operation:
[list]
[*]Place all the even indexed positions together, and then all the odd indexed positions next. Indexing is done starting from $0$.[/*]
[/list]
For example, say our string is ``abcdef''. Performing our operation yields ``abcdef'' $\to$... | \text{ord}_2(2n-1) | 236 | 13 |
math | 2. Let positive real numbers $a, b, c$ satisfy $a^{2}+b^{2}+c^{2}=1$. Then the maximum value of $s=a^{2} b c+a b^{2} c+a b c^{2}$ is $\qquad$ | \frac{1}{3} | 61 | 7 |
math | 3. On the sides $BC$, $CA$, $AB$ of the equilateral $\triangle ABC$, there are internal division points $D$, $E$, $F$ dividing the sides in the ratio $3:(n-3)$ (where $n>6$). The segments $AD$, $BE$, $CF$ intersect to form $\triangle PQR$ (where $BE$ intersects $AD$ at $P$ and $FC$ at $Q$). When the area of $\triangle ... | 8 | 144 | 1 |
math | 318. The diameter of a circle $x$ is measured approximately, with $a \leqslant x \leqslant b$. Considering the diameter as a random variable $X$, uniformly distributed in the interval $(a, b)$, find the mathematical expectation and variance of the area of the circle. | \pi(b^{2}++^{2})/12 | 66 | 13 |
math | 6. Given the sequence $\left\{a_{n}\right\}$ satisfies the recurrence relation
$$
a_{n+1}=2 a_{n}+2^{n}-1\left(n \in \mathbf{N}_{+}\right) \text {, }
$$
and $\left\{\frac{a_{n}+\lambda}{2^{n}}\right\}$ is an arithmetic sequence. Then the value of $\lambda$ is $\qquad$ | -1 | 101 | 2 |
math | On the Island of Misfortune with a population of 96 people, the government decided to carry out five reforms. Each reform is opposed by exactly half of all citizens. A citizen will go to a rally if they are dissatisfied with more than half of all the reforms. What is the maximum number of people the government can expe... | 80 | 71 | 2 |
math | Example 1.12. Find $\int \sin ^{4} x \cos ^{4} x d x$. | \frac{1}{128}(3x-4\sin4x+\frac{1}{8}\sin8x)+C | 27 | 29 |
math | 3. If the positive real numbers $x, y$ satisfy $y=2016 x$, and $x^{y}=y^{x}$, then $\log _{2016} x+\log _{2016} y=$ $\qquad$ | \frac{2017}{2015} | 59 | 13 |
math | 10. (3 points) The expressway from Lishan Town to the provincial capital is 189 kilometers long, passing through the county town. The county town is 54 kilometers away from Lishan Town. At 8:30 AM, a bus departs from Lishan Town to the county town, arriving at 9:15. After a 15-minute stop, it continues to the provincia... | 72 | 166 | 2 |
math | Let $x \neq y$ be positive reals satisfying $x^3+2013y=y^3+2013x$, and let $M = \left( \sqrt{3}+1 \right)x + 2y$. Determine the maximum possible value of $M^2$.
[i]Proposed by Varun Mohan[/i] | 16104 | 81 | 5 |
math | II. (50 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 $a_{n}$;
(2) Find $\sum_{k=2}^{n} \frac{a_{k}}{(k-2)!}$. | \frac{(n+1)!}{2^{n-1}}-2 | 115 | 16 |
math | The equation $x^{2}-7 x+k=0$ has solutions $x=3$ and $x=a$. The equation $x^{2}-8 x+k+1=0$ has solutions $x=b$ and $x=c$. What is the value of $a+b c$ ? | 17 | 62 | 2 |
math | 8. Let $a_{n}$ be the coefficient of the $x$ term in the expansion of $(3-\sqrt{x})^{n}$ $(n=2,3,4, \cdots)$. Then $\lim _{n \rightarrow \infty}\left(\frac{3^{2}}{a_{2}}+\frac{3^{3}}{a_{3}}+\cdots+\frac{3^{n}}{a_{n}}\right)=$ $\qquad$ . | 18 | 105 | 2 |
math | [ Decimal numeral system ]
$[$ Division with remainder $]$
Find all such two-digit numbers that when multiplied by some integer, the resulting number has a second-to-last digit of 5. | Allnotdivisible20 | 39 | 6 |
math | Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as
$$ (x+y+z) P(x, y, z)+(x y+y z+z x) Q(x, y, z)+x y z R(x, y, z) $$
with $P, Q, R \in \mathcal{A}$. Find th... | 4 | 179 | 1 |
math | For a given positive integer $n,$ find
$$\sum_{k=0}^{n} \left(\frac{\binom{n}{k} \cdot (-1)^k}{(n+1-k)^2} - \frac{(-1)^n}{(k+1)(n+1)}\right).$$ | 0 | 69 | 1 |
math | $d(n)$ shows the number of positive integer divisors of positive integer $n$. For which positive integers $n$ one cannot find a positive integer $k$ such that $\underbrace{d(\dots d(d}_{k\ \text{times}} (n) \dots )$ is a perfect square. | n | 64 | 2 |
math | 4. For what values of $x$ and $y$ is the equality $x^{2}+(1-y)^{2}+(x-y)^{2}=1 / 3$ true? | 1/3,2/3 | 42 | 7 |
math | 1. We have sticks of lengths $1, 2, \ldots, 12$, with a sufficient number of each length available. We need to select four sticks (not necessarily of different lengths) from which a tangential quadrilateral with a perimeter of 24 can be formed. In how many ways can this be done? | 21 | 70 | 2 |
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