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 | 7.245. $\log _{1+x}\left(2 x^{3}+2 x^{2}-3 x+1\right)=3$. | 3 | 35 | 1 |
math | $9 \cdot 56$ Find the minimum value of
$$
\begin{aligned}
A & =\sqrt{\left(1264-z_{1}-\cdots-z_{n}\right)^{2}+x_{n}^{2}+y_{n}^{2}}+ \\
& \sqrt{z_{n}^{2}+x_{n-1}^{2}+y_{n-1}^{2}}+\cdots+\sqrt{z_{2}^{2}+x_{1}^{2}+y_{1}^{2}}+ \\
& \sqrt{z_{1}^{2}+\left(948-x_{1}-\cdots-x_{n}\right)^{2}+\left(1185-y_{1}-\cdots-y_{n}\right)... | 1975 | 224 | 4 |
math | For positive real nubers $a,b,c$ find the maximum real number $x$, such that there exist positive numbers $p,q,r$, such that $p+q+r=1$ and $x$ does not exceed numbers $a\frac{p}{q}, b\frac{q}{r}, c\frac{r}{p}$ | \sqrt[3]{abc} | 74 | 8 |
math | We say that a polygon $P$ is [i]inscribed[/i] in another polygon $Q$ when all vertices of $P$ belong to perimeter of $Q$. We also say in this case that $Q$ is [i]circumscribed[/i] to $P$. Given a triangle $T$, let $l$ be the maximum value of the side of a square inscribed in $T$ and $L$ be the minimum value of the side... | \frac{L}{l} \ge 2 | 142 | 12 |
math | 2. Given that $k$ is an integer. If the quadratic equation $k x^{2}+(2 k+3) x+1$ $=0$ has rational roots, then the value of $k$ is $\qquad$ | -2 | 51 | 2 |
math | Let $A B C$ be a triangle in the $x y$ plane, where $B$ is at the origin $(0,0)$. Let $B C$ be produced to $D$ such that $B C: C D=1: 1, C A$ be produced to $E$ such that $C A: A E=1: 2$ and $A B$ be produced to $F$ such that $A B: B F=1: 3$. Let $G(32,24)$ be the centroid of the triangle $A B C$ and $K$ be the centroi... | 48 | 147 | 2 |
math | 7.5. Zya decided to buy a crumblik. In the store, they also sold kryambliks. Zya bought a kryamblik and received coupons worth $50\%$ of the cost of the purchased kryamblik. With these coupons, he was able to pay $20\%$ of the cost of the crumblik. After paying the remaining amount, he bought the crumblik as well. By w... | 20 | 126 | 2 |
math | $\underline{\text { Folklore }}$
Vasya received a list of books for the summer holidays (12 weeks). He set a goal to read them and decided that each week he would read the same number of books. But each week Vasya read one book less than planned, so he completed his plan 3 weeks later than he wanted. How many weeks ea... | 2 | 104 | 1 |
math | 8,9,10,11 |
Author: S $\underline{\text { Saghafian M. }}$.
In the plane, five points are marked. Find the maximum possible number of similar triangles with vertices at these points. | 8 | 50 | 1 |
math | 6. Given the ellipse $C: \frac{x^{2}}{9}+\frac{y^{2}}{8}=1$ with left and right foci $F_{1}$ and $F_{2}$, and left and right vertices $A$ and $B$, the line $l: x=m y+1$ passing through the right focus $F_{2}$ intersects the ellipse $C$ at points $M\left(x_{1}, y_{1}\right)$ and $N\left(x_{2}, y_{2}\right)\left(y_{1}>0,... | \frac{\sqrt{3}}{12} | 158 | 11 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow \frac{\pi}{2}} \frac{e^{\sin 2 x}-e^{\tan 2 x}}{\ln \left(\frac{2 x}{\pi}\right)}$ | -2\pi | 60 | 4 |
math | 7.097. $\log _{3}(x-3)^{2}+\log _{3}|x-3|=3$. | 0;6 | 31 | 3 |
math | 17. Let $f(x)$ be a quadratic function, and $f(0)=12$
$$
\begin{array}{c}
g(x)=2' f(x) . \\
g(x+1)-g(x) \geqslant 2^{x+1} x^{2} .
\end{array}
$$
Find the expressions for $f(x)$ and $g(x)$. | f(x)=2x^{2}-8x+12,\quad(x)=(2x^{2}-8x+12)2^{x} | 87 | 32 |
math | 4. A password lock's password setting involves assigning one of the two numbers, 0 or 1, to each vertex of a regular $n$-sided polygon $A_{1} A_{2} \cdots A_{n}$, and coloring each vertex with one of two colors, red or blue, such that for any two adjacent vertices, at least one of the number or color is the same. How m... | a_{n}=3^{n}+2+(-1)^{n} | 100 | 17 |
math | There is the sequence of numbers $1, a_2, a_3, ...$ such that satisfies $1 \cdot a_2 \cdot a_3 \cdot ... \cdot a_n = n^2$, for every integer $n> 2$. Determine the value of $a_3 + a_5$. | \frac{61}{16} | 71 | 9 |
math | Patricia has a rectangular painting that she wishes to frame. The frame must also be rectangular and will extend $3\text{ cm}$ outward from each of the four sides of the painting. When the painting is framed, the area of the frame not covered by the painting is $108\text{ cm}^2$. What is the perimeter of the painting a... | 24 \text{ cm} | 81 | 7 |
math | ## Problem Statement
Find the coordinates of point $A$, which is equidistant from points $B$ and $C$.
$A(0 ; y ; 0)$
$B(1 ; 6 ; 4)$
$C(5 ; 7 ; 1)$ | A(0;11;0) | 61 | 9 |
math | (1) Given the set $M=\{2,0,11\}$, if $A \varsubsetneqq M$, and $A$ contains at least one even number. Then the number of sets $A$ that satisfy the condition is $\qquad$ . | 5 | 58 | 1 |
math | Example 13. The target consists of 3 pairwise non-intersecting zones. The probability of hitting the first zone with one shot for a given shooter is 0.5. For the second and third zones, this probability is 0.3 and 0.2, respectively. The shooter fires 6 shots at the target. Find the probability that there will be 3 hits... | 0.135 | 101 | 5 |
math | The twelve students in an olympiad class went out to play soccer every day after their math class, forming two teams of six players each and playing against each other. Each day they formed two different teams from those formed in previous days. By the end of the year, they found that every group of five students had p... | 132 | 83 | 3 |
math | II. (This question is worth 40 points) Find all positive integers $n$ that satisfy the following conditions:
(1) $n$ has at least 4 positive divisors:
(2) If $d_{1}<d_{2}<\cdots<d_{k}$ are all the positive divisors of $n$, then $d_{2}-d_{1}, d_{3}-d_{2}, \cdots, d_{k}-d_{k-1}$ form a geometric sequence. | p^{},wherepis\geq3 | 107 | 9 |
math | 192. A metal ball, the temperature of which at the beginning of the experiment was $12^{\circ} \mathrm{C}$, is cooled by a stream of water having a temperature of $0^{\circ}$. After 8 minutes, the ball cooled down to $9^{\circ}$. Assuming the cooling rate is proportional to the difference between the temperature of the... | 15 | 109 | 2 |
math | Determine the minimum value of
$$x^{2014} + 2x^{2013} + 3x^{2012} + 4x^{2011} +\ldots + 2014x + 2015$$
where $x$ is a real number. | 1008 | 73 | 4 |
math | Problem 5. Determine all natural numbers $n$ such that the numbers $3 n-4, 4 n-5$, and $5 n-3$ are prime numbers. | 2 | 38 | 1 |
math | 13. (14 points) To welcome the "World Women's Conference", a school has formed a ceremonial guard consisting of 48 boys and 32 girls. They need to be arranged in $n$ rows $(n>1)$, with the number of boys and girls in each row being equal. How many different ways can they be arranged? How many boys and girls are there i... | 4 | 88 | 1 |
math | 4. Let the function $f: \mathbf{R}_{+} \rightarrow \mathbf{R}_{+}$ satisfy:
(1) If $x \leqslant y$, then $f(x) \leqslant f(y)$;
(2) When $x>0$,
$$
\begin{array}{l}
f\left(x^{4}\right)+f\left(x^{2}\right)+f(x)+f(1) \\
=x^{4}+x^{2}+x+1 .
\end{array}
$$
Find the function $f$. | f(x)=x | 127 | 4 |
math | 3. Find the sum of all four-digit numbers in which the digits $0,3,6,9$ do not appear. | 6479352 | 27 | 7 |
math | 7. $N$ ! ends with exactly 2013 zeros, then the maximum value of $\mathbf{N}$ is
| 8069 | 30 | 4 |
math | 【Question 3】 Divisors of 2015 are called factors of 2015. 1 and 2015 obviously divide 2015, and are called trivial factors of 2015. In addition to the trivial factors, 2015 has some non-trivial factors. Therefore, the sum of all non-trivial factors of 2015 is $\qquad$ . | 672 | 92 | 3 |
math | Example 4 Determine all polynomials $f(x)$ that satisfy the following condition:
$$
f(x+1) \geqslant \frac{1}{2} f[f(x)]+\frac{3}{2}
$$ | f(x)=b,\forallb\geqslant3orf(x)=2x+b,\forallb\leqslant1 | 48 | 28 |
math | For what value of the parameter $p$ does the following inequality hold for all positive $x$:
$$
\lg (x+p)-\frac{1}{2} \geq \lg \sqrt{2 x}
$$ | p\geq5 | 48 | 5 |
math | N2) Determine all triples $(a, b, p)$ of positive integers where $p$ is prime and the equation
$$
(a+b)^{p}=p^{a}+p^{b}
$$
is satisfied. | (1,1,2) | 48 | 7 |
math | What is the least integer a greater than $14$ so that the triangle with side lengths $a - 1$, $a$, and $a + 1$ has integer area? | 52 | 39 | 2 |
math | Task 2. By which smallest natural number should the number 63000 be multiplied so that the resulting product is a perfect square? | 70 | 30 | 2 |
math | 10.085. A circle of radius $R$ is inscribed in an isosceles trapezoid. The upper base of the trapezoid is half the height of the trapezoid. Find the area of the trapezoid. | 5R^{2} | 58 | 5 |
math | Example 4. Solve the Cauchy problem for the system
$$
\left\{\begin{array}{l}
\frac{d x}{d t}=8 y \\
\frac{d y}{d t}=-2 z \\
\frac{d z}{d t}=2 x+8 y-2 z
\end{array}\right.
$$
with initial conditions $x(0)=-4, y(0)=0, z(0)=1$. | -4e^{-2}-2\sin4,\quade^{-2}-\cos4,\quade^{-2}-2\sin4 | 101 | 29 |
math | An alphabet consists of $n$ letters. What is the maximal length of a word if we know that any two consecutive letters $a,b$ of the word are different and that the word cannot be reduced to a word of the kind $abab$ with $a\neq b$ by removing letters. | 2n - 1 | 63 | 7 |
math | Let $a,x,y$ be positive integer such that $a>100,x>100,y>100$ and $y^2-1=a^2(x^2-1)$ . Find the minimum value of $\frac{a}{x}$. | 2 | 59 | 1 |
math | Charles has two six-sided die. One of the die is fair, and the other die is biased so that it comes up six with probability $\frac{2}{3}$ and each of the other five sides has probability $\frac{1}{15}$. Charles chooses one of the two dice at random and rolls it three times. Given that the first two rolls are both sixes... | 167 | 119 | 3 |
math | G1.2 Let $x, y$ and $z$ be positive numbers. Given that $\frac{x+z}{2 z-x}=\frac{z+2 y}{2 x-z}=\frac{x}{y}$. Find the value of $\frac{x}{y}$. | 2 | 59 | 1 |
math | Assume that $f(a+b) = f(a) + f(b) + ab$, and that $f(75) - f(51) = 1230$. Find $f(100)$. | 3825 | 50 | 4 |
math | Find the least positive integer $ a$ such that $ 2001$ divides $ 55^n\plus{}a \cdot 32^n$ for some odd $ n$. | 436 | 40 | 3 |
math | 5. Paul is painting a wall. He knows the area of the wall is 1920 square metres, correct to the nearest ten. He uses tins of paint, each of which can cover 18 square metres, correct to the nearest integer.
He needs to paint the wall completely, and still have at least half a tin of paint left over for any minor repairs... | 111 | 98 | 3 |
math | It is given that there exists a unique triple of positive primes $(p,q,r)$ such that $p<q<r$ and \[\dfrac{p^3+q^3+r^3}{p+q+r} = 249.\] Find $r$. | r = 19 | 58 | 6 |
math | ## Task B-1.4.
Simplify the following fraction to the one that cannot be further reduced
$$
\frac{x^{4}-16}{x^{4}-4 x^{3}+8 x^{2}-16 x+16}
$$ | \frac{x+2}{x-2} | 56 | 10 |
math | 1. (15 points) In a bookstore, there is a rule for "summing" discounts: "if different types of discounts apply to an item, they are applied sequentially one after another." For example, if two discounts A% and B% apply to an item, the first discount is applied to the original price, and the second discount is applied t... | 50 | 178 | 2 |
math | B1. The floor function of any real number $a$ is the integer number denoted by $\lfloor a\rfloor$ such that $\lfloor a\rfloor \leq a$ and $\lfloor a\rfloor>a-1$. For example, $\lfloor 5\rfloor=5,\lfloor\pi\rfloor=3$ and $\lfloor-1.5\rfloor=-2$. Find the difference between the largest integer solution of the equation $\... | 614 | 141 | 3 |
math | Example 12 Let $n$ be a fixed integer, $n \geqslant 2$.
(1) Determine the smallest constant $c$ such that the inequality
$$
\sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leqslant c \cdot\left(\sum_{i=1}^{n} x_{i}\right)^{4}
$$
holds for all non-negative real numbers $x_{1}, x_{2}, \cd... | \frac{1}{8} | 161 | 7 |
math | ## Problem Statement
Find the point of intersection of the line and the plane.
$\frac{x-5}{-2}=\frac{y-2}{0}=\frac{z+4}{-1}$
$2 x-5 y+4 z+24=0$ | (3;2;-5) | 60 | 7 |
math | Determine the GCD of 1000000000 and 1000 000, 005. | 5 | 33 | 1 |
math | 11. Let $f(x)$ be an increasing function defined on $[1,+\infty)$, and the inequality
$$
f\left(k-\cos ^{2} x\right) \leqslant f\left(k^{2}+\sin x\right)
$$
holds for all $x$. Find the range of real numbers $k$. | k \geqslant 2 | 78 | 8 |
math | Task 1. Martin is older than his sisters Ana, Ivana, and Elena by 8, 10, and 13 years, respectively. In one year, the sum of the ages of Martin's sisters will be 39 years more than Martin's age. How old is Martin today? | 34 | 64 | 2 |
math | Let $S$ be the set of natural numbers that cannot be written as the sum of three squares. Legendre's three-square theorem states that $S$ consists of precisely the integers of the form $4^a(8b+7)$ where $a$ and $b$ are nonnegative integers. Find the smallest $n\in\mathbb N$ such that $n$ and $n+1$ are both in $S$. | 111 | 92 | 3 |
math | (1) Let the set $A=\left\{a_{1}, a_{2}, a_{3}, a_{4}\right\}$. If the set $B=\{-1,3,5,8\}$ consists of the sums of all three-element subsets of $A$, then the set $A=$ $\qquad$ . | {-3,0,2,6} | 72 | 9 |
math | 3. (25 points) On a circle, there are $n$ different positive integers $a_{1}$, $a_{2}, \cdots, a_{n}$ placed in a clockwise direction. If for any number $b$ among the ten positive integers $1, 2, \cdots, 10$, there exists a positive integer $i$ such that $a_{i}=b$ or $a_{i}+a_{i+1}=b$, with the convention that $a_{n+1}... | 6 | 129 | 1 |
math | Determine the unique pair of real numbers $(x,y)$ that satisfy the equation
\[(4 x^2+ 6 x + 4)(4 y^2 - 12 y + 25 ) = 28 .\] | \left(-\frac{3}{4}, \frac{3}{2}\right) | 55 | 20 |
math | 7. Find all positive integers $m, n$, such that $2^{m}+3^{n}$ is a perfect square. | (4,2) | 28 | 5 |
math | 4. (8 points) A cruise ship travels from upstream location $A$ to downstream location $B$ in 1 hour. On the return trip, the ship doubles its speed and still takes 1 hour. Therefore, if the cruise ship also doubles its speed when departing from $A$, it will take $\qquad$ minutes to reach $B$. | 36 | 73 | 2 |
math | ## Problem 6
Find all positive integers $n$ such that $3^{n}+5^{n}$ is a multiple of $3^{n-1}+5^{n-1}$. | 1 | 43 | 1 |
math | ## Task Condition
Find the $n$-th order derivative.
$y=x \cdot e^{a x}$ | y^{(n)}=(n+\cdotx)\cdote^{}\cdot^{n-1} | 24 | 21 |
math | 5. Find all pairs of positive numbers $(x, y)$ that satisfy the system of equations $\left\{\begin{array}{l}2 x-\sqrt{x y}-4 \sqrt{\frac{x}{y}}+2=0 \\ 2 x^{2}+x^{2} y^{4}=18 y^{2} .\end{array}\right.$ | (2;2),(\frac{\sqrt[4]{286}}{4};\sqrt[4]{286}) | 79 | 28 |
math | Let $ABCD$ be an isosceles trapezoid with $AB=5$, $CD = 8$, and $BC = DA = 6$. There exists an angle $\theta$ such that there is only one point $X$ satisfying $\angle AXD = 180^{\circ} - \angle BXC = \theta$. Find $\sin(\theta)^2$. | \frac{27}{32} | 84 | 9 |
math | $n$ is composite. $1<a_1<a_2<...<a_k<n$ - all divisors of $n$. It is known, that $a_1+1,...,a_k+1$ are all divisors for some $m$ (except $1,m$). Find all such $n$. | n \in \{4, 8\} | 68 | 11 |
math | Question 69, The quadratic function $f(x)=x^{2}+m x+n$ has real roots, the inequality $s \leq(m-1)^{2}+(n-1)^{2}+$ $(m-n)^{2}$ holds for any quadratic function satisfying the above condition, then the maximum value of $s$ is $\qquad$ _. | \frac{9}{8} | 79 | 7 |
math | 6. (3 points) Person A and Person B work together to process a batch of parts, which can be completed in 8 hours. If Person A works alone, it would take 12 hours to complete the task. Now, Person A and Person B work together for $2 \frac{2}{5}$ hours, after which Person A is reassigned to other work, and Person B conti... | 480 | 105 | 3 |
math | Find the number of subsets of $\{1, 2, 3, 4, 5, 6, 7, 8\}$ that are subsets of neither $\{1, 2, 3, 4, 5\}$ nor $\{4, 5, 6, 7, 8\}$. | 196 | 73 | 3 |
math | 1. Given the number $a=1+3+5+\ldots+4025+2013$.
a) Calculate $[\sqrt{a}]$ where $[x]$ represents the integer part of the number $x$.
b) Prove the inequality:
$$
\sqrt{a+\sqrt{a+\sqrt{a}}}<2014
$$
Prof. Ana Popescu | 2013 | 90 | 4 |
math | 1. Solve the equation $x^{\log _{2}(8 x)}=\frac{x^{7}}{8}$. | 2,8 | 27 | 3 |
math | How much is the integer part of the following number:
$$
\sqrt{6+\sqrt{6+\ldots+\sqrt{6}}}+\sqrt[3]{6+\sqrt[3]{6+\cdots+\sqrt[3]{6}}}
$$
where both the number of square root and cube root symbols is 100? | 4 | 71 | 1 |
math | (7) Let $\left(x^{2}+2 x-2\right)^{6}=a_{0}+a_{1}(x+2)+a_{2}(x+2)^{2}+\cdots+a_{12}(x+$ $2)^{12}$, where $a_{i}(i=0,1,2, \cdots, 12)$ are real constants, then $a_{0}+a_{1}+2 a_{2}+3 a_{3}+\cdots+$ $12 a_{12}=$ $\qquad$ . | 64 | 130 | 2 |
math | 12. In a game activity of a TV entertainment program, each person needs to complete three tasks $A, B, C$. It is known that the probabilities for contestant A to complete tasks $A, B, C$ are $\frac{3}{4}, \frac{3}{4}, \frac{2}{3}$, respectively, and each task is independent.
(1) Contestant A attempts tasks $A, B, C$ on... | \frac{243a}{128} | 256 | 12 |
math | Example 4 The inequality about the real number $x$ is $\left|x-\frac{(a+1)^{2}}{2}\right| \leqslant \frac{(a-1)^{2}}{2}$ and $x^{2}-3(a+1) x+$ $2(3 a+1) \leqslant 0$ (where $a \in \mathbf{R})$. The solution sets of these inequalities are denoted as $A$ and $B$ respectively. Find the range of values for $a$ that makes $... | 1\leqslant\leqslant3or=-1 | 127 | 15 |
math | Let $a, b, c, d$ be the four roots of $X^{4}-X^{3}-X^{2}-1$. Calculate $P(a)+P(b)+P(c)+$ $P(d)$, where $P(X)=X^{6}-X^{5}-X^{4}-X^{3}-X$. | -2 | 70 | 2 |
math | 8. Given the sequence $\left.\mid a_{n}\right\}$, where $a_{n}$ is an integer, and for $n \geqslant 3, n \in \mathbf{N}$, we have $a_{n}=a_{n-1}-$ $a_{n-2}$, if the sum of the first 1985 terms is 1000, and the sum of the first 1995 terms is 4000, then the sum of the first 2002 terms is $\qquad$ _. | 3000 | 127 | 4 |
math | One of Euler's conjectures was disproved in the $1980$s by three American Mathematicians when they showed that there is a positive integer $n$ such that \[n^{5}= 133^{5}+110^{5}+84^{5}+27^{5}.\] Find the value of $n$. | 144 | 77 | 3 |
math | 3. For any sequence, the following operation can be performed: each time select three consecutive terms, denoted as $a, b, c$, and replace them with $b, c, a$, keeping the other terms unchanged. Determine all integers $n \geqslant 3$, such that the sequence $1, 2, \cdots, n$ can be transformed into $n, n-1, \cdots, 1$ ... | n\equiv0or1(\bmod4) | 110 | 11 |
math | 11.31 $|x+1|>2|x+2|$.
Translate the above text into English, keeping the original text's line breaks and format, and output the translation result directly.
11.31 $|x+1|>2|x+2|$. | (-3,-\frac{5}{3}) | 60 | 10 |
math | 12. Write the consecutive odd numbers from 1 to 103 as a single large number: \( A = \) 13579111315171921….9799101103.
Then the number \( \mathrm{a} \) has \_\_\_\_ digits, and the remainder when \( \mathrm{a} \) is divided by 9 is \_\_\_\_ _\. | 101;4 | 101 | 5 |
math | 6. Find all triples $(m, n, p)$ which satisfy the equation
$$
p^{n}+3600=m^{2}
$$
where $p$ is prime and $m, n$ are positive integers. | (61,2,11),(65,4,5),(68,10,2) | 50 | 24 |
math | 6. (15 points) A capacitor with capacitance $C_{1}=10$ μF is charged to a voltage $U_{1}=15$ V. A second capacitor with capacitance $C_{2}=5$ μF is charged to a voltage $U_{2}=10$ V. The capacitors are connected with their oppositely charged plates. Determine the voltage that will establish across the plates. | 6.67\mathrm{~V} | 90 | 10 |
math | 12. There are 900 three-digit numbers (100, 101, 999). If these three-digit numbers are printed on cards, with one number per card, some cards, when flipped, still show a three-digit number, such as 198, which when flipped becomes 861 (1 is still considered 1 when flipped); some cards do not, such as 531, which when fl... | 34 | 126 | 2 |
math | Sure, here is the translated text:
```
Simplify the following expression:
\[
\begin{aligned}
& 1+\frac{a_{1}}{1-a_{1}}+ \frac{a_{2}}{\left(1-a_{1}\right)\left(1-a_{2}\right)}+\frac{a_{3}}{\left(1-a_{1}\right)\left(1-a_{2}\right)\left(1-a_{3}\right)}+ \\
& \frac{a_{4}-a_{1}}{\left(1-a_{1}\right)\left(1-a_{2}\right)\le... | \frac{1}{(1-a_{2})(1-a_{3})(1-a_{4})} | 160 | 22 |
math | 【Question 13】
City A and City B are 55 kilometers apart. Xiao Wang starts from City A to City B, first riding a bicycle for 25 kilometers, then switching to a bus, which travels at twice the speed. Upon arriving in City B, he finds that the time spent cycling is 1 hour more than the time spent on the bus. Xiao Wang's... | 10 | 92 | 2 |
math | Initially given $31$ tuplets
$$(1,0,0,\dots,0),(0,1,0,\dots,0),\dots, (0,0,0,\dots,1)$$
were written on the blackboard. At every move we choose two written $31$ tuplets as $(a_1,a_2,a_3,\dots, a_{31})$ and $(b_1,b_2,b_3,\dots,b_{31})$, then write the $31$ tuplet $(a_1+b_1,a_2+b_2,a_3+b_3,\dots, a_{31}+b_{31})$ to the... | 87 | 223 | 2 |
math | Given an interger $n\geq 2$, determine the maximum value the sum $\frac{a_1}{a_2}+\frac{a_2}{a_3}+...+\frac{a_{n-1}}{a_n}$ may achieve, and the points at which the maximum is achieved, as $a_1,a_2,...a_n$ run over all positive real numers subject to $a_k\geq a_1+a_2...+a_{k-1}$, for $k=2,...n$ | \frac{n}{2} | 118 | 7 |
math | High I.
An irreducible fraction $\frac{a}{b}$ is such that
$$
\frac{a}{b}=\frac{999}{1999}+\frac{999}{1999} \cdot \frac{998}{1998}+\frac{999}{1999} \cdot \frac{998}{1998} \cdot \frac{997}{1997}+\ldots+\frac{999}{1999} \cdot \frac{998}{1998} \cdot \ldots \cdot \frac{1}{1001}
$$
Find $a$ and $b$.
# | =999,b=1001 | 163 | 10 |
math | Example 8 Given that $a$, $b$, and $c$ are all positive integers, and the parabola $y=a x^{2}+b x+c$ intersects the $x$-axis at two distinct points $A$ and $B$. If the distances from $A$ and $B$ to the origin are both less than 1, find the minimum value of $a+b+c$.
$(1996$, National Junior High School Mathematics Leagu... | 11 | 100 | 2 |
math | 1. Calculate: $12345^{2}+12353^{2}-2 \cdot 12349^{2}$. | 32 | 35 | 2 |
math | 4. For positive integer $n$, let $D(n)$ be the eventual value obtained when the digits of $n$ (in base 10) are added up recursively until a one-digit number is obtained. For example $D(4)=4, D(2012)=D(5)=5$ and $D(1997)=D(26)=D(8)=8$. If $x$ denotes the 2012th Fibonacci number (i.e. the 2012th term of the sequence 1, 1... | 6 | 300 | 1 |
math | Example 24. When manufacturing a certain product, its weight $X$ is subject to random fluctuations. The standard weight of the product is 30 g, its standard deviation is 0.7, and the random variable $X$ is normally distributed. Find the probability that the weight of a randomly selected product is within the range of 2... | 0.922 | 79 | 5 |
math | 466. For what number and for which natural $n$ is the fraction $\frac{n^{2}+1}{n+1}$ reducible? Find all solutions. | 2foralloddn | 38 | 4 |
math | 3. How many natural numbers less than 2016 are divisible by 2 or 3, but not by 5? | 1075 | 28 | 4 |
math | 11. The sequence $\left\{\alpha_{n}\right\}$ is an arithmetic sequence, $\beta$ is the common difference, the sequence $\left\{\sin \alpha_{n}\right\}$ is a geometric sequence, with common ratio $q$, and $\alpha_{1}, \beta \in \mathbf{R}$, then $q=$ $\qquad$ | 1or-1 | 80 | 4 |
math | Example 2.61. Calculate the surface area formed by the rotation of the arc of the circle $x^{2}+y^{2}=16(y>0)$ over the segment $-1 \geqslant x \geqslant 1$ around the $O X$ axis. | 16\pi | 65 | 4 |
math | 1. Find all pairs of integers $a, b$ such that the sum $a+b$ is a root of the equation $x^{2}+a x+b=0$. | (-6,8),(-6,9),(0,0),(0,-1) | 38 | 18 |
math | 2. (7 points) Anya multiplied 20 twos, and Vanya multiplied 17 fives. Now they are going to multiply their huge numbers. What will be the sum of the digits of the product? | 8 | 47 | 1 |
math | Example 3. Find the integral $\int \operatorname{sh}^{2} x \operatorname{ch}^{2} x d x$. | \frac{1}{32}\sinh4x-\frac{1}{8}x+C | 32 | 21 |
math | 1. (16 points) If $n$ is a positive integer greater than 2, find the minimum value of
$$
\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2 n}
$$ | \frac{37}{60} | 58 | 9 |
math | Example 3 Find all real numbers $p$ such that the cubic equation $5 x^{3}$ $-5(p+1) x^{2}+(71 p-1) x+1=66 p$ has three roots that are all natural numbers. | 76 | 56 | 2 |
math | Find the greatest integer $d$ that divides $n^{5}-n$ for all integers $n \in \mathbb{Z}$. | 30 | 30 | 2 |
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