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 1. Given the equation of a straight line $\frac{3 x-2}{4}-\frac{2 y-1}{2}=1$, find its
1) general equation;
2) slope-intercept equation;
3) intercept form equation
4) normal equation | 3x-4y-4=0,\quad\frac{3}{4}x-1,\quad\frac{x}{4/3}+\frac{y}{-1}=1,\quad\frac{3}{5}x-\frac{4}{5}y-\frac{4}{5}=0 | 58 | 66 |
math | 5.1. A random number sensor outputs a number $a$ - one of the natural numbers $1,2, \ldots, 100$ (with equal probability). For this value of $a$, we find the maximum possible value $M$ of the function
$$
f(x)=\frac{700}{x^{2}-2 x+2 a}
$$
The probability that $M>10$ is $n$ percent. What is $n$? | 35 | 104 | 2 |
math | Example 4 (2003 China National Training Team) In $\triangle ABC$, $AC > AB$, $P$ is the intersection of the perpendicular bisector of $BC$ and the internal angle bisector of $\angle A$. Draw $PX \perp AB$, intersecting the extension of $AB$ at point $X$, and $PY \perp AC$ intersecting $AC$ at point $Y$, $Z$ is the inte... | 1 | 116 | 1 |
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 | 11. Find the minimum value of the function
$$
y=2 x+\sqrt{4 x^{2}-8 x+3}
$$ | 1 | 31 | 1 |
math | To enter a park, a group with two men, four women, and two children paid 226 reais, while a group with three men, three women, and one child paid 207 reais.
a) How much would a group with 8 men, 10 women, and 4 children pay to enter the park?
b) If the ticket prices are all natural numbers, how many possible prices a... | 640 | 95 | 3 |
math | 5. [4] A sphere is the set of points at a fixed positive distance $r$ from its center. Let $\mathcal{S}$ be a set of 2010dimensional spheres. Suppose that the number of points lying on every element of $\mathcal{S}$ is a finite number $n$. Find the maximum possible value of $n$. | 2 | 77 | 1 |
math | (IMO SL 2018 A1)(M-D) Determine the functions $f: \mathbb{Q}_{+}^{*} \rightarrow \mathbb{Q}_{+}^{*}$ such that
$$
f\left(x^{2} f(y)^{2}\right)=f(x)^{2} f(y)
$$ | f\equiv1 | 74 | 4 |
math | Example 9 Let $a_{1}=1, a_{2}=3$, for all positive integers $n$ we have $a_{n+2}=(n+3) a_{n+1}-(n+$ 2) $a_{n}$, find the values of all numbers divisible by 11. | n=4,n=8orn\geqslant10 | 68 | 14 |
math | 10. Katherine and James are jogging in the same direction around a pond. They start at the same time and from the same place and each jogs at a constant speed. Katherine, the faster jogger, takes 3 minutes to complete one lap and first overtakes James 8 minutes after starting. How many seconds does it take James to com... | 288 | 74 | 3 |
math | Example 2 On the coordinate plane, points with both coordinates as integers are called integer points. For any natural number $n$, connect the origin $O$ with the point $A_{n}(n, n+3)$, and let $f(n)$ denote the number of integer points on the line segment $O A_{n}$, excluding the endpoints. Find $f(1)+f(2)+\cdots+f(19... | 1326 | 95 | 4 |
math | 2. If $n$ positive real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfy the equation
$$
\sum_{k=1}^{n}\left|\lg x_{k}\right|+\sum_{k=1}^{n}\left|\lg \frac{1}{x_{k}}\right|=\left|\sum_{k=1}^{n} \lg x_{k}\right|
$$
then the values of $x_{1}, x_{2}, \cdots, x_{n}$ are $\qquad$ | x_{1}=x_{2}=\cdots=x_{n}=1 | 124 | 16 |
math | 4. Solve the equation $\frac{\cos x}{\sqrt{3}}-\sqrt{\frac{1-\cos 2 x-2 \sin ^{3} x}{6 \sin x-2}}=0$.
# | \frac{\pi}{2}+2\pin,\frac{\pi}{6}+2\pin,n\inZ | 49 | 26 |
math | 7. Let the ellipse $\frac{x^{2}}{m^{2}}+\frac{y^{2}}{n^{2}}=1$ pass through the fixed point $P(1,2)$. Then the minimum value of $m+n$ is $\qquad$ . | \left(1+2^{\frac{2}{3}}\right)^{\frac{3}{2}} | 59 | 24 |
math | 5. Let $a \star b=a b+a+b$ for all integers $a$ and $b$. Evaluate $1 \star(2 \star(3 \star(4 \star \ldots(99 \star 100) \ldots)))$. | 101!-1 | 58 | 6 |
math | [ [Decimal number system ] $[$ Equations in integers $]$
Find a two-digit number that is equal to the sum of the cube of the number of its tens and the square of the number of its units.
# | 24 | 46 | 2 |
math | Example 3 Given $n$ positive integers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying $x_{1}+x_{2}+\cdots+x_{n}=2020$. Find the maximum value of the product $x_{1} x_{2} \cdots x_{n}$ of these $n$ positive integers.
(Adapted from the 2008 National Junior High School Mathematics Competition Tianjin Preliminary Round) | 2^{2}\times3^{672} | 103 | 11 |
math | 11. A three-digit number is thought of, which with any of the numbers 543, 142, and 562, matches one digit in the same place, while the other two do not match. What is the number thought of? | 163 | 56 | 3 |
math | 5. In the set of real numbers, solve the system of equations
$$
\begin{aligned}
& x^{2006}+y^{2006}+z^{2006}=2 \\
& x^{2007}+y^{2007}+z^{2007}=2 \\
& x^{2008}+y^{2008}+z^{2008}=2
\end{aligned}
$$
## County Competition 2006, 4th grade, B variant - solutions to problems
Each correctly solved problem is worth 20 points... | 2,8,14,20,26,32 | 189 | 15 |
math | $7 \cdot 1$ Given $n$ points on a plane, any 3 of which are the 3 vertices of a right-angled triangle, find the maximum value of $n$.
untranslated text retained as requested. | 4 | 48 | 1 |
math | 7.281. $\left\{\begin{array}{l}y \cdot x^{\log _{y} x}=x^{2.5} \\ \log _{3} y \cdot \log _{y}(y-2 x)=1 .\end{array}\right.$ | (3;9) | 65 | 5 |
math | 11. Magic Pen (recommended for 8th grade, 1 point). Katya correctly solves a problem with a probability of $4 / 5$, while the magic pen correctly solves a problem without Katya's help with a probability of $1 / 2$. In the test, there are 20 problems, and to get a B, one needs to solve at least 13 of them correctly. How... | 10 | 124 | 2 |
math | Example 3 Find the maximum value of the function $y=6 \sqrt{x-5}+8 \sqrt{6-x}$. | 10 | 29 | 2 |
math | 5. A toy factory produces cubic building blocks of the same size, with each face painted one of three colors: red, yellow, or blue, and each color is used on exactly 2 faces. When two blocks can be rotated to have the same color faces in the same positions, they are considered the same type of block. Try to explain: wh... | 6 | 86 | 1 |
math | 3. Let $m$ and $n$ be positive integers, and $m>n$. If the last two digits of $9^{m}$ and $9^{n}$ are the same, then the minimum value of $m-n$ is $\qquad$ | 10 | 54 | 2 |
math | The equation $166\times 56 = 8590$ is valid in some base $b \ge 10$ (that is, $1, 6, 5, 8, 9, 0$ are digits in base $b$ in the above equation). Find the sum of all possible values of $b \ge 10$ satisfying the equation. | 12 | 85 | 2 |
math | 1. Given real numbers $x, y$ satisfy $x^{3}+y^{3}=2$. Then the maximum value of $x+y$ is $\qquad$ . | 2 | 38 | 1 |
math | 5.68 Find all quadratic trinomials \( p(x) \), which take a minimum value of \(-\frac{49}{4}\) at \( x = \frac{1}{2} \), and the sum of the fourth powers of its two roots is 337. | p(x)=x^{2}-x-12 | 64 | 11 |
math | G10.2Let $f(x)=x^{9}+x^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1$. When $f\left(x^{10}\right)$ is divided by $f(x)$, the remainder is $b$. Find the value of $b$. | 10 | 87 | 2 |
math | 1. Determine the number of all infinite arithmetic sequences of integers that have both numbers 1 and 2005 among their first ten terms. | 68 | 30 | 2 |
math | Six. (20 points) Let $[x]$ denote the greatest integer not exceeding the real number $x$. Given
$$
\begin{array}{l}
a_{k}=\sum_{i=k^{2}}^{(k+1)^{2}-1} \frac{1}{i}(k=1,2, \cdots) . \\
\text { Find } \sum_{k=1}^{n}\left(\left[\frac{1}{a_{k}}\right]+\left[\frac{1}{a_{k}}+\frac{1}{2}\right]\right) .
\end{array}
$$ | \frac{n(n+1)}{2} | 135 | 10 |
math | 20.1. (New York, 78). The function $f: \mathbf{R} \rightarrow \mathbf{R}$ satisfies the identity
$$
f(x y) \equiv \frac{f(x)+f(y)}{x+y}, \quad x, y \in \mathbf{R}, \quad x+y \neq 0
$$
Does there exist a value $x \in \mathbb{R}$ for which $f(x) \neq 0$? | f(x)=0,x\in{R} | 110 | 10 |
math | There is a prime number $p$ such that $16p+1$ is the cube of a positive integer. Find $p$. | 307 | 30 | 3 |
math | Example 2. Find the zeros of the function $f(z)=1-e^{z}$ and determine their orders. | z_{n}=2n\pii(n=0,\1,\2,\ldots) | 24 | 19 |
math | 14 Let $f(n)$ satisfy $f(0)=0, f(n)=n-f(f(n-1)), n=1$,
$2,3, \cdots$. Determine all real-coefficient polynomials $g(x)$ such that
$$
f(n)=[g(n)], n=0,1,2, \cdots,
$$
where $[g(n)]$ denotes the greatest integer not exceeding $g(n)$. | (x)=\frac{1}{2}(\sqrt{5}-1)(x+1) | 94 | 20 |
math | 5. Let the sequence $a_{1}, a_{2}, a_{3} \cdots a_{n}, \cdots$ satisfy $a_{1}=a_{2}=1, a_{3}=2$, and for any natural number $n$, $a_{n} \cdot a_{n+1} \cdot a_{n+2} \neq$ 1, and also $a_{n} \cdot a_{n+1} \cdot a_{n+2} \cdot a_{n+3}=a_{n}+a_{n+1}+a_{n+2}+a_{n+3}$, then the value of $a_{1}+a_{2}+\cdots+a_{100}$ is $\qquad... | 200 | 168 | 3 |
math | 4. $\cos ^{2} 75^{\circ}+\cos ^{2} 15^{\circ}+\cos 75^{\circ} \cdot \cos 15^{\circ}=$ | \frac{5}{4} | 49 | 7 |
math | 10. The function $f$ has the following property: For any two real numbers $x, f(x)+f(x-1)=x^{2}$. If $f(19)=94$, then the remainder when $f(94)$ is divided by $10^{3}$ is | 561 | 64 | 3 |
math | 10. $[\mathbf{8}]$ Compute
$$
\int_{0}^{\infty} \frac{e^{-x} \sin (x)}{x} d x
$$ | \frac{\pi}{4} | 44 | 7 |
math | 3.1.7 * Given that all terms of the sequence $\left\{a_{n}\right\}$ are positive, and the sum of the first $n$ terms $S_{n}$ satisfies $6 S_{n}=a_{n}^{2}+$ $3 a_{n}+2$. If $a_{2}, a_{4}, a_{9}$ form a geometric sequence, find the general term formula of the sequence. | a_{n}=3n-2 | 94 | 8 |
math | If $x=\sqrt2+\sqrt3+\sqrt6$ is a root of $x^4+ax^3+bx^2+cx+d=0$ where $a,b,c,d$ are integers, what is the value of $|a+b+c+d|$? | 93 | 58 | 2 |
math | 20. A research institute transfers a high-tech technology to a company for production. Both parties agree: the research institute will withdraw 4 million yuan from the company's profit at the end of each year and invest it in other investments, thus earning a profit of 10% annually.
（1）Can the research institute obtain... | 1324 | 119 | 4 |
math | For example, $1 f: A_{3} \rightarrow A_{3}$, find the number of $f$ that satisfy $f^{(3)}=f$, and list all such $f$.
untranslated text remains as:
例 $1 f: A_{3} \rightarrow A_{3}$, 求满足 $f^{(3)}=f$ 的 $f$ 个数, 并作出所有这些 $f$.
However, the requested translation is:
For example, $1 f: A_{3} \rightarrow A_{3}$, find the num... | 19 | 145 | 2 |
math | 2nd Swedish 1962 Problem 3 Find all pairs (m, n) of integers such that n 2 - 3mn + m - n = 0. | (,n)=(0,0)or(0,1) | 38 | 14 |
math | 7. Xiao Dong is 8 years younger than his sister. In 3 years, his sister's age will be twice Xiao Dong's age. His sister is $\qquad$ years old this year. | 13 | 42 | 2 |
math | 10. (20 points) Let $\lambda$ be a positive real number. For any pairwise distinct positive real numbers $a, b, c$, we have
$$
\frac{a^{3}}{(b-c)^{2}}+\frac{b^{3}}{(c-a)^{2}}+\frac{c^{3}}{(a-b)^{2}} \geqslant \lambda(a+b+c) \text {. }
$$
Find the maximum value of $\lambda$. | 1 | 104 | 1 |
math | 8. Let the function $f: \mathbf{R} \rightarrow \mathbf{R}$, satisfy $f(0)=1$, and for any $x, y$ $\in \mathbf{R}$, there is $f(x y+1)=f(x) f(y)-f(y)-x+2$. Then $f(x)=$ $\qquad$ . | x+1 | 81 | 3 |
math | 16*. How many different divisors does the number 86400000 have (including 1 and the number 86400000 itself)? Find the sum of all these divisors. | 264divisors,sum319823280 | 48 | 15 |
math | We use the digits $1,2, \ldots, 9$ once each to form two integers (e.g., 7419 and 82635). What two integers formed in this way have the greatest product? Prove your answer. | 9642\times87531 | 56 | 11 |
math | 10. There are $n$ locks numbered $1,2, \cdots, n$, all of which are initially locked. Now, a series of operations $T_{1}, T_{2}, \cdots, T_{n}$ are performed sequentially, where operation $T_{k}(1 \leqslant k \leqslant n)$ only changes the state of the locks whose numbers are multiples of $k$. After $n$ operations, whi... | k^2 | 102 | 3 |
math | 17 (12 points) Given the ellipse $T: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ and the hyperbola $S$ : $\frac{x^{2}}{m^{2}}-\frac{y^{2}}{n^{2}}=1(m>0, n>0)$ have the same focus $F(2,0)$. Let the asymptote of the hyperbola $S$ in the first quadrant be $l$. If the focus $F$ and the upper vertex $B$ of the ellipse... | \frac{11x^{2}}{60}+\frac{11y^{2}}{16}=1\frac{5x^{2}}{4}-\frac{5y^{2}}{16}=1 | 176 | 51 |
math | Example 3 Let $a$ and $b$ be positive real numbers, and
$$
\begin{array}{l}
\frac{1}{a}-\frac{1}{b}-\frac{1}{a+b}=0 . \\
\text { Then }\left(\frac{b}{a}\right)^{3}+\left(\frac{a}{b}\right)^{3}=
\end{array}
$$ | 2 \sqrt{5} | 91 | 6 |
math | 13.300. The smaller arc between points $A$ and $B$, located on a circle, is 150 m. If the points start moving towards each other along the smaller arc, they will meet after $10 \mathrm{c}$, and if they move along the larger arc, the meeting will occur after $14 \mathrm{c}$. Determine the speeds of the points and the le... | 12 | 125 | 2 |
math | [
Isosceles, inscribed, and circumscribed trapezoids [Theorem on the lengths of a tangent and a secant; the product of the entire secant and its external part
An isosceles trapezoid with bases $A D$ and $B C (A D > B C)$ is circumscribed around a circle, which touches the side $C D$ at point $M$. The segment $A M$ in... | 8k-1 | 130 | 4 |
math | 12. (16 points) Find all positive integers $n$ greater than 1, such that for any positive real numbers $x_{1}, x_{2}, \cdots, x_{n}$, the inequality
$$
\left(x_{1}+x_{2}+\cdots+x_{n}\right)^{2} \geqslant n\left(x_{1} x_{2}+x_{2} x_{3}+\cdots+x_{n} x_{1}\right) \text {. }
$$
holds. | 2, 3, 4 | 119 | 7 |
math | Example 13. There are two urns with balls of three colors. The first contains 2 blue, 3 red, and 5 green balls, while the second contains 4 blue, 2 red, and 4 green balls. One ball is drawn from each urn and their colors are compared. Find the probability that the colors of the drawn balls are the same (event $A$). | 0.34 | 83 | 4 |
math | 6. If 5 consecutive natural numbers are all composite, then this group of numbers is called a "twin 5 composite". So, among the natural numbers not exceeding 100, there are $\qquad$ groups of twin 5 composite. | 10 | 53 | 2 |
math | The integers from 1 to $k$ are concatenated to form the integer $N=123456789101112 \ldots$ Determine the smallest integer value of $k>2019$ such that $N$ is divisible by 9 . | 2024 | 62 | 4 |
math | 76. Given that for any positive integer $\mathrm{n}, 9^{2 n}-8^{2 n}-17$ is always divisible by $\mathrm{m}$, find the largest positive integer $m$.
| 2448 | 46 | 4 |
math | Find all natural $ x $ for which $ 3x+1 $ and $ 6x-2 $ are perfect squares, and the number $ 6x^2-1 $ is prime. | 1 | 42 | 1 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0} \frac{3^{2 x}-5^{3 x}}{\operatorname{arctg} x+x^{3}}$ | \ln\frac{9}{125} | 48 | 11 |
math | Determine the prime numbers $p, q, r$ with the property $\frac {1} {p} + \frac {1} {q} + \frac {1} {r} \ge 1$ | (2, 3, 5) | 46 | 10 |
math | Three, try to find all positive integers $k$, such that for any positive numbers $a, b, c$ satisfying the inequality
$$
k(a b+b c+c a)>5\left(a^{2}+b^{2}+c^{2}\right)
$$
there must exist a triangle with side lengths $a, b, c$. | 6 | 74 | 1 |
math | Suppose $r \ge 2$ is an integer, and let $m_1, n_1, m_2, n_2, \dots, m_r, n_r$ be $2r$ integers such that $$\left|m_in_j-m_jn_i\right|=1$$ for any two integers $i$ and $j$ satisfying $1 \le i<j \le r$. Determine the maximum possible value of $r$.
[i]Proposed by B Sury[/i] | 3 | 107 | 3 |
math | Example 3. For what value of $m$ does the line $y=-x+m$ intersect the ellipse $\frac{x^{2}}{20}+\frac{y^{2}}{5}=1$
(i) intersect;
(ii) be tangent,
(iii) be
separated? (Problem 13, page 132) | m = \pm 5 | 74 | 6 |
math | Example 4 The function $f^{-1}(x)$ is the inverse function of $f(x)=\log _{2}\left(2^{x}+1\right)$.
(1) If the equation $f^{-1}(x)=m+f(x)$ has a solution for $x$ in $[1,2]$, find the range of real number $m$;
(2) If the inequality $f^{-1}(x)>m+f(x)$ has a solution for $x$ in $[1,2]$, find the range of real number $m$;
... | [\log_{2}\frac{1}{3},\log_{2}\frac{3}{5}],(-\infty,\log_{2}\frac{3}{5}),(-\infty,\log_{2}\frac{1}{3}) | 162 | 52 |
math | Problem 6. Inside the magician's hat, there live 100 rabbits: white, blue, and green. It is known that if 81 rabbits are randomly pulled out of the hat, there will definitely be three of different colors among them. What is the minimum number of rabbits that need to be taken out of the hat to ensure that there are defi... | 61 | 81 | 2 |
math | Three, (20 points) Place a real number in each cell of a $4 \times 4$ grid paper,
such that the sum of the four numbers in each row, each column, and each diagonal equals a constant $k$. Find the sum of the numbers in the four corners of this $4 \times 4$ grid paper. | k | 73 | 1 |
math | 8. (10 points) The teacher bought a total of 53 pencils and distributed them among four students, $A$, $B$, $C$, and $D$. The difference between the maximum and minimum number of pencils received is less than 5. If $B$ gives all the pencils to $A$, then $A$ will have twice as many pencils as $C$. If $B$ gives all the p... | 15 | 123 | 2 |
math | Example 1 If numbers $1,2, \cdots, 14$ are taken in ascending order as $a_{1}, a_{2}, a_{3}$, such that both $a_{2}-$ $a_{1} \geqslant 3$ and $a_{3}-a_{2} \geqslant 3$ are satisfied, then the total number of different ways to select the numbers is $\qquad$.
(1989 National High School League Question) | 120 | 109 | 3 |
math | ## C2.
There are 2016 costumers who entered a shop on a particular day. Every customer entered the shop exactly once. (i.e. the customer entered the shop, stayed there for some time and then left the shop without returning back.)
Find the maximal $k$ such that the following holds:
There are $k$ customers such that e... | 45 | 103 | 2 |
math | Problem 8.7. Along an alley, maples and larches were planted in one row, a total of 75 trees. It is known that there are no two maples between which there are exactly 5 trees. What is the maximum number of maples that could have been planted along the alley? | 39 | 66 | 2 |
math | 10,11
A plane intersects the edges $A B, A C, D C$ and $D B$ of the tetrahedron $A B C D$ at points $M, N, P$ and $Q$ respectively, such that $A M$ : $M B=m, A N: N C=n, D P: P C=p$. Find the ratio $B Q / Q D$. | \frac{n}{} | 89 | 5 |
math | [ Inscribed Quadrilaterals [ Angles subtending equal arcs and equal chords $]$
In a convex quadrilateral $A B C D$, it is given that $\angle A B C=116^{\circ}, \angle A D C=64^{\circ}, \angle C A B=35^{\circ}$ and $\angle C A D=52^{\circ}$. Find the angle between the diagonals subtending side $A B$.
# | 81 | 101 | 2 |
math | Problem 7.5. If a platoon of soldiers is divided into brigades of 7 people, then 2 people will not fit into any brigade. If the platoon is divided into brigades of 12 people, then again 2 people will not fit into any brigade. What is the minimum number of soldiers that need to be added to the platoon so that it can be ... | 82 | 101 | 2 |
math | Concentric circles $\Omega_1$ and $\Omega_2$ with radii $1$ and $100$, respectively, are drawn with center $O$. Points $A$ and $B$ are chosen independently at random on the circumferences of $\Omega_1$ and $\Omega_2$, respectively. Denote by $\ell$ the tangent line to $\Omega_1$ passing through $A$, and denote by $P$ t... | 10004 | 124 | 5 |
math | 12.102. The volume of a cone is $V$. A pyramid is inscribed in the cone, with an isosceles triangle as its base, where the angle between the lateral sides is $\alpha$. Find the volume of the pyramid. | \frac{2V}{\pi}\sin\alpha\cos^{2}\frac{\alpha}{2} | 54 | 23 |
math | Example 11 (1996 National High School League Question) Find the range of real numbers $a$ such that for any real number $x$ and any $\theta \in\left[0, \frac{\pi}{2}\right]$, we have $(x+3+2 \sin \theta \cos \theta)^{2}+(x+a \sin \theta+a \cos \theta)^{2} \geqslant \frac{1}{8}$.
| \geqslant\frac{7}{2} | 103 | 12 |
math | B1. What is the maximum possible greatest common divisor of the numbers $a-2b+3, 2a-3b-1$, and $3a+b-2$, if $a$ and $b$ are natural numbers? | 38 | 51 | 2 |
math | 6. If the parabola $C_{m}: y=x^{2}-m x+m+1$ intersects the line segment $A B$ (where $A(0,4), B(4,0)$) at exactly two points, then the range of values for $m$ is $\qquad$ | 3 \leqslant m \leqslant \frac{17}{3} | 66 | 20 |
math | 6. In a checkers tournament, students from 10th and 11th grades participated. Each player played against every other player once. A player received 2 points for a win, 1 point for a draw, and 0 points for a loss. There were 10 times more 11th graders than 10th graders, and together they scored 4.5 times more points tha... | 20 | 113 | 2 |
math | 1. (2 points) In trapezoid $A B C D$ with bases $A D=12$ and $B C=8$, the circles constructed on sides $A B, B C$, and $C D$ as diameters intersect at one point. The length of diagonal $A C$ is 12. Find the length of $B D$. | 16 | 79 | 2 |
math | 7. Let $A B C D E$ be a square pyramid of height $\frac{1}{2}$ with square base $A B C D$ of side length $A B=12$ (so $E$ is the vertex of the pyramid, and the foot of the altitude from $E$ to $A B C D$ is the center of square $A B C D)$. The faces $A D E$ and $C D E$ meet at an acute angle of measure $\alpha$ (so that... | \frac{17}{144} | 131 | 10 |
math | 1. Simplify
$$
\sqrt{1+2 \sin \alpha \cdot \cos \alpha}+\sqrt{1-2 \sin \alpha \cdot \cos \alpha}
$$
$\left(0^{\circ}<\alpha \leqslant 90^{\circ}\right)$ The result is $\qquad$ . | 2 \cos \alpha \text{ or } 2 \sin \alpha | 75 | 16 |
math | 1. The members of the mathematics section in a school agreed that during the New Year's holiday, each of them would write one greeting card to each of the other members. A total of 342 greeting cards were written. How many members were there in the mathematics section of this school? | 19 | 60 | 2 |
math | 6. In space, there are 2017 points. The midpoints of the line segments connecting each pair of points are colored red. The minimum number of red points is | 4031 | 37 | 4 |
math | 4. Let the three-digit number $n=\overline{a b c}$, where the lengths $a, b, c$ can form an isosceles (including equilateral) triangle. Then the number of such three-digit numbers $n$ is $\qquad$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translatio... | 165 | 84 | 3 |
math | $\underline{\text { Folklore }}$
To repair the propeller, Karlson needs to buy three blades and one screw. In the store, blades cost 120 tugriks each and screws cost 9 tugriks each. However, after a purchase of at least 250 tugriks, a $20 \%$ discount is given on all subsequent purchases. Will Karlson be able to repai... | 354 | 104 | 3 |
math | 12. (5 points) If $A$ is a prime number, and $A-4, A-6, A-12, A-18$ are also prime numbers, then $A=$ | 23 | 45 | 2 |
math | ## Task B-1.3.
Solve the equation
$$
\frac{2 x}{x-1}-\frac{x}{x+2}-\frac{5 x}{(x-1)(x+2)}=\frac{4}{x^{2}+x-2}-\frac{2}{x-1}
$$ | 0 | 72 | 1 |
math | 9) If the complex number $z$ satisfies $|z|=1$, and $z^{2}=a+b \mathrm{i}$, where $a, b$ are real numbers, then the maximum value of $a+b$ is $\qquad$ . | \sqrt{2} | 54 | 5 |
math | # Problem 8.
Let $A(n)$ denote the greatest odd divisor of the number $n$. For example, $A(21)=21$, $A(72)=9, A(64)=1$. Find the sum $A(111)+A(112)+\ldots+A(218)+A(219)$. | 12045 | 80 | 5 |
math | 10. Let $S_{n}=1+2+\cdots+n$. Then among $S_{1}, S_{2}$, $\cdots, S_{2015}$, there are. $\qquad$ that are multiples of 2015. | 8 | 58 | 1 |
math | G2.4 Let $d=\frac{1}{2}+\frac{2}{4}+\frac{3}{8}+\frac{4}{16}+\ldots+\frac{10}{2^{10}}$, find the value of $d$. | \frac{509}{256} | 58 | 11 |
math | Example 4 - For a $4n+2$-sided polygon $A_{1} A_{2} A_{3} \cdots A_{4 n+2}$ (where $n$ is a natural number), each interior angle is an integer multiple of $30^{\circ}$.
Given the quadratic equations in $x$
$$
\begin{array}{l}
x^{2}+2 x \sin A_{1}+\sin A_{2}=0, \\
x^{2}+2 x \sin A_{2}+\sin A_{3}=0, \\
x^{2}+2 x \sin A_{... | A_{1}=A_{2}=A_{3}=90^{\circ}, A_{4}=A_{5}=A_{6}=150^{\circ} | 194 | 37 |
math | 117. Find $\lim _{x \rightarrow \infty} \frac{2 x^{3}+x}{x^{3}-1}$. | 2 | 34 | 1 |
math | 338. Solve the system of equations:
$$
\left\{\begin{array}{l}
x+y+z=6 \\
x y+y z+z x=11 \\
x y z=6
\end{array}\right.
$$ | (1;2;3),(1;3;2),(2;1;3),(2;3;1),(3;1;2),(3;2;1) | 52 | 37 |
math | Let $T_1$ be an isosceles triangle with sides of length 8, 11, and 11. Let $T_2$ be an isosceles triangle with sides of length $b$, 1, and 1. Suppose that the radius of the incircle of $T_1$ divided by the radius of the circumcircle of $T_1$ is equal to the radius of the incircle of $T_2$ divided by the radius of the... | \frac{14}{11} | 126 | 9 |
math | 543. Find the integrals:
1) $\int \frac{d x}{2 \sin x - \cos x}$
2) $\int \frac{d x}{5 + 4 \cos a x}$
3) $\int \frac{\tan x \, d x}{1 - \cot^2 x}$;
4) $\int \frac{e^{3 x} \, d x}{e^{2 x} + 1}$. | \begin{aligned}1)&\frac{1}{\sqrt{5}}\ln|\frac{2-\sqrt{5}+\operatorname{tg}\frac{x}{2}}{2+\sqrt{5}+\operatorname{tg}\frac{x}{2}}|+C\\2)&\frac{2}{3}\operatorname{arctg}(\frac{1}{3}\operator | 99 | 85 |
math | Let's determine the last two digits of the number
$$
7^{9^{9^{9}}}
$$ | 7 | 23 | 1 |
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