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 | Balkan Olympiads 2009
Find all functions $f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*}$ such that for all positive integers $m$ and $n$,
$$
f\left(f^{2}(m)+2 f^{2}(n)\right)=m^{2}+2 n^{2}
$$ | f(n)=n | 83 | 4 |
math | 5.2.1 * Given the real-coefficient equation
$$
x^{3}+2(k-1) x^{2}+9 x+5(k-1)=0
$$
has a complex root with a modulus of $\sqrt{5}$. Find the value of $k$, and solve this equation. | -1or3 | 68 | 4 |
math | Problem 6.1. Five consecutive natural numbers are written in a row. The sum of the three smallest of them is 60. What is the sum of the three largest? | 66 | 38 | 2 |
math | Example 8 Determine the largest real number $z$, such that $x+y+z=5, xy+yz+zx=3$, and $x, y$ are also real numbers.
(7th Canadian Competition Question) | \frac{13}{3} | 46 | 8 |
math | 6. Tenth-grader Dima came up with two natural numbers (not necessarily different). Then he found the sum and the product of these numbers. It turned out that one of these numbers coincides with the arithmetic mean of the other three. What numbers did Dima come up with? Find all possible answers and prove that others ar... | (1,2),(4,4),(3,6) | 70 | 13 |
math | Fernando is a cautious person, and the gate of his house has 10 distinct padlocks, each of which can only be opened by its respective key, and each key opens only one padlock. To open the gate, he must have at least one key for each padlock. For security, Fernando has distributed exactly two different keys in the 45 dr... | 37 | 179 | 2 |
math | 18. Find the largest positive integer $n$ such that $n+10$ is a divisor of $n^{3}+2011$. | 1001 | 34 | 4 |
math | # Problem No. 8 (10 points)
A water heater with a power of \( P = 500 \mathrm{W} \) is used to heat a certain amount of water. When the heater is turned on for \( t_{1} = 1 \) minute, the temperature of the water increases by \( \Delta T = 2^{\circ} \mathrm{C} \), and after the heater is turned off, the temperature de... | 2.38 | 167 | 4 |
math | 7.20 A single-player card game has the following rules: Place 6 pairs of different cards into a backpack. The player draws cards randomly from the backpack and returns them, but when a pair is drawn, it is set aside. If the player always draws three cards at a time, and if the three cards drawn are all different (i.e.,... | 394 | 128 | 3 |
math | 13. On a $75 \times 75$ chessboard, the rows and columns are numbered from 1 to 75. Chiara wants to place a pawn on all and only the squares that have one coordinate even and the other a multiple of 3. How many pawns will she place in total on the chessboard? | 1706 | 72 | 4 |
math | Two players take turns breaking a $6 \times 8$ chocolate bar. On a turn, a player is allowed to make a straight break of any of the pieces along a groove. The player who cannot make a move loses. Who will win this game? | 1 | 53 | 1 |
math | 43. Find a four-digit number that is a perfect square, where the digit in the thousands place is the same as the digit in the tens place, and the digit in the hundreds place is 1 more than the digit in the units place. | 8281 | 51 | 4 |
math | 4-4. In the box, there are colored pencils.
Vasya said: "There are at least four blue pencils in the box."
Kolya said: "There are at least five green pencils in the box."
Petya said: "There are at least three blue and at least four green pencils in the box."
Misha said: "There are at least four blue and at least fo... | Kolya | 111 | 3 |
math | ## Task 1.
Let $a=0.365$. The sequence $a_{1}, a_{2}, \ldots, a_{2020}$ is defined by the formulas
$$
a_{1}=a, \quad a_{n+1}=a^{a_{n}} \quad \text { for } \quad n=1, \ldots, 2019
$$
Arrange the numbers $a_{1}, a_{2}, \ldots, a_{2020}$ from the smallest to the largest. | a_{1}<a_{3}<\cdots<a_{2k-1}<a_{2k}<\cdots<a_{2}<1 | 120 | 31 |
math | 6. Given that a line with slope $k$ is drawn through a focus of the ellipse $x^{2}+2 y^{2}=3$, intersecting the ellipse at points $A$ and $B$. If $AB=2$, then $|k|=$ $\qquad$ | \sqrt{1+\sqrt{3}} | 61 | 9 |
math | 2. Through the right focus of the hyperbola $x^{2}-\frac{y^{2}}{2}=1$, a line $l$ intersects the hyperbola at points $A$ and $B$. If a real number $\lambda$ makes $|A B|=\lambda$ such that there are exactly 3 lines $l$, then $\lambda=$
(Proposed by the Problem Committee) | 4 | 87 | 1 |
math | Problem: Place the 2004 positive integers $1, 2, \cdots, 2004$ randomly on a circle. By counting the parity of all adjacent 3 numbers, it is known that there are 600 groups where all 3 numbers are odd, and 500 groups where exactly 2 numbers are odd. How many groups have exactly 1 odd number? How many groups have no odd... | 206, 698 | 115 | 8 |
math | Triangle $ABC$ is inscribed in circle $\omega$ with $AB=5$, $BC=7$, and $AC=3$. The bisector of angle $A$ meets side $\overline{BC}$ at $D$ and circle $\omega$ at a second point $E$. Let $\gamma$ be the circle with diameter $\overline{DE}$. Circles $\omega$ and $\gamma$ meet at $E$ and a second point $F$. Then $AF^2 = ... | 919 | 127 | 3 |
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 | Example 1 Let $X_{n}=\{1,2, \cdots, n\}$, for any non-empty subset $A$ of $X_{n}$, let $T(A)$ be the product of all numbers in $A$. Find $\sum_{A \subseteq X_{n}} T(A)$. | (n+1)!-1 | 68 | 6 |
math | ## Task B-1.4.
Hitting the target, Karlo hit the number nine several times, and the number eight several times. He hit the number seven half as many times as he hit the number nine, and the number of hits on the number five was three less than the number of hits on the number eight. In total, he scored 200 points.
Ho... | 25 | 121 | 2 |
math | Problem 7. (8 points)
The earned salary of the citizen was 23,000 rubles per month from January to June inclusive, and 25,000 rubles from July to December. In August, the citizen, participating in a poetry competition, won a prize and was awarded an e-book worth 10,000 rubles. What amount of personal income tax needs ... | 39540 | 121 | 5 |
math | [ Similar triangles ]
In triangle $ABC$, with sides $a, b$ and $c$ given, a line $MN$ parallel to $AC$ is drawn such that $AM = BN$. Find $MN$.
# | \frac{}{+} | 47 | 6 |
math | Jar A contains four liters of a solution that is 45% acid. Jar B contains five liters of a solution that is 48% acid. Jar C contains one liter of a solution that is $k\%$ acid. From jar C, $\frac{m}{n}$ liters of the solution is added to jar A, and the remainder of the solution in jar C is added to jar B. At the en... | 85 | 132 | 2 |
math | For which values of $p$ and $q$ integers strictly positive does $pq$ divide $3(p-1)(q-1) ?$ | (6,5),(4,9),(2,3) | 31 | 13 |
math | 9.1. Petya wrote 10 integers on the board (not necessarily distinct).
Then he calculated the pairwise products (that is, he multiplied each of the written numbers by each other). Among them, there were exactly 15 negative products. How many zeros were written on the board? | 2 | 62 | 1 |
math | 11. (20 points) Given
function
$$
f(x)=x^{2}+|x+a-1|+(a+1)^{2}
$$
the minimum value is greater than 5. Find the range of $a$. | a \in\left(-\infty, \frac{-1-\sqrt{14}}{2}\right) \cup\left(\frac{\sqrt{6}}{2},+\infty\right) | 54 | 45 |
math | 4. The last two digits of the integer $\left[\frac{10^{93}}{10^{31}+3}\right]$ are $\qquad$ (write the tens digit first, followed by the units digit; where $[x]$ denotes the greatest integer not exceeding $x$). | 8 | 65 | 1 |
math | (9) Let $[x]$ denote the greatest integer not exceeding $x$, then $\left[\log _{2} 1\right]+\left[\log _{2} 2\right]+\left[\log _{2} 3\right]+\cdots$ $+\left[\log _{2} 500\right]=$ $\qquad$ . | 3498 | 79 | 4 |
math | Let $\mathbb{N}$ be the set of all positive integers. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that the number $(f(m)+n)(m+f(n))$ is a square for all $m, n \in \mathbb{N}$. Answer. All functions of the form $f(n)=n+c$, where $c \in \mathbb{N} \cup\{0\}$. | f(n) = n + c | 102 | 7 |
math | Solve the following system of equations:
$$
\begin{aligned}
& x y=500 \\
& x^{\lg y}=25
\end{aligned}
$$ | x_{1}=y_{2}=100x_{2}=y_{1}=5 | 39 | 20 |
math | The function $g$, with domain and real numbers, fulfills the following:
$\bullet$ $g (x) \le x$, for all real $x$
$\bullet$ $g (x + y) \le g (x) + g (y)$ for all real $x,y$
Find $g (1990)$. | 1990 | 74 | 4 |
math | G2.3 If $n \neq 0$ and $s=\left(\frac{20}{2^{2 n+4}+2^{2 n+2}}\right)^{\frac{1}{n}}$, find the value of $s$. | \frac{1}{4} | 57 | 7 |
math | 17. Choose 3 different numbers from 1 to 300, such that the sum of these 3 numbers is exactly divisible by 3. How many ways are there to do this? | 1485100 | 42 | 7 |
math | 4. Determine how many roots the equation has on the interval $[-\pi ; \pi]$
$$
\frac{2 \cos 4 x+1}{2 \cos x-\sqrt{3}}=\frac{2 \sin 4 x-\sqrt{3}}{2 \sin x-1}
$$
and specify these roots | -\frac{5\pi}{6},-\frac{\pi}{3},\frac{\pi}{2},\frac{2\pi}{3} | 72 | 32 |
math | One. (20 points) Let the function
$$
f(x)=\cos x \cdot \cos (x-\theta)-\frac{1}{2} \cos \theta
$$
where, $x \in \mathbf{R}, 0<\theta<\pi$. It is known that when $x=\frac{\pi}{3}$, $f(x)$ achieves its maximum value.
(1) Find the value of $\theta$;
(2) Let $g(x)=2 f\left(\frac{3}{2} x\right)$, find the minimum value of ... | -\frac{1}{2} | 149 | 7 |
math | ## 156. Math Puzzle $5 / 78$
A freshly charged moped battery ( $6 \mathrm{~V} / 4.5 \mathrm{Ah}$ ) is connected to a lamp with the specifications $6 \mathrm{~V}$ and $0.6 \mathrm{~W}$.
How long will the lamp light up if there are no other consumers?
Hint: Ah is the abbreviation for ampere-hour and indicates the char... | 45\mathrm{~} | 105 | 7 |
math | G4.2 If $n$ is an integer, and the unit and tens digits of $n^{2}$ are $u$ and 7, respectively, determine the value of $u$.
If $n$ is an integer, and the unit and tens digits of $n^{2}$ are $u$ and 7, respectively, determine the value of $u$. | 6 | 79 | 1 |
math | Example 3. Solve the Cauchy problem:
$$
\begin{aligned}
x(x-1) y^{\prime}+y & =x^{2}(2 x-1) \\
\left.y\right|_{x=2} & =4
\end{aligned}
$$ | x^{2} | 63 | 4 |
math | 18. There are several students, forming a rectangular array that is exactly eight columns wide. If 120 more students are added to or 120 students are removed from the array, a square array can be formed. How many students are there in the original rectangular array? | 904or136 | 59 | 7 |
math | A jar contains red, blue, and yellow candies. There are $14\%$ more yellow candies than blue candies, and $14\%$ fewer red candies than blue candies. Find the percent of candies in the jar that are yellow. | 38\% | 52 | 4 |
math | 4. Arrange the four numbers $1, 2, 3, 4$ to form a four-digit number, such that this number is a multiple of 11. Then the number of such four-digit numbers is $\qquad$.
| 8 | 51 | 1 |
math | Problem 2. Whenever the owner's wish is granted, the length of the rectangular magic carpet decreases by $\frac{1}{2}$ of its current length, and the width decreases by $\frac{1}{3}$ of its current width. After three granted wishes, the carpet had an area of $18 \mathrm{dm}^{2}$. The initial width of the carpet was $1.... | 4.86\mathrm{~}^{2} | 107 | 12 |
math | ## Problem Statement
Find the coordinates of point $A$, which is equidistant from points $B$ and $C$.
$A(0 ; 0 ; z)$
$B(10 ; 0 ;-2)$
$C(9 ;-2 ; 1)$ | A(0;0;-3) | 60 | 8 |
math | $2 \cdot 4$ For the set $\{1,2,3, \cdots, n\}$ and each of its non-empty subsets, we define the "alternating sum" as follows: arrange the numbers in the subset in decreasing order, then alternately add and subtract the numbers starting from the largest (for example, the alternating sum of $\{1,2,4,6,9\}$ is $9-6+4-2+1=... | 448 | 130 | 3 |
math | We drew all the diagonals in three polygons with different numbers of sides. The second polygon has 3 more sides and 3 times as many diagonals as the first one. The third one has 7 times as many diagonals as the second one. How many more sides does the third polygon have compared to the second one? | 12 | 67 | 2 |
math | XVII OM - I - Problem 1
Present the polynomial $ x^5 + x + 1 $ as a product of two polynomials of lower degree with integer coefficients. | x^5+x+1=(x^2+x+1)(x^3-x^2+1) | 38 | 23 |
math | $5 \cdot 2$ If $n$ is a natural number less than 50, find all values of $n$ such that the values of the algebraic expressions $4n+5$ and $7n+6$ have a common divisor greater than 1.
(Tianjin, China Junior High School Math Competition, 1991) | 7,18,29,40 | 76 | 10 |
math | There are 6 locked suitcases and 6 keys to them. However, it is unknown which key fits which suitcase. What is the minimum number of attempts needed to definitely open all the suitcases? And how many attempts would be needed if there were not 6, but 10 suitcases and keys?
# | 15 | 65 | 2 |
math | 400. Milk Delivery. One morning, a milkman was transporting two 80-liter barrels of milk to his shop when he encountered two women who begged him to sell them 2 liters of milk each right away. Mrs. Green had a 5-liter jug, and Mrs. Brown had a 4-liter jug, while the milkman had nothing to measure the milk with.
How di... | 80&76&2&2 | 159 | 9 |
math | 2. The sequence $\left(a_{n}\right)$ is defined by the following relations: $a_{1}=1, a_{2}=3, a_{n}=a_{n-1}-a_{n-2}+n$ (for $n \geqslant 3$). Find $a_{1000}$. | 1002 | 74 | 4 |
math | ## Task 2 - 140812
Determine all ordered pairs $(x, y)$ of natural numbers $x, y$ for which the equation $13 x+5 y=82$ holds! | (4,6) | 48 | 5 |
math | \section*{Problem 6B - 121236B}
If \(n\) is a natural number greater than 1, then on a line segment \(A B\), points \(P_{1}, P_{2}, P_{3}, \ldots, P_{2 n-1}\) are placed in this order such that they divide the line segment \(A B\) into \(2 n\) equal parts.
a) Give (as a function of \(n\)) the probability that two poi... | \frac{1}{4} | 391 | 7 |
math | 3. Suppose $A B C D$ is a rectangle whose diagonals meet at $E$. The perimeter of triangle $A B E$ is $10 \pi$ and the perimeter of triangle $A D E$ is $n$. Compute the number of possible integer values of $n$. | 47 | 61 | 2 |
math | 346. Solve the equation $x^{2}+p x+q=0$ by the substitution $\boldsymbol{x}=\boldsymbol{y}+\boldsymbol{z}$ and defining $z$ (taking advantage of its arbitrariness) so as to obtain a pure quadratic equation (i.e., one not containing the first power of the unknown). | -\frac{p}{2}\\sqrt{\frac{p^{2}}{4}-q} | 76 | 20 |
math | 18. If $A C$ and $C E$ are two diagonals of a regular hexagon $A B C D E F$, and points $M$ and $N$ internally divide $A C$ and $C E$ such that $A M: A C = C N: C E = r$, if $B$, $M$, and $N$ are collinear, find $r$. | \frac{1}{\sqrt{3}} | 86 | 10 |
math | 6. Given a regular tetrahedron $ABCD$ with edge length 1, $M$ is the midpoint of $AC$, and $P$ lies on the segment $DM$. Then the minimum value of $AP + BP$ is $\qquad$ | \sqrt{1+\frac{\sqrt{6}}{3}} | 55 | 14 |
math | For example, the rules of a "level-passing game" stipulate: In the $n$-th level, a die must be rolled $n$ times. If the sum of the points that appear in these $n$ rolls is greater than $2^n$, then the level is passed. Questions:
( I ) What is the maximum number of levels a person can pass in this game?
( II ) What is t... | \frac{100}{243} | 162 | 11 |
math | 25. A pentagon $A B C D E$ is inscribed in a circle of radius 10 such that $B C$ is parallel to $A D$ and $A D$ intersects $C E$ at $M$. The tangents to this circle at $B$ and $E$ meet the extension of $D A$ at a common point $P$. Suppose $P B=P E=24$ and $\angle B P D=30^{\circ}$. Find $B M$. | 13 | 109 | 2 |
math | 6. If $0^{\circ}<\alpha<30^{\circ}$, and $\sin ^{6} \alpha+\cos ^{6} \alpha=\frac{7}{12}$, then $1998 \cos \alpha=$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 333\sqrt{30} | 82 | 9 |
math | The rodent control task force went into the woods one day and caught $200$ rabbits and $18$ squirrels. The next day they went into the woods and caught $3$ fewer rabbits and two more squirrels than the day before. Each day they went into the woods and caught $3$ fewer rabbits and two more squirrels than the day before.... | 5491 | 104 | 4 |
math | 721. Find the common measure of two segments of length $1 / 5$ m and $1 / 3$ m. | \frac{1}{15} | 29 | 8 |
math | One container of paint is exactly enough to cover the inside of an old rectangle which is three times as long as it is wide. If we make a new rectangle by shortening the old rectangle by $18$ feet and widening it by $8$ feet as shown below, one container of paint is also exactly enough to cover the inside of the new r... | 172 | 259 | 3 |
math | 5. The line $y=k x-2$ intersects the parabola $y^{2}=8 x$ at points $A, B$. If the x-coordinate of the midpoint of segment $A B$ is 2, then the length of segment $A B$ $|A B|=$ $\qquad$ | 2\sqrt{15} | 67 | 7 |
math | Somebody placed digits $1,2,3, \ldots , 9$ around the circumference of a circle in an arbitrary order. Reading clockwise three consecutive digits you get a $3$-digit whole number. There are nine such $3$-digit numbers altogether. Find their sum. | 4995 | 61 | 4 |
math | 13. Given the sequence $\left\{a_{n}\right\}$ satisfies:
$$
a_{1}=a, a_{n+1}=\frac{5 a_{n}-8}{a_{n}-1}\left(n \in \mathbf{Z}_{+}\right) .
$$
(1) If $a=3$, prove that $\left\{\frac{a_{n}-2}{a_{n}-4}\right\}$ is a geometric sequence, and find the general term formula of the sequence $\left\{a_{n}\right\}$;
(2) If for any... | a \in (3, +\infty) | 152 | 11 |
math | Question 82, Given vectors $\overrightarrow{\mathrm{OA}}$ and $\overrightarrow{\mathrm{OB}}$ have an angle of $120^{\circ}$ between them, $|\overrightarrow{\mathrm{OA}}|=2|\overrightarrow{\mathrm{OB}}|=6$. A line segment $\mathrm{PQ}$ of length $2|\overrightarrow{\mathrm{AB}}|$ is drawn through 0, and $\mathrm{P}, \mat... | -9 | 167 | 2 |
math | Problem 2. Does there exist a fraction equal to $\frac{7}{13}$, the difference between the denominator and the numerator of which is 24? | \frac{28}{52} | 35 | 9 |
math | 10. Let $x_{1}, x_{2}, x_{3}$ be non-negative real numbers, satisfying $x_{1}+x_{2}+x_{3}=1$, find the minimum and maximum values of $\left(x_{1}+3 x_{2}+5 x_{3}\right)\left(x_{1}+\frac{x_{2}}{3}+\frac{x_{3}}{5}\right)$. | \frac{9}{5} | 94 | 7 |
math | 6.156. $20\left(\frac{x-2}{x+1}\right)^{2}-5\left(\frac{x+2}{x-1}\right)^{2}+48 \frac{x^{2}-4}{x^{2}-1}=0$. | x_{1}=\frac{2}{3},x_{2}=3 | 63 | 16 |
math | \section*{Problem 6A - 141236A}
A measurement complex \(M\) integrated into an industrial process transmits to a transmission unit \(A_{1}\) exactly one of the two signals \(S_{1}\) or \(S_{2}\), which is then transmitted from \(A_{1}\) to a transmission unit \(A_{2}\), from \(A_{2}\) to a transmission unit \(A_{3}\),... | 0.970596 | 484 | 8 |
math | 9. Given that a line passing through the focus $F$ of the parabola $y^{2}=4 x$ intersects the parabola at points $M$ and $N$, and $E(m, 0)$ is a point on the $x$-axis. The extensions of $M E$ and $N E$ intersect the parabola at points $P$ and $Q$ respectively. If the slopes $k_{1}$ and $k_{2}$ of $M N$ and $P Q$ satisf... | 3 | 134 | 1 |
math | 4.066. In a certain geometric progression containing $2 n$ positive terms, the product of the first term and the last is 1000. Find the sum of the decimal logarithms of all the terms of the progression. | 3n | 51 | 2 |
math | 6.4 Find the third term of an infinite geometric progression with a common ratio $|q|<1$, the sum of which is $\frac{8}{5}$, and the second term is $-\frac{1}{2}$. | \frac{1}{8} | 50 | 7 |
math | Example 2. Solve the system of equations $\left\{\begin{array}{l}\frac{1}{x}+\frac{1}{y+z}=\frac{1}{2}, \\ \frac{1}{y}+\frac{1}{z+x}=\frac{1}{3} \\ \frac{1}{z}+\frac{1}{x+y}=\frac{1}{4} .\end{array}\right.$ | x = \frac{23}{10}, y = \frac{23}{6}, z = \frac{23}{2} | 94 | 31 |
math | ## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow 1}\left(2 e^{x-1}-1\right)^{\frac{x}{x-1}}
$$ | e^2 | 45 | 3 |
math | Two distinct squares of the $8\times8$ chessboard $C$ are said to be adjacent if they have a vertex or side in common.
Also, $g$ is called a $C$-gap if for every numbering of the squares of $C$ with all the integers $1, 2, \ldots, 64$ there exist twoadjacent squares whose numbers differ by at least $g$. Determine the l... | 9 | 99 | 1 |
math | Katka thought of a five-digit natural number. On the first line of her notebook, she wrote the sum of the thought number and half of the thought number. On the second line, she wrote the sum of the thought number and one fifth of the thought number. On the third line, she wrote the sum of the thought number and one nin... | 11250 | 157 | 5 |
math | 6. For the arithmetic sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$, the sums of the first $n$ terms are $S_{n}$ and $T_{n}$, respectively. If for any positive integer $n$, we have
$$
\begin{array}{r}
\frac{S_{n}}{T_{n}}=\frac{5 n-3}{2 n+1}, \\
\text { then } \frac{a_{20}}{b_{7}}=
\end{array}
$$ | \frac{64}{9} | 125 | 8 |
math | 2.2. Gavriila found out that the front tires of the car last for 24000 km, while the rear tires last for 36000 km. Therefore, he decided to swap them at some point to maximize the distance the car can travel. Find this distance (in km). | 28800 | 67 | 5 |
math | Exercise 6. In 5 boxes are found respectively $402, 403, 404, 405$ and 406 stones. The only allowed operation is to take 4 stones from a pile that has at least 4 stones and to put one in each of the other piles. What is the maximum number of stones that can be in a single pile? | 2014 | 84 | 4 |
math | 14. Let the lines $l_{1}: y=\sqrt{3} x, l_{2}: y=-\sqrt{3} x$. Points $A$ and $B$ move on lines $l_{1}$ and $l_{2}$ respectively, and $\overrightarrow{O A} \cdot \overrightarrow{O B}=-2$.
(1) Find the locus of the midpoint $M$ of $A B$;
(2) Let point $P(-2,0)$ have a symmetric point $Q$ with respect to the line $A B$. ... | x^{2}-\frac{y^{2}}{3}=1 | 138 | 15 |
math | ## Task Condition
Find the derivative of the specified order.
$y=\left(4 x^{3}+5\right) e^{2 x+1}, y^{V}=?$ | 32(4x^{3}+30x^{2}+60x+35)e^{2x+1} | 40 | 29 |
math | What is the maximum possible area of a triangle if the sides $a, b, c$ satisfy the following inequalities:
$$
0<a \leqq 1 \leqq b \leqq 2 \leqq c \leqq 3
$$ | 1 | 53 | 1 |
math | 9. (16 points) In the $x O y$ plane, draw two lines $A_{1} B_{1}$ and $A_{2} B_{2}$ through a point $P$ on the $x$-axis. Points $A_{1}$, $B_{1}$, $A_{2}$, and $B_{2}$ are all on the parabola $\Gamma: y^{2}=x$. Let $A_{1} B_{2}$ and $A_{2} B_{1}$ intersect the $x$-axis at points $S$ and $T$, respectively. Compare $\over... | \overrightarrow{OS}\cdot\overrightarrow{OT}=|OP|^{2} | 158 | 19 |
math | 3. On Eeyore's Birthday, Winnie-the-Pooh, Piglet, and Owl came to visit. When Owl left, the average age in this company decreased by 2 years, and when Piglet left, the average age decreased by another 1 year. How many years older is Owl than Piglet?
Answer: Owl is 6 years older than Piglet. | 6 | 79 | 1 |
math | 1. The trousers are cheaper than the jacket, the jacket is cheaper than the coat, the coat is cheaper than the fur, and the fur is cheaper than the diamond necklace by the same percentage. By what percentage is the fur more expensive than the trousers, if the diamond necklace is 6.25 times more expensive than the coat? | 1462.5 | 69 | 6 |
math | 6. (5 points) Teacher Zhang leads the students of Class 6 (1) to plant trees, and the students can be evenly divided into 5 groups. It is known that each teacher and student plants the same number of trees, and they plant a total of 527 trees. Therefore, Class 6 (1) has $\qquad$ students. | 30 | 76 | 2 |
math | Given $a$, $b$, $x$ are positive integers, and $a \neq b$, $\frac{1}{x}=\frac{1}{a^{2}}+\frac{1}{b^{2}}$. Try to find the minimum value of $x$.
---
The above text translated into English, preserving the original text's line breaks and format, is as follows:
Given $a$, $b$, $x$ are positive integers, and $a \neq b$, ... | 20 | 139 | 2 |
math | 1. Find all $x$ for which $x^{2}-10[x]+9=0$, where $[x]$ is the integer part of $x$. | {1;\sqrt{61};\sqrt{71};9} | 35 | 16 |
math | 6. A line with slope $k$ is drawn through one of the foci of the ellipse $x^{2}+2 y^{2}=3$, intersecting the ellipse at points $A, B$. If $|A B|=2$, then $|k|=$ $\qquad$ . | \sqrt{1+\sqrt{3}} | 63 | 9 |
math | Five. (20 points) In a mountain bike race held in a city, two cyclists, A and B, start from point A to point B at the same time. Cyclist A runs $\frac{1}{3}$ of the time at speeds $v_{1}, v_{2}, v_{3}$ respectively; Cyclist B runs $\frac{1}{3}$ of the distance at speeds $v_{1}, v_{2}, v_{3}$ respectively. Who will reac... | t_{1} \leqslant t_{2} | 109 | 13 |
math | 57. The number $123456789(10)(11)(12)(13)(14)$ is written in the base-15 numeral system, i.e., this number is equal to
(14) $+(13) \cdot 15+(12) \cdot 15^{2}+(11) \cdot 15^{3}+\ldots+2 \cdot 15^{12}+15^{13}$. What remainder does it give when divided by 7? | 0 | 122 | 1 |
math | In the sequence
$$
1,4,7,10,13,16,19, \ldots
$$
each term is 3 less than the next term. Find the 1000th term of the sequence. | 2998 | 54 | 4 |
math | 1. If $3^{a}=4^{b}=6^{c}$, then $\frac{1}{a}+\frac{1}{2 b}-\frac{1}{c}=$ $\qquad$ | 0 | 45 | 1 |
math | 3.1. (12 points) The number
$$
\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\ldots+\frac{2017}{2018!}
$$
was written as an irreducible fraction with natural numerator and denominator. Find the last two digits of the numerator. | 99 | 79 | 2 |
math | 9. Find the positive constant $c_{0}$ such that the series
$$
\sum_{n=0}^{\infty} \frac{n!}{(c n)^{n}}
$$
converges for $c>c_{0}$ and diverges for $0<c<c_{0}$. | \frac{1}{e} | 66 | 7 |
math | Example 8 Let $a, b, c \in \mathbf{R}_{+}$, and $abc + a + c = b$. Find the maximum value of
$$
p=\frac{2}{a^{2}+1}-\frac{2}{b^{2}+1}+\frac{3}{c^{2}+1}
$$ | \frac{10}{3} | 77 | 8 |
math | 5. a) How many six-digit numbers can be written using the digits 1, 2, 3, 4, and 5, where not every digit has to be used?
b) How many six-digit numbers can be written using the digits 1, 2, 3, 4, and 5, such that each digit is used at least once?
Each task is scored out of 10 points.
The use of a pocket calculator o... | 15625 | 104 | 5 |
math | 13. (20 points) Let the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ pass through the point $P\left(\frac{\sqrt{6}}{2}, \frac{1}{2}\right)$, with an eccentricity of $\frac{\sqrt{2}}{2}$, and let the moving point $M(2, t)$ $(t>0)$.
(1) Find the standard equation of the ellipse.
(2) Find the equation of the... | \sqrt{2} | 215 | 5 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.