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 | Consider the $4\times4$ array of $16$ dots, shown below.
[asy]
size(2cm);
dot((0,0));
dot((1,0));
dot((2,0));
dot((3,0));
dot((0,1));
dot((1,1));
dot((2,1));
dot((3,1));
dot((0,2));
dot((1,2));
dot((2,2));
dot((3,2));
dot((0,3));
dot((1,3));
dot((2,3));
dot((3,3));
[/asy]
Counting the number of squares whose vertices... | 4 | 233 | 1 |
math | Points $M$, $N$, $P$ are selected on sides $\overline{AB}$, $\overline{AC}$, $\overline{BC}$, respectively, of triangle $ABC$. Find the area of triangle $MNP$ given that $AM=MB=BP=15$ and $AN=NC=CP=25$.
[i]Proposed by Evan Chen[/i] | 150 | 86 | 3 |
math | Let $ABC$ be a triangle and $I$ its incenter. Suppose $AI=\sqrt{2}$, $BI=\sqrt{5}$, $CI=\sqrt{10}$ and the inradius is $1$. Let $A'$ be the reflection of $I$ across $BC$, $B'$ the reflection across $AC$, and $C'$ the reflection across $AB$. Compute the area of triangle $A'B'C'$. | \frac{24}{5} | 93 | 8 |
math | Three. (20 points) Let the moving point $P$ be on the line $l_{1}: y=x-4$. Draw two tangents $PA$ and $PB$ from $P$ to the circle $\odot C: x^{2}+y^{2}=1$, where $A$ and $B$ are the points of tangency. Find the equation of the trajectory of the midpoint $M$ of segment $AB$. | x^{2}+y^{2}-\frac{x}{4}+\frac{y}{4}=0 | 94 | 23 |
math | Example 2 Given the sets
$$
\begin{array}{l}
M=\{(x, y) \mid x(x-1) \leqslant y(1-y)\}, \\
N=\left\{(x, y) \mid x^{2}+y^{2} \leqslant k\right\} .
\end{array}
$$
If $M \subset N$, then the minimum value of $k$ is $\qquad$ .
(2007, Shanghai Jiao Tong University Independent Admission Examination) | 2 | 116 | 1 |
math | There is a lamp on each cell of a $2017 \times 2017$ board. Each lamp is either on or off. A lamp is called [i]bad[/i] if it has an even number of neighbours that are on. What is the smallest possible number of bad lamps on such a board?
(Two lamps are neighbours if their respective cells share a side.) | 1 | 82 | 1 |
math | 20. [10] Triangle $\triangle A B C$ has $A B=21, B C=55$, and $C A=56$. There are two points $P$ in the plane of $\triangle A B C$ for which $\angle B A P=\angle C A P$ and $\angle B P C=90^{\circ}$. Find the distance between them. | \frac{5}{2}\sqrt{409} | 86 | 13 |
math | 1.1. Weights with 1, 2, 3, 4, 8, 16 grams were divided into two piles of equal weight. $B$ in the first one there are two weights, in the second - four weights. Which two weights are in the first pile? | 16+1 | 63 | 4 |
math | 28 Let $\alpha$ and $\beta$ be the roots of $x^{2}-4 x+c=0$, where $c$ is a real number. If $-\alpha$ is a root of $x^{2}+4 x-c=0$, find the value of $\alpha \beta$. | 0 | 64 | 1 |
math | In a small Scottish town stood a school where exactly 1000 students studied. Each of them had a locker for their clothes - a total of 1000 lockers, numbered from 1 to 1000. And in this school lived ghosts - exactly 1000 ghosts. Every student, leaving the school, locked their locker, and at night the ghosts began to pla... | 31 | 256 | 2 |
math | We trace $n$ lines in the plane, no two parallel, no three concurrent. Into how many parts is the plane divided? | u_{n}=\frac{n(n+1)}{2}+1 | 27 | 16 |
math | Let $a, b, c, d \in \mathbb{R}$ such that $a+3 b+5 c+7 d=14$. Find the minimum possible value of $a^{2}+b^{2}+c^{2}+d^{2}$. | \frac{7}{3} | 61 | 7 |
math | 4. Alana, Beatrix, Celine, and Deanna played 6 games of tennis together. In each game, the four of them split into two teams of two and one of the teams won the game. If Alana was on the winning team for 5 games, Beatrix for 2 games, and Celine for 1 game, for how many games was Deanna on the winning team? | 4 | 85 | 1 |
math | 10. The equation $x^{2}+a|x|+a^{2}-3=0$ $(a \in \mathbf{R})$ has a unique real solution. Then $a=$ | \sqrt{3} | 44 | 5 |
math | Two circles $ C_1,C_2$ with different radii are given in the plane, they touch each other externally at $ T$. Consider any points $ A\in C_1$ and $ B\in C_2$, both different from $ T$, such that $ \angle ATB \equal{} 90^{\circ}$.
(a) Show that all such lines $ AB$ are concurrent.
(b) Find the locus of midpoints of ... | O_1O_2 | 102 | 7 |
math | 5. The minimum value of the function $y=\sqrt{x^{2}-2 x+5}+\sqrt{x^{2}-4 x+13}$ is $\qquad$ | \sqrt{26} | 38 | 6 |
math | Find all functions $ f: \mathbb{R} \to \mathbb{R}$ satisfying
\[ f\left(\frac {x \plus{} y}{x \minus{} y}\right) \equal{} \frac {f\left(x\right) \plus{} f\left(y\right)}{f\left(x\right) \minus{} f\left(y\right)}
\]
for all $ x \neq y$. | f(x) = x | 95 | 6 |
math | A game is played with four integers in the following way: add three of these numbers, divide this sum by 3, and add the result to the fourth number. There are four ways to do this game, yielding the following results: 17, 21, 23, and 29. What is the largest of the four numbers? | 21 | 74 | 2 |
math | ## Task 11/73
A train passes a kilometer post with a two-digit kilometer number. After the time $\Delta t_{1}$, it passes another kilometer post with the same digits but in reversed order.
Finally, after the additional time $\Delta t_{2}=\Delta t_{2}$, it encounters a third kilometer post, whose number is the same as... | 16,61,106 | 149 | 9 |
math | 1. If the geometric sequence $\left\{a_{n}\right\}$ satisfies $a_{1}-a_{2}=3, a_{1}-a_{3}=2$, then the common ratio of $\left\{a_{n}\right\}$ is $\qquad$ | -\frac{1}{3} | 60 | 7 |
math | 8. A school's mathematics extracurricular activity group designed a tree planting plan on a coordinate paper for a certain desert as follows: The $k$-th tree is planted at point $P_{k}\left(x_{k}, y_{k}\right)$, where $x_{1}=1, y_{1}=1$, and when $k \geqslant 2$,
\[
\left\{
\begin{array}{l}
x_{k}=x_{k-1}+1-5\left[\fra... | (3,402) | 240 | 7 |
math | 899. Check that the given expression is a total differential of the function $u(x, y)$, and find $u$:
1) $\left.\left(2 x-3 y^{2}+1\right) d x+(2-6 x y) d y ; \quad 2\right)\left(e^{x y}+5\right)(x d y+y d x)$;
2) $(1-\sin 2 x) d y-(3+2 y \cos 2 x) d x$. | y-3x-y\sin2x+C | 113 | 10 |
math | 13.022. On the mathematics entrance exam, $15 \%$ of the applicants did not solve a single problem, 144 people solved problems with errors, and the number of those who solved all problems correctly is to the number of those who did not solve any at all as 5:3. How many people took the mathematics exam that day | 240 | 76 | 3 |
math | 7. Given real numbers $a_{1}, a_{2}, \cdots, a_{18}$ satisfy:
$$
a_{1}=0,\left|a_{k+1}-a_{k}\right|=1(k=1,2, \cdots, 17) \text {. }
$$
Then the probability that $a_{18}=13$ is $\qquad$ | \frac{17}{16384} | 86 | 12 |
math | 17. Let $x, y, z$ be integers, and
$$
x+y+z=3, x^{3}+y^{3}+z^{3}=3 \text {. }
$$
Then $x^{2}+y^{2}+z^{2}=$ $\qquad$ . | 3 \text{ or } 57 | 67 | 9 |
math | 6. Given that $a$ is a real number, and for any $k \in$ $[-1,1]$, when $x \in(0,6]$, we have
$$
6 \ln x+x^{2}-8 x+a \leqslant k x \text {. }
$$
Then the maximum value of $a$ is $\qquad$ . | 6-6\ln6 | 81 | 6 |
math | Find the sum of $$\frac{\sigma(n) \cdot d(n)}{ \phi (n)}$$ over all positive $n$ that divide $ 60$.
Note: The function $d(i)$ outputs the number of divisors of $i$, $\sigma (i)$ outputs the sum of the factors of $i$, and $\phi (i)$ outputs the number of positive integers less than or equal to $i$ that are relatively... | 350 | 100 | 3 |
math | Problem 3. Each student's mentor gave them 2 apples, and 19 apples remained in the basket. How many students and how many apples are there, if their total sum is 100? | 27 | 44 | 2 |
math | Example 25 Let $x, y, z \in(0,1)$, satisfying:
$$\sqrt{\frac{1-x}{y z}}+\sqrt{\frac{1-y}{z x}}+\sqrt{\frac{1-z}{x y}}=2,$$
Find the maximum value of $x y z$. | \frac{27}{64} | 69 | 9 |
math | 1 In an exam, there are 30 multiple-choice questions. Correct answers earn 5 points each, incorrect answers earn 0 points, and unanswered questions earn 1 point each. If person A scores more than 80 points, and tells B the score, B can deduce how many questions A answered correctly. If A's score is slightly lower but s... | 119 | 110 | 3 |
math | Let’s call a positive integer [i]interesting[/i] if it is a product of two (distinct or equal) prime numbers. What is the greatest number of consecutive positive integers all of which are interesting? | 3 | 43 | 1 |
math | Example 5 Let $f(x)$ be a function defined on $\mathbf{R}$, for any $x \in \mathbf{R}$, there are $\hat{j}(x+3) \leqslant f(x)+3$ and $f(x+2) \geqslant f(x)+2$, let $g(x)=f(x)-x$.
(1) Prove that $g(x)$ is a periodic function; (2) If $f(994)=992$, find $f(2008)$.
Translate the above text into English, please keep the o... | 2006 | 146 | 4 |
math | ## Task 1 - 110731
Determine all prime numbers $p$ that simultaneously satisfy the following conditions:
(1) $\mathrm{p}<100$.
(2) $p$ leaves a remainder of 2 when divided by both 3 and 5.
(3) $p$ leaves a remainder of 1 when divided by 4. | 17 | 81 | 2 |
math | Example 4 Let real numbers $a, b$ satisfy
$$
3 a^{2}-10 a b+8 b^{2}+5 a-10 b=0 \text {. }
$$
Find the minimum value of $u=9 a^{2}+72 b+2$. | -34 | 65 | 3 |
math | A triangle has sides of length $48$, $55$, and $73$. A square is inscribed in the triangle such that one side of the square lies on the longest side of the triangle, and the two vertices not on that side of the square touch the other two sides of the triangle. If $c$ and $d$ are relatively prime positive integers suc... | 200689 | 102 | 6 |
math | 1. Find the range of real number $a$ such that the inequality
$$
\sin 2 \theta-(2 \sqrt{2}+\sqrt{2} a) \cdot \sin \left(\theta+\frac{\pi}{4}\right)-\frac{2 \sqrt{2}}{\cos \left(\theta-\frac{\pi}{4}\right)}>-3-2 a \text {, }
$$
holds for all $\theta \in\left[0, \frac{\pi}{2}\right]$. | >3 | 113 | 2 |
math | An express train overtakes a freight train. The speed of the express is as many times greater than that of the freight train as the time it takes for them to pass each other side by side is greater than the time it would take if they were passing each other head-on. What is this ratio? | 1+\sqrt{2} | 61 | 6 |
math | 3. In the Cartesian coordinate system $x O y$, the equation of the ellipse $C$ is $\frac{x^{2}}{9}+\frac{y^{2}}{10}=1, F$ is the upper focus of $C$, $A$ is the right vertex of $C$, and $P$ is a moving point on $C$ located in the first quadrant. Then the maximum value of the area of quadrilateral $O A P F$ is $\qquad$. | \frac{3}{2}\sqrt{11} | 103 | 12 |
math | 2. A natural number is called " $p$ lucky" if it has $p$ digits and is divisible by $p$.
a) What is the smallest "13 lucky" number? (Justify your answer)
b) What is the largest "13 lucky" number? (Justify your answer) | 9999999999990 | 66 | 13 |
math | Problem 7.4. Consider seven-digit natural numbers, in the decimal representation of which each of the digits $1,2,3,4,5,6,7$ appears exactly once.
(a) (1 point) How many of them have the digits from the first to the sixth in ascending order, and from the sixth to the seventh in descending order?
(b) (3 points) How ma... | 15 | 109 | 2 |
math | 3. Some integers, when divided by $\frac{5}{7}, \frac{7}{9}, \frac{9}{11}, \frac{11}{13}$ respectively, yield quotients that, when expressed as mixed numbers, have fractional parts of $\frac{2}{5}, \frac{2}{7}, \frac{2}{9}, \frac{2}{11}$ respectively. The smallest integer greater than 1 that satisfies these conditions ... | 3466 | 104 | 4 |
math | 6. Find a two-digit number if it is known that the sum of its digits is 13, and the difference between the sought number and the number written with the same digits but in reverse order is a two-digit number with 7 units. | 85 | 51 | 2 |
math | ## Task 2 - 330522
Rolf is looking for four-digit numbers in which no two digits are the same. The difference between the tens and the hundreds digit should be 3, and the difference between the hundreds and the thousands digit should be 4.
When calculating these differences, the order of the two digits involved shoul... | 112 | 103 | 3 |
math | Pista forgot her friend's phone number. She remembers that the first digit is 7, the fifth is 2. She knows that the number is six digits long, odd, and when divided by 3, 4, 7, 9, 11, and 13, it gives the same remainder. What is the phone number? | 720721 | 74 | 6 |
math | 3. A large cube is formed by $8 \times 8 \times 8$ unit cubes. How many lines can pass through the centers of 8 unit cubes? | \frac{1}{2}\left(10^{3}-8^{3}\right) | 36 | 20 |
math | 2. It is known that $\frac{\cos x-\sin x}{\sin y}=\frac{2 \sqrt{2}}{5} \tan \frac{x+y}{2}$ and $\frac{\sin x+\cos x}{\cos y}=-\frac{5}{\sqrt{2}} \cot \frac{x+y}{2}$. Find all possible values of the expression $\tan(x+y)$, given that there are at least three. | -1,\frac{20}{21},-\frac{20}{21} | 96 | 20 |
math | 9. Vremyankin and Puteykin simultaneously set out from Utrenneye to Vechernoye. The first of them walked half of the time spent on the journey at a speed of $5 \mathrm{km} / \mathrm{h}$, and then at a speed of $4 \mathrm{km} /$ h. The second, however, walked the first half of the distance at a speed of 4 km/h, and then... | Vremyankin | 117 | 5 |
math | ## Task Condition
Approximately calculate using the differential.
$y=\sqrt{x^{3}}, x=0.98$ | 0.97 | 26 | 4 |
math | 22nd Australian 2001 Problem A1 Let L(n) denote the smallest positive integer divisible by all of 2, 3, ... , n. Find all prime numbers p, q such that q = p + 2 and L(q) > q L(p). Solution | 3,5 | 60 | 3 |
math | 1. If the equality holds
$$
\sqrt{17^{2}+17^{2}+17^{2}+\cdots+17^{2}+17^{2}+17^{2}}=17^{2}+17^{2}+17^{2}
$$
how many times does $17^{2}$ appear as an addend under the root? | 2601 | 89 | 4 |
math | In triangle $ABC$, $AB=125$, $AC=117$ and $BC=120$. The angle bisector of angle $A$ intersects $\overline{BC}$ at point $L$, and the angle bisector of angle $B$ intersects $\overline{AC}$ at point $K$. Let $M$ and $N$ be the feet of the perpendiculars from $C$ to $\overline{BK}$ and $\overline{AL}$, respectively. Find ... | 56 | 110 | 2 |
math | G1.3 A six-digit number 1234xy is divisible by both 8 and 9 . Given that $x+y=c$, find the value of $c$. | 8 | 38 | 1 |
math | 10. (5 points) There is a wooden stick 240 cm long. First, starting from the left end, a line is drawn every 7 cm, then starting from the right end, a line is drawn every 6 cm, and the stick is cut at the lines. Among the small sticks obtained, the number of 3 cm long sticks is $\qquad$.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻... | 12 | 103 | 2 |
math | 3. Two numbers x and y satisfy the equation $280 x^{2}-61 x y+3 y^{2}-13=0$ and are the fourth and ninth terms, respectively, of a decreasing arithmetic progression consisting of integers. Find the common difference of this progression. | -5 | 60 | 2 |
math | Given positive integers $a$, $b$, and $c$ satisfy $a^{2}+b^{2}-c^{2}=2018$.
Find the minimum values of $a+b-c$ and $a+b+c$. | 52 | 50 | 2 |
math | 2. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x, y \in \mathbb{R}$,
$$
f(x)+y-f(x) y=f(x+f(y))-f(x f(y))
$$ | f(x)=x | 60 | 4 |
math | 15. (15 points) Huanhuan, Lele, and Yangyang participated in the final of the Star of Hope competition, where 200 judges voted for them, with each judge casting one vote. If the ratio of votes Huanhuan and Lele received is 3:2, and the ratio of votes Lele and Yangyang received is 6:5, how many votes did Huanhuan, Lele,... | Huanhuan:90 | 101 | 7 |
math | 1532. Find the probability that a randomly taken two-digit number will be divisible by either 2, or 5, or both at the same time. | 0.6 | 34 | 3 |
math | 10、A, B, C are guessing a two-digit number
A says: It has an even number of factors, and it is greater than 50
B says: It is an odd number, and it is greater than 60
C says: It is an even number, and it is greater than 70
If each of them is only half right, then the number is | 64 | 84 | 2 |
math | C1
Let $n>0$ be an integer. We are given a balance and $n$ weights of weight $2^{0}, 2^{1}, \ldots, 2^{n-1}$. In a sequence of $n$ moves we place all weights on the balance. In the first move we choose a weight and put it on the left pan. In each of the following moves we choose one of the remaining weights and we add ... | f(n)=(2n-1)!!=1\cdot3\cdot5\cdot\ldots\cdot(2n-1) | 137 | 30 |
math | In the attached calculation, different letters represent different integers ranging from 1 to 9 .
If the letters $O$ and $J$ represent 4 and 6 respectively, find
\begin{tabular}{rrrrrrr}
& $G$ & $O$ & $L$ & $D$ & $E$ & $N$ \\
$\times$ & & & & & & \\
\hline & $D$ & $E$ & $N$ & $G$ & $O$ & $L$ \\
\hline & 1 & 4 & $L$ & ... | G=1,D=8,L=2,E=5,N=7 | 189 | 15 |
math | Five. (20 points) Let $a_{n}$ be the number of subsets of the set $\{1,2, \cdots, n\}$ $(n \geqslant 3)$ that have the following property: each subset contains at least 2 elements, and the difference (absolute value) between any 2 elements in each subset is greater than 1. Find $a_{10}$. | 133 | 88 | 3 |
math | Problem 5. Vasya and Petya live in the mountains and love to visit each other. When climbing up the mountain, they walk at a speed of 3 km/h, and when descending, they walk at a speed of 6 km/h (there are no horizontal road sections). Vasya calculated that it takes him 2 hours and 30 minutes to get to Petya's, and 3 ho... | 12 | 113 | 2 |
math | Example 2 When does $a$ satisfy the condition that the equation
$$
\left(a^{2}-1\right) x^{2}-6(3 a-1) x+72=0
$$
has roots of opposite signs and the absolute value of the negative root is larger? | \frac{1}{3}<a<1 | 63 | 10 |
math | Solve the following equation:
$$
\sin x + \cos x = \sqrt{\frac{3}{2}}.
$$ | x_{1}=\frac{\pi}{12}\2k\pi,\quadx_{2}=\frac{5\pi}{12}\2k\pi | 27 | 36 |
math | 337. Natural numbers from 1 to 1982 are arranged in some order. A computer scans pairs of adjacent numbers (the first and second, the second and third, etc.) from left to right up to the last pair and swaps the numbers in the scanned pair if the larger number is to the left. Then it scans all pairs from right to left f... | 100 | 131 | 3 |
math | ## Task 3 - 220913
a) Determine all real numbers $x$ for which the term $\frac{4 x-4}{2 x-3}$ is defined.
b) Among the numbers $x$ found in a), determine all those for which $0<\frac{4 x-4}{2 x-3}<1$ holds! | \frac{1}{2}<x<1 | 79 | 10 |
math | 9. Given $A, B \in$
$\left(0, \frac{\pi}{2}\right)$, and $\frac{\sin A}{\sin B}=\sin (A+B)$, find the maximum value of $\tan A$. | \frac{4}{3} | 52 | 7 |
math | Lennart and Eddy are playing a betting game. Lennart starts with $7$ dollars and Eddy starts with $3$ dollars. Each round, both Lennart and Eddy bet an amount equal to the amount of the player with the least money. For example, on the first round, both players bet $3$ dollars. A fair coin is then tossed. If it lands he... | \frac{3}{10} | 133 | 8 |
math | Example 8. Find the differential equation for which the function $y=C_{1} x+C_{2}$, depending on two arbitrary constants, is the general solution. | y^{\\}=0 | 35 | 5 |
math | Let $x_1,x_2, \cdots,x_n$ and $y_1,y_2, \cdots ,y_n$ be arbitrary real numbers satisfying $x_1^2+x_2^2+\cdots+x_n^2=y_1^2+y_2^2+\cdots+y_n^2=1$. Prove that
\[(x_1y_2-x_2y_1)^2 \le 2\left|1-\sum_{k=1}^n x_ky_k\right|\]
and find all cases of equality. | (x_1 y_2 - x_2 y_1)^2 \leq 2 \left| 1 - \sum_{k=1}^n x_k y_k \right| | 126 | 44 |
math | ## Problem Statement
Find the point of intersection of the line and the plane.
$$
\begin{aligned}
& \frac{x+2}{-1}=\frac{y-1}{1}=\frac{z+3}{2} \\
& x+2 y-z-2=0
\end{aligned}
$$ | (-3,2,-1) | 69 | 7 |
math | Problem 9. (12 points)
Andrey lives near the market, and during the summer holidays, he often helped one of the traders lay out fruits on the counter early in the morning. For this, the trader provided Andrey with a $10 \%$ discount on his favorite apples. But autumn came, and the price of apples increased by $10 \%$.... | 99 | 161 | 2 |
math | 1. Given three positive integers $x, y, z$ whose least common multiple is 300, and $\left\{\begin{array}{l}x+3 y-2 z=0, \\ 2 x^{2}-3 y^{2}+z^{2}=0\end{array}\right.$. Then the solution to the system of equations $(x, y, z)=$ $\qquad$ . | (20,60,100) | 90 | 11 |
math | 8. Cut a plastic pipe that is 374 centimeters long into several shorter pipes of 36 centimeters and 24 centimeters (neglecting processing loss), the remaining part of the pipe will be ( ) centimeters. | 2 | 51 | 1 |
math | Let $\{a_n\}$ be a sequence of integers satisfying the following conditions.
[list]
[*] $a_1=2021^{2021}$
[*] $0 \le a_k < k$ for all integers $k \ge 2$
[*] $a_1-a_2+a_3-a_4+ \cdots + (-1)^{k+1}a_k$ is multiple of $k$ for all positive integers $k$.
[/list]
Determine the $2021^{2022}$th term of the sequence $\{a_n\}$.... | 0 | 134 | 1 |
math | Let $ABC$ be an equilateral triangle with side length 16. Three circles of the same radius $r$ are tangent to each other in pairs, and each of these circles is tangent to two sides of the triangle. The radius $r$ can be written as $r=\sqrt{a}-b$ where $a$ and $b$ are integers. Determine $a+b$.
Soit $A B C$ un triangle... | 52 | 259 | 2 |
math | 9. (10 points) The Fifth Plenary Session of the 18th Central Committee of the Communist Party of China, which concluded on October 29, 2015, determined the policy allowing universal second children. Xiao Xiao's father, after seeing the news on that day, told Xiao Xiao: The total age of our family this year is 7 times y... | 11 | 130 | 2 |
math | Four. (50 points) Given that there are $n(n \geqslant 4)$ football teams participating in a round-robin tournament, each pair of teams plays one match. The winning team gets 3 points, the losing team gets 0 points, and in case of a draw, both teams get 1 point. After all the matches, it is found that the total scores o... | n-2 | 128 | 3 |
math | 12. The positive integer solutions $(x, y)$ of the equation $2 x^{2}-x y-3 x+y+2006=0$ are $\qquad$ pairs.
| 4 | 42 | 1 |
math | Let's find the largest number $A$ for which the following statement is true.
No matter how we choose seven real numbers between 1 and $A$, there will always be two of them for which the ratio $h$ satisfies $\frac{1}{2} \leqq h \leqq 2$. | 64 | 64 | 2 |
math | Problem 9.1. Find the largest five-digit number, the product of whose digits is 120. | 85311 | 24 | 5 |
math | 21. Suppose that a function $M(n)$, where $n$ is a positive integer, is defined by
$$
M(n)=\left\{\begin{array}{ll}
n-10 & \text { if } n>100 \\
M(M(n+11)) & \text { if } n \leq 100
\end{array}\right.
$$
How many solutions does the equation $M(n)=91$ have? | 101 | 102 | 3 |
math | 6. A tourist goes on a hike from $A$ to $B$ and back, and completes the entire journey in 3 hours and 41 minutes. The route from $A$ to $B$ first goes uphill, then on flat ground, and finally downhill. Over what distance does the road run on flat ground, if the tourist's speed is 4 km/h when climbing uphill, 5 km/h on ... | 4 | 111 | 1 |
math | # Task 10.2
In the decimal representation of the number ${ }^{1 / 7}$, the 2021st digit after the decimal point was crossed out (and the other digits were not changed). Did the number increase or decrease?
## Number of points 7
# | increased | 63 | 2 |
math | Convex pentagon $ABCDE$ has side lengths $AB=5$, $BC=CD=DE=6$, and $EA=7$. Moreover, the pentagon has an inscribed circle (a circle tangent to each side of the pentagon). Find the area of $ABCDE$. | 60 | 59 | 2 |
math | One, (20 points) If the vertex of the parabola $y=-x^{2}+2 a x+b$ moves on the line $m x-y-2 m+1=0$, and it has a common point with the parabola $y=x^{2}$, find the range of changes for $m$.
| m \leqslant 2-\sqrt{2} \text { or } m \geqslant 2+\sqrt{2} | 72 | 31 |
math | 1.1. For what least natural value of \( b \) does the equation
$$
x^{2}+b x+25=0
$$
have at least one root? | 10 | 41 | 2 |
math | I4.1 If each interior angle of a $n$-sided regular polygon is $140^{\circ}$, find $n$.
I4.2 If the solution of the inequality $2 x^{2}-n x+9<0$ is $k<x<b$, find $b$.
I4.3 If $c x^{3}-b x+x-1$ is divided by $x+1$, the remainder is -7 , find $c$.
I4.4 If $x+\frac{1}{x}=c$ and $x^{2}+\frac{1}{x^{2}}=d$, find $d$. | 9,3,8,62 | 141 | 8 |
math | Example 1 Let $x, y, z \in \mathbf{R}^{+}, \sqrt{x^{2}+y^{2}}+z=1$, try to find the maximum value of $x y+2 x z$.
| \frac{\sqrt{3}}{3} | 53 | 10 |
math | 739*. Can the number 456 be represented as the product of several natural numbers such that the sum of the squares of all these numbers is also equal to 456? | 456=1\cdot1\cdot1\cdot\ldots\cdot1\cdot2\cdot2\cdot2\cdot3\cdot19(74factors,equalto1) | 40 | 45 |
math | 8. Given that the parabola $P$ has the center of the ellipse $E$ as its focus, $P$ passes through the two foci of $E$, and $P$ intersects $E$ at exactly three points, then the eccentricity of the ellipse is $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and ou... | \frac{2\sqrt{5}}{5} | 87 | 12 |
math | fantasticbobob is proctoring a room for the SiSiEyMB with $841$ seats arranged in $29$ rows and $29$ columns. The contestants sit down, take part $1$ of the contest, go outside for a break, and come back to take part $2$ of the contest. fantasticbobob sits among the contestants during part $1$, also goes outside during... | 421 | 156 | 3 |
math | Solve the following system of equations:
$$
x^{\lg y}+y^{\lg \sqrt{x}}=110, \quad x y=1000
$$ | \begin{pmatrix}x_{1}=10,&y_{1}=100\\x_{2}=100,&y_{2}=10\end{pmatrix} | 41 | 41 |
math | Initially 211, try to find the range of real number $m$ that satisfies the existence of two real numbers $a, b$, such that $a \leqslant 0, b \neq 0$, and
$$
\frac{a-b+\sqrt{a b}}{m a+3 b+\sqrt{a b}}=\frac{1}{m} \text {. }
$$ | -3 \leqslant m < -5 + 2\sqrt{5} \text{ or } -5 + 2\sqrt{5} < m < 0 \text{ or } 0 < m < 1 | 88 | 51 |
math | 6.19 Solve the equation
$1+2 x+4 x^{2}+\ldots+(2 x)^{n}+\ldots=3.4-1.2 x$, given that $|x|<0.5$. | \frac{1}{3} | 53 | 7 |
math | 3. (3 points) Teacher Zhang saws a piece of wood into 9 small segments, each 4 meters long. If this piece of wood is sawed into segments that are 3 meters long, how many times in total will it need to be sawed? $\qquad$ | 11 | 60 | 2 |
math | 5. In an equilateral triangle $A B C$, points $A_{1}$ and $A_{2}$ are chosen on side $B C$ such that $B A_{1}=A_{1} A_{2}=A_{2} C$. On side $A C$, a point $B_{1}$ is chosen such that $A B_{1}: B_{1} C=1: 2$. Find the sum of the angles $\angle A A_{1} B_{1}+\angle A A_{2} B_{1}$. | 30 | 117 | 2 |
math | 3. Let $O(0,0), A(1,0), B(0,1)$, point $P$ is a moving point on line segment $AB$, $\overrightarrow{A P}=\lambda \overrightarrow{A B}$, if $\overrightarrow{O P} \cdot \overrightarrow{A B} \geq \overrightarrow{P A} \cdot \overrightarrow{P B}$, then the range of the real number $\lambda$ is | [1-\frac{\sqrt{2}}{2},1] | 102 | 14 |
math | How many $6$-tuples $(a, b, c, d, e, f)$ of natural numbers are there for which $a>b>c>d>e>f$ and $a+f=b+e=c+d=30$ are simultaneously true? | 364 | 53 | 3 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.