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 | Task 4. Let $n \geq 3$ be an integer and consider an $n \times n$ board, divided into $n^{2}$ unit squares. We have for every $m \geq 1$ arbitrarily many $1 \times m$ rectangles (type I) and arbitrarily many $m \times 1$ rectangles (type II) available. We cover the board with $N$ of these rectangles, which do not overl... | 2n-1 | 164 | 4 |
math | Task B-1.2. The sum of the digits of the natural number $x$ is $y$, and the sum of the digits of the number $y$ is $z$. Determine all numbers $x$ for which
$$
x+y+z=60
$$ | x\in{44,47,50} | 58 | 13 |
math | Find all prime numbers $p, q$ such that $p q$ divides $2^{p}+2^{q}$.
## - Exercise Solutions - | (2,2),(2,3),(3,2) | 32 | 13 |
math | Let $A, B, C$ be unique collinear points$ AB = BC =\frac13$. Let $P$ be a point that lies on the circle centered at $B$ with radius $\frac13$ and the circle centered at $C$ with radius $\frac13$ . Find the measure of angle $\angle PAC$ in degrees. | 30^\circ | 75 | 4 |
math | Karcsinak has 10 identical balls, among which 5 are red, 3 are white, and 2 are green, and he also has two boxes, one of which can hold 4 balls, and the other can hold 6 balls. In how many ways can he place the balls into the two boxes? (The arrangement of the balls within the boxes is not important.) | 11 | 81 | 2 |
math | ## Task 6
All 21 students in class 2b participated in a waste collection. 15 students collected waste paper and 18 students collected glass.
How many students collected both waste paper and glass? | 12 | 46 | 2 |
math | ## problem statement
Write the canonical equations of the line.
$6 x-5 y+3 z+8=0$
$6 x+5 y-4 z+4=0$ | \frac{x+1}{5}=\frac{y-\frac{2}{5}}{42}=\frac{z}{60} | 40 | 31 |
math | # Task 8.2
For a natural number $N$, all its divisors were listed, and then the sum of digits for each of these divisors was calculated. It turned out that among these sums, all numbers from 1 to 9 were found. Find the smallest value of $\mathrm{N}$.
## Number of points 7 | 288 | 72 | 3 |
math | Example 23 Given an ellipse with eccentricity $e=\frac{2}{5} \sqrt{5}$, passing through the point $(1,0)$ and tangent to the line $l: 2 x-y+3=0$ at point $P\left(-\frac{2}{3}\right.$, $\left.\frac{5}{3}\right)$, with the major axis parallel to the $y$-axis. Find the equation of this ellipse. | x^{2}+\frac{1}{5}y^{2}=1 | 100 | 16 |
math | ## Task 1 - 250821
Determine the twelve consecutive integers that have the property that the sum of the two largest of these numbers is equal to the sum of the remaining ten! | -3,-2,-1,0,1,2,3,4,5,6,7,8 | 43 | 24 |
math | $A$ and $B$ - although one lives $3 \mathrm{~km}$ farther from the city than the other - arrived in the city at the same time, with $A$ in a car and $B$ in a truck. The coachman and the driver picked them up along the way. Both $A$ and $B$ started walking from their homes at the same time, and both had covered half of ... | 13.5 | 175 | 4 |
math | The terms of a number sequence are:
$$
\begin{aligned}
& a_{0}=1, \quad a_{1}=1-\left(\frac{1}{2}\right)^{2} a_{0}, \quad a_{2}=1-\left(\frac{2}{3}\right)^{2} a_{1} \\
& a_{3}=1-\left(\frac{3}{4}\right)^{2} a_{2}, \ldots, \quad a_{n}=1-\left(\frac{n}{n+1}\right)^{2} a_{n-1}
\end{aligned}
$$
Write an expression for $a... | a_{n}=\frac{n+2}{2(n+1)},\quada_{0}a_{1}a_{2}\ldotsa_{n}=\frac{n+2}{2^{n+1}} | 193 | 47 |
math | Let $ABC$ be a triangle with $|AB|=|AC|=26$, $|BC|=20$. The altitudes of $\triangle ABC$ from $A$ and $B$ cut the opposite sides at $D$ and $E$, respectively. Calculate the radius of the circle passing through $D$ and tangent to $AC$ at $E$. | \frac{65}{12} | 76 | 9 |
math | If two points are picked randomly on the perimeter of a square, what is the probability that the distance between those points is less than the side length of the square? | \frac{1}{4} + \frac{\pi}{8} | 33 | 15 |
math | For all real numbers $r$, denote by $\{r\}$ the fractional part of $r$, i.e. the unique real number $s\in[0,1)$ such that $r-s$ is an integer. How many real numbers $x\in[1,2)$ satisfy the equation $\left\{x^{2018}\right\} = \left\{x^{2017}\right\}?$ | 2^{2017} | 94 | 7 |
math | 1.2. Inside a rectangular parallelepiped $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$, there are two spheres such that the first one touches all three faces containing $A$, and the second one touches all three faces containing $C^{\prime}$. Additionally, the spheres touch each other. The radii of the spheres a... | 41 | 131 | 2 |
math | 5. Find the number of such sequences: of length $n$, each term is $0$, $1$ or 2, and 0 is neither the preceding term of 2 nor the following term of 2. | \frac{1}{2}[(1+\sqrt{2})^{n+1}+(1-\sqrt{2})^{n+1}] | 46 | 31 |
math | 327. Find $\left(\cos ^{3} x\right)^{\prime}$. | -3\cos^{2}x\sinx | 21 | 11 |
math | The sequence of numbers $t_{1}, t_{2}, t_{3}, \ldots$ is defined by
$$
\left\{\begin{array}{l}
t_{1}=2 \\
t_{n+1}=\frac{t_{n}-1}{t_{n}+1}
\end{array}\right.
$$
Suggestion: Calculate the first five terms of the sequence.
for each positive integer $n$. Find $\mathrm{t}_{2011}$. | -\frac{1}{2} | 105 | 7 |
math | 3. (16 points) Mitya, Anton, Gosha, and Boris bought a lottery ticket for 20 rubles. Mitya paid $24\%$ of the ticket's cost, Anton - 3 rubles 70 kopecks, Gosha - 0.21 of the ticket's cost, and Boris contributed the remaining amount. The boys agreed to divide the winnings in proportion to their contributions. The ticket... | 365 | 115 | 3 |
math | 9. In the sequence $\left\{a_{n}\right\}$, $a_{1}=-1, a_{2}=1, a_{3}=-2$. If for all $n \in \mathbf{N}_{+}$, $a_{n} a_{n+1} a_{n+2} a_{n+3}=a_{n}+a_{n+1}+$ $a_{n+2}+a_{n+3}$, and $a_{n+1} a_{n+2} a_{n+3} \neq 1$, then the sum of the first 4321 terms $S_{4321}$ of the sequence is | -4321 | 154 | 5 |
math | 2. In the domain of integers, solve the system of equations
$$
\begin{aligned}
& x(y+z+1)=y^{2}+z^{2}-5, \\
& y(z+x+1)=z^{2}+x^{2}-5, \\
& z(x+y+1)=x^{2}+y^{2}-5 .
\end{aligned}
$$ | (-5,-5,-5),(-2,1,0) | 83 | 14 |
math | 8. In triangle $A B C$ with $\angle B=120^{\circ}$, the angle bisectors $A A_{1}, B B_{1}, C C_{1}$ are drawn. Segment $A_{1} B_{1}$ intersects the angle bisector $C C_{1}$ at point M. Find $\angle C B M$.
# | 30 | 79 | 2 |
math | A* Find all integer triples $(x, y, z)$ that satisfy $8^{x}+15^{y}=17^{z}$. | (x,y,z)=(2,2,2) | 32 | 10 |
math | 2. In a regular tetrahedron $ABCD$, let
$$
\overrightarrow{AE}=\frac{1}{4} \overrightarrow{AB}, \overrightarrow{CF}=\frac{1}{4} \overrightarrow{CD} \text {, }
$$
$\vec{U} \overrightarrow{DE}$ and $\overrightarrow{BF}$ form an angle $\theta$. Then $\cos \theta=$ | -\frac{4}{13} | 92 | 8 |
math | 10. $\mathrm{U}$ knows
$$
a+3 b \perp 7 a-5 b \text {, and } a-4 b \perp 7 a-2 \text {. }
$$
Then the angle between $a$ and $b$ is $\qquad$ . | \frac{\pi}{3} | 66 | 7 |
math | 9.2 Find all real numbers $x$ that satisfy the inequality
$$\sqrt{3-x}-\sqrt{x+1}>\frac{1}{2}$$ | -1 \leqslant x<1-\frac{\sqrt{31}}{8} | 36 | 21 |
math | The mumbo-jumbo tribe's language consists of $k$ different words. The words only contain two letters, $A$ and $O$. No word can be obtained from another by removing some letters from the end. What is the minimum number of letters contained in the complete dictionary of the language? | kH+(2^{H}-k) | 61 | 9 |
math | Example 2 The solution set of the inequality $(m-1) x<\sqrt{4 x-x^{2}}$ about $x$ is $\{x \mid 0<x<2\}$, find the value of the real number $m$.
| 2 | 55 | 1 |
math | 2. If $\sqrt{7 x^{2}+9 x+13}+\sqrt{7 x^{2}-5 x+13}=$ $7 x$, then $x=$ $\qquad$ | \frac{12}{7} | 45 | 8 |
math | 1. Let $x>1$, if
$$
\log _{2}\left(\log _{4} x\right)+\log _{4}\left(\log _{16} x\right)+\log _{16}\left(\log _{2} x\right)=0 \text {, }
$$
then $\log _{2}\left(\log _{16} x\right)+\log _{16}\left(\log _{4} x\right)+\log _{4}\left(\log _{2} x\right)=$
$\qquad$ | -\frac{1}{4} | 133 | 7 |
math | 355. Find the equation of the sphere with center at point $C(a ; b ; c)$ and radius $R$. | (x-)^{2}+(y-b)^{2}+(z-)^{2}=R^{2} | 27 | 24 |
math | 5. Compute
$$
\int_{0}^{1} \frac{d x}{\sqrt{x}+\sqrt[3]{x}}
$$ | 5-6\ln2 | 32 | 6 |
math | Example 7 Let $x_{1}, x_{2}, \cdots, x_{7}$ all be integers, and
$$
\begin{array}{l}
x_{1}+4 x_{2}+9 x_{3}+16 x_{4}+25 x_{5}+36 x_{6} \\
+49 x_{7}=1, \\
4 x_{1}+9 x_{2}+16 x_{3}+25 x_{4}+36 x_{5}+49 x_{6} \\
+64 x_{7}=12, \\
9 x_{1}+16 x_{2}+25 x_{3}+36 x_{4}+49 x_{5}+64 x_{6} \\
+81 x_{7}=123 .
\end{array}
$$
Find $1... | 334 | 256 | 3 |
math | ## Task B-1.5.
Ivo and Ana both drank lemonade at the cinema and watched a movie. Ivo took a medium size, and Ana a large one, which is $50\%$ larger than the medium. After both had drunk $\frac{3}{4}$ of their lemonade, Ana gave Ivo one third of what was left to her and an additional 0.5 dl. After the movie ended and... | 10 | 129 | 2 |
math | Example 4 For all $a, b, c \in \mathbf{R}^{+}$, find the minimum value of $f(a, b, c)=\frac{a}{\sqrt{a^{2}+8 b c}}+\frac{b}{\sqrt{b^{2}+8 a c}}+\frac{c}{\sqrt{c^{2}+8 a b}}$. | 1 | 88 | 1 |
math | Let's determine $m$ such that the expression
$$
x^{4}+2 x^{3}-23 x^{2}+12 x+m
$$
is identically equal to the product of the trinomials $x^{2}+a x+c$ and $x^{2}+b x+c$. Determine $a$, $b$, and $c$, as well as the values of $x$ for which the given polynomial is equal to 0. | x_{1}=2,\quadx_{2}=3,\quadx_{3}=-1,\quadx_{4}=-6 | 102 | 28 |
math | Let $n$ be a nonzero natural number, and $x_1, x_2,..., x_n$ positive real numbers that $ \frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}= n$. Find the minimum value of the expression $x_1 +\frac{x_2^2}{2}++\frac{x_3^3}{3}+...++\frac{x_n^n}{n}$. | 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} | 104 | 29 |
math | Three real numbers $x, y, z$ are such that $x^2 + 6y = -17, y^2 + 4z = 1$ and $z^2 + 2x = 2$. What is the value of $x^2 + y^2 + z^2$? | 14 | 69 | 2 |
math | Example 2: With $2009^{12}$ as one of the legs, and all three sides being integers, the number of different right-angled triangles (congruent triangles are considered the same) is $\qquad$.
(2009, International Mathematics Tournament of the Cities for Young Mathematicians) | 612 | 68 | 3 |
math | 12. (22 points) Find all pairs of positive integers $(m, n)$ that satisfy $1 \leqslant m^{n}-n^{m} \leqslant m n$. | {(,1),(2,5),(3,2)\mid\geqslant2} | 44 | 20 |
math | ## 1. Jaja
Baka Mara has four hens. The first hen lays one egg every day. The second hen lays one egg every other day. The third hen lays one egg every third day. The fourth hen lays one egg every fourth day. If on January 1, 2023, all four hens laid one egg, how many eggs in total will Baka Mara's hens lay throughout... | 762 | 108 | 3 |
math | 5. Given $f(x)=\left(x^{2}+3 x+2\right)^{\cos \pi x}$. Then the sum of all $n$ that satisfy the equation
$$
\left|\sum_{k=1}^{n} \log _{10} f(k)\right|=1
$$
is . $\qquad$ | 21 | 77 | 2 |
math | 33rd Putnam 1972 Problem B2 A particle moves in a straight line with monotonically decreasing acceleration. It starts from rest and has velocity v a distance d from the start. What is the maximum time it could have taken to travel the distance d? Solution | 2d/v | 58 | 3 |
math | 4. In the unit cube $A B C D-A^{\prime} B^{\prime} C^{\prime} D^{\prime}$, there is a point $M$ on the side face $A A^{\prime} B^{\prime} B$ such that the distances from $M$ to the lines $A B$ and $B^{\prime} C^{\prime}$ are equal. The minimum distance from a point on the trajectory of $M$ to $C^{\prime}$ is $\qquad$ | \frac{\sqrt{5}}{2} | 112 | 10 |
math | Question 161: In the Cartesian coordinate system, there is an ellipse with its two foci at $(9,20)$ and $(49,55)$, and the ellipse is tangent to the $x$-axis. Then the length of the major axis of the ellipse is $\qquad$.
| 85 | 66 | 2 |
math | 4- 123 Let $a, b, c$ be given positive real numbers, try to determine all positive real numbers $x, y, z$ that satisfy the system of equations
$$
\left\{\begin{array}{l}
x+y+z=a+b+c, \\
4 x y z-\left(a^{2} x+b^{2} y+c^{2} z\right)=a b c .
\end{array}\right.
$$ | \frac{b+}{2},\frac{+}{2},\frac{+b}{2} | 97 | 23 |
math | The workers in a factory produce widgets and whoosits. For each product, production time is constant and identical for all workers, but not necessarily equal for the two products. In one hour, $100$ workers can produce $300$ widgets and $200$ whoosits. In two hours, $60$ workers can produce $240$ widgets and $300$ whoo... | 450 | 119 | 3 |
math | In triangle $\vartriangle ABC$ with orthocenter $H$, the internal angle bisector of $\angle BAC$ intersects $\overline{BC}$ at $Y$ . Given that $AH = 4$, $AY = 6$, and the distance from $Y$ to $\overline{AC}$ is $\sqrt{15}$, compute $BC$. | 4\sqrt{35} | 77 | 7 |
math | $$
17 x^{19}-4 x^{17}-17 x^{15}+4=0
$$
all positive real solutions. | 1 | 34 | 1 |
math | 【Question 8】
Given that $A$ is a prime number less than 100, and $A+10, A-20, A+30, A+60, A+70$ are all prime numbers, then $A=$ $\qquad$.(List all possible numbers) | 37,43,79 | 68 | 8 |
math | Example 8 If $a, b, c$ are the length, width, and height of a rectangular prism, and $a+b-$ $c=1$, it is known that the length of the diagonal of the rectangular prism is 1, and $a>b$, try to find the range of values for the height $c$.
The above text is translated into English, please retain the original text's line ... | 0<c<\frac{1}{3} | 96 | 10 |
math | ## Task 2 - 150822
Determine all natural numbers $n \geq 1$ for which among the six numbers $n+1, n+2, n+3$, $n+4, n+5, n+6$ a pair can be found where the first number of the pair is a proper divisor of the second number of the pair!
State (for each such $n$) all such pairs! | (2;4),(2;6),(3;6),(3;6),(4;8),(4;8),(5;10) | 95 | 30 |
math | Let ABCD be a trapezoid with $AB \parallel CD, AB = 5, BC = 9, CD = 10,$ and $DA = 7$. Lines $BC$ and $DA$ intersect at point $E$. Let $M$ be the midpoint of $CD$, and let $N$ be the intersection of the circumcircles of $\triangle BMC$ and $\triangle DMA$ (other than $M$). If $EN^2 = \tfrac ab$ for relatively prime pos... | 90011 | 126 | 5 |
math | Solve the following equation in the set of integers:
$$
x^{3}+y^{3}=8^{30}
$$ | (2^{30},0)(0,2^{30}) | 28 | 15 |
math | A derangement is a permutation of size $n$ such that for all $1 \leq i \leq n$ we have $p(i) \neq i$.
How many derangements of size $n$ are there? | n!(\frac{1}{2!}-\frac{1}{3!}+\cdots+\frac{(-1)^{n}}{n!}) | 51 | 34 |
math | 9. Let $\triangle A B C$ have internal angles $A, B, C$ with opposite sides $a, b, c$ respectively, and $A-C=\frac{\pi}{2}, a, b, c$ form an arithmetic sequence, then the value of $\cos B$ is | \frac{3}{4} | 62 | 7 |
math | 1. Given $A=\left\{x \mid \log _{3}\left(x^{2}-2 x\right) \leqslant 1\right\}, B=(-\infty, a] \cup(b,+\infty)$, where $a<b$, if $A \cup B=\mathbf{R}$, then the minimum value of $a-b$ is $\qquad$ . | -1 | 89 | 2 |
math | B3. In a class, there are 23 students, each of whom has chosen exactly one foreign language, either German or French. There are a total of 10 girls in the class, and there are 11 students in total who are taking French. The number of girls who have chosen French plus the number of boys who have chosen German is 16.
Ho... | 7 | 85 | 1 |
math | 5.75 Find all values of $a$ such that the polynomial
$$
x^{3}-6 x^{2}+a x+a
$$
has roots $x_{1}, x_{2}, x_{3}$ satisfying
$$
\left(x_{1}-3\right)^{3}+\left(x_{2}-3\right)^{3}+\left(x_{3}-3\right)^{3}=0 .
$$ | -9 | 95 | 2 |
math | The number of routes she chooses $f(m, n)$ satisfies $f(m, n) \leqslant 2^{m n}$.
13. Let the set $S=\{1,2,3, \cdots, 1990\}, A \subset S$, and $|A|=26$. If the sum of all elements in $A$ is a multiple of 5, then $A$ is called a 26-element "good subset" of $S$. Find the number of 26-element "good subsets" of $S$. | \frac{1}{5}C_{1990}^{26} | 123 | 18 |
math | Task B-4.3. Determine the complex number $z^{3}$, if the imaginary part of the number $z$ is equal to $-\frac{1}{3}$ and if the argument of the number $z$ is equal to $\frac{7 \pi}{6}$. | -\frac{8}{27}i | 61 | 9 |
math | 2. Find all even natural numbers $n$ for which the number of divisors (including 1 and $n$ itself) is equal to $\frac{n}{2}$. (For example, the number 12 has 6 divisors: $\left.1,2,3,4,6,12.\right)$ | {8,12} | 70 | 6 |
math | How many permutations have only one cycle? | (n-1)! | 8 | 4 |
math | 17. Find the value of $\left(\log _{\sqrt{2}}\left(\cos 20^{\circ}\right)+\log _{\sqrt{2}}\left(\cos 40^{\circ}\right)+\log _{\sqrt{2}}\left(\cos 80^{\circ}\right)\right)^{2}$. | 36 | 78 | 2 |
math | 12. Let the function $f(x)$ be a differentiable function defined on the interval $(-\infty, 0)$, with its derivative being $f^{\prime}(x)$, and $2 f(x) + x f^{\prime}(x) > x^{2}$. Then
$$
(x+2017)^{2} f(x+2017)-f(-1)>0
$$
The solution set is $\qquad$ . | (-\infty,-2018) | 102 | 10 |
math | 7. For a given positive integer $k$, let $f_{1}(k)$ denote the square of the sum of the digits of $k$, and set $f_{n+1}(k)=f_{1}\left(f_{n}\right.$ $(k))(n \geqslant 1)$. Find the value of $f_{2005}\left(2^{2006}\right)$. | 169 | 90 | 3 |
math | 14. (15 points) Build a highway. If A, B, and C work together, it can be completed in 90 days; if A, B, and D work together, it can be completed in 120 days; if C and D work together, it can be completed in 180 days. If A and B work together for 36 days, and the remaining work is completed by A, B, C, and D working tog... | 60 | 112 | 2 |
math | Arrange the positive integers into two lines as follows:
\begin{align*} 1 \quad 3 \qquad 6 \qquad\qquad\quad 11 \qquad\qquad\qquad\qquad\quad\ 19\qquad\qquad32\qquad\qquad 53\ldots\\
\mbox{\ \ } 2 \quad 4\ \ 5 \quad 7\ \ 8\ \ 9\ \ 10\quad\ 12\ 13\ 14\ 15\ 16\ 17\ 18\quad\ 20 \mbox{ to } 31\quad\ 33 \mbox{ to } 52\quad\... | a_n = F_{n+3} - 2 | 291 | 12 |
math | Task 4. Given a natural number $n$, we define $\tau(n)$ as the number of natural numbers that divide $n$, and we define $\sigma(n)$ as the sum of these divisors. Find all natural numbers $n$ for which
$$
\sigma(n)=\tau(n) \cdot\lceil\sqrt{n}\rceil
$$
For a real number $x$, the notation $\lceil x\rceil$ means the smal... | 1,3,5,6 | 105 | 7 |
math | Given two natural numbers $ w$ and $ n,$ the tower of $ n$ $ w's$ is the natural number $ T_n(w)$ defined by
\[ T_n(w) = w^{w^{\cdots^{w}}},\]
with $ n$ $ w's$ on the right side. More precisely, $ T_1(w) = w$ and $ T_{n+1}(w) = w^{T_n(w)}.$ For example, $ T_3(2) = 2^{2^2} = 16,$ $ T_4(2) = 2^{16} = 65536,$ and $... | 1988 | 222 | 4 |
math | (13) For any positive integer $n$, the sequence $\left\{a_{n}\right\}$ satisfies $\sum_{i=1}^{n} a_{i}=n^{3}$, then $\sum_{i=2}^{2009} \frac{1}{a_{i}-1}=$ | \frac{2008}{6027} | 70 | 13 |
math | 8. Let the three vertices of $\triangle A B C$ in the complex plane correspond to the complex numbers $z_{1}, z_{2}, z_{3}$, and $\left|z_{1}\right|=r(r>0$ is a constant), $z_{2}=\overline{z_{1}}, z_{3}=\frac{1}{z_{1}}$, find the maximum value of the area of $\triangle A B C$. | \frac{1}{2}(r^{2}-1) | 96 | 13 |
math | 10.2. Solve the equation:
$1+\frac{3}{x+3}\left(1+\frac{2}{x+2}\left(1+\frac{1}{x+1}\right)\right)=x$. | 2 | 50 | 1 |
math | 1. a) Solve in $Z$ the equation: $5 \cdot(2 \cdot|3 x-4|+4)-30=10$ | 2 | 35 | 1 |
math | Example 8 Try to find
$$
\begin{aligned}
p= & (1-1993)\left(1-1993^{2}\right) \cdots\left(1-1993^{1993}\right)+1993\left(1-1993^{2}\right)\left(1-1993^{3}\right) \cdots\left(1-1993^{1993}\right)+ \\
& 1993^{2}\left(1-1993^{3}\right) \cdots\left(1-1993^{1993}\right)+1993^{3}\left(1-1993^{4}\right) \cdots\left(1-1993^{19... | 1 | 242 | 1 |
math | Example 2 Given that $a$, $b$, and $c$ are all integers, and for all real numbers $x$, we have
$$
(x-a)(x-2005)-2=(x-b)(x-c)
$$
holds. Find all such ordered triples $(a, b, c)$.
Analysis: $(x-a)(x-2005)-2=(x-b)$. $(x-c)$ always holds, which means $x^{2}-(a+2005) x+$ $2005 a-2=(x-b)(x-c)$ always holds, indicating that
... | (2004,2003,2006),(2004,2006,2003),(2006,2004,2007),(2006,2007,2004) | 170 | 61 |
math | 3. $P$ is a moving point on the plane of the equilateral $\triangle A B C$ with side length 2, and $P A^{2}+P B^{2}+P C^{2}=16$. Then the trajectory of the moving point $P$ is $\qquad$ . | a circle with the center of \triangle ABC and a radius of 2 | 66 | 15 |
math | 731. Calculate the approximate value:
1) $1.08^{3.96}$;
2) $\frac{\sin 1.49 \cdot \operatorname{arctg} 0.07}{2^{2.95}}$. | 0.01 | 58 | 4 |
math | A tetrahedron $ABCD$ satisfies the following conditions: the edges $AB,AC$ and $AD$ are pairwise orthogonal, $AB=3$ and $CD=\sqrt2$. Find the minimum possible value of
$$BC^6+BD^6-AC^6-AD^6.$$ | 1998 | 65 | 4 |
math | 4. Choose any two numbers from $2,4,6,7,8,11,12,13$ to form a fraction. Then, there are $\qquad$ irreducible fractions among these fractions. | 36 | 47 | 2 |
math | A fair die is rolled six times. Find the mathematical expectation of the number of different faces that appear.
# | \frac{6^{6}-5^{6}}{6^{5}} | 22 | 16 |
math | Zaslavsky A.A.
A line passing through the center of the circumscribed circle and the intersection point of the altitudes of a non-equilateral triangle $ABC$ divides its perimeter and area in the same ratio. Find this ratio. | 1:1 | 50 | 3 |
math | 3. When dividing the number $a$ by 7, we get a remainder of 3, and when dividing the number $b$ by 7, we get a remainder of 4. What is the remainder when the square of the sum of the numbers $a$ and $b$ is divided by 7? Justify your answer. | 0 | 72 | 1 |
math | 88. $\int \cos 5 x d x$
88. $\int \cos 5x \, dx$ | \frac{1}{5}\sin5x+C | 27 | 11 |
math | 13.095. On a flat horizontal platform, two masts stand 5 m apart from each other. At a height of $3.6 \mathrm{M}$ above the platform, one end of a wire segment $13 \mathrm{M}$ long is attached to each mast. The wire is stretched in the plane of the masts and is attached to the platform, as shown in Fig. 13.7. At what d... | 2.7\mathrm{~} | 111 | 8 |
math | 16. 1.1 $\star$ Calculate the sum of digits for each number from 1 to $10^{9}$, resulting in $10^{9}$ new numbers, then find the sum of digits for each of these new numbers; continue this process until all numbers are single digits. In the final set of numbers, are there more 1s or 2s? | 1 | 81 | 1 |
math | 9.22 Thirty people are divided into three groups (I, II, and III) with 10 people in each. How many different group compositions are possible? | \frac{30!}{(10!)^{3}} | 35 | 14 |
math | Someone in 1893 is as old as the sum of the digits of their birth year; so how old are they? | 24 | 27 | 2 |
math | Given an integer $n>1$, let $a_{1}, a_{2}, \cdots, a_{n}$ be distinct non-negative real numbers, and define the sets
$$
A=\left\{a_{i}+a_{j} \mid 1 \leqslant i \leqslant j \leqslant n\right\}, B=\left\{a_{i} a_{j} \mid 1 \leqslant i \leqslant j \leqslant n\right\} .
$$
Find the minimum value of $\frac{|A|}{|B|}$. Here... | \frac{2(2n-1)}{n(n+1)} | 155 | 16 |
math | 1. In the set of real numbers, solve the system of equations
$$
\left\{\begin{array}{l}
\log _{y-x^{3}}\left(x^{3}+y\right)=2^{y-x^{3}} \\
\frac{1}{9} \log _{x^{3}+y}\left(y-x^{3}\right)=6^{x^{3}-y}
\end{array}\right.
$$ | \sqrt[3]{7},9 | 97 | 8 |
math | Under which condition does the following equality sequence hold:
$$
\frac{a+\frac{a b c}{a-b c+b}}{b+\frac{a b c}{a-a c+b}}=\frac{a-\frac{a b}{a+2 b}}{b-\frac{a b}{2 a+b}}=\frac{\frac{2 a b}{a-b}+a}{\frac{2 a b}{a-b}-b}=\frac{a}{b}
$$ | 0,\quadb\neq0,\quad\neq1 | 104 | 14 |
math | Frankin 5.P.
Let $p$ be a prime number. How many natural numbers $n$ exist such that $p n$ is divisible by $p+n$? | 1 | 37 | 1 |
math | 3. Replacing the larger number of two different natural numbers with the difference between these two numbers is called one operation. For example, for 18 and 42, such operations can be performed continuously. Thus, we have: $18,42 \rightarrow 18,24 \rightarrow 18,6 \rightarrow 12$, $6 \rightarrow 6,6$, until the two n... | 1000510020 | 134 | 10 |
math | The first question: Let real numbers $a_{1}, a_{2}, \cdots, a_{2016}$ satisfy $9 a_{i}>11 a_{i+1}^{2}(i=1,2, \cdots, 2015)$. Find
$$
\left(a_{1}-a_{2}^{2}\right)\left(a_{2}-a_{3}^{2}\right) \cdots\left(a_{2015}-a_{2016}^{2}\right)\left(a_{2016}-a_{1}^{2}\right)
$$
the maximum value. | \frac{1}{4^{2016}} | 142 | 12 |
math | Example 6 What is the smallest positive integer that can be expressed as the sum of 9 consecutive integers, the sum of 10 consecutive integers, and the sum of 11 consecutive integers?
(11th American Invitational Mathematics Examination (AIME))
| 495 | 54 | 3 |
math | Transform the following expression into a product:
$$
x^{7}+x^{6} y+x^{5} y^{2}+x^{4} y^{3}+x^{3} y^{4}+x^{2} y^{5}+x y^{6}+y^{7}
$$ | (x+y)(x^{2}+y^{2})(x^{4}+y^{4}) | 67 | 21 |
math | 2. A box contains 3 red balls and 3 white balls, all of the same size and shape. Now, a fair die is rolled, and the number of balls taken from the box is equal to the number rolled. What is the probability that the number of red balls taken is greater than the number of white balls taken? $\qquad$ . | \frac{19}{60} | 73 | 9 |
math | 1. $\left(7\right.$ points) Calculate $\frac{(2009 \cdot 2029+100) \cdot(1999 \cdot 2039+400)}{2019^{4}}$. | 1 | 59 | 1 |
math | $\begin{array}{l}\text { 1. (5 points) } \frac{5}{16}-\frac{3}{16}+\frac{7}{16}= \\ \frac{3}{12}-\frac{4}{12}+\frac{6}{12}= \\ 64+27+81+36+173+219+136= \\ 2-\frac{8}{9}-\frac{1}{9}+1 \frac{98}{99}= \\\end{array}$ | \frac{9}{16},\frac{5}{12},736,2\frac{98}{99} | 126 | 30 |
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