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 | $605$ spheres of same radius are divided in two parts. From one part, upright "pyramid" is made with square base. From the other part, upright "pyramid" is made with equilateral triangle base. Both "pyramids" are put together from equal numbers of sphere rows. Find number of spheres in every "pyramid" | n = 10 | 73 | 6 |
math | 35th Putnam 1974 Problem A1 S is a subset of {1, 2, 3, ... , 16} which does not contain three integers which are relatively prime in pairs. How many elements can S have? Solution | 11 | 54 | 2 |
math | How many ways are there to place three rooks on an $8 \times 8$ chessboard such that the rooks are in different columns and different rows? | 18816 | 34 | 5 |
math | Problem 9-5. A divisor of a natural number is called proper if it is different from 1 and the number itself. Find all natural numbers for which the difference between the sum of the two largest proper divisors and the sum of the two smallest proper divisors is a prime number. | 12 | 60 | 2 |
math | 3. Consider a square on the complex plane, whose 4 vertices correspond to the 4 roots of a certain monic quartic equation with integer coefficients $x^{4}+p x^{3}+q x^{2}+r x+s=0$. Find the minimum value of the area of such a square.
| 2 | 67 | 1 |
math | 19. (3 points) A male and a female track and field athletes are practicing running on a 110-meter slope (the top of the slope is $A$, the bottom is $B$). Both start from point $A$ at the same time, running back and forth between $A$ and $B$ continuously. If the male athlete's uphill speed is 3 meters per second and dow... | 47\frac{1}{7} | 137 | 9 |
math | 1. The positive numbers $b_{1}, b_{2}, \cdots, b_{60}$ are arranged in sequence and satisfy $\frac{b_{2}}{b_{1}}=\frac{b_{3}}{b_{2}}=\cdots=\frac{b_{60}}{b_{59}}$.
Determine the value of $\log _{b_{11} b_{50}}\left(b_{1} b_{2} \cdots b_{60}\right)$. | 30 | 112 | 2 |
math | Task 1. For the natural numbers $a, b, c, d$ it is known that
$$
a c+a d+b c+d b=68 \text { and } c+d=4
$$
Calculate the value of the expression $a+b+c+d$. | 21 | 59 | 2 |
math | ## Problem Statement
Find the derivative.
$y=\frac{1}{4} \cdot \ln \frac{x-1}{x+1}-\frac{1}{2} \cdot \operatorname{arctg} x$ | \frac{1}{x^{4}-1} | 50 | 11 |
math | 10. All positive integer solutions to the equation $5(x y+y z+z x)=4 x y z$, with $x \leqslant y \leqslant z$ are
$\qquad$ . | (2,5,10),(2,4,20) | 47 | 15 |
math | INan * Find: The maximum distance between two points, one on the surface of a sphere with center at $(-2,-10,5)$ and radius 19, and the other on the surface of a sphere with center at $(12,8,-16)$ and radius 87. | 137 | 64 | 3 |
math | 3.30. The lateral edge of a regular triangular prism is equal to the side of the base. Find the angle between the side of the base and the non-intersecting diagonal of the lateral face. | \arccos\frac{\sqrt{2}}{4} | 43 | 14 |
math | Example 18 Let positive real numbers $x, y, z$ satisfy $x^{2}+y^{2}+z^{2}=1$. Find the minimum value of $\frac{x}{1-x^{2}}+\frac{y}{1-y^{2}}+\frac{z}{1-z^{2}}$. | \frac{3\sqrt{3}}{2} | 68 | 12 |
math | ## 34. Family Breakfast
Every Sunday, a married couple has breakfast with their mothers. Unfortunately, each spouse's relationship with their mother-in-law is quite strained: both know that there are two out of three chances of getting into an argument with their mother-in-law upon meeting. In the event of a conflict,... | \frac{4}{9} | 144 | 7 |
math | 11. Find all prime pairs $(p, q)$ such that $p q \mid\left(5^{p}-2^{p}\right)\left(5^{q}-2^{q}\right)$. | (3,3),(3,13),(13,3) | 45 | 15 |
math | 3. The equation $x^{2}+a x+8=0$ has two distinct roots $x_{1}$ and $x_{2}$; in this case,
$$
x_{1}-\frac{64}{17 x_{2}^{3}}=x_{2}-\frac{64}{17 x_{1}^{3}}
$$
Find all possible values of $a$. | \12 | 88 | 3 |
math | 8-29 Let the sequence $a_{1}, a_{2}, a_{3}, \cdots$ satisfy: $a_{1}=1$,
$$a_{n+1}=\frac{1}{16}\left(1+4 a_{n}+\sqrt{1+24 a_{n}}\right), n=1,2, \cdots$$
Find its general term formula. | a_{n}=\frac{1}{3}+\left(\frac{1}{2}\right)^{n}+\frac{1}{3}\left(\frac{1}{2}\right)^{2 n-1} | 89 | 48 |
math | In a mathematics competition, 30 students participated. Among them,
the 1st problem was solved by 20,
the 2nd problem was solved by 16,
the 3rd problem was solved by 10,
the 1st and 2nd problems were solved by 11,
the 1st and 3rd problems were solved by 7,
the 2nd and 3rd problems were solved by 5,
all three pro... | 3 | 120 | 1 |
math | 5. Let the set $I=\{1,2, \cdots, 2020\}$. We define
$$
\begin{array}{l}
W=\{w(a, b)=(a+b)+a b \mid a, b \in I\} \cap I, \\
Y=\{y(a, b)=(a+b) \cdot a b \mid a, b \in I\} \cap I, \\
X=W \cap Y,
\end{array}
$$
as the "Wu" set, "Yue" set, and "Xizi" set, respectively. Find the sum of the largest and smallest numbers in th... | 2020 | 146 | 4 |
math | Solve the following equation:
$$
\log _{3}(3-x)=\sqrt{\sqrt{x-1}-\frac{1}{\sqrt{x-1}}}
$$ | 2 | 38 | 1 |
math | 9. (3 points) Cars A and B start from locations $A$ and $B$ simultaneously and travel back and forth between $A$ and $B$ at a constant speed. If after the first meeting, Car A continues to drive for 4 hours to reach $B$, while Car B only drives for 1 hour to reach $A$, then when the two cars meet for the 15th time (mee... | 86 | 110 | 2 |
math | Problem 5. A group of tourists is dividing cookies. If they evenly distribute two identical packs, one extra cookie will remain. But if they evenly distribute three such packs, 13 extra cookies will remain. How many tourists are in the group? [7 points] (I.V. Raskina) | 23 | 63 | 2 |
math | Example 1 Calculate: $\lg ^{2} 2 \cdot \lg 250+\lg ^{2} 5 \cdot \lg 40$. | 1 | 37 | 1 |
math | ## 36. San Salvador Embankment
Would you come to have dinner with me tonight? I live in one of the eleven houses on San Salvador Embankment; however, to find out which one, you will have to think.
When, from my home, I look at the sea and multiply the number of houses to my left by the number of houses to my right, I... | 4 | 103 | 1 |
math | ## Task A-4.1.
Determine all numbers $a, b \in \mathbb{N}_{0}$ and $n \in \mathbb{N}$ for which
$$
2^{a}+3^{b}+1=n!
$$ | (,b,n)=(1,1,3),(2,0,3) | 56 | 17 |
math | (15) Find the smallest positive real number $k$, such that the inequality
$$
a b+b c+c a+k\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right) \geqslant 9,
$$
holds for all positive real numbers $a, b, c$. | 2 | 75 | 1 |
math | 12.1. Find the smallest natural number consisting of identical digits and divisible by 18.
$$
(5-6 \text { cl.) }
$$ | 666 | 34 | 3 |
math | 10. (20 points) In the first month, there is a pair of adult rabbits. Every month, they breed 3 male rabbits and 2 female rabbits. The young female rabbits grow up and also breed 3 male rabbits and 2 female rabbits every other month. Question: How many rabbits will there be in the $n$ $\left(n \in \mathbf{Z}_{+}\right)... | \frac{5\times2^{n+2}-5(-1)^{n}-3}{6} | 90 | 23 |
math | 1.36. Inside a right angle, there is a point $M$, the distances from which to the sides of the angle are 4 and $8 \mathrm{~cm}$. A line passing through point $M$ cuts off a triangle from the right angle with an area of $100 \mathrm{~cm}^{2}$. Find the legs of the triangle. | 40 | 81 | 2 |
math | 9.2 Find all real numbers $x$ that satisfy the inequality
$$
\sqrt{3-x}-\sqrt{x+1}>\frac{1}{2}
$$ | -1\leqslantx<1-\frac{\sqrt{31}}{8} | 37 | 21 |
math | 10. A florist received between 300 and 400 roses for a celebration. When he arranged them in vases with 21 roses in each, 13 roses were left. But when arranging them in vases with 15 roses in each, 8 roses were missing. How many roses were there in total? | 307 | 73 | 3 |
math | The lines with equations $x+y=3$ and $2 x-y=0$ meet at point $A$. The lines with equations $x+y=3$ and $3 x-t y=4$ meet at point $B$. The lines with equations $2 x-y=0$ and $3 x-t y=4$ meet at point $C$. Determine all values of $t$ for which $A B=A C$. | -\frac{1}{2},9\3\sqrt{10} | 90 | 16 |
math | In a right triangle, the altitude from a vertex to the hypotenuse splits the hypotenuse into two segments of lengths $a$ and $b$. If the right triangle has area $T$ and is inscribed in a circle of area $C$, find $ab$ in terms of $T$ and $C$. | \frac{\pi T^2}{C} | 67 | 10 |
math | For all $n \in \mathbb{N}$, we denote $u_{n}$ as the number of ways to tile a grid of size $2 \times n$ using dominoes of size $2 \times 1$. Provide a formula for $u_{n}$. | u_{n}=u_{n-1}+u_{n-2} | 60 | 17 |
math | 2. Given vectors $\boldsymbol{m}=(\cos x, -\sin x), \boldsymbol{n}=(\cos x, \sin x - 2 \sqrt{3} \cos x), x \in \mathbf{R}$, then the range of $f(x)=$ $m \cdot n, x \in\left[\frac{\pi}{4}, \frac{\pi}{2}\right]$ is $\qquad$. | [-1,\sqrt{3}] | 96 | 7 |
math | 4. Given $O$ as the origin, $M$ as a point on the moving chord $AB$ of the parabola $y^{2}=2 p x$ $(p>0)$. If $O A \perp O B, O M \perp A B$, denote the area of $\triangle O A B$ as $S$, and $O M=h$. Then the range of $\frac{S}{h}$ is . $\qquad$ | [2 p,+\infty) | 98 | 8 |
math | Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$, such that $af(a)^3+2abf(a)+bf(b)$ is a perfect square for all positive integers $a,b$. | f(n) = n | 48 | 6 |
math | [Example 2.2.2] Let $P(x)$ be an $n$-degree polynomial, and for $k=0,1,2, \cdots, n$, $P(k)=\frac{k}{k+1}$.
Find the value of $P(n+1)$. | \frac{n+1+(-1)^{n+1}}{n+2} | 64 | 19 |
math | 3. In the Cartesian coordinate system $x O y$, $F_{1}$ and $F_{2}$ are the left and right foci of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{4}=1(a>0)$, respectively. The line $l$ passes through the right vertex $A$ of the hyperbola and the point $B(0,2)$. If the sum of the distances from points $F_{1}$ and $F_{2}... | (2 \sqrt{2}, 0) | 145 | 10 |
math | 118. The random variable $X$ is given by the distribution function:
$$
F(x)=\left\{\begin{array}{ccc}
0 & \text { if } & x \leq -c \\
\frac{1}{2}+\frac{1}{\pi} \arcsin \frac{x}{c} & \text { if } & -c < x \leq c \\
1 & \text { if } & x > c
\end{array}\right.
$$
( arcsine law ).
Find the mathematical expectation of th... | 0 | 122 | 1 |
math | A $20 \times 20 \times 20$ cube is divided into 8000 unit cubes. A number is written in each unit cube. In each row and in each column of 20 small cubes, parallel to one of the cube's edges, the sum of the numbers is 1. In one of the small cubes, the number written is 10. Through this small cube pass three layers $1 \t... | 333 | 287 | 3 |
math | In the plane, there are 2020 points, some of which are black and the rest are green.
For each black point, there are exactly two green points that are at a distance of 2020 from this black point.
Determine the smallest possible number of green points.
(Walther Janous)
Answer. The smallest possible number is 45 gree... | 45 | 80 | 2 |
math | 29. Let $p$ be a prime, $a, b \in \mathbf{N}^{*}$, satisfying: $p>a>b>1$. Find the largest integer $c$, such that for all $(p, a, b)$ satisfying the conditions, we have
$$p^{c} \mid\left(\mathrm{C}_{a p}^{b p}-\mathrm{C}_{a}^{b}\right)$$ | 3 | 95 | 1 |
math | 4. (10 points) Person A leaves location $A$ to find person B, and after walking 80 kilometers, arrives at location $B$. At this point, person B had left for location $C$ half an hour earlier. Person A has been away from location $A$ for 2 hours, so A continues to location $C$ at twice the original speed. After another ... | 64 | 110 | 2 |
math | Task 6.4. On a computer keyboard, the key with the digit 1 is not working. For example, if you try to type the number 1231234, only the number 23234 will be printed.
Sasha tried to type an 8-digit number, but only 202020 was printed. How many 8-digit numbers meet this condition? | 28 | 87 | 2 |
math | 4. Let $H$ be the orthocenter of acute triangle $A B C$, given that $\angle A=60^{\circ}, B C=3$, then $A H=$ | \sqrt{3} | 41 | 5 |
math | 1. Let $\alpha, \beta$ be a pair of conjugate complex numbers. If $|\alpha-\beta|=2 \sqrt{3}$, and $\frac{\alpha}{\beta^{2}}$ is a real number, then $|\alpha|=$ | 2 | 55 | 1 |
math | 9. (16 points) Given the function
$$
f(x)=a \sin x-\frac{1}{2} \cos 2 x+a-\frac{3}{a}+\frac{1}{2} \text {, }
$$
where $a \in \mathbf{R}$, and $a \neq 0$.
(1) If for any $x \in \mathbf{R}$, $f(x) \leqslant 0$, find the range of values for $a$.
(2) If $a \geqslant 2$, and there exists $x \in \mathbf{R}$ such that $f(x) ... | [2,3] | 165 | 5 |
math | Consider the non-decreasing sequence of odd integers
$\left(a_{1}, a_{2}, a_{3}, \cdots\right)=(1,3,3,5,5,5,5,5, \cdots)$, where each positive odd integer $k$ appears $k$ times. It is known that there exist integers $b$, $c$, $d$ such that for all positive integers $n$, $a_{n}=b[\sqrt{n+c}]+d$. Find $b, c, d$. | b=2,=-1,=1 | 112 | 9 |
math | Question 40: Find all pairs of real numbers $(a, b)$ such that the function $\mathrm{f}(\mathrm{x})=\mathrm{ax}+\mathrm{b}$ satisfies for any real numbers $\mathrm{x}, \mathrm{y} \in$ $[0,1]$, we have $\mathrm{f}(\mathrm{x}) \mathrm{f}(\mathrm{y})+\mathrm{f}(\mathrm{x}+\mathrm{y}-\mathrm{xy}) \leq 0$. | (,b)\mid-1\leqb\leq0,-(b+1)\leq\leq-b | 109 | 26 |
math | 4. let $q(n)$ be the sum of the digits of the natural number $n$. Determine the value of
$$
q\left(q\left(q\left(2000^{2000}\right)\right)\right)
$$
## Solution | 4 | 57 | 1 |
math | 10. Let $P=\left(3^{1}+1\right)\left(3^{2}+1\right)\left(3^{3}+1\right) \ldots\left(3^{2020}+1\right)$. Find the largest value of the integer $n$ such that $2^{n}$ divides $P$. | 3030 | 80 | 4 |
math | 10. Given the function $f: \mathbf{R} \rightarrow \mathbf{R}$, if for any real numbers $x, y, z$, we have $\frac{1}{3} f(x y)+\frac{1}{3} f(x z)-f(x) f(y z) \geqslant \frac{1}{9}$, find $\sum_{i=1}^{100}[i f(i)]$. Here $[x]$ represents the greatest integer not exceeding $x$. | 1650 | 112 | 4 |
math | 131 On each face of an opaque cube, a natural number is written. If several (one, two, or three) faces of the cube can be seen at the same time, then find the sum of the numbers on these faces. Using this method, the maximum number of different sums that can be obtained is $\qquad$. | 26 | 69 | 2 |
math | 11. (20 points) Given the sequence $\left\{a_{n}\right\}$ satisfies:
$a_{1}=\sqrt{2}+1$,
$\frac{a_{n}-\sqrt{n}}{a_{n-1}-\sqrt{n-1}}=\sqrt{1+\frac{1}{n}}(n=2,3, \cdots)$.
Let $b_{n}=\frac{1}{a_{n}}$, and let the sum of the first $n$ terms of the sequence $\left\{b_{n}\right\}$ be $S_{n}$.
(1) Find the general term formu... | \frac{\sqrt{6}}{2}\leqslant\lambda\leqslant20 | 229 | 23 |
math | 6. The reading on a factory's electricity meter is 52222 kilowatts. After several days, the meter reading (a five-digit number) again shows four identical digits. How much electricity, in kilowatts, did the factory use at least in these days?
untranslated portion: $\qquad$ | 333 | 66 | 3 |
math | Exercise 12. For all integers $n \geqslant 1$, determine all $n$-tuples of real numbers $\left(x_{1}, \ldots, x_{n}\right)$ such that
$$
\sqrt{x_{1}-1^{2}}+2 \sqrt{x_{2}-2^{2}}+\ldots+n \sqrt{x_{n}-n^{2}}=\frac{1}{2}\left(x_{1}+\ldots+x_{n}\right)
$$ | x_{i}=2i^{2} | 108 | 9 |
math | 8.1. A dandelion blooms in the morning, remains yellow for this and the next day, turns white on the third morning, and by evening of the third day, it has lost its petals. Yesterday afternoon, there were 20 yellow and 14 white dandelions on the meadow, and today there are 15 yellow and 11 white. How many yellow dandel... | 25 | 108 | 2 |
math | 52. The median line of a trapezoid divides it into two quadrilaterals. The difference in the perimeters of these two quadrilaterals is 24, and the ratio of their areas is $\frac{20}{17}$. If the height of the trapezoid is 2, then the area of this trapezoid is $\qquad$ _. | 148 | 82 | 3 |
math | Through the midpoint of the hypotenuse of a right triangle, a perpendicular is drawn to it. The segment of this perpendicular, enclosed within the triangle, is equal to c, and the segment enclosed between one leg and the extension of the other is equal to 3c. Find the hypotenuse.
# | 4c | 63 | 2 |
math | Example 1. Find $\lim _{x \rightarrow 0} \frac{\ln (1+\alpha x)}{\sin \beta x}$. | \frac{\alpha}{\beta} | 32 | 8 |
math | 6. Solve the system of equations: $\left\{\begin{array}{l}x^{3}+y^{3}=3 y+3 z+4, \\ y^{3}+z^{3}=3 z+3 x+4, \\ z^{3}+x^{3}=3 x+3 y+4 .\end{array}\right.$
(2014, German Mathematical Olympiad) | (x,y,z)=(-1,-1,-1),(2,2,2) | 90 | 17 |
math | Three, (25 points) If the sum, difference, product, and quotient of two unequal natural numbers add up to a perfect square, then such two numbers are called a "wise pair" (for example, $(8,2)$ is a wise pair, since $\left.(8+2)+(8-2)+8 \times 2+\frac{8}{2}=36=6^{2}\right)$.
If both of these natural numbers do not exce... | 53 | 110 | 2 |
math | Example 14 Let $x_{1}, x_{2}, x_{3}, x_{4}$ all be positive numbers, and $x_{1}+x_{2}+x_{3}+x_{4}=\pi$, find the minimum value of the expression $\left(2 \sin ^{2} x_{1}+\right.$ $\left.\frac{1}{\sin ^{2} x_{1}}\right)\left(2 \sin ^{2} x_{2}+\frac{1}{\sin ^{2} x_{2}}\right)\left(2 \sin ^{2} x_{3}+\frac{1}{\sin ^{2} x_{... | 81 | 189 | 2 |
math | 7. Let $f(n)$ be the integer closest to $\sqrt[4]{n}$, then $\sum_{k=1}^{2017} \frac{1}{f(k)}=$ | \frac{2822}{7} | 43 | 10 |
math | # Task 11.3. (12 points)
How many distinct roots does the equation $f(f(f(x)))=1$ have, if $f(x)=x-\frac{2}{x}$. | 8 | 45 | 1 |
math | 6. Given that $n, k$ are positive integers, $n>k$. Given real numbers $a_{1}, a_{2}, \cdots, a_{n} \in(k-1, k)$. Let positive real numbers $x_{1}, x_{2}$, $\cdots, x_{n}$ satisfy that for any $k$-element subset $I$ of $\{1,2, \cdots, n\}$, we have $\sum_{i \in I} x_{i} \leqslant \sum_{i \in I} a_{i}$. Find the maximum ... | a_{1} a_{2} \cdots a_{n} | 159 | 15 |
math | Example 3 Given $|z|=1, k$ is a real number, $z$ is a complex number. Find the maximum value of $\left|z^{2}+k z+1\right|$. | \left|z^{2}+k z+1\right|_{\max }=\left\{\begin{array}{ll}k+2, & k \geqslant 0 \text {; } \\ 2-k, & k<0 .\end{array}\right.} | 46 | 65 |
math | ## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0} \frac{3^{x+1}-3}{\ln \left(1+x \sqrt{1+x e^{x}}\right)}
$$ | 3\ln3 | 54 | 4 |
math | ## Task 28/82
Determine all prime pairs $(p ; q)$ for which $\binom{p}{q}$ is also a prime number! | (p,q)=(3,2) | 35 | 7 |
math | 95. Find the mathematical expectation of the number of points that fall when throwing a dice. | 3.5 | 19 | 3 |
math | 5. (3 points) If the square of a two-digit number has only the tens digit as 0, then there are $\qquad$ such two-digit numbers. | 9 | 35 | 1 |
math | 6 - 130 Let the function
$$
f_{k}(x, y)=\frac{x^{k}+y^{k}+(-1)^{k}(x+y)^{k}}{k}, k \in Z, k \neq 0 .
$$
Find all non-zero integer pairs $(m, n)$ such that $m \leqslant n, m+n \neq 0$, and
$$
f_{m}(x, y) f_{n}(x, y) \equiv f_{m+n}(x, y),
$$
where $x, y \in R, x y(x+y) \neq 0$.
Hint: The pairs $m=2, n=3$ and $m=2, n=5$... | (2,3),(2,5) | 173 | 9 |
math | 1. One sixth of the total quantity of a certain commodity is sold at a profit of $20 \%$, and half of the total quantity of the same commodity is sold at a loss of $10 \%$. By what percentage profit should the remainder of the commodity be sold to cover the loss? | 5 | 61 | 1 |
math | 6.9 An arithmetic progression has the following property: for any $n$, the sum of its first $n$ terms is equal to $5 n^{2}$. Find the common difference of this progression and its first three terms. | 5;15;25;=10 | 48 | 11 |
math | Solve the following inequality:
$$
\frac{2+7 x-15 x^{2}}{5-x+6 x^{2}}>0
$$ | -\frac{1}{5}<x<\frac{2}{3} | 34 | 16 |
math | For each positive integer, define a function \[ f(n)=\begin{cases}0, &\text{if n is the square of an integer}\\ \\ \left\lfloor\frac{1}{\{\sqrt{n}\}}\right\rfloor, &\text{if n is not the square of an integer}\end{cases}. \] Find the value of $\sum_{k=1}^{200} f(k)$. | 629 | 96 | 3 |
math | For $n$ distinct positive integers all their $n(n-1)/2$ pairwise sums are considered. For each of these sums Ivan has written on the board the number of original integers which are less than that sum and divide it. What is the maximum possible sum of the numbers written by Ivan? | \frac{(n-1)n(n+1)}{6} | 61 | 14 |
math | 5. (6 points) Two identical air capacitors with capacitance $C$ each are charged to a voltage $U$. One of them is submerged in a dielectric liquid with permittivity $\varepsilon$, after which the capacitors are connected in parallel. Determine the amount of heat released upon connecting the capacitors.
Possible soluti... | \frac{CU^{2}(\varepsilon-1)^{2}}{2\varepsilon(\varepsilon+1)} | 380 | 29 |
math | Example 2.5 Let $g_{n}$ denote the number of different permutations of $2 n(n \geqslant 2)$ distinct elements $a_{1}, a_{2}, \cdots, a_{n}, b_{1}$, $b_{2}, \cdots, b_{n}$ such that $a_{k}$ and $b_{k}(k=1,2, \cdots, n)$ are not adjacent. Find the counting formula for $g_{n}$. | g_{n}=\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}2^{k}\cdot(2n-k)! | 107 | 36 |
math | Task A-1.5. (4 points)
How many elements at least need to be removed from the set $\{2,4,6,8,10,12,14,16\}$ so that the product of the remaining elements is a square of a natural number? | 3 | 62 | 1 |
math | 1. Solve the equation
$$
\sqrt{x^{2}+x}+\sqrt{1+\frac{1}{x^{2}}}=\sqrt{x+3}
$$ | -1 | 37 | 2 |
math | 6. Five cards have the numbers $101,102,103,104$ and 105 on their fronts.
105
On the reverse, each card has a statement printed as follows:
101: The statement on card 102 is false
102: Exactly two of these cards have true statements
103: Four of these cards have false statements
104: The statement on card 101 is false
... | 206 | 144 | 3 |
math | What is the multiplicity of the root 1 of $X^{2 n}-n X^{n+1}+n X^{n}-X^{2}$ for $n \in \mathbb{N}_{\geq 2}$? | 1 | 52 | 1 |
math | 11. If the distances from the three vertices of a regular $\triangle A B C$ with side length 6 to the plane $\alpha$ are $1,2,3$, then the distance from the centroid $G$ of $\triangle A B C$ to the plane $\alpha$ is $\qquad$ . | {0,\frac{2}{3},\frac{4}{3},2} | 66 | 18 |
math | Three, (Full marks 10 points) The numbers A, B, and C are 312, $270$, and 211, respectively. When these three numbers are divided by a natural number $A$, the remainder of dividing A is twice the remainder of dividing B, and the remainder of dividing B is twice the remainder of dividing C. Find this natural number $A$.... | 19 | 151 | 2 |
math | 4. In square $A B C D$ with side 2, point $A_{1}$ lies on $A B$, point $B_{1}$ lies on $B C$, point $C_{1}$ lies on $C D$, point $D_{1}$ lies on $D A$. Points $A_{1}, B_{1}, C_{1}, D_{1}$ are the vertices of the square of the smallest possible area. Find the area of triangle $A A_{1} D_{1} .(\mathbf{1 1}$ points) | 0.5 | 119 | 3 |
math | 3. In a convex quadrilateral $A B C D: A B=A C=A D=B D$ and $\angle B A C=\angle C B D$. Find $\angle A C D$. | 70 | 41 | 2 |
math | Problem 2. a) Determine the natural numbers $a$ for which
$$
\frac{1}{4}<\frac{1}{a+1}+\frac{1}{a+2}+\frac{1}{a+3}<\frac{1}{3}
$$
b) Prove that for any natural number $p \geq 2$ there exist $p$ consecutive natural numbers $a_{1}, a_{2}, \ldots, a_{p}$ such that
$$
\frac{1}{p+1}<\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots... | =8,=9,=10 | 156 | 9 |
math | 1. Find the smallest natural number, which when multiplied by 127008 gives a cube of a natural number. Explain your answer! | 126 | 31 | 3 |
math | ## Task A-1.5.
How many four-digit numbers divisible by 3 do not contain the digits $2, 4, 6$ or 9 in their decimal representation? | 360 | 39 | 3 |
math | 13.387 A motorboat departed from point A upstream, and simultaneously a raft set off downstream from point B. They met after a hours and continued moving without stopping. Upon reaching point B, the boat turned back and caught up with the raft at point A. The boat's own speed remained constant throughout. How long were... | (1+\sqrt{2}) | 77 | 7 |
math | 32. Let $A$ be an angle such that $\tan 8 A=\frac{\cos A-\sin A}{\cos A+\sin A}$. Suppose $A=x^{\circ}$ for some positive real number $x$. Find the smallest possible value of $x$. | 5 | 59 | 1 |
math | 10. Let $\mathrm{f}$ be a real-valued function with the rule $\mathrm{f}(x)=x^{3}+3 x^{2}+6 x+14$ defined for all real value of $x$. It is given that $a$ and $b$ are two real numbers such that $\mathrm{f}(a)=1$ and $\mathrm{f}(b)=19$. Find the value of $(a+b)^{2}$. | 4 | 100 | 1 |
math | ## Task B-1.6.
The hotel owners at the beginning of the tourist season bought new blankets, towels, and pillowcases for 4000 kn. They paid 120 kn for each blanket, 50 kn for each towel, and 25 kn for each pillowcase. If they bought a total of 100 items, how many blankets, how many towels, and how many pillowcases were... | 5 | 102 | 1 |
math | Let ${{\left( {{a}_{n}} \right)}_{n\ge 1}}$ an increasing sequence and bounded.Calculate
$\underset{n\to \infty }{\mathop{\lim }}\,\left( 2{{a}_{n}}-{{a}_{1}}-{{a}_{2}} \right)\left( 2{{a}_{n}}-{{a}_{2}}-{{a}_{3}} \right)...\left( 2{{a}_{n}}-{{a}_{n-2}}-{{a}_{n-1}} \right)\left( 2{{a}_{n}}-{{a}_{n-1}}-{{a}_{1}} \right... | 0 | 153 | 1 |
math | Example 2. Investigate the analytical properties of the function $w=|z|^{2}$ and find its derivative. | f^{\}(0)=0 | 26 | 7 |
math | 2. Given a function $f(x)$ defined on $\mathbf{R}$ that satisfies
$$
\begin{array}{l}
f(x+1)=f(-x), \\
f(x)=\left\{\begin{array}{ll}
1, & -1<x \leqslant 0 \\
-1, & 0<x \leqslant 1 .
\end{array}\right.
\end{array}
$$
Then $f(f(3.5))=$ $\qquad$ | -1 | 109 | 2 |
math | 11. Find the value of $\tan \left(\tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{1}{2 \times 2^{2}}+\tan ^{-1} \frac{1}{2 \times 3^{2}}+\cdots+\tan ^{-1} \frac{1}{2 \times 2009^{2}}\right)$.
(2 marks)求 $\tan \left(\tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{1}{2 \times 2^{2}}+\tan ^{-1} \frac{1}{2 \times 3^{2}}+\cdots+\tan ^{-1} ... | \frac{2009}{2010} | 182 | 13 |
math | The $ 2^N$ vertices of the $ N$-dimensional hypercube $ \{0,1\}^N$ are labelled with integers from $ 0$ to $ 2^N \minus{} 1$, by, for $ x \equal{} (x_1,x_2,\ldots ,x_N)\in \{0,1\}^N$,
\[v(x) \equal{} \sum_{k \equal{} 1}^{N}x_k2^{k \minus{} 1}.\]
For which values $ n$, $ 2\leq n \leq 2^n$ can the vertices with labels in... | n = 4, 6, 8, \ldots, 2^N | 196 | 20 |
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