task_type stringclasses 4
values | problem stringlengths 21 5.23k | answer stringlengths 1 8.29k | problem_tokens int64 11 1.16k | answer_tokens int64 1 2.04k |
|---|---|---|---|---|
math | 17. (12 points) In the sequence $\left\{a_{n}\right\}$, $a_{1}=1$, and its first $n$ terms sum $S_{n}$ satisfies the relation
$$
3 t S_{n}-(2 t+3) S_{n-1}=3 t(t>0, n=2,3, \cdots) \text {. }
$$
(1) Prove that the sequence $\left\{a_{n}\right\}$ is a geometric sequence;
(2) Let the common ratio of the sequence $\left\{a_... | -\frac{8}{9} n^{2}-\frac{4}{3} n | 272 | 19 |
math | Example 20 (1998 National High School League Question) If the ellipse $x^{2}+4(y-a)^{2}=4$ intersects the parabola $x^{2}=2 y$, then the range of real number $a$ is $\qquad$ . | -1\leqslant\leqslant\frac{17}{8} | 61 | 20 |
math | Find all sets of four real numbers $x_1, x_2, x_3, x_4$ such that the sum of any one and the product of the other three is equal to 2. | (1, 1, 1, 1) | 44 | 13 |
math | Suppose $a, b$ are integers and $a+b$ is a root of $x^2 +ax+b = 0$. What is the maximum possible value of $b^2$? | 81 | 42 | 2 |
math | Example 3 If a positive integer has eight positive divisors, and the sum of these eight positive divisors is 3240, then this positive integer is called a "good number". For example, 2006 is a good number, because the sum of its divisors 1, $2,17,34,59,118,1003,2006$ is 3240. Find the smallest good number. ${ }^{[3]}$
(... | 1614 | 120 | 4 |
math | Find the greatest natural number $n$ such that $n\leq 2008$ and \[(1^2+2^2+3^2+\cdots + n^2)\left[(n+1)^2+(n+2)^2+(n+3)^2+\cdots + (2n)^2\right]\] is a perfect square. | 1921 | 80 | 4 |
math | 4. In a chess tournament, every two players play exactly one game against each other. The winner of each game gets 2 points, the loser gets 0 points, and in the case of a draw, both players get 1 point. Now, there are 4 scorers who have tallied the total points of the tournament, but due to some being careless, their d... | 45 | 128 | 2 |
math | During the first eleven days, 700 people responded to the survey question. Each of them chose exactly one of the three offered options. The frequency ratio of the individual responses was $4: 7: 14$. On the twelfth day, several more people participated in the survey, changing the frequency ratio of the responses to $6:... | 75 | 104 | 2 |
math | ## 4. Sum of Perimeters
What is the sum of the perimeters of all mutually incongruent triangles whose perimeter is a natural number, and the lengths of two sides are 7 and 12? | 403 | 45 | 3 |
math | 41st Putnam 1980 Problem A3 Find ∫ 0 π/2 f(x) dx, where f(x) = 1/(1 + tan √2 x). Solution | \frac{\pi}{4} | 43 | 7 |
math | Example 1 Given that $a$ and $b$ are real numbers, and $a^{2} + ab + b^{2} = 3$. If the maximum value of $a^{2} - ab + b^{2}$ is $m$, and the minimum value is $n$, find the value of $m + n$. ${ }^{\text {[2] }}$ | 10 | 81 | 2 |
math | Determine the number of pairs of positive integers $x,y$ such that $x\le y$, $\gcd (x,y)=5!$ and $\text{lcm}(x,y)=50!$. | 16384 | 43 | 5 |
math | 26. (1) Find all prime sequences $p_{1}<p_{2}<\cdots<p_{n}$, such that
$$\left(1+\frac{1}{p_{1}}\right)\left(1+\frac{1}{p_{2}}\right) \cdot \cdots \cdot\left(1+\frac{1}{p_{n}}\right)$$
is an integer; $\square$
(2) Does there exist $n$ distinct positive integers $a_{1}, a_{2}, \cdots, a_{n}, n \in \mathbf{N}^{\vee}$, g... | (2,3) | 216 | 5 |
math | Let $n$ be a positive integer. All numbers $m$ which are coprime to $n$ all satisfy $m^6\equiv 1\pmod n$. Find the maximum possible value of $n$. | 504 | 47 | 3 |
math | A number is said to be a palindrome if reading from right to left is the same as reading from left to right. For example, the numbers 23432 and 18781 are palindromes. How many 4-digit palindrome numbers are divisible by 9? | 10 | 61 | 2 |
math | ## Task B-2.4.
In how many ways can we choose two different numbers from the set $\{1,2, \ldots, 2022,2023\}$ so that their sum is divisible by 5? | 409051 | 53 | 6 |
math | For any positive integer $a$, define $M(a)$ to be the number of positive integers $b$ for which $a+b$ divides $ab$. Find all integer(s) $a$ with $1\le a\le 2013$ such that $M(a)$ attains the largest possible value in the range of $a$. | 1680 | 73 | 4 |
math | (Inspired by IMO 1972 Longlist 1, treated)
Find all integers $x$ such that $x^{4}+x^{3}+x^{2}+x+1$ is a perfect square. | -1,0,3 | 50 | 6 |
math | ## Task A-1.4.
On the board, there are the first $n$ natural numbers $(n \geqslant 3)$. Ante repeats the following procedure: first, he arbitrarily selects two numbers on the board, and then increases them by the same arbitrary amount.
Determine all natural numbers $n$ for which Ante, by repeating this procedure, can... | n\neq4k+2 | 89 | 8 |
math | If points $P_{1}\left(x_{1}, y_{1}\right), P_{2}\left(x_{2}, y_{2}\right), P_{3}\left(x_{3}, y_{3}\right)$ are on the curve $x y=1$, then the area of triangle $P_{1} P_{2} P_{3}$ is:
$$
t=\frac{\left(x_{1}-x_{2}\right)\left(x_{2}-x_{3}\right)\left(x_{3}-x_{1}\right)}{2 x_{1} x_{2} x_{3}}
$$ | \frac{(x_{1}-x_{2})(x_{2}-x_{3})(x_{3}-x_{1})}{2x_{1}x_{2}x_{3}} | 132 | 41 |
math | In how many ways can pawns be placed on a $4 \times 4$ chessboard such that each row and each column contains exactly two pawns? | 90 | 33 | 2 |
math | $ P(x)$ is a quadratic trinomial. What maximum number of terms equal to the sum of the two preceding terms can occur in the sequence $ P(1)$, $ P(2)$, $ P(3)$, $ \dots?$
[i]Proposed by A. Golovanov[/i] | 2 | 66 | 1 |
math | 8.254. $\sin ^{2} 2 x \cos \left(\frac{3 \pi}{2}-2 x\right)+3 \sin 2 x \sin ^{2}\left(\frac{3 \pi}{2}+2 x\right)+2 \cos ^{3} 2 x=0$. | x_{1}=\frac{\pi}{8}(4k-1);x_{2}=\frac{1}{2}\operatorname{arctg}2+\frac{\pin}{2},k,n\inZ | 74 | 47 |
math | 40th Putnam 1979 Problem B4 Find a non-trivial solution of the differential equation F(y) ≡ (3x 2 + x - 1)y'' - (9x 2 + 9x - 2)y' + (18x + 3)y = 0. y = f(x) is the solution of F(y) = 6(6x + 1) such that f(0) = 1, and ( f(-1) - 2)( f(1) - 6) = 1. Find a relation of the form ( f(-2) - a)( f(2) - b) = c. Solution | (f(-2)-6)(f(2)-14)=1 | 147 | 14 |
math | 129. In a parallelogram, there are two circles of radius 1, each touching the other and three sides of the parallelogram. It is also known that one of the segments of a side of the parallelogram from a vertex to the point of tangency is $\sqrt{3}$. Find the area of the parallelogram. | \frac{4}{3}(2\sqrt{3}+3) | 74 | 16 |
math | 13. Let $n$ be the smallest positive integer of 4 digits greater than or equal to 2016 that has the following property: there exists a positive integer $S$ such that
$$
S=\sqrt{a+\sqrt{b+\sqrt{c+\sqrt{d+S}}}}
$$
where $a, b, c, d$ are, in order, the thousands, hundreds, tens, and units digits of $n$. What is the valu... | 2167 | 105 | 4 |
math | 11.5. Find the set of values of the function $y=\sqrt{x}-\sqrt{2-x}+2 \sin x$. | [-\sqrt{2};\sqrt{2}+2\sin2] | 31 | 17 |
math | ## Task 2 - 050812
For which real numbers $a$ and $b$ is the equation
$$
\frac{1}{a}+\frac{1}{b}=\frac{(a+b) \cdot(a-b)}{a b} \quad \text{ satisfied? }
$$ | +b=0or=b+1 | 68 | 7 |
math | Among the right-angled triangles with equal perimeter, determine the one into which the largest incircle can be drawn. What is the radius of the circle if the perimeter of the triangle is $k$? | \frac{k}{2}(3-2\sqrt{2}) | 41 | 14 |
math | 12. A sphere is placed inside a regular octahedron of side length 6 . Find the greatest possible radius of the sphere.
(2 marks)
一個球體被放進一個邊長為 6 的正八面體內。求該球體半徑的最大可能值。
(2 分) | \sqrt{6} | 66 | 5 |
math | 2. For real numbers $a, b, c$ and a positive number $\lambda$, such that
$$
f(x)=x^{3}+a x^{2}+b x+c
$$
has three real roots $x_{1}, x_{2}, x_{3}$, and satisfies:
(1) $x_{2}-x_{1}=\lambda$;
(2) $x_{3}>\frac{1}{2}\left(x_{1}+x_{2}\right)$.
Find the maximum value of $\frac{2 a^{3}+27 c-9 a b}{\lambda^{3}}$. | \frac{3 \sqrt{3}}{2} | 137 | 12 |
math | Solve the equations:
a) $x^{4}+x^{3}-3 a^{2} x^{2}-2 a^{2} x+2 a^{4}=0$
b) $x^{3}-3 x=a^{3}+a^{-3}$. | x_{1,2}=\\sqrt{2},x_{3,4}=\frac{-1\\sqrt{1+4^{2}}}{2};\quad+^{-1}for\neq0,\1;\quadx_{1}=2,x_{2}=-1for=1;\quadx_{1}=-2,x | 59 | 71 |
math | # Problem 1.
It is known that the numbers $\frac{x}{2}, 2 x-3, \frac{18}{x}+1$, taken in the given order, form a geometric progression. Find the common ratio of this progression. Round your answer to two decimal places. | 2.08 | 61 | 4 |
math | Let $ABCD$ be a rectangle with side lengths $AB = CD = 5$ and $BC = AD = 10$. $W, X, Y, Z$ are points on $AB, BC, CD$ and $DA$ respectively chosen in such a way that $WXYZ$ is a kite, where $\angle ZWX$ is a right angle. Given that $WX = WZ = \sqrt{13}$ and $XY = ZY$, determine the length of $XY$. | \sqrt{65} | 105 | 6 |
math | There are $n$ football teams participating in a league, where each team plays a match against every other team exactly once. It is stipulated that a win earns 3 points, a draw earns 1 point, and a loss earns 0 points. After the league ends, some teams may be disqualified, and thus their match results will also be cance... | n(n-1) | 214 | 5 |
math | The sum of the squares of three positive numbers is $160$. One of the numbers is equal to the sum of the other two. The difference between the smaller two numbers is $4.$ What is the difference between the cubes of the smaller two numbers?
[i]Author: Ray Li[/i]
[hide="Clarification"]The problem should ask for the pos... | 320 | 80 | 3 |
math | 152. Find the smallest natural number that is written with identical digits and is divisible by 18. | 666 | 23 | 3 |
math | Let $\mathbb{N}=\{1,2,3, \ldots\}$ be the set of all positive integers. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for any positive integers $a$ and $b$, the following two conditions hold:
(1) $f(a b)=f(a) f(b)$, and
(2) at least two of the numbers $f(a), f(b)$ and $f(a+b)$ are equal.
Proposed... | f(n)=a^{v_{p}(n)} | 172 | 11 |
math | Determine all real polynomials $P$ such that $P(0)=0$ and $P\left(X^{2}+1\right)=P(X)^{2}+1$. | P(X)=X | 41 | 4 |
math | Example 5 Let $a, b, c, x, y, z$ be real numbers, and
$$a^{2}+b^{2}+c^{2}=25, x^{2}+y^{2}+z^{2}=36, a x+b y+c z=30 .$$
Find the value of $\frac{a+b+c}{x+y+z}$. | \frac{5}{6} | 86 | 7 |
math | Example 4. Find the inverse function of $y=\pi-\arcsin (3-x)$, $(2 \leqslant x \leqslant 4)$. | y=3-\sin x, x \in\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right] | 39 | 30 |
math | A $2.9 \, \text{cm}$ diameter measuring cylinder is filled with water to a height of $4 \, \text{cm}$, then dense copper cubes with an edge length of $2 \, \text{cm}$ are placed into the cylinder. What is the maximum number of cubes that the water in the cylinder can submerge? | 5 | 74 | 1 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty}\left(\frac{1+3+5+7+\ldots+(2 n-1)}{n+1}-\frac{2 n+1}{2}\right)
$$ | -\frac{3}{2} | 63 | 7 |
math | 13. 1. 3 * The vertices of $\triangle A B C$ are $A(0,0), B(0,420), C(560,0)$, and a die has its six faces marked with $A, A, B, B, C, C$. A point $P_{1}=(k, m)$ is taken inside $\triangle A B C$, and points $P_{2}, P_{3}, P_{4}, \cdots$ are generated according to the following rule: If $P_{n}$ is already determined, r... | 344 | 188 | 3 |
math | 10. In quadrilateral $A B C D$, $A B=B C=C D=$ $26, A D=30 \sqrt{3}, A C$ and $B D$ intersect at point $O, \angle A O B=$ $60^{\circ}$. Then $S_{\text {quadrilateral } A B C D}=$ $\qquad$ | 506 \sqrt{3} | 83 | 8 |
math | Example 1. Let $x+y=1, x^{2}+y^{2}=2$, find the value of $x^{7}+y^{7}$. (Japanese University Entrance Exam Question, 1979) | \frac{71}{8} | 50 | 8 |
math | Calculate the following indefinite integrals.
[1] $\int \frac{\sin x\cos x}{1+\sin ^ 2 x}dx$
[2] $\int x\log_{10} x dx$
[3] $\int \frac{x}{\sqrt{2x-1}}dx$
[4] $\int (x^2+1)\ln x dx$
[5] $\int e^x\cos x dx$ | \frac{e^x (\sin x + \cos x)}{2} + C | 98 | 19 |
math | 10.1. Find the smallest of the solutions to the inequality
$$
\frac{-\log _{2}(120-2 x \sqrt{32-2 x})^{2}+\left|\log _{2} \frac{120-2 x \sqrt{32-2 x}}{\left(x^{2}-2 x+8\right)^{3}}\right|}{5 \log _{7}(71-2 x \sqrt{32-2 x})-2 \log _{2}(120-2 x \sqrt{32-2 x})} \geqslant 0
$$ | -13-\sqrt{57} | 144 | 9 |
math | 2B. Determine all solutions ( $x, y$ ) of the equation $\frac{x+6}{y}+\frac{13}{x y}=\frac{4-y}{x}$, in the set of real numbers. | (-3,2) | 49 | 5 |
math | 4. If the acute angle $\alpha$ satisfies
$$
\frac{1}{\sqrt{\tan \frac{\alpha}{2}}}=\sqrt{2 \sqrt{3}} \cdot \sqrt{\tan 10^{\circ}}+\sqrt{\tan \frac{\alpha}{2}},
$$
then $\alpha=$ $\qquad$ | 50^{\circ} | 72 | 6 |
math | Example 4. Find the locus of the midpoint of the chord passing through the imaginary vertex $B(0,-b)$ of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$, | \frac{\left(y+\frac{b}{2}\right)^{2}}{\frac{b^{2}}{4}}-\frac{x^{2}}{\frac{a^{2}}{4}}=1 | 56 | 45 |
math | 2. The sum of three numbers is 1281. From the first number, 329 was subtracted, and to the second number, 401 was added. What needs to be done with the third number so that the sum does not change? Will the solution to the problem change if the word "sum" is replaced with "difference"? | 72 | 76 | 2 |
math | 11. (15 points) There is a strange four-digit number (the first digit is not 0), which is a perfect square, and the sum of its digits is also a perfect square. When this four-digit number is divided by the sum of its digits, the result is still a perfect square, and the number of its divisors is exactly equal to the su... | 2601 | 107 | 4 |
math | 8. (1997 National High School Competition Question) Given the complex number $z$ satisfies $\left|2 z+\frac{1}{z}\right|=1$, then the principal value range of the argument of $z$ is
$\qquad$ . | k\pi+\frac{\pi}{2}-\frac{1}{2}\arccos\frac{3}{4}\leqslant\theta\leqslantk\pi+\frac{\pi}{2}+\frac{1}{2}\arccos\frac{3}{4}(k=0,1) | 56 | 71 |
math | 7. Given a line passing through the origin (excluding the $x$-axis) intersects the circle $x^{2}+y^{2}=1$ at points $A$ and $B$, and a fixed point $C(2,0)$, then the minimum value of the area of the circumcircle of $\triangle A B C$ is $\qquad$ . | \frac{25 \pi}{16} | 78 | 11 |
math | ## Task B-2.1.
Determine the quadratic function with rational coefficients that has a minimum value of -4, and one of its roots is $\sqrt{2}-1$. | f(x)=2x^{2}+4x-2 | 38 | 13 |
math | 287. $y=x^{2}+\frac{2}{x^{4}}-\sqrt[3]{x}$
287. $y=x^{2}+\frac{2}{x^{4}}-\sqrt[3]{x}$ | 2x-\frac{8}{x^{5}}-\frac{1}{3\sqrt[3]{x^{2}}} | 53 | 26 |
math | Example 8 Find a natural number $N$, such that it is divisible by 5 and 49, and including 1 and $N$, it has a total of 10 divisors. | 5\cdot7^{4} | 42 | 7 |
math | Example 1 Let $S$ be a subset of the set $\{1,2,3, \cdots, 50\}$, and the sum of any two elements in $S$ cannot be divisible by 7. What is the maximum number of elements in $S$? | 23 | 61 | 2 |
math | Subject (2). Consider the following natural numbers:
$$
a=1 \cdot 3 \cdot 5 \cdot 7 \cdots 27 \cdot 29 \cdot 31 \text { and } b=1 \cdot 3 \cdot 5 \cdot 7 \cdots 27 \cdot 29
$$
a) Prove that the number $a$ is divisible by 2015.
b) Find the largest natural number $n$ such that the number $a+b$ is divisible by $10^n$.
... | 4 | 145 | 1 |
math | 2. Given positive numbers $a, b, c, d$. Find the minimum value of the expression
$$
A=\left(\frac{a^{2}+b^{2}}{c d}\right)^{4}+\left(\frac{b^{2}+c^{2}}{a d}\right)^{4}+\left(\frac{c^{2}+d^{2}}{a b}\right)^{4}+\left(\frac{d^{2}+a^{2}}{b c}\right)^{4}
$$ | 64 | 118 | 2 |
math | One, (20 points) Given the quadratic equation $x^{2}+a x+b=0$ has two consecutive integer roots, and the quadratic equation $x^{2}+b x+a=0$ has integer roots. Find the values of $a, b$.
| (a, b) \text{ are } (-1,0), (-3,2), (5,6) | 59 | 24 |
math | 5. Determine all natural numbers $n$ and prime numbers $p$ for which $p^{2}+7^{n}$ is a perfect square.
## Third grade - B category | p=3n=1 | 38 | 6 |
math | ## Task A-3.5.
For a pair of numbers $\{a, b\}$, we say it has a weight of $|a-b|$. In how many ways can the set $\{1,2, \ldots, 12\}$ be divided into six pairs such that the total sum of the weights of these pairs is 30? | 360+384+360 | 77 | 11 |
math | 5. (7 points) Three schoolgirls entered a store. Anya bought 2 pens, 7 pencils, and 1 notebook, Varya - 5 pens, 6 pencils, and 5 notebooks, Sasha - 8 pens, 4 pencils, and 9 notebooks. They all paid equally, but one of them used a discount when paying. Who? (Explain your answer). | Varya | 83 | 2 |
math | 1. If $f(x)=\sqrt{x+27}+\sqrt{13-x}+\sqrt{x}$, then the maximum value of $f(x)$ is $\qquad$ . | 11 | 41 | 2 |
math | 2. Simplify the expression
$$
A=\left(1+\frac{1+i}{2}\right)\left(1+\left(\frac{1+i}{2}\right)^{2}\right)\left(1+\left(\frac{1+i}{2}\right)^{4}\right)\left(1+\left(\frac{1+i}{2}\right)^{8}\right)
$$
( $i$ is the imaginary unit) | \frac{255}{256}(1+i) | 95 | 14 |
math | Two points, $A$ and $B$, lie on two lines that form a $60^{\circ}$ angle with each other. The distance between the points is $31 \mathrm{~m}$. If point $A$ moves $20 \mathrm{~m}$ closer to the intersection point of the lines, then the distance between the points becomes $21 \mathrm{~m}$. How far are the points from the... | MB=24\mathrm{~},MA=35\mathrm{~} | 99 | 18 |
math | 15. There are 40 sets of CASIO cards, each set consisting of C, A, S, I, O five cards stacked in the order of C, A, S, I, O from top to bottom. Now, these 40 sets of cards are stacked together from top to bottom, and then the first card is discarded, the second card is placed at the bottom, the third card is discarded,... | 22, \text{I of the 29th set} | 151 | 15 |
math | 229. $\log _{3}(12 x+4)-\log _{3}(x-7)=\log _{3} 9$. | Nosolution | 35 | 2 |
math | 8. A cube consists of eight smaller identical cubes. Three of the smaller cubes were replaced with ones of the same size but with a density three times greater. Determine the ratio of the final to the initial density of the large cube. (10 points) | 1.75 | 52 | 4 |
math | 6. In the geometric sequence $\left\{a_{n}\right\}$, if for any positive integer $n, a_{1}+a_{2}+\cdots+a_{n}=2^{n}-1$, then $a_{1}^{3}+a_{2}^{3}+\cdots+a_{n}^{3}=$ $\qquad$ | \frac{1}{7}(8^{n}-1) | 80 | 13 |
math | I OM - B - Task 4
Find two natural numbers $ a $ and $ b $ given their greatest common divisor
$ D=12 $ and least common multiple $ M=432 $. Provide a method for finding
solutions in the general case. | (12, | 56 | 4 |
math | 4. Baron Munchausen placed a horse in some cells of an $N \times N$ board. He claims that no one will find two different $4 \times 4$ squares on this board (with sides along the grid lines) with the same number of horses. For what largest $N$ can his words be true?
# | 7 | 71 | 1 |
math | Example 5 Let $S_{n}$ denote some subsets of the set of positive integers $\{1,2,3, \cdots, 100\}$ that satisfy the condition: no number is twice another. What is the maximum number of elements such a subset can contain? | 67 | 60 | 2 |
math | Example 6 Suppose it is known that 2 is a primitive root of $p=13$, try to find all the primitive roots of $p$. | 2, 6, 7, 11 | 32 | 11 |
math | Problem 6.5. On a line, 5 points $P, Q, R, S, T$ are marked, exactly in that order. It is known that the sum of the distances from $P$ to the other 4 points is 67, and the sum of the distances from $Q$ to the other 4 points is 34. Find the length of the segment $P Q$. | 11 | 87 | 2 |
math | 5. At $17-00$ the speed of the racing car was 30 km/h. Every 5 minutes thereafter, the speed increased by 6 km/h. Determine the distance traveled by the car from $17-00$ to $20-00$ of the same day. | 405 | 66 | 3 |
math | 16. Two bullets are placed in two consecutive chambers of a 6-chamber pistol. The cylinder is then spun. The pistol is fired but the first shot is a blank. Let $p$ denote the probability that the second shot is also a blank if the cylinder is spun after the first shot and let $\mathrm{q}$ denote the probability that th... | 89 | 118 | 2 |
math | 9. Given an arithmetic sequence $\left\{a_{n}\right\}$ with $2 n+1$ terms, the sum of the odd-numbered terms is 2002, and the sum of the even-numbered terms is 2000. Find the middle term and the number of terms. | a_{n+1}=2,ofterms=2001 | 67 | 15 |
math | 18. In how many different ways can we rearrange the twelve integers 1 to 12 on the face of a clock so that the sum of any three adjacent integers after the rearrangement is divisible by 3 ?
(2 marks)
有多少種不同的方法可把鐘面上 1 至 12 等 12 個整數重新排列, 使得排列後任意三個相鄰的整數之和皆可被 3 整除？
（２分） | 82944 | 105 | 5 |
math | B. Originally planned to spend 1500 yuan to buy $x$ pieces of product A and $y$ pieces of product B, but the price of product A increased by 1.5 yuan per piece, and the price of product B increased by 1 yuan per piece. Despite buying 10 fewer pieces of product A, the total amount was still 29 yuan more. If the price of... | x=76,y=55 | 213 | 8 |
math | A natural number $n$ was alternately divided by $29$, $41$ and $59$. The result was three nonzero remainders, the sum of which equals $n$. Find all such $n$ | 79 \text{ and } 114 | 47 | 11 |
math | Example 7 (1982 Kyiv Mathematical Olympiad) Find the natural number $N$, such that it is divisible by 5 and 49, and including 1 and $N$, it has a total of 10 divisors. | 5\cdot7^{4} | 53 | 7 |
math | 8. Find $a$ such that the sum of the squares of the real roots of the equation $x^{4}+a x^{2}-$ $2017=0$ is 4. | 1006.5 | 44 | 6 |
math | 4. There are 28 students in the class. On March 8th, each boy gave each girl one flower - a tulip, a rose, or a daffodil. How many roses were given, if it is known that there were 4 times as many roses as daffodils, but 3 times fewer than tulips?
(A. A. Tesler) | 44 | 81 | 2 |
math | 7. On a circle, 2017 different points $A_{1}, \ldots, A_{2017}$ are marked, and all possible chords connecting these points pairwise are drawn. A line is drawn through the point $A_{1}$, not passing through any of the points $A_{2}, \ldots, A_{2017}$. Find the maximum possible number of chords that can have at least on... | 1018080 | 111 | 7 |
math | Shmarov V.
On a circle, $2 N$ points are marked ($N$ is a natural number). It is known that through any point inside the circle, no more than two chords with endpoints at the marked points pass. We will call a matching a set of $N$ chords with endpoints at the marked points such that each marked point is the endpoint ... | 1 | 116 | 1 |
math | 5. Find the maximum value of the expression $(\sin 2 x+\sin y+\sin 3 z)(\cos 2 x+\cos y+\cos 3 z)$. (15 points) | 4.5 | 44 | 3 |
math | 8-6 Let the sequence $a_{0}, a_{1}, a_{2}, \cdots$ satisfy
$$
a_{0}=a_{1}=11, a_{m+n}=\frac{1}{2}\left(a_{2 m}+a_{2 n}\right)-(m-n)^{2}, m, n \geqslant 0 .
$$
Find $a_{45}$. | 1991 | 91 | 4 |
math | If the sum of the points that appear in these $n$ dice rolls is greater than $2^{n}$, then it counts as passing. Ask:
(1) What is the maximum number of levels a person can pass in this game?
(2) What is the probability that he can pass the first three levels consecutively?
(Note: A die is a uniform cube with 1, 2, 3, 4... | \frac{100}{243} | 129 | 11 |
math | 4.5.1 ** Find the smallest real number $m$, such that for any positive real numbers $a, b, c$ satisfying $a+b+c=1$, we have $m\left(a^{3}+b^{3}+c^{3}\right) \geqslant 6\left(a^{2}+b^{2}+c^{2}\right)+1$. | 27 | 85 | 2 |
math | 4. Consider triangle $ABC$, where $AC = BC$, $m(ACB) = 90^{\circ}$, and triangle $DAB$, where $DA = DB$, located in perpendicular planes. Let $\quad M \in (BC), \quad BM = 2CM, \quad N \in (AC)$, $AC = 3AN, P \in MN \cap AB$, $T$ be the midpoint of segment $[AB]$, and $G$ be the centroid of triangle $DAB$. Calculate th... | \sqrt{6} | 169 | 5 |
math | 19. Suppose $x, y, z$ and $\lambda$ are positive real numbers such that
$$
\begin{aligned}
y z & =6 \lambda x \\
x z & =6 \lambda y \\
x y & =6 \lambda z \\
x^{2}+y^{2}+z^{2} & =1
\end{aligned}
$$
Find the value of $(x y z \lambda)^{-1}$. | 54 | 96 | 2 |
math | (2) Let the sequence $\left\{a_{n}\right\}$ satisfy: $a_{1}=\frac{1}{2}, a_{n+1}=\frac{1+a_{n}}{1-a_{n}} \quad(n \geqslant 1)$, then $a_{2008}=$ | -\frac{1}{3} | 73 | 7 |
math | Let's find all positive integers $n$ for which $2^{n}-1$ and $2^{n+2}-1$ are both prime, and $2^{n+1}-1$ is not divisible by 7. | 3 | 49 | 1 |
math | I4.3 Given that there are $R$ odd numbers in the digits of the product of the two $Q$-digit numbers 1111...11 and $9999 \ldots 99$, find the value of $R$. | 12 | 56 | 2 |
math | ## Task 3 - 140813
Given a circle $k_{1}$ with radius $r_{1}$ and center $M$. Around $M$, a circle $k_{2}$ is to be drawn such that the area of the annulus between $k_{1}$ and $k_{2}$ is three times the area of the circle $k_{1}$.
Calculate the radius $r_{2}$ of the circle $k_{2}$! | r_{2}=2r_{1} | 99 | 9 |
math | 1. In the field of real numbers, solve the equation $2^{|x+1|}-2^{x}=1+\left|2^{x}-1\right|$. | -2anyx\geq0 | 38 | 8 |
math | 【Question 4】A natural number can be expressed as the sum of 5 consecutive natural numbers, and also as the sum of 7 consecutive natural numbers. Therefore, if we arrange the natural numbers that meet the above conditions in ascending order, the first 3 numbers are $\qquad$ - | 35,70,105 | 61 | 9 |
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