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 | 2. Given positive integer $n=a b c<10000, a, b, c$ are all prime numbers, and $2 a+3 b=c, 4 a+c+1=4 b$. Find the value of $n$.
| 1118 | 55 | 4 |
math | 16. Given the function $y=\log _{3} \frac{m x^{2}+8 x+n}{x^{2}+1}$ defined on $\mathbf{R}$, its maximum value is 2 and its minimum value is 0. Find the values of the real numbers $m$ and $n$. | =n=5 | 71 | 3 |
math | 8. In an $m$ row by 10 column grid, fill each cell with either 0 or 1, such that each column contains exactly three 1s. Let the sum of the numbers in the $i$-th row ($i=1,2, \cdots, m$) be denoted as $x_{i}$, and for any two columns, there exists a row where the cells intersecting with these two columns are both filled... | 5 | 148 | 1 |
math | 6.3. Find all solutions to the puzzle: ТУК + ТУК + ТУК + ТУК + ТУК = СТУК. The same letters correspond to the same digits, different letters correspond to different digits. | 250+\ldots+250=1250750+\ldots+750=3750 | 52 | 30 |
math | B3. Find all real solutions of the equation $\left(x^{2}+x+3\right)^{2}+3\left(x^{2}+x-1\right)=28$. | x_{1}=-2,x_{2}=1 | 44 | 11 |
math | Let $\triangle ABC$ be a triangle with $AB=85$, $BC=125$, $CA=140$, and incircle $\omega$. Let $D$, $E$, $F$ be the points of tangency of $\omega$ with $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ respectively, and furthermore denote by $X$, $Y$, and $Z$ the incenters of $\triangle AEF$, $\triangle BFD$, and $\t... | 30 | 138 | 2 |
math | 123. Decompose the sequence $\Phi_{11}^{0}$
$0,1,1,2,3,5,8,2,10,1,0,1,1,2,3,5,8,2,10,1, \ldots$
into the sum of two 11 -Fibonacci sequences, which are also geometric progressions ${ }^{1}$ ). | c_{n}=3\cdot8^{n}+8\cdot4^{n} | 91 | 19 |
math | 1.200 people stand in a circle, some of whom are honest people, and some are liars. Liars always tell lies, while honest people tell the truth depending on the situation. If both of his neighbors are honest people, he will definitely tell the truth; if at least one of his neighbors is a liar, he may sometimes tell the ... | 150 | 129 | 3 |
math | 22nd CanMO 1990 Problem 2 n(n + 1)/2 distinct numbers are arranged at random into n rows. The first row has 1 number, the second has 2 numbers, the third has 3 numbers and so on. Find the probability that the largest number in each row is smaller than the largest number in each row with more numbers. Solution | \frac{2^n}{(n+1)!} | 79 | 12 |
math | 66. In the workshop, seven men and three women work. By their employee numbers, three people are randomly selected. Find the probability that all the selected individuals will be men. | \frac{7}{24} | 37 | 8 |
math | 11. (10 points) Among the 6 moments, $6: 30, 6: 31, 6: 32, 6: 33, 6: 34, 6: 35$, the $\qquad$ moment the hour and minute hands are closest, and the $\qquad$ moment the hour and minute hands are farthest. | 6:33,6:30 | 86 | 9 |
math | What relationship exists between the variables $x$ and $y$ if they depend on the parameter $z$ as follows:
$$
x=\frac{z^{4}+z^{3}+z^{2}+z+1}{z^{2}+1}, \quad y=\frac{z^{3}+z^{2}+z+1}{z^{2}+1}
$$ | (y^{2}-2y+2)(x+y-y^{2})-1=0 | 85 | 19 |
math | 1. Given $x \in[0, \pi]$. Then the range of the function
$$
f(x)=2 \sin 3 x+3 \sin x+3 \sqrt{3} \cos x
$$
is $\qquad$ . | [-3\sqrt{3},8] | 56 | 9 |
math | Example 7 When $m=$ $\qquad$, the polynomial
$$
12 x^{2}-10 x y+2 y^{2}+11 x-5 y+m
$$
can be factored into the product of two linear factors.
(1992, Zhengzhou City Junior High School Mathematics Competition) | 2 | 70 | 1 |
math | Positive real numbers are arranged in the form:
$ 1 \ \ \ 3 \ \ \ 6 \ \ \ 10 \ \ \ 15 ...$
$ 2 \ \ \ 5 \ \ \ 9 \ \ \ 14 ...$
$ 4 \ \ \ 8 \ \ \ 13 ...$
$ 7 \ \ \ 12 ...$
$ 11 ...$
Find the number of the line and column where the number 2002 stays. | (15, 49) | 109 | 8 |
math | Example 14. A die is rolled 10 times. What is the probability that the faces $1,2,3,4,5,6$ will appear respectively $2,3,1,1,1,2$ times (event $A$)? | 0.002 | 57 | 5 |
math | 812. Solve in integers $x, y$ and $z$ the equation
$$
5 x^{2} + y^{2} + 3 z^{2} - 2 y z = 30
$$ | (1;5;0),(1;-5;0),(-1;5;0),(-1;-5;0) | 49 | 27 |
math | Condition of the problem
Find the differential $d y$.
$$
y=\ln \left|x^{2}-1\right|-\frac{1}{x^{2}-1}
$$ | \frac{2x^{3}}{(x^{2}-1)^{2}}\cdot | 40 | 20 |
math | 10. Given a sequence $\left\{a_{n}\right\}$ with 100 terms, satisfying $a_{1}=0, a_{100}=475$,
and $\left|a_{k+1}-a_{k}\right|=5(k=1,2, \cdots, 99)$.
Then the number of different sequences that meet the conditions is $\qquad$. | 4851 | 90 | 4 |
math | 13. Given the parabola $C: y=a x^{2}(a>0)$, the line $y=x+2$ intersects the parabola at points $A$ and $B$, and $M$ is the midpoint of segment $AB$. A vertical line through $M$ intersects the parabola $C$ at point $N$.
(1) Prove: The tangent line $l$ to the parabola $C$ at point $N$ is parallel to $AB$.
(2) Does there ... | a=\frac{7}{8} | 161 | 8 |
math | $10 x$ is a positive integer, the last two digits of $3^{x}$ form a prime number, then the two values of $x$ closest to 2008 are $\qquad$ and $\qquad$ . | 20082009 | 51 | 8 |
math | Solve the following equation:
$$
(x+y)^{2}=(x+1)(y-1)
$$ | -1,1 | 24 | 4 |
math | 12 bags contain 8 white balls and 2 red balls. Each time, one ball is randomly taken out, and then 1 white ball is put back. The probability that exactly all red balls are taken out by the 4th draw is $\qquad$ . | 0.0434 | 56 | 6 |
math | In a chess tournament, all players faced each other exactly once. In each match, a player earns 1 point for a win, $1 / 2$ for a draw, and 0 points for a loss. At the end of the tournament, a reporter summed up the scores of all the players and obtained 190 points. In this type of tournament, the winner is the one who ... | 20 | 127 | 2 |
math | A marble is placed on each $33$ unit square of a $10*10$ chessboard. After that, the number of marbles in the same row or column with that square is written on each of the remaining empty unit squares. What is the maximum sum of the numbers written on the board?
| 438 | 65 | 5 |
math | 308. The scale division value of the amperemeter is 0.1 A. The readings of the amperemeter are rounded to the nearest whole division. Find the probability that an error exceeding $0.02 \mathrm{~A}$ will be made during the reading. | 0.6 | 62 | 3 |
math | 5.6. (Yugoslavia, 83). Find all values of $n \in \mathbf{N}$ that have the following property: if the numbers $n^{3}$ and $n^{4}$ are written next to each other (in the decimal system), then in the resulting record each of the 10 digits $0,1, \ldots, 9$ will appear exactly once. | 18 | 89 | 2 |
math | Three. (25 points) If positive numbers $a, b, c$ satisfy
$$
\left(\frac{b^{2}+c^{2}-a^{2}}{2 b c}\right)^{2}+\left(\frac{c^{2}+a^{2}-b^{2}}{2 c a}\right)^{2}+\left(\frac{a^{2}+b^{2}-c^{2}}{2 a b}\right)^{2}=3 \text {, }
$$
find the value of the algebraic expression
$$
\frac{b^{2}+c^{2}-a^{2}}{2 b c}+\frac{c^{2}+a^{2}-... | 1 | 188 | 1 |
math | Determine the number of positive integers $n$ satisfying:
[list]
[*] $n<10^6$
[*] $n$ is divisible by 7
[*] $n$ does not contain any of the digits 2,3,4,5,6,7,8.
[/list] | 104 | 65 | 3 |
math | 1. In the four-digit number 4753 , three two-digit numbers are formed by successive pairs of digits $(47,75,53)$. Exactly two of these two-digit numbers are prime. Find all four-digit numbers in which all four digits are prime, and all three two-digit numbers formed by successive digits are prime. | 2373,3737,5373,7373 | 71 | 19 |
math | 4. (6 points) The older brother and the younger brother each bought several apples. The older brother said to the younger brother: “If I give you one apple, we will have the same number of apples.” The younger brother thought for a moment and said to the older brother: “If I give you one apple, your number of apples wi... | 12 | 91 | 2 |
math | . Determine all polynomials $p(x)$ with real coefficients such that
$$
p\left((x+1)^{3}\right)=(p(x)+1)^{3}
$$
and
$$
p(0)=0
$$
Answer: $p(x)=x$. | p(x)=x | 60 | 4 |
math | IMO 1964 Problem B2 5 points in a plane are situated so that no two of the lines joining a pair of points are coincident, parallel or perpendicular. Through each point lines are drawn perpendicular to each of the lines through two of the other 4 points. Determine the maximum number of intersections these perpendiculars... | 315 | 70 | 3 |
math | 11. Two concentric circles have radii 2006 and 2007. $A B C$ is an equilateral triangle inscribed in the smaller circle and $P$ is a point on the circumference of the larger circle. Given that a triangle with side lengths equal to $P A, P B$ and $P C$ has area $\frac{a \sqrt{b}}{c}$, where $a, b, c$ are positive intege... | 4020 | 287 | 4 |
math | 4 Let $n \in \mathbf{N}^{*}$, and let $d(n)$ denote the number of all positive divisors of $n$, and $\varphi(n)$ denote the number of integers in $1,2, \cdots, n$ that are coprime to $n$.
Find all non-negative integers $c$, such that there exists a positive integer $n$, satisfying
$$
d(n)+\varphi(n)=n+c,
$$
and for ea... | 0 | 122 | 1 |
math | Four, (20 points) If the hyperbola $y^{2}-x^{2}=1$ and $\frac{x y-x-y+1}{x^{2}-3 x+2}=k$ have a unique common point, find all possible values of $k$.
untranslated part:
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
Note: The note at the end is not part of the translation and is provided for context. The actual translation i... | 1, -1, 0, \frac{4}{5}, 1+\sqrt{2}, 1-\sqrt{2} | 111 | 29 |
math | Numbers $x,y,z$ are positive integers and satisfy the equation $x+y+z=2013$. (E)
a) Find the number of the triplets $(x,y,z)$ that are solutions of the equation (E).
b) Find the number of the solutions of the equation (E) for which $x=y$.
c) Find the solution $(x,y,z)$ of the equation (E) for which the product $xyz$ b... | (671, 671, 671) | 97 | 15 |
math | 36. Given $\frac{x}{1 \times 2}+\frac{x}{2 \times 3}+\frac{x}{3 \times 4}+\ldots+\frac{x}{999 \times 1000}=999$. Then $x=$ | 1000 | 60 | 4 |
math | 6. A circle passes through vertices $A$ and $B$ of triangle $ABC$ and intersects its sides $AC$ and $BC$ at points $Q$ and $N$ respectively, such that $AQ: QC = 5: 2$ and $CN: NB = 5: 2$. Find $AB$, if $QN = 5 \sqrt{2}$. | 7\sqrt{5} | 82 | 6 |
math | Determine the number of pairs of integers $(a, b)$ such that $1 \leqslant a \leqslant 30, 3 \leqslant b \leqslant 30$ and such that $a$ is divisible by $b$ and by $b-2$. | 22 | 68 | 2 |
math | 12. The vertex of a surface of revolution that is conical coincides with the origin $O$ of a rectangular coordinate system $O x y z$, the axis of rotation coincides with the positive direction of the $O z$ axis, and the ray, by the rotation of which the surface is obtained, forms an angle $\varphi$ with the $O z$ axis.... | x^{2}+y^{2}-z^{2}\operatorname{tg}^{2}\varphi=0 | 89 | 25 |
math | Find the sum of the real roots of the polynomial \[ \prod_{k=1}^{100} \left( x^2-11x+k \right) = \left( x^2-11x+1 \right)\left( x^2-11x+2 \right)\dots\left(x^2-11x+100\right). \][i]Proposed by Evan Chen[/i] | 330 | 97 | 3 |
math | 4. For any positive integer $n$, let $a_{n}$ be the smallest positive integer such that $n \mid a_{n}$!. If $\frac{a_{n}}{n}=\frac{2}{5}$, then $n=$ $\qquad$ . | 25 | 59 | 2 |
math | 7. Ice Cream (from 6th grade, 2 points.) Anya wants to buy an ice cream that costs 19 rubles. In her pocket, she has two 10-ruble coins, two 5-ruble coins, and one 2-ruble coin. Anya takes out three coins without looking. Find the probability that the coins she took out will be enough to pay for the ice cream. | 0.5 | 88 | 3 |
math | Example 2. Find $\lim _{x \rightarrow 0} \frac{\ln (1+x)}{x}$. | 1 | 27 | 1 |
math | Find all numbers with the following properties:
- the number is even,
- the number is two-digit,
- the sum of its digits is greater than 6,
- the digit in the tens place is at least 4 greater than the digit in the units place.
(M. Smitková) | 90,80,70,92,82,72,62,94,84 | 61 | 26 |
math | 1.50 For what integer $k$ does the equation
$$
\left(k^{2}-1\right) x^{2}-6(3 k-1) x+72=0
$$
have two distinct positive integer roots?
(Xining, Qinghai Province, China Junior High School Mathematics Competition, 1983) | 2 | 73 | 1 |
math | Solve the system of equations
$$
\left\{\begin{array}{l}
x+y+z=0 \\
x y z=-1 \\
x^{2}+y^{2}+z^{2}=4
\end{array}\right.
$$ | x,y,z=1,\frac{-1+\sqrt{5}}{2},\frac{-1-\sqrt{5}}{2} | 55 | 29 |
math | 5. [5] The function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfies $f\left(x^{2}\right) f^{\prime \prime}(x)=f^{\prime}(x) f^{\prime}\left(x^{2}\right)$ for all real $x$. Given that $f(1)=1$ and $f^{\prime \prime \prime}(1)=8$, determine $f^{\prime}(1)+f^{\prime \prime}(1)$. | 6 | 112 | 1 |
math | G4.4 Let $[x]$ be the largest integer not greater than $x$, for example, $[2.5]=2$.
If $a=1+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\cdots+\frac{1}{2004^{2}}$ and $S=[a]$, find the value of $a$. | 1 | 84 | 1 |
math | 2. Let $p$ be a positive real number. Among all pairs of positive real numbers $(x, y)$ that satisfy the equation $x y(x+y)=p$, find the one for which the value of the expression $x^{3}+y^{3}$ is the smallest. | (\frac{p}{2})^{\frac{1}{3}} | 60 | 15 |
math | 232. $\frac{1}{12} \lg ^{2} x=\frac{1}{3}-\frac{1}{4} \lg x$. | x_{1}=0.0001,x_{2}=10 | 37 | 16 |
math | 22. The base of a right prism is a right triangle. All edges of this prism have natural lengths, and the areas of some two of its faces are 13 and 30. Find the sides of the base of this prism. | 5,12,13 | 51 | 7 |
math | 7. If the equation about $x$
$$
x^{3}+a x^{2}+b x-4=0\left(a 、 b \in \mathbf{N}_{+}\right)
$$
has a positive integer solution, then $|a-b|=$ | 1 | 61 | 1 |
math | 1. For a finite set $A$, there exists a function $f: \mathbf{N}^{*} \rightarrow A$, with the following property: if $i, j \in \mathbf{N}^{*}$, and $|i-j|$ is a prime number, then $f(i) \neq f(j)$. How many elements must the set $A$ have at least? | 4 | 87 | 1 |
math | Example 13 Determine the largest real number $z$, such that $x+y+z=5, xy+yz+zx=3$, and $x, y$ are also real numbers.
(7th Canadian Intermediate Mathematics Competition) | \frac{13}{3} | 48 | 8 |
math | Subject (3). Determine the natural numbers $m, n$ such that $85^{m}-n^{4}=4$. | =1,n=3 | 27 | 5 |
math | 7.9 A gardener is going to plant three maple trees, four oak trees, and five birch trees in a row. He randomly determines the order of these trees, and all different arrangements are equally probable. Use $\frac{m}{n}$ to represent the probability that no two birch trees are adjacent (simplified to the lowest terms), a... | 106 | 79 | 3 |
math | Agakhanov N.K.
Find all angles $\alpha$ for which the set of numbers $\sin \alpha, \sin 2 \alpha, \sin 3 \alpha$ coincides with the set $\cos \alpha, \cos 2 \alpha, \cos 3 \alpha$. | \alpha=\frac{\pi}{8}+\frac{\pik}{2} | 62 | 17 |
math | 1. Let $f(n)=3 n^{2}-3 n+1$. Find the last four digits of $f(1)+f(2)+\cdots+f(2010)$.
(1 mark)設 $f(n)=3 n^{2}-3 n+1$ 。求 $f(1)+f(2)+\cdots+f(2010)$ 的最後四位數字。 | 1000 | 89 | 4 |
math | 1-195 1 to 1982 natural numbers are arranged in a certain order, one after another. The computer reads two adjacent numbers from left to right (the 1st and the 2nd, the 2nd and the 3rd, etc.), until the last two numbers, and if the larger number is on the left, the computer swaps their positions. Then the computer read... | 100 | 137 | 3 |
math | 11. Let $x, y, z \in \mathbf{R}^{+}$, and satisfy $x y z(x+y+z)=1$, find the minimum value of $(x+y)(x+z)$. | 2 | 47 | 1 |
math | 2. Given an isosceles triangle $ABC, \overline{AB}=\overline{AC}$. The angle bisector of $\angle ABC$ intersects side $AC$ at point $D$. If $\overline{BC}=\overline{AB}+2 \overline{AD}$, calculate the measures of the angles of the triangle. | \measuredangleABC=\measuredangleACB=30,\measuredangleCAB=120 | 76 | 24 |
math | Example 3: If $a^{2}+b^{2}+c^{2}+3 \leqslant a b+3 b+2 c$, find the values of $a$, $b$, and $c$.
(1992, Hungarian Mathematical Olympiad) | a=1, b=2, c=1 | 62 | 11 |
math | Let $ x,y$ are real numbers such that $x^2+2cosy=1$. Find the ranges of $x-cosy$. | [-1, 1 + \sqrt{3}] | 32 | 12 |
math | 1. The number of proper subsets of the set $\left\{x \left\lvert\,-1 \leqslant \log _{\frac{1}{x}} 10<-\frac{1}{2}\right., x \in \mathbf{N}\right\}$ is | 2^{90}-1 | 64 | 6 |
math | 4.069 Find a three-digit number whose digits form a geometric progression. If 792 is subtracted from this number, the result is a number written with the same digits but in reverse order. If 4 is subtracted from the digit representing the hundreds, and the other digits of the desired number are left unchanged, the resu... | 931 | 81 | 3 |
math | (i) Find all integers $n \geqslant 1$ such that $n$ divides $2^{n}-1$.
(ii) Find all odd integers $n \geqslant 1$ such that $n$ divides $3^{n}+1$.
## - Solution - | 1 | 64 | 1 |
math | Let $a$ and $b$ be real numbers such that $a+b=\log_2( \log_2 3)$. What is the minimum value of $2^a + 3^b$ ? | \log_2 6 | 46 | 6 |
math | 1. Calculate the value of the expression $a^{2}-c^{2}+b^{2}+2 a b$ if $a=786, b=389$ and $c=175$. | 1350000 | 49 | 7 |
math |
C2. Fix integers $n \geq k \geq 2$. We call a collection of integral valued coins $n$-diverse if no value occurs in it more than $n$ times. Given such a collection, a number $S$ is $n$-reachable if that collection contains $n$ coins whose sum of values equals $S$. Find the least positive integer $D$ such that for any ... | n+k-1 | 128 | 4 |
math | 10. Consider the expression $1 * 2 * 3 * 4 * 5 * 6$.
Each star in the expression is to be replaced with either ' + ' or ' $x$ '.
$N$ is the largest possible value of the expression. What is the largest prime factor of $N$ ? | 103 | 69 | 3 |
math | $\underline{\text { templesovv }}$ D:
Two pirates are dividing a treasure consisting of two bags of coins and a diamond, acting according to the following rules. First, the first pirate takes some coins from any bag and moves the same number of coins from this bag to the other. Then the second pirate does the same (ch... | If\the\\of\\in\the\bags\differs\\no\\than\1,\the\diamond\will\go\to\the\\pirate,\otherwise\to\the\first | 137 | 43 |
math | 9. (16 points) Let point $F(1,0)$, line $l: x=-1$, line $l^{\prime} \perp l$ at point $P$, the perpendicular bisector of segment $P F$ intersects $l^{\prime}$ at point $Q$.
(1) Find the equation of the locus $C$ of point $Q$;
(2) Given point $H(1,2)$, a line passing through point $F$ and not perpendicular to the $x$-ax... | (-3,0)(1,0) | 171 | 9 |
math | ## Problem Statement
Calculate the volume of the tetrahedron with vertices at points \( A_{1}, A_{2}, A_{3}, A_{4} \) and its height dropped from vertex \( A_{4} \) to the face \( A_{1} A_{2} A_{3} \).
\( A_{1}(5 ; 2 ; 0) \)
\( A_{2}(2 ; 5 ; 0) \)
\( A_{3}(1 ; 2 ; 4) \)
\( A_{4}(-1 ; 1 ; 1) \) | 2\sqrt{3} | 128 | 6 |
math | 1. Let the function $y=\left|\log _{2} \frac{x}{2}\right|$ have a domain of $[m, n]$, and a range of $[0,2]$. Then the minimum length of the interval $[m, n]$ is | \frac{3}{2} | 59 | 7 |
math | Three. (50 points) Let $S=\{1,2, \cdots, 100\}$. Find the largest integer $k$, such that the set $S$ has $k$ distinct non-empty subsets with the property: for any two different subsets among these $k$ subsets, if their intersection is non-empty, then the smallest element in their intersection is different from the larg... | 2^{99}-1 | 103 | 6 |
math | 34. Consider the set $S=\{1,2,3, \cdots, 2010,2011\}$. A subset $T$ of $S$ is said to be a $k$-element RP-subset if $T$ has exactly $k$ elements and every pair of elements of $T$ are relatively prime. Find the smallest positive integer $k$ such that every $k$-element RP-subset of $S$ contains at least one prime number.... | 16 | 142 | 2 |
math | [ Combinatorics (miscellaneous).]
$[$ Estimation + example $]$
In a pond, 30 pikes were released, which gradually eat each other. A pike is considered full if it has eaten at least three pikes (full or hungry). What is the maximum number of pikes that can become full? | 9 | 70 | 1 |
math | 12. Let $f: \mathbf{N}^{*} \rightarrow \mathbf{N}^{*}$, and for all $m, n \in \mathbf{N}^{*}$, we have $f(f(m)+f(n))=m+n$, then $f(2005)=$ | 2005 | 71 | 4 |
math | 1. Solve the system of equations in the set of real numbers
$$
\left\{\begin{array}{l}
x^{2}+y^{2}+z=2 \\
x+y^{2}+z^{2}=2 \\
x^{2}+y+z^{2}=2
\end{array}\right.
$$ | (\frac{-1\\sqrt{17}}{4},\frac{-1\\sqrt{17}}{4},\frac{-1\\sqrt{17}}{4}),(1,0,1),(-\frac{1}{2},\frac{3}{2},-\frac{1}{2}),(0,1,1),(\frac{3}{2}, | 72 | 82 |
math | 13.413 An alloy consists of tin, copper, and zinc. If 20 g is separated from this alloy and melted with 2 g of tin, then in the newly obtained alloy, the mass of copper will be equal to the mass of tin. If, however, 30 g is separated from the original alloy and 9 g of zinc is added, then in this new alloy, the mass of ... | 50,40,10 | 107 | 8 |
math | For every positive integer $n$, let $s(n)$ be the sum of the exponents of $71$ and $97$ in the prime factorization of $n$; for example, $s(2021) = s(43 \cdot 47) = 0$ and $s(488977) = s(71^2 \cdot 97) = 3$. If we define $f(n)=(-1)^{s(n)}$, prove that the limit
\[ \lim_{n \to +\infty} \frac{f(1) + f(2) + \cdots+ f(n)}{n... | \frac{20}{21} | 156 | 10 |
math | 11.6. Solve the equation
$$
\cos 2x - \sin 2x + 2 \cos x + 1 = 0
$$ | x_{0}+2k\pi,k\in{Z},x_{0}\in{-\frac{\pi}{2},\frac{\pi}{2},\pi} | 36 | 38 |
math | 20. Suppose $x_{1}, x_{2}, \ldots, x_{2011}$ are positive integers satisfying
$$
x_{1}+x_{2}+\cdots+x_{2011}=x_{1} x_{2} \cdots x_{2011}
$$
Find the maximum value of $x_{1}+x_{2}+\cdots+x_{2011}$. | 4022 | 96 | 4 |
math | 23rd CanMO 1991 Problem 2 Find the sum of all positive integers which have n 1s and n 0s when written in base 2. | (2n-1)!/(n!(n-1)!)2^{2n-1}+(2n-2)!/(n!(n-2)!)(2^{2n-1}-1) | 38 | 44 |
math | [ Equations in integers ]
Solve the equation $x^{2}-5 y^{2}=1$ in integers.
# | \(x_{n},\y_{n}),wherex_{n}+y_{n}\sqrt{5}=(9+4\sqrt{5})^{n},n=0,1,2,3,\ldots | 26 | 49 |
math | Problem 14. In triangle $ABC$, the lengths of its heights are known: $h_{a} ; h_{b} ; h_{c}$. Find the area of this triangle. | \frac{1}{\sqrt{(\frac{1}{h_{}}+\frac{1}{h_{b}}+\frac{1}{h_{}})(\frac{1}{h_{}}+\frac{1}{h_{b}}-\frac{1}{h_{}})(\frac{1}{h_{}}+\frac{1}{h_{}}-\frac{1}{h_{} | 41 | 84 |
math | 13. (15 points) Let each Chinese character in the four idioms “一家之言”、“言扬行举”、“举世皆知”、“知行合一” represent one of 11 consecutive non-zero natural numbers, with the same character representing the same number and different characters representing different numbers. If the sum of the numbers represented by the four characters i... | 8 | 114 | 1 |
math | Let $f(x) = x^3 - 3x + b$ and $g(x) = x^2 + bx -3$, where $b$ is a real number. What is the sum of all possible values of $b$ for which the equations $f(x)$ = 0 and $g(x) = 0$ have a common root? | 0 | 76 | 1 |
math | A diagonal of a regular 2006-gon is called odd if its endpoints divide the boundary into two parts, each composed of an odd number of sides. Sides are also regarded as odd diagonals. Suppose the 2006-gon has been dissected into triangles by 2003 nonintersecting diagonals. Find the maximum possible number of isosceles t... | 1003 | 91 | 4 |
math | # Problem 5. Maximum 15 points
The great-grandfather-banker left a legacy to his newborn great-grandson. According to the agreement with the bank, the amount in the great-grandson's account increases. Every year, on the day after the birthday, the current amount is increased by 1 million rubles more than in the previo... | 36 | 176 | 2 |
math | Together, the products by four and by five of an integer use each digit from 1 to 9 once, and only once. What is this number? | 2469 | 32 | 4 |
math | ## Task A-3.2.
Determine all pairs of natural numbers $x$ and $y$ for which $\frac{x y^{2}}{x+y}$ is a prime number. | (2,2),(6,2) | 40 | 9 |
math | 10. (This question is worth 20 points) The sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=\frac{\pi}{6}, a_{n+1}=\arctan \left(\sec a_{n}\right)\left(n \in \mathbf{N}^{*}\right)$. Find the positive integer $m$, such that
$$
\sin a_{1} \cdot \sin a_{2} \cdots \cdot \sin a_{m}=\frac{1}{100} .
$$ | 3333 | 123 | 4 |
math | 9. If the equation $x^{2}+(m+2) x+m+5=0$ has only positive roots, then the range of real number $m$ is $\qquad$ . | -5<\leqslant-4 | 42 | 10 |
math | Example 7 Determine the smallest natural number $k$, such that for any $a \in[0,1]$ and any $n \in \mathbf{N}$, we have $a^{k}(1-a)^{n}<\frac{1}{(n+1)^{3}}$. | 4 | 64 | 1 |
math | XII OM - II - Task 4
Find the last four digits of the number $ 5^{5555} $. | 8125 | 28 | 4 |
math | [Divisibility of numbers. General properties] $[$ Prime numbers and their properties ]
Find all natural numbers $n$ such that the number $(n-1)!$ does not divide $n^2$.
# | 8,9,p,2p | 46 | 7 |
math | 5. Given a real number $x$ between 0 and 1, consider its decimal representation $0, c_{1} c_{2} c_{3} \ldots$ We call $B(x)$ the set of different subsequences of six consecutive digits that appear in the sequence $c_{1} c_{2} c_{3} \ldots$
For example, $B(1 / 22)=\{045454,454545,545454\}$.
Determine the minimum numbe... | 7 | 160 | 1 |
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