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 | 2. A positive integer $n$ is said to be 'good' if $n^{2}-1$ can be written as the product of three distinct prime numbers. Find the sum of the five smallest 'good' integers.
(1 mark)
對於正整數 $n$, 若 $n^{2}-1$ 可寫成三個不同質數之積, 則 $n$ 稱為「好數」。求最小的五個「好數」之和。 | 104 | 104 | 3 |
math | 3. Let $X$ be a 100-element set. Find the smallest positive integer $n$ such that for any sequence $A_{1}, A_{2}, \cdots, A_{n}$ of subsets of $X$, there exist $1 \leqslant i<j<k \leqslant n$ satisfying $A_{i} \subseteq A_{j} \subseteq A_{k}$ or $A_{i} \supseteq A_{j} \supseteq A_{k}$.
(Zhai Zhenhua) | \mathrm{C}_{102}^{51}+1 | 117 | 15 |
math | 1. Given that $x$ and $y$ are positive integers, and satisfy $x y + x + y = 71$, $x^{2} y + x y^{2} = 880$. Then $x^{2} + y^{2} =$ $\qquad$ . | 146 | 64 | 3 |
math | 16. Given that $S_{n}$ is the sum of the first $n$ terms of the sequence $\left\{a_{n}\right\}$, for any positive integer $n$, we have
$$
(1-b) S_{n}=-b a_{n}+4^{n}(b>0)
$$
(1) Find the general term formula of the sequence $\left\{a_{n}\right\}$;
(2) Let $c_{n}=\frac{a_{n}}{4^{n}}\left(n \in \mathbf{N}_{+}\right)$, if... | \left(0, \frac{5}{2}\right] | 167 | 14 |
math | 12.18 A pedestrian set out for a walk from point $A$ at a speed of $v$ km/h. After he had walked 6 km from $A$, a cyclist set out after him from $A$ at a speed 9 km/h greater than the pedestrian's speed. When the cyclist caught up with the pedestrian, they turned back and returned to $A$ together at a speed of 4 km/h. ... | 6 | 107 | 1 |
math | 4. The sum of three natural numbers is 3053. If we divide the first number by half of the second, we get a quotient of 3 and a remainder of 5, and if we divide the third number by a quarter of the second, we get a quotient of 5 and a remainder of 3.
Find the numbers. | 1223,812,1018 | 73 | 13 |
math | 6. (9th "Hope Cup" Invitational Competition Question) Let $\alpha, \beta$ be the roots of the equations $\log _{2} x+x+2=0$ and $2^{x}+x+2=0$ respectively, then the value of $\alpha+\beta$ is equal to
保留源文本的换行和格式，翻译结果如下：
6. (9th "Hope Cup" Invitational Competition Question) Let $\alpha, \beta$ be the roots of the equ... | -2 | 148 | 2 |
math | 4. Problem: Find all triples $(x, p, n)$ of non-negative integers such that $p$ is prime and
$$
2 x(x+5)=p^{n}+3(x-1) \text {. }
$$ | (2,5,2)(0,3,1) | 50 | 13 |
math | A Vinczlér. A poor Vinczlér was just milking, when the landowner approached and asked for 4 messzely of must to drink. The poor Vinczlér only had two measuring vessels, one 3 messzely, the other 5 messzely, the question is, how did he still manage to give 4 messzely of drink to the lord? 1[^0]
[^0]: 1* We are happ... | 4 | 116 | 1 |
math | Find all functions $f\colon \mathbb{R}\to\mathbb{R}$ that satisfy $f(x+y)-f(x-y)=2y(3x^2+y^2)$ for all $x,y{\in}R$
______________________________________
Azerbaijan Land of the Fire :lol: | f(x) = x^3 + a | 67 | 10 |
math | 2. On four cards, four numbers are written, the sum of which is 360. You can choose three cards with the same number written on them. There are two cards, one of which has a number three times larger than the other. What numbers could be written on the cards? | 36,108,108,108or60,60,60,180 | 61 | 27 |
math | 15. In the 6 circles below, fill in $1,2,3,4,5,6$, using each number only once, so that the sum of the three numbers on each side is equal. | 9 | 45 | 1 |
math | \section*{Exercise 2 - 081022}
Determine all real numbers \(x\) that satisfy the condition \(\log _{2}\left[\log _{2}\left(\log _{2} x\right)\right]=0\)! | 4 | 58 | 1 |
math | Given a triangle with sides $A B=2, B C=3, A C=4$. A circle is inscribed in it, and the point $M$ where the circle touches side $B C$ is connected to point $A$. Circles are inscribed in triangles $A M B$ and $A M C$. Find the distance between the points where these circles touch the line $A M$. | 0 | 85 | 1 |
math | \section*{Exercise 1 - 011121}
\(3,4,5\) is a so-called Pythagorean triplet, as \(3^{2}+4^{2}=5^{2}\).
It is the only such triplet whose elements differ by only 1. Are there other triplets for the equation \(a^{2}+b^{2}=c^{2}\) where \(c=b+1\)?
What pattern can you recognize here? Try to find an expression that allo... | ^{2}+b^{2}=^{2} | 115 | 11 |
math | 1. Cyclists Petya, Vlad, and Timur simultaneously started a warm-up race on a circular cycling track. Their speeds are 27 km/h, 30 km/h, and 32 km/h, respectively. After what shortest time will they all be at the same point on the track again? (The length of the cycling track is 400 meters.) | 24 | 80 | 2 |
math | Example 3 The function $f$ is defined on the set of positive integers, and satisfies
$$
f(x)=\left\{\begin{array}{l}
n-3, \quad n \geqslant 1000 \\
f(f(n+5)), 1 \leqslant n<1000 .
\end{array}\right.
$$
Find $f(84)$. | 997 | 90 | 3 |
math | Let it be known that all roots of a certain equation $x^{3} + p x^{2} + q x + r = 0$ are positive. What additional condition must the coefficients $p, q$, and $r$ satisfy so that a triangle can be formed from segments whose lengths are equal to these roots? | p^{3}-4pq+8r>0 | 67 | 11 |
math | 16. Given a function $f(x)$ defined on $(0,+\infty)$ that satisfies:
(1) For any $x \in(0, \infty)$, it always holds that $f(2 x)=2 f(x)$;
(2) When $x \in(1,2]$, $f(x)=2-x$.
The following conclusions are given: (1) For any $m \in \mathbf{Z}$, $f\left(2^{m}\right)=0$;
(2) The range of the function $f(x)$ is $[0,+\infty... | (1)(2)(4) | 243 | 7 |
math | G6.1 If $12345 \times 6789=a \times 10^{p}$ where $p$ is a positive integer and $1 \leq a<10$, find $p$. | 7 | 50 | 1 |
math |
Zadatak B-4.1. Ako je
$$
f\left(\log _{3} x\right)=\frac{\log _{3} \frac{9}{x^{4}}}{\log _{0 . \dot{3}} x-\log _{\sqrt{3}} x} \quad \text { i } \quad(f \circ g)(x)=e^{x}
$$
koliko je $g(\ln 2)$ ?
| -1 | 102 | 2 |
math | Six circles form a ring with with each circle externally tangent to two circles adjacent to it. All circles are internally tangent to a circle $C$ with radius $30$. Let $K$ be the area of the region inside circle $C$ and outside of the six circles in the ring. Find $\lfloor K \rfloor$. | 942 | 69 | 3 |
math | 3. Set $A=\left\{(x, y) \left\lvert\,\left\{\begin{array}{l}y=\sqrt{1-x}, \\ y=1-x^{2}\end{array}\right\}\right.\right.$ has the number of subsets as $\qquad$ | 8 | 65 | 1 |
math | 3B. In the set of real numbers, solve the system of equations
$$
\left\{\begin{array}{l}
x+y+z=\frac{13}{3} \\
\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{13}{3} \\
x y z=1
\end{array}\right.
$$ | (1,3,\frac{1}{3}) | 84 | 11 |
math | 3. Given $0<\alpha \leqslant \beta \leqslant \gamma, \alpha+\beta+\gamma=\pi$. Then the range of $\min \left\{\frac{\sin \beta}{\sin \alpha}, \frac{\sin \gamma}{\sin \beta}\right\}$ is . $\qquad$ | \left[1, \frac{\sqrt{5}+1}{2}\right) | 74 | 19 |
math | 6. 22 For a given positive integer $k$, define $f_{1}(k)$ as the square of the sum of the digits of $k$, and let
$$f_{n+1}(k)=f_{1}\left(f_{n}(k)\right),$$
Find: $f_{1991}\left(2^{1990}\right)$.
| 256 | 83 | 3 |
math | 2. Kolya is twice as old as Olya was when Kolya was as old as Olya is now. And when Olya is as old as Kolya is now, their combined age will be 36 years. How old is Kolya now?
ANS: 16 years. | 16 | 65 | 2 |
math | IMO 1981 Problem A3 Determine the maximum value of m 2 + n 2 , where m and n are integers in the range 1, 2, ... , 1981 satisfying (n 2 - mn - m 2 ) 2 = 1. | 1597^2+987^2 | 62 | 12 |
math | 8. Magic Pies (6th - 11th grades, 1 point). Alice has six magic pies in her pocket - two enlarging (eat one and you grow), and the rest are shrinking (eat one and you shrink). When Alice met Mary Ann, she took out three pies from her pocket without looking and gave them to Mary. Find the probability that one of the gir... | 0.4 | 88 | 3 |
math | 192. Solve the system of equations in integers:
$$
\left\{\begin{array}{l}
3 x-2 y+4 z+2 t=19 \\
5 x+6 y-2 z+3 t=23
\end{array}\right.
$$ | x=16z-18y-11,\quad=28y-26z+26 | 62 | 25 |
math | 11. A, B, and C together have 100 yuan. If A's money becomes 6 times the original, B's money becomes $1 / 3$ of the original, and C's money remains unchanged, then the three of them still have a total of 100 yuan. C's money does not exceed 30 yuan. Question: How much money do A, B, and C each have? | x=10, y=75, z=15 | 91 | 14 |
math | Problem 6.6. Several oranges (not necessarily of equal weight) were picked from a tree. When they were weighed, it turned out that the weight of any three oranges taken together is less than $5 \%$ of the total weight of the remaining oranges. What is the smallest number of oranges that could have been picked? | 64 | 67 | 2 |
math | (British MO) Determine the greatest value that the expression
$$
x^{2} y+y^{2} z+z^{2} x-x y^{2}-y z^{2}-z x^{2}
$$
can take for $0 \leqslant x, y, z \leqslant 1$. | \frac{1}{4} | 70 | 7 |
math | Question 173, Given that a square has three vertices on the parabola $y=x^{2}$, find the minimum value of the area of such a square.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.
(Note: The note itself is not part of the tra... | 2 | 122 | 1 |
math | Find the smallest positive integer $n$ that satisfies the following:
We can color each positive integer with one of $n$ colors such that the equation $w + 6x = 2y + 3z$ has no solutions in positive integers with all of $w, x, y$ and $z$ having the same color. (Note that $w, x, y$ and $z$ need not be distinct.) | 4 | 89 | 1 |
math | 61st Putnam 2000 Problem A3 An octagon is incribed in a circle. One set of alternate vertices forms a square area 5. The other set forms a rectangle area 4. What is the maximum possible area for the octagon? Solution | 3\sqrt{5} | 57 | 6 |
math | For a fixed positive integer $n \geq 3$ we are given a $n$ $\times$ $n$ board with all unit squares initially white. We define a [i]floating plus [/i]as a $5$-tuple $(M,L,R,A,B)$ of unit squares such that $L$ is in the same row and left of $M$, $R$ is in the same row and right of $M$, $A$ is in the same column and abov... | 4n-4 | 191 | 6 |
math | [ Decimal numeral system ]
Try to find all natural numbers that are 5 times larger than their last digit.
# | 25 | 23 | 2 |
math | 3. One drilling rig on an oil drilling rig can complete a job in 20 days. The second drilling rig would complete the same job in 30 days. If the first and second drilling rigs are joined by a third, all three together would finish the job in 7 days. How long would the third drilling rig take to complete the job on the ... | 16.8 | 79 | 4 |
math | Example 22 Simplify
$$
\frac{\left(z-z_{1}\right)\left(z^{\prime}-z_{1}\right)}{\left(z_{1}-z_{2}\right)\left(z_{1}-z_{3}\right)}+
$$
$$
\frac{\left(z-z_{2}\right)\left(z^{\prime}-z_{2}\right)}{\left(z_{2}-z_{3}\right)\left(z_{2}-z_{1}\right)}+\frac{\left(z-z_{3}\right)\left(z^{\prime}-z_{3}\right)}{\left(z_{3}-z_{1}\... | 1 | 154 | 1 |
math | Let $S=\{1,2,3,...,12\}$. How many subsets of $S$, excluding the empty set, have an even sum but not an even product?
[i]Proposed by Gabriel Wu[/i] | 31 | 49 | 2 |
math | Let $S$ be a subset of $\{0,1,2,\ldots,98 \}$ with exactly $m\geq 3$ (distinct) elements, such that for any $x,y\in S$ there exists $z\in S$ satisfying $x+y \equiv 2z \pmod{99}$. Determine all possible values of $m$. | m = 3, 9, 11, 33, 99 | 86 | 20 |
math | 5. The force with which the airflow acts on the sail can be calculated using the formula
$F=\frac{A S \rho\left(v_{0}-v\right)^{2}}{2}$, where $A-$ is the aerodynamic force coefficient, $S-$ is the area of the sail $S$ $=4 \mathrm{M}^{2} ; \rho-$ is the density of air, $v_{0}$ - is the wind speed $v_{0}=4.8 \mathrm{~m... | 1.6\mathrm{M}/\mathrm{} | 229 | 11 |
math | 11. Given the function $f(x)=-x^{2}+x+m+2$, if the solution set of the inequality $f(x) \geqslant|x|$ contains exactly one integer, then the range of the real number $m$ is $\qquad$ . | [-2,-1) | 60 | 5 |
math | 10.4. Find all functions $f(x)$ defined on the entire real line that satisfy the equation $f(y-f(x))=1-x-y$ for any $x$ and $y$. | f(x)=\frac{1}{2}-x | 42 | 11 |
math | (solved by François Caddet and Jean-Alix David). Is 1000000027 a prime number? | 1000000027=(1000+3)\cdot(1000^{2}-1000\cdot3+3^{2}) | 29 | 39 |
math | Seven cases of a chessboard $(8 \times 8)$ are on fire. At each step, a case catches fire if at least two of its neighbors (by a side, not a corner) are on fire. Is it possible for the fire to spread everywhere? If not, how many cases initially on fire are needed at a minimum for the fire to be able to spread everywher... | 8 | 79 | 1 |
math | Find all functions $f: R \to R ,g: R \to R$ satisfying the following equality $f(f(x+y))=xf(y)+g(x)$ for all real $x$ and $y$.
I. Gorodnin | (f(x), g(x)) = (k, k(1 - x)) \quad \forall x | 52 | 21 |
math | 10.5. After watching the movie, viewers rated it one by one with an integer score from 0 to 10. At any given time, the movie's rating was calculated as the sum of all the given scores divided by their number. At some point in time $T$, the rating became an integer, and then with each new voting viewer, it decreased by ... | 5 | 109 | 1 |
math | 1. (a) Determine the average of the six integers $22,23,23,25,26,31$.
(b) The average of the three numbers $y+7,2 y-9,8 y+6$ is 27 . What is the value of $y$ ?
(c) Four positive integers, not necessarily different and each less than 100, have an average of 94 . Determine, with explanation, the minimum possible value fo... | 25,7,79 | 110 | 7 |
math | A hemisphere with radius $r=1$ is to be divided into two equal volume parts by a plane parallel to the base. How high is each of the two parts? | 0.6527 | 35 | 6 |
math | 7.2. On a line, several points were marked, after which two points were placed between each pair of adjacent points, and then the same procedure (with the entire set of points) was repeated again. Could there be 82 points on the line as a result? | 10 | 57 | 2 |
math | Jack's room has $27 \mathrm{~m}^{2}$ of wall and ceiling area. To paint it, Jack can use 1 can of paint, but 1 liter of paint would be left over, or 5 gallons of paint, but 1 liter would also be left over, or 4 gallons plus 2.8 liters of paint.
a) What is the ratio between the volume of a can and the volume of a gallo... | 1.5\mathrm{~}^{2}/\ell | 133 | 13 |
math | ## Subject III. (20 points)
A bathtub would be filled by the cold water tap in 5 minutes, and by the hot water tap in 7 minutes. The bathtub is equipped with a drain hole, through which, when the bathtub is full, the water can drain in 3 minutes. If we open both taps and do not close the drain hole, will the bathtub f... | 105 | 113 | 3 |
math | $14 \cdot 37$ The general term of the sequence is $a_{n}=b[\sqrt{n+c}]+d$, and the terms are calculated successively as
$$
1,3,3,3,5,5,5,5,5, \cdots
$$
where each positive odd number $m$ appears exactly $m$ times consecutively. The integers $b, c, d$ are to be determined. Find the value of $b+c+d$.
(Advanced Math Test... | 2 | 123 | 1 |
math | 17. [10] Let $p(x)=x^{2}-x+1$. Let $\alpha$ be a root of $p(p(p(p(x)))$. Find the value of
$$
(p(\alpha)-1) p(\alpha) p(p(\alpha)) p(p(p(\alpha))
$$ | -1 | 65 | 2 |
math | 2. [5 points] Solve the equation $\frac{\cos 4 x}{\cos 3 x - \sin 3 x} + \frac{\sin 4 x}{\cos 3 x + \sin 3 x} = \sqrt{2}$. | \frac{\pi}{52}+\frac{2\pik}{13},k\neq13p-5,k\in\mathbb{Z},p\in\mathbb{Z} | 57 | 46 |
math | Task 10/61 In a photo equipment store, a customer asks: "How much does this lens cost?" The saleswoman replies: "With leather case 115.00 DM, sir!" - "And how much does the lens cost without the case?" the customer asks further. "Exactly 100.00 DM more than the case!" says the saleswoman with a smile. How much does the... | 107.50\mathrm{DM} | 93 | 11 |
math | 13.155. In the first week of their vacation trip, the friends spent 60 rubles less than $2 / 5$ of the amount of money they brought with them; in the second week, $1 / 3$ of the remainder and another 12 rubles on theater tickets; in the third week, $3 / 5$ of the new remainder and another 31 rubles 20 kopecks on boat r... | 2330 | 124 | 4 |
math | For any real number $x$, $[x]$ represents the greatest integer not exceeding $x$, and $\{x\}$ represents the fractional part of $x$. Then
$$
\left\{\frac{2014}{2015}\right\}+\left\{\frac{2014^{2}}{2015}\right\}+\cdots+\left\{\frac{2014^{2014}}{2015}\right\}
$$
$=$ | 1007 | 111 | 4 |
math | 4. A cube with an edge length of a certain integer is cut into 99 smaller cubes, 98 of which are unit cubes with an edge length of 1, and the other cube also has an integer edge length. Then its edge length is $\qquad$ | 3 | 57 | 1 |
math | 20. For any positive integer $n$, let $f(n)$ denote the index of the highest power of 2 which divides $n!$, e.g. $f(10)=8$ since $10!=2^{8} \times 3^{4} \times 5^{2} \times 7$. Find the value of $f(1)+f(2)+\cdots+f(1023)$.
(3 marks)
對於任何正整數 $n$, 設 $f(n)$ 為可整除 $n$ ! 的 2 的最高乘冪的指數。例如：因為 $10!=2^{8} \times 3^{4} \times 5^{2... | 518656 | 200 | 6 |
math | Example 9 Let $f(x)$ be a monotonic continuous function defined on the set of real numbers $\mathrm{R}$. Solve the functional equation $f(x) \cdot f(y)=f(x+y)$. | f(x)=^xwhere0\neq1 | 45 | 11 |
math | Tsar Gvidon had 5 sons. Among his descendants, 100 each had exactly 3 sons, and the rest died childless.
How many descendants did Tsar Gvidon have
# | 305 | 44 | 3 |
math | Jamie, Linda, and Don bought bundles of roses at a flower shop, each paying the same price for each
bundle. Then Jamie, Linda, and Don took their bundles of roses to a fair where they tried selling their
bundles for a fixed price which was higher than the price that the flower shop charged. At the end of the
fair, Jamie, ... | 252 | 182 | 3 |
math | Let's determine the smallest value of $\left|36^{m}-5^{n}\right|$, where $m$ and $n$ are positive integers. | 11 | 34 | 2 |
math | ## Task 3 - 120613
In a room with a rectangular floor area of $11 \mathrm{~m}$ width and $36 \mathrm{~m}$ length, 6 machines with the following base areas are standing:
| Machine A: $15^{2}$ | Machine D: $60 \mathrm{~m}^{2}$ |
| :--- | :--- |
| Machine B: $5 \mathrm{~m}^{2}$ | Machine E: $18 \mathrm{~m}^{2}$ |
| Mach... | 3\mathrm{~} | 400 | 6 |
math | Example 3 Find all $f: \mathbf{R} \rightarrow \mathbf{R}$, satisfying the following inequality: $f(x+y)+f(y+z)+f(z+x) \geqslant 3 f(x+2 y+3 z)$, for all $x, y, z \in \mathbf{R}$. | f()=f(0) | 75 | 7 |
math | 3. Xiao Li and Xiao Zhang are running at a constant speed on a circular track. They start from the same place at the same time. Xiao Li runs clockwise and completes a lap every 72 seconds, while Xiao Zhang runs counterclockwise and completes a lap every 80 seconds. At the start, Xiao Li has a relay baton, and each time... | 720 | 120 | 3 |
math | 2. Four numbers $a_{1}, a_{2}, a_{3}, a_{4}$ form an arithmetic progression. The product of the first and fourth terms is equal to the larger root of the equation
$$
x^{1+\log x}=0.001^{-\frac{2}{3}}
$$
and the sum of the squares of the second and third terms is five times the number of the term in the expansion of
... | -10,-7,-4,-1or10,7,4,1 | 136 | 18 |
math | Anička and Blanka each wrote a two-digit number starting with a seven. The girls chose different numbers. Then each inserted a zero between the two digits, creating a three-digit number. From this, each subtracted their original two-digit number. The result surprised them.
Determine how their results differed.
(L. Ho... | 630 | 81 | 3 |
math | In which base number systems is it true that the rule for divisibility by four is the same as the rule for divisibility by nine in the decimal system, and the rule for divisibility by nine is the same as the rule for divisibility by four in the decimal system? | 12n+9 | 56 | 5 |
math | ## Problem Statement
Find the coordinates of point $A$, which is equidistant from points $B$ and $C$.
$A(0 ; 0 ; z)$
$B(-13 ; 4 ; 6)$
$C(10 ;-9 ; 5)$ | A(0;0;7.5) | 62 | 10 |
math | 34. Vova is as many times older than his sister Galya as he is younger than his grandmother Katya. How old is each of them, if Galya is not yet 6 years old, and Vova together with his grandmother is already 112 years old? | Gala:2,Vova:14,Katya:98 | 61 | 15 |
math | Suppose $x$ and $y$ are real numbers which satisfy the system of equations \[x^2-3y^2=\frac{17}x\qquad\text{and}\qquad 3x^2-y^2=\frac{23}y.\] Then $x^2+y^2$ can be written in the form $\sqrt[m]{n}$, where $m$ and $n$ are positive integers and $m$ is as small as possible. Find $m+n$. | 821 | 111 | 3 |
math | 3. Find the minimum value of the expression
$$
A=\left(2\left(\sin x_{1}+\ldots+\sin x_{n}\right)+\cos x_{1}+\ldots+\cos x_{n}\right) \cdot\left(\sin x_{1}+\ldots+\sin x_{n}-2\left(\cos x_{1}+\ldots+\cos x_{n}\right)\right)
$$ | -\frac{5n^{2}}{2} | 95 | 11 |
math | \section*{Problem 4 - 331034}
\begin{tabular}{ccccc}
& \(\mathrm{E}\) & \(\mathrm{I}\) & \(\mathrm{N}\) & \(\mathrm{S}\) \\
+ & \(\mathrm{E}\) & \(\mathrm{I}\) & \(\mathrm{N}\) & \(\mathrm{S}\) \\
+ & \(\mathrm{E}\) & \(\mathrm{I}\) & \(\mathrm{N}\) & \(\mathrm{S}\) \\
+ & \(\mathrm{E}\) & \(\mathrm{I}\) & \(\mathrm{... | 1049+1049+1049+1049+1049=5245 | 365 | 29 |
math | 16. Given $a+b+c=0$, and $a, b, c$ are all non-zero. Then simplify $a\left(\frac{1}{b}+\frac{1}{c}\right)+b\left(\frac{1}{a}+\frac{1}{c}\right)+c\left(\frac{1}{a}+\frac{1}{b}\right)$ to | -3 | 85 | 2 |
math | 1. The solution set of the inequality $\left|x^{2}-2\right| \leqslant 2 x+1$ is | [\sqrt{2}-1,3] | 30 | 9 |
math | ## 22. Math Puzzle $3 / 67$
Peter wants to use a balance scale, whose beam lengths $a$ and $b$ are no longer exactly equal. He first places a $5 \mathrm{~kg}$-"weight" on the left pan and weighs, and then the $5 \mathrm{~kg}$ piece on the right pan and weighs the rest of the apples.
Is the weighed amount heavier or l... | G>10 | 102 | 4 |
math | 8. Given $a b=1$, and $\frac{1}{1-2^{x} a}+\frac{1}{1-2^{y+1} b}=1$, then the value of $x+y$ is $\qquad$. | -1 | 53 | 2 |
math | 1.4. Let initially each island is inhabited by one colony, and let one of the islands have $d$ neighboring islands. What can the maximum possible number of colonies that can settle on this island be equal to? | +1 | 45 | 2 |
math | Let $P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$ be a polynomial of degree $n\geq 3.$ Knowing that $a_{n-1}=-\binom{n}{1}$ and $a_{n-2}=\binom{n}{2},$ and that all the roots of $P$ are real, find the remaining coefficients. Note that $\binom{n}{r}=\frac{n!}{(n-r)!r!}.$ | P(x) = (x-1)^n | 111 | 11 |
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 | Given $\sin \alpha=\frac{5}{7}$, $\cos (\alpha+\beta)=\frac{11}{14}$, and $\alpha, \beta$ are acute angles, find $\cos \beta$. | \frac{1}{98}(22 \sqrt{6}+25 \sqrt{3}) | 47 | 23 |
math | 3. We will call a natural number $n$ amusing if for any of its natural divisors $d$, the number $d+2$ is prime.
( a ) What is the maximum number of divisors an amusing number can have?
(b) Find all amusing numbers with the maximum number of divisors.
Answer: a maximum of 8 divisors, only one amusing number $3^{3} \c... | 3^{3}\cdot5 | 88 | 6 |
math | 11. (6 points) Natural numbers $1,2,3, \cdots$ are written down consecutively to form a number $123456789101112 \cdots$. When a certain number is reached, the formed number is exactly divisible by 72 for the first time. This number is $\qquad$ _. $\qquad$ | 36 | 84 | 2 |
math | 18. $\left(\frac{1}{a}-\frac{n}{x}\right)+\left(\frac{1}{a}-\frac{n-1}{x}\right)+\left(\frac{1}{a}-\frac{n-2}{x}\right)+\ldots+\left(\frac{1}{a}-\frac{1}{x}\right)$. | n(\frac{1}{}-\frac{n+1}{2x}) | 80 | 16 |
math | 18. Solve the equation $\sqrt{x^{2}-4}+2 \sqrt{x^{2}-1}=\sqrt{7} x$. | \frac{2}{3}\sqrt{21} | 31 | 12 |
math | 5. Solution: Let Masha have $x$ rubles, and Petya have $y$ rubles, then
$n(x-3)=y+3$
$x+n=3(y-n)$
Express $x$ from the second equation and substitute into the first:
$n(3 y-4 n-3)=y+3$,
$3 n y-y=4 n^{2}+3 n+3$,
$y=\left(4 n^{2}+3 n+3\right):(3 n-1)$
For $y$ to be an integer, $\left(4 n^{2}+3 n+3\right)$ must be ... | 1;2;3;7 | 344 | 7 |
math | Find the number of positive integers x satisfying the following two conditions:
1. $x<10^{2006}$
2. $x^{2}-x$ is divisible by $10^{2006}$ | 3 | 48 | 1 |
math | On a hundred-seat airplane, a hundred people are traveling, each with a pre-assigned seat. The first passenger, however, disregards this and randomly sits in one of the hundred seats. Thereafter, each passenger tries to sit in their assigned seat, or if it is occupied, randomly selects another one. What is the probabil... | \frac{1}{2} | 91 | 7 |
math | 9. Xiao Ming set off from home to school at exactly 7:00 AM at a speed of 52 meters per minute. When he arrived at school, the hour and minute hands on his watch were on either side of the number 7 and equidistant from it. Given that Xiao Ming walked for less than an hour, the distance from Xiao Ming's home to the scho... | 1680 | 105 | 4 |
math | 16th Mexico 2002 Problem B2 A trio is a set of three distinct integers such that two of the numbers are divisors or multiples of the third. Which trio contained in {1, 2, ... , 2002} has the largest possible sum? Find all trios with the maximum sum. | (,2002-,2002),where1,2,7,11,13,14,22,26,77,91,143,154,182,286 | 69 | 55 |
math | Two sequences $\{a_i\}$ and $\{b_i\}$ are defined as follows: $\{ a_i \} = 0, 3, 8, \dots, n^2 - 1, \dots$ and $\{ b_i \} = 2, 5, 10, \dots, n^2 + 1, \dots $. If both sequences are defined with $i$ ranging across the natural numbers, how many numbers belong to both sequences?
[i]Proposed by Isabella Grabski[/i] | 0 | 116 | 1 |
math | 8. Let $z$ be a complex number, $z^{7}=1$. Then $z+z^{2}+z^{4}=$ | \frac{-1\\sqrt{7}\mathrm{i}}{2} | 31 | 15 |
math | Question 11: Given the mapping f: $\{1,2,3,4,5\} \rightarrow\{1,2,3,4,5\}$ is both injective and surjective, and satisfies $f(x)+f(f(x))=6$, then $f(1)=$ $\qquad$ _. | 5 | 71 | 1 |
math | 9.030. $\frac{1}{x+2}<\frac{3}{x-3}$. | x\in(-\frac{9}{2};-2)\cup(3;\infty) | 25 | 21 |
math | 7.2. A city held three rounds of Go tournaments, with the same participants in each round. It is known that every 2 participants in the three rounds of the tournament won 1 time each, drew 1 time each. A person came in last place in the first two rounds. Question: What place did he get in the third round? | 1 | 72 | 1 |
math | Question 17: Let the function $f(x)$ be defined on $[0,1]$, satisfying: $f(0)=f(1)$, and for any $x, y \in[0,1]$ there is $|f(x)-f(y)|<|x-y|$. Try to find the smallest real number $m$, such that for any $f(x)$ satisfying the above conditions and any $x, y \in[0,1]$, we have $|f(x)-f(y)|<m$. | \frac{1}{2} | 113 | 7 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.