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 | ## Task A-4.4.
Tamara wrote an even natural number on the board. Then, one by one, she wrote twelve more numbers such that each number was 5 greater than the square of the previously written number. Determine which digit the last written number can end with. | 0or6 | 58 | 3 |
math | ## 8. Wooden Numbers
Little Marko is playing with numbers made of wood. He has two number 1s, seven number 2s, and one number 3. He wants to string them together one by one so that the numbers 1 and 3 are not adjacent. How many different ten-digit numbers can Marko get this way?
Result: $\quad \mathbf{2 2 4}$ | 224 | 87 | 3 |
math | 3. Find the function $f: \mathbf{R} \rightarrow \mathbf{R}$ that satisfies the condition: $f\left(\frac{1+x}{x}\right)=\frac{x^{2}+1}{x^{2}}+\frac{1}{x}$. | f(x)=x^{2}-x+1 | 62 | 10 |
math | 17. Find the digits $a$ and $b$ in the number $\overline{42 a 4 b}$, given that this number is divisible by 72. | =8,b=0or=0,b=8 | 39 | 11 |
math | Given an integer $n>1$, let $a_{1}, a_{2}, \cdots, a_{n}$ be distinct non-negative real numbers, and define the sets
$$
A=\left\{a_{i}+a_{j} \mid 1 \leqslant i \leqslant j \leqslant n\right\}, B=\left\{a_{i} a_{j} \mid 1 \leqslant i \leqslant j \leqslant n\right\} .
$$
Find the minimum value of $\frac{|A|}{|B|}$. Here... | \frac{2(2n-1)}{n(n+1)} | 155 | 16 |
math | Example 7 (Problem from the 28th Russian Mathematical Olympiad): There is a red card box and $k$ blue card boxes ($k>1$), and a deck of cards, totaling $2n$ cards, numbered from 1 to $2n$. Initially, these cards are stacked in the red card box in any order. From any card box, the top card can be taken out, placed in an... | k-1 | 128 | 3 |
math | 9. (16 points) Given $a, b \in [1,3], a+b=4$.
Find the maximum value of $f(a, b)=\left|\sqrt{a+\frac{1}{b}}-\sqrt{b+\frac{1}{a}}\right|$. | 2-\frac{2}{\sqrt{3}} | 64 | 11 |
math | 32. (MON 3) Determine the volume of the body obtained by cutting the ball of radius $R$ by the trihedron with vertex in the center of that ball if its dihedral angles are $\alpha, \beta, \gamma$. | \frac{1}{3}R^{3}(\alpha+\beta+\gamma-\pi) | 53 | 20 |
math | 3. Find all pairs of positive integers $(n, k)$ such that $(n+1)^{k} -1=n!$.
untranslated portion:
-1=n!.
(Note: The last part "-1=n!." was already in English and thus not translated.) | (1,1),(2,1),(4,2) | 56 | 13 |
math | Let's determine all prime numbers $p$ and $q$ for which, for some $n>1$,
$$
\frac{p^{2 n+1}-1}{p-1}=\frac{q^{3}-1}{q-1}
$$ | p=2,q=5,n=2 | 56 | 9 |
math | 1. Janica had several candies. She gave half of the candies to her brother, and ate a sixth of the remaining candies. She lost 2 or 3 candies on the way, and gave 6 candies to her friend Petra. In the end, she had 21 candies left. How many candies did Janica have? | 72 | 69 | 2 |
math | 1. Solve the inequality $\lg \left(5^{x}+x-20\right)>x-x \lg 2$. | x>20 | 29 | 4 |
math | \section*{Problem 2 - 161242}
Given a natural number \(n \geq 1\).
Determine the number of different ways to distribute \(2 n\) red, \(2 n\) green, and \(2 n\) black balls into two containers \(Q_{1}\) and \(Q_{2}\) such that each container contains \(3 n\) balls.
Hint:
Two distribution possibilities are considered... | 3n^{2}+3n+1 | 139 | 10 |
math | 4. Let $a_{n}=\frac{1}{(n+1) \sqrt{n}+n \sqrt{n+1}}, n=1,2,3, \cdots$, then the value of $a_{1}+a_{2}+\cdots+a_{99}$ is | \frac{9}{10} | 65 | 8 |
math | [Periodicity and Aperiodicity]
Let's start counting the fingers on the right hand: the first is the pinky, the second is the ring finger, the third is the middle finger, the fourth is the index finger, the fifth is the thumb, the sixth is the index finger again, the seventh is the middle finger again, the eighth is th... | Index | 109 | 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 | Example 1. Find the equation of the line passing through the intersection points of the two curves
$$
3 x^{2}+3 y^{2}+2 x+y=0 \text { and } x^{2}+y^{2}+3 x
$$
$-\mathrm{y}=0$. | 7 x-4 y=0 | 67 | 7 |
math | Given a square $A B C D$ in the plane, a line is to be constructed such that the orthogonal projections of the square's vertices onto this line are arranged in such a way that the three consecutive distances determined by the projection points are equal. | \inftyinitelymanylinesperpendicularto4directions | 51 | 15 |
math | 5. Let $N>1$ be a positive integer, and $m$ denote the largest divisor of $N$ that is less than $N$. If $N+m$ is a power of 10, find $N$.
| 75 | 50 | 2 |
math | We are given five equal-looking weights of pairwise distinct masses. For any three weights $A$, $B$, $C$, we can check by a measuring if $m(A) < m(B) < m(C)$, where $m(X)$ denotes the mass of a weight $X$ (the answer is [i]yes[/i] or [i]no[/i].) Can we always arrange the masses of the weights in the increasing order ... | 9 | 102 | 1 |
math | 3. Determine the maximum possible value of the expression
$$
27 a b c+a \sqrt{a^{2}+2 b c}+b \sqrt{b^{2}+2 c a}+c \sqrt{c^{2}+2 a b}
$$
where $a, b, c$ are positive real numbers such that $a+b+c=\frac{1}{\sqrt{3}}$. | \frac{2}{3\sqrt{3}} | 90 | 11 |
math | 3 Theorem Application Example
Question Given the sequence $\left\{a_{n}\right\}$ satisfies:
$$
a_{1}=a_{2}=a_{3}=1, a_{4}=-1, a_{5}=0 \text {, }
$$
and for any $n \in \mathbf{Z}_{+}$, we have
$$
a_{n+5}=3 a_{n+4}-4 a_{n+3}+4 a_{n+2}-3 a_{n+1}+a_{n} \text {. }
$$
Find the general term formula for the sequence $\left\{... | a_{n}=\frac{-3+7\mathrm{i}}{8}\mathrm{i}^{n}+\frac{-3-7\mathrm{i}}{8}(-\mathrm{i})^{n}+\frac{3n^{2}-19n+27}{4} | 145 | 61 |
math | There are eight identical Black Queens in the first row of a chessboard and eight identical White Queens in the last row. The Queens move one at a time, horizontally, vertically or diagonally by any number of squares as long as no other Queens are in the way. Black and White Queens move alternately. What is the minimal... | 24 | 87 | 2 |
math | 3 [Mutual arrangement of altitudes, medians, bisectors, etc.]
In triangle $ABC$, the bisector drawn from vertex $A$, the altitude drawn from vertex $B$, and the perpendicular bisector of side $AB$ intersect at one point. Find the angle at vertex $A$. | 60 | 62 | 2 |
math | 1. $\left(20 \frac{94}{95} \times 1.65-20 \frac{94}{95}+\frac{7}{20} \times 20 \frac{94}{95}\right) \times 47.5 \times 0.8 \times 2.5$. | 1994 | 79 | 4 |
math | 1. Compute the value and the successive convergents of the following finite continued fractions:
(i) $\langle 1,2,3\rangle$;
(ii) $\langle 0,1,2,3\rangle$;
(iii) $\langle 3,2,1\rangle$;
(iv) $\langle 2,1,1,4,1,1\rangle$;
(v) $\langle-4,2,1,7,8\rangle$;
(vi) $\langle-1,1 / 2,1 / 3\rangle$;
(vii) $\langle 1 / 2,1 / 4,1 /... | \frac{1193}{322} | 151 | 12 |
math | ## 31. Child Psychology
The brightest hours of a little child are spent drawing. He depicts either Indians or Eskimos and draws a dwelling for each of them. Thus, next to the Eskimos, he usually draws an igloo, and next to the Indians - wigwams. But sometimes the child makes mistakes, and not infrequently, for example... | \frac{7}{8} | 195 | 7 |
math | ## 27. Large Families
The Martens have more children than the Duponts. Suppose the difference of the squares of these two numbers is 24 and that both families have more than one child. How many children do the Martens have?
70 | 7 | 55 | 1 |
math | There is a $40\%$ chance of rain on Saturday and a $30\%$ of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive intege... | 107 | 91 | 3 |
math | 5. Let $n$ and $k$ be positive integers, $k \geqslant n$, and $k-n$ is an even number. $2 n$ lamps are sequentially numbered as $1,2, \cdots, 2 n$, and each lamp can be either "on" or "off". Initially, all lamps are off. An operation can be performed on these lamps: each operation changes the state of one lamp (i.e., o... | 2^{k-n} | 302 | 5 |
math | 1. The solution set of the inequality $(x+1)^{3}\left(x^{3}+5 x\right)<10(x+1)^{2}$ +8 is | (-2,1) | 39 | 5 |
math | 3. Compute the sum
$$
\frac{1}{1 \cdot 4}+\frac{1}{4 \cdot 7}+\ldots+\frac{1}{(3 n-2)(3 n+1)}
$$ | \frac{n}{3n+1} | 50 | 9 |
math | B1. Brothers Janko and Peter sold a herd of 25 cows at the fair. For each cow, they received as many gold coins as there were cows in the herd. With the proceeds, they bought a herd of sheep, paying 20 gold coins for each sheep. They had just enough gold coins left to buy one lamb. They then divided the livestock so th... | 7.5 | 157 | 3 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0} \frac{3^{2 x}-7^{x}}{\arcsin 3 x-5 x}$ | \ln\sqrt{\frac{7}{9}} | 44 | 11 |
math | 8,9
In triangle $ABC$, the altitude $AM$ is not less than $BC$, and the altitude $BH$ is not less than $AC$. Find the angles of triangle $ABC$. | 45,45,90 | 42 | 8 |
math | 15. (16 points) Let $x_{1}, x_{2}, x_{3} \in [0,12]$,
$$
x_{1} x_{2} x_{3}=\left(\left(12-x_{1}\right)\left(12-x_{2}\right)\left(12-x_{3}\right)\right)^{2} \text {. }
$$
Find the maximum value of $f=x_{1} x_{2} x_{3}$. | 729 | 109 | 3 |
math | 8.146. $\operatorname{tg} x+\operatorname{tg} 50^{\circ}+\operatorname{tg} 70^{\circ}=\operatorname{tg} x \operatorname{tg} 50^{\circ} \operatorname{tg} 70^{\circ}$. | 60+180n,n\inZ | 73 | 11 |
math | ## Task A-2.3.
A wire of length $10 \mathrm{~m}$ needs to be cut into two parts, and from one part a square should be made, and from the other part an equilateral triangle. At what point should the wire be cut so that the total area of the square and the equilateral triangle is minimized? | \frac{40(3\sqrt{3}-4)}{11} | 73 | 18 |
math | Dedalo buys a finite number of binary strings, each of finite length and made up of the binary digits 0 and 1. For each string, he pays $(\frac{1}{2})^L$ drachmas, where $L$ is the length of the string. The Minotaur is able to escape the labyrinth if he can find an infinite sequence of binary digits that does not conta... | c = 75 | 249 | 6 |
math | 23. In $\triangle A B C, A B: A C=4: 3$ and $M$ is the midpoint of $B C$. $E$ is a point on $A B$ and $F$ is a point on $A C$ such that $A E: A F=2: 1$. It is also given that $E F$ and $A M$ intersect at $G$ with $G F=72 \mathrm{~cm}$ and $G E=x \mathrm{~cm}$. Find the value of $x$. | 108 | 121 | 3 |
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 | Suppose that $m$ and $n$ are positive integers with $m < n$ such that the interval $[m, n)$ contains more multiples of $2021$ than multiples of $2000$. Compute the maximum possible value of $n - m$. | 1999 | 59 | 4 |
math | 3. Consider the equation $0 . a b c d+0 . e f g h=1$, where each letter stands for a digit from 1 to 8 inclusive.
(a) Suppose each letter stands for a different digit. Prove that there are no solutions.
(b) Suppose instead that digits may be repeated. How many solutions are there? (You may give your final answer as a p... | 1792 | 132 | 4 |
math | Task 2. Determine all digits $a, b$ for which the number $\overline{a 13 b}$ is divisible by 45. | 5130,9135 | 33 | 9 |
math | 7*. In a model of a regular quadrilateral pyramid, all faces are made of uniform plates of the same material (and have the same thickness). The lateral edges of the pyramid are twice as long as the sides of the base. Determine the position of the center of mass of the pyramid, given the radius vectors of the vertex $S$... | \vec{P}=\frac{4\vec{S}\sqrt{15}+(2\sqrt{15}+3)(\overrightarrow{A_{1}}+\overrightarrow{A_{2}}+\overrightarrow{A_{3}}+\overrightarrow{A_{4}})}{12(\sqrt{15}+1)} | 94 | 75 |
math | 12. Delia is joining three vertices of a square to make four right-angled triangles.
She can create four triangles doing this, as shown.
How many right-angled triangles can Delia make by joining three vertices of a regular polygon with 18 sides? | 144 | 55 | 3 |
math | 41. Find all geometric progressions in which each term, starting from the third, is equal to the sum of the two preceding ones. | \frac{1\\sqrt{5}}{2} | 29 | 12 |
math | In the figure on the right, $O$ is the center of the circle, $OK$ and $OA$ are perpendicular to one another, $M$ is the midpoint of $OK$, $BN$ is parallel to $OK$, and $\angle AMN=\angle NMO$. Determine the measure of $\angle A B N$ in degrees.
[asy]
defaultpen(linewidth(0.7)+fontsize(10));
pair O=origin, A=dir(90), K... | 36^\circ | 239 | 4 |
math | 11.6*. The perimeter of triangle $A B C$ is $2 p$. Points $M$ and $N$ are taken on sides $A B$ and $A C$ such that $M N \| B C$ and $M N$ is tangent to the inscribed circle of triangle $A B C$. Find the maximum value of the length of segment $M N$. | \frac{p}{4} | 81 | 7 |
math | Determine all pairs $ (n,p)$ of positive integers, where $ p$ is prime, such that $ 3^p\minus{}np\equal{}n\plus{}p$. | (n, p) = (6, 3) | 40 | 13 |
math | 3.2005 real numbers $x_{1}, x_{2}, \cdots, x_{2000}$ satisfy
$$
\begin{array}{l}
\left|x_{1}-x_{2}\right|+\left|x_{2}-x_{3}\right|+\cdots+ \\
\left|x_{2004}-x_{2000}\right|+\left|x_{2000}-x_{1}\right|=1 .
\end{array}
$$
Then $\left|x_{1}\right|+\left|x_{2}\right|+\cdots+\left|x_{2000}\right|$ has a minimum value of $\... | \frac{1}{2} | 150 | 7 |
math | Q. A light source at the point $(0, 16)$ in the co-ordinate plane casts light in all directions. A disc(circle along ith it's interior) of radius $2$ with center at $(6, 10)$ casts a shadow on the X-axis. The length of the shadow can be written in the form $m\sqrt{n}$ where $m, n$ are positive integers and $n$ is squar... | 21 | 99 | 2 |
math | 7.079. $2 \log _{x} 27-3 \log _{27} x=1$. | \frac{1}{27};9 | 30 | 9 |
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 | Let $ P(x) \in \mathbb{Z}[x]$ be a polynomial of degree $ \text{deg} P \equal{} n > 1$. Determine the largest number of consecutive integers to be found in $ P(\mathbb{Z})$.
[i]B. Berceanu[/i] | n | 66 | 2 |
math | 10. Given $p, q(q \neq 0)$ are real numbers, the equation $x^{2}-p x+q=0$ has two real roots $\alpha, \beta$, and the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=p, a_{2}=p^{2}-q, a_{n}=p a_{n-1}-q a_{n-2}(n=3,4, \cdots)$.
(1) Find the general term formula of the sequence $\left\{a_{n}\right\}$ (expressed in terms... | S_{n}=3-\frac{n+3}{2^{n}} | 180 | 15 |
math | Denote by $d(n)$ be the biggest prime divisor of $|n|>1$. Find all polynomials with integer coefficients satisfy;
$$P(n+d(n))=n+d(P(n)) $$
for the all $|n|>1$ integers such that $P(n)>1$ and $d(P(n))$ can be defined. | P(n) = n | 73 | 6 |
math | 3. Given an equilateral triangle ABC. Point $D$ is chosen on the extension of side $A B$ beyond point $A$, point $E$ is on the extension of $B C$ beyond point $C$, and point $F$ is on the extension of $A C$ beyond point $C$ such that $C F=A D$ and $A C+E F=D E$. Find the angle BDE. (A. Kuznetsov) | 60 | 97 | 2 |
math | Let $x, y$ be two positive integers, with $x> y$, such that $2n = x + y$, where n is a number two-digit integer. If $\sqrt{xy}$ is an integer with the digits of $n$ but in reverse order, determine the value of $x - y$ | 66 | 66 | 2 |
math | Call a natural number $n{}$ [i]interesting[/i] if any natural number not exceeding $n{}$ can be represented as the sum of several (possibly one) pairwise distinct positive divisors of $n{}$.
[list=a]
[*]Find the largest three-digit interesting number.
[*]Prove that there are arbitrarily large interesting numbers other ... | 992 | 96 | 5 |
math | Example 12 Given that $x, y, z$ are 3 non-negative rational numbers, and satisfy $3x+2y+z=5, x+y-z=2$. If $s=2x+y-z$, then what is the sum of the maximum and minimum values of $s$?
(1996, Tianjin Junior High School Mathematics Competition) | 5 | 78 | 1 |
math | Find the smallest $\lambda \in \mathbb{R}$ such that for all $n \in \mathbb{N}_+$, there exists $x_1, x_2, \ldots, x_n$ satisfying $n = x_1 x_2 \ldots x_{2023}$, where $x_i$ is either a prime or a positive integer not exceeding $n^\lambda$ for all $i \in \left\{ 1,2, \ldots, 2023 \right\}$. | \frac{2}{2024} | 117 | 10 |
math | The positive integer $d$ has the property that each of 468,636, and 867 has the same remainder, $r$, when divided by $d$. What is the largest possible value of $d+r$ ? | 27 | 52 | 2 |
math | 1. We call polynomials $A(x)=a_{n} x^{n}+\ldots+a_{1} x+a_{0}$ and $B(x)=b_{m} x^{m}+\ldots+b_{1} x+b_{0}$ $\left(a_{n} b_{m} \neq 0\right)$ similar if the following conditions hold:
(i) $n=m$;
(ii) There is a permutation $\pi$ of the set $\{0,1, \ldots, n\}$ such that $b_{i}=a_{\pi(i)}$ for each $i \in\{0,1, \ldots,... | 1 | 215 | 1 |
math | 2. (5 points) Two different natural numbers end with 7 zeros and have exactly 72 divisors. Find their sum.
# | 70000000 | 29 | 8 |
math | Problem 5. The sum of two natural numbers is 2018. If the digit in the units place of one is crossed out, the other number will be obtained. Find all such numbers. | 1835 | 42 | 4 |
math | Example 3 In the Cartesian coordinate system $x O y$, given two points $M(-1,2)$ and $N(1,4)$, point $P$ moves on the $x$-axis. When $\angle M P N$ takes the maximum value, find the x-coordinate of point $P$.
(2004 National High School Mathematics Competition Problem) | 1 | 79 | 1 |
math | Example 2. Approximately calculate the increase in the volume of a cylinder with height $H=40 \mathrm{~cm}$ and base radius $R=30 \mathrm{~cm}$ when the radius is increased by $0.5 \mathrm{~cm}$. | \DeltaV=1200\pi\mathrm{}^{3} | 58 | 16 |
math | João's scientific calculator has a special key that transforms any number $x$ displayed on the screen and that is different from 1 into the number $\frac{1}{1-x}$.
a) What happens if the number 2 is displayed on the screen and we press the special key three times?
b) What happens if the number 2 is displayed on the s... | \frac{1}{2} | 114 | 7 |
math | Let $a$ and $b$ be real numbers bigger than $1$. Find maximal value of $c \in \mathbb{R}$ such that $$\frac{1}{3+\log _{a} b}+\frac{1}{3+\log _{b} a} \geq c$$
| \frac{1}{3} | 67 | 8 |
math | 4. The distance between the foci of the conic section
$$
(3 x+4 y-13)(7 x-24 y+3)=200
$$
is $\qquad$ | 2\sqrt{10} | 46 | 7 |
math | 5. Eight strangers are preparing to play bridge. How many ways can they be grouped into two bridge games - that is, into unordered pairs of unordered pairs of people? | 315 | 34 | 3 |
math | # Problem 2.
Solve the system of equations
$$
\begin{cases}2\left[x_{1}\right]+x_{2} & =3 / 2 \\ 3\left[x_{1}\right]-2 x_{2} & =4\end{cases}
$$
Here $[a]$ denotes the integer part of the number $a$.
# | x_{1}\in[1;2),x_{2}=-\frac{1}{2} | 81 | 22 |
math | 16. Let $f(x)=k\left(x^{2}-x+1\right)-x^{4}(1-x)^{4}$. If for any $x \in[0,1]$, we have $f(x) \geqslant 0$, then the minimum value of $k$ is $\qquad$ . | \frac{1}{192} | 73 | 9 |
math | Example 4.37. How many times can the trigonometric series $\sum_{k=1}^{\infty} \frac{\cos k x}{k^{4}}$ be differentiated term by term? | 2 | 46 | 1 |
math | Example 3 Calculate $\tan \frac{\pi}{7} \cdot \tan \frac{2 \pi}{7} \cdot \tan \frac{3 \pi}{7}$.
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | \sqrt{7} | 65 | 5 |
math | I1.1 解方程 $\log _{5} a+\log _{3} a=\log _{5} a \cdot \log _{3} a$, 其中 $a>1$ 為實數。 | 15 | 52 | 2 |
math | 3. Calculate $\frac{1}{1 \times 3 \times 5}+\frac{1}{3 \times 5 \times 7}+\frac{1}{5 \times 7 \times 9}+\cdots+\frac{1}{2001 \times 2003 \times 2005}$. | \frac{1004003}{12048045} | 76 | 20 |
math | 10.1. Find all four-digit numbers $\overline{x y z t}$, where all digits $x, y, z, t$ are distinct and not equal to 0, such that the sum of all four-digit numbers obtained from $\overline{x y z t}$ by all possible permutations of the digits is 10 times the number $\overline{x x x x}$. | 9123 | 82 | 4 |
math | 10.2. In triangle $A B C$, the measure of angle $A$ is 30 degrees, and the length of the median drawn from vertex $B$ is equal to the length of the altitude drawn from vertex $C$. Find the measures of angles $B$ and $C$. | \angleB=90,\angleC=60 | 63 | 12 |
math | Task 4. (20 points) It is known that the function $f(x)$ for each value of $x \in(-\infty ;+\infty)$ satisfies the equation $f(x)-(x-0.5) f(-x-1)=1$. Find all such functions $f(x)$. | f(x)={\begin{pmatrix}\frac{1}{0.5+x},x\neq-0.5,\\0.5,-0.50\end{pmatrix}.} | 66 | 44 |
math | 13.146. Find three numbers, the first of which is as many times greater than the second as the second is greater than the third. If the first number is decreased by the sum of the other two, the result is 2, and if the first number is increased by half the difference of the second and the third, the result is 9. | 8;4;2or-6.4;11.2;-19.6 | 76 | 20 |
math | 1. (5 points) Find the value of the function $f(x)$ at the point $x_{0}=3000$, if $f(0)=1$ and for any $x$ the equality $f(x+2)=f(x)+3 x+2$ holds. | 6748501 | 61 | 7 |
math | 8.108. $2 \sin ^{2} z+\tan ^{2} z=2$. | \frac{\pi}{4}(2k+1),k\inZ | 25 | 16 |
math | 448. $\frac{\sin 7 \alpha-\sin 5 \alpha}{\sin 7 \alpha+\sin 5 \alpha}$. | \operatorname{tg}\alpha\operatorname{ctg}6\alpha | 33 | 17 |
math | We say that a number is [i]superstitious [/i] when it is equal to $13$ times the sum of its digits . Find all superstitious numbers. | 117, 156, 195 | 38 | 13 |
math | (solved by Lucas Boczkowski). Find all $n \in\{1,2, \ldots, 999\}$ such that $n^{2}$ is equal to the cube of the sum of the digits of $n$. | 1,27 | 53 | 4 |
math | 74(1184). Find all pairs of natural numbers that satisfy the equation: a) $x^{2}-y^{2}=105$; b) $2 x^{2}+5 x y-12 y^{2}=28$. | (53,52),(19,16),(13,8),(11,4) | 57 | 23 |
math | Example 3: A certain area currently has 10,000 hectares of arable land. It is planned that in 10 years, the grain yield per unit area will increase by $22\%$, and the per capita grain possession will increase by $10\%$. If the annual population growth rate is $1\%$, try to find the maximum number of hectares by which t... | 4 | 95 | 1 |
math | 6.1. Replace each asterisk in the example of adding decimal fractions
$$
0, * *+0, * *+0, * *+0, * *=1
$$
with the digit 2 or the digit 3 so that the equation is correct. | 0.32+0.22+0.23+0.23=1 | 58 | 21 |
math | 24. (2004 National College Entrance Examination Chongqing Paper) Suppose a car has to pass through 4 intersections on its way forward. The probability of the car encountering a green light (allowing passage) at each intersection is $\frac{3}{4}$, and the probability of encountering a red light (prohibiting passage) is ... | \frac{525}{256} | 153 | 11 |
math | ## Task A-2.3. (4 points)
A wire of length $10 \mathrm{~km}$ is to be used to fence a rectangular plot of land that has a straight wall on one side (the wire must be used for the remaining three sides), so that the area of this plot is as large as possible. What is the area of the thus enclosed plot of land? | 12.5\mathrm{~}^2 | 81 | 11 |
math | 7. Given $\frac{1}{1-2^{x}}+\frac{1}{1-2^{y+1}}=1, x+y=$
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。
translated part:
7. Given $\frac{1}{1-2^{x}}+\frac{1}{1-2^{y+1}}=1, x+y=$ | -1 | 96 | 2 |
math | Let $t$ be TNYWR.
The sum of the even integers from 2 to $2 k$ inclusive equals $t$ for some positive integer $k$. That is,
$$
2+4+6+\cdots+(2 k-2)+2 k=t
$$
What is the value of $k$ ? | 11 | 69 | 2 |
math | EST-
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ for which
$$
f(x+y)=f(x-y)+f(f(1-x y))
$$
holds for all real numbers $x$ and $y$.
Answer: $f(x) \equiv 0$. | f(x)\equiv0 | 68 | 5 |
math | 46th Putnam 1985 Problem A4 Let a n be the sequence defined by a 1 = 3, a n+1 = 3 k , where k = a n . Let b n be the remainder when a n is divided by 100. Which values b n occur for infinitely many n? Solution | 87 | 71 | 2 |
math | Two cyclists set out at the same time from the same place on a trip. They traveled the same route and returned together. On the way, both took breaks. The first cycled twice as long as the second rested. The second cycled four times as long as the first rested. Who of them cycles faster and by how many times?
(Černek)... | 1.5 | 75 | 3 |
math | Let $z_1,z_2,z_3,\dots,z_{12}$ be the 12 zeroes of the polynomial $z^{12}-2^{36}$. For each $j$, let $w_j$ be one of $z_j$ or $i z_j$. Then the maximum possible value of the real part of $\displaystyle\sum_{j=1}^{12} w_j$ can be written as $m+\sqrt{n}$ where $m$ and $n$ are positive integers. Find $m+n$. | 784 | 116 | 3 |
math | Sequence {$x_n$} satisfies: $x_1=1$ , ${x_n=\sqrt{x_{n-1}^2+x_{n-1}}+x_{n-1}}$ ( ${n>=2}$ )
Find the general term of {$x_n$} | \frac{1}{\sqrt[2^{n-1}]{2} - 1} | 60 | 21 |
math | 7. (10 points) There are 11, 12, and 17 balls of red, yellow, and blue colors respectively. Each operation can replace 2 balls of different colors with 2 balls of the third color. During the operation process, the maximum number of red balls can be $\qquad$. | 39 | 68 | 2 |
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