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 | 3. In triangle $ABC$, the angle bisectors $BK$ and $CL$ are drawn. A point $N$ is marked on segment $BK$ such that $LN \parallel AC$. It turns out that $NK = LN$. Find the measure of angle $ABC$. (A. Kuznetsov) | 120 | 66 | 3 |
math | 8. Find the largest real number $a$ such that for all positive integers $n$ and all real numbers $x_{0}, x_{1}, \cdots, x_{n}\left(0=x_{0}<x_{1}<\cdots\right.$ $\left.<x_{n}\right)$, we have
$$
\sum_{i=1}^{n} \frac{1}{x_{i}-x_{i-1}} \geqslant a \sum_{i=1}^{n} \frac{i+1}{x_{i}} .
$$ | \frac{4}{9} | 125 | 7 |
math | The surface area of a spherical segment is $n$ times the lateral surface area of the cone that can be inscribed in the corresponding slice. What is the ratio of the height of the segment to the diameter of the sphere? | \frac{}{2r}=\frac{n^2-1}{n^2} | 46 | 19 |
math | \section*{Task 3 - 320923}
When refueling an oldtimer with a two-stroke engine that requires a fuel-oil mixture of \(1: 50\), 7 liters of fuel without oil were mistakenly added first.
How many liters of the still available mixture with a ratio of \(1: 33\) need to be added now to achieve the correct mixture ratio of ... | 14 | 108 | 2 |
math | Given are $100$ positive integers whose sum equals their product. Determine the minimum number of $1$s that may occur among the $100$ numbers. | 95 | 35 | 4 |
math | For what $n$ can the following system of inequalities be solved?
$$
1<x<2 ; \quad 2<x^{2}<3 ; \quad \ldots, \quad n<x^{n}<n+1
$$ | 1,2,3,4 | 49 | 7 |
math | 8. Let the function $f(x)$ be defined on $\mathbf{R}$, for any $x \in \mathbf{R}$, $f(x+1006)=\frac{1}{2}+\sqrt{f(x)-f^{2}(x)}$,
and $f(-1005)=\frac{3}{4}$.
Then $f(2013)=$ | \frac{1}{2}+\frac{\sqrt{3}}{4} | 89 | 17 |
math | ## Task B-1.5.
In the new exhibition of a certain museum, each exhibit is numbered in sequence with the numbers $1, 2, 3, 4, \ldots$ If a total of 2022 digits were used to number all the exhibits, how many exhibits are there in the new exhibition of the museum? | 710 | 73 | 3 |
math | 1.6. Find the distance between two intersecting medians of the faces of a regular tetrahedron with edge 1. (Consider all possible arrangements of the medians.)
## § 2. Angles between lines and planes | \sqrt{2/35} | 49 | 8 |
math | B1. Find all real numbers $x$ that satisfy the equation
$$
\log _{2}(10 x)+\log _{4}(100 x)+\log _{8}(1000 x)-2 \log _{64} x=9
$$
Write the result in the form of a simplified fraction.
$$
\text { (6 points) }
$$ | \frac{16}{25} | 86 | 9 |
math | 7. $A_{1}(i=1,2,3,4)$ are subsets of the set $S=\{1,2, \cdots, 2005\}$, $F$ is the set of all ordered quadruples $\left(A_{1}, A_{2}, A_{3}, A_{1}\right)$, then the value of $\sum_{1}\left|A_{1} \cup A_{2} \cup A_{3} \cup A_{4}\right|$ is $\qquad$ . | 2^{8016}\times2005\times15 | 115 | 16 |
math | ## Aufgabe $3 / 72$
Gegeben seien die vier Scheitelpunkte einer Ellipse. Man konstruiere unter ausschließlicher Verwendung von Zirkel und Lineal das der Ellipse umbeschriebene Quadrat.
| \sqrt{^{2}+b^{2}} | 55 | 11 |
math | The temperatures $ f^\circ \text{F}$ and $ c^\circ \text{C}$ are equal when $ f \equal{} \frac {9}{5}c \plus{} 32$. What temperature is the same in both $ ^\circ \text{F}$ and $ ^\circ \text{C}$? | -40 | 71 | 3 |
math | 2. When point $P(x, y)$ is any point on the line $l$, point $Q(4 x+2 y, x+3 y)$ is also a point on this line. Then the equation of the line $l$ is $\qquad$ . | x+y=0 \text{ or } x-2 y=0 | 57 | 15 |
math | 2. For a natural number ending not in zero, one of its digits (not the most significant) was erased. As a result, the number decreased by 9 times. How many numbers exist for which this is possible? | 28 | 46 | 2 |
math | 11. a) How many integers less than 1000 are not divisible by 5 or 7?
b) How many of these numbers are not divisible by 3, 5, or 7? | 457 | 46 | 3 |
math | 7. The solution set of the inequality $|x|^{3}-2 x^{2}-4|x|+3<0$ is $\qquad$ | x\in(-3,\frac{1-\sqrt{5}}{2})\cup(\frac{-1+\sqrt{5}}{2},3) | 33 | 33 |
math | For every positive integer $ n$ consider
\[ A_n\equal{}\sqrt{49n^2\plus{}0,35n}.
\]
(a) Find the first three digits after decimal point of $ A_1$.
(b) Prove that the first three digits after decimal point of $ A_n$ and $ A_1$ are the same, for every $ n$. | 024 | 83 | 5 |
math | 5. The minimum value of the function $y=-4 x+3 \sqrt{4 x^{2}+1}$ is | \sqrt{5} | 27 | 5 |
math | ## Task A-3.3.
How many ordered pairs of natural numbers $(m, n)$ satisfy the equation $m^{2}-n^{2}=$ $2^{2013} ?$ | 1006 | 43 | 4 |
math | 3. Let $a_{1}, a_{2}, \cdots, a_{n}$ represent any permutation of the integers $1,2, \cdots, n$. Let $f(n)$ be the number of such permutations such that
(i) $a_{1}=1$;
(ii) $\left|a_{i}-a_{i+1}\right| \leqslant 2 . i=1,2, \cdots, n-1$.
Determine whether $f(1996)$ is divisible by 3. | 1 | 118 | 1 |
math | 5. (5 points) The average height of boys in Class 5-1 is 149 cm, and the average height of girls is 144 cm. The average height of the entire class is 147 cm. Therefore, the number of boys in the class is $\qquad$ times the number of girls. | \frac{3}{2} | 71 | 7 |
math | 3.1. (14 points) Svetlana, Katya, Olya, Masha, and Tanya attend a math club, in which more than $60 \%$ of the students are boys. What is the smallest number of schoolchildren that can be in this club? | 13 | 61 | 2 |
math | Shapovalov A.V.
55 boxers participated in a tournament with a "loser leaves" system. The fights proceeded sequentially. It is known that in each match, the number of previous victories of the participants differed by no more than 1. What is the maximum number of fights the tournament winner could have conducted? | 8 | 66 | 1 |
math | Problem 7.1. Sasha and Vanya are playing a game. Sasha asks Vanya questions. If Vanya answers correctly, Sasha gives him 7 candies. If Vanya answers incorrectly, he gives Sasha 3 candies. After Sasha asked 50 questions, it turned out that each of them had as many candies as they had at the beginning. How many questions... | 15 | 82 | 2 |
math | 5. (3 points) There are 211 students and four different types of chocolate. Each type of chocolate has more than 633 pieces. It is stipulated that each student can take at most three chocolates, or they can take none. If the students are grouped according to the types and quantities of chocolates they take, then the la... | 7 | 83 | 1 |
math | 7. (3 points) The number of four-digit numbers that satisfy the following two conditions is $\qquad$.
(1) The sum of any two adjacent digits is no more than 2;
(2) The sum of any three adjacent digits is no less than 3. | 1 | 58 | 1 |
math | $1 \cdot 66$ number $\sin \frac{\pi}{18} \sin \frac{3 \pi}{18} \sin \frac{5 \pi}{18} \sin \frac{7 \pi}{18} \sin \frac{9 \pi}{18}$ is it a rational number? | \frac{1}{16} | 73 | 8 |
math | 7. Xiao Ming, Xiao Hua, and Xiao Gang are dividing 363 cards among themselves, deciding to distribute them according to their age ratio. If Xiao Ming takes 7 cards, Xiao Hua should take 6 cards; if Xiao Gang takes 8 cards, Xiao Ming should take 5 cards. In the end, Xiao Ming took $\qquad$ cards; Xiao Hua took $\qquad$ ... | 105,90,168 | 97 | 10 |
math | Task 3. (15 points) Solve the system of equations
$$
\left\{\begin{array}{l}
x^{2}-22 y-69 z+703=0 \\
y^{2}+23 x+23 z-1473=0 \\
z^{2}-63 x+66 y+2183=0
\end{array}\right.
$$ | (20;-22;23) | 92 | 10 |
math | Example 12. Find $\int \frac{\sqrt{4-x^{2}}}{x^{2}} d x$. | -\operatorname{ctg}(\arcsin\frac{x}{2})-\arcsin\frac{x}{2}+C | 26 | 29 |
math | Exercise 4. Find all real numbers $a$ such that $a+\frac{2}{3}$ and $\frac{1}{a}-\frac{3}{4}$ are integers. | \frac{4}{3} | 40 | 7 |
math | ## Task 2 - 010712
During a voluntary potato collection, three groups of 7th-grade students held a small competition. Together, they collected a total of $52 \mathrm{dt}$ of potatoes. The second group collected $1 \frac{1}{2}$ times as much as the first, and the third group collected $3 \mathrm{dt}$ more than the firs... | 14 | 100 | 2 |
math | Solve the following equation:
$$
\sqrt{x+6}+\sqrt{x+1}=\sqrt{7 x+4} .
$$ | x_1=3 | 30 | 5 |
math | II. (50 points) Among the subsets of the set $S_{n}=\{1,2, \cdots, n\}$, those that do not contain two consecutive natural numbers are called "good subsets". How many good subsets are there in $S_{n}$? | a_{n}=\frac{1}{\sqrt{5}}[(\frac{1+\sqrt{5}}{2})^{n+2}-(\frac{1-\sqrt{5}}{2})^{n+2}] | 60 | 50 |
math | Example 1. Find the volume of the parallelepiped constructed on the vectors $\vec{a}\{1,2,3\}, \vec{b}\{0,1,1\}, \vec{c}\{2,1,-1\}$. | 4 | 56 | 1 |
math | There were seven boxes. In some of them, seven more boxes (not nested within each other) were placed, and so on. In the end, there were 10 non-empty boxes.
How many boxes are there in total? | 77 | 48 | 2 |
math | 77. Find the equation of the line passing through the point $(2 ; 2)$ and making an angle of $60^{\circ}$ with the $O x$ axis. | \sqrt{3}x+2(1-\sqrt{3}) | 39 | 15 |
math | 5. In a regular 1000-gon, all diagonals are drawn. What is the maximum number of diagonals that can be selected such that among any three of the selected diagonals, at least two have the same length? | 2000 | 50 | 4 |
math | 3. Let $n \geqslant 3$ be a positive integer. Then
$$
\sum_{k=1}^{n} \arctan \frac{1}{k^{2}+k+1}=
$$
$\qquad$ | \arctan(n+1)-\frac{\pi}{4} | 57 | 15 |
math | 2. What is the maximum number of different numbers from 1 to 1000 that can be chosen so that the difference between any two chosen numbers is not equal to any of the numbers 4, 5, 6. | 400 | 49 | 3 |
math | 8. The number $\left(2^{222}\right)^{5} \times\left(5^{555}\right)^{2}$ is $Q$ digits long. What is the largest prime factor of $Q$ ? | 101 | 52 | 3 |
math | Assume that $x_1,x_2,\ldots,x_7$ are real numbers such that
\begin{align*} x_1 + 4x_2 + 9x_3 + 16x_4 + 25x_5 + 36x_6 + 49x_7 &= 1, \\ 4x_1 + 9x_2 + 16x_3 + 25x_4 + 36x_5 + 49x_6 + 64x_7 &= 12, \\ 9x_1 + 16x_2 + 25x_3 + 36x_4 + 49x_5 + 64x_6 + 81x_7 &= 123. \end{align*}
Find the value of $16x_1+25x_2+36x_3+49x_4+64x_5+8... | 334 | 236 | 3 |
math | 958*. Solve the inequality in integers
$$
\frac{1}{\sqrt{x-2 y+z+1}}+\frac{2}{\sqrt{2 x-y+3 z-1}}+\frac{3}{\sqrt{3 y-3 x-4 z+3}}>x^{2}-4 x+3
$$ | (3;1;-1) | 73 | 7 |
math | Let $S_{n}=\{1,n,n^{2},n^{3}, \cdots \}$, where $n$ is an integer greater than $1$. Find the smallest number $k=k(n)$ such that there is a number which may be expressed as a sum of $k$ (possibly repeated) elements in $S_{n}$ in more than one way. (Rearrangements are considered the same.) | k(n) = n + 1 | 89 | 9 |
math | 7.5. The children went to the forest to pick mushrooms. If Anya gives half of her mushrooms to Vitya, all the children will have the same number of mushrooms, and if instead Anya gives all her mushrooms to Sasha, Sasha will have as many mushrooms as all the others combined. How many children went to pick mushrooms | 6 | 69 | 1 |
math | 240. Find $y^{\prime}$, if $y=\sin ^{2} 4 x$. | 4\sin8x | 25 | 5 |
math | 7. Let $x, y, z > 0$, and $x + 2y + 3z = 6$, then the maximum value of $xyz$ is $\qquad$ | \frac{4}{3} | 41 | 7 |
math | 68. When Vasya Verkhoglyadkin was given this problem: "Two tourists set out from $A$ to $B$ simultaneously. The first tourist walked half of the total time at a speed of $5 \mathrm{km} /$ h, and the remaining half of the time at a speed of 4 km/h. The second tourist walked the first half of the distance at a speed of 5... | \frac{2a}{9}<\frac{9a}{40} | 137 | 17 |
math | 5. (3 points) In a room, knights who always tell the truth and liars who always lie have gathered (both are definitely present). They were asked the question: "How many liars are in the room?". To this question, all possible answers from 1 to 200 were received (some possibly multiple times). How many liars could there ... | 199or200 | 79 | 7 |
math | Find all positive integers $m$ for which $2001\cdot S (m) = m$ where $S(m)$ denotes the sum of the digits of $m$. | 36018 | 38 | 5 |
math | 5. In isosceles $\triangle A B C$, it is known that $A C=B C=\sqrt{5}$, points $D, E, F$ are on sides $A B, B C, C A$ respectively, and $A D=D B=E F=1$. If $\overrightarrow{D E} \cdot \overrightarrow{D F} \leqslant \frac{25}{16}$, then the range of $\overrightarrow{E F} \cdot \overrightarrow{B A}$ is | \left[\frac{4}{3}, 2\right] | 115 | 14 |
math | The twelfth question: Find all pairs of positive integers $(k, n)$ that satisfy $k!=\prod_{\mathrm{i}=0}^{\mathrm{n}-1}\left(2^{\mathrm{n}}-2^{\mathrm{i}}\right)$.
---
Here is the translation, maintaining the original text's line breaks and format. | (k,n)=(1,1),(3,2) | 72 | 11 |
math | 4. $[x]$ is the greatest integer not exceeding the real number $x$. It is known that the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{1}=\frac{3}{2}, a_{n+1}=a_{n}^{2}-a_{n}+1\left(n \in \mathbf{N}_{+}\right) \text {. }
$$
Then $m=\left[\sum_{k=1}^{2011} \frac{1}{a_{k}}\right]$ is . $\qquad$ | 1 | 126 | 1 |
math | 1. There are candies in five bags. The first has 2, the second has 12, the third has 12, the fourth has 12, and the fifth has 12. Any number of candies can be moved from any bag to any other bag. What is the minimum number of moves required to ensure that all bags have an equal number of candies? | 4 | 79 | 1 |
math | [ Law of Cosines
The center of the circle inscribed in a right triangle is at distances $\sqrt{5}$ and $\sqrt{10}$ from the ends of the hypotenuse. Find the legs.
# | 3\times4 | 46 | 4 |
math | Example 41 (12th CMO Question) Let real numbers $x_{1}, x_{2}, \cdots, x_{1997}$ satisfy the following conditions:
(1) $-\frac{1}{\sqrt{3}} \leqslant x_{i} \leqslant \sqrt{3}(i=1,2, \cdots, 1997)$;
(2) $x_{1}+x_{2}+\cdots+x_{1997}=-318 \sqrt{3}$, find the maximum value of $x_{1}^{12}+x_{2}^{12}+\cdots+x_{1997}^{12}$. | 189548 | 161 | 6 |
math | ## Task 1 - 200821
Mr. Schäfer had bought two dogs. However, he had to sell them soon. For each dog, he received 180 Marks.
As Mr. Schäfer found out, with one of the dogs, he had made a profit of $20 \%$ of its original purchase price, while he sold the other dog at a loss of $20 \%$ of its original purchase price.
... | 15 | 142 | 2 |
math | Three, given that $a_{1}, a_{2}, \cdots, a_{11}$ are 11 distinct positive integers, and their total sum is less than 2007. On the blackboard, the numbers $1, 2, \cdots, 2007$ are written in sequence. Define a sequence of 22 consecutive operations as an operation group: the $i$-th operation can select any number from th... | \prod_{i=1}^{11} a_{i} | 327 | 15 |
math | 12. Given the sequence $\left\{a_{n}\right\}$, where $a_{1}=1, a_{2}=2, a_{n+1}=3 a_{n}-2 a_{n-1}, a_{2002}=$ | 2^{2001} | 58 | 7 |
math | 3. A book has a total of 61 pages, sequentially numbered as 1, 2, ..., 61. Someone, while adding these numbers, mistakenly reversed the digits of two two-digit page numbers (a two-digit number of the form $\overline{a b}$ was treated as $\overline{b a}$), resulting in a total sum of 2008. Therefore, the maximum sum of ... | 68 | 227 | 2 |
math | Problem 1. Consider three collinear points $A, B$, and $C$ such that $\frac{A B}{B C}=\frac{3}{5}$ and $A C=40 \text{ cm}$. Let $M$ and $P$ be the midpoints of segments $[A B]$, respectively $[B C]$.
a) Determine the lengths of segments $[A B]$ and $[B C]$.
b) Determine the value of the ratio $\frac{A P}{M C}$. | \frac{11}{13} | 113 | 9 |
math | 741*. Which years of the $XX$ century can be represented in the form $2^{n}-2^{k}$, where $n$ and $k$ are natural numbers? Find all such years. | 1984,1920 | 45 | 9 |
math | Problem 2. On each of the two gardens, Grandpa planted the same number of turnips. If Granddaughter enters the garden, she pulls out exactly $1 / 3$ of the turnips present at that moment. If Doggy enters, she pulls out $1 / 7$ of the turnips, and if Mousey enters, she pulls out only $1 / 12$ of the turnips. By the end ... | Yes | 120 | 1 |
math | Example 14. Find $\lim _{x \rightarrow 0}\left(1+\operatorname{tg}^{2} x\right)^{2 \operatorname{ctg}^{2} x}$. | e^{2} | 47 | 4 |
math | Let $p(x)$ be a polynomial with integer coefficients such that $p(0) = 0$ and $0 \le p(1) \le 10^7$. Suppose that there exist positive integers $a,b$ such that $p(a) = 1999$ and $p(b) = 2001$. Determine all possible values of $p(1)$.
(Note: $1999$ is a prime number.)
| 1, 1999, 3996001, 7992001 | 101 | 25 |
math | 96 Let the sequence $\left\{a_{1}, a_{2}, \cdots,\right\}=\left\{\frac{1}{1}, \frac{2}{1}, \frac{1}{2}, \frac{3}{1}, \frac{2}{2}, \frac{1}{3}, \frac{4}{1}, \frac{3}{2}, \frac{2}{3}, \frac{1}{4}, \cdots\right\}$, then the 1988th term of this sequence $a_{1988}=$ $\qquad$ . | \frac{29}{35} | 131 | 9 |
math | 1. A bank clerk exchanged the euros and cents when cashing a check for a customer, giving cents instead of euros and euros instead of cents. After buying a newspaper for 5 cents, the customer noticed that the remaining value was exactly twice the value of his check. What was the value of the check? | 31 | 63 | 2 |
math | You know that the binary function $\diamond$ takes in two non-negative integers and has the following properties:
\begin{align*}0\diamond a&=1\\ a\diamond a&=0\end{align*}
$\text{If } a<b, \text{ then } a\diamond b\&=(b-a)[(a-1)\diamond (b-1)].$
Find a general formula for $x\diamond y$, assuming that $y\gex>0$. | (y - x)^x | 105 | 7 |
math | Problem 21. In a triangle with a perimeter of $2 \sqrt{3}$, the product of its three angle bisectors is 1, and the radius of the inscribed circle is $\frac{1}{3}$. Find the angles of the triangle. | 60;60;60 | 56 | 8 |
math | 12.17. Find all solutions to the equation $x^{y}=y^{x}$:
a) in natural numbers;
b) in rational numbers.
## 12.3. Finding some solutions | 4,2 | 44 | 3 |
math | 1.4. Find the greatest negative root of the equation $\sin 2 \pi x=\sqrt{2} \cos \pi x$. Solution. The roots of the equation are the following series of values: $x=\frac{1}{2}+n, x=\frac{1}{4}+2 n, x=$ $\frac{3}{4}+2 n, n \in \mathbb{Z}$. Therefore, the greatest negative root of the equation is $-\frac{1}{2}$. | -\frac{1}{2} | 110 | 7 |
math | ## Task 2 - 330712
A six-digit natural number should, when read from left to right, have the digits $3, a, 3, b, 2, c$.
Determine all possibilities for choosing the digits $a, b, c$ such that the specified number is divisible by 90! | \begin{pmatrix}\mathrm{}&0&1&1&2&3&4&5&6&7&8&9\\\mathrm{~b}&1&0&9&8&7&6&5&4&3&2&1\\\mathrm{} | 73 | 60 |
math | Find the maximum integral value of $k$ such that $0 \le k \le 2019$ and $|e^{2\pi i \frac{k}{2019}} - 1|$ is maximal. | 1010 | 50 | 4 |
math | 145. In a regular hexagonal pyramid, the center of the circumscribed sphere lies on the surface of the inscribed sphere. Find the ratio of the radii of the circumscribed and inscribed spheres. | \frac{3+\sqrt{21}}{3} | 46 | 13 |
math | 2. The real solutions $(x, y)=$ of the equation $\sin (3 \cos x)-2 \sqrt{\sin (3 \cos x)} \cdot \sin (2 \cos y)+1=0$ | \left(2 k \pi \pm \arccos \frac{\pi}{6}, 2 k \pi \pm \arccos \frac{\pi}{4}, k \in Z\right) | 47 | 45 |
math | 2.227. $\frac{\sqrt{2 a+2 \sqrt{a^{2}-9}}}{\sqrt{2 a-2 \sqrt{a^{2}-9}}}$.
2.227. $\frac{\sqrt{2 a+2 \sqrt{a^{2}-9}}}{\sqrt{2 a-2 \sqrt{a^{2}-9}}}$. | \frac{+\sqrt{^{2}-9}}{3} | 85 | 14 |
math | 10. Two students, A and B, are selecting courses from five subjects. They have exactly one course in common, and A selects more courses than B. The number of ways A and B can select courses to meet the above conditions is $\qquad$ | 155 | 53 | 3 |
math | Kanel-Belov A.Y.
A cube with a side of 20 is divided into 8000 unit cubes, and a number is written in each cube. It is known that in each column of 20 cubes, parallel to the edge of the cube, the sum of the numbers is 1 (columns in all three directions are considered). In a certain cube, the number 10 is written. Thro... | 333 | 127 | 3 |
math | 【Question 4】
A cloth bag contains 10 small balls each of red, yellow, and green colors, all of the same size. The red balls are marked with the number "4", the yellow balls are marked with the number "5", and the green balls are marked with the number "6". Xiao Ming draws 8 balls from the bag, and their sum of numbers ... | 4 | 95 | 1 |
math | 2. What is the greatest possible value of the expression $x y - x^{3} y - x y^{3}$, where $x$, $y$ are positive real numbers? For which $x$, $y$ is this value achieved?
(Mária Dományová, Patrik Bak) | \frac{1}{8} | 63 | 7 |
math | 2. Given $\left\{\begin{array}{l}4^{x}+y=4030, \\ 17 \times 2^{x}+2 y=2019 .\end{array}\right.$ Find all real roots of the system of equations. | \log_{2}\frac{17+\sqrt{48617}}{4},\quad\frac{7787-17\sqrt{48617}}{8} | 62 | 45 |
math | Three given natural numbers are arranged in order of magnitude. Determine them based on the following information:
- the arithmetic mean of the given three numbers is equal to the middle one,
- the difference between some two of the given numbers is 321,
- the sum of some two of the given numbers is 777.
(L. Šimünek) | 228,549,870 | 75 | 11 |
math | 134. Three numbers are consecutive terms of some geometric progression. If 2 is added to the second number, they will form an arithmetic progression. If after that 16 is added to the third number, they will again form a geometric progression. Find these numbers. | 1,3,9 | 56 | 5 |
math | 5. Given a natural number $x=9^{n}-1$, where $n$ is a natural number. It is known that $x$ has exactly three distinct prime divisors, one of which is 11. Find $x$. | 59048 | 51 | 5 |
math | At the end of the quarter, Vovochka wrote down his current grades for singing in a row and placed a multiplication sign between some of them. The product of the resulting numbers turned out to be 2007. What grade does Vovochka get for singing in the quarter? ("Kol" is not given by the singing teacher.)
# | 3 | 74 | 1 |
math | ## Task 3 - 330723
Participants report on their four-day cycling tour:
Michael: "On the second day, we covered exactly $7 \mathrm{~km}$ more than on the third day."
Martin: "On the second and third day, we drove a total of $105 \mathrm{~km}$."
Matthias: "On the first day, exactly 516 and on the fourth day exactly 1... | 75,56,49,60 | 133 | 11 |
math | 82. Using the Gaussian method, solve the system of equations
$$
\left\{\begin{array}{l}
3 x+2 y-z=4 \\
2 x-y+3 z=9 \\
x-2 y+2 z=3
\end{array}\right.
$$ | (1;2;3) | 62 | 7 |
math | 3. Given the complex number sequence $\left\{z_{n}\right\}$ satisfies $z_{1}=1, z_{n+1}=\overline{z_{n}}+1+n i(n=1,2, \cdots)$, then $z_{2015}=$ | 2015+1007\mathrm{i} | 65 | 13 |
math | 3. Find all finite sets $S$ of positive integers with at least two elements, such that if $m>n$ are two elements of $S$, then
$$
\frac{n^{2}}{m-n}
$$
is also an element of $S$.
Answer: $S=\{s, 2 s\}$, for a positive integer $s$. | {,2s} | 78 | 5 |
math | 5. Which of the fractions is greater: $\frac{199719973}{199719977}$ or $\frac{199819983}{199819987}$? | \frac{199719973}{199719977}<\frac{199819983}{199819987} | 57 | 46 |
math | 12. The route from location A to location B consists only of uphill and downhill sections, with a total distance of 21 kilometers. If the speed uphill is 4 kilometers/hour, and the speed downhill is 6 kilometers/hour, the journey from A to B takes 4.25 hours, then the journey from B to A would take ( ) hours. | 4.5 | 78 | 3 |
math | 10.338. A curvilinear triangle is formed by three equal mutually tangent arcs of circles of radius $R$. Find the area of this triangle. | \frac{R^{2}(2\sqrt{3}-\pi)}{2} | 34 | 19 |
math | 3. Let $\mathbb{N}$ be the set of positive integers. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that:
$$
n+f(m) \text { divides } f(n)+n f(m)
$$
for all $m, n \in \mathbb{N}$.
(Albania) | f(x)\equivx^{2}f(x)\equiv1 | 79 | 13 |
math | 10. Among the students in the second year of junior high school, 32 students participated in the math competition, 27 students participated in the English competition, and 22 students participated in the Chinese competition. Among them, 12 students participated in both math and English, 14 students participated in both... | 30 | 111 | 2 |
math | Let $p$ be an odd prime of the form $p=4n+1$. [list=a][*] Show that $n$ is a quadratic residue $\pmod{p}$. [*] Calculate the value $n^{n}$ $\pmod{p}$. [/list] | 1 | 62 | 3 |
math | Folklore
Compare: $\sin 3$ and $\sin 3^{\circ}$. | \sin3>\sin3 | 21 | 6 |
math | 3. Determine the function $f: R \rightarrow R$ that satisfies the conditions
a) $f$ is differentiable with a continuous derivative;
b) $f(0)=2013$;
c) $f^{\prime}(\mathrm{x})-\mathrm{f}(\mathrm{x})=e^{x} \cos x$, for any real number $x$.
Teodor Trișcă and Daniela Vicol, Botoșani, G.M Supplement | f(x)=e^{x}\sinx+2013e^{x},x\in\mathbb{R} | 102 | 27 |
math | Given $0<a<1,0<b<1$, and $a b=\frac{1}{36}$. Find the minimum value of $u=\frac{1}{1-a}+\frac{1}{1-b}$. | \frac{12}{5} | 49 | 8 |
math | A quadratic polynomial $p(x)$ with integer coefficients satisfies $p(41) = 42$. For some integers $a, b > 41$, $p(a) = 13$ and $p(b) = 73$. Compute the value of $p(1)$.
[i]Proposed by Aaron Lin[/i] | 2842 | 73 | 4 |
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