task_type stringclasses 1
value | problem stringlengths 23 3.94k | answer stringlengths 1 231 | problem_tokens int64 10 1.39k | answer_tokens int64 1 98 |
|---|---|---|---|---|
math | 8.4. In triangle $A B C$, the bisector $A M$ is perpendicular to the median $B K$. Find the ratios $B P: P K$ and $A P: P M$, where $P$ is the point of intersection of the bisector and the median. | BP:PK=1,AP:PM=3:1 | 62 | 13 |
math | The $25$ member states of the European Union set up a committee with the following rules:
1) the committee should meet daily;
2) at each meeting, at least one member should be represented;
3) at any two different meetings, a different set of member states should be represented;
4) at $n^{th}$ meeting, for every $k<n$, ... | 2^{24} | 110 | 7 |
math | Trapezoid $ABCD^{}_{}$ has sides $AB=92^{}_{}$, $BC=50^{}_{}$, $CD=19^{}_{}$, and $AD=70^{}_{}$, with $AB^{}_{}$ parallel to $CD^{}_{}$. A circle with center $P^{}_{}$ on $AB^{}_{}$ is drawn tangent to $BC^{}_{}$ and $AD^{}_{}$. Given that $AP^{}_{}=\frac mn$, where $m^{}_{}$ and $n^{}_{}$ are relatively prime positive... | 164 | 138 | 3 |
math | Let $f(x) = 1 + 2x + 3x^2 + 4x^3 + 5x^4$ and let $\zeta = e^{2\pi i/5} = \cos \frac{2\pi}{5} + i \sin \frac{2\pi}{5}$. Find the value of the following expression: $$f(\zeta)f(\zeta^2)f(\zeta^3)f(\zeta^4).$$
| 125 | 106 | 3 |
math | Example 1. Find $\frac{1}{1-\frac{1}{1-\cdots \frac{1}{1-\frac{355}{113}}}}$. | \frac{355}{113} | 39 | 11 |
math | 10. In the Cartesian coordinate system, let point $A(0,4)$, $B(3,8)$. If point $P(x, 0)$ makes $\angle A P B$ maximum, then $x=$ $\qquad$ | 5 \sqrt{2}-3 | 53 | 7 |
math | 4. Let the function $y=f(x)$ be defined on $\mathbf{R}$, and have an inverse function $f^{-1}(x)$. It is known that the inverse function of $y=f(x+1)-2$ is $y=f^{-1}(2 x+1)$, and $f(1)=4$. If $n \in \mathbf{N}_{+}$, then $f(n)=$ $\qquad$ | 3+\left(\frac{1}{2}\right)^{n-1} | 95 | 17 |
math | 4. Six numbers are written in a row on the board. It is known that each number, starting from the third, is equal to the product of the two preceding numbers, and the fifth number is equal to 108. Find the product of all six numbers in this row. | 136048896 | 59 | 9 |
math | 35. By writing down 6 different numbers, none of which is 1, in ascending order and multiplying them, Olya got the result 135135. Write down the numbers that Olya multiplied. | 3\cdot5\cdot7\cdot9\cdot11\cdot13=135135 | 47 | 25 |
math | ## SUBJECT I
The sequence of numbers is given: $1,4,7,10,13,16,19$
a) Determine whether the 22nd term of the sequence is a perfect square.
b) Find the 2015th term of the sequence. | 6043 | 63 | 4 |
math | 5. In $\triangle A B C$, $A B=B C>A C, A H$ and $A M$ are the altitude and median from vertex $A$ to side $B C$, respectively, and $\frac{S_{\triangle A M H}}{S_{\triangle A B C}}=\frac{3}{8}$. Determine the value of $\cos \angle B A C$. | \frac{1}{4} | 84 | 7 |
math | Example 1 Find all integers $x$ such that $1+5 \times 2^{x}$ is the square of a rational number. ${ }^{[1]}$
(2008, Croatia National Training (Grade 2)) | x=-2 \text{ or } 4 | 51 | 10 |
math | 26th Putnam 1965 Problem A1 How many positive integers divide at least one of 10 40 and 20 30 ? Solution | 2301 | 37 | 4 |
math | 4. 188 Find all three-digit numbers that satisfy the following condition: the quotient when divided by 11 equals the sum of the squares of its digits. | 550 \text{ and } 803 | 35 | 12 |
math | 3. In the complex plane, the points $0, z, \frac{1}{z}, z+\frac{1}{z}$ form a parallelogram with an area of $\frac{4}{5}$. Then the minimum value of $\left|z+\frac{1}{z}\right|$ is | \frac{2\sqrt{5}}{5} | 65 | 12 |
math | 61. Another task about a piece of land
- Here's another task,- said the Black Queen. Another farmer had a piece of land. On one third of his land, he grew pumpkins, on one fourth he planted peas, on one fifth he sowed beans, and the remaining twenty-six acres he allocated for corn.
How many acres of land did the farm... | 120 | 100 | 3 |
math | Lirfisches. $\underline{\text {... }}$.
Yura laid out 2001 coins of 1, 2, and 3 kopecks in a row. It turned out that between any two 1-kopeck coins there is at least one coin, between any two 2-kopeck coins there are at least two coins, and between any two 3-kopeck coins there are at least three coins. How many 3-kope... | 500or501 | 106 | 7 |
math | Jeff has a deck of $12$ cards: $4$ $L$s, $4$ $M$s, and $4$ $T$s. Armaan randomly draws three cards without replacement. The probability that he takes $3$ $L$s can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m +n$. | 56 | 86 | 2 |
math | Example 5 If real numbers $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$ satisfy the system of equations
$$
\left\{\begin{array}{l}
x_{1} x_{2}+x_{1} x_{3}+x_{1} x_{4}+x_{1} x_{5}=-1, \\
x_{2} x_{1}+x_{2} x_{3}+x_{2} x_{4}+x_{2} x_{5}=-1, \\
x_{3} x_{1}+x_{3} x_{2}+x_{3} x_{4}+x_{3} x_{5}=-1, \\
x_{4} x_{1}+x_{4} x_{2}+x_{4} x_{3... | \pm \sqrt{2}, \pm \frac{\sqrt{2}}{2} | 274 | 19 |
math | Problem 2.3. A square $C$ is completely covered with a whole number of unit squares, without overlaps. If one places as many squares as possible of area 2 inside $C$, with sides parallel to the sides of $C$, without overlaps, it is possible to cover eight ninths of the area of the square. Determine all possible dimensi... | \begin{pmatrix}n^{\}=^{\}=1&n=2&=3\\n^{\}=^{\}=2&n=4&=6\\n^{\}=^{\}=3&n=6&=9\\n^{\}=^{\}=4&n=8&=12\\n^{\}=^{\} | 77 | 78 |
math | Given a circle, point $C$ on it and point $A$ outside the circle. The equilateral triangle $ACP$ is constructed on the segment $AC$. Point $C$ moves along the circle. What trajectory will the point $P$ describe? | \text{The trajectory of point } P \text{ is a circle with center } \left( \frac{a - \sqrt{3} b}{2}, \frac{\sqrt{3} a + b}{2} \right) \text{ and radius } r. | 53 | 59 |
math | 2. In a correspondence mathematics olympiad, out of 500 participants, exactly 30 did not like the problem conditions, exactly 40 did not like the organization of the event, and finally, exactly 50 did not like the method of determining the winners of the olympiad. We will call an olympiad participant "significantly dis... | 60 | 121 | 2 |
math | In a group of $n$ people, there are $k$ individuals who each have exactly two acquaintances among the present. Among the remaining $(n-k)$ members of the group, no two know each other.
What is the maximum number of handshakes that can occur if the strangers introduce themselves to each other? (We consider the acquaint... | S_{\max}=\frac{1}{2}n(n-1)-k | 87 | 18 |
math | 5. In a regular tetrahedron $ABCD$, $E$ and $F$ are on edges $AB$ and $AC$ respectively, satisfying $BE=3$, $EF=4$, and $EF$ is parallel to plane $BCD$. Then the area of $\triangle DEF$ is $\qquad$. | 2\sqrt{33} | 68 | 7 |
math | 2. On the shores of a circular island (viewed from above), there are cities $A, B, C$, and $D$. The straight asphalt road $A C$ divides the island into two equal halves. The straight asphalt road $B D$ is shorter than road $A C$ and intersects it. The speed of a cyclist on any asphalt road is 15 km/h. The island also h... | 450 | 157 | 3 |
math | ## Task B-4.3.
Determine all three-digit numbers that in base 9 have the representation $\overline{x y z}_{9}$, and in base 11 have the representation $\overline{z y x}_{11}$. | 302_{9}=203_{11},604_{9}=406_{11} | 53 | 26 |
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 | Folkolo
Find the maximum value of the expression $ab + bc + ac + abc$, if $a + b + c = 12$ (where $a, b$, and $c$ are non-negative numbers). | 112 | 48 | 3 |
math | 908*. Does the equation
$$
x^{2}+y^{2}+z^{2}=2 x y z
$$
have a solution in non-negative integers? | (0,0,0) | 39 | 7 |
math | # Task No. 5.1
## Condition:
A Dog, a Cat, and a Mouse are running around a circular lake. They all started simultaneously in the same direction from the same point and finished at the same time, each running at a constant speed. The Dog ran 12 laps, the Cat ran 6 laps, and the Mouse ran 4 laps. How many total overta... | 13 | 119 | 2 |
math | Let's draw the circles that touch the sides and pass through the vertices of a right-angled triangle. What are the lengths of the sides of the triangle if the radius of the first circle is $8 \mathrm{~cm}$ and the radius of the second circle is $41 \mathrm{~cm}$? | \overline{AB}=18 | 65 | 8 |
math | 9.75 A rectangular prism is constructed from small cubes of the same size. Three faces that share a common vertex are painted. As a result, exactly half of the small cubes have at least one face painted. How many small cubes have at least one face painted?
The rectangular prism is constructed from small cubes of the s... | 60,72,84,90,120 | 106 | 15 |
math | $$
\begin{array}{l}
\text { 2. If } f(g(x))=\sin 2 x, \\
g(x)=\tan \frac{x}{2}(0<x<\pi),
\end{array}
$$
then $f\left(\frac{\sqrt{2}}{2}\right)=$ $\qquad$ | \frac{4 \sqrt{2}}{9} | 74 | 12 |
math | 2B. Determine all solutions to the equation
$$
3^{x}+3^{y}+3^{z}=21897
$$
such that $x, y, z \in \mathbb{N}$ and $x<y<z$. | 3,7,9 | 57 | 5 |
math | How many three-digit numbers exist in which the digits 1, 2, 3 appear exactly once each?
# | 6 | 24 | 1 |
math | ## Task 4 - 060514
We are looking for a natural number with the following properties:
If you divide 100 by this number, the remainder is 4, and if you divide 90 by this number, the remainder is 18.
What is the number we are looking for? | 24 | 69 | 2 |
math | ## Task 3 - 241213
Determine all real numbers $x$ for which $2 x-3,5 x-14$ and $\frac{2 x-3}{5 x-14}$ are integers. | 3 | 53 | 1 |
math | Given $n$ integers $a_{1}=1, a_{2}, a_{3}, \ldots, a_{n}$, and $a_{i} \leq a_{i+1} \leq 2 a_{i}(i=1,2, \ldots, n-1)$, and the sum of all numbers is even. Can these numbers be divided into two groups such that the sums of the numbers in these groups are equal? | S_{1}=S_{2} | 98 | 8 |
math | 11.2. In a row, 21 numbers are written sequentially: from 2000 to 2020 inclusive. Enthusiastic numerologists Vova and Dima performed the following ritual: first, Vova erased several consecutive numbers, then Dima erased several consecutive numbers, and finally, Vova erased several consecutive numbers (at each step, the... | 2009,2010,2011 | 145 | 14 |
math | 1. Vasya's dad is good at math, but on the way to the garage, he forgot the code for the digital lock on the garage. In his memory, he recalls that all the digits of the code are different and their sum is 28. How many different codes does dad need to try to definitely open the garage, if the opening mechanism of the l... | 48 | 90 | 2 |
math | 2. Let $n$ be a positive integer. If $n$ is divisible by 2010 and exactly one of the digits of $n$ is even, find the smallest possible value of $n$.
(1 mark)
Let $n$ be a positive integer. If $n$ is divisible by 2010 and exactly one of the digits of $n$ is even, find the smallest possible value of $n$. | 311550 | 93 | 6 |
math | 5. Find all positive integers $x, y$ such that $\frac{x^{3}+y^{3}-x^{2} y^{2}}{(x+y)^{2}}$ is a non-negative integer. | 2 | 46 | 1 |
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 | 1. The sum of all real numbers $x$ that satisfy the equation $\sqrt{3 x-4}+\sqrt[3]{5-3 x}=1$ is $\qquad$ . | \frac{22}{3} | 41 | 8 |
math | For $0<x<1,$ let $f(x)=\int_0^x \frac{dt}{\sqrt{1-t^2}}\ dt$
(1) Find $\frac{d}{dx} f(\sqrt{1-x^2})$
(2) Find $f\left(\frac{1}{\sqrt{2}}\right)$
(3) Prove that $f(x)+f(\sqrt{1-x^2})=\frac{\pi}{2}$ | \frac{\pi}{2} | 105 | 7 |
math | 3. If the length, width, and height of a rectangular prism are all prime numbers, and the sum of the areas of two adjacent sides is 341, then the volume of this rectangular prism $V=$ $\qquad$ . | 638 | 50 | 3 |
math | 14. Solve the inequality $\log _{2}\left(x^{12}+3 x^{10}+5 x^{8}+3 x^{6}+1\right)<1+\log _{2}\left(x^{4}+1\right)$. | x\in(-\sqrt{\frac{\sqrt{5}-1}{2}},\sqrt{\frac{\sqrt{5}-1}{2}}) | 60 | 31 |
math | 2. Given: $b_{1}, b_{2}, b_{3}, b_{4}$ are positive integers, the polynomial $g(z)=(1-z)^{b_{1}}\left(1-z^{2}\right)^{b_{2}}\left(1-z^{3}\right)^{b_{3}}\left(1-z^{4}\right)^{b_{4}}$ when expanded and terms higher than 4th degree are omitted, becomes $1-2 z$. Also, $\alpha$ is the largest root of the polynomial $f(x)=x^... | 5 | 180 | 1 |
math | 1. The 6 -digit number $739 A B C$ is divisible by 7,8 , and 9 . What values can $A, B$, and $C$ take? | (A,B,C)\in{(3,6,8),(8,7,2)} | 42 | 18 |
math | 5. For any $x \in\left[-\frac{\pi}{6}, \frac{\pi}{2}\right]$, the inequality
$$
\sin ^{2} x+a \sin x+a+3 \geqslant 0
$$
always holds. Then the range of the real number $a$ is $\qquad$ | a \geqslant -2 | 75 | 8 |
math | ## Task 5 - 030715
How many zeros does the product of all natural numbers from 1 to 40 end with? (Justification!) | 9 | 37 | 1 |
math | 11.082. The height of the cone and its slant height are 4 and 5 cm, respectively. Find the volume of a hemisphere inscribed in the cone, with its base lying on the base of the cone. | \frac{1152}{125}\pi\mathrm{}^{3} | 50 | 19 |
math | ## Task B-4.4.
The line $t_{1}$ touches the left, and the line $t_{2}$, which is parallel to it, touches the right branch of the hyperbola $x^{2}-y^{2}=$ 1. If the lines $t_{1}$ and $t_{2}$ intersect the $x$-axis at an angle of $60^{\circ}$, calculate their mutual distance. | \sqrt{2} | 94 | 5 |
math | 1. If from one page of a book three letters are removed from each line and then two such lines are removed, the number of all letters will decrease by 145. If, on the other hand, we add four letters to each line and write three such lines, then the number
of all letters will increase by 224. How many lines are there on... | 29,32 | 90 | 5 |
math | Problem 4. A rectangle and a square have equal areas. The numerical values of their dimensions are natural numbers from the first ten. If the width of the rectangle is $2 \mathrm{~cm}$, determine the dimensions of the rectangle and the square. | 4\, | 53 | 3 |
math | Find every real values that $a$ can assume such that
$$\begin{cases}
x^3 + y^2 + z^2 = a\\
x^2 + y^3 + z^2 = a\\
x^2 + y^2 + z^3 = a
\end{cases}$$
has a solution with $x, y, z$ distinct real numbers. | \left(\frac{23}{27}, 1\right) | 85 | 16 |
math | Example 6 Find the equation of the curve $E^{\prime}$ symmetric to the curve $E: x^{2}+2 x y+y^{2}+3 x+y=$ 0 with respect to the line $l: 2 x-y-1=0$. | x^{2}+14xy+49y^{2}-21x+103y+54=0 | 58 | 28 |
math | A right-angled triangle has side lengths that are integers, and the measurement of its perimeter is equal to the measurement of its area. Which triangle is this? | 6,8,10 | 32 | 6 |
math | 28. In response to a question about his age, the grandfather answered: “The number expressing my age in years is a two-digit number equal to the sum of the number of its tens and the square of its units.” How old is the grandfather? | 89 | 52 | 2 |
math | 4. Find all five-digit numbers consisting of non-zero digits such that each time we erase the first digit, we get a divisor of the previous number.
## Solutions
Problem 1. | 91125,53125,95625 | 38 | 17 |
math | What is the largest integer $n$ for which
$$
\frac{\sqrt{7}+2 \sqrt{n}}{2 \sqrt{7}-\sqrt{n}}
$$
is an integer? | 343 | 43 | 3 |
math | 9. (16 points) Let the constant $a \in \mathbf{R}$, and the function
$$
f(x)=(a-x)|x|
$$
has an inverse function $f^{-1}(x)$. If the inequality
$$
f^{-1}\left(x^{2}+m\right)<f(x)
$$
holds for all $x \in[-2,2]$, find the range of the real number $m$. | \in(12,+\infty) | 97 | 10 |
math | 11.056. The diagonal of a rectangular parallelepiped is 10 cm and forms an angle of $60^{\circ}$ with the base plane. The area of the base is $12 \mathrm{~cm}^{2}$. Find the lateral surface area of the parallelepiped. | 70\sqrt{3} | 67 | 7 |
math | 16. Let $n$ be a given positive integer, $S_{n} \subseteq\left\{\alpha \mid \alpha=\left(p_{1}, p_{2}, \cdots, p_{n}\right), p_{k} \in\{0,1\}, k=1,2, \cdots, n\right\}$. For any elements $\beta=\left(x_{1}, x_{2}, \cdots, x_{n}\right)$ and $\gamma=\left(y_{1}, y_{2}, \cdots, y_{n}\right)$ in the set $S_{n}$, the follow... | 5 | 259 | 1 |
math | 3. Let $A B C$ be a triangle such that $A B=7$, and let the angle bisector of $\angle B A C$ intersect line $B C$ at $D$. If there exist points $E$ and $F$ on sides $A C$ and $B C$, respectively, such that lines $A D$ and $E F$ are parallel and divide triangle $A B C$ into three parts of equal area, determine the numbe... | 13 | 107 | 2 |
math | A coin is tossed $10$ times. Compute the probability that two heads will turn up in succession somewhere in the sequence of throws. | \frac{55}{64} | 28 | 9 |
math | Let $a, b, c, d \in \mathbb{R}$ such that $a+3 b+5 c+7 d=14$. Find the minimum possible value of $a^{2}+b^{2}+c^{2}+d^{2}$. | \frac{7}{3} | 61 | 7 |
math | For how many integer values of $m$,
(i) $1\le m \le 5000$
(ii) $[\sqrt{m}] =[\sqrt{m+125}]$
Note: $[x]$ is the greatest integer function | 72 | 55 | 2 |
math | Three distinct real numbers form (in some order) a 3-term arithmetic sequence, and also form (in possibly a different order) a 3-term geometric sequence. Compute the greatest possible value of the common ratio of this geometric sequence. | -2 | 49 | 2 |
math | 2. (5 points) Xiao Fou buys 3 erasers and 5 pencils for 10.6 yuan. If he buys 4 erasers and 4 pencils of the same type, he needs to pay 12 yuan. Then the price of one eraser is $\qquad$ yuan. | 2.2 | 64 | 3 |
math | At an Antarctic station, there are $n$ polar explorers, all of different ages. With probability $p$, any two polar explorers establish friendly relations, independently of other sympathies or antipathies. When the wintering period ends and it's time to return home, in each pair of friends, the older one gives a friendl... | \frac{1}{p}(1-(1-p)^{n}) | 90 | 15 |
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 | Let $n$ be a natural number divisible by $4$. Determine the number of bijections $f$ on the set $\{1,2,...,n\}$ such that $f (j )+f^{-1}(j ) = n+1$ for $j = 1,..., n.$ | \frac{(\frac{n}{2})!}{(\frac{n}{4})!} | 64 | 19 |
math | Let $ABC$ be a triangle with $\angle C=90^\circ$, and $A_0$, $B_0$, $C_0$ be the mid-points of sides $BC$, $CA$, $AB$ respectively. Two regular triangles $AB_0C_1$ and $BA_0C_2$ are constructed outside $ABC$. Find the angle $C_0C_1C_2$. | 30^\circ | 90 | 4 |
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 | Determine the number of integers $a$ with $1\leq a\leq 1007$ and the property that both $a$ and $a+1$ are quadratic residues mod $1009$. | 251 | 49 | 3 |
math | 5. (20 points) Professor K., wishing to be known as a wit, plans to tell no fewer than two but no more than three different jokes at each of his lectures. At the same time, the sets of jokes told at different lectures should not coincide. How many lectures in total will Professor K. be able to give if he knows 8 jokes? | 84 | 75 | 2 |
math | ## 16. How many of you were there, children?
If you had asked me such a question, I would have answered you only that my mother dreamed of having no fewer than 19 children, but she did not manage to fulfill her dream; however, I had three times as many sisters as cousins, and brothers - half as many as sisters. How ma... | 10 | 82 | 2 |
math | 1. Find all ordered pairs $(a, b)$ of positive integers such that $a^{2}+b^{2}+25=15 a b$ and $a^{2}+a b+b^{2}$ is prime. | (1,2)(2,1) | 51 | 9 |
math | Find all prime numbers $p$ for which there exist positive integers $x, y$ and $z$ such that the number
$$
x^{p}+y^{p}+z^{p}-x-y-z
$$
is a product of exactly three distinct prime numbers. | p = 2, 3, 5 | 59 | 10 |
math | ## Problem Statement
Find the derivative $y_{x}^{\prime}$.
$$
\left\{\begin{array}{l}
x=\frac{3 t^{2}+1}{3 t^{3}} \\
y=\sin \left(\frac{t^{3}}{3}+t\right)
\end{array}\right.
$$ | -^{4}\cdot\cos(\frac{^{3}}{3}+) | 75 | 17 |
math | ## Problem Statement
Find the derivative.
$$
y=(\tan x)^{\ln (\tan x) / 4}
$$ | (\tanx)^{(\ln(\tanx)/4)}\cdot\frac{\ln(\tanx)}{\sin2x} | 27 | 28 |
math | 2. Two bottles of equal volume are filled with a mixture of water and juice. In the first bottle, the ratio of the quantities of water and juice is $2: 1$, and in the second bottle, it is $4: 1$. If we pour the contents of both bottles into a third bottle, what will be the ratio of the quantities of water and juice in ... | 11:4 | 80 | 4 |
math | 1. Let $A D, B E, C F$ be the three altitudes of an acute triangle $A B C$, with the coordinates of $D, E, F$ being $(4,0),\left(\frac{80}{17}, \frac{20}{17}\right),\left(\frac{5}{2}, \frac{5}{2}\right)$, respectively. Find the coordinates of $A, B, C$. | A(4,4) | 98 | 6 |
math | Let $s(a)$ denote the sum of digits of a given positive integer $a$. The sequence $a_{1}, a_{2}, \ldots a_{n}, \ldots$ of positive integers is such that $a_{n+1}=a_{n}+s\left(a_{n}\right)$ for each positive integer $n$. Find the greatest possible $n$ for which it is possible to have $a_{n}=2008$.
Let $s(a)$ denote the... | 6 | 197 | 1 |
math | 1. If real numbers $x, y$ satisfy $4 x^{2}+y^{2}=1$, then the minimum value of $\frac{4 x y}{2 x+y-1}$ is . $\qquad$ | 1-\sqrt{2} | 48 | 6 |
math | 1. Given arrays $\left(a_{1}, a_{2}, \cdots, a_{n}\right)$ and $\left(b_{1}, b_{2}, \cdots, b_{n}\right)$ are both permutations of $1,2, \cdots, n$. Then
$$
a_{1} b_{1}+a_{2} b_{2}+\cdots+a_{n} b_{n}
$$
the maximum value is | \frac{n(n+1)(2 n+1)}{6} | 98 | 15 |
math | 3. The number of positive integers $m$ that make $m^{2}+m+7$ a perfect square is $\qquad$ . | 2 | 31 | 1 |
math | 23. The line $l$ passing through point $E\left(-\frac{p}{2}, 0\right)$ intersects the parabola $C: y^{2}=2 p x(p>0)$ at points $A$ and $B$, and the inclination angle of line $l$ is $\alpha$. Then the range of $\alpha$ is $\qquad$; $F$ is the focus of the parabola, and the area of $\triangle A B F$ is $\qquad$ (expresse... | \left(0, \frac{\pi}{4}\right) \cup\left(\frac{3 \pi}{4}, \pi\right), \frac{p^{2} \sqrt{\cos 2 \alpha}}{\sin \alpha} | 123 | 53 |
math | ## Problem Statement
Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
$M_{1}(1 ; 0 ; 2)$
$M_{2}(1 ; 2 ;-1)$
$M_{3}(2 ;-2 ; 1)$
$M_{0}(-5 ;-9 ; 1)$ | \sqrt{77} | 88 | 6 |
math | 2. Let $\mathrm{i}=\sqrt{-1}$ be the imaginary unit, then $\mathrm{i}+2 \mathrm{i}^{2}+3 \mathrm{i}^{3}+\cdots+2013 \mathrm{i}^{2013}=$ | 1006+1007\mathrm{i} | 59 | 13 |
math | The polynomial
$$
Q(x_1,x_2,\ldots,x_4)=4(x_1^2+x_2^2+x_3^2+x_4^2)-(x_1+x_2+x_3+x_4)^2
$$
is represented as the sum of squares of four polynomials of four variables with integer coefficients.
[b]a)[/b] Find at least one such representation
[b]b)[/b] Prove that for any such representation at least one of the four polyn... | Q(x_1, x_2, x_3, x_4) = (x_1 + x_2 - x_3 - x_4)^2 + (x_1 - x_2 + x_3 - x_4)^2 + (x_1 - x_2 - x_3 + x_4)^2 | 127 | 75 |
math | There are $100$ white points on a circle. Asya and Borya play the following game: they alternate, starting with Asya, coloring a white point in green or blue. Asya wants to obtain as much as possible pairs of adjacent points of distinct colors, while Borya wants these pairs to be as less as possible. What is the maxima... | 50 | 95 | 2 |
math | 1. Let positive numbers $x, y, z$ satisfy
$$
\frac{1}{x^{3}}=\frac{8}{y^{3}}=\frac{27}{z^{3}}=\frac{k}{(x+y+z)^{3}} \text {. }
$$
Then $k=$ $\qquad$ | 216 | 70 | 3 |
math | 3. (48th Slovenian Mathematical Olympiad) Mathieu first wrote down all the numbers from 1 to 10000 in order, then erased those numbers that are neither divisible by 5 nor by 11. Among the remaining numbers, what is the number at the 2004th position? | 7348 | 69 | 4 |
math | 9.12. Calculate in the same way the sums $S_{2}(n)=1^{2}+$ $+2^{2}+\ldots+n^{2}$ and $S_{3}(n)=1^{3}+2^{3}+\ldots+n^{3}$. | S_{2}(n)=\frac{n(n+1)(2n+1)}{6},\quadS_{3}(n)=(\frac{n(n+1)}{2})^2 | 62 | 41 |
math | 11. Let real numbers $x_{1}, x_{2}, \cdots, x_{2014}$ satisfy
$$
\left|x_{1}\right|=99,\left|x_{n}\right|=\left|x_{n-1}+1\right| \text {, }
$$
where, $n=2,3, \cdots, 2014$. Find the minimum value of $x_{1}+x_{2}+\cdots+x_{2014}$. | -5907 | 113 | 5 |
math | 29th IMO 1988 shortlist Problem 20 Find the smallest n such that if {1, 2, ... , n} is divided into two disjoint subsets then we can always find three distinct numbers a, b, c in the same subset with ab = c. | 96 | 60 | 2 |
math | [Coordinate method in space]
Find the angle between the line passing through points $A(-3 ; 0 ; 1)$ and $B(2 ; 1 ;-1)$, and the line passing through points $C(-2 ; 2 ; 0)$ and $D(1 ; 3 ; 2)$. | \arccos\frac{2\sqrt{105}}{35} | 68 | 19 |
math | 2. (6 points) Convert $\frac{15}{37}$ to a decimal, the 2016th digit from left to right in the decimal part is | 5 | 37 | 1 |
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