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 | 1. (7 points) 45 candies cost as many rubles as can be bought for 20 rubles. How many candies can be bought for 50 rubles?
Answer: 75 candies. | 75 | 46 | 2 |
math | 13. Given $a_{1}, a_{2}, \cdots, a_{n}$ are positive real numbers, and $a_{1}+a_{2}+\cdots+a_{n}=1, k$ is a positive integer, find the minimum value of $\frac{1}{a_{1}^{k}\left(1+a_{1}\right)}+\frac{1}{a_{2}^{k}\left(1+a_{2}\right)}+\cdots+\frac{1}{a_{n}^{k}\left(1+a_{n}\right)}$. | \frac{n^{k+2}}{n+1} | 125 | 13 |
math | In the cuboid $ABCDA'B'C'D'$ with $AB=a$, $AD=b$ and $AA'=c$ such that $a>b>c>0$, the points $E$ and $F$ are the orthogonal projections of $A$ on the lines $A'D$ and $A'B$, respectively, and the points $M$ and $N$ are the orthogonal projections of $C$ on the lines $C'D$ and $C'B$, respectively. Let $DF\cap BE=\{G\}$ an... | \frac{a^2 - b^2}{\sqrt{a^2 + b^2}} | 194 | 23 |
math | 1. How many natural numbers $x$ satisfy the inequality $2014<\sqrt{x}<2015$? | 4028 | 28 | 4 |
math | 1. Usually, schoolboy Gavriila takes a minute to go up a moving escalator by standing on its step. But if Gavriila is late, he runs up the working escalator and saves 36 seconds this way. Today, there are many people at the escalator, and Gavriila decides to run up the adjacent non-working escalator. How much time will... | 40 | 102 | 2 |
math | 17th Iberoamerican 2002 Problem A1 The numbers 1, 2, ... , 2002 are written in order on a blackboard. Then the 1st, 4th, 7th, ... , 3k+1th, ... numbers in the list are erased. Then the 1st, 4th, 7th, ... 3k+1th numbers in the remaining list are erased (leaving 3, 5, 8, 9, 12, ... ). This process is carried out repeated... | 1598 | 141 | 4 |
math | 2. There are 1993 boxes arranged in a row from left to right, each containing some small balls. If the leftmost box contains 7 balls, and every four adjacent boxes contain a total of 30 balls, then the rightmost box contains $\qquad$ balls. | 7 | 61 | 1 |
math | 6.211. $\left\{\begin{array}{l}x y-\frac{x}{y}=\frac{16}{3}, \\ x y-\frac{y}{x}=\frac{9}{2} .\end{array}\right.$ | (2;3),(-2;-3) | 57 | 10 |
math | 4. For any real numbers $x, y$, define the operation $x * y$ as $x * y=a x+b y+c x y$, where $a, b, c$ are constants, and the operations on the right side of the equation are the usual real number addition and multiplication. It is known that $1 * 2=3, 2 * 3=4$, and there is a non-zero real number $d$, such that for an... | 4 | 118 | 1 |
math | Let $x_1,x_2, \cdots,x_n$ and $y_1,y_2, \cdots ,y_n$ be arbitrary real numbers satisfying $x_1^2+x_2^2+\cdots+x_n^2=y_1^2+y_2^2+\cdots+y_n^2=1$. Prove that
\[(x_1y_2-x_2y_1)^2 \le 2\left|1-\sum_{k=1}^n x_ky_k\right|\]
and find all cases of equality. | (x_1 y_2 - x_2 y_1)^2 \leq 2 \left| 1 - \sum_{k=1}^n x_k y_k \right| | 126 | 44 |
math | 7.1.1. (12 points) Find the greatest negative root of the equation
$$
\frac{\sin \pi x - \cos 2 \pi x}{(\sin \pi x - 1)^2 + \cos^2 \pi x - 1} = 0
$$ | -0.5 | 65 | 4 |
math | Let $a, b$ be real numbers, and there exists a complex number $z$, such that $|z| \leqslant 1$, and $z+\bar{z}|z|=a+b \mathrm{i}$. Then the maximum value of $a b$ is $\qquad$
$\qquad$. | \frac{1}{8} | 68 | 7 |
math | 8. (10 points) Two small rulers form a set of rulers, and the small rulers can slide along the large ruler. Each unit on the large ruler is marked with a natural number. The first small ruler divides 11 units on the large ruler into 10, and the second small ruler divides 9 units on the large ruler into 10. The starting... | 7 | 218 | 1 |
math | 4. (3 points) Warehouses A and B originally each stored whole bags of grain. If 90 bags are transferred from Warehouse A to Warehouse B, then the number of bags in Warehouse B will be twice that in Warehouse A. If a certain number of bags are transferred from Warehouse B to Warehouse A, then the number of bags in Wareh... | 153 | 109 | 3 |
math | 55.There were two positive numbers. One of them was increased by $1 \%$, the other by $4 \%$. Could their sum have increased by $3 \%$? | 2x | 36 | 2 |
math | \section*{Problem 2 - 171232}
For every integer \(a\), determine all real solutions \(x\) of the equation
\[
x^{4}+x^{3}+a^{2} x^{2}+x+1=0
\] | -1 | 63 | 2 |
math | 7、The arithmetic sequence $a_{1}, a_{2}, \cdots, a_{19}$ has a total of 19 terms. It is known that $a_{1}+a_{7}+a_{14}+a_{18}=120$, then $a_{1}+a_{2}+a_{3} \cdots+a_{19}=$ $\qquad$ | 570 | 91 | 3 |
math | Find all pairs $(x, y)$ of integers that satisfy
$$
x^{2}+y^{2}+3^{3}=456 \sqrt{x-y} .
$$ | (30,21), (-21,-30) | 39 | 14 |
math | Find the smallest positive integer $j$ such that for every polynomial $p(x)$ with integer coefficients and for every integer $k,$ the integer
\[p^{(j)}(k)=\left. \frac{d^j}{dx^j}p(x) \right|_{x=k}\]
(the $j$-th derivative of $p(x)$ at $k$) is divisible by $2016.$
| 8 | 91 | 1 |
math | Example 8 Find the maximum value of the function $y=\sqrt{5-2 x}+\sqrt{3+2 x}$.
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。
This part is not translated as it contains instructions for the translation task itself. Here is the requested translation above. | 4 | 77 | 1 |
math | 5. Given an arithmetic sequence $\left\{a_{n}\right\}$ satisfies:
$$
a_{1}<0, a_{100} \geqslant 74, a_{200}<200 \text {, }
$$
and the number of terms of this sequence in the interval $\left(\frac{1}{2}, 8\right)$ is 2 less than the number of terms in the interval $\left[14, \frac{43}{2}\right]$. Then the general term ... | a_{n}=\frac{3}{4} n-1 | 130 | 14 |
math | ## Task Condition
Find the derivative.
$$
y=\frac{1}{(x+2) \sqrt{x^{2}+4 x+5}}
$$ | -\frac{2x^{2}+8x+9}{(x+2)^{2}\cdot\sqrt{(x^{2}+4x+5)^{3}}} | 34 | 39 |
math | Example 4 Try to find the minimum value of the function $f(x, y)=6\left(x^{2}+y^{2}\right)(x+y)-4\left(x^{2}+x y+y^{2}\right)-3(x+y)+5$ in the region $D=\{(x, y) \mid x>0, y>0\}$. | 2 | 80 | 1 |
math | Example 3 (Olympiad Training Problem from "Intermediate Mathematics" Issue 5, 2004) Let $a \in \mathbf{R}, A=\left\{x \mid 2^{1+x}+2^{1-x}=a\right\}, B=\{\sin \theta \mid$ $\theta \in \mathbf{R}\}$. If $A \cap B$ contains exactly one element, find the range of values for $a$. | 4 | 102 | 1 |
math | 10.9. When dividing the polynomial $x^{1951}-1$ by $x^{4}+x^{3}+2 x^{2}+$ $+x+1$, a quotient and a remainder are obtained. Find the coefficient of $x^{14}$ in the quotient.
## 10.4. Vieta's Theorem | -1 | 77 | 2 |
math | Example 1.8. Find $\int\left(\frac{1-\cos x}{1+\cos x}\right)^{2} \frac{d x}{3 \sin x+4 \cos x+5}$. | 2(\frac{1}{3}\operatorname{tg}^{3}\frac{x}{2}-3\operatorname{tg}^{2}\frac{x}{2}+27\operatorname{tg}\frac{x}{2}-108\ln|\operatorname{tg}\frac{x}{2}+3|-\frac{81}{\operatorname{tg}\frac{x}{2} | 48 | 86 |
math | ## Task Condition
Find the point $M^{\prime}$ symmetric to the point $M$ with respect to the line.
$M(2 ; 1 ; 0)$
$\frac{x-2}{0}=\frac{y+1.5}{-1}=\frac{z+0.5}{1}$ | M^{\}(2;-2;-3) | 69 | 10 |
math | 7. Let $f(x)$ satisfy the equation $f(x)-2 f\left(\frac{1}{x}\right)=x$. Then the range of $f(x)$ is $\qquad$ . | \left(-\infty,-\frac{2 \sqrt{2}}{3}\right] \cup\left[\frac{2 \sqrt{2}}{3},+\infty\right) | 43 | 43 |
math | 10. Vера and Аnya attend a math club, in which there are more than $91 \%$ boys. Find the smallest possible number of club participants. | 23 | 35 | 2 |
math | 10. (10 points) A school provides students with three types of fruits: apples, bananas, and pears, to be used as snacks during breaks. Each student chooses at least one type, and can choose more than one. The statistics show: $70 \%$ of the students chose apples, $40 \%$ chose bananas, and $30 \%$ chose pears. What is ... | 20 | 97 | 2 |
math | 8.112. $\operatorname{ctg}^{3} x+\sin ^{-2} x-3 \operatorname{ctg} x-4=0$.
8.112. $\cot^{3} x+\csc ^{2} x-3 \cot x-4=0$. | x_{1}=\frac{3\pi}{4}+\pin;x_{2}=\\frac{\pi}{6}+\pik,n,k\inZ | 70 | 34 |
math | 13.187. A team of workers was supposed to manufacture 8000 identical parts within a certain period. In fact, the work was completed 8 days ahead of schedule because the team produced 50 more parts daily than planned. What was the original deadline for completing the work, and what was the daily percentage of overachiev... | 40;25 | 73 | 5 |
math | Question 164, Point $\mathrm{P}$ moves on the circle $(\mathrm{x}-2)^{2}+(\mathrm{y}-1)^{2}=1$, vector $\overrightarrow{\mathrm{PO}}$ (where $\mathrm{O}$ is the origin of coordinates) rotates counterclockwise by $90^{\circ}$ around point $\mathrm{P}$ to get $\overrightarrow{\mathrm{PQ}}$, then the trajectory equation o... | (x-3)^{2}+(y+1)^{2}=2 | 110 | 16 |
math | 5. Find all natural numbers whose proper divisors can be divided into pairs such that in each pair the numbers differ by 545. A proper divisor of a natural number is a natural divisor other than 1 and the number itself.
# | 2\cdot547 | 50 | 6 |
math | 10.4. Ten chess players over nine days played a full round-robin tournament, during which each of them played exactly one game with each other. Each day, exactly five games were played, with each chess player involved in exactly one of them. For what maximum $n \leq 9$ can it be claimed that, regardless of the schedule... | 5 | 112 | 1 |
math | The quadrilateral $ABCD$ is inscribed in the parabola $y=x^2$. It is known that angle $BAD=90$, the dioganal $AC$ is parallel to the axis $Ox$ and $AC$ is the bisector of the angle BAD.
Find the area of the quadrilateral $ABCD$ if the length of the dioganal $BD$ is equal to $p$. | \frac{p^2 - 4}{4} | 89 | 12 |
math | 45. Find the largest real number $\alpha$, such that there exists an infinite sequence of positive integers $\left\{a_{n}\right\}_{n=1}^{+\infty}$, satisfying:
(1) For any $n \in \mathbf{N}^{*}$, we have $a_{n}>2008^{n}$;
(2) For any $n \in \mathbf{N}^{*}, n \geqslant 2$, the number $a_{n}^{\alpha}$ does not exceed the... | \frac{1}{2} | 166 | 7 |
math | 4. Let $1 \leqslant k<n$, consider all finite sequences of positive integers that sum to $n$, find the number of sequences with $k$ terms, denoted as $T(n, k)$. | C_{n-1}^{k-1} | 47 | 11 |
math | 1. Calculate $\left(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots+\frac{1}{99}-\frac{1}{100}\right) \div\left(\frac{1}{51 \times 100}+\frac{1}{52 \times 99}+\cdots+\frac{1}{75 \times 76}\right)$ | 151 | 99 | 3 |
math | 2. Given the line $l: 2 x+y=10$, draw a line $l^{\prime} \perp$ $l$ through the point $(-10,0)$. The coordinates of the intersection point of $l^{\prime}$ and $l$ are $\qquad$ | (2,6) | 65 | 5 |
math | ## Task 4 - 060524
Hans is participating in the training of the track and field section of his school sports community. One of the exercises consists of rhythmic walking followed by rebounding in a standing position.
The length of the exercise track is $30 \mathrm{~m}$. At the beginning and end, there are flagpoles. ... | 176 | 149 | 3 |
math | 4. Triangle $A B C$ is similar to the triangle formed by its altitudes. Two sides of triangle $A B C$ are 4 cm and 9 cm. Find the third side. | 6 | 42 | 1 |
math | 1. What is the last digit of $17^{103}+5$ ? | 8 | 20 | 1 |
math | 1. If $x+y+z=0$, simplify the expression
$$
\frac{x^{7}+y^{7}+z^{7}}{x y z\left(x^{4}+y^{4}+z^{4}\right)}
$$
Hint: first calculate $(x+y)^{4}$ and $(x+y)^{6}$. | \frac{7}{2} | 76 | 7 |
math | Example 14. Solve the inequality
$$
10^{7 x-1}+6 \cdot 10^{1-7 x}-5 \leqslant 0
$$ | \frac{1}{7}(1+\lg2)\leqslantx\leqslant\frac{1}{7}(1+\lg3) | 42 | 34 |
math | 3. Arrange $r$ boxes in a row, and put all $n$ identical balls into these arranged boxes. How many different ways are there to do this? | C_{n+r-1}^{r-1} | 34 | 12 |
math |
Problem 1. Find all pairs $(a, b)$ of positive integers such that
$$
11 a b \leq a^{3}-b^{3} \leq 12 a b
$$
| (5,2) | 47 | 5 |
math | Example 2 Let a sequence be
$2,12,36,80,150, \cdots$.
Determine the type of its general term formula and find the analytical expression. | P(x)=x^{3}+x^{2} | 44 | 12 |
math | 12. Maddie has a paper ribbon of length $36 \mathrm{~cm}$. She divides it into four rectangles of different lengths. She draws two lines joining the centres of two adjacent rectangles as shown.
What is the sum of the lengths of the lines that she draws? | 18\mathrm{~} | 59 | 7 |
math | 35th Putnam 1974 Problem A1 S is a subset of {1, 2, 3, ... , 16} which does not contain three integers which are relatively prime in pairs. How many elements can S have? Solution | 11 | 54 | 2 |
math | 4. The minimum value of the function $f(x)=\sqrt{x^{2}+4}+\sqrt{x^{2}-4 x+5}$ is $\qquad$ . | \sqrt{13} | 38 | 6 |
math | Let $ X$ be the set of all positive integers greater than or equal to $ 8$ and let $ f: X\rightarrow X$ be a function such that $ f(x\plus{}y)\equal{}f(xy)$ for all $ x\ge 4, y\ge 4 .$ if $ f(8)\equal{}9$, determine $ f(9) .$ | 9 | 80 | 1 |
math | 7. Let non-zero distinct complex numbers $x, y$ satisfy $x^{2}+x y+$ $y^{2}=0$. Then the value of the expression
$$
\left[\frac{x y}{(x+y)(x-y)^{2}}\right]^{2000}\left(x^{2006}+y^{2006}\right)
$$
is $\qquad$ . | -\frac{1}{3^{2006}} | 91 | 12 |
math | 1. Let real numbers $x, y$ satisfy $4 x^{2}-5 x y+4 y^{2}=5$. If $S=x^{2}+y^{2}$, denote the maximum and minimum values of $S$ as $p$ and $q$, respectively, then $\frac{1}{p}+\frac{1}{q}=$ $\qquad$ | \frac{8}{5} | 81 | 7 |
math | There are three equal-radius, tangent circles. What is the area of the lobe between the circles if the radius of each circle is $r$? | r^{2}(\sqrt{3}-\frac{\pi}{2}) | 31 | 16 |
math | 6. (15 points) Every day, Ivan Ivanovich is taken to work by a company car. One day, Ivan Ivanovich decided to walk and left home one and a half hours earlier than usual. On the way, he met the company car and finished the journey in it. As a result, he arrived at work 20 minutes earlier than the usual time. How long d... | 80 | 85 | 2 |
math | 4. Let $n$ be a natural number. If 2005 can be written as the sum of $n$ positive odd composite numbers, then $n$ is called a "good number". The number of such good numbers is $\qquad$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result di... | 111 | 80 | 3 |
math | 4. Find the set of values of the expression $x-y+1$ under the condition
$$
(x-y)^{2}=2|2 y-x|+x+15
$$ | (-\infty;-2]\cup[6;+\infty) | 41 | 15 |
math | 1. Given that $a$, $b$, and $c$ are non-zero rational numbers, and satisfy
$$
\begin{array}{l}
a b^{2}=\frac{c}{a}-b . \\
\text { Then }\left(\frac{a^{2} b^{2}}{c^{2}}-\frac{2}{c}+\frac{1}{a^{2} b^{2}}+\frac{2 a b}{c^{2}}-\frac{2}{a b c}\right) \div \\
\left(\frac{2}{a b}-\frac{2 a b}{c}\right) \div \frac{101}{c}=
\end... | -\frac{1}{202} | 153 | 9 |
math | [ Quadratic Equations and Systems of Equations ]
Find all positive solutions of the system of equations
$$
\left\{\begin{array}{l}
x_{1}+x_{2}=x_{3}^{2} \\
x_{2}+x_{3}=x_{4}^{2} \\
x_{3}+x_{4}=x_{5}^{2} \\
x_{4}+x_{5}=x_{1}^{2} \\
x_{5}+x_{1}=x_{2}^{2}
\end{array}\right.
$$ | (2,2,2,2,2) | 125 | 11 |
math | Let $d(n)$ denote the number of positive divisors of a positive integer $n$. What is the smallest real value of $c$ such that $d(n) \leq$ $c \cdot \sqrt{n}$ holds for all positive integers $n$? | \sqrt{3} | 56 | 5 |
math | [ Motion Problems ]
Three runners - Anton, Seryozha, and Tolya - are participating in a 100 m race. When Anton finished, Seryozha was 10 meters behind him, and when Seryozha finished, Tolya was 10 meters behind him. At what
| distance from each other were Tolya and Anton when Anton finished? (It is assumed that all b... | 19\mathrm{~} | 104 | 7 |
math | ## Problem 1.
Consider the set $A$ of four-digit numbers that are at most equal to 2014. Determine the maximum number of elements of a subset of $A$ that contains only perfect squares, any two of which are coprime. | 6 | 55 | 1 |
math | # Problem 3. (3 points)
It is known that $a^{2} b+a^{2} c+b^{2} a+b^{2} c+c^{2} a+c^{2} b+3 a b c=30$ and $a^{2}+b^{2}+c^{2}=13$.
Find $a+b+c$. | 5 | 79 | 1 |
math | 1. Find the number of points in the plane $x O y$ that have natural coordinates $(x, y)$ and lie on the parabola $y=-\frac{x^{2}}{3}+98$ | 5 | 47 | 1 |
math | Let $(F_n)_{n\in{N^*}}$ be the Fibonacci sequence defined by
$F_1=1$, $F_2=1$, $F_{n+1}=F_n+F_{n-1}$ for every $n\geq{2}$. Find
the limit: \[ \lim_{n \to \infty}(\sum_{i=1}^n{\frac{F_i}{2^i}}) \] | 2 | 101 | 1 |
math | \section*{Problem 5 - 021235}
Given a line segment \(A B\) and a point \(M\) on it.
Construct squares \(A M D E\) and \(M B G H\) on the same side of the line segment \(A B\)! The centers of the two squares are \(R\) and \(S\).
What is the geometric locus of the midpoints of the segment \(R S\) ? | K(\frac{}{4},\frac{}{4})\quad;\quadL(\frac{3}{4},\frac{}{4}) | 93 | 31 |
math | ## Task A-2.3.
Determine all ordered triples $(m, n, p)$ where $m$ and $n$ are natural numbers, and $p$ is a prime number, for which the following holds:
$$
25^{n}+2 \cdot 5^{n}=p^{m}+8
$$ | (,n,p)=(3,1,3) | 71 | 11 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty}\left(\frac{3 n^{2}-6 n+7}{3 n^{2}+20 n-1}\right)^{-n+1}
$$ | e^{\frac{26}{3}} | 59 | 10 |
math | Consider the sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, ...$ Find $n$ such that the first $n$ terms sum up to $2010.$ | 1027 | 73 | 4 |
math | 14. Let $f(x)$ be an odd function defined on $\mathbf{R}$, and for any $x \in \mathbf{R}$, we have
$$
\begin{aligned}
f(x+2) & =f(x)+2, \\
\text { then } \sum_{k=1}^{2014} f(k) & =
\end{aligned}
$$ | 2029105 | 87 | 7 |
math | 4. Considering $\boldsymbol{x}$ and $\boldsymbol{y}$ as integers, solve the system of equations (11 points):
$$
\left\{\begin{array}{l}
8^{x^{2}-2 x y+1}=(z+4) 5^{|y|-1} \\
\sin \frac{3 \pi z}{2}=-1
\end{array}\right.
$$ | (-1;-1;-3),(1;1;-3) | 89 | 13 |
math | 17. (10 points) Let the function
$$
f(x)=\log _{2}\left(a^{x}-b^{x}\right) \text {, }
$$
and $f(1)=1, f(2)=\log _{2} 12$.
(1) Find the values of $a$ and $b$;
(2) When $x \in[1,2]$, find the maximum value of $f(x)$. | 2+\log _{2} 3 | 103 | 9 |
math | 5. (3 points) Three circles with radii 1, 1, and \(2 \sqrt{\frac{13-6 \sqrt{3}}{13}}\) are arranged such that the triangle formed by their centers is equilateral with a side length of \(\sqrt{3}\). Find the radius of the circumcircle of the triangle, each vertex of which is the point of intersection of two of these cir... | 4\sqrt{3}-6 | 111 | 7 |
math | Example 5 Find all positive integers $a, b, c$ such that $1 < a < b < c$ and $(a-1)(b-1)(c-1)$ is a divisor of $abc-1$.
(33rd IMO) | (3,5,15) \text{ and } (2,4,8) | 55 | 20 |
math | 5. Given a natural number $x=6^{n}+1$, where $n-$ is an odd natural number. It is known that $x$ has exactly three distinct prime divisors, one of which is 11. Find $x$. | 7777 | 53 | 4 |
math | Find the maximum value of
$$\int^1_0|f'(x)|^2|f(x)|\frac1{\sqrt x}dx$$over all continuously differentiable functions $f:[0,1]\to\mathbb R$ with $f(0)=0$ and
$$\int^1_0|f'(x)|^2dx\le1.$$ | \frac{2}{3} | 81 | 7 |
math | 8. Expand the binomial $\left(\sqrt{x}+\frac{1}{2 \sqrt[4]{x}}\right)^{n}$ in descending powers of $x$. If the coefficients of the first three terms form an arithmetic sequence, then the number of terms in the expansion where the power of $x$ is an integer is $\qquad$ . | 3 | 75 | 1 |
math | (3) For the regular triangular pyramid $V-ABC$, the side edge length is 3, and the base edge length is 2. A section through the base edge $AB$ intersects the side edge $VC$ at point $D$. The minimum value of the area of the section $\triangle ABD$ is $\qquad$. | \frac{\sqrt{23}}{3} | 70 | 11 |
math | In a Cartesian coordinate system, project the origin $O$ onto those lines for which
$$
\frac{1}{\overline{O A}^{2}}+\frac{1}{\overline{O R}^{2}}=\text{ constant. }
$$
$O A$ and $O B$ represent the segments that the line intercepts on the coordinate axes. Determine the geometric locus of the projection of point $O$! | x^{2}+y^{2}=k^{2} | 93 | 13 |
math | 5. Sides $A B$ and $A D$ of parallelogram $A B C D$ are equal to 1 and $a$, respectively, $\angle B A D=\alpha$, and triangle $A B D$ is acute. For which $a$ and $\alpha$ will four circles of radius 1 with centers at the vertices of the parallelogram completely cover the parallelogram? | \leq\cos\alpha+\sqrt{3}\sin\alpha | 85 | 15 |
math | 6. (3 points) On a $3 \times 3$ chessboard, there are knights, who always tell the truth, and liars, who always lie. Each of them said: "Among my neighbors, there are exactly three liars." How many liars are on the board?
Neighbors are considered to be people on cells that share a common side. | 5 | 76 | 1 |
math | A positive integer is called fancy if it can be expressed in the form
$$
2^{a_{1}}+2^{a_{2}}+\cdots+2^{a_{100}}
$$
where $a_{1}, a_{2}, \ldots, a_{100}$ are non-negative integers that are not necessarily distinct.
Find the smallest positive integer $n$ such that no multiple of $n$ is a fancy number.
Answer: The answe... | 2^{101}-1 | 270 | 7 |
math | 2. $42 N$ is the set of all positive integers. For a subset $S$ of $N$ and $n \in N$, define
$$S \oplus\{n\}=\{s+n \mid s \in S\}$$
Additionally, define the subset $S_{k}$ as follows:
$$S_{1}=\{1\}, S_{k}=\left\{S_{k-1} \oplus\{k\}\right\} \cup\{2 k-1\}, k=2,3,4 \cdots .$$
(1) Find $N-\bigcup_{k=1}^{\infty} S_{k}$.
(2... | 500 | 178 | 3 |
math | Example 1 Let the sequence $\left\{a_{n}\right\}$ have the sum of the first $n$ terms $S_{n}=2 a_{n}-1(n=1,2, \cdots)$, and the sequence $\left\{b_{n}\right\}$ satisfies $b_{1}=3, b_{k+1}=b_{k}+a_{k}(k=1,2, \cdots)$, find the sum of the first $n$ terms of the sequence $\left\{b_{n}\right\}$. | 2n+2^n-1 | 122 | 7 |
math | 50th Putnam 1989 Problem B5 A quadrilateral is inscribed in a circle radius 1. Two opposite sides are parallel. The difference between their lengths is d > 0. The distance from the intersection of the diagonals to the center of the circle is h. Find sup d/h and describe the cases in which it is attained. | 2 | 75 | 1 |
math | If we multiply the sum of two sides of a triangle by the third side, we get the following numbers: 96, 91, 75. a) How many $m$ is each side? b) How many $m^{2}$ is the area of the triangle? | =5,b=7,=8,=10\sqrt{3} | 61 | 17 |
math | 2B. Solve the equation:
$$
\log _{\frac{1}{8}}(2 x)-4 \log _{\frac{1}{4}} x \cdot \log _{8} x=0
$$ | x_{1}=\frac{1}{\sqrt{2}},x_{2}=2 | 48 | 19 |
math | Example 13 (1989 Irish Mathematical Olympiad) The function $f$ is defined on the set of natural numbers and satisfies the following conditions:
(1) $f(1)=1$;
(2) $f(2 n)=f(n), f(2 n+1)=f(2 n)+1(n \geqslant 1)$.
For $1 \leqslant n \leqslant 1989$, find the maximum value $u$ of $f(n)$, and determine how many $n(1 \leqsla... | 10 | 144 | 2 |
math | [ Combinations and Permutations ] $[$ Polygons (other) ]
In a convex polygon with an odd number of vertices, equal to $2 n+1$, two random diagonals are chosen independently of each other.
Find the probability that these diagonals intersect inside the polygon. | \frac{n(2n-1)}{3(2n^{2}-n-2)} | 58 | 21 |
math | 5. Given planar vectors $\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}$ satisfy:
$$
\begin{array}{l}
|\boldsymbol{a}|=|\boldsymbol{b}|=|\boldsymbol{c}|=2, \boldsymbol{a}+\boldsymbol{b}+\boldsymbol{c}=\mathbf{0} . \\
\text { If } 0 \leqslant x \leqslant \frac{1}{2} \leqslant y \leqslant 1 \text {, then } \\
|x(\boldsym... | \frac{1}{2} | 178 | 7 |
math | $12 \cdot 91$ Try to find all natural number solutions of the equation $7^{x}-3 \cdot 2^{y}=1$.
(16th All-Russian Mathematical Olympiad, 1990) | (x,y)=(1,1),(2,4) | 51 | 11 |
math | In the addition below, the same letters represent the same digit, and different letters represent different digits. Find the number $A B C D E$.
$A B C D E$ $B C D E$ $C D E$ $D E$ $A A A A A$ | 52487 | 60 | 5 |
math | Find all positive integer $n$, such that
$$\left[\frac{n}{2^0}\right]\left[\frac{n}{2^1}\right]\ldots\left[\frac{n}{2^k}\right]+2\cdot 4^{[\frac{k}{2}]}$$
is a square, where $k$ is the non-negative integer satisfying $2^k\leq n<2^{k+1}$. | n = 2, 4 | 94 | 7 |
math | In the convex quadrilateral $ABCD$, point $X$ is selected on side $AD$, and the diagonals intersect at point $E$. It is known that $AC = BD$, $\angle ABX = \angle AX B = 50^o$, $\angle CAD = 51^o$, $\angle AED = 80^o$. Find the value of angle $\angle AXC$.
| 80^\circ | 87 | 4 |
math | Find $3x^2 y^2$ if $x$ and $y$ are integers such that $y^2 + 3x^2 y^2 = 30x^2 + 517$. | 588 | 48 | 3 |
math | Two of the numbers $a+b, a-b, ab, a/b$ are positive, the other two are negative. Find the sign of $b$ | b < 0 | 32 | 5 |
math | 10.1. Kolya wrote a ten-digit number on the board, consisting of different digits. Sasha added one digit so that the resulting number would be divisible by 9. Which digit could Sasha have added? | 0or9 | 45 | 3 |
math | ## Task A-4.5.
In a certain country, there are three cities $A, B$, and $C$. Between each pair of cities, there are several roads (at least one), and all roads are bidirectional. In addition to direct road connections between two cities, there are also indirect connections. An indirect road connection between cities $... | 97or23 | 166 | 5 |
math | 8. In jar A, there are 6 balls, of which 4 are red and 2 are white; in jar B, there are 4 balls, of which 3 are red and 1 is white; in jar C, there are 5 balls, of which 2 are red and 3 are white. One ball is drawn from each of the jars A, B, and C. Find:
(1) the probability that exactly 2 of the balls are white;
(2) t... | \frac{1}{3} | 117 | 7 |
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