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 | (12) Given the arithmetic sequence $\left\{a_{n}\right\}$, the sum of the first 15 terms $S_{15}=30$, then $a_{1}+a_{8}+a_{15}=$
$\qquad$ . | 6 | 62 | 1 |
math | 2. Given the line $l: y=x-a$ intersects the parabola $C$: $x^{2}=2 p y(p>0)$ at points $M$ and $N$, and the circle passing through $M$ and $N$ intersects the parabola $C$ at two other distinct points $E$ and $F$. Then the cosine of the inclination angle of the line $E F$ is $\qquad$. | -\frac{\sqrt{2}}{2} | 92 | 10 |
math | Example 20 (IMO-29 Preliminary Question) The Fibonacci numbers are defined as
$$
a_{0}=0, a_{1}=a_{2}=1, a_{n+1}=a_{n}+a_{n-1} \quad(n \geqslant 1) .
$$
Find the greatest common divisor of the 1960th and 1988th terms. | 317811 | 91 | 6 |
math | 8.3. In triangle $A B C$, the median $A M$ is perpendicular to the bisector $B D$. Find the perimeter of the triangle, given that $A B=1$, and the lengths of all sides are integers. | 5 | 51 | 1 |
math | 4. [4] Let $P$ be a fourth degree polynomial, with derivative $P^{\prime}$, such that $P(1)=P(3)=P(5)=P^{\prime}(7)=0$. Find the real number $x \neq 1,3,5$ such that $P(x)=0$. | \frac{89}{11} | 72 | 9 |
math | Let $P$ be an interior point of triangle $ABC$. Let $a,b,c$ be the sidelengths of triangle $ABC$ and let $p$ be it's semiperimeter. Find the maximum possible value of
$$ \min\left(\frac{PA}{p-a},\frac{PB}{p-b},\frac{PC}{p-c}\right)$$
taking into consideration all possible choices of triangle $ABC$ and of point $P$.
by... | \frac{2}{\sqrt{3}} | 108 | 11 |
math | ## Problem Statement
Find the derivative $y_{x}^{\prime}$.
$$
\left\{\begin{array}{l}
x=\left(1+\cos ^{2} t\right)^{2} \\
y=\frac{\cos t}{\sin ^{2} t}
\end{array}\right.
$$ | \frac{1}{4\sin^{4}\cdot\cos} | 71 | 15 |
math | **Let $N$ be an even number not divisible by 10. What is the tens digit of $N^{20}$? What is the hundreds digit of $N^{200}$?** | 7 | 44 | 1 |
math | [Decimal numeral system]
How many two-digit numbers exist where the digit in the tens place is greater than the digit in the units place?
# | 45 | 29 | 2 |
math | 7.1. Append one digit to the left and one digit to the right of the number 2016 so that the resulting six-digit number is divisible by 72 (provide all solutions). | 920160120168 | 42 | 12 |
math | 26th IMO 1985 shortlist Problem 25 34 countries each sent a leader and a deputy leader to a meeting. Some of the participants shook hands before the meeting, but no leader shook hands with his deputy. Let S be the set of all 68 participants except the leader of country X. Every member of S shook hands with a different ... | 33 | 107 | 2 |
math | 12 If the sum of the digits of a natural number $a$ equals 7, then $a$ is called a "lucky number". Arrange all "lucky numbers" in ascending order as $a_{1}, a_{2}, a_{3}, \cdots$, if $a_{n}=2005$, then $a_{5 n}=$ $\qquad$ | 52000 | 82 | 5 |
math | 2. Let the complex numbers be
$$
\begin{array}{l}
z_{1}=(6-a)+(4-b) \mathrm{i}, \\
z_{2}=(3+2 a)+(2+3 b) \mathrm{i}, \\
z_{3}=(3-a)+(3-2 b) \mathrm{i},
\end{array}
$$
where, $a, b \in \mathbf{R}$.
When $\left|z_{1}\right|+\left|z_{2}\right|+\left|z_{3}\right|$ reaches its minimum value, $3 a+4 b$ $=$ | 12 | 134 | 2 |
math | 4. (7 points) The numbers $a, b, c, d$ belong to the interval $[-11.5,11.5]$. Find the maximum value of the expression $a+2 b+c+2 d-a b-b c-c d-d a$. | 552 | 59 | 3 |
math | $3 \cdot 13$ Fill all integers from 1 to $n^{2}$ into an $n \times n$ square grid according to the following rules: 1 can be placed in any cell, the row of 2 should have the same column index as the column of 1, the row of 3 should have the same column index as the column of 2, and so on. What is the difference between... | n(n-1) | 121 | 5 |
math | Find the values of $ t\in{[0,\ 2]}$ for which $ \int_{t\minus{}3}^{2t} 2^{x^2}\ dx$ is maximal and minimal. | t = -1 + \frac{\sqrt{33}}{3} | 46 | 17 |
math | ## Task B-2.1.
If the square and double the cube of a number are added, the result is three times the fourth power of the same number. Determine all real numbers for which this is true. | x_1=1,x_2=-\frac{1}{3},x_3=0 | 44 | 21 |
math | 6.14. (NPR, 68). Find all values of $n \in \mathbf{N}$ for which there exists a set of positive numbers $x_{1}, \ldots, x_{\mu}$, satisfying the system
$$
\left\{\begin{array}{l}
x_{1}+x_{2}+\ldots+x_{n}=9 \\
\frac{1}{x_{1}}+\frac{1}{x_{2}}+\ldots+\frac{1}{x_{n}}=1
\end{array}\right.
$$
Indicate all such sets for eac... | 2,3 | 142 | 3 |
math | For a non-empty integer set $A$, if it satisfies $a \in A, a-1 \notin A, a+1 \notin A$, then $a$ is called an isolated element of set $A$. Question: For the set
$$
M=\{1,2, \cdots, n\}(n \geqslant 3)
$$
how many $k(k \geqslant 3)$-element subsets of $M$ have no isolated elements? | P_{k}=\sum_{l=1}^{r} \mathrm{C}_{k-l-1}^{l-1} \mathrm{C}_{n-k+1}^{l} | 104 | 42 |
math | 17. Let the complex number $z$ satisfy $|z|=1$. Try to find the maximum and minimum values of $\left|z^{3}-3 z-2\right|$. | 0 | 41 | 1 |
math | Let the sum of a set of numbers be the sum of its elements. Let $S$ be a set of positive integers, none greater than 15. Suppose no two disjoint subsets of $S$ have the same sum. What is the largest sum a set $S$ with these properties can have? | 61 | 63 | 2 |
math | 4. Solve the equation $2^{x}+2^{y}=6^{t}$ in integers. | (x,y,)=(-1,-1,0),(1,2,1),(2,5,2) | 22 | 23 |
math | Example 7. Find $\lim _{x \rightarrow 0}[(x-\sin x) \ln x]$. | 0 | 26 | 1 |
math | 1. Let $x, y, z$ be real numbers, $3 x, 4 y, 5 z$ form a geometric sequence, and $\frac{1}{x}$, $\frac{1}{y}$, $\frac{1}{z}$ form an arithmetic sequence, then the value of $\frac{x}{z}+\frac{z}{x}$ is $\qquad$ | \frac{34}{15} | 82 | 9 |
math | In a convex quadrilateral $A B C D$, the diagonals $A C$ and $B D$ are equal to $a$ and $b$ respectively. Points $E, F, G$ and $H$ are the midpoints of sides $A B, B C, C D$ and $D A$ respectively. The area of quadrilateral $E F G H$ is $S$. Find the diagonals $E G$ and $H F$ of quadrilateral $E F G H$. | \frac{1}{2}\sqrt{^{2}+b^{2}\2\sqrt{^{2}b^{2}-16S^{2}}} | 107 | 34 |
math | ## Task 4
How many minutes are there from 8:35 PM to 9:10 PM, from 7:55 AM to 8:45 AM, from 11:55 AM to 12:05 PM. | 35,50,10 | 57 | 8 |
math | Example 2.79. Determine the amount of heat $Q$ generated by the current $I=5+4 t$ in a conductor with resistance $R=40$ over time $t=10$, given that the amount of heat generated per unit time by a constant current flowing through a conductor with constant resistance is equal to the product of the square of the current ... | 303750 | 82 | 6 |
math | 2. Find all pairs of integers $(x, y)$ that satisfy the equation $x^{2}-x y-6 y^{2}-11=0$. For each pair $(x, y)$ found, calculate the product $x y$. In the answer, write the sum of these products. | 8 | 62 | 1 |
math | 39. How many six-digit numbers are there with the second-to-last digit being 1, which are divisible by 4? | 18000 | 27 | 5 |
math | 9. (16 points) If the function
$$
f(x)=256 x^{9}-576 x^{7}+432 x^{5}-120 x^{3}+9 x \text {, }
$$
find the range of the function $f(x)$ for $x \in[-1,1]$ | [-1,1] | 75 | 5 |
math | 1. (6 points) Calculate: $\frac{2 \frac{1}{4}+0.25}{2 \frac{3}{4}-\frac{1}{2}}+\frac{2 \times 0.5}{2 \frac{1}{5}-\frac{2}{5}}=$ | \frac{5}{3} | 67 | 7 |
math | Example 3. Compute the integral $\int_{C} e^{\bar{z}} d z$, where $C$ is the line segment $y=-x$, connecting the points $z_{1}=0$ and $z_{2}=\pi-i \pi$. | (e^{\pi}+1)i | 57 | 8 |
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 an... | 2^{101}-1 | 111 | 7 |
math | 8. Given the function $f(x)=a x^{2}+b x(a b \neq 0)$, if $f\left(x_{1}\right)=f\left(x_{2}\right)\left(x_{1} \neq x_{2}\right)$, then the value of $f\left(x_{1}+x_{2}\right)$ is $\qquad$ . | 0 | 86 | 1 |
math | 6. Given the sequence: $\frac{2}{3}, \frac{2}{9}, \frac{4}{9}, \frac{6}{9}, \frac{8}{9}, \frac{2}{27}, \frac{4}{27}, \cdots$, $\frac{26}{27}, \cdots, \frac{2}{3^{n}}, \frac{4}{3^{n}}, \cdots, \frac{3^{n}-1}{3^{n}}, \cdots$. Then $\frac{2018}{2187}$ is the $\qquad$th term of the sequence. | 1552 | 139 | 4 |
math | 8. There is no less than 10 liters of milk in the bucket. How can you pour exactly 6 liters of milk from it using an empty nine-liter bucket and a five-liter bucket? | 6 | 41 | 1 |
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 | The base of a right parallelepiped is a rhombus. A plane, passing through one side of the lower base and the opposite side of the upper base, forms an angle of $45^{\circ}$ with the base plane. The area of the resulting section is $Q$. Find the lateral surface area of the parallelepiped. | 2Q\sqrt{2} | 71 | 7 |
math | $4 \cdot 59$ The equation $z^{6}+z^{3}+1=0$ has a complex root, on the complex plane this root has an argument $\theta$ between $90^{\circ}$ and $180^{\circ}$, find the degree measure of $\theta$.
| 160 | 69 | 3 |
math | A coin that comes up heads with probability $p > 0$ and tails with probability $1 - p > 0$ independently on each flip is flipped $8$ times. Suppose that the probability of three heads and five tails is equal to $\frac {1}{25}$ of the probability of five heads and three tails. Let $p = \frac {m}{n}$, where $m$ and $n$ a... | 11 | 100 | 2 |
math | 1. For a natural number $n$, the smallest divisor $a$, different from 1, and the next largest divisor $b$ were taken. It turned out that $n=a^{a}+b^{b}$. Find $n$. | 260 | 51 | 3 |
math | Example 8 For a positive integer $n$, find the smallest integer $k$ such that for any given real numbers $a_{1}, a_{2}, \cdots, a_{d}$, we have
$$
\begin{array}{l}
a_{1}+a_{2}+\cdots+a_{d}=n, \\
0 \leqslant a_{i} \leqslant 1(i=1,2, \cdots, d),
\end{array}
$$
these real numbers can be divided into $k$ groups (allowing ... | 2n-1 | 150 | 4 |
math | 1. 178 Find all three-digit numbers $A$ such that the arithmetic mean of all numbers obtained by rearranging the digits of $A$ is still equal to $A$. | 111, 222, \cdots, 999, 407, 518, 629, 370, 481, 592 | 39 | 47 |
math | 3.84. All lateral faces of the pyramid form the same angle with the base plane. Find this angle if the ratio of the total surface area of the pyramid to the area of the base is $k$. For what values of $k$ does the problem have a solution? | \alpha=\arccos\frac{1}{k-1},wherek>2 | 58 | 19 |
math | 6. Through the midpoints of sides $A B$ and $A D$ of the base of a regular quadrilateral pyramid $S A B C D$, a plane is drawn parallel to the median of the lateral face $S D C$, drawn from vertex $D$. Find the area of the section of the pyramid by this plane, if the side of the base of the pyramid is 2, and
the later... | \frac{15\sqrt{2}}{4} | 91 | 13 |
math | Find the product of all real $x$ for which \[ 2^{3x+1} - 17 \cdot 2^{2x} + 2^{x+3} = 0. \] | -3 | 47 | 2 |
math | 4. For any two points $P_{1}\left(x_{1}, y_{1}\right), P_{2}\left(x_{2}, y_{2}\right)$ in a Cartesian coordinate system, we call $d\left(P_{1}, P_{2}\right)=$ $\left|x_{1}-x_{2}\right|+\left|y_{1}-y_{2}\right|$ the Manhattan distance between $P_{1}$ and $P_{2}$. If $P_{0}\left(x_{0}, y_{0}\right)$ is a fixed point and ... | \frac{3}{4} | 213 | 7 |
math | $1 \mathrm{~kg}$ of pure carbon is burned to carbon dioxide $\left(\mathrm{CO}_{2}\right)$, using exactly the amount of oxygen required for the reaction. The gas is collected in a container with a volume of $v=30$ liters and a constant temperature of $t=25^{\circ} \mathrm{C}$. The pressure $p$ exerted by the gas on the... | 67.6\mathrm{~atm} | 222 | 11 |
math | 3. (2 points) A poor student wrote the following incorrect formulas for the sine and cosine of the difference: $\sin (\alpha-\beta)=$ $\sin \alpha-\sin \beta$ and $\cos (\alpha-\beta)=\cos \alpha-\cos \beta$. In his defense, he said that for some $\alpha$ and $\beta$ his formulas are still correct. Find all such pairs ... | \alpha=2\pin,n\in\mathbb{Z} | 91 | 15 |
math | 41st Putnam 1980 Problem B3 Define a n by a 0 = α, a n+1 = 2a n - n 2 . For which α are all a n positive? Solution | \alpha\geq3 | 47 | 6 |
math | 9. Given that $x, y, z$ are three non-negative rational numbers, and satisfy $3 x$ $+2 y+z=5, x+y-z=2$. If $S=2 x+y-z$, then what is the sum of the maximum and minimum values of $S$? | 5 | 63 | 1 |
math | ## Task 3 - 340823
a) How many times in total do the hour and minute hands of a clock stand perpendicular to each other over the course of 24 hours (from 0:00 to 24:00)?
b) In particular, calculate all such times between 4:00 and 5:00! Give these times as they would be displayed on a digital clock, assuming that the ... | 44,04:05:27,04:38:10 | 119 | 20 |
math | 7. A regular tetrahedron frame with edge length 3 contains a solid sphere with radius 1. Then the maximum volume of the common part of the tetrahedron and the sphere is $\qquad$ | (\frac{7\sqrt{6}}{8}-\frac{4}{3})\pi | 45 | 21 |
math | 6. For two numbers $a, b$, a new operation is defined as $a \triangle b=3 \times a+2 \times b, a \nabla b=2 \times a+3 \times b$. Therefore, $2 \triangle(3 \nabla 4)$ $=$ $\qquad$ | 42 | 70 | 2 |
math | A certain rectangular plot is almost square, as its width and length measure integers that differ exactly by one unit of measurement. The area of this plot, in square units, is a four-digit number, with the thousands and hundreds digits being the same, as well as the tens and units digits. What are the possible dimensi... | 33\times34,66\times67,99\times100 | 68 | 21 |
math | 3. In $\triangle A B C$, it is known that
$$
\sin A=10 \sin B \cdot \sin C, \cos A=10 \cos B \cdot \cos C \text {. }
$$
Then $\tan A=$ $\qquad$ | 11 | 60 | 2 |
math | Find all pairs of integers $(x, y)$ such that $y^{2}=x^{3}+16$ (Factorization) | (x,y)=(0,4) | 29 | 7 |
math | 2. Let $n$ be a three-digit positive integer that does not contain the digit 0. If any permutation of the units, tens, and hundreds digits of $n$ does not form a three-digit number that is a multiple of 4, find the number of such $n$.
(54th Ukrainian Mathematical Olympiad) | 283 | 70 | 3 |
math | 11.117. The base of a right prism is an isosceles trapezoid $A B C D ; A B=C D=13$ cm, $B C=11$ cm, $A D=21$ cm. The area of its diagonal section is $180 \mathrm{~cm}^{2}$. Calculate the total surface area of the prism. | 906 | 87 | 3 |
math | Example 4 (2004 Slovenia Mathematical Olympiad Selection Test) Find all positive integer solutions $a, b, c$ for which the expression $\left(b-\frac{1}{a}\right)\left(c-\frac{1}{b}\right)\left(a-\frac{1}{c}\right)$ is an integer. | (1,1,),(1,,1),(,1,1),(2,3,5),(2,5,3),(3,2,5),(3,5,2),(5,2,3),(5,3,2) | 69 | 51 |
math | Let $M$ be a finite set of numbers. It is known that among any three of its elements, there are two whose sum belongs to $M$.
What is the maximum number of elements that can be in $M$? | 7 | 48 | 1 |
math | ## Task 29/81
We are looking for all (proper) three-digit numbers where the sum of the $i$-th powers of the $i$-th digit (counted from left to right) equals the original number. | 135,175,518,598 | 52 | 15 |
math | 21. Your national football coach brought a squad of 18 players to the 2010 World Cup, consisting of 3 goalkeepers, 5 defenders, 5 midfielders and 5 strikers. Midfielders are versatile enough to play as both defenders and midfielders, while the other players can only play in their designated positions. How many possible... | 2250 | 100 | 4 |
math | A tree has 10 pounds of apples at dawn. Every afternoon, a bird comes and eats x pounds of apples. Overnight, the amount of food on the tree increases by 10%. What is the maximum value of x such that the bird can sustain itself indefinitely on the tree without the tree running out of food? | \frac{10}{11} | 69 | 9 |
math | 1. Add the same integer $a(a>0)$ to the numerator and denominator of $\frac{2008}{3}$, making the fraction an integer. Then the integer $a$ added has $\qquad$ solutions. | 3 | 49 | 1 |
math | Problem 1. Consider the parallelogram $A B C D$, whose diagonals intersect at $O$. The angle bisectors of $\angle D A C$ and $\angle D B C$ intersect at $T$. It is known that $\overrightarrow{T D}+\overrightarrow{T C}=\overrightarrow{T O}$. Determine the measures of the angles of triangle $A B T$. | 60 | 82 | 2 |
math | ## Exercise 3
## Statement.
Find all integers $x, y, z$ that satisfy: $1<x<y<z$ and $x+y+z+xy+yz+zx+xyz=2009$. | (2,9,66),(2,4,133),(4,5,66) | 47 | 23 |
math | Problem No. 5 (15 points)
Two people are moving in the same direction. At the initial moment, the distance between them is $S_{0}=100 \text{ m}$. The speed of the first, faster, pedestrian is $v_{1}=8 \text{ m} / \text{s}$. Determine the speed $v_{2}$ of the second, if it is known that after $t=5$ min the distance bet... | 7.5or7.83 | 110 | 8 |
math | Example 2. Find the domain of convergence of the series
$$
\sum_{n=1}^{\infty} \frac{4^{n}}{n^{3}\left(x^{2}-4 x+7\right)^{n}}
$$ | (-\infty,1]\cup[3,+\infty) | 55 | 15 |
math | 4 Given $M=\{x \mid 1 \leqslant x \leqslant a, a>1\}$, if the domain and range of the function $y=\frac{1}{2} x^{2}-x+\frac{3}{2}$ are both the set $M$, then $a=$ $\qquad$ . | 3 | 75 | 1 |
math | 10. (10 points) During the Spring Festival promotion, customers receive a 50 yuan voucher for every 100 yuan paid in cash. These vouchers cannot be exchanged for cash but can be used to purchase goods, with the following rules: vouchers received in a single purchase cannot be used in the same purchase; the cash paid fo... | 2300 | 129 | 4 |
math | ```
Given real numbers $a, b, c$ are all not equal to 0, and
\[
\begin{array}{l}
a+b+c=m, a^{2}+b^{2}+c^{2}=\frac{m^{2}}{2} . \\
\text { Find } \frac{a(m-2 a)^{2}+b(m-2 b)^{2}+c(m-2 c)^{2}}{a b c}
\end{array}
\]
``` | 12 | 113 | 2 |
math | 4. The board has the number 5555 written in an even base $r$ ($r \geqslant 18$). Petya found out that the $r$-ary representation of $x^2$ is an eight-digit palindrome, where the difference between the fourth and third digits is 2. (A palindrome is a number that reads the same from left to right and from right to left).... | 24 | 99 | 2 |
math | The sum of sides $AB$ and $BC$ of triangle $ABC$ is 11, angle $B$ is $60^{\circ}$, and the radius of the inscribed circle is $\frac{2}{\sqrt{3}}$. It is also known that side $AB$ is greater than side $BC$. Find the height of the triangle dropped from vertex
A.
# | 4\sqrt{3} | 83 | 6 |
math | ## Task 16/76
Given the equation $x^{3}-4 x^{2}-17 x+a_{0}=0$, it is known that the sum of two of its solutions is 1. The goal is to find $a_{0}$. | 60 | 57 | 2 |
math | Task B-3.3. Determine the real numbers $a, b$ and $c$ for which the function
$$
f(x)=a x^{2}+b x-c \sin x \cos x
$$
is odd. | 0,\quadb\in\mathbb{R},\quad\in\mathbb{R} | 51 | 22 |
math | Example 16 Given that $\alpha, \beta$ are acute angles, and $\alpha-\beta=\frac{\pi}{3}$, try to find the value of $\sin ^{2} \alpha+\cos ^{2} \beta-\sqrt{3} \sin \alpha \cdot \cos \beta$.
---
The original text has been translated into English while preserving the original formatting and line breaks. | \frac{1}{4} | 86 | 7 |
math | 1. On an island, there live only knights, who always tell the truth, and liars, who always lie, and there are at least two knights and at least two liars. One fine day, each islander, in turn, pointed to each of the others and said one of two phrases: "You are a knight!" or "You are a liar!" The phrase "You are a liar!... | 526 | 107 | 3 |
math | ## Task 6 - 320736
The following information was given about a swimming pool:
The pool can be filled through two separate water pipes. From the second pipe, 50 cubic meters more flow out per minute than from the first. To completely fill the pool, 48 minutes are required if both pipes are opened simultaneously; in co... | 12000 | 115 | 5 |
math | 【Question 8】
Weiwei is 8 years old this year, and his father is 34 years old. In $\qquad$ years, his father's age will be three times Weiwei's age. | 5 | 46 | 1 |
math | 8. Let the function $f(x)=a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{n} x^{n}$, where $a_{0}, a_{1}, a_{2}, \cdots, a_{n}$ are non-negative integers. Given that $f(1)=4$, $f(5)=152$, then $f(6)=$ $\qquad$ | 254 | 95 | 3 |
math | 9. Given four points $O, A, B, C$ on a plane, satisfying
$$
O A=4, O B=3, O C=2, \overrightarrow{O B} \cdot \overrightarrow{O C}=3 \text {. }
$$
Then the maximum area of $\triangle A B C$ is $\qquad$ | 2 \sqrt{7}+\frac{3 \sqrt{3}}{2} | 76 | 18 |
math | 2. Solve the inequality $\sqrt{\frac{x-4}{x+3}}-\sqrt{\frac{x+3}{x-4}}<\frac{7}{12}$. | x\in(-\infty;-12)\cup(4;+\infty) | 39 | 19 |
math | In $\triangle ABC$, points $E$ and $F$ lie on $\overline{AC}, \overline{AB}$, respectively. Denote by $P$ the intersection of $\overline{BE}$ and $\overline{CF}$. Compute the maximum possible area of $\triangle ABC$ if $PB = 14$, $PC = 4$, $PE = 7$, $PF = 2$.
[i]Proposed by Eugene Chen[/i] | 84 | 100 | 2 |
math | 7.2. A square with a side of $100 \mathrm{~cm}$ was drawn on the board. Alexei crossed it with two lines parallel to one pair of sides of the square. Then Danil crossed the square with two lines parallel to the other pair of sides of the square. As a result, the square was divided into 9 rectangles, and it turned out t... | 2400 | 112 | 4 |
math | Example 1 Set $A=\left\{(x, y) \mid y=x^{2}+m x+2\right\}, B=\{(x, y) \mid x-y+1=0$ and $0 \leqslant x \leqslant 2\}$. If $A \cap B \neq \varnothing$, find the range of values for $m$. | \leqslant-1 | 87 | 7 |
math | Calculate the sum of the digits of the sum of the digits of the sum of the digits of $A:=4444^{4444}$. | 7 | 33 | 1 |
math | Example 10 Let real numbers $s, t$ satisfy $19 s^{2}+99 s+1=0, t^{2}+99 t+19=0$, and $s t \neq 1$. Find the value of $\frac{s t+4 s+1}{t}$. | -5 | 70 | 2 |
math | 7. Calculate:
$$
(1000+15+314) \times(201+360+110)+(1000-201-360-110) \times(15+314)=
$$
$\qquad$ | 1000000 | 68 | 7 |
math | Five. (Full marks 20 points) Given the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ $(a>b>0)$ intersects the positive direction of the $y$-axis at point $B$. Find the number of isosceles right triangles inscribed in the ellipse with point $B$ as the right-angle vertex.
---
Please note that the translation pres... | 2 | 101 | 1 |
math | 111. How many five-digit natural numbers can be formed using the digits 1 and 0 if the digit 1 appears exactly three times in each number? | 6 | 34 | 1 |
math | 1412.1 ** The general term formula for the sequence $101,104,116, \cdots$ is
$$
a_{n}=100+n^{2}, n=1,2,3, \cdots
$$
For each $n$, let $d_{n}$ denote the greatest common divisor of $a_{n}$ and $a_{n+1}$. Find the maximum value of $d_{n}$ when $n$ takes all positive integers. | 401 | 110 | 3 |
math | For the numbers $a, b, c$, it holds that
$$
\frac{-a+b+c}{a}=\frac{a-b+c}{b}=\frac{a+b-c}{c}
$$
What values can the expression
$$
p=\frac{(a+b)(b+c)(c+a)}{a b c}
$$
take? | -1or8 | 75 | 4 |
math | Let's consider the sum of the digits of all numbers from 1 to 1000000 inclusive. For the resulting numbers, we will again consider the sum of the digits, and so on, until we get a million single-digit numbers. Which are more numerous among them - ones or twos? | 1 | 65 | 1 |
math | How many nonzero coefficients can a polynomial $ P(x)$ have if its coefficients are integers and $ |P(z)| \le 2$ for any complex number $ z$ of unit length? | 2 | 39 | 1 |
math | Find $\min |3+2 i-z|$ for $|z| \leq 1$.
# | \sqrt{13}-1 | 23 | 7 |
math | 183. To determine the side of a regular dodecagon inscribed in a circle.
Al-Karhi gives a verbal expression, according to which:
$$
a_{12}^{2}=\left[\frac{d}{2}-\sqrt{\left(\frac{d}{2}\right)^{2}-\left(\frac{d}{4}\right)^{2}}\right]^{2}+\left(\frac{d}{4}\right)^{2}
$$
Verify the validity of this expression.
Note. P... | a_{12}=r\sqrt{2-\sqrt{3}} | 160 | 15 |
math | 14-22 Solve the equation $\left[x^{3}\right]+\left[x^{2}\right]+[x]=\{x\}-1$.
(Kiev Mathematical Olympiad, 1972) | -1 | 46 | 2 |
math | 3.1. Find the sum
$$
\sqrt[7]{(-7)^{7}}+\sqrt[8]{(-8)^{8}}+\sqrt[9]{(-9)^{9}}+\ldots+\sqrt[100]{(-100)^{100}}
$$
(Each term is of the form $\left.\sqrt[k]{(-k)^{k}}\right)$ | 47 | 87 | 2 |
math | We divide a segment into four parts with three randomly chosen points. What is the probability that these four sub-segments can be the four sides of some quadrilateral? | \frac{1}{2} | 33 | 7 |
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