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 | 6.065. $\sqrt{2 x+5}+\sqrt{5 x+6}=\sqrt{12 x+25}$. | 2 | 33 | 1 |
math | Example 2. Calculate the static moments of the tetrahedron bounded by the coordinate planes and the plane $x+y+z=1$, if the density of the tetrahedron is $\rho=x y$. | M_{xy}=\frac{1}{720},M_{yz}=\frac{1}{360},M_{xz}=\frac{1}{360} | 44 | 39 |
math | Example 3 If the digits of a four-digit number are reversed to form a new four-digit number, the new number is exactly four times the original number. Find the original number.
(1988, Nanjing Mathematical Olympiad Selection Contest) | 2178 | 51 | 4 |
math | 1. For any three-digit number, we determine its remainders when divided by the numbers 2, 3, 4, ..., 10 and then sum the nine resulting numbers. Find the smallest possible value of such a sum. | 3 | 49 | 1 |
math | How long does it take for a freely falling body to fall so that in the subsequent 1.3 seconds it falls $49.34 \mathrm{~m}$? 1[^0]
[^0]: ${ }^{1}$ The acceleration due to free fall: $g=980.8 \frac{\mathrm{cm}}{\mathrm{sec}^{2}}$ (according to Gruber's determination, in Budapest $g=980.837 \frac{\mathrm{cm}}{\mathrm... | 3.22 | 119 | 4 |
math | 3. (12 points) Four friends went to the forest to pick mushrooms. Upon returning, every pair of them counted the total number of mushrooms they had collected. The numbers obtained were $6,7,9,9,11,12$ How many mushrooms did each collect? | 2,4,5,7 | 60 | 7 |
math | ## Problem Statement
Find the coordinates of point $A$, which is equidistant from points $B$ and $C$.
$A(0 ; 0 ; z)$
$B(10 ; 0 ;-2)$
$C(9 ;-2 ; 1)$ | A(0;0;-3) | 60 | 8 |
math | Find all functions $f: N_0 \to N_0$, (where $N_0$ is the set of all non-negative integers) such that $f(f(n))=f(n)+1$ for all $n \in N_0$ and the minimum of the set $\{ f(0), f(1), f(2) \cdots \}$ is $1$. | f(n) = n + 1 | 83 | 8 |
math | 5. Given $0<a<b<c<d<300$,
$$
a+d=b+c, b c-a d=91 \text {. }
$$
Then the number of ordered quadruples of positive integers $(a, b, c, d)$ that satisfy the above conditions is $\qquad$. | 486 | 64 | 3 |
math | 7. There are 100 chess pieces, and two people take turns to take the pieces. Each time, you are allowed to take 1 or 2 pieces. The one who takes the last piece wins. If you go first, how many pieces should you take the first time to ensure a win? | 1 | 64 | 1 |
math | 3.323. $\sin \alpha \sin ^{2}\left(\alpha-270^{\circ}\right)\left(1+\operatorname{tg}^{2} \alpha\right)+\cos \alpha \cos ^{2}\left(\alpha+270^{\circ}\right)\left(1+\operatorname{ctg}^{2} \alpha\right)$. | \sqrt{2}\sin(45+\alpha) | 88 | 12 |
math | 12. Given $S$ as a binary string of $10^{4}$ bits containing only $0$ and $1$, a positive integer $k \leqslant 10^{4}$, a $k$-block of $S$ is a substring of $S$ consisting of $k$ consecutive bits. Two $k$-blocks $a_{1} a_{2} \cdots a_{k}=b_{1} b_{2} \cdots b_{k}$ are equal if and only if $a_{i}=b_{i}(i=1$, $2, \cdots, ... | 504 | 176 | 3 |
math | 1. Determine all triples $(x, y, z)$ of real numbers for which
$$
x^{2}+y^{2}+z^{2} \leqq 6+\min \left\{x^{2}-\frac{8}{x^{4}}, y^{2}-\frac{8}{y^{4}}, z^{2}-\frac{8}{z^{4}}\right\}
$$
(J. Švrček) | (x,y,z)=(\varepsilon_{1}\sqrt{2},\varepsilon_{2}\sqrt{2},\varepsilon_{3}\sqrt{2}),\quad\text{where}\varepsilon_{i}\in{-1;1}\text{for}i=1,2,3 | 98 | 67 |
math | Task 3. A master and an apprentice need to make a concrete slab with dimensions $4 m, 3 m$ and $20 \mathrm{~cm}$, as well as 4 fence posts with dimensions $20 \mathrm{~cm}, 3 \mathrm{~m}$ and $25 \mathrm{~cm}$. The master told the apprentice to calculate and order the exact amount of concrete needed. The apprentice cal... | 3\mathrm{~}^{3} | 117 | 9 |
math | Place real numbers at the vertices of a pentagon so that the sum of the numbers at the ends of one side is equal to 1, at the ends of another side is equal to 2, ..., and at the ends of the last side is equal to 5.
# | \frac{3}{2},-\frac{1}{2},\frac{5}{2},\frac{1}{2},\frac{7}{2} | 57 | 35 |
math | 4. In Rt $\triangle A B C$, $C D$ is the altitude on the hypotenuse $A B$, the three sides $a, b, c$ of $\triangle A B C$ are all positive integers, $B D$ $=27$. Then $\cos B=$ $\qquad$ | \frac{3}{5} | 66 | 7 |
math | 66. In the box, there are 100 balls of different colors: 28 red balls, 20 green balls, 12 yellow balls, 20 blue balls, 10 white balls, and 10 black balls. If you randomly draw balls from the box, how many balls must you draw to ensure that you have 15 balls of the same color? $\qquad$ | 75 | 88 | 2 |
math | ## 16. Navigation
A ship is sailing along a river. After 6 hours, it returns to the starting point, having traveled a distance of 36 km on the map (naturally, the ship had to move in different directions at different times).
What is the speed of the ship, assuming it did not spend any time turning around, and the spe... | 7.24 | 96 | 4 |
math | Problem 4. The bases $AB$ and $CD$ of trapezoid $ABCD$ are equal to 65 and 31, respectively, and its lateral sides are perpendicular to each other. Find the scalar product of vectors $\overrightarrow{AC}$ and $\overrightarrow{BD}$. | -2015 | 65 | 5 |
math | Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f\left(a^{2}\right)-f\left(b^{2}\right) \leqslant(f(a)+b)(a-f(b)), \quad \text { for all } a, b \in \mathbb{R}
$$ | f(x)=x \text{ or } f(x)=-x | 77 | 14 |
math | Example 6 Let $S=\{1,2,3, \cdots, 280\}$. Find the smallest natural number $n$ such that every subset of $S$ with $n$ elements contains 5 pairwise coprime numbers. | 217 | 55 | 3 |
math | Example 1. Find $\int e^{x^{2}} \cdot x d x$. | \frac{1}{2}e^{x^{2}}+C | 19 | 15 |
math | 144(1172). Two athletes are running on the same closed track. The speed of each is constant, and one takes 5 seconds less than the other to run the entire track. If they start running simultaneously from the same starting line in the same direction, they will be side by side after 30 seconds. How long will it take for ... | 6 | 92 | 1 |
math | 10. Given that the circumcenter of $\triangle A B C$ is $O$, and
$$
2 \overrightarrow{O A}+3 \overrightarrow{O B}+4 \overrightarrow{O C}=\mathbf{0} \text {. }
$$
Then $\cos \angle B A C=$ $\qquad$ | \frac{1}{4} | 74 | 7 |
math | 1. Consider the set $A=\{100,101,102, \ldots, 200\}$.
a) Determine the number of elements in $A$ that are divisible by 10.
b) Determine the elements of $A$ that, when divided by 21 give a remainder of 12, and when divided by 28 give a remainder of 5. | 117 | 90 | 3 |
math | \section*{Problem 4 - 111044}
Determine all triples \((m, x, y)\) consisting of a real number \(m\), a negative integer \(x\), and a positive integer \(y\) that satisfy the following system of equations (1), (2)!
\[
-2 x+3 y=2 m \quad(1) \quad ; \quad x-5 y=-11
\] | (4,-1,2)(\frac{15}{2},-6,1) | 97 | 20 |
math | Suppose that $n$ people each know exactly one piece of information, and all $n$ pieces are different. Every time person $A$ phones person $B$, $A$ tells $B$ everything that $A$ knows, while $B$ tells $A$ nothing. What is the minimum number of phone calls between pairs of people needed for everyone to know everything? P... | 2n - 2 | 86 | 7 |
math | Example 4.3. Find the general integral of the equation
$$
\cos ^{2} y \operatorname{ctg} x d x+\sin ^{2} x \operatorname{tg} y d y=0
$$ | \operatorname{tg}^{2}y-\operatorname{ctg}^{2}x=C | 53 | 22 |
math | Example 3. Find the residues of the function $f(z)=\frac{1}{z^{4}+1}$ at its singular points. | \begin{aligned}&\operatorname{res}f(z_{1})=\frac{1}{4}e^{-i3\pi/4}\\&\operatorname{res}f(z_{2})=\frac{1}{4}e^{-i9\pi/4}\\&\operatorname{res}f(z_{3})=\frac{1}{4} | 31 | 79 |
math | Three, (20 points) Given a real number $k$, determine all functions $f: \mathbf{R} \rightarrow \mathbf{R}$, such that for any $x, y \in \mathbf{R}$, we have $f\left(x^{2}+2 x y+y^{2}\right)=(x+y)(f(x)+f(y))$ and $|f(x)-k x| \leqslant\left|x^{2}-x\right|$. | f(x)=k x | 107 | 5 |
math | 1. Let the sets be
$$
\begin{aligned}
A & =\{n(n+1) \mid n=1,2, \cdots\}, \\
B & =\{3 m-1 \mid m=1,2, \cdots\} .
\end{aligned}
$$
If the elements of the set $A \cap B$ are arranged in ascending order to form a sequence $\left\{a_{k}\right\}$, then the general term formula for the sequence $\left\{a_{k}\right\}$ is $a_... | 9k^{2}-9k+2 | 131 | 9 |
math | 6. (4 points) With the number written on the board, one of the following operations is allowed:
1) Replace the original number with the difference between the number obtained by removing the last four digits and the number formed by the last four digits (possibly written in an improper form - with leading zeros; the di... | 80or66 | 177 | 5 |
math | ## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0} \ln \left(\left(e^{x^{2}}-\cos x\right) \cos \left(\frac{1}{x}\right)+\tan\left(x+\frac{\pi}{3}\right)\right)
$$ | \ln\sqrt{3} | 71 | 7 |
math | We are looking for an important document in our desk. We know that $p_{j}$ is the probability that the document is in the $j$-th drawer $\left(j=1,2, \ldots, n, \sum_{j=0}^{n} p j=1\right)$, and we also know that it takes $T_{j}$ time to check the $j$-th drawer. Before we start searching, we make a plan, that is, we gi... | \frac{p_{i_{1}}}{T_{i_{1}}}\geq\frac{p_{i_{2}}}{T_{i_{2}}}\geq\ldots\geq\frac{p_{i_{n}}}{T_{i_{n}}} | 301 | 60 |
math | 5. A natural number $n$ is called cubish if $n^{3}+13 n-273$ is a cube of a natural number. Find the sum of all cubish numbers. | 29 | 44 | 2 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \frac{\sqrt{n+1}-\sqrt[3]{n^{3}+1}}{\sqrt[4]{n+1}-\sqrt[5]{n^{5}+1}}$ | 1 | 65 | 1 |
math | 6. (10 points) Use 40 yuan to buy three types of exercise books priced at 2 yuan, 5 yuan, and 11 yuan each. You must buy at least one of each type, and the money must be exactly spent. How many different ways are there to buy the exercise books? $\qquad$ kinds. | 5 | 72 | 1 |
math | One. (20 points) The equation concerning $x$
$$
\sqrt{x^{2}-m}+2 \sqrt{x^{2}-1}=x
$$
has exactly one real root. Find the range of real values for $m$. | 0\leqslant\leqslant\frac{4}{3} | 53 | 18 |
math | 5. Given the sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ with the general terms $a_{n}=2^{n}, b_{n}=3 n+2$, arrange the common terms of $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ in ascending order to form a new sequence $\left\{c_{n}\right\}$. The general term of $\left\{c_{n}\right\}$ is $\qqua... | 2^{2n+1} | 118 | 7 |
math | $\triangle A B C$ has vertices $A(-1,2), B(5,2)$ and $C(-4,-3)$. What is the area of $\triangle A B C$ ? | 15 | 42 | 2 |
math | Fifteen identical sheets of paper placed on top of each other, I folded them all at once. This resulted in a "booklet" whose pages I numbered in sequence from 1 to 60. Which three other numbers are written on the same sheet of paper as the number $25?$
(L. Šimünek) | 26,35,36 | 71 | 8 |
math | 1. How many distinct permutations of the letters of the word REDDER are there that do not contain a palindromic substring of length at least two? (A substring is a contiguous block of letters that is part of the string. A string is palindromic if it is the same when read backwards.) | 6 | 64 | 1 |
math | 1 Let positive real numbers $x, y, z$ satisfy $x+y+z=xyz$. Find the minimum value of $x^{7}(yz-1)+y^{7}(zx-1)+z^{7}(xy-1)$. | 162\sqrt{3} | 51 | 8 |
math | Let $ ABC$ be an acute triangle, $ CC_1$ its bisector, $ O$ its circumcenter. The perpendicular from $ C$ to $ AB$ meets line $ OC_1$ in a point lying on the circumcircle of $ AOB$. Determine angle $ C$. | 60^\circ | 60 | 4 |
math | 6. Given $\boldsymbol{m}$ is a non-zero vector, $n$ is a unit vector, $\boldsymbol{m} \neq \boldsymbol{n}$, the angle between $\boldsymbol{m}$ and $\boldsymbol{m}-\boldsymbol{n}$ is $60^{\circ}$, $|\boldsymbol{m}| \in(0, a]$, then the minimum value of $a$ is $\qquad$ | \frac{2\sqrt{3}}{3} | 96 | 12 |
math | 13.155. In the first week of their vacation trip, the friends spent 60 rubles less than $2 / 5$ of the amount of money they brought with them; in the second week, $1 / 3$ of the remainder and another 12 rubles on theater tickets; in the third week, $3 / 5$ of the new remainder and another 31 rubles 20 kopecks on boat r... | 2330 | 124 | 4 |
math | 4. A merchant accidentally mixed candies of the 1st grade (at 3 rubles per pound) and candies of the 2nd grade (at 2 rubles per pound). At what price should this mixture be sold to receive the same amount of money, given that initially the total cost of all candies of the 1st grade was equal to the total cost of all ca... | 2.4 | 86 | 3 |
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. Let $t=\left(\frac{1}{2}\right)^{x}+\left(\frac{2}{3}\right)^{x}+\left(\frac{5}{6}\right)^{x}$. Then the sum of all real solutions of the equation $(t-1)(t-2)(t-3)=0$ with respect to $x$ is $\qquad$ | 4 | 84 | 1 |
math | 1. Given real numbers $x, y$ satisfy $\left(x^{2}+6 x+12\right)\left(5 y^{2}+2 y+1\right)=\frac{12}{5}$. Then the value of $x y$ is $\qquad$ . | \frac{3}{5} | 64 | 7 |
math | ## Task B-1.1.
For real numbers $x$ and $y$, if $x-y=6$ and $x^{2}+y^{2}=22$, what is $x^{3}-y^{3}$? | 90 | 51 | 2 |
math | 19. Football-2. (from 10th grade. 6 points) A regular football match is in progress. A draw is possible. The waiting time for the next goal does not depend on previous events in the match. It is known that the expected number of goals in football matches between these teams is 2.8. Find the probability that an even num... | 0.502 | 86 | 5 |
math | 2. The little squirrel dried mushrooms: $85 \%$ were white mushrooms, and the rest were russulas. Then he ate some of the white mushrooms, and now the russulas made up $30 \%$ of the remaining mushrooms. What part of the mushrooms did the little squirrel eat? | \frac{1}{2} | 61 | 7 |
math | ## Task 2.
A natural number $n$ is good if we can color each side and diagonal of a regular $n$-gon in some color so that for any two vertices $A$ and $B$, there is exactly one vertex $C$, different from $A$ and $B$, such that the segments $\overline{A B}, \overline{B C}$, and $\overline{C A}$ are colored the same col... | 7 | 126 | 1 |
math | Example 7, Find the minimum and maximum values of the function $y=\frac{x^{2}+2 x+5}{x^{2}+4 x+5}$. | y_{\text {minimum }}=3-\sqrt{5}, y_{\text {maximum }}=3+\sqrt{5} | 38 | 28 |
math | Is the number $6^{1962}+1$ divisible by 31? | 2 | 20 | 1 |
math | Example 4 - Let $\mu^{2}, \gamma^{2}, \rho^{2}$ be pairwise distinct, and the parameters $\mu, \gamma, \rho, b, c$ be chosen such that the following system of equations has a solution:
$$
\begin{array}{l}
x^{2}>0, y^{2}>0, z^{2}>0, \\
\frac{x^{2}}{\mu^{2}}+\frac{y^{2}}{\mu^{2}-b^{2}}+-\frac{z^{2}}{\mu^{2}-c^{2}}=1, \\
... | x^{2}+y^{2}+z^{2}=\mu^{2}+\gamma^{2}+\rho^{2}-b^{2}-c^{2} | 249 | 37 |
math | A cube with side length $100cm$ is filled with water and has a hole through which the water drains into a cylinder of radius $100cm.$ If the water level in the cube is falling at a rate of $1 \frac{cm}{s} ,$ how fast is the water level in the cylinder rising? | \frac{1}{\pi} | 70 | 8 |
math | 3. Polycarp instead of the usual multiplication of three-digit numbers decided to simply "glue the numbers", appending one number to the other. The result turned out to be 7 times larger than the usual. What numbers was Polycarp multiplying? | 143143 | 52 | 6 |
math | The number sequence $1,4,5,7,12,15,16,18,23, \ldots$ was written based on a certain rule. Continue the number sequence until it exceeds the number 50.
Continue the number sequence until it exceeds the number 50. | 51 | 65 | 2 |
math |
99.1. The function $f$ is defined for non-negative integers and satisfies the condition
$$
f(n)= \begin{cases}f(f(n+11)), & \text { if } n \leq 1999 \\ n-5, & \text { if } n>1999\end{cases}
$$
Find all solutions of the equation $f(n)=1999$.
| 1999=f(6n),ifonlyifn=1,2,\ldots,334 | 94 | 24 |
math | In $\triangle A B C$, the sides opposite to $\angle A, \angle B, \angle C$ are $a, b, c$ respectively, and
$$
a=5, b=4, \cos (A-B)=\frac{31}{32} \text {. }
$$
Then the area of $\triangle A B C$ is $\qquad$ | \frac{15\sqrt{7}}{4} | 81 | 13 |
math | Find all pairs of prime numbers $(p, q)$ for which
$$
p\left(p^{2}-p-1\right)=q(2 q+3)
$$ | (13,31) | 37 | 7 |
math | The second question: Let the three sides of a triangle be integers $l$, $m$, and $n$, and $l>m>n$. It is known that $\left\{\frac{3^{l}}{10^{4}}\right\}=\left\{\frac{3^{m}}{10^{4}}\right\}=$ $\left\{\frac{3^{n}}{10^{4}}\right\}$, where $\{x\}=x-[x]$, and $[x]$ represents the greatest integer not exceeding $x$. Find the... | 3003 | 129 | 4 |
math | There are several teacups in the kitchen, some with handles and the others without handles. The number of ways of selecting two cups without a handle and three with a handle is exactly $1200$. What is the maximum possible number of cups in the kitchen? | 29 | 55 | 2 |
math | Example: 15 Given positive integers $x, y, z$ satisfy $x^{3}-y^{3}-z^{3}=3 x y z, x^{2}=2(y+z)$. Find the value of $x y+y z+z x$.
---
The translation is provided as requested, maintaining the original text's line breaks and format. | 5 | 74 | 1 |
math | 15 Positive integers $a, b, c$ satisfy: $[a, b]=1000,[b, c]=2000,[c, a]=2000$. Find the number of such ordered positive integer triples $(a, b, c)$. | 70 | 59 | 2 |
math | 7. (10 points) For a natural number $N$, if at least eight of the nine natural numbers from $1$ to $9$ can divide $N$, then $N$ is called an "Eight Immortals Number". Among the natural numbers greater than 2000, the smallest "Eight Immortals Number" is | 2016 | 72 | 4 |
math | Example 1 If a set does not contain three numbers $x, y, z$ that satisfy $x+y=z$, then it is called simple. Let $M=\{1,2, \cdots, 2 n+1\}, A$ be a simple subset of $M$, find the maximum value of $|A|$. (1982 | n+1 | 76 | 3 |
math | Problem 11.5. In a chess tournament, a team of schoolchildren and a team of students, each consisting of 15 people, are competing against each other. During the tournament, each schoolchild must play against each student exactly once, and each person must play no more than one game per day. The number of games played o... | 120 | 145 | 3 |
math | Problem 2. Find all primes $p \geq 3$ such that $p-\left[\frac{p}{q}\right] q$ is a square-free integer for any prime $q<p$. | 3,5,7,13 | 44 | 8 |
math | 6-175 Let $f$ be a function from $R \rightarrow R$, and
(1) For any $x, y \in R$,
$$f(x)+f(y)+1 \geqslant f(x+y) \geqslant f(x)+f(y)$$
(2) For any $x \in[0,1)$, $f(0) \geqslant f(x)$;
(3) $-f(-1)=f(1)=1$.
Find all functions $f$ that satisfy the conditions. | f(x)=[x] | 121 | 6 |
math | 10. A florist received between 300 and 400 roses for a celebration. When he arranged them in vases with 21 roses in each, 13 roses were left. But when arranging them in vases with 15 roses in each, 8 roses were missing. How many roses were there in total? | 307 | 73 | 3 |
math | 5. In triangle $\mathrm{ABC}$, the sides $A B=4, B C=6$. Point $M$ lies on the perpendicular bisector of segment $A B$, and lines $A M$ and $A C$ are perpendicular. Find $M A$, if the radius of the circumscribed circle around triangle $A B C$ is 9. | 6 | 78 | 1 |
math | 12. Let $(X, Y)$ be a Gaussian system, where $Y$ is a random variable in $\mathbb{R}$, and $X$ is a random vector in $\mathbb{R}^{d}$. Determine the structure of the conditional expectations $\mathrm{E}\left(Y^{n} \mid X=x\right), n \geqslant 1$. | \mathrm{E}(Y^{n}\midX=x)=\sum_{k\leqslantn/2}C_{n}^{2k}(\mathrm{E}Z+\mathrm{E}YX^{*}\cdot(\mathrm{D}X)^{+}x)^{n-2k}\frac{(2k)!}{2^{k}\cdotk!}(\mathrm{D} | 82 | 88 |
math | 2. Solve the equation $\frac{15}{x\left(\sqrt[3]{35-8 x^{3}}\right)}=2 x+\sqrt[3]{35-8 x^{3}}$. Write the sum of all obtained solutions in the answer.
(5 points)
# | 2.5 | 63 | 3 |
math | 6. Divide a rectangle with side lengths of positive integers $m, n$ into several squares with side lengths of positive integers, with each square's sides parallel to the corresponding sides of the rectangle. Try to find the minimum value of the sum of the side lengths of these squares. | f(,n)=+n-(,n) | 57 | 11 |
math | 1. Given complex numbers $z_{1}, z_{2}$ corresponding to points $A, B$ on the complex plane, and $\left|z_{1}\right|=2, z_{1}^{2}-2 z_{1} z_{2}+4 z_{2}^{2}=0, O$ is the origin, then the perimeter of $\triangle O A B$ is $\qquad$ . | 3+\sqrt{3} | 88 | 6 |
math | 14. (2004 National College Entrance Examination - Liaoning Paper) Let the universal set $U=\mathbf{R}$.
(1) Solve the inequality for $x$: $|x-1|+a-1 \geqslant 0(a \in \mathbf{R})$
(2) Let $A$ be the solution set of the inequality in (1). The set $B=\left\{x \left\lvert\, \sin \left(\pi x-\frac{\pi}{3}\right)+\sqrt{3} \... | -1<\leqslant0 | 176 | 9 |
math | 8. In $\triangle A B C$, the medians $A E, B F$, and $C D$ intersect at $M$. Given that $E, C, F$, and $M$ are concyclic, and $C D=n$. Find the length of segment $A B$.
(18th All-Russian Competition Problem) | AB=\frac{2}{3}\sqrt{3}n | 71 | 13 |
math | The diagonals $AC$ and $BD$ of a convex cyclic quadrilateral $ABCD$ intersect at point $E$. Given that $AB = 39, AE = 45, AD = 60$ and $BC = 56$, determine the length of $CD.$ | \frac{91}{5} | 62 | 8 |
math | 6. (5 points) The length, width, and height of a rectangular prism (all greater than 1) are three mutually prime natural numbers. If the volume of this rectangular prism is 665, then its surface area is
| 526 | 50 | 3 |
math | 12. Find all values of integers $x$ and $y$ satisfying $2^{3 x}+5^{3 y}=189$. | 2,1 | 32 | 3 |
math | A deck of $ 2n\plus{}1$ cards consists of a joker and, for each number between 1 and $ n$ inclusive, two cards marked with that number. The $ 2n\plus{}1$ cards are placed in a row, with the joker in the middle. For each $ k$ with $ 1 \leq k \leq n,$ the two cards numbered $ k$ have exactly $ k\minus{}1$ cards between t... | n = 3, 4, 7, 8 | 133 | 14 |
math | The sequence $ (a_n)$ satisfies $ a_0 \equal{} 0$ and $ \displaystyle a_{n \plus{} 1} \equal{} \frac85a_n \plus{} \frac65\sqrt {4^n \minus{} a_n^2}$ for $ n\ge0$. Find the greatest integer less than or equal to $ a_{10}$. | 983 | 84 | 3 |
math | Let $(a_n)\subset (\frac{1}{2},1)$. Define the sequence $x_0=0,\displaystyle x_{n+1}=\frac{a_{n+1}+x_n}{1+a_{n+1}x_n}$. Is this sequence convergent? If yes find the limit. | 1 | 70 | 3 |
math | Determine the positive integers expressible in the form $\frac{x^2+y}{xy+1}$, for at least $2$ pairs $(x,y)$ of positive integers | 1 | 36 | 1 |
math | 5. If $\log _{4}(x+2 y)+\log _{4}(x-2 y)=1$, then the minimum value of $|x|-|y|$ is
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | \sqrt{3} | 66 | 5 |
math | 4. (3 points) Point $C$ is located on the segment $A E$. On one side of the line $A E$, points $B$ and $D$ are marked such that $A B C$ is an equilateral triangle, and $C D E$ is an isosceles right triangle with a right angle at $D$. It turns out that triangle $B C D$ is isosceles with base $B C$. Find the angle $A D E... | 105 | 103 | 3 |
math | [ Examples and counterexamples. Constructions ] [ Partitions into pairs and groups; bijections ] [ Decimal number system ]
A subset $X$ of the set of "two-digit" numbers $00,01, \ldots, 98,99$ is such that in any infinite sequence of digits, there are two adjacent digits that form a number in $X$. What is the smalle... | 55 | 96 | 2 |
math | ## Task 4 - 040914
It is known about the natural numbers $p$ and $q$ that $0<p<q$.
a) Order the numbers $1, \frac{p}{q}$, and $\frac{q}{p}$ by size! Start with the smallest number!
b) Determine which of the two numbers $\frac{p}{q}$ and $\frac{q}{p}$ is closer to 1! | \frac{p}{q}<1<\frac{q}{p}1-\frac{p}{q}<\frac{q}{p}-1 | 97 | 32 |
math | 6. (1994 Bulgarian Mathematical Olympiad) Find all integers $k$ such that there exists an integer $x$ satisfying the equation $\sqrt{39-6 \sqrt{12}}+\sqrt{k x(k x+\sqrt{12})+3}=2 k$. | 3or6 | 61 | 3 |
math | ## Task A-2.5.
Let $A$ be the number of six-digit numbers whose product of digits is 105, and $B$ be the number of six-digit numbers whose product of digits is 147. Determine the ratio $A: B$. | 2:1 | 58 | 3 |
math | 420. What should two geometric progressions be so that the series formed by the sums of their corresponding terms is also a geometric progression? (Instead of the sum, what more general function can be taken here?)
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。 | (p-q)^{2}=0 | 66 | 7 |
math | Let
$C=\{ (i,j)|i,j$ integers such that $0\leq i,j\leq 24\}$
How many squares can be formed in the plane all of whose vertices are in $C$ and whose sides are parallel to the $X-$axis and $Y-$ axis? | 4900 | 67 | 4 |
math | 2.39 Form a quadratic equation with roots $\frac{1}{x_{1}}$ and $\frac{1}{x_{2}}$, where $x_{1}$ and $x_{2}$ are the roots of the equation $a x^{2}+b x+c=0$. | ^{2}++=0 | 62 | 6 |
math | Solve the following equation:
$$
3125^{\frac{x+1}{x+2}} \cdot 15625^{-\frac{x+2}{x+3}}=0.2
$$ | 3 | 48 | 1 |
math | 7. A rectangular sheet of iron was divided into two parts such that the first part was four times larger than the second. What is the area of the entire sheet if the first part is $2208 \mathrm{~cm}^{2}$ larger than the second? | 3680\mathrm{~}^{2} | 57 | 12 |
math | $\left[\begin{array}{l}{[\text { Problems on maximum and minimum (miscellaneous). }} \\ {[\quad \underline{\text { Rectangular parallelepipeds }}]}\end{array}\right]$
Consider all possible rectangular parallelepipeds, each with a volume of 4, and whose bases are squares. Find among them the parallelepiped with the sma... | 6 | 90 | 1 |
math | 79. In a triangle, three lines parallel to its sides and tangent to the inscribed circle have been drawn. They cut off three triangles from the given one. The radii of the circumcircles of these triangles are $R_{1}, R_{2}, R_{3}$. Find the radius of the circumcircle of the given triangle. | R_{1}+R_{2}+R_{3} | 72 | 14 |
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