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 | 2.014. $\frac{x-y}{x^{3 / 4}+x^{1 / 2} y^{1 / 4}} \cdot \frac{x^{1 / 2} y^{1 / 4}+x^{1 / 4} y^{1 / 2}}{x^{1 / 2}+y^{1 / 2}} \cdot \frac{x^{1 / 4} y^{-1 / 4}}{x^{1 / 2}-2 x^{1 / 4} y^{1 / 4}+y^{1 / 2}}$. | \frac{\sqrt[4]{x}+\sqrt[4]{y}}{\sqrt[4]{x}-\sqrt[4]{y}} | 132 | 30 |
math | 4. Mom asked Xiaoming to boil water and make tea for the guests. It takes 1 minute to clean the kettle, 15 minutes to boil water, 1 minute to clean the teapot, 1 minute to clean the teacup, and 2 minutes to get the tea leaves. Xiaoming estimated that it would take 20 minutes to complete these tasks. To make the guests ... | 16 | 109 | 2 |
math | 22. Yura left the house for school 5 minutes later than Lena, but walked twice as fast as she did. How many minutes after leaving will Yura catch up to Lena? | 5 | 39 | 1 |
math | 6.052.
$$
\frac{1}{x-\sqrt{x^{2}-x}}-\frac{1}{x+\sqrt{x^{2}-x}}=\sqrt{3}
$$ | 4 | 42 | 1 |
math | 3. Find all three-digit numbers that decrease by 5 times after the first digit is erased. | 125,250,375 | 20 | 11 |
math | Let $x_0=1$, and let $\delta$ be some constant satisfying $0<\delta<1$. Iteratively, for $n=0,1,2,\dots$, a point $x_{n+1}$ is chosen uniformly form the interval $[0,x_n]$. Let $Z$ be the smallest value of $n$ for which $x_n<\delta$. Find the expected value of $Z$, as a function of $\delta$. | 1 - \ln \delta | 98 | 6 |
math | 6. Let $n=\sum_{a_{1}=0}^{2} \sum_{a_{2}=0}^{a_{1}} \cdots \sum_{a_{2} 012=0}^{a_{2} 011}\left(\prod_{i=1}^{2012} a_{i}\right)$. Then the remainder when $n$ is divided by 1000 is . $\qquad$ | 191 | 100 | 3 |
math | Solve the equation
$$
x^{3}+\left(\frac{x}{2 x-1}\right)^{3}=\frac{243}{64}
$$ | x_{1}=\frac{3}{4}\quad | 38 | 12 |
math | Example 2 Let $a$ and $b$ both be positive numbers, and $a+b=1$. Find
$$
E(a, b)=3 \sqrt{1+2 a^{2}}+2 \sqrt{40+9 b^{2}}
$$
the minimum value. | 5 \sqrt{11} | 62 | 7 |
math | 10.203. From a point on the circumference, two chords of lengths 9 and $17 \mathrm{~cm}$ are drawn. Find the radius of the circle if the distance between the midpoints of these chords is 5 cm. | 10\frac{5}{8} | 54 | 9 |
math | 4. Points $A, B, C, D$ lie on the circumference of a circle, and $B C=D C=4, A E=6$, the lengths of segments $B E$ and $D E$ are both integers, find the length of $B D$.
| 7 | 60 | 1 |
math | 1. A thread is strung with 75 blue, 75 red, and 75 green beads. We will call a sequence of five consecutive beads good if it contains exactly 3 green beads and one each of red and blue. What is the maximum number of good quintets that can be on this thread? | 123 | 67 | 3 |
math | 6 If the equation $\lg k x=2 \lg (x+1)$ has only one real root, then the range of values for $k$ is $\qquad$ . | k<0ork=4 | 38 | 6 |
math | Find all positive integers $N$ such that an $N\times N$ board can be tiled using tiles of size $5\times 5$ or $1\times 3$.
Note: The tiles must completely cover all the board, with no overlappings. | N \ne 1, 2, 4 | 56 | 12 |
math | [ Recurrence relations ] $[\quad \underline{\text { C }}$ mean values $]$
Thirteen turkey chicks pecked at grains. The first turkey chick pecked 40 grains; the second - 60, each subsequent chick pecked the average number of grains pecked by all previous chicks. How many grains did the 10th turkey chick peck? | 50 | 84 | 2 |
math | 2. From five positive integers $a, b, c, d, e$, any four are taken to find their sum, resulting in the set of sums $\{44,45,46,47\}$, then $a+b+c+d+e=$ $\qquad$ | 57 | 61 | 2 |
math | Example 4.37. How many times can the trigonometric series $\sum_{k=1}^{\infty} \frac{\cos k x}{k^{4}}$ be differentiated term by term? | 2 | 46 | 1 |
math | 1.2. [4] On the sheet, the numbers 1 and $x$, which is not a natural number, are written. In one move, the reciprocal of any of the already written numbers or the sum or difference of two of the written numbers can be written on the sheet. Is it possible for the number $x^{2}$ to appear on the sheet after several moves... | x^2 | 86 | 3 |
math | 11. Xiao Ming brought 24 handmade souvenirs to sell at the London Olympics. In the morning, each souvenir was sold for 7 pounds, and the number of souvenirs sold was less than half of the total. In the afternoon, he discounted the price of each souvenir, and the discounted price was still an integer. In the afternoon, ... | 8 | 108 | 1 |
math | 6. Given a fixed point $P(0,1)$, a moving point $Q$ satisfies that the perpendicular bisector of line segment $P Q$ is tangent to the parabola $y=x^{2}$. Then the equation of the locus of point $Q$ is $\qquad$ . | (2y-1)x^{2}+2(y+1)(y-1)^{2}=0(-1\leqslanty<\frac{1}{2}) | 64 | 39 |
math | From point A, 100 planes (the flagship and 99 additional planes) take off simultaneously. With a full tank of fuel, a plane can fly 1000 km. In flight, planes can transfer fuel to each other. A plane that has transferred fuel to others makes a gliding landing. How should the flight be organized so that the flagship fli... | 5187 | 82 | 4 |
math | 28. There are three pairs of real numbers $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)$, and $\left(x_{3}, y_{3}\right)$ that satisfy both $x^{3}-3 x y^{2}=2005$ and $y^{3}-3 x^{2} y=2004$. Compute $\left(1-\frac{x_{1}}{y_{1}}\right)\left(1-\frac{x_{2}}{y_{2}}\right)\left(1-\frac{x_{3}}{y_{3}}\right)$. | \frac{1}{1002} | 139 | 10 |
math | [ Processes and Operations ]
Between neighboring camps, it takes 1 day to travel. The expedition needs to transfer 1 can of food to the camp located 5 days away from the base camp and return. At the same time:
- each member of the expedition can carry no more than 3 cans of food;
- in 1 day, he consumes 1 can of food... | 243 | 110 | 3 |
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 | ## Task 4 - 190614
Three pioneers from a school, Karin, Dieter, and Frank, were delegated to the district mathematics olympiad and won a first, a second, and a third prize (each of the three pioneers exactly one of these prizes). Later, Anette inquired about the performance of the three olympiad participants. She was ... | Karin:2,Dieter:3,Frank:1 | 162 | 13 |
math | $3+$ [ Identical Transformations $]$
The number $x$ is such that among the four numbers $a=x-\sqrt{2}, b=x-\frac{1}{x}, c=x+\frac{1}{x}, d=x^{2}+2 \sqrt{2}$ exactly one is not an integer.
Find all such $x$. | \sqrt{2}-1 | 74 | 6 |
math | 6. (10 points) There are 2015 beauties, each of whom is either an angel or a demon; angels always tell the truth, demons sometimes tell the truth and sometimes lie.
The 1st says: There is exactly 1 angel among us.
The 3rd says: There are exactly 3 angels among us.
The 2013th says: There are exactly 2013 angels among u... | 3 | 177 | 1 |
math | 4. Solve the equation $3^{x}+4^{y}=5^{z}$ in natural numbers. | x=y=z=2 | 23 | 5 |
math | A-3. For a given positive integer $m$, find all positive integer triples $(n, x, y)$, where $m, n$ are coprime, and satisfy
$$
\left(x^{2}+y^{2}\right)^{m}=(x y)^{n} .
$$ | (n, x, y)=\left(m+1,2^{\frac{m}{2}}, 2^{\frac{m}{2}}\right) | 66 | 34 |
math | 15. If there exists a positive integer $m$ such that $m!$ ends with exactly $n$ zeros, then the positive integer $n$ is called a "factorial tail number." How many non-"factorial tail number" positive integers are there less than 1992? | 396 | 63 | 3 |
math | 3. The number of positive integer pairs $(x, y)$ that satisfy the equation
$$
\begin{array}{l}
x \sqrt{y}+y \sqrt{x}-\sqrt{2006 x}-\sqrt{2006 y}+\sqrt{2006 x y} \\
\quad=2006
\end{array}
$$ | 8 | 82 | 1 |
math | Determine all rational numbers $a$ for which the matrix
$$\begin{pmatrix}
a & -a & -1 & 0 \\
a & -a & 0 & -1 \\
1 & 0 & a & -a\\
0 & 1 & a & -a
\end{pmatrix}$$
is the square of a matrix with all rational entries.
[i]Proposed by Daniël Kroes, University of California, San Diego[/i] | a = 0 | 112 | 5 |
math | 9. The sum of the first $n$ terms of an arithmetic sequence is 2000, the common difference is 2, the first term is an integer, and $n>1$, the sum of all possible values of $n$ is $\qquad$ . | 4835 | 58 | 4 |
math | Let $ABCD$ be a quadrilateral with $AD = 20$ and $BC = 13$. The area of $\triangle ABC$ is $338$ and the area of $\triangle DBC$ is $212$. Compute the smallest possible perimeter of $ABCD$.
[i]Proposed by Evan Chen[/i] | 118 | 76 | 3 |
math | 19.2.5 * Given that $n$ is a positive integer. Let $A_{n}=(7+4 \sqrt{3})^{n}$. Simplify $1+\left[A_{n}\right]-A_{n}$. | (7-4\sqrt{3})^{n} | 53 | 12 |
math | 7.034. $\lg \left(3^{x}-2^{4-x}\right)=2+0.25 \lg 16-0.5 x \lg 4$. | 3 | 43 | 1 |
math | Let $p, q \in \mathbf{R}_{+}, x \in\left(0, \frac{\pi}{2}\right)$. Find the minimum value of the function $f(x)=\frac{p}{\sqrt{\sin x}}+\frac{q}{\sqrt{\cos x}}$. | \left(p^{\frac{4}{5}}+q^{\frac{4}{5}}\right)^{\frac{5}{4}} | 67 | 31 |
math | 4.1. Given an arithmetic progression $a_{1}, a_{2}, \ldots, a_{100}$. It is known that $a_{3}=9.5$, and the common difference of the progression $d=0.6$. Find the sum $\left\{a_{1}\right\}+\left\{a_{2}\right\}+\ldots+\left\{a_{100}\right\}$. The notation $\{x\}$ represents the fractional part of the number $x$, i.e., t... | 50 | 170 | 2 |
math | Example 5. The vector $\vec{d}=\{-9,2,25\}$ is to be decomposed into the vectors $\vec{a}=$ $=\{1,1,3\}, \vec{b}=\{2,-1,-6\}$ and $\vec{c}=\{5,3,-1\}$. | \vec{}=2\vec{}-3\vec{b}-\vec{} | 75 | 18 |
math | Example 3. Calculate the value of the derivative of the implicit function $x y^{2}=4$ at the point $M(1,2)$. | -1 | 33 | 2 |
math | 4. Given $f(x)=\frac{x^{2}}{2 x+1}, f_{1}(x)=f(x)$, $f_{n}(x)=\underbrace{f(\cdots f}_{n \uparrow}(x) \cdots)$.
Then $f_{6}(x)=$ | \frac{1}{(1+\frac{1}{x})^{64}-1} | 67 | 20 |
math | 8. (10 points) The number of pages in the 5 books, The Book of Songs, The Book of Documents, The Book of Rites, The Book of Changes, and The Spring and Autumn Annals, are all different: The Book of Songs and The Book of Documents differ by 24 pages. The Book of Documents and The Book of Rites differ by 17 pages. The Bo... | 34 | 173 | 2 |
math | Read the results of a survey conducted in Pec pod Snezkou, where 1240 people were approached:
"46% of the respondents believe in the existence of Krkonose (rounded to the nearest whole number), 31% do not believe in its existence (rounded to the nearest whole number). The rest of the respondents refused to respond to ... | 565 | 130 | 3 |
math | 4. For the geometric sequence $a_{1}, a_{2}, a_{3}, a_{4}$, it satisfies $a_{1} \in(0,1), a_{2} \in(1,2), a_{3} \in(2,3)$, then the range of $a_{4}$ is
$\qquad$ | (2\sqrt{2},9) | 76 | 9 |
math | 6. Determine $x, y, z$ if
$$
\frac{a y+b x}{x y}=\frac{b z+c y}{y z}=\frac{c x+a z}{z x}=\frac{4 a^{2}+4 b^{2}+4 c^{2}}{x^{2}+y^{2}+z^{2}}, \quad a, b, c \in \mathbb{R}
$$ | 2a,2b,2c | 98 | 8 |
math | Define the set $M_q=\{x \in \mathbb{Q} \mid x^3-2015x=q \}$ , where $q$ is an arbitrary rational number.
[b]a)[/b] Show that there exists values for $q$ such that the set is null as well as values for which it has exactly one element.
[b]b)[/b] Determine all the possible values for the cardinality of $M_q$ | |M_q| \in \{0, 1\} | 97 | 15 |
math | 236. A cube with edge $a$ is standing on a plane. The light source is located at a distance $b(b>a)$ from the plane. Find the minimum value of the area of the shadow cast by the cube on the plane. | (\frac{}{b-})^{2} | 52 | 10 |
math | 8.3. Given an acute-angled triangle $A B C$. Point $M$ is the intersection point of its altitudes. Find the angle $A$, if it is known that $A M=B C$.
---
The text has been translated while preserving the original formatting and line breaks. | 45 | 61 | 2 |
math | 8. Let the sequence $a_{n}=\left[(\sqrt{2}+1)^{n}+\left(\frac{1}{2}\right)^{n}\right], n \geq 0$, where $[x]$ denotes the greatest integer less than or equal to $x$. Then
$$
\sum_{n=1}^{\infty} \frac{1}{a_{n-1} a_{n+1}}=
$$ | \frac{1}{8} | 99 | 7 |
math | 7. Given a composite number $k(1<k<100)$. If the sum of the digits of $k$ is a prime number, then the composite number $k$ is called a "pseudo-prime". The number of such pseudo-primes is . $\qquad$ | 23 | 60 | 2 |
math | 7. (4 points) Find what $x+y$ can be equal to, given that $x^{3}+6 x^{2}+16 x=-15$ and $y^{3}+6 y^{2}+16 y=-17$. | -4 | 58 | 2 |
math | On a circular running track, two people are running in the same direction at constant speeds. At a certain moment, runner $A$ is 10 meters ahead of runner $B$, but after $A$ runs 22 meters, $B$ catches up.
How many points on the track can $B$ later overtake $A$? | 5 | 72 | 1 |
math | ## Task Condition
Find the derivative.
$$
y=x-\ln \left(1+e^{x}\right)-2 e^{-\frac{x}{2}} \operatorname{arctan} e^{\frac{x}{2}}
$$ | x\cdote^{-\frac{x}{2}}\cdot\operatorname{arctg}e^{\frac{x}{2}} | 51 | 29 |
math | Highimiy and.
In Anchuria, a day can be either clear, when the sun shines all day, or rainy, when it rains all day. And if today is not like yesterday, the Anchurians say that the weather has changed today. Once, Anchurian scientists established that January 1 is always clear, and each subsequent day in January will b... | 2047 | 138 | 4 |
math | 11. (This question is worth 20 points) Determine all complex numbers $\alpha$ such that for any complex numbers $z_{1}, z_{2}\left(\left|z_{1}\right|,\left|z_{2}\right|<1, z_{1} \neq z_{2}\right)$, we have
$$
\left(z_{1}+\alpha\right)^{2}+\alpha \overline{z_{1}} \neq\left(z_{2}+\alpha\right)^{2}+\alpha \overline{z_{2}}... | {\alpha|\alpha\in{C},|\alpha\mid\geq2} | 129 | 18 |
math | [Example 4.5.1] Nine bags contain $9,12,14,16,18,21,24,25,28$ balls respectively. A takes several bags, B also takes several bags, and in the end, only one bag is left. It is known that the total number of balls A takes is twice the total number of balls B takes. How many balls are in the remaining bag? | 14 | 94 | 2 |
math | We wrote to ten of our friends and put the letters into the addressed envelopes randomly. What is the probability that exactly 5 letters end up with the person they were intended for? | \frac{11}{3600} | 36 | 11 |
math | ## Task A-3.1. (4 points)
Solve the equation $\quad \sin x \cdot \cos 2 x \cdot \cos 4 x=1$. | \frac{3\pi}{2}+2k\pi | 38 | 14 |
math | 6. Find all pairs of real numbers $a$ and $b$ for which the system of equations
$$
\frac{x+y}{x^{2}+y^{2}}=a, \quad \frac{x^{3}+y^{3}}{x^{2}+y^{2}}=b
$$
with unknowns $x$ and $y$ has a solution in the set of real numbers.
(J. Šimša) | 0\leqq\leqq\frac{9}{8} | 97 | 14 |
math | 58(1186). Find for any natural $n$ the value of the expression
$$
\sqrt[3]{\frac{1 \cdot 2 \cdot 4+2 \cdot 4 \cdot 8+\ldots+n \cdot 2 n \cdot 4 n}{1 \cdot 3 \cdot 9+2 \cdot 6 \cdot 18+\ldots+n \cdot 3 n \cdot 9 n}}
$$ | \frac{2}{3} | 101 | 7 |
math | G7.1 Find $3+6+9+\ldots+45$. | 360 | 18 | 3 |
math | 3. For any natural numbers $m, n$ satisfying $\frac{m}{n}<\sqrt{7}$, the inequality $7-\frac{m^{2}}{n^{2}} \geqslant \frac{\lambda}{n^{2}}$ always holds. Find the maximum value of $\lambda$.
| 3 | 68 | 1 |
math | 15. Find the maximum value of the positive real number $A$ such that for any real numbers $x, y, z$, the inequality
$$
\begin{array}{l}
x^{4}+y^{4}+z^{4}+x^{2} y z+x y^{2} z+x y z^{2}- \\
A(x y+y z+z x)^{2} \geqslant 0
\end{array}
$$
holds. | \frac{2}{3} | 102 | 7 |
math | 9. (40 points) Given the following numbers: 20172017, 20172018, 20172019, 20172020, and 20172021. Is there a number among them that is coprime with all the others? If so, which one? | 20172017,20172019 | 85 | 17 |
math | Exercise 1. What is the number of integers between 1 and 10000 that are divisible by 7 and not by 5? | 1143 | 32 | 4 |
math | Four bronze weights together weigh 40 kilograms; with their help, any whole number weight from 1 to 40 kilograms can be measured; what are the weights? | 1,3,9,u=27 | 35 | 9 |
math | (a) Determine the point of intersection of the lines with equations $y=4 x-32$ and $y=-6 x+8$.
(b) Suppose that $a$ is an integer. Determine the point of intersection of the lines with equations $y=-x+3$ and $y=2 x-3 a^{2}$. (The coordinates of this point will be in terms of $a$.)
(c) Suppose that $c$ is an integer. ... | ()(4,-16),(b)(1+^2,2-^2),()(3,3-3c^2),()1,-1,2,-2 | 173 | 36 |
math | [Mathematical logic (other)]
In the room, there are 85 balloons - red and blue. It is known that: 1) at least one of the balloons is red, 2) in every arbitrarily chosen pair of balloons, at least one is blue. How many red balloons are in the room?
# | 1 | 66 | 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 | Seven, for a cylindrical can with a fixed volume, when the relationship between the diameter of the base and the height is what, the least amount of iron sheet is used?
When the volume of a cylindrical can is fixed, the relationship between the diameter of the base and the height that results in the least amount of ... | h=2r | 70 | 4 |
math | A [i]string of length $n$[/i] is a sequence of $n$ characters from a specified set. For example, $BCAAB$ is a string of length 5 with characters from the set $\{A,B,C\}$. A [i]substring[/i] of a given string is a string of characters that occur consecutively and in order in the given string. For example, the string $CA... | 963 | 364 | 3 |
math | C2. The integers $a, b, c, d, e, f$ and $g$, none of which is negative, satisfy the following five simultaneous equations:
$$
\begin{array}{l}
a+b+c=2 \\
b+c+d=2 \\
c+d+e=2 \\
d+e+f=2 \\
e+f+g=2 .
\end{array}
$$
What is the maximum possible value of $a+b+c+d+e+f+g$ ? | 6 | 103 | 1 |
math | Determine all positive integers $n \geq 2$ for which there exists a positive divisor $m \mid n$ such that
$$
n=d^{3}+m^{3},
$$
where $d$ is the smallest divisor of $n$ greater than 1. | 16, 72, 520 | 60 | 11 |
math | ## Task 4 - 170514
A distance of $240 \mathrm{~m}$ is divided into four segments such that the following conditions are met:
(1) The first segment is twice as long as the second segment.
(2) The second segment is as long as the sum of the lengths of the third and fourth segments.
(3) The third segment is one-fifth ... | 120 | 100 | 3 |
math | A certain triangle has two sides of $8 \mathrm{dm}$ and $5 \mathrm{dm}$; the angle opposite the first is twice as large as the one opposite the second. What is the length of the third side of the triangle? | 7.8\mathrm{} | 51 | 6 |
math | Start with three numbers $1,1,1$, each operation replaces one of the numbers with the sum of the other two. After 10 operations, what is the maximum possible value of the largest number among the three obtained? | 144 | 47 | 3 |
math | 15. Given the function $f(x)=a_{1} x+a_{2} x^{2}+\cdots+a_{n} x^{n}$
$\left(n \in \mathbf{N}_{+}\right), a_{1}, a_{2}, \cdots, a_{n}$ are terms of a sequence. If $f(1)=n^{2}+1$, (1) find the general term formula of the sequence $\left\{a_{n}\right\}$; (2) find $\lim _{n \rightarrow \infty}\left(1-\frac{1}{a_{n}}\right)... | e^{-\frac{1}{2}} | 142 | 9 |
math | a) Name the first 10 natural numbers that have an odd number of divisors (including one and the number itself).
b) Try to formulate and prove a rule that allows finding the next such numbers. | 1,4,9,16,25,36,49,64,81,100 | 43 | 27 |
math | 2. Given that the first $n$ terms sum of the arithmetic sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ are $S_{n}$ and $T_{n}$ respectively, and for all positive integers $n$, $\frac{a_{n}}{b_{n}}=\frac{2 n-1}{3 n+1}$. Then $\frac{S_{6}}{T_{5}}=$ $\qquad$ . | \frac{18}{25} | 105 | 9 |
math | 1. Upon entering the Earth's atmosphere, the asteroid heated up significantly and exploded near the surface, breaking into a large number of fragments. Scientists collected all the fragments and divided them into groups based on size. It was found that one-fifth of all fragments had a diameter of 1 to 3 meters, another... | 70 | 112 | 2 |
math | 1. If positive numbers $a, b$ satisfy $2+\log _{2} a=3+\log _{3} b=\log _{6}(a+b)$, then the value of $\frac{1}{a}+\frac{1}{b}$ is $\qquad$ | 108 | 62 | 3 |
math | 7. Given that $a$ is a root of the equation $x^{2}-3 x+1=0$. Then the value of the fraction $\frac{2 a^{6}-6 a^{4}+2 a^{5}-a^{2}-1}{3 a}$ is $\qquad$ - | -1 | 65 | 2 |
math | 7. In order to cater to customers' psychology, a mall bundles two items with different cost prices for sale. Both items are sold for 216 yuan, with one item at a loss of $20 \%$, but the overall profit is $20 \%$. The cost price of the other item should be $\qquad$ yuan. | 90 | 71 | 2 |
math | $8 \cdot 74$ Find the value of the smallest term in the following sequence:
$$a_{1}=1993^{1994^{1995}}, a_{n+1}=\left\{\begin{array}{ll}
\frac{1}{2} a_{n}, & \text { if } a_{n} \text { is even, } \\
a_{n}+7, & \text { if } a_{n} \text { is odd. }
\end{array}\right.$$ | 1 | 116 | 1 |
math | 8. Lei Lei bought some goats and sheep. If she bought 2 more goats, the average price per sheep would increase by 60 yuan. If she bought 2 fewer goats, the average price per sheep would decrease by 90 yuan. Lei Lei bought $\qquad$ sheep in total. | 10 | 63 | 2 |
math | . Find all functions $f: \mathbb{Q}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that
$$
f(x y) \cdot \operatorname{gcd}\left(f(x) f(y), f\left(\frac{1}{x}\right) f\left(\frac{1}{y}\right)\right)=x y f\left(\frac{1}{x}\right) f\left(\frac{1}{y}\right)
$$
for all $x, y \in \mathbb{Q}_{>0}$, where $\operatorname{gcd}(a, b)$ ... | f(\frac{}{n})=\quad\text{forall},n\in\mathbb{Z}_{>0}\text{suchthat}\operatorname{gcd}(,n)=1 | 147 | 40 |
math | For real numbers $x$, $y$ and $z$, solve the system of equations:
$$x^3+y^3=3y+3z+4$$ $$y^3+z^3=3z+3x+4$$ $$x^3+z^3=3x+3y+4$$ | x = y = z = 2 | 67 | 9 |
math | G7.2 Find $b$, if $b$ is the remainder when $1998^{10}$ is divided by $10^{4}$. | 1024 | 35 | 4 |
math | Determine all $6$-digit numbers $(abcdef)$ such that $(abcdef) = (def)^2$ where $(x_1x_2...x_n)$ is not a multiplication but a number of $n$ digits. | 390625 \text{ and } 141376 | 48 | 18 |
math | 5. Is the number
$$
\frac{1}{2+\sqrt{2}}+\frac{1}{3 \sqrt{2}+2 \sqrt{3}}+\frac{1}{4 \sqrt{3}+3 \sqrt{4}}+\ldots+\frac{1}{100 \sqrt{99}+99 \sqrt{100}}
$$
rational? Justify! | \frac{9}{10} | 90 | 8 |
math | 6. (8 points) Let for positive numbers $x, y, z$ the following system of equations holds:
$$
\left\{\begin{array}{l}
x^{2}+x y+y^{2}=108 \\
y^{2}+y z+z^{2}=64 \\
z^{2}+x z+x^{2}=172
\end{array}\right.
$$
Find the value of the expression $x y+y z+x z$. | 96 | 103 | 2 |
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 | Find all the pairs of positive integers $(x,y)$ such that $x\leq y$ and \[\frac{(x+y)(xy-1)}{xy+1}=p,\]where $p$ is a prime number. | (x, y) = (3, 5) | 49 | 13 |
math | Example 3. Find the variance of a random variable $X$ that has a geometric distribution. | \frac{1-p}{p^{2}} | 20 | 10 |
math | On 6. Person A and Person B simultaneously solve the radical equation $\sqrt{x+a}+$ $\sqrt{x+b}=7$. When copying, A mistakenly copies it as $\sqrt{x-a}+\sqrt{x+b}=7$, and ends up solving one of its roots as 12. B mistakenly copies it as $\sqrt{x+a}+\sqrt{x+d}$ $=7$, and ends up solving one of its roots as 13. It is kno... | a=12, b=37; a=3, b=4; a=-4, b=-3; a=-13, b=-8 | 156 | 34 |
math | ## Task 3 - 230813
On a $22.5 \mathrm{~km}$ long tram line, trains are to run in both directions from 8:00 AM to 4:00 PM at ten-minute intervals, starting with the departure time exactly at 8:00 AM at both terminal stops. The average speed of the trains is $18 \frac{\mathrm{km}}{\mathrm{h}}$. Each train, upon arrival ... | 11:00 | 213 | 5 |
math | ## Task A-4.5.
In a circle, a finite number of real numbers are arranged. Each number is colored red, white, or blue. Each red number is twice as small as the sum of its two neighboring numbers, each white number is equal to the sum of its two neighboring numbers, and each blue number is twice as large as the sum of i... | \frac{b}{p}=-\frac{3}{2} | 124 | 15 |
math | Three, (50 points) If the three sides of a triangle are all rational numbers, and one of its interior angles is also a rational number, find all possible values of this interior angle.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result direc... | 60^{\circ}, 90^{\circ}, 120^{\circ} | 65 | 21 |
math | A standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 187 | 54 | 3 |
math | 7. Without rearranging the digits on the left side of the equation, insert two "+" signs between them so that the equation $8789924=1010$ becomes true. | 87+899+24=1010 | 42 | 14 |
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