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 | 35. (1985 American Mathematical Invitation Contest) Let the four vertices of a regular tetrahedron be $A, B, C, D$, with each edge length being 1 meter. A small insect starts from point $A$ and moves according to the following rule: at each vertex, it chooses one of the three edges connected to that vertex with equal p... | 182 | 130 | 3 |
math | 1. Determine which of the numbers is greater
$$
\operatorname{arctg}(2+\sqrt{5})+\operatorname{arcctg}(2-\sqrt{5}) \quad \text { or } \quad \frac{5 \sqrt{7}}{4} .
$$ | \frac{5\sqrt{7}}{4}>\pi | 63 | 14 |
math | 13. (i) (Grade 11) In the arithmetic sequence $\left\{a_{n}\right\}: a_{n}=4 n -1\left(n \in \mathbf{N}_{+}\right)$, after deleting all numbers that can be divided by 3 or 5, the remaining numbers are arranged in ascending order to form a sequence $\left\{b_{n}\right\}$. Find the value of $b_{2006}$.
(ii) (Grade 12) Gi... | 15043 | 217 | 5 |
math | 24. (POL 5) For points $A_{1}, \ldots, A_{5}$ on the sphere of radius 1 , what is the maximum value that $\min _{1 \leq 1, j \leq 5} A_{i} A_{j}$ can take? Determine all configurations for which this maximum is attained. (Or: determine the diameter of any set $\left\{A_{1}, \ldots, A_{5}\right\}$ for which this maximum... | \sqrt{2} | 111 | 5 |
math | For each positive integer $n$, let $g(n)$ be the sum of the digits when $n$ is written in binary. For how many positive integers $n$, where $1\leq n\leq 2007$, is $g(n)\geq 3$? | 1941 | 63 | 4 |
math | Mr. Rychlý and Mr. Louda set out on the same hiking tour at the same time, but Mr. Rychlý started from the mountain hut and Mr. Louda from the bus at the bottom of the town, heading up to the hut. They met on the trail at 10 o'clock. Mr. Rychlý was in a hurry and reached his destination by 12 o'clock. On the other hand... | 6 | 146 | 1 |
math | 5. Given a natural number $x=2^{n}-32$, where $n-$ is a natural number. It is known that $x$ has exactly three distinct prime divisors, one of which is 3. Find $x$. | 480or2016 | 51 | 8 |
math | 4. In a chess tournament, there are $n$ female players and $9 n$ male players. Each player plays one game against each of the other $10 n-1$ players. The scoring system is as follows: the winner gets 2 points, the loser gets 0 points, and in the case of a draw, each player gets 1 point. After the tournament, it was fou... | 1 | 118 | 1 |
math | ## Task B-1.2.
Determine the last three digits of the number
$$
2^{2015}-2^{2013}+2^{2010}
$$ | 600 | 43 | 3 |
math | 7.067. $2,5^{\frac{4+\sqrt{9-x}}{\sqrt{9-x}}} \cdot 0,4^{1-\sqrt{9-x}}=5^{10} \cdot 0,1^{5}$. | -7,8 | 57 | 4 |
math | 1. The speed of light is 300,000 kilometers per second, and the distance from the Sun to the Earth is 150 million kilometers. Question: How many minutes does it take for light to travel from the Sun to the Earth (round the answer to one decimal place)? | 8.3 | 63 | 3 |
math | 4.1.1. (12 points) From point $A$ to point $B$, a bus and a cyclist departed simultaneously at 13:00. After arriving at point $B$, the bus, without stopping, headed back and met the cyclist at point $C$ at 13:10. Upon returning to point $A$, the bus again, without stopping, headed to point $B$ and caught up with the cy... | 40 | 155 | 2 |
math | 1. There are 800 marbles in a bag. Each marble is colored with one of 100 colors, and there are eight marbles of each color. Anna draws one marble at a time from the bag, without replacement, until she gets eight marbles of the same color, and then she immediately stops.
Suppose Anna has not stopped after drawing 699 m... | \frac{99}{101} | 100 | 10 |
math | 19.4.3 ** For any positive integer $q_{0}$, consider the sequence $q_{1}, q_{2}, \cdots, q_{n}$ defined by $q_{i}=\left(q_{i-1}-1\right)^{3}+3, i=1,2, \cdots, n$. If each $q_{i}(i=1,2, \cdots, n)$ is a power of a prime, find the largest possible value of $n$. | 2 | 109 | 1 |
math | 2. Task: Calculate $\sqrt[3]{\frac{x}{2015+2016}}$, where $x$ is the harmonic mean of the numbers
$a=\frac{2016+2015}{2016^{2}+2016 \cdot 2015+2015^{2}}$ and $b=\frac{2016-2015}{2016^{2}-2016 \cdot 2015+2015^{2}}$.
The harmonic mean of two positive numbers $a$ and $b$ is the number $c$ such that $\frac{1}{c}=\frac{1}... | \frac{1}{2016} | 176 | 10 |
math | Ann and Max play a game on a $100 \times 100$ board.
First, Ann writes an integer from 1 to 10 000 in each square of the board so that each number is used exactly once.
Then Max chooses a square in the leftmost column and places a token on this square. He makes a number of moves in order to reach the rightmost column... | 500000 | 182 | 6 |
math | 4. (15 points) The efficiency of an ideal heat engine is $40 \%$. What will it become if the temperature of the heater is increased by $40 \%$, and the temperature of the cooler is decreased by $40 \%$? | 74 | 53 | 2 |
math | 7. Let $a, b>0$, satisfy: the equation $\sqrt{|x|}+\sqrt{|x+a|}=b$ has exactly three distinct real solutions $x_{1}, x_{2}, x_{3}$, and $x_{1}<x_{2}<x_{3}=b$, then the value of $a+b$ is $\qquad$ . | 144 | 79 | 3 |
math | $1 \cdot 49$ integers $1,2, \cdots, n$ are arranged in a permutation such that each number is either greater than all the numbers before it, or less than all the numbers before it. How many such permutations are there? | 2^{n-1} | 55 | 6 |
math | Initially, the level markers for internet users are stars, moons, and suns. The rule is: if the cumulative online time from 0:00 to 24:00 in a day is full 2 hours, it counts as 1 day of internet use. Starting from the first day of internet use, after 5 days, the level marker is 1 star; after another 7 days, the level m... | 2 \text{ suns, 2 moons, and 2 stars; 10 days} | 213 | 21 |
math | Let $p$ be a prime number satisfying $p \equiv 3(\bmod 8)$. Find all integer solutions $(x, y)$ to the equation $y^{2}=x^{3}-p^{2} x$. (2007, Indian National Team Selection Exam) | (0,0),(p, 0),(-p, 0) | 61 | 16 |
math | ## Task A-2.2.
Two circles with radii 1 and 3 touch each other externally at point $A$, and their external common tangent touches them at points $B$ and $C$. Determine the sum of the squares of the lengths of the sides of triangle $A B C$. | 24 | 62 | 2 |
math | 3. $A B C D$ is a square and $X$ is a point on the side $D A$ such that the semicircle with diameter $C X$ touches the side $A B$. Find the ratio $A X: X D$. | AX:XD=1:3 | 54 | 7 |
math | The product of three natural numbers is 600. If we decreased one of the factors by 10, the product would decrease by 400. If we increased that one factor by 5 instead, the product would double. Which three natural numbers have this property?
(L. Hozová)
Hint. From each declarative sentence in the problem, one factor ... | 5,8,15 | 83 | 6 |
math | $8.27 \quad \log _{2}\left(1+\log _{\frac{1}{9}} x-\log _{9} x\right)<1$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
$8.27 \quad \log _{2}\left(1+\log _{\frac{1}{9}} x-\log _{9} x\right)<1$. | (\frac{1}{3},3) | 104 | 9 |
math | Example 6 Let $M=\{1,2, \cdots, 20\}, A_{1}, A_{2}, \cdots A_{n}$ be distinct non-empty subsets of $M$, such that when $i \neq j$, $A_{i} \cap A_{j}$ has at most two elements. Find the maximum value of $n$.
untranslated text remains the same as the source text in terms of line breaks and formatting. | 1350 | 98 | 4 |
math | 6. (10 points) An Englishman was the owner of a plot of land in Russia. He knows that, in the units familiar to him, the size of his plot is three acres. The cost of the land is 250000 rubles per hectare. It is known that 1 acre $=4840$ square yards, 1 yard $=0.9144$ meters, 1 hectare $=10000 m^{2}$. Calculate how much... | 303514 | 122 | 6 |
math | For each integer $n\geq3$, let $f(n)$ be the number of $3$-element subsets of the vertices of the regular $n$-gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of $n$ such that $f(n+1)=f(n)+78$. | 245 | 77 | 3 |
math | Example 3. Find the order of the zero $z_{0}=0$ for the function
$$
f(z)=\frac{z^{8}}{z-\sin z}
$$ | 5 | 40 | 1 |
math | 12. Let $\mathbb{N}$ be the set of all positive integers. A function $f: \mathbb{N} \rightarrow \mathbb{N}$ satisfies $f(m+$ $n)=f(f(m)+n)$ for all $m, n \in \mathbb{N}$, and $f(6)=2$. Also, no two of the values $f(6), f(9), f(12)$ and $f(15)$ coincide. How many three-digit positive integers $n$ satisfy $f(n)=f(2005)$ ... | 225 | 126 | 3 |
math | 3.469 Given: $\cos \left(x-\frac{3 \pi}{2}\right)=-\frac{4}{5}, 0<x<\frac{\pi}{2}$. Find $A=\sin \frac{x}{2} \cdot \cos \frac{5 x}{2}$. | -\frac{38}{125} | 67 | 10 |
math | 19. Find all prime numbers $p$ such that the numbers $p+4$ and $p+8$ are also prime. | 3 | 29 | 1 |
math | 7. Given the sequence $\left\{a_{n}\right\}$ satisfies:
$$
\begin{array}{l}
a_{1}=2, a_{2}=6, \\
a_{n+1}=\frac{a_{n}^{2}-2 a_{n}}{a_{n-1}}(n=2,3, \cdots) .
\end{array}
$$
Then $\lim _{n \rightarrow \infty}\left\{\sqrt{a_{n}+n}\right\}=$ $\qquad$ | 1 | 119 | 1 |
math | 11. (5 points) A team of 23 people participated in the 16th Guangzhou Asian Games. They were arranged in descending order of height. The average height of the top 5 players is 3 cm more than the average height of the top 8 players; the average height of the last 15 players is 0.5 cm less than the average height of the ... | 8 | 115 | 1 |
math | Find the smallest positive integer $n$ with the following property: there does not exist an arithmetic progression of $1999$ real numbers containing exactly $n$ integers. | 70 | 36 | 2 |
math | 7. Find all three-digit numbers in the decimal system that are equal to one third of the number with the same representation in another number system. | 116,120,153,195,236,240,356,360,476,480,596 | 29 | 43 |
math | 1.014. $\frac{\left(4.5 \cdot 1 \frac{2}{3}-6.75\right) \cdot \frac{2}{3}}{\left(3 \frac{1}{3} \cdot 0.3+5 \frac{1}{3} \cdot \frac{1}{8}\right): 2 \frac{2}{3}}+\frac{1 \frac{4}{11} \cdot 0.22: 0.3-0.96}{\left(0.2-\frac{3}{40}\right) \cdot 1.6}$. | 1 | 141 | 1 |
math | For a positive real number $ [x] $ be its integer part. For example, $[2.711] = 2, [7] = 7, [6.9] = 6$. $z$ is the maximum real number such that [$\frac{5}{z}$] + [$\frac{6}{z}$] = 7. Find the value of$ 20z$. | 30 | 91 | 2 |
math | 4. Four cars $A, B, C$, and $D$ start simultaneously from the same point on a circular track. $A$ and $B$ drive clockwise, while $C$ and $D$ drive counterclockwise. All cars move at constant (but pairwise different) speeds. Exactly 7 minutes after the start of the race, $A$ meets $C$ for the first time, and at the same... | 371 | 143 | 3 |
math | # Problem 3. (3 points)
The sequence $\left\{a_{n}\right\}$ is defined by the conditions $a_{1}=1$ and $a_{n}=a_{1}+a_{2}+\ldots+a_{n-1}+n$ for $n \geqslant 2$. Find the explicit formula for this sequence. | a_{n}=2^{n}-1 | 80 | 9 |
math | 5. Find all natural numbers $n$ such that $2^{n}-1$ is divisible by 7. | 3k(k\inZ^{+}) | 24 | 9 |
math | Joãozinho collects natural numbers whose unit digit is the sum of the other digits. For example, he collected 10023, because $1+0+0+2=3$.
a) In Joãozinho's collection, there is a number that has 4 digits and whose unit digit is 1. What is this number?
b) What is the largest number without the digit 0 that can appear ... | 1001,1111111119,62109 | 110 | 21 |
math | 8. Given: when $x \in\left(0, \frac{\pi}{2}\right)$, $\sin x<x<\tan x$. Compare $x$ with $\frac{1}{2}(\sin x+\tan x)$, where $x \in$ $\left(0, \frac{\pi}{2}\right)$ | x<\frac{1}{2}(\sinx+\tanx) | 73 | 16 |
math | 2. Given $\boldsymbol{a}=\left(\cos \frac{2}{3} \pi, \sin \frac{2}{3} \pi\right), \overrightarrow{O A}=\boldsymbol{a}-$ $\boldsymbol{b}, \overrightarrow{O B}=\boldsymbol{a}+\boldsymbol{b}$, if $\triangle O A B$ is an isosceles right triangle with $O$ as the right-angle vertex, then the area of $\triangle O A B$ is $\qq... | 1 | 117 | 1 |
math | Example 1. Given $f(n)=n^{4}+n^{3}+n^{2}+n+1$, find the remainder of $f\left(2^{5}\right)$ divided by $f(2)$. | 5 | 51 | 1 |
math | Example 12. There are three urns with balls. The first contains 5 blue and 3 red balls, the second - 4 blue and 4 red, and the third - 8 blue. One of the urns is randomly chosen, and a ball is randomly drawn from it. What is the probability that it will be red (event $A$). | \frac{7}{24} | 77 | 8 |
math | 13.035. A shoe factory completed $20 \%$ of the monthly plan in the first week, produced $120 \%$ of the amount of products made in the first week in the second week, and produced $60 \%$ of the products made in the first two weeks combined in the third week. What is the monthly production plan for shoes, if it is know... | 5000 | 108 | 4 |
math | Three, (50 points) Let $A=\{1,2, \cdots, 30\}$. Find the smallest positive integer $n$, such that for any 11 subsets of $A$, if the union of any 5 of them has at least $n$ elements, then there must exist 3 of these 11 subsets whose intersection is non-empty. | 22 | 81 | 2 |
math | Magozinov A.
A straight stick 2 meters long was sawn into $N$ sticks, the length of each of which is expressed as an integer number of centimeters. For what smallest $N$ can it be guaranteed that, using all the resulting sticks, one can, without breaking them, form the contour of some rectangle? | 102 | 70 | 3 |
math | $12 \cdot 58$ Find the smallest positive integer $n$ (where $n>1$) such that the quadratic mean of the first $n$ natural numbers is an integer.
(Note: The quadratic mean of $n$ numbers $a_{1}, a_{2}, \cdots, a_{n}$ is $\left.\sqrt{\frac{a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}}{n}}.\right)$
(15th United States of America M... | 337 | 127 | 3 |
math | 7.1. (14 points) Find the greatest integer value of $a$ for which the equation
$$
(x-a)(x-7)+3=0
$$
has at least one integer root. | 11 | 45 | 2 |
math | 6. The expression contains the following products:
$(2 \cdot 5) \cdot(10) \cdot(12 \cdot 15) \cdot(25 \cdot 4) \cdot(20 \cdot 30) \cdot(35 \cdot 14) \ldots \Rightarrow$ since each of these grouped products ends in 0, the number will end in 8 zeros $\Rightarrow B=0$
Let's break down the expression into the following p... | A=2,B=0,C=1,D=4 | 629 | 12 |
math | Problem 2. Gosha entered a natural number into the calculator. Then he performed the following operation, consisting of two actions, three times: first, he extracted the square root, and then took the integer part of the obtained number. In the end, he got the number 1. What is the largest number that Gosha could have ... | 255 | 92 | 3 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 2 \pi}(\cos x)^{\frac{\operatorname{ctg} 2 x}{\sin 3 x}}$ | e^{-\frac{1}{12}} | 47 | 10 |
math | A convex pentagon $ ABCDE$ is inscribed in a circle. The distances of $ A$ from the lines $ BC,CD,DE$ are $ a,b,c,$ respectively. Compute the distance of $ A$ from the line $ BE$. | \frac{a \cdot c}{b} | 52 | 10 |
math | 1. Given $z_{1}, z_{2}$ are conjugate complex numbers, if $\left|z_{1}-z_{2}\right|=4 \sqrt{3}, \frac{z_{1}}{z_{2}^{2}} \in \mathbf{R}$, then $\left|z_{1}\right|=$ | 4 | 73 | 1 |
math | The height of a straight pyramid is $m$; this pyramid is to be divided into three equal volume parts by two planes parallel to the base. What are the heights of these parts? | \frac{}{3}\sqrt[3]{9},\\frac{}{3}(\sqrt[3]{18}-\sqrt[3]{9}),\\frac{}{3}(3-\sqrt[3]{18}) | 38 | 48 |
math | Example 2 A glasses workshop in a factory has received a batch of tasks, requiring the processing of 6000 type $A$ parts and 2000 type $B$ parts. This workshop has 214 workers, each of whom can process 3 type $B$ parts in the time it takes to process 5 type $A$ parts. These people are divided into two groups, both work... | 137 | 117 | 3 |
math | 4.46. Find a four-digit number that, when divided by 131, gives a remainder of 112, and when divided by 132, gives a remainder of 98. | 1946 | 46 | 4 |
math | 12.61 There is a group of children, the sum of whose ages is 50 years. The oldest is 13 years old, and one of them is 10 years old. Excluding the 10-year-old child, the ages of the remaining children form an arithmetic sequence. How many children are there, and how old is each child?
(China Beijing High School Mathemat... | 5childrenwithages13,11,10,9,7 | 91 | 16 |
math | 1. At Cornthwaite H.S., many students enroll in an after-school arts program. The program offers a drama class and a music class. Each student enrolled in the program is in one class or both classes.
(a) This year, 41 students are in the drama class and 28 students are in the music class. If 15 students are in both cla... | 54,9,95 | 233 | 7 |
math | Example 6. If $u, v$ satisfy
$$
v=\sqrt{\frac{2 u-v}{4 u+3 v}}+\sqrt{\frac{v-2 u}{4 u+3 v}}+\frac{3}{2} \text {. }
$$
then $u^{2}-\cdots v+v^{2}=$ | \frac{27}{16} | 74 | 9 |
math | Let $\ell$ be a line and let points $A$, $B$, $C$ lie on $\ell$ so that $AB = 7$ and $BC = 5$. Let $m$ be the line through $A$ perpendicular to $\ell$. Let $P$ lie on $m$. Compute the smallest possible value of $PB + PC$.
[i]Proposed by Ankan Bhattacharya and Brandon Wang[/i] | 19 | 92 | 2 |
math | 1. Points $A$ and $B$ lie on a circle with center $O$ and radius 6, and point $C$ is equidistant from points $A, B$, and $O$. Another circle with center $Q$ and radius 8 is circumscribed around triangle $A C O$. Find $B Q$. Answer: 10 | 10 | 76 | 2 |
math | 4. In the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, $A B$ $=2, A A_{1}=A D=1$, points $E$, $F$, and $G$ are the midpoints of edges $A A_{1}$, $C_{1} D_{1}$, and $B C$ respectively. Then, the volume of the tetrahedron $B_{1}-E F G$ is $\qquad$ . | \frac{3}{8} | 112 | 7 |
math | Four, (50 points) Find all positive integers $n$, such that $\frac{n+1}{2}$ and $\frac{n^{2}+1}{2}$ are both perfect squares.
untranslated part:
(50 points) Find all positive integers $n$, such that $\frac{n+1}{2}$ and $\frac{n^{2}+1}{2}$ are both perfect squares. | 1,7 | 83 | 3 |
math | 10. Let $[a]$ denote the greatest integer not exceeding $a$, then the maximum positive integer solution of the equation $\left[\frac{x}{7}\right]=\left[\frac{x}{8}\right]+1$ is $\qquad$ . | 104 | 54 | 3 |
math | 4. (10 points) Write a three-digit number after 1220 to get a seven-digit number; if this seven-digit number is a multiple of 2014, then this three-digit number is $\qquad$ . | 484 | 51 | 3 |
math | Find all such integers $x$ that $x \equiv 3(\bmod 7), x^{2} \equiv 44\left(\bmod 7^{2}\right), x^{3} \equiv 111\left(\bmod 7^{3}\right)$. | x\equiv17(\bmod343) | 64 | 12 |
math | 5. When $1 \leqslant x \leqslant 2$, simplify
$$
\sqrt{x+2 \sqrt{x-1}}+\sqrt{x-2 \sqrt{x-1}}=
$$
$\qquad$ . | 2 | 53 | 1 |
math | 2. If
$$
a^{2}=1 \underbrace{00 \ldots 005}_{1990} \cdot \underbrace{11 \ldots 11}_{1991}+1
$$
determine the number $a$. | \underbrace{33\ldots334}_{1990} | 62 | 18 |
math | We have a deck containing 52 cards. Each card has a value among "1, 2, 3, $4,5,6,7,8,9,10$, jack, queen, king" and a suit among "hearts, diamonds, spades, clubs", such that, for each value and each suit, the deck contains a unique card with this value and this suit. A hand of 5 cards is a selection of 5 cards from this... | 624 | 136 | 3 |
math | 8. Given a tetrahedron $ABCD$ where $AB=CD=2a$, $AC=BD=BC=AD=\sqrt{10}$, then the range of values for $a$ is $\qquad$. | 0<<\sqrt{5} | 50 | 7 |
math | 34. A bag of fruit contains 10 fruits, with an even number of apples, at most two oranges, a multiple of 3 bananas, and at most one pear. How many types of these bagged fruits are there? | 11 | 49 | 2 |
math | 158. Find $x$ and $y$ from the equality:
a) $3 y + 5 x i = 15 - 7 i$;
b) $(2 x + 3 y) + (x - y) i = 7 + 6 i$. | -\frac{7}{5},5 | 62 | 8 |
math | 4. Person A and Person B agreed to meet at a restaurant for dinner. Due to the restaurant's popularity, when A arrived, they took a waiting number and waited for B. B arrived a while later but didn't see A, so they also took a waiting number. While waiting, B saw A, and they both showed their waiting numbers. They foun... | 35 | 141 | 2 |
math | Problem 3. Sasha's collection consists of coins and stickers, with fewer coins than stickers, but at least one coin is present. Sasha chose a positive number $t>1$ (not necessarily an integer). If he increases the number of coins by a factor of $t$ and leaves the number of stickers the same, his collection will have 10... | 34or66 | 130 | 5 |
math | Let's inscribe a cone in a sphere, such that the volume of the cone is maximum. | \frac{4}{3}R,\quad\frac{2}{3}R\sqrt{2} | 20 | 23 |
math | 1. Given the sets
$$
\begin{array}{c}
M=\{x, x y, \lg (x y)\} \\
N=\{0,|x|, y\},
\end{array}
$$
and
and $M=N$, then
$\left(x+\frac{1}{y}\right)+\left(x^{2}+\frac{1}{y^{2}}\right)+\left(x^{3}+\frac{1}{y^{3}}\right)+\cdots+\left(x^{2001}+\frac{1}{y^{2001}}\right)$ is equal to . $\qquad$ | -2 | 142 | 2 |
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 | Task B-1.8. Determine all three-digit natural numbers that are divisible by 7, and when divided by 9 give a remainder of 5. | n\in{140,203,266,329,392,455,518,581,644,707,770,833,896,959} | 33 | 60 |
math | $\textbf{Problem 4.}$ The number of perfect inhabitants of a city was a perfect square, in other words, a whole number squared. with $100$ people plus the new number of inhabitants turned out to be a perfect square plus one. Now, with another increase of $100$ people, the number of inhabitants is again a perfect square... | 49^2 | 86 | 4 |
math | Example 2 Find all positive integers $n$ such that $n=d(n)^{2}$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | n=1 \text{ or } 9 | 46 | 10 |
math | 11. (China Mathematical Olympiad) Let $S=\{1,2,3, \cdots, 98\}$, find the smallest natural number $n$, such that in any $n$-element subset of $S$, one can always select 10 numbers, and no matter how these 10 numbers are divided into two groups, there is always a number in one group that is coprime with the other four n... | 50 | 114 | 2 |
math | 3. (7 points) Given an arithmetic progression. The sum of its first 10 terms is 60, and the sum of its first 20 terms is 320. What can the 15th term of this progression be? | 25 | 54 | 2 |
math | 6・151 Let $k$ be a positive integer, find all polynomials
$$p(x)=a_{0}+a_{1} x+\cdots \cdot+a_{n} x^{n}$$
such that $a_{i}$ are real numbers $(i=0,1,2, \cdots, n)$, and satisfy the equation
$$p(p(x))=[p(x)]^{k}$$ | p(x)=x^{k} \quad(k \geqslant 1) | 92 | 18 |
math | 1. Find all pairs of natural numbers $a, b$ for which the set equality holds
$$
\{a \cdot[a, b], b \cdot(a, b)\}=\{45,180\}
$$
where $(x, y)$ and $[x, y]$ denote the greatest common divisor and the least common multiple of numbers $x$ and $y$, respectively. | =2,b=45=6,b=15 | 85 | 12 |
math | 11. A. Given the equation in terms of $x$
$$
\frac{x^{2}+k x+3}{x-1}=3 x+k
$$
has only one positive real solution. Find the range of real values for $k$. | k=-\frac{33}{8}ork=-4ork\geqslant-3 | 55 | 21 |
math | Three. (20 points) Place the 2004 positive integers 1, 2, $\cdots$, 2004 randomly on a circle. By counting the parity of all adjacent 3 numbers, it is found that there are 600 groups where all 3 numbers are odd, and 500 groups where exactly 2 numbers are odd. How many groups have exactly 1 odd number? How many groups h... | 206, 698 | 100 | 8 |
math | Example 3 Let $g(k)$ denote the greatest odd divisor of the positive integer $k$
$$
\begin{aligned}
\text { (for example, } g(3) & =3, g(20)=5) . \text { Find } \\
f(n) & =g(1)+g(2)+g(3)+\cdots+g\left(2^{n}\right) .
\end{aligned}
$$ | \frac{4^{n}+2}{3} | 95 | 12 |
math | 89. A white ball is placed into an urn containing two balls, after which one ball is randomly drawn from it. Find the probability that the drawn ball will be white, if all possible assumptions about the initial composition of the balls (by color) are equally likely. | \frac{2}{3} | 55 | 7 |
math | 1. The system of equations in $x, y, z$
$$
\left\{\begin{array}{l}
x y+y z+z x=1, \\
5 x+8 y+9 z=12
\end{array}\right.
$$
all real solutions $(x, y, z)$ are $\qquad$ | \left(1, \frac{1}{2}, \frac{1}{3}\right) | 72 | 21 |
math | 3B. Solve the inequality:
$$
\frac{1}{2^{2 x}+3} \geq \frac{1}{2^{x+2}-1}
$$ | x\in(-\infty,-2)\cup{1} | 39 | 14 |
math | 1. The infantry column stretched out to 1 km. Sergeant Kim, riding out on a gyro-scooter from the end of the column, reached its beginning and returned to the end. The infantrymen walked $4 / 3$ km during this time. How far did the sergeant travel during this time? | \frac{8}{3} | 64 | 7 |
math | 230. Mental Arithmetic. In order to test his pupils' ability in mental arithmetic, Rakebrain one morning asked them to do the following:
- Find two whole numbers (each less than 10), the sum of whose squares plus their product would give a perfect square.
The answer was soon found. | (3,5)(7,8) | 64 | 9 |
math | Example 1 Remove the big and small jokers from a deck of cards, and randomly draw five cards from the remaining 52 cards. The probability that at least two of the cards have the same number (or letter $K, Q, J, A$) is $\qquad$ (require calculating the numerical value of this probability, accurate to 0.01). | 0.49 | 78 | 4 |
math | 1. Solve the inequality
$$
\frac{x^{2}-4 x+4}{x^{2}-6 x+9}+\frac{x-2}{x-3}-12<0
$$
in the set of real numbers $\mathbb{R}$ | (-\infty,\frac{14}{5})\cup(\frac{7}{2},+\infty) | 57 | 25 |
math | Find the maximal $x$ such that the expression $4^{27} + 4^{1000} + 4^x$ is the exact square.
| x = 1972 | 36 | 8 |
math | 1. Find all even natural numbers $n$ for which
$$
-53<\frac{2009}{53-n}<53-n
$$ | 2,4,6,8,92,94,96 | 36 | 16 |
math | Four, (50 points) Find all positive integers $a, b$ such that the polynomial
$$
f(x)=\frac{x^{5}+a}{b}
$$ | b=1, a \text{ being any integer, or } b=11, a \text{ being any positive integer that is congruent to } \pm 1 \text{ modulo } b | 38 | 43 |
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