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(7) Let $z \in \mathbf{C}$, and satisfy $\|z-\mathrm{i}\| \leqslant 1, A=\operatorname{Re}(z)\left(\|z-\mathrm{i}\|^{2}-1\right)$. Find $\max A$ and $\min A$.	<think> Okay, so I have this problem here: Let z be a complex number such that \|z - i\| ≤ 1. Then A is defined as Re(z) multiplied by (\|z - i\|² - 1). I need to find the maximum and minimum values of A. Hmm, complex numbers and their moduli... Let me break this down step by step. First, let's recall that any complex num...	[ { "from": "human", "value": "(7) Let $z \\in \\mathbf{C}$, and satisfy $\|z-\\mathrm{i}\| \\leqslant 1, A=\\operatorname{Re}(z)\\left(\|z-\\mathrm{i}\|^{2}-1\\right)$. Find $\\max A$ and $\\min A$." }, { "from": "gpt", "value": "<think>\nOkay, so I have this problem here: Let z be a complex number s...
(3) For a positive integer $n$, if $n=p \cdot q\left(p \geqslant q, p, q \in \mathbf{N}^{*}\right)$, when $p-q$ is the smallest, we call $p \cdot q$ the "best factorization" of $n$, and define $f(n)=\frac{q}{p}$. For example, the factorizations of 12 are $12 \times 1, 6 \times 2, 4 \times 3$, where $4 \times 3$ is the ...	<think> Okay, let's see. I need to check which of the four statements about the function f(n) are correct. The function f(n) is defined as q/p where p and q are factors of n such that p ≥ q and p - q is minimized. This is called the "best factorization" of n. The example given is 12, where the best factorization is 4×3...	[ { "from": "human", "value": "(3) For a positive integer $n$, if $n=p \\cdot q\\left(p \\geqslant q, p, q \\in \\mathbf{N}^{*}\\right)$, when $p-q$ is the smallest, we call $p \\cdot q$ the \"best factorization\" of $n$, and define $f(n)=\\frac{q}{p}$. For example, the factorizations of 12 are $12 \\times 1,...
3. Before the trial run of one of the units of the hydropower station under construction, it was found that a fishing net is located $S$ km upstream from the dam. The river current speed is $v$ km/h. The hydropower station workers decided to go there by boat. Removing the net will take 5 minutes. What should be the boa...	<think> Okay, let's see. So the problem is about a hydropower station needing to remove a fishing net that's S kilometers upstream from the dam. The river current speed is v km/h. The workers are taking a boat, and they need to go up to the net, remove it (which takes 5 minutes), and come back, all within 45 minutes. W...	[ { "from": "human", "value": "3. Before the trial run of one of the units of the hydropower station under construction, it was found that a fishing net is located $S$ km upstream from the dam. The river current speed is $v$ km/h. The hydropower station workers decided to go there by boat. Removing the net wi...
Four. (30 points) Given the sequence $\left\{a_{n}\right\}$ satisfies $$ a_{1}=\frac{1}{2}, a_{n}=2 a_{n} a_{n+1}+3 a_{n+1}\left(n \in \mathbf{N}_{+}\right) \text {. } $$ (1) Find the general term formula of the sequence $\left\{a_{n}\right\}$; (2) If the sequence $\left\{b_{n}\right\}$ satisfies $b_{n}=1+\frac{1}{a_{n...	<think> Okay, let's see. I need to solve this problem about the sequence {a_n}. There are two parts: first, finding the general term formula, and second, dealing with another sequence {b_n} related to {a_n} and finding the maximum integer m such that a certain inequality holds for all n ≥ 2. Let me start with part (1)....	[ { "from": "human", "value": "Four. (30 points) Given the sequence $\\left\\{a_{n}\\right\\}$ satisfies\n$$\na_{1}=\\frac{1}{2}, a_{n}=2 a_{n} a_{n+1}+3 a_{n+1}\\left(n \\in \\mathbf{N}_{+}\\right) \\text {. }\n$$\n(1) Find the general term formula of the sequence $\\left\\{a_{n}\\right\\}$;\n(2) If the sequ...
In a unit cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, if we use the planes $A B_{1} C$, $B C_{1} D$, $C D_{1} A$, $D A_{1} B$, $A_{1} B C_{1}$, $B_{1} C D_{1}$, $C_{1} D A_{1}$, and $D_{1} A B_{1}$ to cut this unit cube, then the volume of the part containing the center of the cube is $\qquad$ .	<think> Okay, let's try to figure out this cube problem. So, we have a unit cube, right? That means all edges are 1 unit long. The cube is labeled ABCD-A₁B₁C₁D₁. I need to visualize this first. Let me sketch a cube in my mind with the base ABCD and the top face A₁B₁C₁D₁. Each vertex is connected vertically, so A is con...	[ { "from": "human", "value": "In a unit cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, if we use the planes $A B_{1} C$, $B C_{1} D$, $C D_{1} A$, $D A_{1} B$, $A_{1} B C_{1}$, $B_{1} C D_{1}$, $C_{1} D A_{1}$, and $D_{1} A B_{1}$ to cut this unit cube, then the volume of the part containing the center of the cube ...
Example 24 (Problem from the 48th Slovenian Mathematical Olympiad) Find all five-digit numbers $\overline{a b c d e}$, such that the number is divisible by 9, and $\overline{a c e}-\overline{b d a}=760$.	<think> Alright, let's tackle this problem step by step. The goal is to find all five-digit numbers \(\overline{a b c d e}\) that are divisible by 9 and satisfy the condition \(\overline{a c e} - \overline{b d a} = 760\). Hmm, okay. Let's break this down. First, let's recall that a number is divisible by 9 if the sum ...	[ { "from": "human", "value": "Example 24 (Problem from the 48th Slovenian Mathematical Olympiad) Find all five-digit numbers $\\overline{a b c d e}$, such that the number is divisible by 9, and $\\overline{a c e}-\\overline{b d a}=760$." }, { "from": "gpt", "value": "<think>\nAlright, let's tackl...
Example 6 Let real numbers $a$, $b$, $c$ satisfy $$ \left\{\begin{array}{l} a^{2}-b c-8 a+7=0, \\ b^{2}+c^{2}+b c-6 a+6=0 . \end{array}\right. $$ Find the range of values for $a$. (1995, Jilin Province Junior High School Mathematics Competition)	<think> Okay, so I need to find the range of possible values for the real number \( a \) given the system of equations: \[ \begin{cases} a^2 - bc - 8a + 7 = 0, \\ b^2 + c^2 + bc - 6a + 6 = 0. \end{cases} \] Hmm, let's see. The problem involves three variables \( a \), \( b \), and \( c \), but we only need to find th...	[ { "from": "human", "value": "Example 6 Let real numbers $a$, $b$, $c$ satisfy\n$$\n\\left\\{\\begin{array}{l}\na^{2}-b c-8 a+7=0, \\\\\nb^{2}+c^{2}+b c-6 a+6=0 .\n\\end{array}\\right.\n$$\n\nFind the range of values for $a$.\n(1995, Jilin Province Junior High School Mathematics Competition)" }, { "f...
Ahn chooses a two-digit integer, subtracts it from 200, and doubles the result. What is the largest number Ahn can get? $\text{(A)}\ 200 \qquad \text{(B)}\ 202 \qquad \text{(C)}\ 220 \qquad \text{(D)}\ 380 \qquad \text{(E)}\ 398$	<think> Okay, let me try to figure out this problem step by step. So, Ahn chooses a two-digit integer. Let's denote that two-digit integer as \( n \). Then, she subtracts it from 200 and doubles the result. We need to find the largest number she can get from this operation. The answer choices are given from A to E, wit...	[ { "from": "human", "value": "Ahn chooses a two-digit integer, subtracts it from 200, and doubles the result. What is the largest number Ahn can get?\n$\\text{(A)}\\ 200 \\qquad \\text{(B)}\\ 202 \\qquad \\text{(C)}\\ 220 \\qquad \\text{(D)}\\ 380 \\qquad \\text{(E)}\\ 398$" }, { "from": "gpt", ...
Example 4.28 Use red, blue, and green to color a $1 \times n$ chessboard, with each cell colored in one color. Find the number of coloring methods $e_{n}$ such that the number of cells colored red and blue are both even.	<think> Okay, so I need to figure out how many ways there are to color a 1xN chessboard using red, blue, and green, such that the number of red cells and the number of blue cells are both even. Hmm. Let me try to break this down step by step. First, let me understand the problem. Each cell in the chessboard can be col...	[ { "from": "human", "value": "Example 4.28 Use red, blue, and green to color a $1 \\times n$ chessboard, with each cell colored in one color. Find the number of coloring methods $e_{n}$ such that the number of cells colored red and blue are both even." }, { "from": "gpt", "value": "<think>\nOkay,...
Example 5 (IMO-31 Shortlist) The integer 9 can be expressed as the sum of two consecutive integers: $9=4+5$. At the same time, 9 can be written as the sum of consecutive natural numbers in two different ways: $9=4+5=2+3+4$. Is there a natural number $N$ such that: (1) It is the sum of 1990 consecutive integers; (2) It ...	<think> Okay, so I need to figure out if there's a natural number N that can be expressed both as the sum of 1990 consecutive integers and also can be expressed as the sum of consecutive integers in exactly 1990 different ways. Hmm, let's start by understanding the problem step by step. First, let's recall how express...	[ { "from": "human", "value": "Example 5 (IMO-31 Shortlist) The integer 9 can be expressed as the sum of two consecutive integers: $9=4+5$. At the same time, 9 can be written as the sum of consecutive natural numbers in two different ways: $9=4+5=2+3+4$.\nIs there a natural number $N$ such that:\n(1) It is th...
Find all prime number $p$ such that the number of positive integer pair $(x,y)$ satisfy the following is not $29$. [list] []$1\le x,y\le 29$ []$29\mid y^2-x^p-26$ [/list]	<think> Okay, let's try to tackle this problem step by step. The question is asking for all prime numbers \( p \) such that the number of positive integer pairs \( (x, y) \) satisfying the following conditions is not 29. The conditions are: 1. \( 1 \leq x, y \leq 29 \) 2. \( 29 \) divides \( y^2 - x^p - 26 \), which c...	[ { "from": "human", "value": "Find all prime number $p$ such that the number of positive integer pair $(x,y)$ satisfy the following is not $29$.\n[list]\n[]$1\\le x,y\\le 29$\n[]$29\\mid y^2-x^p-26$\n[/list]\n" }, { "from": "gpt", "value": "<think>\nOkay, let's try to tackle this problem step b...
Problem 5.4. At the end-of-the-year school dance, there were twice as many boys as girls. Masha counted that there were 8 fewer girls, besides herself, than boys. How many boys came to the dance?	<think> Okay, let's see. So, the problem is about the end-of-the-year school dance. There were twice as many boys as girls. Masha counted that there were 8 fewer girls, besides herself, than boys. We need to find how many boys came to the dance. Hmm, first, let me parse this problem step by step. Let me restate it to ...	[ { "from": "human", "value": "Problem 5.4. At the end-of-the-year school dance, there were twice as many boys as girls. Masha counted that there were 8 fewer girls, besides herself, than boys. How many boys came to the dance?" }, { "from": "gpt", "value": "<think>\nOkay, let's see. So, the proble...
Problem 6.8. In the country of Dragonia, there live red, green, and blue dragons. Each dragon has three heads, each of which always tells the truth or always lies. Moreover, each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each of them said: - 1st head: “To my ...	<think> Okay, so I need to figure out the maximum number of red dragons that could be at the table in Dragonia. Let's start by understanding the problem step by step. First, each dragon has three heads, and each head is either a truth-teller or a liar. Also, each dragon has at least one truthful head. The dragons are ...	[ { "from": "human", "value": "Problem 6.8. In the country of Dragonia, there live red, green, and blue dragons. Each dragon has three heads, each of which always tells the truth or always lies. Moreover, each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, an...
Suppose the side lengths of triangle $ABC$ are the roots of polynomial $x^3 - 27x^2 + 222x - 540$. What is the product of its inradius and circumradius?	<think> Okay, let's see. The problem says that the side lengths of triangle ABC are the roots of the polynomial x³ - 27x² + 222x - 540. We need to find the product of its inradius and circumradius. Hmm, okay. So first, I need to recall some formulas related to triangles, inradius, circumradius, and maybe Vieta's formul...	[ { "from": "human", "value": "Suppose the side lengths of triangle $ABC$ are the roots of polynomial $x^3 - 27x^2 + 222x - 540$. What is the product of its inradius and circumradius?" }, { "from": "gpt", "value": "<think>\nOkay, let's see. The problem says that the side lengths of triangle ABC ar...
Example 1 For all real numbers $p$ satisfying $0 \leqslant p \leqslant 4$, the inequality $x^{2}+p x>4 x+p-3$ always holds. Try to find the range of values for $x$.	<think> Okay, let's see. The problem says that for all real numbers p between 0 and 4, inclusive, the inequality x² + p x > 4x + p - 3 always holds. I need to find the range of x values that satisfy this inequality no matter what p is in that interval. Hmm, okay. So, first, maybe I should rewrite the inequality to make...	[ { "from": "human", "value": "Example 1 For all real numbers $p$ satisfying $0 \\leqslant p \\leqslant 4$, the inequality $x^{2}+p x>4 x+p-3$ always holds. Try to find the range of values for $x$." }, { "from": "gpt", "value": "<think>\nOkay, let's see. The problem says that for all real numbers ...
5. The number of $n$ that makes each interior angle of a regular $n$-sided polygon an even number of degrees is ( ). (A) 15 (B) 16 (C) 17 (D) 18	<think> Okay, let me try to figure out this problem. So the question is asking for the number of regular n-sided polygons where each interior angle is an even number of degrees. The options are 15, 16, 17, or 18. Hmm. Alright, let's start by recalling the formula for the interior angle of a regular n-sided polygon. I ...	[ { "from": "human", "value": "5. The number of $n$ that makes each interior angle of a regular $n$-sided polygon an even number of degrees is ( ).\n(A) 15\n(B) 16\n(C) 17\n(D) 18" }, { "from": "gpt", "value": "<think>\nOkay, let me try to figure out this problem. So the question is asking for the...
61. Given $a \neq 0, S_{1}=a, S_{2}=\frac{2}{S_{1}}, \cdots, S_{2018}=\frac{2}{S_{2017}}$, express $S_{2018}=$ in terms of $a$	<think> Okay, let's see. I need to find S_{2018} in terms of a for this sequence where each term is defined by S_{n} = 2 / S_{n-1}, starting with S_1 = a. Hmm, let's break it down step by step. First, let's write out the first few terms to see if there's a pattern. S₁ = a. That's given. Then S₂ = 2 / S₁ = 2 / a. ...	[ { "from": "human", "value": "61. Given $a \\neq 0, S_{1}=a, S_{2}=\\frac{2}{S_{1}}, \\cdots, S_{2018}=\\frac{2}{S_{2017}}$, express $S_{2018}=$ in terms of $a$" }, { "from": "gpt", "value": "<think>\nOkay, let's see. I need to find S_{2018} in terms of a for this sequence where each term is defi...
Let $A$, $B$, $C$, $D$ be four points on a line in this order. Suppose that $AC = 25$, $BD = 40$, and $AD = 57$. Compute $AB \cdot CD + AD \cdot BC$. [i]Proposed by Evan Chen[/i]	<think> Okay, let's see. I have four points on a line: A, B, C, D in that order. The distances given are AC = 25, BD = 40, and AD = 57. I need to compute AB · CD + AD · BC. Hmm. Alright, let's try to visualize this first. Since they're on a line in the order A-B-C-D, the distances between them should add up in certain ...	[ { "from": "human", "value": "Let $A$, $B$, $C$, $D$ be four points on a line in this order. Suppose that $AC = 25$, $BD = 40$, and $AD = 57$. Compute $AB \\cdot CD + AD \\cdot BC$.\n\n[i]Proposed by Evan Chen[/i]" }, { "from": "gpt", "value": "<think>\nOkay, let's see. I have four points on a l...
Francisco has 3 daughters: Alina, Valentina, and Civela. An interesting fact is that all three daughters were born on March 18. Today, March 18, 2014, is their birthday. Noting another curious fact, Francisco says: - Alina, your age is now double the age of Valentina. a) Show that this could never have happened befor...	<think> Okay, let's try to tackle part a) first. So Francisco has three daughters: Alina, Valentina, and Civela. All three were born on March 18, and today is March 18, 2014, their birthday. Francisco says that Alina's age is now double Valentina's age. We need to show that this could never have happened before and tha...	[ { "from": "human", "value": "Francisco has 3 daughters: Alina, Valentina, and Civela. An interesting fact is that all three daughters were born on March 18. Today, March 18, 2014, is their birthday. Noting another curious fact, Francisco says:\n\n- Alina, your age is now double the age of Valentina.\n\na) S...
16. 2.3 * In $\{1000,1001, \cdots, 2000\}$, how many pairs of consecutive integers can be added without carrying over?	<think> Alright, let's try to figure out how to solve this problem: In the set of integers from 1000 to 2000, how many pairs of consecutive integers can be added without carrying over? Hmm, okay. So we need to look at each consecutive pair, like (1000, 1001), (1001, 1002), ..., up to (1999, 2000), and determine how man...	[ { "from": "human", "value": "16. 2.3 * In $\\{1000,1001, \\cdots, 2000\\}$, how many pairs of consecutive integers can be added without carrying over?" }, { "from": "gpt", "value": "<think>\nAlright, let's try to figure out how to solve this problem: In the set of integers from 1000 to 2000, how...
6. Let's find the range of the function $f(x)=g\left(g^{2}(x)\right)$, where $g(x)=\frac{3}{x^{2}-4 x+5}$. The function $g(x)=\frac{3}{x^{2}-4 x+5}=\frac{3}{(x-2)^{2}+1}$ is defined for all real numbers and takes all values from the interval $(0 ; 3]$. The function $g(x)$ reaches its maximum value at the point $x=2$, ...	<think> Okay, let's try to figure out the range of the function \( f(x) = g(g^{2}(x)) \), where \( g(x) = \frac{3}{x^2 - 4x + 5} \). Hmm, first, I need to understand the components involved here. Let me break it down step by step. First, let's analyze the function \( g(x) \). The problem statement mentions that \( g(x...	[ { "from": "human", "value": "6. Let's find the range of the function $f(x)=g\\left(g^{2}(x)\\right)$, where $g(x)=\\frac{3}{x^{2}-4 x+5}$.\n\nThe function $g(x)=\\frac{3}{x^{2}-4 x+5}=\\frac{3}{(x-2)^{2}+1}$ is defined for all real numbers and takes all values from the interval $(0 ; 3]$. The function $g(x)...
1. Let $a, b, c, d$ be integers, and $a<2b, b<3c, c<4d$. Given $d<100$, then the maximum possible value of $a$ is ( ). (A) 2367 (B) 2375 (C) 2391 (D) 2399	<think> Okay, so I have this problem here: There are four integers a, b, c, d. The constraints are a < 2b, b < 3c, c < 4d, and d < 100. I need to find the maximum possible value of a from the given options. The options are 2367, 2375, 2391, and 2399. Let me think through how to approach this step by step. First, since...	[ { "from": "human", "value": "1. Let $a, b, c, d$ be integers, and $a<2b, b<3c, c<4d$. Given $d<100$, then the maximum possible value of $a$ is ( ).\n(A) 2367\n(B) 2375\n(C) 2391\n(D) 2399" }, { "from": "gpt", "value": "<think>\nOkay, so I have this problem here: There are four integers a, b, c, ...
\section*{Problem 4 - 261014} Jürgen claims that there is a positional system with base \(m\) in which the following calculation is correct: \begin{tabular}{lllllll} & 7 & 0 & 1 &. & 3 & 4 \\ \hline 2 & 5 & 0 & 3 & & & \\ & 3 & 4 & 0 & 4 & & \\ \hline 3 & 0 & 4 & 3 & 4 & & \end{tabular} Determine all natural number...	<think> Okay, so I have this problem where Jürgen claims there's a positional system with base \( m \) where a certain addition is correct. The addition is written out in columns, like how we do addition in base 10, but here the base is unknown. My job is to figure out all natural numbers \( m \) for which this calcula...	[ { "from": "human", "value": "\\section*{Problem 4 - 261014}\n\nJürgen claims that there is a positional system with base \\(m\\) in which the following calculation is correct:\n\n\\begin{tabular}{lllllll} \n& 7 & 0 & 1 &. & 3 & 4 \\\\\n\\hline 2 & 5 & 0 & 3 & & & \\\\\n& 3 & 4 & 0 & 4 & & \\\\\n\\hline 3 & ...
5. At a fair, there is a very inviting game, as it can be played for free; the winner gets a prize. The agreed prize for the first four games is one coin, for the fifth it is two coins. Nicola has a probability of $\frac{2}{3}$ of winning the prize in each game and decides to play 5 games. What is the probability that ...	<think> Okay, so let me try to figure out this probability problem. Alright, Nicola is playing 5 games at a fair. The prize for the first four games is 1 coin each, and the fifth game gives 2 coins. Nicola has a 2/3 chance of winning each game, and we need to find the probability that she wins at least 4 coins. The ans...	[ { "from": "human", "value": "5. At a fair, there is a very inviting game, as it can be played for free; the winner gets a prize. The agreed prize for the first four games is one coin, for the fifth it is two coins. Nicola has a probability of $\\frac{2}{3}$ of winning the prize in each game and decides to p...
A softball team played ten games, scoring $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$ runs. They lost by one run in exactly five games. In each of their other games, they scored twice as many runs as their opponent. How many total runs did their opponents score? $\textbf{(A)}\ 35 \qquad\textbf{(B)}\ 40 \qqu...	<think> Alright, let's tackle this problem step by step. So, we have a softball team that played ten games. In each game, they scored runs ranging from 1 up to 10, right? That means in the first game, they scored 1 run, in the second game 2 runs, all the way up to the tenth game where they scored 10 runs. Now, the pr...	[ { "from": "human", "value": "A softball team played ten games, scoring $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$ runs. They lost by one run in exactly five games. In each of their other games, they scored twice as many runs as their opponent. How many total runs did their opponents score?\n$\\...
22. Consider a list of six numbers. When the largest number is removed from the list, the average is decreased by 1 . When the smallest number is removed, the average is increased by 1 . When both the largest and the smallest numbers are removed, the average of the remaining four numbers is 20 . Find the product of the...	<think> Okay, so I need to solve this problem about a list of six numbers. Let me read it again and make sure I understand all the parts. The problem says: When the largest number is removed, the average decreases by 1. When the smallest number is removed, the average increases by 1. When both the largest and smalles...	[ { "from": "human", "value": "22. Consider a list of six numbers. When the largest number is removed from the list, the average is decreased by 1 . When the smallest number is removed, the average is increased by 1 . When both the largest and the smallest numbers are removed, the average of the remaining fou...
17. (10 points) There are small sticks of lengths 1 cm, 2 cm, 3 cm, 4 cm, 5 cm, 6 cm, 7 cm, 8 cm, and 9 cm, with one stick of each length. Choose several sticks to form a square (no breaking allowed). How many different ways can this be done? Please draw four of these ways and label the lengths of the sticks.	<think> Okay, so I need to figure out how many different ways I can form a square using the sticks of lengths 1 cm to 9 cm, each one exactly once. Then I have to draw four of these ways and label the lengths. Hmm, let's start by understanding the problem. First, forming a square means that the total length of the stic...	[ { "from": "human", "value": "17. (10 points) There are small sticks of lengths 1 cm, 2 cm, 3 cm, 4 cm, 5 cm, 6 cm, 7 cm, 8 cm, and 9 cm, with one stick of each length. Choose several sticks to form a square (no breaking allowed). How many different ways can this be done? Please draw four of these ways and l...
7. Highway (from 7th grade, 3 points). A highway running from west to east intersects with $n$ equal roads, numbered from 1 to $n$ in order. Cars travel on these roads from south to north and from north to south. The probability that a car will approach the highway from each of these roads is $\frac{1}{n}$. Similarly, ...	<think> Okay, let me try to figure out this probability problem. So, there's a highway that goes from west to east, and it intersects with n equal roads. These roads are numbered from 1 to n in order, right? Cars are coming from both south to north and north to south on these roads. The probability that a car enters th...	[ { "from": "human", "value": "7. Highway (from 7th grade, 3 points). A highway running from west to east intersects with $n$ equal roads, numbered from 1 to $n$ in order. Cars travel on these roads from south to north and from north to south. The probability that a car will approach the highway from each of ...
$9 \cdot 23$ The largest of the following four numbers is (A) $\operatorname{tg} 48^{\circ}+\operatorname{ctg} 48^{\circ}$. (B) $\sin 48^{\circ}+\cos 48^{\circ}$. (C) $\operatorname{tg} 48^{\circ}+\cos 48^{\circ}$. (D) $\operatorname{ctg} 48^{\circ}+\sin 48^{\circ}$. (Chinese Junior High School Mathematics League, 1988...	<think> Okay, let's see. The problem is asking which of the four options (A to D) is the largest number. All options involve trigonometric functions of 48 degrees. The options are: (A) tan 48° + cot 48° (B) sin 48° + cos 48° (C) tan 48° + cos 48° (D) cot 48° + sin 48° Hmm. So, I need to compare these four expressions...	[ { "from": "human", "value": "$9 \\cdot 23$ The largest of the following four numbers is\n(A) $\\operatorname{tg} 48^{\\circ}+\\operatorname{ctg} 48^{\\circ}$.\n(B) $\\sin 48^{\\circ}+\\cos 48^{\\circ}$.\n(C) $\\operatorname{tg} 48^{\\circ}+\\cos 48^{\\circ}$.\n(D) $\\operatorname{ctg} 48^{\\circ}+\\sin 48^{...
$17 \cdot 143$ In $\triangle A B C$, $E$ is the midpoint of side $B C$, and $D$ is on side $A C$. If the length of $A C$ is $1, \angle B A C=60^{\circ}, \angle A B C=100^{\circ}, \angle A C B=20^{\circ}, \angle D E C=80^{\circ}$, then the area of $\triangle A B C$ plus twice the area of $\triangle C D E$ equals (A) $\f...	<think> Okay, so I need to solve this geometry problem. Let me start by understanding all the given information and what's being asked. We have triangle ABC. Here's the breakdown of the given data: - E is the midpoint of side BC. - D is a point on side AC. - The length of AC is 1. - Angles: ∠BAC = 60°, ∠ABC = 100°, ∠...	[ { "from": "human", "value": "$17 \\cdot 143$ In $\\triangle A B C$, $E$ is the midpoint of side $B C$, and $D$ is on side $A C$. If the length of $A C$ is $1, \\angle B A C=60^{\\circ}, \\angle A B C=100^{\\circ}, \\angle A C B=20^{\\circ}, \\angle D E C=80^{\\circ}$, then the area of $\\triangle A B C$ plu...
Through the midpoints $M$ and $N$ of the edges $A A_1$ and $C_1 D_1$ of the parallelepiped $A B C D A_1 B_1 C_1 D_1$, a plane parallel to the diagonal $B D$ of the base is drawn. Construct the section of the parallelepiped by this plane. In what ratio does it divide the diagonal $A_1 C$?	<think> Okay, let's try to figure out how to construct this section and find the ratio. First, I need to visualize the parallelepiped. A parallelepiped is like a 3D figure with six faces, each of which is a parallelogram. The base is ABCD, and the top face is A1B1C1D1. The edges connecting the base to the top face are ...	[ { "from": "human", "value": "Through the midpoints $M$ and $N$ of the edges $A A_1$ and $C_1 D_1$ of the parallelepiped $A B C D A_1 B_1 C_1 D_1$, a plane parallel to the diagonal $B D$ of the base is drawn. Construct the section of the parallelepiped by this plane. In what ratio does it divide the diagonal...
1. On a very long road, a race was organized. 20 runners started at different times, each running at a constant speed. The race continued until each runner had overtaken all the slower ones. The speeds of the runners who started first and last were the same, and the speeds of the others were different from theirs and d...	<think> Okay, so I need to figure out the possible number of overtakes in this race scenario. Let me start by understanding the problem again. There are 20 runners on a very long road. They all start at different times, each running at a constant speed. The race continues until each runner has overtaken all the slower ...	[ { "from": "human", "value": "1. On a very long road, a race was organized. 20 runners started at different times, each running at a constant speed. The race continued until each runner had overtaken all the slower ones. The speeds of the runners who started first and last were the same, and the speeds of th...
Example 4 As shown in Figure 2, there is a rectangular plot of land $A B C D$, and a rectangular flower bed $E F G H$ is to be built in the center, with its area being half of the area of this plot, and the width of the paths around the flower bed being equal. Now, without any measuring tools, only a sufficiently long ...	<think> Okay, let's see. We have a rectangular plot ABCD, and we need to build a rectangular flower bed EFGH in the center. The flower bed's area should be half of the plot's area. The paths around the flower bed have equal width, and we need to find that width using only a rope with equal markings, no measuring tools....	[ { "from": "human", "value": "Example 4 As shown in Figure 2, there is a rectangular plot of land $A B C D$, and a rectangular flower bed $E F G H$ is to be built in the center, with its area being half of the area of this plot, and the width of the paths around the flower bed being equal. Now, without any m...
3) Pietro and Paolo are celebrating their name day at a pizzeria with their friends. At the end of the dinner, the bill is divided equally among all those present, and each person should pay 12 Euros. However, with great generosity, the friends decide to treat Pietro and Paolo to dinner; the bill is divided equally amo...	<think> Okay, let me try to solve this problem step by step. So, Pietro and Paolo are celebrating their name day with friends at a pizzeria. When the bill comes, initially, everyone is supposed to split it equally, and each person pays 12 Euros. But then their friends decide to treat them, so only the friends pay, each...	[ { "from": "human", "value": "3) Pietro and Paolo are celebrating their name day at a pizzeria with their friends. At the end of the dinner, the bill is divided equally among all those present, and each person should pay 12 Euros. However, with great generosity, the friends decide to treat Pietro and Paolo t...
## 128. Math Puzzle $1 / 76$ A circus gave 200 performances in the last season, all of which were sold out. The number of seats in the circus tent is three times the fourth part of the number of performances given. a) How many program leaflets were printed if one fourth of the visitors bought a leaflet? b) How many ...	<think> Okay, let's tackle this math puzzle step by step. First, I need to understand what's being asked in each part. Let's start with part a). Problem a): We need to find out how many program leaflets were printed. The key information here is that one fourth of the visitors bought a leaflet. So, to find the tota...	[ { "from": "human", "value": "## 128. Math Puzzle $1 / 76$\n\nA circus gave 200 performances in the last season, all of which were sold out. The number of seats in the circus tent is three times the fourth part of the number of performances given.\n\na) How many program leaflets were printed if one fourth of...
10. On a plane, 2011 points are marked. We will call a pair of marked points $A$ and $B$ isolated if all other points are strictly outside the circle constructed on $A B$ as its diameter. What is the smallest number of isolated pairs that can exist?	<think> Okay, so I have this problem here: On a plane, 2011 points are marked. We need to find the smallest number of isolated pairs. An isolated pair A and B is defined as a pair where all the other points are strictly outside the circle constructed on AB as its diameter. Hmm, interesting. First, let me make sure I u...	[ { "from": "human", "value": "10. On a plane, 2011 points are marked. We will call a pair of marked points $A$ and $B$ isolated if all other points are strictly outside the circle constructed on $A B$ as its diameter. What is the smallest number of isolated pairs that can exist?" }, { "from": "gpt", ...
4. Given that $a$ is an integer, the equation concerning $x$ $$ \frac{x^{2}}{x^{2}+1}-\frac{4\|x\|}{\sqrt{x^{2}+1}}+2-a=0 $$ has real roots. Then the possible values of $a$ are	<think> Okay, let's see. I need to find the possible integer values of 'a' such that the given equation has real roots. The equation is: \[ \frac{x^{2}}{x^{2}+1} - \frac{4\|x\|}{\sqrt{x^{2}+1}} + 2 - a = 0 \] Hmm, first, maybe I should simplify this equation. Let me think. The equation involves terms with x squared and...	[ { "from": "human", "value": "4. Given that $a$ is an integer, the equation concerning $x$\n$$\n\\frac{x^{2}}{x^{2}+1}-\\frac{4\|x\|}{\\sqrt{x^{2}+1}}+2-a=0\n$$\n\nhas real roots. Then the possible values of $a$ are" }, { "from": "gpt", "value": "<think>\nOkay, let's see. I need to find the possibl...
$[$ [Arithmetic. Mental calculation, etc.] $]$ Authors: Gaityerri G.A., Grierenko D.: 2002 is a palindrome year, which means it reads the same backward as forward. The previous palindrome year was 11 years earlier (1991). What is the maximum number of non-palindrome years that can occur consecutively (between 1000 an...	<think> Okay, so I need to figure out the maximum number of consecutive non-palindrome years between 1000 and 9999. Hmm, palindrome years read the same forwards and backwards, like 2002 or 1991. The problem says the previous one before 2002 was 1991, which is 11 years earlier. So sometimes there's an 11-year gap betwee...	[ { "from": "human", "value": "$[$ [Arithmetic. Mental calculation, etc.] $]$\n\nAuthors: Gaityerri G.A., Grierenko D.:\n\n2002 is a palindrome year, which means it reads the same backward as forward. The previous palindrome year was 11 years earlier (1991). What is the maximum number of non-palindrome years ...
6. (1999 American High School Mathematics Exam) A regular tetrahedron has two spheres, one inscribed and one circumscribed. Additionally, there are four smaller spheres between each face of the tetrahedron and the circumscribed sphere, each tangent to the center of the face. Point $P$ is a point inside the circumscribe...	<think> Okay, let's see. The problem is about a regular tetrahedron with an inscribed sphere (inradius) and a circumscribed sphere (circumradius). Then there are four smaller spheres between each face and the circumscribed sphere, each tangent to the center of the face. We need to find the probability that a random poi...	[ { "from": "human", "value": "6. (1999 American High School Mathematics Exam) A regular tetrahedron has two spheres, one inscribed and one circumscribed. Additionally, there are four smaller spheres between each face of the tetrahedron and the circumscribed sphere, each tangent to the center of the face. Poi...
3.40 A red pencil costs 27 kopecks, a blue one - 23 kopecks. No more than 9 rubles 40 kopecks can be spent on purchasing pencils. It is necessary to purchase the maximum possible total number of red and blue pencils. At the same time, the number of red pencils should be as few as possible, but the number of blue pencil...	<think> Okay, let's try to figure out this problem step by step. So, we need to buy red and blue pencils with certain cost constraints, maximize the total number of pencils, but also make sure that the number of red pencils is as few as possible. Additionally, the difference between blue and red pencils shouldn't be mo...	[ { "from": "human", "value": "3.40 A red pencil costs 27 kopecks, a blue one - 23 kopecks. No more than 9 rubles 40 kopecks can be spent on purchasing pencils. It is necessary to purchase the maximum possible total number of red and blue pencils. At the same time, the number of red pencils should be as few a...