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A $4\times4\times4$ cube is composed of $64$ unit cubes. The faces of $16$ unit cubes are to be coloured red. A colouring is called interesting if there is exactly $1$ red unit cube in every $1\times1\times 4$ rectangular box composed of $4$ unit cubes. Determine the number of interesting colourings.	<think> Alright, so I have this problem about a 4x4x4 cube, made up of 64 unit cubes. The task is to color the faces of 16 unit cubes red such that in every 1x1x4 rectangular box (which contains 4 unit cubes), there's exactly one red cube. I need to find the number of such "interesting" colorings. First, let me try to...	[ { "from": "human", "value": "A $4\\times4\\times4$ cube is composed of $64$ unit cubes. The faces of $16$ unit cubes are to be coloured red. A colouring is called interesting if there is exactly $1$ red unit cube in every $1\\times1\\times 4$ rectangular box composed of $4$ unit cubes. Determine the number ...
2. In $\triangle A B C$, $\angle A B C=60^{\circ}$, points $O$ and $H$ are the circumcenter and orthocenter of $\triangle A B C$, respectively. Points $D$ and $E$ are on sides $B C$ and $A B$, respectively, such that $B D=B H, B E=B O$. Given that $B O=1$, find the area of $\triangle B D E$.	<think> Okay, let's try to figure out how to solve this problem. So, we have triangle ABC with angle ABC equal to 60 degrees. The circumcenter is O and the orthocenter is H. Points D and E are on sides BC and AB respectively, such that BD equals BH and BE equals BO. We're given that BO is 1, and we need to find the are...	[ { "from": "human", "value": "2. In $\\triangle A B C$, $\\angle A B C=60^{\\circ}$, points $O$ and $H$ are the circumcenter and orthocenter of $\\triangle A B C$, respectively. Points $D$ and $E$ are on sides $B C$ and $A B$, respectively, such that $B D=B H, B E=B O$. Given that $B O=1$, find the area of $...
6. (15 points) The radar received the signal reflected from the target after 6 microseconds. Determine the distance to the target, given that the speed of the radar signal is $300000 \mathrm{k} \nu / \mathrm{c}$. Note that one microsecond is one millionth of a second.	<think> Okay, so I need to figure out the distance to the target based on the time it took for the radar signal to bounce back. Hmm, the problem says the radar received the reflected signal after 6 microseconds. And the speed of the radar signal is given as 300,000 km per second. Wait, let me make sure I got the units ...	[ { "from": "human", "value": "6. (15 points) The radar received the signal reflected from the target after 6 microseconds. Determine the distance to the target, given that the speed of the radar signal is $300000 \\mathrm{k} \\nu / \\mathrm{c}$. Note that one microsecond is one millionth of a second." }, ...
12. After removing one element from the set $\{1!, 2!, \cdots, 24!\}$, the product of the remaining elements is exactly a perfect square.	<think> Alright, let's tackle this problem. So, we have the set of factorials from 1! up to 24!, and we need to remove one element such that the product of the remaining elements is a perfect square. Hmm, interesting. I need to figure out which factorial to remove. First off, let's recall that a perfect square is a n...	[ { "from": "human", "value": "12. After removing one element from the set $\\{1!, 2!, \\cdots, 24!\\}$, the product of the remaining elements is exactly a perfect square." }, { "from": "gpt", "value": "<think>\nAlright, let's tackle this problem. So, we have the set of factorials from 1! up to 24...
The function $\mathrm{f}$ on the positive integers satisfies $\mathrm{f}(1)=1, \mathrm{f}(2 \mathrm{n}+1)=\mathrm{f}(2 \mathrm{n})+1$ and $\mathrm{f}(2 \mathrm{n})=3$ $f(n)$. Find the set of all $m$ such that $m=f(n)$ for some $n$.	<think> Alright, let's try to tackle this problem step by step. The function f is defined on positive integers with some recursive rules. Let me write down what's given: 1. f(1) = 1 2. If n is a positive integer, then f(2n + 1) = f(2n) + 1 3. And f(2n) = 3f(n) We need to find all m such that m = f(n) for some n. So, ...	[ { "from": "human", "value": "The function $\\mathrm{f}$ on the positive integers satisfies $\\mathrm{f}(1)=1, \\mathrm{f}(2 \\mathrm{n}+1)=\\mathrm{f}(2 \\mathrm{n})+1$ and $\\mathrm{f}(2 \\mathrm{n})=3$ $f(n)$. Find the set of all $m$ such that $m=f(n)$ for some $n$." }, { "from": "gpt", "value...
19. (6 points) Use rectangular wooden blocks that are 5 cm long, 4 cm wide, and 3 cm high to stack into a cube. At least how many of these rectangular wooden blocks are needed. To maintain the original format and line breaks, the translation is as follows: 19. (6 points) Use rectangular wooden blocks that are 5 cm lo...	<think> Okay, so I need to figure out the minimum number of these wooden blocks required to stack into a cube. Each block is 5 cm long, 4 cm wide, and 3 cm high. Hmm, let's start by understanding what the problem is asking for. We need to create a cube using these rectangular blocks, which means all sides of the cube m...	[ { "from": "human", "value": "19. (6 points) Use rectangular wooden blocks that are 5 cm long, 4 cm wide, and 3 cm high to stack into a cube. At least how many of these rectangular wooden blocks are needed.\n\nTo maintain the original format and line breaks, the translation is as follows:\n\n19. (6 points) U...
1. Given the polynomial $$ P(x)=x^{2019}+2018 x^{2017}+2016 x+2015 $$ Find all integers $n$ such that $$ P(n) \mid P(P(n)+1) $$	<think> Okay, let's try to figure out this problem. So we have the polynomial P(x) = x²⁰¹⁹ + 2018x²⁰¹⁷ + 2016x + 2015, and we need to find all integers n such that P(n) divides P(P(n) + 1). Hmm, that seems a bit complex, but let's take it step by step. First, I remember that when we have something like P(n) dividing P...	[ { "from": "human", "value": "1. Given the polynomial\n\n$$\nP(x)=x^{2019}+2018 x^{2017}+2016 x+2015\n$$\n\nFind all integers $n$ such that\n\n$$\nP(n) \\mid P(P(n)+1)\n$$" }, { "from": "gpt", "value": "<think>\nOkay, let's try to figure out this problem. So we have the polynomial P(x) = x²⁰¹⁹ + ...
## 29. Leonie and Cats When old lady Leonie is asked how many cats she has, she melancholically replies: “Four fifths of my cats plus four fifths of a cat.” How many cats does she have? ![](https://cdn.mathpix.com/cropped/2024_05_21_fe999c0fe2ad81fc5164g-072.jpg?height=663&width=916&top_left_y=1459&top_left_x=570)	<think> Okay, let's try to figure out how many cats Leonie has. The problem says she answered with "Four fifths of my cats plus four fifths of a cat." Hmm, that sounds a bit tricky, but maybe if I break it down, I can solve it. First, let's translate her words into a mathematical equation. Let me denote the number of ...	[ { "from": "human", "value": "## 29. Leonie and Cats\n\nWhen old lady Leonie is asked how many cats she has, she melancholically replies: “Four fifths of my cats plus four fifths of a cat.” How many cats does she have?\n\n![](https://cdn.mathpix.com/cropped/2024_05_21_fe999c0fe2ad81fc5164g-072.jpg?height=663...
3. In astronomy, "parsec" is commonly used as a unit of distance. If in a right triangle $\triangle ABC$, $$ \angle ACB=90^{\circ}, CB=1.496 \times 10^{8} \text{ km}, $$ i.e., the length of side $CB$ is equal to the average distance between the Moon $(C)$ and the Earth $(B)$, then, when the size of $\angle BAC$ is 1 s...	<think> Okay, so I need to figure out what 1 parsec is in kilometers, right? Let me start by recalling the problem statement. We have a right triangle ABC with a right angle at C. The side CB is given as 1.496×10^8 km, which is the average distance between the Moon (C) and the Earth (B). Then, when the angle at A, whic...	[ { "from": "human", "value": "3. In astronomy, \"parsec\" is commonly used as a unit of distance. If in a right triangle $\\triangle ABC$,\n$$\n\\angle ACB=90^{\\circ}, CB=1.496 \\times 10^{8} \\text{ km},\n$$\n\ni.e., the length of side $CB$ is equal to the average distance between the Moon $(C)$ and the Ea...
$10 \cdot 25$ is the sum of an arithmetic sequence with an even number of terms. The sums of the odd-numbered terms and the even-numbered terms are 24 and 30, respectively. If the last term exceeds the first term by 10.5, then the number of terms in the arithmetic sequence is (A) 20. (B) 18. (C) 12. (D) 10. (E) 8. (24t...	<think> Okay, so I need to solve this problem about an arithmetic sequence. Let me read it again carefully. The problem states: 10 ⋅ 25 is the sum of an arithmetic sequence with an even number of terms. The sums of the odd-numbered terms and the even-numbered terms are 24 and 30, respectively. If the last term exceeds...	[ { "from": "human", "value": "$10 \\cdot 25$ is the sum of an arithmetic sequence with an even number of terms. The sums of the odd-numbered terms and the even-numbered terms are 24 and 30, respectively. If the last term exceeds the first term by 10.5, then the number of terms in the arithmetic sequence is\n...
Third question Given ten points in space, where no four points lie on the same plane. Connect some of the points with line segments. If the resulting figure contains no triangles and no spatial quadrilaterals, determine the maximum number of line segments that can be drawn.	<think> Okay, so I have this problem here: Given ten points in space, where no four points lie on the same plane. We need to connect some of the points with line segments such that the resulting figure contains no triangles and no spatial quadrilaterals. Then, we have to determine the maximum number of line segments th...	[ { "from": "human", "value": "Third question Given ten points in space, where no four points lie on the same plane. Connect some of the points with line segments. If the resulting figure contains no triangles and no spatial quadrilaterals, determine the maximum number of line segments that can be drawn." }...
3. Let $a, b \in \mathbf{C}, a^{2}+b^{2}=7, a^{3}+b^{3}=10$, and let $m, n$ be the maximum and minimum real values that $a+b$ can take, respectively. Then $m+n=$ $\qquad$	<think> Okay, so I need to find the maximum and minimum real values that \( a + b \) can take, given that \( a \) and \( b \) are complex numbers satisfying \( a^2 + b^2 = 7 \) and \( a^3 + b^3 = 10 \). Then, I have to add those two values, \( m + n \), to get the answer. Alright, let's start breaking this down step by...	[ { "from": "human", "value": "3. Let $a, b \\in \\mathbf{C}, a^{2}+b^{2}=7, a^{3}+b^{3}=10$, and let $m, n$ be the maximum and minimum real values that $a+b$ can take, respectively. Then $m+n=$ $\\qquad$" }, { "from": "gpt", "value": "<think>\nOkay, so I need to find the maximum and minimum real ...
For any two coprime positive integers $p, q$, define $f(i)$ to be the remainder of $p\cdot i$ divided by $q$ for $i = 1, 2,\ldots,q -1$. The number $i$ is called a[b] large [/b]number (resp. [b]small[/b] number) when $f(i)$ is the maximum (resp. the minimum) among the numbers $f(1), f(2),\ldots,f(i)$. Note that $1$ is ...	<think> Okay, let's tackle this problem step by step. The problem is about finding the least possible value of \( q \) given that there are exactly \( a \) large numbers and \( b \) small numbers among the integers from 1 to \( q-1 \). Here, \( p \) and \( q \) are coprime positive integers, and for each \( i \), \( f(...	[ { "from": "human", "value": "For any two coprime positive integers $p, q$, define $f(i)$ to be the remainder of $p\\cdot i$ divided by $q$ for $i = 1, 2,\\ldots,q -1$. The number $i$ is called a[b] large [/b]number (resp. [b]small[/b] number) when $f(i)$ is the maximum (resp. the minimum) among the numbers ...
Let $(a_n)_{n\geq 1}$ be a sequence such that $a_1=1$ and $3a_{n+1}-3a_n=1$ for all $n\geq 1$. Find $a_{2002}$. $\textbf{(A) }666\qquad\textbf{(B) }667\qquad\textbf{(C) }668\qquad\textbf{(D) }669\qquad\textbf{(E) }670$	<think> Okay, so I need to find the 2002nd term of this sequence. The sequence is defined by a1 equals 1, and then there's a recurrence relation: 3a_{n+1} - 3a_n = 1 for all n ≥ 1. The answer choices are from A to E, with numbers around 666 to 670. Let me try to figure this out step by step. First, let me write down w...	[ { "from": "human", "value": "Let $(a_n)_{n\\geq 1}$ be a sequence such that $a_1=1$ and $3a_{n+1}-3a_n=1$ for all $n\\geq 1$. Find $a_{2002}$.\n\n$\\textbf{(A) }666\\qquad\\textbf{(B) }667\\qquad\\textbf{(C) }668\\qquad\\textbf{(D) }669\\qquad\\textbf{(E) }670$" }, { "from": "gpt", "value": "<t...
1. Consider the sequence 10, 13, 16, 19, .... Calculate the 2016th term of the sequence.	<think> Okay, so I need to find the 2016th term of the sequence 10, 13, 16, 19, .... Let me think. First, let me recall what kind of sequence this is. The terms are increasing by 3 each time: 10, then 13 (which is 10 + 3), then 16 (13 + 3), then 19 (16 + 3), and so on. So this is an arithmetic sequence because the diff...	[ { "from": "human", "value": "1. Consider the sequence 10, 13, 16, 19, .... Calculate the 2016th term of the sequence." }, { "from": "gpt", "value": "<think>\nOkay, so I need to find the 2016th term of the sequence 10, 13, 16, 19, .... Let me think. First, let me recall what kind of sequence this...
A5. The steps are made of cubes. How many do we need if we want to make 9 steps? (A) 40 (B) 42 (C) 45 (D) 48 (E) 50 ![](https://cdn.mathpix.com/cropped/2024_06_07_c7870cece3ffb80afd9eg-2.jpg?height=189&width=197&top_left_y=2204&top_left_x=1720)	<think> Okay, let me try to figure out how many cubes are needed to make 9 steps. The problem says the steps are made of cubes, so I need to visualize what each step looks like. Maybe each step is like a staircase where each subsequent step has one more cube than the previous one? Hmm, not sure. Let me think. If it's ...	[ { "from": "human", "value": "A5. The steps are made of cubes. How many do we need if we want to make 9 steps?\n(A) 40\n(B) 42\n(C) 45\n(D) 48\n(E) 50\n\n![](https://cdn.mathpix.com/cropped/2024_06_07_c7870cece3ffb80afd9eg-2.jpg?height=189&width=197&top_left_y=2204&top_left_x=1720)" }, { "from": "gpt...
Let $ f : \mathbf{R} \to \mathbf{R} $ be a smooth function such that $f'(x)=f(1-x)$ for all $x$ and $f(0)=1$. Find $f(1)$.	<think> Okay, let's see. I need to find f(1) given that f'(x) = f(1 - x) for all real numbers x, and that f(0) = 1. Hmm. Alright, let's start by understanding what's given here. So, the function f is smooth, which means it's infinitely differentiable, right? That's good because we might have to take derivatives multi...	[ { "from": "human", "value": "Let $ f : \\mathbf{R} \\to \\mathbf{R} $ be a smooth function such that $f'(x)=f(1-x)$ for all $x$ and $f(0)=1$. Find $f(1)$." }, { "from": "gpt", "value": "<think>\nOkay, let's see. I need to find f(1) given that f'(x) = f(1 - x) for all real numbers x, and that f(0...
4. Let $a$ be a real number such that the graph of the function $y=f(x)=a \sin 2x + \cos 2x + \sin x + \cos x$ is symmetric about the line $x=-\pi$. Let the set of all such $a$ be denoted by $S$. Then $S$ is ( ). (A) empty set (B) singleton set (contains only one element) (C) finite set with more than one element (D) i...	<think> Okay, so I need to figure out for what real numbers \( a \) the graph of the function \( y = a \sin 2x + \cos 2x + \sin x + \cos x \) is symmetric about the line \( x = -\pi \). The options are whether the set \( S \) of such \( a \) is empty, a singleton, finite with more than one element, or infinite. Hmm. F...	[ { "from": "human", "value": "4. Let $a$ be a real number such that the graph of the function $y=f(x)=a \\sin 2x + \\cos 2x + \\sin x + \\cos x$ is symmetric about the line $x=-\\pi$. Let the set of all such $a$ be denoted by $S$. Then $S$ is ( ).\n(A) empty set\n(B) singleton set (contains only one element)...
Line segment $A D$ is divided into three segments by points $B$ and $C$, so that $A B: B C=1: 2$ and $B C: C D=6: 5$. The length of $A D$ is 56 units. What is the length of $A B$ ? ![](https://cdn.mathpix.com/cropped/2024_04_17_03a8986aaed00af3ee97g-1.jpg?height=87&width=762&top_left_y=2426&top_left_x=736)	<think> Alright, let's see. The problem is about a line segment AD divided into three parts by points B and C. The ratios given are AB:BC = 1:2 and BC:CD = 6:5. The total length of AD is 56 units, and we need to find the length of AB. Hmm, okay. So first, let me visualize this. AD is a straight line with points B and ...	[ { "from": "human", "value": "Line segment $A D$ is divided into three segments by points $B$ and $C$, so that $A B: B C=1: 2$ and $B C: C D=6: 5$. The length of $A D$ is 56 units. What is the length of $A B$ ?\n\n![](https://cdn.mathpix.com/cropped/2024_04_17_03a8986aaed00af3ee97g-1.jpg?height=87&width=762&...
25.38*. Can a regular triangle be cut into 1000000 convex polygons such that any straight line intersects no more than 40 of them?	<think> Okay, so the problem is: Can a regular triangle be cut into 1,000,000 convex polygons such that any straight line intersects no more than 40 of them? Hmm, that's a mouthful. Let me try to break it down step by step. First off, a regular triangle is just an equilateral triangle, right? All sides equal, all angl...	[ { "from": "human", "value": "25.38*. Can a regular triangle be cut into 1000000 convex polygons such that any straight line intersects no more than 40 of them?" }, { "from": "gpt", "value": "<think>\nOkay, so the problem is: Can a regular triangle be cut into 1,000,000 convex polygons such that ...
1. For any real number $x$, the quadratic trinomial $x^{2}+3 m x+m^{2}-$ $m+\frac{1}{4}$ is a perfect square, then $m=$ $\qquad$.	<think> Okay, so I have this quadratic trinomial: x² + 3mx + m² - m + 1/4. The problem says that for any real number x, this trinomial is a perfect square. I need to find the value of m. Hmm, let me think. First, if a quadratic is a perfect square, it should be expressible in the form (x + a)², right? Expanding that ...	[ { "from": "human", "value": "1. For any real number $x$, the quadratic trinomial $x^{2}+3 m x+m^{2}-$ $m+\\frac{1}{4}$ is a perfect square, then $m=$ $\\qquad$." }, { "from": "gpt", "value": "<think>\nOkay, so I have this quadratic trinomial: x² + 3mx + m² - m + 1/4. The problem says that for an...
204. Divide the number 10 into 2 parts, the difference of which is 5.	<think> Okay, so I need to divide the number 10 into two parts such that their difference is 5. Hmm, let me think. Let me start by understanding the problem. I have to split 10 into two numbers, let's call them x and y. The conditions are that when you add them together, they equal 10, and when you subtract one from th...	[ { "from": "human", "value": "204. Divide the number 10 into 2 parts, the difference of which is 5." }, { "from": "gpt", "value": "<think>\nOkay, so I need to divide the number 10 into two parts such that their difference is 5. Hmm, let me think. Let me start by understanding the problem. I have ...
$19 \cdot 28$ convex polygon, the interior angles less than $120^{\circ}$ cannot be more than (A) 2. (B) 3. (C) 4. (D) 5. (China Zhejiang Ningbo Junior High School Mathematics Competition, 1987)	<think> Okay, so I need to figure out the maximum number of interior angles less than 120 degrees in a convex polygon with 19 sides. The options are from A to D, with the answer likely being one of them. Let me start by recalling some properties of convex polygons. First, in any convex polygon, the sum of the interior...	[ { "from": "human", "value": "$19 \\cdot 28$ convex polygon, the interior angles less than $120^{\\circ}$ cannot be more than\n(A) 2.\n(B) 3.\n(C) 4.\n(D) 5.\n(China Zhejiang Ningbo Junior High School Mathematics Competition, 1987)" }, { "from": "gpt", "value": "<think>\nOkay, so I need to figure...
In the diagram, the radius of the larger circle is 3 times the radius of the smaller circle. What fraction of the area of the larger circle is not shaded? (A) $\frac{8}{9}$ (B) $\frac{2}{3}$ (D) $\frac{7}{9}$ (E) $\frac{1}{3}$ (C) $\frac{5}{6}$ ![](https://cdn.mathpix.com/cropped/2024_04_20_6027bc27089ed4fc493cg-012.j...	<think> Okay, let's see. The problem says that the radius of the larger circle is 3 times the radius of the smaller circle. We need to find what fraction of the area of the larger circle is not shaded. The answer choices are given, so maybe I can figure it out step by step. First, let me visualize the diagram. There's...	[ { "from": "human", "value": "In the diagram, the radius of the larger circle is 3 times the radius of the smaller circle. What fraction of the area of the larger circle is not shaded?\n(A) $\\frac{8}{9}$\n(B) $\\frac{2}{3}$\n(D) $\\frac{7}{9}$\n(E) $\\frac{1}{3}$\n(C) $\\frac{5}{6}$\n\n![](https://cdn.mathp...
Let $S_n$ be the sum of the reciprocals of the non-zero digits of the integers from 1 to $10^n$ inclusive. Find the smallest positive integer $n$ for which $S_n$ is an integer.	<think> Okay, so I need to find the smallest positive integer n such that Sₙ is an integer. Sₙ is defined as the sum of the reciprocals of the non-zero digits of all integers from 1 to 10ⁿ inclusive. Hmm, that sounds a bit complicated, but let's break it down step by step. First, let me make sure I understand the prob...	[ { "from": "human", "value": "Let $S_n$ be the sum of the reciprocals of the non-zero digits of the integers from 1 to $10^n$ inclusive. Find the smallest positive integer $n$ for which $S_n$ is an integer." }, { "from": "gpt", "value": "<think>\nOkay, so I need to find the smallest positive inte...
$13 \cdot 10$ On the coordinate plane, a point with both coordinates as integers is called an integer point. For any natural number $n$, connect the origin $O$ with the point $A_{n}(n, n+3)$. Let $f(n)$ denote the number of integer points on the line segment $O A_{n}$, excluding the endpoints. Find the value of $f(1)+f...	<think> Okay, so I need to solve this problem where for each natural number n from 1 to 1990, we connect the origin O(0,0) to the point A_n(n, n+3). Then, for each n, f(n) is the number of integer points on the line segment OA_n, excluding the endpoints. The task is to find the sum f(1) + f(2) + ... + f(1990). First,...	[ { "from": "human", "value": "$13 \\cdot 10$ On the coordinate plane, a point with both coordinates as integers is called an integer point. For any natural number $n$, connect the origin $O$ with the point $A_{n}(n, n+3)$. Let $f(n)$ denote the number of integer points on the line segment $O A_{n}$, excludin...
14. The graph of $\left(x^{2}+y^{2}-2 x\right)^{2}=2\left(x^{2}+y^{2}\right)^{2}$ meets the $x$-axis in $p$ different places and meets the $y$-axis in $q$ different places. What is the value of $100 p+100 q$ ?	<think> Okay, so I need to figure out how many times the graph of the equation (x² + y² - 2x)² = 2(x² + y²)² intersects the x-axis and y-axis. Then, multiply those counts p and q by 100 each and add them together. Let me start by understanding how to find these intersection points. First, when a graph intersects the x...	[ { "from": "human", "value": "14. The graph of $\\left(x^{2}+y^{2}-2 x\\right)^{2}=2\\left(x^{2}+y^{2}\\right)^{2}$ meets the $x$-axis in $p$ different places and meets the $y$-axis in $q$ different places.\nWhat is the value of $100 p+100 q$ ?" }, { "from": "gpt", "value": "<think>\nOkay, so I n...
4. If $\log _{a} \frac{2}{3}<1$, then the range of values for $a$ is $(\quad)$. A. $\left(\frac{2}{3}, 1\right)$ B. $\left(\frac{2}{3},+\infty\right)$ C. $\left(0, \frac{2}{3}\right) \cup(1,+\infty)$ D. $\left(0, \frac{2}{3}\right) \cup\left(\frac{2}{3},+\infty\right)$	<think> Okay, let's see. The problem is: If log base a of (2/3) is less than 1, then what is the range of values for a? The options are A through D. Hmm, logarithms can be tricky because the base affects the inequality direction depending on whether it's greater than 1 or between 0 and 1. First, I need to recall the ...	[ { "from": "human", "value": "4. If $\\log _{a} \\frac{2}{3}<1$, then the range of values for $a$ is $(\\quad)$.\nA. $\\left(\\frac{2}{3}, 1\\right)$\nB. $\\left(\\frac{2}{3},+\\infty\\right)$\nC. $\\left(0, \\frac{2}{3}\\right) \\cup(1,+\\infty)$\nD. $\\left(0, \\frac{2}{3}\\right) \\cup\\left(\\frac{2}{3},...
1. On a meadow, there are 3 anthills: $A, B$, and $C$. The distance between anthills $A$ and $B$ is 260 mm, and the distance between anthills $B$ and $C$ is 1200 mm, with $\measuredangle A B C=60^{\circ}$. An ant $a$ starts from anthill $A$ towards anthill $B$, moving in a straight line at a speed of $1 \frac{\text { m...	<think> Okay, so there's this problem about two ants starting from anthills A and B, moving towards anthills B and C respectively. I need to find the time when the distance between them is the smallest. Let me try to visualize this first. First, the setup: three anthills A, B, and C. The distance between A and B is 26...	[ { "from": "human", "value": "1. On a meadow, there are 3 anthills: $A, B$, and $C$. The distance between anthills $A$ and $B$ is 260 mm, and the distance between anthills $B$ and $C$ is 1200 mm, with $\\measuredangle A B C=60^{\\circ}$. An ant $a$ starts from anthill $A$ towards anthill $B$, moving in a str...
3.303. $1-\sin ^{2} \alpha-\sin ^{2} \beta+2 \sin \alpha \sin \beta \cos (\alpha-\beta)$.	<think> Okay, let me try to figure out how to simplify this expression: 1 - sin²α - sin²β + 2 sinα sinβ cos(α - β). Hmm, trigonometric identities can be tricky, but let's take it step by step. First, I remember that 1 - sin²θ is equal to cos²θ. Maybe I can apply that here. Let's rewrite the expression using that ident...	[ { "from": "human", "value": "3.303. $1-\\sin ^{2} \\alpha-\\sin ^{2} \\beta+2 \\sin \\alpha \\sin \\beta \\cos (\\alpha-\\beta)$." }, { "from": "gpt", "value": "<think>\nOkay, let me try to figure out how to simplify this expression: 1 - sin²α - sin²β + 2 sinα sinβ cos(α - β). Hmm, trigonometric...
9. Euler's conjecture was refuted by American mathematicians in 1960, who proved that there exists a positive integer $n$ such that $133^{5}+110^{5}+84^{5}+27^{5}=n^{5}$. Find $n$ when it is satisfied.	<think> Okay, so I need to find the value of n such that 133^5 + 110^5 + 84^5 + 27^5 equals n^5. This problem is related to Euler's conjecture, which was proven false by this example. Hmm, interesting. I remember that Euler conjectured that at least n nth powers are needed to sum to another nth power, but here they hav...	[ { "from": "human", "value": "9. Euler's conjecture was refuted by American mathematicians in 1960, who proved that there exists a positive integer $n$ such that $133^{5}+110^{5}+84^{5}+27^{5}=n^{5}$. Find $n$ when it is satisfied." }, { "from": "gpt", "value": "<think>\nOkay, so I need to find t...
The graph shows the fuel used per $100 \mathrm{~km}$ of driving for five different vehicles. Which vehicle would travel the farthest using 50 litres of fuel? (A) $U$ (B) $V$ (C) $W$ (D) $X$ (E) $Y$ ![](https://cdn.mathpix.com/cropped/2024_04_20_6027bc27089ed4fc493cg-059.jpg?height=431&width=380&top_left_y=1297&top_lef...	<think> Okay, let's see. The question is about which vehicle can travel the farthest using 50 liters of fuel, based on a graph showing fuel used per 100 km for five different vehicles. The options are U, V, W, X, Y. Hmm, since the graph isn't here, I need to figure out the reasoning process even without the visual data...	[ { "from": "human", "value": "The graph shows the fuel used per $100 \\mathrm{~km}$ of driving for five different vehicles. Which vehicle would travel the farthest using 50 litres of fuel?\n(A) $U$\n(B) $V$\n(C) $W$\n(D) $X$\n(E) $Y$\n\n![](https://cdn.mathpix.com/cropped/2024_04_20_6027bc27089ed4fc493cg-059...
14. Concept. In Anchuria, there are $K$ laws and $N$ ministers. The probability that a randomly chosen minister knows a randomly chosen law is $p$. One day, the ministers gathered for a council to write the Concept. If at least one minister knows a law, then this law will be considered in the Concept; otherwise, this l...	<think> Okay, so I need to solve part a) and b) of this probability problem about Anchuria's laws and ministers. Let me start by understanding the problem clearly. First, the setup: There are K laws and N ministers. Each minister has a probability p of knowing any given law. The events of a minister knowing different ...	[ { "from": "human", "value": "14. Concept. In Anchuria, there are $K$ laws and $N$ ministers. The probability that a randomly chosen minister knows a randomly chosen law is $p$. One day, the ministers gathered for a council to write the Concept. If at least one minister knows a law, then this law will be con...
Four. (20 points) Let $n$ be a positive integer greater than 1. If there exists a positive integer $i (i < n)$ such that $$ \sum_{a=1}^{i} a^{2}-\sum_{b=i+1}^{n} b^{2} $$ is a perfect square, then $n$ is called a "TI number." For example, by $\left(1^{2}+2^{2}+\cdots+9^{2}\right)-\left(10^{2}+11^{2}\right)=8^{2}$, we ...	<think> Okay, let's tackle this problem step by step. We need to determine if 46, 36, and 26 are TI numbers. A TI number is defined as a positive integer n > 1 for which there exists some positive integer i < n such that the difference between the sum of squares from 1 to i and the sum of squares from i+1 to n is a per...	[ { "from": "human", "value": "Four. (20 points) Let $n$ be a positive integer greater than 1. If there exists a positive integer $i (i < n)$ such that\n$$\n\\sum_{a=1}^{i} a^{2}-\\sum_{b=i+1}^{n} b^{2}\n$$\n\nis a perfect square, then $n$ is called a \"TI number.\" For example, by $\\left(1^{2}+2^{2}+\\cdots...
If $x=\frac{1}{4}$, which of the following has the largest value? (A) $x$ (B) $x^{2}$ (C) $\frac{1}{2} x$ (D) $\frac{1}{x}$ (E) $\sqrt{x}$	<think> Okay, so I need to figure out which of the options A through E has the largest value when x is 1/4. Let me take it step by step. First, let's write down what each option is: (A) x (B) x squared (C) half of x (D) reciprocal of x (E) square root of x Given that x = 1/4, I'll substitute this value into each op...	[ { "from": "human", "value": "If $x=\\frac{1}{4}$, which of the following has the largest value?\n(A) $x$\n(B) $x^{2}$\n(C) $\\frac{1}{2} x$\n(D) $\\frac{1}{x}$\n(E) $\\sqrt{x}$" }, { "from": "gpt", "value": "<think>\nOkay, so I need to figure out which of the options A through E has the largest ...
[ Inscribed and Circumscribed Circles ] [ Inscribed Angle Subtended by a Diameter ] A circle with radius 1 is circumscribed around triangle $A P K$. The extension of side $A P$ beyond vertex $P$ intercepts a segment $B K$ from the tangent to the circle at vertex $K$, and $B K$ is equal to 7. Find the area of triangle ...	<think> Okay, let's try to tackle this geometry problem. Hmm, so we have a triangle APK with a circumscribed circle of radius 1. The extension of side AP beyond point P intercepts a segment BK from the tangent at K, and BK is 7 units long. Also, angle ABK is arctg(2/7). We need to find the area of triangle APK. First,...	[ { "from": "human", "value": "[ Inscribed and Circumscribed Circles ] [ Inscribed Angle Subtended by a Diameter ]\n\nA circle with radius 1 is circumscribed around triangle $A P K$. The extension of side $A P$ beyond vertex $P$ intercepts a segment $B K$ from the tangent to the circle at vertex $K$, and $B K...
To [i]dissect [/i] a polygon means to divide it into several regions by cutting along finitely many line segments. For example, the diagram below shows a dissection of a hexagon into two triangles and two quadrilaterals: [img]https://cdn.artofproblemsolving.com/attachments/0/a/378e477bcbcec26fc90412c3eada855ae52b45.png...	<think> Okay, so I need to figure out for which integers n it's possible to dissect a regular n-sided polygon into integer-ratio right triangles. Hmm. Let me start by understanding the problem thoroughly. First, a regular n-gon. Dissecting it means cutting it into several regions using straight line segments. The key ...	[ { "from": "human", "value": "To [i]dissect [/i] a polygon means to divide it into several regions by cutting along finitely many line segments. For example, the diagram below shows a dissection of a hexagon into two triangles and two quadrilaterals:\n[img]https://cdn.artofproblemsolving.com/attachments/0/a/...
Let $a, b, c,$ and $d$ be positive integers such that $\gcd(a, b)=24$, $\gcd(b, c)=36$, $\gcd(c, d)=54$, and $70<\gcd(d, a)<100$. Which of the following must be a divisor of $a$? $\textbf{(A)} \text{ 5} \qquad \textbf{(B)} \text{ 7} \qquad \textbf{(C)} \text{ 11} \qquad \textbf{(D)} \text{ 13} \qquad \textbf{(E)} \tex...	<think> Okay, let's try to figure out this problem step by step. So, we have four positive integers: a, b, c, and d. The problem gives us some greatest common divisors (gcds) between pairs of these numbers and tells us that the gcd of d and a is between 70 and 100. Then we need to determine which of the given options m...	[ { "from": "human", "value": "Let $a, b, c,$ and $d$ be positive integers such that $\\gcd(a, b)=24$, $\\gcd(b, c)=36$, $\\gcd(c, d)=54$, and $70<\\gcd(d, a)<100$. Which of the following must be a divisor of $a$?\n\n$\\textbf{(A)} \\text{ 5} \\qquad \\textbf{(B)} \\text{ 7} \\qquad \\textbf{(C)} \\text{ 11} ...
There are 20 people - 10 boys and 10 girls. How many ways are there to form a company where the number of boys and girls is equal? #	<think> Okay, let's see. The problem is about forming a company from 20 people—10 boys and 10 girls—where the number of boys and girls is equal. I need to figure out how many different ways this can be done. Hmm, so the company must have an equal number of boys and girls, but how many people in total? Wait, the problem...	[ { "from": "human", "value": "There are 20 people - 10 boys and 10 girls. How many ways are there to form a company where the number of boys and girls is equal?\n\n#" }, { "from": "gpt", "value": "<think>\nOkay, let's see. The problem is about forming a company from 20 people—10 boys and 10 girls...
6. A rectangular piece of land enclosed by fences has a length and width of $52 \mathrm{~m}$ and $24 \mathrm{~m}$, respectively. An agricultural science technician wants to divide this land into several congruent square test plots. The land must be fully divided, and the sides of the squares must be parallel to the bou...	<think> Okay, so I have this problem here where there's a rectangular piece of land that's 52 meters long and 24 meters wide. They want to divide this land into congruent square test plots. The squares have to be aligned with the sides of the rectangle, so no rotating them or anything. The catch is that there's only 20...	[ { "from": "human", "value": "6. A rectangular piece of land enclosed by fences has a length and width of $52 \\mathrm{~m}$ and $24 \\mathrm{~m}$, respectively. An agricultural science technician wants to divide this land into several congruent square test plots. The land must be fully divided, and the sides...