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Three vertices of parallelogram $PQRS$ are $P(-3,-2), Q(1,-5), R(9,1)$ with $P$ and $R$ diagonally opposite. The sum of the coordinates of vertex $S$ is: $\textbf{(A)}\ 13 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 11 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 9$	<think> Okay, so I need to find the coordinates of vertex S in the parallelogram PQRS, where three vertices are given: P(-3,-2), Q(1,-5), and R(9,1). P and R are diagonally opposite, which means S and Q must also be diagonally opposite. The question asks for the sum of the coordinates of S once we find them. The answer...	[ { "from": "human", "value": "Three vertices of parallelogram $PQRS$ are $P(-3,-2), Q(1,-5), R(9,1)$ with $P$ and $R$ diagonally opposite. \nThe sum of the coordinates of vertex $S$ is:\n$\\textbf{(A)}\\ 13 \\qquad \\textbf{(B)}\\ 12 \\qquad \\textbf{(C)}\\ 11 \\qquad \\textbf{(D)}\\ 10 \\qquad \\textbf{(E)}...
In an equilateral triangle $A B C$ with a side length of $8 \mathrm{~cm}$, point $D$ is the midpoint of side $B C$ and point $E$ is the midpoint of side $A C$. Point $F$ lies on segment $B C$ such that the area of triangle $A B F$ is equal to the area of quadrilateral $A B D E$. Calculate the length of segment $B F$. ...	<think> Okay, let's tackle this geometry problem. So, we have an equilateral triangle ABC with each side being 8 cm. Point D is the midpoint of BC, and point E is the midpoint of AC. We need to find the length of segment BF, where point F is on BC such that the area of triangle ABF is equal to the area of quadrilateral...	[ { "from": "human", "value": "In an equilateral triangle $A B C$ with a side length of $8 \\mathrm{~cm}$, point $D$ is the midpoint of side $B C$ and point $E$ is the midpoint of side $A C$. Point $F$ lies on segment $B C$ such that the area of triangle $A B F$ is equal to the area of quadrilateral $A B D E$...
Three generous friends, each with some money, redistribute the money as followed: Amy gives enough money to Jan and Toy to double each amount has. Jan then gives enough to Amy and Toy to double their amounts. Finally, Toy gives enough to Amy and Jan to double their amounts. If Toy had 36 dollars at the beginning and 3...	<think> Okay, let's see. There are three friends: Amy, Jan, and Toy. Toy started with $36 and ended with $36. The problem is asking for the total amount of money all three have together. The answer choices are numbers like 108, 180, 216, etc. So, the total is one of those, and we need to figure it out based on the tran...	[ { "from": "human", "value": "Three generous friends, each with some money, redistribute the money as followed:\nAmy gives enough money to Jan and Toy to double each amount has.\nJan then gives enough to Amy and Toy to double their amounts. \nFinally, Toy gives enough to Amy and Jan to double their amounts.\...
Solve the following system of equations: $$ \sqrt[3]{a^{y}}=\sqrt{\frac{a^{13}}{a^{x}}} $$ $$ \sqrt[7]{a^{3 x}} \cdot \sqrt[9]{a^{4 y}}=a^{7} $$	<think> Okay, let me try to solve this system of equations. Hmm, the equations involve exponents and radicals, so maybe I can simplify them using properties of exponents. Let me start with the first equation: the cube root of a to the power y equals the square root of (a^13 divided by a^x). Let me write that down more ...	[ { "from": "human", "value": "Solve the following system of equations:\n\n$$\n\\sqrt[3]{a^{y}}=\\sqrt{\\frac{a^{13}}{a^{x}}}\n$$\n\n$$\n\\sqrt[7]{a^{3 x}} \\cdot \\sqrt[9]{a^{4 y}}=a^{7}\n$$" }, { "from": "gpt", "value": "<think>\nOkay, let me try to solve this system of equations. Hmm, the equat...
5. In the sequence $\left\{a_{n}\right\}$, $a_{1}=2, a_{n}+a_{n+1}=1\left(n \in \mathbf{N}_{+}\right)$, let $S_{n}$ be the sum of the first $n$ terms of the sequence $a_{n}$, then the value of $S_{2017}-$ $2 S_{2018}+S_{2019}$ is $\qquad$	<think> Okay, let's see. I need to solve this problem involving a sequence and some sums. Let me start by understanding what's given and what's being asked. First, the sequence {a_n} has the first term a₁ = 2. Then there's a recursive relation: aₙ + a_{n+1} = 1 for any positive integer n. So, each term plus the next t...	[ { "from": "human", "value": "5. In the sequence $\\left\\{a_{n}\\right\\}$, $a_{1}=2, a_{n}+a_{n+1}=1\\left(n \\in \\mathbf{N}_{+}\\right)$, let $S_{n}$ be the sum of the first $n$ terms of the sequence $a_{n}$, then the value of $S_{2017}-$ $2 S_{2018}+S_{2019}$ is $\\qquad$" }, { "from": "gpt", ...
From a barrel, we pour out 4 liters of wine and replace it with 4 liters of water. From the resulting mixture, we again pour out 4 liters and replace it with 4 liters of water. We repeat this process a total of three times, and in the final result, there is 2.5 liters more water than wine. How many liters of wine were ...	<think> Okay, so I have this problem where we start with a barrel of wine. Then, we pour out 4 liters of wine and replace it with 4 liters of water. We do this three times total, and after the third time, there's 2.5 liters more water than wine. The question is, how many liters of wine were originally in the barrel? H...	[ { "from": "human", "value": "From a barrel, we pour out 4 liters of wine and replace it with 4 liters of water. From the resulting mixture, we again pour out 4 liters and replace it with 4 liters of water. We repeat this process a total of three times, and in the final result, there is 2.5 liters more water...
XLVII OM - I - Problem 2 A palindromic number is defined as a natural number whose decimal representation read from left to right is the same as when read from right to left. Let $ (x_n) $ be the increasing sequence of all palindromic numbers. Determine all prime numbers that are divisors of at least one of the differ...	<think> Okay, so I have this problem about palindromic numbers and their differences. The question is asking for all prime numbers that can divide at least one of the differences between consecutive palindromic numbers. Hmm, let's start by recalling what a palindromic number is. It's a number that reads the same forwar...	[ { "from": "human", "value": "XLVII OM - I - Problem 2\n\nA palindromic number is defined as a natural number whose decimal representation read from left to right is the same as when read from right to left. Let $ (x_n) $ be the increasing sequence of all palindromic numbers. Determine all prime numbers that...
4. The area of a rectangle is 180 units $^{2}$ and the perimeter is 54 units. If the length of each side of the rectangle is increased by six units, what is the area of the resulting rectangle?	<think> Okay, so I have this problem here about a rectangle. The area is 180 units squared and the perimeter is 54 units. Then, each side is increased by six units, and I need to find the new area. Hmm, let me think about how to approach this. First, I remember that for a rectangle, the area is length times width, and...	[ { "from": "human", "value": "4. The area of a rectangle is 180 units $^{2}$ and the perimeter is 54 units. If the length of each side of the rectangle is increased by six units, what is the area of the resulting rectangle?" }, { "from": "gpt", "value": "<think>\nOkay, so I have this problem here...
160. Solve the equation in integers $$ x^{2}+x=y^{4}+y^{3}+y^{2}+y $$	<think> Okay, so I need to solve the equation \( x^{2} + x = y^{4} + y^{3} + y^{2} + y \) for integers x and y. Hmm, let's see. Since both sides are polynomials, maybe I can factor them or find some bounds to limit possible values of y and x. First, let me try to understand the right-hand side. The right side is \( y^...	[ { "from": "human", "value": "160. Solve the equation in integers\n\n$$\nx^{2}+x=y^{4}+y^{3}+y^{2}+y\n$$" }, { "from": "gpt", "value": "<think>\nOkay, so I need to solve the equation \\( x^{2} + x = y^{4} + y^{3} + y^{2} + y \\) for integers x and y. Hmm, let's see. Since both sides are polynomia...
6. For natural numbers $a, b, c$, define the operation * such that it satisfies $$ \begin{array}{l} (a * b) * c=a (b c), \\ (a b)(a * c)=a (b+c) . \end{array} $$ Then the value of $3 4$ is ( ). (A) 12 (B) 64 (C) 81 (D) None of the above	<think> Okay, let me try to figure out this problem. So, we have an operation * defined for natural numbers a, b, c. There are two properties given: 1. (a * b) * c = a * (b c) 2. (a * b)(a * c) = a * (b + c) And we need to find the value of 3 * 4, with the options being 12, 64, 81, or none of the above. First, let ...	[ { "from": "human", "value": "6. For natural numbers $a, b, c$, define the operation * such that it satisfies\n$$\n\\begin{array}{l}\n(a * b) * c=a (b c), \\\\\n(a b)(a * c)=a (b+c) .\n\\end{array}\n$$\n\nThen the value of $3 4$ is ( ).\n(A) 12\n(B) 64\n(C) 81\n(D) None of the above" }, { "from...
7. In a free-throw test, a person only needs to make 3 shots to pass and does not need to continue shooting, but each person can shoot at most 5 times. The probability that a player with a shooting accuracy of $\frac{2}{3}$ passes the test is . $\qquad$	<think> Okay, so here's this problem about a free-throw test. Let me read it again: A person needs to make 3 shots to pass and doesn't need to continue shooting after that, but each person can shoot at most 5 times. The player has a shooting accuracy of 2/3. We need to find the probability that the player passes the te...	[ { "from": "human", "value": "7. In a free-throw test, a person only needs to make 3 shots to pass and does not need to continue shooting, but each person can shoot at most 5 times. The probability that a player with a shooting accuracy of $\\frac{2}{3}$ passes the test is . $\\qquad$" }, { "from": "...
$4 \cdot 103$ real numbers $\alpha, \beta$ satisfy the system of equations $$ \left\{\begin{array}{l} \alpha^{3}-3 \alpha^{2}+5 \alpha-17=0 \\ \beta^{3}-3 \beta^{2}+5 \beta+11=0 . \end{array}\right. $$ Find $\alpha+\beta$.	<think> Okay, let me see. So I have this problem where there are two real numbers, α and β, that satisfy these two cubic equations: First equation: α³ - 3α² + 5α - 17 = 0 Second equation: β³ - 3β² + 5β + 11 = 0 And I need to find α + β. Hmm, interesting. Both equations are cubic and look very similar except for the ...	[ { "from": "human", "value": "$4 \\cdot 103$ real numbers $\\alpha, \\beta$ satisfy the system of equations\n$$\n\\left\\{\\begin{array}{l}\n\\alpha^{3}-3 \\alpha^{2}+5 \\alpha-17=0 \\\\\n\\beta^{3}-3 \\beta^{2}+5 \\beta+11=0 .\n\\end{array}\\right.\n$$\n\nFind $\\alpha+\\beta$." }, { "from": "gpt", ...
20. Ruth and Sarah decide to have a race. Ruth runs around the perimeter of the pool shown in the diagram while Sarah swims lengths of the pool. Ruth runs three times as fast as Sarah swims. Sarah swims six lengths of the pool in the same time Ruth runs around the pool five times. How wide is the pool? A $25 \mathrm{~m...	<think> Okay, let's try to figure out the width of the pool. Hmm, the problem says Ruth runs around the perimeter of the pool while Sarah swims lengths of it. Ruth runs three times as fast as Sarah swims. Also, Sarah swims six lengths in the same time Ruth runs around the pool five times. The question is asking for the...	[ { "from": "human", "value": "20. Ruth and Sarah decide to have a race. Ruth runs around the perimeter of the pool shown in the diagram while Sarah swims lengths of the pool.\nRuth runs three times as fast as Sarah swims. Sarah swims six lengths of the pool in the same time Ruth runs around the pool five tim...
## Task 1 - 220831 On a day in 1981, Cathrin asks her grandfather about his birth year. The grandfather, a friend of puzzle questions, replied: "I am older than 65 years, but younger than 100 years. The year of my birth is not divisible by 2, 3, or 5. The remainder when this year is divided by 60 is not a prime numbe...	<think> Okay, let's try to figure out the grandfather's birth year based on the clues given. So, Cathrin asked her grandfather in 1981, and he gave some hints about his birth year. Let me break down each clue step by step. First, the grandfather says he's older than 65 but younger than 100. Since this conversation is ...	[ { "from": "human", "value": "## Task 1 - 220831\n\nOn a day in 1981, Cathrin asks her grandfather about his birth year. The grandfather, a friend of puzzle questions, replied:\n\n\"I am older than 65 years, but younger than 100 years. The year of my birth is not divisible by 2, 3, or 5. The remainder when t...
## Task A-1.5. How many four-digit numbers divisible by 3 do not contain the digits $2, 4, 6$ or 9 in their decimal representation?	<think> Okay, so I need to figure out how many four-digit numbers are divisible by 3 and don't contain the digits 2, 4, 6, or 9. Let me start by breaking down the problem step by step. First, a four-digit number ranges from 1000 to 9999. The first digit can't be zero, right? So the thousands place has to be from 1 to ...	[ { "from": "human", "value": "## Task A-1.5.\n\nHow many four-digit numbers divisible by 3 do not contain the digits $2, 4, 6$ or 9 in their decimal representation?" }, { "from": "gpt", "value": "<think>\nOkay, so I need to figure out how many four-digit numbers are divisible by 3 and don't conta...
7. The volume of a cube in cubic metres and its surface area in square metres is numerically equal to four-thirds of the sum of the lengths of its edges in metres. What is the total volume in cubic metres of twenty-seven such cubes?	<think> Okay, let's try to figure out this cube problem. So, the problem says that the volume of a cube in cubic meters is numerically equal to four-thirds of the sum of the lengths of its edges. Also, the surface area in square meters is equal to the same four-thirds of the sum of the edges. Wait, actually, let me che...	[ { "from": "human", "value": "7. The volume of a cube in cubic metres and its surface area in square metres is numerically equal to four-thirds of the sum of the lengths of its edges in metres.\nWhat is the total volume in cubic metres of twenty-seven such cubes?" }, { "from": "gpt", "value": "<t...
Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for all real numbers \( x, y \), we have: \[ f\left(x^{2}-y^{2}\right)=x f(x)-y f(y) \] (USAMO 2002)	<think> Okay, so I need to find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy the equation \( f(x^{2} - y^{2}) = x f(x) - y f(y) \) for all real numbers \( x \) and \( y \). Hmm, this is a functional equation problem. Let me start by recalling some strategies for solving such problems. Usually, ...	[ { "from": "human", "value": "Find all functions \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) such that for all real numbers \\( x, y \\), we have:\n\n\\[\nf\\left(x^{2}-y^{2}\\right)=x f(x)-y f(y)\n\\]\n\n(USAMO 2002)" }, { "from": "gpt", "value": "<think>\nOkay, so I need to find all functio...
1. Using this picture we can observe that $$ 1+3+5+7=4 \times 4 \text {. } $$ What is the value of $$ 1+3+5+7+9+11+13+15+17+19+21 ? $$	<think> Okay, let's see. The problem here is about finding the sum of a series of odd numbers: 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21. The example given is 1 + 3 + 5 + 7, which equals 4 times 4, so 16. That makes sense because 1+3 is 4, then +5 is 9, then +7 is 16. Wait, that's actually the squares of the numb...	[ { "from": "human", "value": "1. Using this picture we can observe that\n$$\n1+3+5+7=4 \\times 4 \\text {. }\n$$\n\nWhat is the value of\n$$\n1+3+5+7+9+11+13+15+17+19+21 ?\n$$" }, { "from": "gpt", "value": "<think>\nOkay, let's see. The problem here is about finding the sum of a series of odd num...
A [i]normal magic square[/i] of order $n$ is an arrangement of the integers from $1$ to $n^2$ in a square such that the $n$ numbers in each row, each column, and each of the two diagonals sum to a constant, called the [i]magic sum[/i] of the magic square. Compute the magic sum of a normal magic square of order $8$.	<think> Okay, so I need to find the magic sum of a normal magic square of order 8. Hmm, let me think. A magic square is a square grid where the numbers in each row, column, and both main diagonals add up to the same number, called the magic sum. Since it's a normal magic square, it uses the numbers from 1 to n², where ...	[ { "from": "human", "value": "A [i]normal magic square[/i] of order $n$ is an arrangement of the integers from $1$ to $n^2$ in a square such that the $n$ numbers in each row, each column, and each of the two diagonals sum to a constant, called the [i]magic sum[/i] of the magic square. Compute the magic sum o...
Task B-4.3. (20 points) What are the minimum and maximum of the function $$ y=\frac{\sin ^{2} x-\sin x+1}{\sin ^{2} x+\sin x+1} $$ For which $x \in[0,2 \pi]$ does the function take its minimum, and for which its maximum value?	<think> Okay, so I need to find the minimum and maximum values of the function y = (sin²x - sinx + 1)/(sin²x + sinx + 1) for x in [0, 2π]. Hmm, let's see. First, since both the numerator and denominator are expressions involving sinx, maybe I can simplify this by letting t = sinx. That way, the function becomes y = (t²...	[ { "from": "human", "value": "Task B-4.3. (20 points) What are the minimum and maximum of the function\n\n$$\ny=\\frac{\\sin ^{2} x-\\sin x+1}{\\sin ^{2} x+\\sin x+1}\n$$\n\nFor which $x \\in[0,2 \\pi]$ does the function take its minimum, and for which its maximum value?" }, { "from": "gpt", "val...
(9 Given the function $f(x)=\cos ^{2} \theta x+\cos \theta x \sin \theta x$ has the smallest positive period of $\frac{\pi}{2}$, then the maximum value of $\theta f(x)$ is $\qquad$ .	<think> Okay, let's see. I need to find the maximum value of θ times f(x), given that the smallest positive period of the function f(x) = cos²(θx) + cos(θx)sin(θx) is π/2. Hmm, first, maybe I should start by understanding the period of the given function. The function f(x) is a combination of cos squared and a product...	[ { "from": "human", "value": "(9 Given the function $f(x)=\\cos ^{2} \\theta x+\\cos \\theta x \\sin \\theta x$ has the smallest positive period of $\\frac{\\pi}{2}$, then the maximum value of $\\theta f(x)$ is $\\qquad$ ." }, { "from": "gpt", "value": "<think>\nOkay, let's see. I need to find th...
$\frac{1}{4}$ of 100 is equal to (A) $20 \%$ of 200 (B) $10 \%$ of 250 (C) $15 \%$ of 100 (D) $25 \%$ of 50 (E) $5 \%$ of 300	<think> Okay, let's see. The problem is asking me to figure out which of the options (A) through (E) is equal to 1/4 of 100. Alright, let's start by calculating what 1/4 of 100 is. So, 1/4 of 100... Well, 100 divided by 4 is 25. Right? Because 4 times 25 is 100. So, 1/4 of 100 is 25. Got that down. Now, I need to che...	[ { "from": "human", "value": "$\\frac{1}{4}$ of 100 is equal to\n(A) $20 \\%$ of 200\n(B) $10 \\%$ of 250\n(C) $15 \\%$ of 100\n(D) $25 \\%$ of 50\n(E) $5 \\%$ of 300" }, { "from": "gpt", "value": "<think>\nOkay, let's see. The problem is asking me to figure out which of the options (A) through (...
## 162. Math Puzzle $11 / 78$ Assume that a fly lays 120 eggs at the beginning of summer, on June 21st, and after 20 days, fully developed insects emerge from these eggs, each of which then lays 120 eggs. How many "descendants" would this fly have in total by the beginning of autumn?	<think> Okay, so I need to figure out how many descendants the original fly has by the beginning of autumn, which is September 21st, right? Let me break this down step by step. First, the timeline. The fly lays 120 eggs on June 21st. Then, after 20 days, those eggs hatch into adult flies, each of which also lays 120 ...	[ { "from": "human", "value": "## 162. Math Puzzle $11 / 78$\n\nAssume that a fly lays 120 eggs at the beginning of summer, on June 21st, and after 20 days, fully developed insects emerge from these eggs, each of which then lays 120 eggs. How many \"descendants\" would this fly have in total by the beginning ...
3. Let $a_{1}, a_{2}, \cdots, a_{n}$ represent any permutation of the integers $1,2, \cdots, n$. Let $f(n)$ be the number of such permutations such that (i) $a_{1}=1$; (ii) $\left\|a_{i}-a_{i+1}\right\| \leqslant 2 . i=1,2, \cdots, n-1$. Determine whether $f(1996)$ is divisible by 3.	<think> Okay, let's try to figure out whether f(1996) is divisible by 3. So first, let's understand what the problem is asking for. We need to find the number of permutations of the integers from 1 to n (where n is 1996 in this case) that satisfy two conditions: (i) the first element is 1, and (ii) the absolute differe...	[ { "from": "human", "value": "3. Let $a_{1}, a_{2}, \\cdots, a_{n}$ represent any permutation of the integers $1,2, \\cdots, n$. Let $f(n)$ be the number of such permutations such that\n(i) $a_{1}=1$;\n(ii) $\\left\|a_{i}-a_{i+1}\\right\| \\leqslant 2 . i=1,2, \\cdots, n-1$.\nDetermine whether $f(1996)$ is div...
Given an integer $\mathrm{n} \geq 3$, let $\mathrm{A}_{1}, \mathrm{~A}_{2}, \ldots, \mathrm{~A}_{2 \mathrm{n}}$ be pairwise distinct non-empty subsets of the set $\{1,2, \ldots, \mathrm{n}\}$, and let $A_{2 n+1}=A_{1}$. Find the maximum value of $\sum_{i=1}^{2 n} \frac{\left\|A_{i} \cap A_{i+1}\right\|}{\left\|A_{i}\right...	<think> Okay, so I need to find the maximum value of this sum: the sum from i=1 to 2n of (\|A_i ∩ A_{i+1}\|)/(\|A_i\| * \|A_{i+1}\|), where each A_i is a distinct non-empty subset of {1,2,...,n}, and A_{2n+1} is set to A_1. Hmm. Alright, let me break this down. First, the problem is about arranging 2n distinct non-empty sub...	[ { "from": "human", "value": "Given an integer $\\mathrm{n} \\geq 3$, let $\\mathrm{A}_{1}, \\mathrm{~A}_{2}, \\ldots, \\mathrm{~A}_{2 \\mathrm{n}}$ be pairwise distinct non-empty subsets of the set $\\{1,2, \\ldots, \\mathrm{n}\\}$, and let $A_{2 n+1}=A_{1}$. Find the maximum value of $\\sum_{i=1}^{2 n} \\f...
3. In the regular triangular frustum $A^{\prime} B^{\prime} C^{\prime}-A B C$, $A^{\prime} B^{\prime} : A B = 5 : 7$, the sections $A B C^{\prime}$ and $A^{\prime} B C^{\prime}$ divide the frustum into three pyramids $C^{\prime}-A B C$, $B-A^{\prime} B^{\prime} C^{\prime}$, and $C^{\prime}-A A B A^{\prime}$. The volume...	<think> Alright, let's tackle this problem step by step. First, I need to visualize the regular triangular frustum. A frustum is like a pyramid with the top cut off by a plane parallel to the base. Since it's a regular triangular frustum, both the base ABC and the top face A'B'C' are equilateral triangles. The ratio of...	[ { "from": "human", "value": "3. In the regular triangular frustum $A^{\\prime} B^{\\prime} C^{\\prime}-A B C$, $A^{\\prime} B^{\\prime} : A B = 5 : 7$, the sections $A B C^{\\prime}$ and $A^{\\prime} B C^{\\prime}$ divide the frustum into three pyramids $C^{\\prime}-A B C$, $B-A^{\\prime} B^{\\prime} C^{\\p...
3.20. Calculate $$ \int_{1}^{i} \frac{\ln ^{2} z}{z} d z $$	<think> Okay, let's see. I need to compute the integral from 1 to i of (ln z)^2 / z dz. Hmm. Alright, complex integrals. So first, I remember that when integrating complex functions, the path of integration matters unless the function is analytic in a simply connected domain containing the path. Wait, is 1/z analytic? ...	[ { "from": "human", "value": "3.20. Calculate\n\n$$\n\\int_{1}^{i} \\frac{\\ln ^{2} z}{z} d z\n$$" }, { "from": "gpt", "value": "<think>\nOkay, let's see. I need to compute the integral from 1 to i of (ln z)^2 / z dz. Hmm. Alright, complex integrals. So first, I remember that when integrating com...
Example 6 Solve the system of equations $\left\{\begin{array}{l}x+y+z=3, \\ x^{2}+y^{2}+z^{2}=3,(\text{Example 4 in [1]}) \\ x^{5}+y^{5}+z^{5}=3 .\end{array}\right.$	<think> Okay, let's try to solve this system of equations. So, we have three equations: 1. \( x + y + z = 3 \) 2. \( x^2 + y^2 + z^2 = 3 \) 3. \( x^5 + y^5 + z^5 = 3 \) Hmm, this looks interesting. The first equation is linear, the second is quadratic, and the fifth is a fifth-degree equation. I remember from algebra...	[ { "from": "human", "value": "Example 6 Solve the system of equations $\\left\\{\\begin{array}{l}x+y+z=3, \\\\ x^{2}+y^{2}+z^{2}=3,(\\text{Example 4 in [1]}) \\\\ x^{5}+y^{5}+z^{5}=3 .\\end{array}\\right.$" }, { "from": "gpt", "value": "<think>\nOkay, let's try to solve this system of equations. ...
6. Given $a, b, c \in [0,1]$. Then $$ \frac{a}{bc+1}+\frac{b}{ca+1}+\frac{c}{ab+1} $$ the range of values is	<think> Okay, so I have this problem here where I need to find the range of the expression \(\frac{a}{bc+1} + \frac{b}{ca+1} + \frac{c}{ab+1}\) given that \(a\), \(b\), and \(c\) are all in the interval \([0,1]\). Hmm, let's see. I need to figure out the minimum and maximum values this expression can take. First off,...	[ { "from": "human", "value": "6. Given $a, b, c \\in [0,1]$. Then\n$$\n\\frac{a}{bc+1}+\\frac{b}{ca+1}+\\frac{c}{ab+1}\n$$\n\nthe range of values is" }, { "from": "gpt", "value": "<think>\nOkay, so I have this problem here where I need to find the range of the expression \\(\\frac{a}{bc+1} + \\fr...
17. What is the value of the sum: $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots+\frac{1}{1024}$ ?	<think> Alright, so I need to find the value of the sum: 1 + 1/2 + 1/4 + 1/8 + ... + 1/1024. Hmm, let's see. I remember that this looks like a geometric series. Let me recall what a geometric series is. It's a series where each term after the first is found by multiplying the previous term by a constant called the comm...	[ { "from": "human", "value": "17. What is the value of the sum: $1+\\frac{1}{2}+\\frac{1}{4}+\\frac{1}{8}+\\ldots+\\frac{1}{1024}$ ?" }, { "from": "gpt", "value": "<think>\nAlright, so I need to find the value of the sum: 1 + 1/2 + 1/4 + 1/8 + ... + 1/1024. Hmm, let's see. I remember that this lo...
9. As shown in the figure, in the right triangle $A B C$, $\angle A C B=\frac{\pi}{2}, A C=B C=2$, point $P$ is a point on the hypotenuse $A B$, and $B P=2 P A$, then $\overrightarrow{C P} \cdot \overrightarrow{C A}+\overrightarrow{C P} \cdot \overrightarrow{C B}=$ $\qquad$ .	<think> Okay, let me try to figure out this problem. So, we have a right triangle ABC where angle ACB is π/2, which means it's a right angle at point C. The sides AC and BC are both 2 units long. So, triangle ABC is an isosceles right triangle with legs of length 2. The hypotenuse AB must then be 2√2 because in a right...	[ { "from": "human", "value": "9. As shown in the figure, in the right triangle $A B C$, $\\angle A C B=\\frac{\\pi}{2}, A C=B C=2$, point $P$ is a point on the hypotenuse $A B$, and $B P=2 P A$, then $\\overrightarrow{C P} \\cdot \\overrightarrow{C A}+\\overrightarrow{C P} \\cdot \\overrightarrow{C B}=$ $\\q...
## Task 4 - 240624 Rita is experimenting with a balance scale. (With such a scale, it can be determined whether the content of one pan weighs as much as the content of the other pan or which of the two contents weighs more than the other.) Rita has 17 balls, 6 cubes, and 1 pyramid. She observes: (1) Each ball weigh...	<think> Okay, let's see. Rita has a balance scale and some objects: 17 balls, 6 cubes, and 1 pyramid. The problem states that all balls weigh the same, all cubes weigh the same, and we have two equations relating the weights of these objects. Rolf wants to know how many balls weigh the same as the pyramid. The question...	[ { "from": "human", "value": "## Task 4 - 240624\n\nRita is experimenting with a balance scale.\n\n(With such a scale, it can be determined whether the content of one pan weighs as much as the content of the other pan or which of the two contents weighs more than the other.)\n\nRita has 17 balls, 6 cubes, an...
2. (a) In Figure 1, trapezoid $A B E D$ is formed by joining $\triangle B C E$ (which is rightangled at $C$ ) to rectangle $A B C D$. If $A B=3, C E=6$ and the area of trapezoid $A B E D$ is 48 , determine the length of $B E$. (b) In Figure 2, circles with centres $P$ and $Q$ touch a line $\ell$ (that is, are tangent t...	<think> Okay, so I need to solve part (a) of this problem. Let me read it again to make sure I understand. We have a trapezoid ABED formed by joining triangle BCE, which is right-angled at C, to rectangle ABCD. Given that AB is 3 units, CE is 6 units, and the area of trapezoid ABED is 48. We need to find the length of...	[ { "from": "human", "value": "2. (a) In Figure 1, trapezoid $A B E D$ is formed by joining $\\triangle B C E$ (which is rightangled at $C$ ) to rectangle $A B C D$. If $A B=3, C E=6$ and the area of trapezoid $A B E D$ is 48 , determine the length of $B E$.\n(b) In Figure 2, circles with centres $P$ and $Q$ ...
Natural numbers $k, l,p$ and $q$ are such that if $a$ and $b$ are roots of $x^2 - kx + l = 0$ then $a +\frac1b$ and $b + \frac1a$ are the roots of $x^2 -px + q = 0$. What is the sum of all possible values of $q$?	<think> Okay, let me try to work through this problem step by step. So, we have two quadratic equations. The first one is \( x^2 - kx + l = 0 \) with roots \( a \) and \( b \). Then, the second quadratic equation is \( x^2 - px + q = 0 \) with roots \( a + \frac{1}{b} \) and \( b + \frac{1}{a} \). We need to find the s...	[ { "from": "human", "value": "Natural numbers $k, l,p$ and $q$ are such that if $a$ and $b$ are roots of $x^2 - kx + l = 0$ then $a +\\frac1b$ and $b + \\frac1a$ are the roots of $x^2 -px + q = 0$. What is the sum of all possible values of $q$?" }, { "from": "gpt", "value": "<think>\nOkay, let me...
1. [4] A hundred friends, including Petya and Vasya, live in several cities. Petya learned the distance from his city to the city of each of the remaining 99 friends and added these 99 numbers. Vasya did the same. Petya got 1000 km. What is the largest number Vasya could have obtained? (Consider the cities as points on...	<think> Okay, let's see. So there are 100 friends, including Petya and Vasya, living in several cities. The cities are points on a plane, and if two people are in the same city, the distance between them is zero. Petya calculated the sum of distances from his city to each of the other 99 friends, and got 1000 km. Now w...	[ { "from": "human", "value": "1. [4] A hundred friends, including Petya and Vasya, live in several cities. Petya learned the distance from his city to the city of each of the remaining 99 friends and added these 99 numbers. Vasya did the same. Petya got 1000 km. What is the largest number Vasya could have ob...
10.243. Inside an angle of $60^{\circ}$, there is a point located at distances of $\sqrt{7}$ and $2 \sqrt{7}$ cm from the sides of the angle. Find the distance from this point to the vertex of the angle.	<think> Okay, so there's this geometry problem here. Let me read it again: Inside an angle of 60 degrees, there's a point. This point is sqrt(7) cm away from one side of the angle and 2*sqrt(7) cm away from the other side. We need to find the distance from this point to the vertex of the angle. Hmm, okay. Let me visual...	[ { "from": "human", "value": "10.243. Inside an angle of $60^{\\circ}$, there is a point located at distances of $\\sqrt{7}$ and $2 \\sqrt{7}$ cm from the sides of the angle. Find the distance from this point to the vertex of the angle." }, { "from": "gpt", "value": "<think>\nOkay, so there's thi...
3. [7 points] Let $M$ be a figure on the Cartesian plane consisting of all points $(x, y)$ such that there exists a pair of real numbers $a, b$ for which the system of inequalities is satisfied $$ \left\{\begin{array}{l} (x-a)^{2}+(y-b)^{2} \leqslant 20 \\ a^{2}+b^{2} \leqslant \min (8 a-4 b ; 20) \end{array}\right. $...	<think> Okay, so I need to find the area of figure M, which consists of all points (x, y) such that there exist real numbers a and b satisfying two conditions: 1. The point (x, y) lies inside or on the circle centered at (a, b) with radius sqrt(20). 2. The point (a, b) itself lies inside or on the circle defined by a...	[ { "from": "human", "value": "3. [7 points] Let $M$ be a figure on the Cartesian plane consisting of all points $(x, y)$ such that there exists a pair of real numbers $a, b$ for which the system of inequalities is satisfied\n\n$$\n\\left\\{\\begin{array}{l}\n(x-a)^{2}+(y-b)^{2} \\leqslant 20 \\\\\na^{2}+b^{2...
An airplane traces out a regular $n$-sided polygon, and does so at speeds of $v_{1}, v_{2}, \ldots v_{n}$ along each side. What is the average speed of the airplane?	<think> Okay, let's see. The problem is about an airplane tracing out a regular n-sided polygon, and it has different speeds on each side: v1, v2, ..., vn. We need to find the average speed of the airplane. Hmm, average speed usually involves total distance divided by total time, right? So maybe that's the approach her...	[ { "from": "human", "value": "An airplane traces out a regular $n$-sided polygon, and does so at speeds of $v_{1}, v_{2}, \\ldots v_{n}$ along each side. What is the average speed of the airplane?" }, { "from": "gpt", "value": "<think>\nOkay, let's see. The problem is about an airplane tracing ou...
6. What is the last digit of $1^{1}+2^{2}+3^{3}+\cdots+100^{100}$ ?	<think> Okay, so I need to find the last digit of the sum \(1^{1} + 2^{2} + 3^{3} + \cdots + 100^{100}\). Hmm, that seems like a pretty big sum, but since we're only interested in the last digit, maybe there's a pattern or a cycle in the last digits of these exponents that we can exploit. Let me think through this step...	[ { "from": "human", "value": "6. What is the last digit of $1^{1}+2^{2}+3^{3}+\\cdots+100^{100}$ ?" }, { "from": "gpt", "value": "<think>\nOkay, so I need to find the last digit of the sum \\(1^{1} + 2^{2} + 3^{3} + \\cdots + 100^{100}\\). Hmm, that seems like a pretty big sum, but since we're on...
1 Convex quadrilateral $E F G H$ has vertices $E, F, G, H$ on the sides $A B, B C, C D, D A$ of convex quadrilateral $A B C D$, respectively, satisfying $$ \frac{A E}{E B} \cdot \frac{B F}{F C} \cdot \frac{C G}{G D} \cdot \frac{D H}{H A}=1 $$ and points $A, B, C, D$ are on the sides $H_{1} E_{1}, E_{1} F_{1}, F_{1} G_...	<think> Okay, so I have this problem about convex quadrilaterals and some ratios. Let me try to parse through it step by step. First, the problem states: There's a convex quadrilateral EFGH with vertices E, F, G, H on the sides AB, BC, CD, DA of another convex quadrilateral ABCD, respectively. These points satisfy the...	[ { "from": "human", "value": "1 Convex quadrilateral $E F G H$ has vertices $E, F, G, H$ on the sides $A B, B C, C D, D A$ of convex quadrilateral $A B C D$, respectively, satisfying\n$$\n\\frac{A E}{E B} \\cdot \\frac{B F}{F C} \\cdot \\frac{C G}{G D} \\cdot \\frac{D H}{H A}=1\n$$\n\nand points $A, B, C, D$...