answer：Okay, so I have this equation: ( x^3y - 2x^2y + xy - 100y = 1061500 ), and I know that ( x = 103 ). I need to find the value of ( y ). Hmm, let me think about how to approach this.First, since I know the value of ( x ), I can substitute ( x = 103 ) into the equation. That should make it easier to solve for ( y ). Let me write that out:( (103)^3y - 2(103)^2y + 103y - 100y = 1061500 )Okay, so I need to calculate each of these terms. Let me compute each part step by step.Starting with ( 103^3 ). I know that ( 103^2 = 10609 ) because ( 100^2 = 10000 ) and then adding the cross terms and the square of 3. So, ( 103^3 ) would be ( 103 times 10609 ). Let me compute that:( 103 times 10609 ). Hmm, breaking it down: ( 100 times 10609 = 1,060,900 ) and ( 3 times 10609 = 31,827 ). Adding those together: ( 1,060,900 + 31,827 = 1,092,727 ). So, ( 103^3 = 1,092,727 ).Next, ( 2 times 103^2 ). Since ( 103^2 = 10,609 ), multiplying by 2 gives ( 21,218 ).Now, substituting these back into the equation:( 1,092,727y - 21,218y + 103y - 100y = 1,061,500 )Let me combine like terms. First, combine the coefficients of ( y ):( 1,092,727y - 21,218y = (1,092,727 - 21,218)y = 1,071,509y )Then, ( 103y - 100y = 3y )So, the equation simplifies to:( 1,071,509y + 3y = 1,061,500 )Combining those terms:( (1,071,509 + 3)y = 1,061,500 )Which is:( 1,071,512y = 1,061,500 )Now, to solve for ( y ), I need to divide both sides by ( 1,071,512 ):( y = frac{1,061,500}{1,071,512} )Hmm, let me see if this fraction can be simplified. I'll check if both numerator and denominator have a common factor. Let's see:First, I can note that both numbers are even, so they are divisible by 2:( frac{1,061,500 ÷ 2}{1,071,512 ÷ 2} = frac{530,750}{535,756} )Still, both are even, so divide by 2 again:( frac{530,750 ÷ 2}{535,756 ÷ 2} = frac{265,375}{267,878} )Now, checking if 265,375 and 267,878 have any common factors. Let's see:265,375 ends with 5, so it's divisible by 5. Let me check 267,878: it ends with 8, so not divisible by 5. So, 5 is not a common factor.Next, let's try 7. Let me check 265,375 ÷ 7: 7 × 37,910 = 265,370, which is 5 less than 265,375, so not divisible by 7.How about 13? 13 × 20,413 = 265,369, which is 6 less than 265,375. Not divisible by 13.Maybe 17? 17 × 15,600 = 265,200, which is 175 less. 17 × 15,600 + 17 × 10 = 265,200 + 170 = 265,370, which is still 5 less. So, not divisible by 17.Perhaps 25? 25 × 10,615 = 265,375. So, 265,375 is 25 × 10,615. Let me check if 267,878 is divisible by 25. It ends with 78, which isn't divisible by 25, so no.What about 10,615? Let me see if 267,878 is divisible by 10,615. 10,615 × 25 = 265,375, which is less than 267,878. 267,878 - 265,375 = 2,503. 10,615 doesn't divide into 2,503 evenly. So, no common factor there.Maybe I should check for smaller factors. Let's see:265,375: prime factors. It's 25 × 10,615, as I saw earlier. 10,615: let's see, 10,615 ÷ 5 = 2,123. So, 265,375 = 5 × 5 × 5 × 2,123. Now, 2,123: let's check if it's prime. Dividing by small primes: 2,123 ÷ 7 = 303.28... Not integer. 2,123 ÷ 11 = 193, which is exact because 11 × 193 = 2,123. So, 265,375 = 5^3 × 11 × 193.Now, let's factor 267,878. It's even, so 2 × 133,939. 133,939: let's check divisibility. 133,939 ÷ 7 = 19,134.14... Not integer. 133,939 ÷ 11 = 12,176.27... Not integer. 133,939 ÷ 13 = 10,303. So, 13 × 10,303 = 133,939. Now, 10,303: checking divisibility. 10,303 ÷ 7 = 1,471.85... Not integer. 10,303 ÷ 11 = 936.63... Not integer. 10,303 ÷ 13 = 792.53... Not integer. It might be prime. So, 267,878 = 2 × 13 × 10,303.Looking at the prime factors, 265,375 = 5^3 × 11 × 193 and 267,878 = 2 × 13 × 10,303. There are no common prime factors between numerator and denominator, so the fraction is already in its simplest form.Therefore, ( y = frac{265,375}{267,878} ). But let me double-check my calculations to make sure I didn't make any mistakes.Wait a second, earlier when I combined the terms, I had:( 1,092,727y - 21,218y + 103y - 100y )Which I simplified to:( (1,092,727 - 21,218)y + (103 - 100)y = 1,071,509y + 3y = 1,071,512y )But let me verify that subtraction:1,092,727 - 21,218: Let's compute 1,092,727 - 20,000 = 1,072,727, then subtract 1,218 more: 1,072,727 - 1,218 = 1,071,509. That seems correct.Then, 103 - 100 = 3. So, total is 1,071,509y + 3y = 1,071,512y. That seems right.So, 1,071,512y = 1,061,500. Therefore, y = 1,061,500 / 1,071,512.Wait, but in my earlier simplification, I divided numerator and denominator by 2 twice, getting to 265,375 / 267,878. Let me confirm that:1,061,500 ÷ 2 = 530,7501,071,512 ÷ 2 = 535,756Then, 530,750 ÷ 2 = 265,375535,756 ÷ 2 = 267,878Yes, that's correct. So, y = 265,375 / 267,878.But wait, the original equation was 1061500, which is 1,061,500. So, my calculations seem consistent.Is there a way to simplify this fraction further? From the prime factorization, it seems not, as there are no common factors. So, y = 265,375 / 267,878.Alternatively, I can express this as a decimal to check its approximate value. Let me compute 265,375 ÷ 267,878.Dividing 265,375 by 267,878: since 265,375 is less than 267,878, it's approximately 0.9907.So, y ≈ 0.9907. But since the question didn't specify the form, and fractions are exact, I should present it as 265,375/267,878.Wait, but let me check if I made any mistake in the initial substitution or calculations. Let me go back step by step.Given x = 103, substitute into the equation:( x^3y - 2x^2y + xy - 100y = 1061500 )Compute each term:( x^3 = 103^3 = 1,092,727 )( 2x^2 = 2*(103)^2 = 2*10,609 = 21,218 )So, the equation becomes:( 1,092,727y - 21,218y + 103y - 100y = 1,061,500 )Combine like terms:1,092,727y - 21,218y = (1,092,727 - 21,218)y = 1,071,509y103y - 100y = 3ySo, total: 1,071,509y + 3y = 1,071,512yThus, 1,071,512y = 1,061,500So, y = 1,061,500 / 1,071,512Simplify by dividing numerator and denominator by 2: 530,750 / 535,756Divide by 2 again: 265,375 / 267,878Yes, that's correct. So, y = 265,375 / 267,878.I think that's the simplest form. Alternatively, I can check if 265,375 and 267,878 have any common factors beyond what I've already considered.Let me compute the greatest common divisor (GCD) of 265,375 and 267,878.Using the Euclidean algorithm:GCD(267,878, 265,375)267,878 - 265,375 = 2,503Now, GCD(265,375, 2,503)265,375 ÷ 2,503 = let's see, 2,503 * 100 = 250,300265,375 - 250,300 = 15,075Now, GCD(2,503, 15,075)15,075 ÷ 2,503 = 6 times (2,503*6=15,018)15,075 - 15,018 = 57Now, GCD(2,503, 57)2,503 ÷ 57 = 43 times (57*43=2,451)2,503 - 2,451 = 52GCD(57, 52)57 ÷ 52 = 1 with remainder 5GCD(52, 5)52 ÷ 5 = 10 with remainder 2GCD(5, 2)5 ÷ 2 = 2 with remainder 1GCD(2,1) = 1So, the GCD is 1. Therefore, the fraction cannot be simplified further.Thus, y = 265,375 / 267,878.Alternatively, I can write this as a decimal, but since it's a fraction, it's better to present it as is unless specified otherwise.So, my final answer is y = 265,375/267,878.

answer：Okay, so I have this problem here where three positive integers are each divided by 30, and they leave remainders of 7, 11, and 23 respectively. I need to find the remainder when the sum of these three integers is divided by 30. Hmm, let's break this down step by step.First, I remember that when a number is divided by another number, say 30, it can be expressed as 30 times some quotient plus a remainder. So, if I have a number that leaves a remainder of 7 when divided by 30, I can write it as 30k + 7, where k is some integer. Similarly, the other two numbers can be written as 30m + 11 and 30n + 23, where m and n are integers as well.So, let me denote the three numbers as:- a = 30k + 7- b = 30m + 11- c = 30n + 23Now, the problem asks for the remainder when the sum of these three numbers is divided by 30. So, I need to find (a + b + c) mod 30.Let me compute the sum first:a + b + c = (30k + 7) + (30m + 11) + (30n + 23)Combining like terms:= 30k + 30m + 30n + 7 + 11 + 23= 30(k + m + n) + (7 + 11 + 23)Calculating the sum of the remainders:7 + 11 = 1818 + 23 = 41So, the total sum is:30(k + m + n) + 41Now, when this sum is divided by 30, the term 30(k + m + n) will be completely divisible by 30, leaving no remainder. The remainder will come from the 41 part.So, I need to find 41 mod 30. To do this, I can subtract 30 from 41:41 - 30 = 11Therefore, the remainder is 11.Wait, let me double-check that. If I add the remainders: 7 + 11 + 23, that's 41. Then, 41 divided by 30 is 1 with a remainder of 11. Yep, that seems right.I think I've got it. The remainder is 11.

answer：Okay, so I have this function f(x) = sin(x - π/6)cosx + 1, and I need to find its smallest positive period and then determine its maximum and minimum values on the interval [π/12, π/2]. Hmm, let me start with part (i).First, I remember that the period of a function is the smallest positive number T such that f(x + T) = f(x) for all x. Since f(x) is a combination of sine and cosine functions, I should probably look at the periods of each component.The function f(x) is sin(x - π/6)cosx + 1. The "1" at the end is just a constant, so it doesn't affect the period. The main part is sin(x - π/6)cosx. I think I can use a trigonometric identity to simplify this product. Maybe the product-to-sum formulas?Let me recall: sin A cos B = [sin(A + B) + sin(A - B)] / 2. So, applying that here, where A is (x - π/6) and B is x.So, sin(x - π/6)cosx = [sin((x - π/6) + x) + sin((x - π/6) - x)] / 2.Simplifying inside the sine functions:First term: (x - π/6) + x = 2x - π/6Second term: (x - π/6) - x = -π/6So, sin(x - π/6)cosx = [sin(2x - π/6) + sin(-π/6)] / 2.But sin(-π/6) is just -sin(π/6) = -1/2. So, substituting that in:sin(x - π/6)cosx = [sin(2x - π/6) - 1/2] / 2 = (1/2)sin(2x - π/6) - 1/4.Therefore, f(x) = (1/2)sin(2x - π/6) - 1/4 + 1 = (1/2)sin(2x - π/6) + 3/4.Okay, so f(x) simplifies to (1/2)sin(2x - π/6) + 3/4. Now, to find the period of this function. The general form is A sin(Bx + C) + D, where the period is 2π / |B|. Here, B is 2, so the period should be 2π / 2 = π.But wait, is that the smallest positive period? Let me think. The function inside the sine is 2x - π/6, which is a linear function with a coefficient of 2 on x. So, the period of sin(2x - π/6) is π, as I thought. Adding a constant or scaling the amplitude doesn't change the period, so the period of f(x) is indeed π.So, part (i) answer is π.Now, moving on to part (ii). I need to find the maximum and minimum values of f(x) on the interval [π/12, π/2].Given that f(x) = (1/2)sin(2x - π/6) + 3/4, I can analyze the behavior of this function on the given interval.First, let me find the range of the argument of the sine function, which is 2x - π/6, when x is in [π/12, π/2].Calculating the lower bound: when x = π/12, 2x - π/6 = 2*(π/12) - π/6 = π/6 - π/6 = 0.Calculating the upper bound: when x = π/2, 2x - π/6 = 2*(π/2) - π/6 = π - π/6 = 5π/6.So, the argument 2x - π/6 ranges from 0 to 5π/6 as x goes from π/12 to π/2.Now, the sine function, sinθ, has a maximum value of 1 and a minimum value of -1. However, since θ is between 0 and 5π/6, which is less than π, the sine function is increasing from 0 to π/2 and then decreasing from π/2 to 5π/6.Therefore, the maximum value of sinθ in this interval occurs at θ = π/2, which is 1. The minimum value occurs at the endpoints, either θ = 0 or θ = 5π/6. Let's compute sin(0) and sin(5π/6):sin(0) = 0sin(5π/6) = sin(π - π/6) = sin(π/6) = 1/2So, the minimum value of sinθ in this interval is 0, and the maximum is 1.Therefore, sin(2x - π/6) ranges from 0 to 1 on the interval x ∈ [π/12, π/2].Thus, f(x) = (1/2)sin(2x - π/6) + 3/4 will range from (1/2)*0 + 3/4 = 3/4 to (1/2)*1 + 3/4 = 5/4.So, the minimum value of f(x) is 3/4, and the maximum value is 5/4.But wait, let me verify if these extrema are actually attained within the interval. For the maximum, sin(2x - π/6) = 1 when 2x - π/6 = π/2, which implies 2x = π/2 + π/6 = (3π/6 + π/6) = 4π/6 = 2π/3. So, x = π/3.Is π/3 within [π/12, π/2]? Yes, because π/12 ≈ 0.2618, π/3 ≈ 1.0472, and π/2 ≈ 1.5708. So, π/3 is indeed in the interval.For the minimum, sin(2x - π/6) = 0 when 2x - π/6 = 0, which implies x = π/12, which is the left endpoint of the interval. So, the minimum is attained at x = π/12.Therefore, the maximum value is 5/4 at x = π/3, and the minimum value is 3/4 at x = π/12.I think that's it. Let me just recap:(i) The function simplifies to (1/2)sin(2x - π/6) + 3/4, which has a period of π.(ii) On the interval [π/12, π/2], the function attains a maximum of 5/4 at x = π/3 and a minimum of 3/4 at x = π/12.Yeah, that seems consistent. I don't think I made any mistakes in the calculations.**Final Answer**(i) The smallest positive period is boxed{pi}.(ii) The maximum value is boxed{dfrac{5}{4}} and the minimum value is boxed{dfrac{3}{4}}.

answer：First, I need to understand the structure of the octagon and its surrounding grid. The octagon is drawn on a grid with dots, and there are four shaded triangles outside the octagon.Each shaded triangle has a base of 1 unit and a height of 4 units. The area of one such triangle can be calculated using the formula for the area of a triangle: ( frac{1}{2} times text{base} times text{height} ).Calculating the area of one triangle:[text{Area of one triangle} = frac{1}{2} times 1 times 4 = 2 text{ square units}]Since there are four such triangles, the total area of the shaded triangles is:[text{Total area of shaded triangles} = 4 times 2 = 8 text{ square units}]Next, I consider the rectangular region that encompasses both the octagon and the shaded triangles. The rectangle has a width of 5 units and a height of 8 units. The area of this rectangle is:[text{Area of the rectangle} = 5 times 8 = 40 text{ square units}]Finally, to find the area of the octagon, I subtract the total area of the shaded triangles from the area of the rectangle:[text{Area of the octagon} = 40 - 8 = 32 text{ square units}]

answer：Okay, so I'm trying to figure out whether there are any blue frogs on the Island of Colorful Frogs. There are three inhabitants: Bre, Ke, and Keks. Each of them makes a statement, and I need to determine if there are blue frogs based on their statements.First, let's list out what each person says:- Bre says: "There are no blue frogs on our island."- Ke says: "Bre is a liar. He himself is a blue frog!"- Keks says: "Of course, Bre is a liar. But he is a red frog."Alright, so the first thing I notice is that Ke and Keks both accuse Bre of being a liar. However, they contradict each other about Bre's color: Ke says Bre is a blue frog, while Keks says Bre is a red frog. This contradiction suggests that at least one of them is lying.Now, let's consider the possibilities. There are two types of inhabitants on the island: truth-tellers and liars. So, if Bre is telling the truth, then his statement "There are no blue frogs on our island" would be true. That would mean Ke and Keks are lying when they say Bre is a liar. But if Ke and Keks are lying about Bre being a liar, does that mean Bre is actually telling the truth? That seems consistent.But wait, if Bre is telling the truth, then Ke and Keks are lying. But Ke says Bre is a liar and a blue frog, while Keks says Bre is a liar and a red frog. If both are lying, then Bre is not a liar, and Bre is not a blue frog or a red frog? That doesn't make sense because Bre must be either a truth-teller or a liar, and his color must be either blue or red or something else.Hold on, maybe I'm overcomplicating this. Let's break it down step by step.1. Assume Bre is telling the truth. - Then, "There are no blue frogs on our island" is true. - Therefore, Ke and Keks are lying when they say Bre is a liar. - If Ke is lying about Bre being a liar, then Bre is not a liar, which aligns with our assumption. - Similarly, Keks is lying about Bre being a liar, which also aligns with our assumption. - However, Ke says Bre is a blue frog, and Keks says Bre is a red frog. Since both are lying, Bre cannot be a blue frog or a red frog? That doesn't make sense because Bre must be one color or another.2. Assume Bre is lying. - Then, "There are no blue frogs on our island" is false, meaning there are blue frogs on the island. - Therefore, Ke and Keks are telling the truth when they say Bre is a liar. - Now, Ke says Bre is a blue frog, and Keks says Bre is a red frog. But if both are telling the truth, then Bre would have to be both blue and red, which is impossible. - This leads to a contradiction because Bre cannot be both colors at the same time.Hmm, so assuming Bre is lying leads to a contradiction because Ke and Keks cannot both be telling the truth about Bre's color. On the other hand, assuming Bre is telling the truth also leads to an inconsistency because Ke and Keks cannot both be lying about Bre's color.Wait, maybe I need to consider that not all inhabitants are either truth-tellers or liars exclusively. But the problem states that some always tell the truth and others always lie, so each person is either a truth-teller or a liar.Let me try another approach. Let's consider the statements about Bre's color separately.- Ke says Bre is a blue frog.- Keks says Bre is a red frog.These two statements cannot both be true. Therefore, at least one of them is lying. If Bre is a truth-teller, then Ke and Keks are lying about Bre being a liar, but they are also lying about Bre's color. If Bre is a liar, then Ke and Keks are telling the truth about Bre being a liar, but they cannot both be telling the truth about Bre's color.This seems like a paradox. Maybe there's a way to resolve it by considering that the color statements are separate from the truthfulness statements.If Bre is a truth-teller:- Bre's statement: "There are no blue frogs on our island" is true.- Therefore, Ke and Keks are lying about Bre being a liar.- Ke's statement: "Bre is a liar. He himself is a blue frog!" is a lie. So, Bre is not a liar, and Bre is not a blue frog.- Keks's statement: "Of course, Bre is a liar. But he is a red frog." is also a lie. So, Bre is not a liar, and Bre is not a red frog.- Wait, this suggests that Bre is neither blue nor red, but the problem doesn't specify that frogs can only be blue or red. Maybe there are other colors, but the problem doesn't mention them. This is confusing.If Bre is a liar:- Bre's statement: "There are no blue frogs on our island" is false, so there are blue frogs.- Ke's statement: "Bre is a liar. He himself is a blue frog!" is true, so Bre is a liar and a blue frog.- Keks's statement: "Of course, Bre is a liar. But he is a red frog." is partially true (Bre is a liar) but false about the color, so Keks is lying.- This seems consistent because Ke is telling the truth, and Keks is lying. So, Bre is a liar and a blue frog, and there are blue frogs on the island.Wait, this seems to resolve the contradiction. If Bre is a liar, then Ke is telling the truth about Bre being a liar and a blue frog, and Keks is lying about Bre being a red frog. Therefore, there are blue frogs on the island.But earlier, I thought assuming Bre is lying leads to a contradiction because Ke and Keks both say Bre is a liar, but one says blue and the other says red. But if Bre is a liar, then Ke is telling the truth about both parts, and Keks is lying about the color. So, it's okay for Ke to be truthful and Keks to be lying.Therefore, the conclusion is that there are blue frogs on the island.

answer：Okay, so I have this problem here about negating a proposition. The original statement is: "There exists an ( x in mathbb{R} ) such that ( x^2 + 4x + 5 leq 0 )." I need to find its negation from the given options. Let me think about how to approach this.First, I remember that negating a proposition involves changing the quantifier and the statement. The original statement uses the existential quantifier "there exists," so its negation should use the universal quantifier "for all." Also, the inequality ( leq 0 ) should be negated to ( > 0 ).So, the negation should be: "For all ( x in mathbb{R} ), ( x^2 + 4x + 5 > 0 )." Let me check the options:A: There exists an ( x in mathbb{R} ) such that ( x^2 + 4x + 5 > 0 ).B: There exists an ( x in mathbb{R} ) such that ( x^2 + 4x + 5 leq 0 ).C: For all ( x in mathbb{R} ), ( x^2 + 4x + 5 > 0 ).D: For all ( x in mathbb{R} ), ( x^2 + 4x + 5 leq 0 ).Hmm, option C matches what I thought the negation should be. But just to be sure, maybe I should verify if the original statement is true or not. If the original statement is true, then its negation should be false, and vice versa.Let me analyze the quadratic expression ( x^2 + 4x + 5 ). To see if it can ever be less than or equal to zero, I can find its discriminant. The discriminant ( D ) of a quadratic ( ax^2 + bx + c ) is ( D = b^2 - 4ac ).Calculating the discriminant:( D = 4^2 - 4(1)(5) = 16 - 20 = -4 ).Since the discriminant is negative, the quadratic does not have real roots, which means it doesn't cross the x-axis. The coefficient of ( x^2 ) is positive, so the parabola opens upwards. Therefore, the quadratic is always positive for all real ( x ).Wait, that means the original statement "There exists an ( x in mathbb{R} ) such that ( x^2 + 4x + 5 leq 0 )" is actually false because the quadratic is always positive. So, its negation should be true.Looking back at the options, option C is "For all ( x in mathbb{R} ), ( x^2 + 4x + 5 > 0 )," which is true based on the discriminant analysis. Therefore, option C is indeed the correct negation.I think I got it right. The negation involves changing the existential quantifier to a universal one and flipping the inequality. Also, verifying the original statement's truth value helped confirm that option C is correct.