answer：Okay, so I have this problem here: I need to determine the range of the real number ( m ) such that the point corresponding to the complex number ( z = (m + 3) + (m - 1)i ) lies in the third quadrant of the complex plane. Hmm, let's break this down step by step.First, I remember that in the complex plane, any complex number ( z = a + bi ) can be represented as a point with coordinates ( (a, b) ). The complex plane is divided into four quadrants, just like the standard Cartesian plane. The third quadrant is where both the real part and the imaginary part of the complex number are negative. So, for ( z ) to be in the third quadrant, both ( a ) and ( b ) must be less than zero.Given ( z = (m + 3) + (m - 1)i ), the real part is ( m + 3 ) and the imaginary part is ( m - 1 ). Therefore, I need both ( m + 3 < 0 ) and ( m - 1 < 0 ) to be true simultaneously.Let me write down these inequalities:1. ( m + 3 < 0 )2. ( m - 1 < 0 )Now, I'll solve each inequality for ( m ).Starting with the first inequality:( m + 3 < 0 )Subtracting 3 from both sides:( m < -3 )Okay, so ( m ) must be less than -3.Now, the second inequality:( m - 1 < 0 )Adding 1 to both sides:( m < 1 )So, ( m ) must also be less than 1.Now, I need to find the values of ( m ) that satisfy both conditions. That is, ( m ) has to be less than -3 and less than 1. But wait, if ( m ) is less than -3, it is automatically less than 1 because -3 is less than 1. So, the more restrictive condition is ( m < -3 ).Therefore, the range of ( m ) that satisfies both inequalities is all real numbers less than -3.Let me double-check to make sure I didn't make a mistake. If ( m ) is less than -3, then ( m + 3 ) is negative, which is good for the real part. Also, ( m - 1 ) would be less than -4, which is also negative, so the imaginary part is negative as required. So, both conditions are satisfied when ( m < -3 ).Is there any other consideration I need to make? Hmm, I don't think so. The problem is straightforward because both inequalities are linear and only involve ( m ). There are no quadratic terms or absolute values that could complicate things.Just to visualize, if I pick a value of ( m ) less than -3, say ( m = -4 ), then the real part becomes ( -4 + 3 = -1 ) and the imaginary part becomes ( -4 - 1 = -5 ). So, the point would be ( (-1, -5) ), which is indeed in the third quadrant. If I pick ( m = -2 ), which is greater than -3, the real part is ( -2 + 3 = 1 ) and the imaginary part is ( -2 - 1 = -3 ). So, the point would be ( (1, -3) ), which is in the fourth quadrant, not the third. That confirms that ( m ) must be less than -3.Another test: ( m = -3 ). Then, the real part is ( -3 + 3 = 0 ) and the imaginary part is ( -3 - 1 = -4 ). The point is ( (0, -4) ), which lies on the negative imaginary axis, not in any quadrant. So, ( m ) cannot be equal to -3 either; it has to be strictly less than -3.Therefore, after all this checking, I'm confident that the range of ( m ) is all real numbers less than -3.**Final Answer**The range of ( m ) is boxed{(-infty, -3)}.

answer：Okay, so I have this multiple-choice question here, and I need to figure out which proposition is incorrect. The options are A, B, C, and D. Let me go through each one step by step.Starting with option A: "A parallelogram with equal diagonals is a rectangle." Hmm, I remember that in a parallelogram, the diagonals bisect each other. But do equal diagonals make it a rectangle? I think so. Because in a rectangle, the diagonals are equal in length, which is one of its properties. So, if a parallelogram has equal diagonals, it must be a rectangle. That seems correct. So, A is probably correct.Moving on to option B: "An isosceles triangle with one angle of 60° is an equilateral triangle." Okay, an isosceles triangle has two equal sides and two equal angles. If one of the angles is 60°, does that make all angles 60°? Let me think. If it's isosceles and one angle is 60°, then the other two angles must be equal. Since the sum of angles in a triangle is 180°, if one angle is 60°, the other two must add up to 120°. If they are equal, each would be 60°, making all three angles 60°, which means it's an equilateral triangle. So, B seems correct too.Now, option C: "The height to the hypotenuse of a right triangle is equal to half of the hypotenuse." Hmm, this one sounds a bit tricky. I know that in a right triangle, the altitude to the hypotenuse has a special relationship with the sides. Let me recall the formula. The area of the triangle can be expressed in two ways: one as (1/2)*base*height, and another as (1/2)*leg1*leg2. If I set these equal, I can solve for the height. So, if the legs are 'a' and 'b', and the hypotenuse is 'c', then the area is (1/2)*a*b. The height 'h' to the hypotenuse would satisfy (1/2)*c*h = (1/2)*a*b, so h = (a*b)/c. Is this equal to c/2? That would mean (a*b)/c = c/2, which implies that a*b = c²/2. But in a right triangle, by Pythagoras, a² + b² = c². So, unless a*b = c²/2, which isn't generally true, the height isn't half the hypotenuse. For example, take a 3-4-5 triangle. The area is (1/2)*3*4 = 6. The hypotenuse is 5, so the height h would be (2*6)/5 = 12/5 = 2.4, which is not half of 5 (which is 2.5). So, in this case, h is 2.4, which is close but not exactly half. Wait, actually, 12/5 is 2.4, and half of 5 is 2.5, so it's not equal. Therefore, the height to the hypotenuse isn't necessarily half the hypotenuse. So, C seems incorrect.Finally, option D: "The two diagonals of a square are equal in length and bisect each other perpendicularly." I know that in a square, the diagonals are equal in length and they bisect each other at 90 degrees. That's one of the properties of a square. So, D is correct.So, after going through each option, it seems like option C is the incorrect one because the height to the hypotenuse isn't necessarily half the hypotenuse in a right triangle. It depends on the specific lengths of the sides. So, I think the answer is C.

answer：Alright, so I have this problem here that says: For any pair of functions ( f(x) ) and ( g(x) ) defined on the closed interval ([0,1]), there exist ( 0 leq x leq 1 ) and ( 0 leq y leq 1 ) such that[|f(x) + g(y) - xy| geq frac{1}{4}]Hmm, okay. So I need to prove that no matter what functions ( f ) and ( g ) I pick on ([0,1]), I can always find some ( x ) and ( y ) in that interval where the absolute value of ( f(x) + g(y) - xy ) is at least ( frac{1}{4} ).Let me think about how to approach this. Maybe I can consider specific points in the interval ([0,1]) and evaluate the expression ( f(x) + g(y) - xy ) there. If I can show that at least one of these points meets the required inequality, that would solve the problem.What are the simplest points to consider? Probably the endpoints: ( x = 0 ), ( x = 1 ), ( y = 0 ), and ( y = 1 ). Let me evaluate the expression at these points.So, let's define:- ( a = f(0) + g(0) )- ( b = f(0) + g(1) )- ( c = f(1) + g(0) )- ( d = f(1) + g(1) - 1 )Wait, why subtract 1 in the case of ( d )? Because when ( x = 1 ) and ( y = 1 ), the term ( xy ) becomes ( 1 times 1 = 1 ), so we subtract that from ( f(1) + g(1) ).Now, let's see what these expressions represent. Each of them is ( f(x) + g(y) - xy ) evaluated at the corners of the unit square ([0,1] times [0,1]). So, ( a ) is at (0,0), ( b ) at (0,1), ( c ) at (1,0), and ( d ) at (1,1).If I can show that at least one of these four values ( a, b, c, d ) has an absolute value of at least ( frac{1}{4} ), then the proof is complete.But how can I relate these four values? Maybe by considering their sum or some combination. Let me try adding ( b ) and ( c ) and subtracting ( a ) and ( d ):[b + c - a - d = (f(0) + g(1)) + (f(1) + g(0)) - (f(0) + g(0)) - (f(1) + g(1) - 1)]Simplifying this:[b + c - a - d = f(0) + g(1) + f(1) + g(0) - f(0) - g(0) - f(1) - g(1) + 1]A lot of terms cancel out:- ( f(0) ) cancels with ( -f(0) )- ( f(1) ) cancels with ( -f(1) )- ( g(0) ) cancels with ( -g(0) )- ( g(1) ) cancels with ( -g(1) )So, we're left with:[b + c - a - d = 1]Interesting! So, the sum ( b + c - a - d ) equals 1. Now, since this sum is 1, and we have four terms ( b, c, -a, -d ), perhaps we can apply the pigeonhole principle here.The pigeonhole principle states that if you have more pigeons than pigeonholes, at least one pigeonhole must contain more than one pigeon. In this context, if we have four terms adding up to 1, at least one of them must be at least ( frac{1}{4} ) in absolute value.Wait, is that correct? Let me think. If the sum of four numbers is 1, does it imply that at least one of them is at least ( frac{1}{4} )?Actually, yes. Because if all four numbers were less than ( frac{1}{4} ) in absolute value, their sum could not reach 1. For example, if each term is at most ( frac{1}{4} ), then the maximum sum would be ( 4 times frac{1}{4} = 1 ), but that's only if all are positive and equal. However, since some terms could be negative, the total sum could be less than 1. But in our case, the sum is exactly 1, so at least one term must be at least ( frac{1}{4} ).Therefore, at least one of ( b, c, -a, -d ) must be greater than or equal to ( frac{1}{4} ) in absolute value. This means that at least one of the expressions ( f(x) + g(y) - xy ) evaluated at the corners of the unit square must satisfy the inequality ( |f(x) + g(y) - xy| geq frac{1}{4} ).So, this shows that there exist ( x ) and ( y ) in ([0,1]) such that the absolute value of ( f(x) + g(y) - xy ) is at least ( frac{1}{4} ).Wait, but does this cover all possible functions ( f ) and ( g )? I mean, what if the maximum occurs somewhere inside the interval, not just at the corners? Hmm, but the problem only requires the existence of some ( x ) and ( y ), not necessarily at the corners. However, by evaluating at the corners, we've already shown that at least one of them must satisfy the inequality. So, regardless of what happens inside the interval, we've found specific points where the inequality holds.Therefore, the proof is complete.**Final Answer**boxed{frac{1}{4}}

answer：Okay, so I have this problem where a parabola with the equation ( y = x^2 + bx + c ) passes through two points: ( (2, 12) ) and ( (-2, 8) ). I need to find the value of ( c ). Hmm, let's see. I remember that if a point lies on a parabola, its coordinates satisfy the equation of the parabola. So, I can plug in these points into the equation and get two equations with two variables, ( b ) and ( c ). Then, I can solve this system of equations to find ( c ).Alright, let's start with the first point ( (2, 12) ). Plugging ( x = 2 ) and ( y = 12 ) into the equation:( 12 = (2)^2 + b(2) + c )Calculating ( (2)^2 ) gives me 4, so:( 12 = 4 + 2b + c )Subtracting 4 from both sides:( 8 = 2b + c )Okay, so that's my first equation: ( 2b + c = 8 ).Now, moving on to the second point ( (-2, 8) ). Plugging ( x = -2 ) and ( y = 8 ) into the equation:( 8 = (-2)^2 + b(-2) + c )Calculating ( (-2)^2 ) also gives me 4, so:( 8 = 4 - 2b + c )Subtracting 4 from both sides:( 4 = -2b + c )That's my second equation: ( -2b + c = 4 ).Now I have a system of two equations:1. ( 2b + c = 8 )2. ( -2b + c = 4 )I need to solve for ( c ). Hmm, maybe I can add these two equations together to eliminate ( b ). Let's try that.Adding equation 1 and equation 2:( (2b + c) + (-2b + c) = 8 + 4 )Simplifying the left side:( 2b - 2b + c + c = 12 )Which simplifies to:( 0b + 2c = 12 )So, ( 2c = 12 ). Dividing both sides by 2:( c = 6 )Wait, so ( c ) is 6? Let me just check my steps to make sure I didn't make a mistake.First, plugging in ( (2, 12) ):( 12 = 4 + 2b + c ) leads to ( 2b + c = 8 ). That seems right.Then, plugging in ( (-2, 8) ):( 8 = 4 - 2b + c ) leads to ( -2b + c = 4 ). That also looks correct.Adding the two equations:( 2b + c = 8 )( -2b + c = 4 )Adding them gives ( 2c = 12 ), so ( c = 6 ). Yep, that checks out.I think that's the correct value for ( c ). I don't see any errors in my calculations, so I'm confident that ( c = 6 ).

answer：Alright, so I have this problem here about a sequence of distinct positive numbers, {a_n}, where their reciprocals form an arithmetic sequence. I need to find the value of the expression:frac{a_{1} a_{2} + a_{2} a_{3} + cdots + a_{2014} a_{2015}}{a_{1} a_{2015}}Hmm, okay. Let me try to unpack this step by step.First, the problem says that the reciprocals of the sequence {a_n} form an arithmetic sequence. So, that means if I take 1/a₁, 1/a₂, 1/a₃, ..., 1/a_{2015}, these terms form an arithmetic progression. An arithmetic sequence has a common difference, right? So, if I denote the common difference as d, then:frac{1}{a_{n+1}} - frac{1}{a_n} = d quad text{for all } nSo, this difference is constant throughout the sequence. That gives me a relationship between consecutive terms of the sequence {a_n}.Let me write that out explicitly for the first few terms:frac{1}{a_2} - frac{1}{a_1} = d frac{1}{a_3} - frac{1}{a_2} = d vdots frac{1}{a_{2015}} - frac{1}{a_{2014}} = dSo, each of these differences is equal to d. That seems useful.Now, the expression I need to evaluate is the sum of the products of consecutive terms in the sequence {a_n}, divided by the product of the first and last terms, a₁a_{2015}.Let me denote the numerator as S:S = a_1 a_2 + a_2 a_3 + cdots + a_{2014} a_{2015}So, the problem is asking for S divided by a₁a_{2015}.I need to find S / (a₁a_{2015}).Hmm, okay. Maybe I can find a relationship between the terms a_n and their reciprocals to express S in terms that can be simplified.Since the reciprocals form an arithmetic sequence, I can express 1/a_n in terms of 1/a₁ and the common difference d.In an arithmetic sequence, the nth term is given by:frac{1}{a_n} = frac{1}{a_1} + (n - 1)dSo, that gives me:a_n = frac{1}{frac{1}{a_1} + (n - 1)d}Alternatively, I can write this as:a_n = frac{a_1}{1 + (n - 1)a_1 d}Hmm, but I'm not sure if that's immediately helpful. Maybe I can think about the differences between the reciprocals.Given that:frac{1}{a_{n+1}} - frac{1}{a_n} = dI can rearrange this to express a_{n+1} in terms of a_n:frac{1}{a_{n+1}} = frac{1}{a_n} + d Rightarrow a_{n+1} = frac{1}{frac{1}{a_n} + d} Rightarrow a_{n+1} = frac{a_n}{1 + a_n d}So, each term is related to the previous term by this formula.Hmm, maybe I can use this recursive relationship to find a pattern or express a_{n} in terms of a₁.Alternatively, perhaps I can consider the sum S.Let me write S as:S = sum_{k=1}^{2014} a_k a_{k+1}So, S is the sum of the products of consecutive terms from a₁a₂ up to a_{2014}a_{2015}.I need to find S / (a₁a_{2015}).Hmm, perhaps I can find a telescoping sum or some pattern that allows me to express S in terms of a₁ and a_{2015}.Let me think about the expression for d.From the arithmetic sequence of reciprocals, we have:frac{1}{a_{n+1}} - frac{1}{a_n} = dWhich can be rewritten as:frac{a_n - a_{n+1}}{a_n a_{n+1}}} = dSo,a_n - a_{n+1} = d a_n a_{n+1}Hmm, that's interesting. So, each term a_n minus the next term a_{n+1} equals d times the product a_n a_{n+1}.So, if I rearrange this, I get:a_n a_{n+1} = frac{a_n - a_{n+1}}{d}Therefore, each product a_k a_{k+1} can be expressed as (a_k - a_{k+1}) / d.So, substituting this into the sum S:S = sum_{k=1}^{2014} a_k a_{k+1} = sum_{k=1}^{2014} frac{a_k - a_{k+1}}{d}So, S is equal to 1/d times the sum of (a_k - a_{k+1}) from k=1 to 2014.But this is a telescoping sum! Because when we expand it, most terms will cancel out.Let me write out the sum:sum_{k=1}^{2014} (a_k - a_{k+1}) = (a_1 - a_2) + (a_2 - a_3) + cdots + (a_{2014} - a_{2015})All the intermediate terms cancel out, leaving:a_1 - a_{2015}So, the sum simplifies to a₁ - a_{2015}.Therefore, S = (a₁ - a_{2015}) / d.So, now, going back to the expression we need to find:frac{S}{a_1 a_{2015}} = frac{(a_1 - a_{2015}) / d}{a_1 a_{2015}} = frac{a_1 - a_{2015}}{d a_1 a_{2015}}Now, I need to find the value of this expression.Let me recall that the reciprocals form an arithmetic sequence with common difference d. So, the total number of terms is 2015, since we go from a₁ to a_{2015}.In an arithmetic sequence, the nth term is given by:frac{1}{a_n} = frac{1}{a_1} + (n - 1)dSo, for n = 2015:frac{1}{a_{2015}} = frac{1}{a_1} + (2015 - 1)d = frac{1}{a_1} + 2014dSo, rearranging this:frac{1}{a_{2015}} - frac{1}{a_1} = 2014dWhich can be written as:frac{a_1 - a_{2015}}{a_1 a_{2015}}} = 2014dSo, from this equation, we have:frac{a_1 - a_{2015}}{a_1 a_{2015}}} = 2014d Rightarrow frac{a_1 - a_{2015}}{d a_1 a_{2015}}} = 2014Wait a second, that's exactly the expression we had earlier for S / (a₁a_{2015})!So,frac{S}{a_1 a_{2015}} = 2014Therefore, the value of the given expression is 2014.I think that makes sense. Let me recap to make sure I didn't make any mistakes.1. Recognized that the reciprocals form an arithmetic sequence with common difference d.2. Expressed the difference between reciprocals as d, leading to a relationship between consecutive terms.3. Expressed each product a_k a_{k+1} in terms of the difference a_k - a_{k+1} divided by d.4. Summed these products, recognizing the telescoping nature of the sum, which simplified to a₁ - a_{2015}.5. Substituted back into the expression to find S / (a₁a_{2015}).6. Used the arithmetic sequence formula to relate a₁ and a_{2015}, finding that the expression equals 2014.Everything seems to check out. I don't see any errors in the logic or calculations.**Final Answer**boxed{2014}

answer：First, I need to determine the total cost of purchasing the 48 backpacks, which is 576.Next, I'll calculate the revenue from each sales transaction:- Selling 17 backpacks at 18 each: 17 × 18 = 306- Selling 10 backpacks at 25 each: 10 × 25 = 250- Selling the remaining 21 backpacks at 22 each: 21 × 22 = 462Adding up all the revenues: 306 + 250 + 462 = 1018Finally, to find the profit, I'll subtract the total cost from the total revenue: 1018 - 576 = 442