💡Okay, so I've got this problem here about two ponds with frogs. Let me try to understand it step by step. The problem says that Pond A has twice as many frogs as Pond B. And it also tells me that there are 32 frogs in Pond A. I need to find out how many frogs are there in both ponds combined.Alright, first, let's break down what "twice as many" means. If Pond A has twice as many frogs as Pond B, that means the number of frogs in Pond A is two times the number in Pond B. So, if I let the number of frogs in Pond B be some number, say, X, then the number of frogs in Pond A would be 2 times X, which is 2X.But wait, the problem already gives me the number of frogs in Pond A, which is 32. So, if 2X equals 32, I can solve for X to find out how many frogs are in Pond B. Let me write that down:2X = 32To find X, I need to divide both sides of the equation by 2:X = 32 / 2X = 16So, Pond B has 16 frogs.Now, the question asks for the total number of frogs in both ponds combined. That means I need to add the number of frogs in Pond A and Pond B together.Number of frogs in Pond A = 32Number of frogs in Pond B = 16Total number of frogs = 32 + 16 = 48Let me double-check to make sure I didn't make any mistakes. If Pond B has 16 frogs, then twice that would be 32, which matches the number given for Pond A. Adding them together gives 48 frogs in total. That seems right.I think I've got it. There are 48 frogs in both ponds combined.

💡Alright, so I've got this problem here: For any positive integer ( n ), the sequence ( a_n ) is defined as ( a_n = n^2 + lambda n ). The question is asking for the range of the real number ( lambda ) such that the sequence ( {a_n} ) is increasing. Hmm, okay, let's break this down step by step.First, I need to recall what it means for a sequence to be increasing. If a sequence is increasing, each term must be greater than the previous one. In other words, for all positive integers ( n ), ( a_{n+1} > a_n ). That makes sense. So, I need to ensure that ( a_{n+1} - a_n > 0 ) for all ( n geq 1 ).Alright, let's compute ( a_{n+1} - a_n ). Given ( a_n = n^2 + lambda n ), then:[a_{n+1} = (n+1)^2 + lambda(n+1)]So, subtracting ( a_n ) from ( a_{n+1} ):[a_{n+1} - a_n = [(n+1)^2 + lambda(n+1)] - [n^2 + lambda n]]Let me expand ( (n+1)^2 ):[(n+1)^2 = n^2 + 2n + 1]So, plugging that back in:[a_{n+1} - a_n = [n^2 + 2n + 1 + lambda n + lambda] - [n^2 + lambda n]]Now, let's simplify this expression by subtracting term by term:- ( n^2 - n^2 = 0 )- ( 2n - 0 = 2n )- ( 1 - 0 = 1 )- ( lambda n - lambda n = 0 )- ( lambda - 0 = lambda )So, putting it all together:[a_{n+1} - a_n = 2n + 1 + lambda]Alright, so for the sequence to be increasing, this difference must be greater than zero for all positive integers ( n ):[2n + 1 + lambda > 0]Which can be rewritten as:[lambda > -2n - 1]Now, this inequality must hold true for all positive integers ( n ). So, ( lambda ) has to be greater than ( -2n - 1 ) for every ( n geq 1 ). Let me think about this. Since ( n ) is a positive integer, the smallest value ( n ) can take is 1. Plugging ( n = 1 ) into the inequality:[lambda > -2(1) - 1 = -3]So, ( lambda > -3 ).But wait, what about larger values of ( n )? For example, if ( n = 2 ):[lambda > -2(2) - 1 = -5]But ( -5 ) is less than ( -3 ), so if ( lambda > -3 ), it automatically satisfies ( lambda > -5 ) because ( -3 > -5 ). Similarly, for ( n = 3 ):[lambda > -2(3) - 1 = -7]Again, ( -7 ) is less than ( -3 ), so ( lambda > -3 ) still holds. It seems that as ( n ) increases, the lower bound for ( lambda ) becomes more negative, but since we're looking for ( lambda ) that satisfies the inequality for all ( n ), the most restrictive condition is when ( n ) is the smallest, which is ( n = 1 ).Therefore, the condition ( lambda > -3 ) ensures that ( a_{n+1} - a_n > 0 ) for all positive integers ( n ). Let me just double-check this reasoning. If ( lambda = -3 ), then:[a_{n+1} - a_n = 2n + 1 - 3 = 2n - 2]For ( n = 1 ):[2(1) - 2 = 0]So, ( a_{2} - a_1 = 0 ), which means the sequence isn't strictly increasing at ( n = 1 ). Therefore, ( lambda ) must be strictly greater than ( -3 ), not equal to it.What if ( lambda ) is less than ( -3 )? Let's say ( lambda = -4 ). Then:[a_{n+1} - a_n = 2n + 1 - 4 = 2n - 3]For ( n = 1 ):[2(1) - 3 = -1 < 0]Which means ( a_2 < a_1 ), so the sequence isn't increasing. Hence, ( lambda ) must indeed be greater than ( -3 ).Is there an upper bound on ( lambda )? The problem doesn't specify any constraints on how large ( lambda ) can be, just that the sequence must be increasing. If ( lambda ) is very large, say ( lambda = 1000 ), then:[a_{n+1} - a_n = 2n + 1 + 1000 = 2n + 1001]Which is clearly positive for all ( n geq 1 ). So, there's no upper limit on ( lambda ); it can be any real number greater than ( -3 ).Therefore, the range of ( lambda ) is all real numbers greater than ( -3 ), which is written as ( (-3, +infty) ).Just to recap:1. Start with the definition of an increasing sequence: ( a_{n+1} > a_n ) for all ( n ).2. Compute the difference ( a_{n+1} - a_n ).3. Simplify the difference to find an inequality involving ( lambda ).4. Determine the most restrictive condition on ( lambda ) by considering the smallest ( n ).5. Verify the result by checking edge cases.This seems solid. I don't think I've missed anything here. The key was realizing that the inequality must hold for all ( n ), so the smallest ( n ) gives the tightest bound on ( lambda ).**Final Answer**The range of the real number ( lambda ) is boxed{(-3, +infty)}.

💡Okay, so I have this problem where I need to find the probability that a randomly chosen lattice point inside a square will result in a line segment from (0, 2010) to that point containing an even number of lattice points, excluding the endpoints. Hmm, that sounds a bit complicated, but let me break it down step by step.First, I know that a lattice point is a point where both the x and y coordinates are integers. The square in question has vertices at (0,0), (101,0), (101,101), and (0,101). So, the lattice points inside this square would have x and y coordinates ranging from 1 to 100, right? Because we're excluding the boundary, so 0 and 101 are not included.Now, the line segment in question goes from (0, 2010) to some point (a, b) inside the square. I need to figure out when this line segment has an even number of lattice points, excluding the endpoints. I remember that the number of lattice points on a line segment between two points can be determined using the greatest common divisor (GCD) of the differences in their coordinates.Specifically, the formula for the number of lattice points on the segment between (x1, y1) and (x2, y2), excluding the endpoints, is GCD(|x2 - x1|, |y2 - y1|) - 1. So, applying that here, the number of lattice points on the segment from (0, 2010) to (a, b) would be GCD(a, |2010 - b|) - 1.We want this number to be even. So, GCD(a, |2010 - b|) - 1 should be even, which implies that GCD(a, |2010 - b|) should be odd. Because subtracting 1 from an odd number gives an even number.Now, when is the GCD of two numbers odd? Well, the GCD is odd if and only if both numbers are odd. Because if either number is even, the GCD would be even. So, for GCD(a, |2010 - b|) to be odd, both a and |2010 - b| must be odd.Let's think about |2010 - b|. Since 2010 is an even number, subtracting b from it will result in an even number if b is even, and an odd number if b is odd. So, |2010 - b| is odd if and only if b is odd.Therefore, for GCD(a, |2010 - b|) to be odd, both a and b must be odd. Because if a is odd and b is odd, then |2010 - b| is even minus odd, which is odd, and GCD of two odd numbers is odd. Wait, hold on, 2010 is even, so 2010 - b is even minus odd, which is odd. So, yes, if b is odd, |2010 - b| is odd. So, if both a and |2010 - b| are odd, then their GCD is odd.But wait, if a is even, then GCD(a, |2010 - b|) will be even regardless of |2010 - b|, right? Because if a is even, then the GCD will have at least a factor of 2. So, to have GCD(a, |2010 - b|) odd, a must be odd, and |2010 - b| must be odd, which as we saw, requires b to be odd.So, both a and b must be odd for the GCD to be odd, which in turn makes the number of lattice points even. Therefore, the number of favorable points is the number of points where both a and b are odd.Now, how many such points are there? Since a and b each range from 1 to 100, there are 100 choices for each. Half of them are odd, so 50 odd numbers for a and 50 odd numbers for b. Therefore, the number of favorable points is 50 * 50 = 2500.The total number of possible points is 100 * 100 = 10,000. So, the probability is 2500 / 10,000, which simplifies to 1/4. Wait, but hold on, the problem asks for the probability that the number of lattice points is even. But according to this, the probability is 1/4. But wait, I think I might have made a mistake here.Because earlier, I concluded that both a and b must be odd for the GCD to be odd, which would make the number of lattice points even. But actually, let's double-check that. If both a and b are odd, then |2010 - b| is odd, as 2010 is even. So, GCD(a, |2010 - b|) is GCD(odd, odd), which could be odd or even? Wait, no, GCD of two odd numbers is always odd because odd numbers don't have 2 as a factor. So, yes, GCD(a, |2010 - b|) is odd, which makes the number of lattice points even.But wait, the problem is asking for the probability that the number of lattice points is even. So, the favorable cases are when both a and b are odd, which is 2500 cases. So, the probability is 2500 / 10,000 = 1/4. But wait, that seems low. Let me think again.Wait, no, actually, if both a and b are odd, then the number of lattice points is even. But what if one of them is even? Let's see. If a is even and b is odd, then |2010 - b| is odd, so GCD(even, odd) is 1, which is odd. So, the number of lattice points would be 1 - 1 = 0, which is even. Wait, that's also even. Hmm, so maybe my earlier conclusion was wrong.Wait, let me clarify. If a is even and b is odd, then GCD(a, |2010 - b|) is GCD(even, odd). Since 2010 is even, |2010 - b| is odd when b is odd. So, GCD(even, odd) is 1, which is odd. Therefore, the number of lattice points is 1 - 1 = 0, which is even. Similarly, if a is odd and b is even, then |2010 - b| is even, because b is even, so 2010 - b is even. So, GCD(odd, even) is 1, which is odd. So, the number of lattice points is 1 - 1 = 0, which is even.Wait, so actually, the number of lattice points is even in two cases: when both a and b are odd, and when one is even and the other is odd. The only case when the number of lattice points is odd is when both a and b are even. Because then, |2010 - b| is even, so GCD(even, even) is at least 2, which is even. So, the number of lattice points is GCD(a, |2010 - b|) - 1, which is even - 1 = odd.Therefore, the number of lattice points is even in all cases except when both a and b are even. So, the number of favorable cases is total cases minus the cases where both a and b are even.Total cases: 100 * 100 = 10,000.Number of cases where both a and b are even: 50 * 50 = 2500.Therefore, the number of favorable cases is 10,000 - 2500 = 7500.So, the probability is 7500 / 10,000 = 3/4.Wait, so earlier I thought it was 1/4, but now I see it's actually 3/4. That makes more sense because the only case when the number of lattice points is odd is when both a and b are even, which is 1/4 of the cases. Therefore, the probability that the number of lattice points is even is 1 - 1/4 = 3/4.Let me verify this with an example. Suppose a = 1 (odd) and b = 1 (odd). Then, the line segment from (0, 2010) to (1,1). The difference in x is 1, difference in y is 2009. GCD(1, 2009) = 1, so number of lattice points is 0, which is even.If a = 2 (even) and b = 1 (odd). Then, difference in x is 2, difference in y is 2009. GCD(2, 2009) = 1, so number of lattice points is 0, which is even.If a = 1 (odd) and b = 2 (even). Difference in x is 1, difference in y is 2008. GCD(1, 2008) = 1, so number of lattice points is 0, which is even.If a = 2 (even) and b = 2 (even). Difference in x is 2, difference in y is 2008. GCD(2, 2008) = 2, so number of lattice points is 2 - 1 = 1, which is odd.So, yes, only when both a and b are even do we get an odd number of lattice points. Therefore, the probability is 3/4.I think that makes sense now. So, the final answer should be 3/4.

💡First, I'll define the work rates of a, b, and c as ( A ), ( B ), and ( C ) respectively, where these rates are in units of work per day.Based on the problem, I have the following equations for the combined work rates of each pair:1. ( A + B = frac{1}{3} ) (since a and b can complete the work in 3 days)2. ( B + C = frac{1}{4} ) (since b and c can complete the work in 4 days)3. ( C + A = frac{1}{6} ) (since c and a can complete the work in 6 days)To find the time it takes for c alone to complete the work, I need to determine ( C ). I'll start by adding all three equations together:[(A + B) + (B + C) + (C + A) = frac{1}{3} + frac{1}{4} + frac{1}{6}]Simplifying the left side gives:[2A + 2B + 2C]On the right side, I'll find a common denominator:[frac{4}{12} + frac{3}{12} + frac{2}{12} = frac{9}{12} = frac{3}{4}]So, the equation becomes:[2(A + B + C) = frac{3}{4}]Dividing both sides by 2:[A + B + C = frac{3}{8}]Now, I'll subtract the first equation (( A + B = frac{1}{3} )) from this result to find ( C ):[C = frac{3}{8} - frac{1}{3}]To subtract these fractions, I'll find a common denominator:[C = frac{9}{24} - frac{8}{24} = frac{1}{24}]Thus, ( C ), the work rate of c, is ( frac{1}{24} ) of the work per day. To find the time it takes for c to complete the work alone, I'll take the reciprocal of the work rate:[text{Time for c to complete the work} = frac{1}{frac{1}{24}} = 24 text{ days}]

💡First, I need to determine the quadrant for the angle (-950^circ 12'). To simplify, I'll convert the minutes to degrees. Since (12') is (frac{12}{60} = 0.2^circ), the angle becomes (-950.2^circ).Next, I'll find a coterminal angle by adding multiples of (360^circ) until the result is between (0^circ) and (360^circ). Adding (3 times 360^circ = 1080^circ) gives:[-950.2^circ + 1080^circ = 129.8^circ]This angle, (129.8^circ), falls between (90^circ) and (180^circ), placing it in the second quadrant.For the second angle, (-1575^circ), I'll find a coterminal angle by adding multiples of (360^circ). Adding (5 times 360^circ = 1800^circ) results in:[-1575^circ + 1800^circ = 225^circ]The angle (225^circ) is between (180^circ) and (270^circ), which means it lies in the third quadrant.

💡Alright, so I've got this problem about three quadratic polynomials: P(x), Q(x), and R(x). Each of them has positive leading coefficients and two distinct roots. The problem says that if I substitute the roots of R(x) into P(x) + Q(x), I get equal values. Similarly, substituting the roots of P(x) into Q(x) + R(x) gives equal values, and the same thing happens when substituting the roots of Q(x) into P(x) + R(x). I need to prove that the sums of the roots of these three polynomials are equal.Hmm, okay. Let me start by recalling some properties of quadratic polynomials. A quadratic polynomial can be written as P(x) = a(x - r1)(x - r2), where a is the leading coefficient, and r1 and r2 are the roots. The sum of the roots is (r1 + r2) = -b/a, where the polynomial is in the form ax² + bx + c. Since all leading coefficients are positive, the parabolas open upwards.Given that each polynomial has two distinct roots, their discriminants must be positive. That means for each polynomial, the square of the coefficient of x is greater than four times the product of the leading coefficient and the constant term.Now, the problem mentions substituting the roots of one polynomial into the sum of the other two. Let's break that down.First, let's denote the roots:- Let the roots of P(x) be α and β.- Let the roots of Q(x) be γ and δ.- Let the roots of R(x) be ε and ζ.So, P(x) = a(x - α)(x - β), Q(x) = b(x - γ)(x - δ), R(x) = c(x - ε)(x - ζ), where a, b, c are positive.The first condition is that substituting the roots of R(x) into P(x) + Q(x) yields equal values. So, P(ε) + Q(ε) = P(ζ) + Q(ζ). Similarly, substituting the roots of P(x) into Q(x) + R(x) gives Q(α) + R(α) = Q(β) + R(β). And substituting the roots of Q(x) into P(x) + R(x) gives P(γ) + R(γ) = P(δ) + R(δ).I need to translate these conditions into equations involving the roots and coefficients.Let me first compute P(ε) + Q(ε). Since ε is a root of R(x), R(ε) = 0. So, P(ε) + Q(ε) is just the value of P at ε plus the value of Q at ε.Similarly, P(ζ) + Q(ζ) is the value of P at ζ plus the value of Q at ζ. And these two are equal.So, P(ε) + Q(ε) = P(ζ) + Q(ζ).Similarly, for the other conditions.I think it might help to express these polynomials in standard form.Let me write P(x) = a x² + b x + c, Q(x) = d x² + e x + f, R(x) = g x² + h x + k.Wait, but that might get complicated with so many variables. Maybe it's better to stick with the factored form.Alternatively, perhaps I can use the fact that for a quadratic polynomial, the sum of the roots is -b/a, and the product is c/a.So, for P(x), sum of roots is α + β = -b/a, and product is αβ = c/a.Similarly for Q(x) and R(x).But I need to relate the conditions given.Wait, another thought: if I have a quadratic polynomial S(x) = P(x) + Q(x), then S(x) is also a quadratic polynomial, since the leading coefficients are positive and we're adding them.Similarly, S(x) = P(x) + Q(x) will have its own roots, but in this problem, we are evaluating S(x) at the roots of R(x) and getting equal values.So, S(ε) = S(ζ). Since S(x) is quadratic, if it takes the same value at two different points, ε and ζ, then those points are symmetric with respect to the vertex of the parabola S(x). That is, the axis of symmetry of S(x) is exactly halfway between ε and ζ.Similarly, when we evaluate Q(x) + R(x) at the roots of P(x), we get equal values, so those roots are symmetric with respect to the axis of symmetry of Q(x) + R(x). And the same for the other condition.So, perhaps the axes of symmetry of these sums are related to the roots of the third polynomial.Let me formalize this.Let me denote S(x) = P(x) + Q(x). Then, S(ε) = S(ζ). Since S(x) is quadratic, this implies that the axis of symmetry of S(x) is at x = (ε + ζ)/2.Similarly, let me denote T(x) = Q(x) + R(x). Then, T(α) = T(β), so the axis of symmetry of T(x) is at x = (α + β)/2.And similarly, let U(x) = P(x) + R(x). Then, U(γ) = U(δ), so the axis of symmetry of U(x) is at x = (γ + δ)/2.But S(x) = P(x) + Q(x), T(x) = Q(x) + R(x), and U(x) = P(x) + R(x). So, each of these sums is a quadratic polynomial, and their axes of symmetry are determined by the roots of the third polynomial.But since S(x), T(x), and U(x) are all quadratic polynomials, their axes of symmetry are determined by their coefficients.Wait, for a quadratic polynomial Ax² + Bx + C, the axis of symmetry is at x = -B/(2A).So, for S(x) = P(x) + Q(x), which is (a + d)x² + (b + e)x + (c + f), the axis of symmetry is at x = -(b + e)/(2(a + d)).Similarly, for T(x) = Q(x) + R(x), the axis of symmetry is at x = -(e + h)/(2(d + g)).And for U(x) = P(x) + R(x), the axis of symmetry is at x = -(b + h)/(2(a + g)).But from the earlier reasoning, the axis of symmetry of S(x) is at (ε + ζ)/2, which is the average of the roots of R(x). Similarly, the axis of symmetry of T(x) is at (α + β)/2, the average of the roots of P(x), and the axis of symmetry of U(x) is at (γ + δ)/2, the average of the roots of Q(x).Therefore, we have:1. -(b + e)/(2(a + d)) = (ε + ζ)/22. -(e + h)/(2(d + g)) = (α + β)/23. -(b + h)/(2(a + g)) = (γ + δ)/2But (ε + ζ) is the sum of the roots of R(x), which is equal to -h/g, since R(x) = g x² + h x + k, so sum of roots is -h/g.Similarly, (α + β) = -b/a, (γ + δ) = -e/d.So, substituting these into the equations:1. -(b + e)/(2(a + d)) = (-h/g)/22. -(e + h)/(2(d + g)) = (-b/a)/23. -(b + h)/(2(a + g)) = (-e/d)/2Simplify each equation by multiplying both sides by 2:1. -(b + e)/(a + d) = -h/g2. -(e + h)/(d + g) = -b/a3. -(b + h)/(a + g) = -e/dMultiply both sides by -1:1. (b + e)/(a + d) = h/g2. (e + h)/(d + g) = b/a3. (b + h)/(a + g) = e/dSo, now we have three equations:1. (b + e)/(a + d) = h/g2. (e + h)/(d + g) = b/a3. (b + h)/(a + g) = e/dThese equations relate the coefficients of the polynomials P(x), Q(x), and R(x). Our goal is to show that the sums of the roots are equal, which are:- For P(x): α + β = -b/a- For Q(x): γ + δ = -e/d- For R(x): ε + ζ = -h/gWe need to show that -b/a = -e/d = -h/g.Let me denote S_P = -b/a, S_Q = -e/d, S_R = -h/g. We need to show that S_P = S_Q = S_R.From the first equation: (b + e)/(a + d) = h/g.Let me express h in terms of b, e, a, d: h = g*(b + e)/(a + d).Similarly, from the second equation: (e + h)/(d + g) = b/a.Substitute h from the first equation into the second equation:(e + [g*(b + e)/(a + d)]) / (d + g) = b/aLet me simplify the numerator:e + [g*(b + e)/(a + d)] = [e*(a + d) + g*(b + e)] / (a + d)So, the entire expression becomes:[ [e*(a + d) + g*(b + e)] / (a + d) ] / (d + g) = b/aSimplify:[ e*(a + d) + g*(b + e) ] / [ (a + d)(d + g) ) ] = b/aMultiply both sides by (a + d)(d + g):e*(a + d) + g*(b + e) = (b/a)*(a + d)(d + g)Simplify the left side:e*a + e*d + g*b + g*e = e*a + e*d + g*b + g*eRight side:(b/a)*(a + d)(d + g) = b*(1 + d/a)*(d + g)Wait, maybe expand the right side:(b/a)*(a + d)(d + g) = (b/a)*(a*d + a*g + d² + d*g) = b*d + b*g + (b d²)/a + (b d g)/aSo, putting it all together:e*a + e*d + g*b + g*e = b*d + b*g + (b d²)/a + (b d g)/aLet me collect like terms:Left side: e*a + e*d + g*b + g*eRight side: b*d + b*g + (b d²)/a + (b d g)/aSubtract right side from both sides:e*a + e*d + g*b + g*e - b*d - b*g - (b d²)/a - (b d g)/a = 0Simplify:e*a + e*d + g*e - b*d - (b d²)/a - (b d g)/a = 0Factor terms:e*a + e*d + e*g - b*d - (b d²)/a - (b d g)/a = 0Factor e from the first three terms:e*(a + d + g) - b*d - (b d²)/a - (b d g)/a = 0Factor b from the last three terms:e*(a + d + g) - b*(d + (d²)/a + (d g)/a) = 0Let me factor d from the terms inside the parentheses:e*(a + d + g) - b*d*(1 + d/a + g/a) = 0Notice that 1 + d/a + g/a = (a + d + g)/aSo, we have:e*(a + d + g) - b*d*(a + d + g)/a = 0Factor out (a + d + g):(a + d + g)*(e - (b d)/a) = 0Since a, d, g are positive coefficients, (a + d + g) ≠ 0. Therefore:e - (b d)/a = 0 => e = (b d)/aSo, e = (b d)/aOkay, that's one relationship: e is equal to (b d)/a.Let me keep that in mind.Now, let's look at the third equation: (b + h)/(a + g) = e/dWe already have h from the first equation: h = g*(b + e)/(a + d)And we have e = (b d)/aSo, substitute e into h:h = g*(b + (b d)/a)/(a + d) = g*(b(1 + d/a))/(a + d) = g*b*(a + d)/a / (a + d) ) = g*b/aSo, h = (g b)/aSo, h = (g b)/aNow, substitute e and h into the third equation:(b + h)/(a + g) = e/dWe have h = (g b)/a, e = (b d)/aSo, left side: (b + (g b)/a)/(a + g) = b*(1 + g/a)/(a + g) = b*(a + g)/a / (a + g) ) = b/aRight side: e/d = (b d)/a / d = b/aSo, both sides equal b/a, which is consistent.So, that checks out.So, from the first two equations, we've deduced that e = (b d)/a and h = (g b)/a.Now, let's recall what we need to show: that the sums of the roots are equal.Sum of roots for P(x): S_P = -b/aSum of roots for Q(x): S_Q = -e/dSum of roots for R(x): S_R = -h/gWe have e = (b d)/a, so S_Q = -e/d = - (b d)/a / d = -b/aSimilarly, h = (g b)/a, so S_R = -h/g = - (g b)/a / g = -b/aTherefore, S_P = S_Q = S_R = -b/aHence, the sums of the roots of the three polynomials are equal.So, that's the conclusion.**Final Answer**The sums of the roots of the three polynomials are equal, so boxed{a_1 + a_2 = b_1 + b_2 = c_1 + c_2}.