answer：Okay, so I have this function here: ( f(x) = left( frac{1}{2} right)^{x^2 - 2x + 3} ). I need to find the interval where this function is monotonically increasing. Hmm, let me think about how to approach this.First, I remember that when dealing with functions of the form ( a^{g(x)} ), the behavior of the function depends on the base ( a ) and the exponent ( g(x) ). Since the base here is ( frac{1}{2} ), which is between 0 and 1, I recall that exponential functions with bases in this range are decreasing functions. That means as the exponent increases, the overall function decreases, and vice versa.So, if I let ( t = x^2 - 2x + 3 ), then the function becomes ( f(x) = left( frac{1}{2} right)^t ). Since ( frac{1}{2} ) is less than 1, ( f(x) ) will increase when ( t ) decreases and decrease when ( t ) increases. Therefore, to find where ( f(x) ) is increasing, I need to find where ( t ) is decreasing.Now, ( t ) is a quadratic function: ( t = x^2 - 2x + 3 ). Quadratic functions have a parabola shape, and their direction (opening upwards or downwards) depends on the coefficient of ( x^2 ). In this case, the coefficient is 1, which is positive, so the parabola opens upwards. That means the function ( t ) has a minimum point and is decreasing to the left of this minimum and increasing to the right.To find the vertex of the parabola, which gives the minimum point, I can use the formula for the vertex of a quadratic ( ax^2 + bx + c ). The x-coordinate of the vertex is at ( x = -frac{b}{2a} ). Here, ( a = 1 ) and ( b = -2 ), so:[x = -frac{-2}{2 times 1} = frac{2}{2} = 1]So, the vertex is at ( x = 1 ). Since the parabola opens upwards, the function ( t ) is decreasing for all ( x ) less than 1 and increasing for all ( x ) greater than 1.Therefore, ( t ) is decreasing on the interval ( (-infty, 1) ). As established earlier, since ( f(x) = left( frac{1}{2} right)^t ) increases when ( t ) decreases, the function ( f(x) ) is increasing on the same interval where ( t ) is decreasing.So, putting it all together, the interval where ( f(x) ) is monotonically increasing is ( (-infty, 1) ).**Final Answer**The interval of monotonic increase for ( f(x) ) is boxed{(-infty, 1)}.

answer：Okay, so I have this problem about two triangles, ABC and ADE, with areas 4000 and 12000 respectively. The coordinates given are B=(0,0), C=(400,0), D=(1000,500), and E=(1010,515). I need to find the sum of all possible x-coordinates of point A.First, I remember that the area of a triangle can be calculated using the formula: (1/2)*base*height. For triangle ABC, the base is BC, which is from (0,0) to (400,0), so that's 400 units long. The area is 4000, so I can set up the equation:4000 = (1/2)*400*heightSolving for height, I get:height = (4000*2)/400 = 20So, the height from point A to base BC must be 20 units. Since BC is along the x-axis, the y-coordinate of point A must be either 20 or -20. That means point A lies somewhere on the lines y=20 or y=-20.Now, moving on to triangle ADE. The area is 12000. I need to relate this to the coordinates of A. To find the area of triangle ADE, I can use the formula involving the determinant, which is:Area = (1/2)| (x_A(y_D - y_E) + x_D(y_E - y_A) + x_E(y_A - y_D) ) |Plugging in the coordinates D=(1000,500) and E=(1010,515), let me denote A as (a, b). Then the area becomes:12000 = (1/2)| a*(500 - 515) + 1000*(515 - b) + 1010*(b - 500) |Simplify the terms inside the absolute value:500 - 515 = -15515 - b = 515 - bb - 500 = b - 500So, substituting back:12000 = (1/2)| a*(-15) + 1000*(515 - b) + 1010*(b - 500) |Let me compute each term:First term: -15aSecond term: 1000*(515 - b) = 1000*515 - 1000b = 515000 - 1000bThird term: 1010*(b - 500) = 1010b - 1010*500 = 1010b - 505000Now, combine all terms:-15a + 515000 - 1000b + 1010b - 505000Simplify:-15a + (515000 - 505000) + (-1000b + 1010b)Which is:-15a + 10000 + 10bSo, the equation becomes:12000 = (1/2)| -15a + 10b + 10000 |Multiply both sides by 2:24000 = | -15a + 10b + 10000 |So, we have two cases:Case 1: -15a + 10b + 10000 = 24000Case 2: -15a + 10b + 10000 = -24000Simplify both cases:Case 1: -15a + 10b = 14000Case 2: -15a + 10b = -34000Now, since we know from triangle ABC that b is either 20 or -20, let's substitute these values into both cases.First, let's take b = 20.Case 1 with b=20:-15a + 10*20 = 14000-15a + 200 = 14000-15a = 14000 - 200 = 13800a = 13800 / (-15) = -920Case 2 with b=20:-15a + 10*20 = -34000-15a + 200 = -34000-15a = -34000 - 200 = -34200a = (-34200)/(-15) = 2280Now, take b = -20.Case 1 with b=-20:-15a + 10*(-20) = 14000-15a - 200 = 14000-15a = 14000 + 200 = 14200a = 14200 / (-15) ≈ -946.666...Wait, that's a fractional value, but maybe it's okay. Let me check my calculations again.Wait, 14200 divided by -15 is indeed -946.666..., which is -946 and 2/3. Hmm, but let me see if I made a mistake earlier.Wait, when I substituted b=-20 into Case 1:-15a + 10*(-20) = 14000-15a - 200 = 14000-15a = 14000 + 200 = 14200a = 14200 / (-15) = -946.666...Similarly, Case 2 with b=-20:-15a + 10*(-20) = -34000-15a - 200 = -34000-15a = -34000 + 200 = -33800a = (-33800)/(-15) ≈ 2253.333...Wait, but these are fractional x-coordinates. Is that acceptable? Well, coordinates can be fractions, so maybe that's okay.But let me double-check my initial setup because I might have made a mistake in the area formula.Alternatively, maybe I should approach this differently. Instead of using the determinant formula, perhaps I can use vectors or the shoelace formula.Wait, another approach is to find the equation of line DE and then compute the distance from point A to line DE, since the area can also be expressed as (1/2)*base*height, where base is DE and height is the distance from A to DE.Let me try that approach.First, find the equation of line DE. Points D=(1000,500) and E=(1010,515).The slope of DE is (515 - 500)/(1010 - 1000) = 15/10 = 3/2.So, the equation of DE can be written as y - 500 = (3/2)(x - 1000).Multiplying both sides by 2 to eliminate the fraction:2(y - 500) = 3(x - 1000)2y - 1000 = 3x - 3000Rearranging:-3x + 2y + 2000 = 0Or, 3x - 2y - 2000 = 0Wait, let me check that again.Starting from y - 500 = (3/2)(x - 1000)Multiply both sides by 2:2y - 1000 = 3x - 3000Bring all terms to one side:-3x + 2y + 2000 = 0Alternatively, 3x - 2y - 2000 = 0.Yes, that's correct.Now, the distance from point A=(a,b) to line DE is given by the formula:Distance = |3a - 2b - 2000| / sqrt(3^2 + (-2)^2) = |3a - 2b - 2000| / sqrt(13)The length of DE can be calculated using the distance formula:DE = sqrt[(1010 - 1000)^2 + (515 - 500)^2] = sqrt[10^2 + 15^2] = sqrt[100 + 225] = sqrt[325] = 5*sqrt(13)So, the area of triangle ADE is (1/2)*DE*height = (1/2)*(5*sqrt(13))*(|3a - 2b - 2000| / sqrt(13)) ) = (1/2)*5*|3a - 2b - 2000|Set this equal to 12000:(5/2)*|3a - 2b - 2000| = 12000Multiply both sides by 2/5:|3a - 2b - 2000| = (12000*2)/5 = 4800So, 3a - 2b - 2000 = ±4800Which gives two equations:1) 3a - 2b = 68002) 3a - 2b = -2800Now, since from triangle ABC, we know that b is either 20 or -20, let's substitute these values into both equations.First, b=20:Equation 1: 3a - 2*20 = 6800 → 3a - 40 = 6800 → 3a = 6840 → a = 2280Equation 2: 3a - 2*20 = -2800 → 3a - 40 = -2800 → 3a = -2760 → a = -920Now, b=-20:Equation 1: 3a - 2*(-20) = 6800 → 3a + 40 = 6800 → 3a = 6760 → a = 6760/3 ≈ 2253.333...Equation 2: 3a - 2*(-20) = -2800 → 3a + 40 = -2800 → 3a = -2840 → a = -2840/3 ≈ -946.666...Wait, so these are the same results as before, but expressed as fractions. So, the x-coordinates are 2280, -920, 2253.333..., and -946.666...But the problem asks for the sum of all possible x-coordinates of A. So, let's add them up:2280 + (-920) + 2253.333... + (-946.666...) =First, 2280 - 920 = 1360Then, 2253.333... - 946.666... = 1306.666...Adding these together: 1360 + 1306.666... = 2666.666...Wait, that's approximately 2666.666..., but let me check if I did the addition correctly.Wait, 2280 - 920 is indeed 1360.2253.333... - 946.666... is 2253.333 - 946.666 = 1306.666...So, 1360 + 1306.666... = 2666.666...But 2666.666... is 2666 and 2/3, which is 8000/3.Wait, but that doesn't seem to match the initial thought of 2960.Wait, maybe I made a mistake in the area calculation.Wait, let me go back.When I used the determinant method, I got equations leading to a sum of 2960, but when I used the distance method, I got a different sum.Wait, perhaps I made a mistake in the determinant approach.Wait, let me check the determinant approach again.The area formula using determinant is:Area = (1/2)|x_A(y_D - y_E) + x_D(y_E - y_A) + x_E(y_A - y_D)|Plugging in the values:= (1/2)|a*(500 - 515) + 1000*(515 - b) + 1010*(b - 500)|= (1/2)|a*(-15) + 1000*(515 - b) + 1010*(b - 500)|= (1/2)|-15a + 515000 - 1000b + 1010b - 505000|= (1/2)|-15a + 10000 + 10b|Set equal to 12000:(1/2)|-15a + 10b + 10000| = 12000Multiply both sides by 2:|-15a + 10b + 10000| = 24000So, -15a + 10b + 10000 = ±24000Which gives:-15a + 10b = 14000 or -15a + 10b = -34000Now, with b=20:Case 1: -15a + 200 = 14000 → -15a = 13800 → a = -920Case 2: -15a + 200 = -34000 → -15a = -34200 → a = 2280With b=-20:Case 1: -15a - 200 = 14000 → -15a = 14200 → a = -14200/15 ≈ -946.666...Case 2: -15a - 200 = -34000 → -15a = -33800 → a = 33800/15 ≈ 2253.333...So, the x-coordinates are -920, 2280, -946.666..., and 2253.333...Adding them up:-920 + 2280 = 1360-946.666... + 2253.333... = 1306.666...Total sum: 1360 + 1306.666... = 2666.666...But earlier, when I used the distance method, I got the same result.Wait, but the initial thought process in the problem statement got 2960. So, where is the discrepancy?Wait, perhaps I made a mistake in the distance method.Wait, let me re-examine the distance method.The area of triangle ADE is (1/2)*DE*height, where height is the distance from A to DE.DE length: sqrt[(1010-1000)^2 + (515-500)^2] = sqrt[10^2 + 15^2] = sqrt[100 + 225] = sqrt[325] = 5*sqrt(13)So, area = (1/2)*5*sqrt(13)*distance = 12000Thus, (5/2)*sqrt(13)*distance = 12000So, distance = (12000*2)/(5*sqrt(13)) = (24000)/(5*sqrt(13)) = 4800/sqrt(13)But distance from A to DE is |3a - 2b - 2000|/sqrt(13) = 4800/sqrt(13)Thus, |3a - 2b - 2000| = 4800So, 3a - 2b - 2000 = ±4800Which gives 3a - 2b = 6800 or 3a - 2b = -2800Now, with b=20:3a - 40 = 6800 → 3a = 6840 → a=22803a - 40 = -2800 → 3a = -2760 → a=-920With b=-20:3a +40 = 6800 → 3a=6760 → a=6760/3≈2253.333...3a +40 = -2800 → 3a=-2840 → a=-2840/3≈-946.666...So, same results as before.Thus, the sum is 2280 + (-920) + 2253.333... + (-946.666...) = 2280 - 920 + 2253.333 - 946.666Calculating step by step:2280 - 920 = 13602253.333 - 946.666 = 1306.6661360 + 1306.666 = 2666.666...Which is 8000/3 ≈ 2666.666...Wait, but the initial problem statement's solution got 2960. So, where is the mistake?Wait, perhaps I made a mistake in the initial setup of the determinant formula.Wait, let me double-check the determinant formula.The area of triangle ADE can be calculated using the formula:Area = (1/2)| (x_A(y_D - y_E) + x_D(y_E - y_A) + x_E(y_A - y_D) ) |Plugging in the values:= (1/2)|a*(500 - 515) + 1000*(515 - b) + 1010*(b - 500)|= (1/2)|a*(-15) + 1000*(515 - b) + 1010*(b - 500)|= (1/2)|-15a + 515000 - 1000b + 1010b - 505000|= (1/2)|-15a + 10000 + 10b|So, that's correct.Set equal to 12000:(1/2)|-15a + 10b + 10000| = 12000Multiply by 2:|-15a + 10b + 10000| = 24000So, -15a + 10b + 10000 = ±24000Which gives:-15a + 10b = 14000 or -15a + 10b = -34000Now, solving for a when b=20:Case 1: -15a + 200 = 14000 → -15a = 13800 → a = -920Case 2: -15a + 200 = -34000 → -15a = -34200 → a = 2280When b=-20:Case 1: -15a - 200 = 14000 → -15a = 14200 → a = -14200/15 ≈ -946.666...Case 2: -15a - 200 = -34000 → -15a = -33800 → a = 33800/15 ≈ 2253.333...So, the x-coordinates are -920, 2280, -946.666..., and 2253.333...Summing them up:-920 + 2280 = 1360-946.666... + 2253.333... = 1306.666...Total sum: 1360 + 1306.666... = 2666.666...Which is 8000/3.Wait, but 8000/3 is approximately 2666.666..., but the initial solution got 2960.Wait, perhaps the initial solution made a mistake in the equation setup.Wait, in the initial solution, the user wrote:"Next, the line DE has equation x - 2y - 850 = 0."Wait, that's different from what I got. I got 3x - 2y - 2000 = 0.Wait, let me check the equation of DE again.Points D=(1000,500) and E=(1010,515).Slope is (515-500)/(1010-1000) = 15/10 = 3/2.So, point-slope form: y - 500 = (3/2)(x - 1000)Multiply both sides by 2: 2y - 1000 = 3x - 3000Bring all terms to left: -3x + 2y + 2000 = 0 or 3x - 2y - 2000 = 0.So, the equation is 3x - 2y - 2000 = 0.But the initial solution said x - 2y - 850 = 0. That must be incorrect.Wait, let me see how they got x - 2y - 850 = 0.Wait, perhaps they made a mistake in calculating the slope or the equation.Wait, if they thought the slope was 1/2 instead of 3/2, that would change the equation.Because if slope is 1/2, then equation would be y - 500 = (1/2)(x - 1000)Multiply by 2: 2y - 1000 = x - 1000Bring all terms to left: -x + 2y = 0 → x - 2y = 0But that's not matching. Wait, no.Wait, let me recalculate.If slope is 1/2, then:y - 500 = (1/2)(x - 1000)Multiply by 2: 2y - 1000 = x - 1000Bring all terms to left: -x + 2y = 0 → x - 2y = 0But that's not correct because the slope is actually 3/2, not 1/2.So, the initial solution must have made a mistake in calculating the slope, leading to an incorrect equation for DE.Therefore, their subsequent calculations are based on an incorrect equation, leading to the wrong sum of 2960.So, the correct equation is 3x - 2y - 2000 = 0, leading to the sum of x-coordinates being 8000/3 ≈ 2666.666...But since the problem asks for the sum, and 8000/3 is an exact value, perhaps we can express it as a fraction.But let me see if there's another way to approach this without fractions.Wait, perhaps I can express the sum of the roots without solving for each a.Looking back at the equations:From triangle ABC, b=20 or b=-20.From triangle ADE, we have two equations for each b:For b=20:-15a + 10*20 = 14000 → -15a = 14000 - 200 = 13800 → a = -920-15a + 10*20 = -34000 → -15a = -34000 - 200 = -34200 → a = 2280For b=-20:-15a + 10*(-20) = 14000 → -15a - 200 = 14000 → -15a = 14200 → a = -14200/15 = -2840/3 ≈ -946.666...-15a + 10*(-20) = -34000 → -15a - 200 = -34000 → -15a = -33800 → a = 33800/15 = 6760/3 ≈ 2253.333...So, the four x-coordinates are:a1 = -920a2 = 2280a3 = -2840/3a4 = 6760/3Now, sum them up:Sum = a1 + a2 + a3 + a4 = (-920) + 2280 + (-2840/3) + (6760/3)Combine the fractions:(-2840/3 + 6760/3) = (6760 - 2840)/3 = 3920/3Now, combine with the integers:(-920 + 2280) = 1360So, total sum = 1360 + 3920/3Convert 1360 to thirds: 1360 = 4080/3So, total sum = 4080/3 + 3920/3 = (4080 + 3920)/3 = 8000/3Which is approximately 2666.666...So, the exact sum is 8000/3.But the initial solution got 2960, which is incorrect because of the wrong equation for DE.Therefore, the correct sum of all possible x-coordinates of A is 8000/3.But wait, the problem might expect an integer answer, so perhaps I made a mistake somewhere.Wait, let me check the determinant approach again.Wait, in the determinant approach, I got:Area = (1/2)|-15a + 10b + 10000| = 12000Which led to |-15a + 10b + 10000| = 24000So, -15a + 10b = 14000 or -34000But when I solved for a with b=20 and b=-20, I got fractional a's when b=-20.But perhaps there's another way to interpret the problem.Wait, maybe the area formula using vectors or shoelace formula is more accurate.Alternatively, perhaps the initial solution's equation for DE was correct, and I made a mistake.Wait, let me recalculate the equation of DE.Points D=(1000,500) and E=(1010,515).Slope m = (515 - 500)/(1010 - 1000) = 15/10 = 3/2.So, equation is y - 500 = (3/2)(x - 1000)Multiply both sides by 2: 2y - 1000 = 3x - 3000Bring all terms to left: -3x + 2y + 2000 = 0Or, 3x - 2y - 2000 = 0So, the initial solution's equation was x - 2y - 850 = 0, which is incorrect.Therefore, their subsequent calculations are wrong.Thus, the correct sum is 8000/3.But 8000/3 is approximately 2666.666..., which is not an integer, but the problem might accept it as a fraction.Alternatively, perhaps I made a mistake in interpreting the area formula.Wait, let me try another approach.Using vectors, the area of triangle ADE can be calculated as (1/2)| (E - D) × (A - D) |.Where × denotes the cross product.Compute vector DE: E - D = (1010-1000, 515-500) = (10,15)Vector DA: A - D = (a - 1000, b - 500)Cross product in 2D is scalar: (10)(b - 500) - (15)(a - 1000) = 10b - 5000 -15a + 15000 = -15a + 10b + 10000So, area is (1/2)|-15a + 10b + 10000| = 12000Which is the same as before.Thus, |-15a + 10b + 10000| = 24000So, same equations.Thus, the sum of x-coordinates is 8000/3.But the initial solution got 2960, which is incorrect.Therefore, the correct answer is 8000/3, which is approximately 2666.666...But since the problem asks for the sum, and it's likely expecting an exact value, I'll present it as 8000/3.But wait, let me check if 8000/3 can be simplified or if I made a mistake in the sum.Wait, the four x-coordinates are:a1 = -920a2 = 2280a3 = -2840/3a4 = 6760/3Sum = (-920) + 2280 + (-2840/3) + (6760/3)Combine the fractions:(-2840 + 6760)/3 = 3920/3Combine the integers:(-920 + 2280) = 1360Convert 1360 to thirds: 1360 = 4080/3Total sum: 4080/3 + 3920/3 = 8000/3Yes, that's correct.Therefore, the sum of all possible x-coordinates of A is 8000/3.But the initial solution got 2960, which is incorrect because of an error in the equation of DE.So, the correct answer is 8000/3, which is approximately 2666.666...But since the problem might expect an integer, perhaps I made a mistake in interpreting the problem.Wait, let me check the initial problem statement again."Triangles ABC and ADE have areas 4000 and 12000, respectively, with B=(0,0), C=(400,0), D=(1000,500), and E=(1010,515). What is the sum of all possible x-coordinates of A?"Wait, perhaps the initial solution was correct, and I made a mistake in the equation of DE.Wait, let me recalculate the equation of DE.Points D=(1000,500) and E=(1010,515).Slope m = (515 - 500)/(1010 - 1000) = 15/10 = 3/2.So, equation is y - 500 = (3/2)(x - 1000)Multiply both sides by 2: 2y - 1000 = 3x - 3000Bring all terms to left: -3x + 2y + 2000 = 0Or, 3x - 2y - 2000 = 0So, the initial solution's equation was x - 2y - 850 = 0, which is incorrect.Thus, their calculations are wrong.Therefore, the correct sum is 8000/3.But 8000/3 is approximately 2666.666..., which is not an integer.Wait, but the initial solution got 2960, which is an integer.Perhaps I made a mistake in the determinant approach.Wait, let me try another method.Using the shoelace formula for triangle ADE.Coordinates:A=(a,b), D=(1000,500), E=(1010,515)Shoelace formula:Area = (1/2)| (a*500 + 1000*515 + 1010*b) - (b*1000 + 500*1010 + 515*a) |Simplify:= (1/2)|500a + 515000 + 1010b - 1000b - 505000 - 515a|= (1/2)|(500a - 515a) + (1010b - 1000b) + (515000 - 505000)|= (1/2)|(-15a) + (10b) + 10000|Which is the same as before.Thus, the area is (1/2)|-15a + 10b + 10000| = 12000So, |-15a + 10b + 10000| = 24000Thus, same equations.Therefore, the sum is 8000/3.So, the correct answer is 8000/3.But since the problem might expect an integer, perhaps I made a mistake in interpreting the problem.Alternatively, perhaps the initial solution's equation was correct, and I made a mistake.Wait, let me check the initial solution's equation of DE.They said: "Next, the line DE has equation x - 2y - 850 = 0."Let me check if that's correct.If the equation is x - 2y - 850 = 0, then plugging in D=(1000,500):1000 - 2*500 - 850 = 1000 - 1000 - 850 = -850 ≠ 0So, that's incorrect. Therefore, their equation is wrong.Thus, their solution is incorrect.Therefore, the correct sum is 8000/3.But since the problem might expect an integer, perhaps I made a mistake in the area calculation.Wait, let me check the area using the shoelace formula again.Wait, perhaps I missed a term.Wait, shoelace formula for three points A, D, E:Area = (1/2)|x_A(y_D - y_E) + x_D(y_E - y_A) + x_E(y_A - y_D)|Which is what I used before.Yes, that's correct.Thus, the sum is 8000/3.Therefore, the answer is 8000/3.But since the problem asks for the sum, and it's likely expecting an exact value, I'll present it as 8000/3.But wait, 8000/3 is approximately 2666.666..., which is not an integer, but the problem might accept it as a fraction.Alternatively, perhaps I made a mistake in the initial setup.Wait, perhaps the area of triangle ADE is 12000, which is three times the area of ABC, which is 4000.But that might not directly help.Alternatively, perhaps the initial solution's mistake was in the equation of DE, leading to incorrect a values.Thus, the correct sum is 8000/3.Therefore, the sum of all possible x-coordinates of A is 8000/3.

answer：Alright, so I've got this problem here about balloons in a room. There are 85 balloons total, and they're either red or blue. The problem gives two pieces of information: first, that there's at least one red balloon, and second, that if you pick any two balloons, at least one of them is blue. I need to figure out how many red balloons there are.Okay, let's break this down. There are 85 balloons, some red and some blue. The first condition is straightforward: there's at least one red balloon. So, the number of red balloons, let's call that R, is at least 1. The second condition is a bit trickier: in every pair of balloons, at least one is blue. That means if I pick any two balloons, they can't both be red. So, there can't be two red balloons in the room because if there were, I could pick those two, and both would be red, which violates the condition.Wait, so if there can't be two red balloons, that means there can only be one red balloon. Because if there were two or more, then I could pick those two, and both would be red. So, that must mean there's only one red balloon. Let me check that logic again.If there's only one red balloon, then no matter which two balloons I pick, one of them has to be blue because there's only one red one. So, that satisfies the second condition. And since there's at least one red balloon, that also satisfies the first condition. So, that makes sense.But wait, what if there were zero red balloons? Then all 85 balloons would be blue, and any pair I pick would definitely have at least one blue balloon. But the problem says there's at least one red balloon, so zero red balloons isn't an option. Therefore, the only possibility is one red balloon and 84 blue balloons.Let me think if there's any other way this could work. Suppose there were two red balloons. Then, if I pick those two, both would be red, which violates the second condition. So, two red balloons don't work. What about three? Same problem. Any number greater than one would allow for the possibility of picking two red balloons, which isn't allowed. So, one red balloon is the only number that fits both conditions.Therefore, the number of red balloons must be one.

answer：First, I need to calculate the values of (7^3) and (9^3).Calculating (7^3):[7^3 = 7 times 7 times 7 = 343]Calculating (9^3):[9^3 = 9 times 9 times 9 = 729]Next, I add these two results together:[343 + 729 = 1072]Now, I need to find the cube root of 1072:[sqrt[3]{1072}]To determine which integer is closest to this cube root, I consider the cubes of nearby integers:[10^3 = 1000 quad text{and} quad 11^3 = 1331]Since 1072 is closer to 1000 than to 1331, the cube root of 1072 is closer to 10.

answer：Alright, so I have this triangle ABC, and I know a few things about it. First off, the length of side BC is 6 units, and the length of side AC is 4 units. Also, I'm given that the sine of angle A is 3/4. My goal is to find the measure of angle B. Hmm, okay, let's break this down step by step.First, I remember that in any triangle, the sum of the angles is 180 degrees, or π radians. So, if I can find two angles, I can find the third one by subtracting the sum from π. But right now, I only know one angle's sine, which is angle A. Maybe I can use the Law of Sines here. The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. So, that is:[frac{a}{sin A} = frac{b}{sin B} = frac{c}{sin C}]In this case, side BC is opposite angle A, and side AC is opposite angle B. So, if I let side BC be 'a' and side AC be 'b', then according to the Law of Sines:[frac{BC}{sin A} = frac{AC}{sin B}]Plugging in the values I have:[frac{6}{sin A} = frac{4}{sin B}]I know that (sin A = frac{3}{4}), so substituting that in:[frac{6}{frac{3}{4}} = frac{4}{sin B}]Simplifying the left side:[6 div frac{3}{4} = 6 times frac{4}{3} = 8]So now the equation is:[8 = frac{4}{sin B}]To solve for (sin B), I can rearrange the equation:[sin B = frac{4}{8} = frac{1}{2}]Okay, so (sin B = frac{1}{2}). Now, I need to find angle B such that its sine is 1/2. I remember that (sin 30^circ = frac{1}{2}) and (sin 150^circ = frac{1}{2}). So, angle B could be either 30 degrees or 150 degrees. But wait, can angle B be 150 degrees?Let me think about the triangle. The sum of angles in a triangle is 180 degrees. If angle B were 150 degrees, then angle A and angle C would have to add up to 30 degrees. But angle A is already given with a sine of 3/4. Let me calculate angle A first to see if that makes sense.Since (sin A = frac{3}{4}), angle A is (arcsinleft(frac{3}{4}right)). Calculating that, (arcsin(0.75)) is approximately 48.59 degrees. So, angle A is roughly 48.59 degrees.If angle B were 150 degrees, then angle C would be 180 - 48.59 - 150 = negative 18.59 degrees, which doesn't make sense because angles can't be negative. Therefore, angle B can't be 150 degrees. So, angle B must be 30 degrees.Wait a second, let me double-check this. If angle B is 30 degrees, then angle C would be 180 - 48.59 - 30 = approximately 101.41 degrees. That seems reasonable because all angles would be positive and add up to 180 degrees.But just to be thorough, let me also consider the sides. In a triangle, the larger side is opposite the larger angle. Here, side BC is 6 units, which is longer than side AC, which is 4 units. So, angle A, opposite side BC, should be larger than angle B, opposite side AC. Since angle A is approximately 48.59 degrees, angle B being 30 degrees makes sense because 48.59 > 30. If angle B were 150 degrees, it would be larger than angle A, which contradicts the side lengths because side BC is longer than AC, so angle A should be larger. Therefore, angle B must be 30 degrees.So, after considering both the angle sum and the side lengths, I'm confident that angle B is 30 degrees, which is π/6 radians.**Final Answer**The measure of angle ( B ) is boxed{dfrac{pi}{6}}.

answer：Okay, so I have this equation here: 4x minus 5y equals 9. I need to express y in terms of x. Hmm, that means I need to solve for y, right? Let me think about how to do that.First, I remember that to solve for a variable, I need to isolate it on one side of the equation. So, I want to get y by itself. The equation is 4x minus 5y equals 9. Maybe I can start by moving the 4x to the other side to get rid of it from the left side.If I subtract 4x from both sides, that should do it. Let me write that down:4x - 5y = 9Subtracting 4x from both sides gives:-5y = -4x + 9Wait, is that right? Let me check. Yes, subtracting 4x from both sides cancels it out on the left, leaving -5y, and on the right, it's -4x plus 9.Now, I have -5y equals -4x plus 9. I need to get y by itself, so I should divide both sides by -5 to solve for y. Let me do that:Dividing both sides by -5:y = (-4x + 9) / (-5)Hmm, that looks a bit messy with the negatives. Maybe I can simplify that. If I factor out a negative from the numerator, I get:y = (-1)(4x - 9) / (-5)Now, the negatives in the numerator and denominator cancel each other out:y = (4x - 9) / 5Okay, that looks better. So, y equals (4x minus 9) divided by 5. Let me double-check my steps to make sure I didn't make a mistake.Starting with 4x - 5y = 9.Subtract 4x from both sides: -5y = -4x + 9.Divide both sides by -5: y = (-4x + 9)/(-5).Simplify the negatives: y = (4x - 9)/5.Yes, that seems correct. So, y is equal to (4x minus 9) over 5. I think that's the answer they're looking for.