answer：Alright, so I've got this problem here where Fedya deposited some money into his mobile phone account through a payment terminal, and after a commission was applied, he ended up with 847 rubles. The commission is a natural percentage, meaning it's a whole number, and it's less than 30%. I need to figure out how much money Fedya originally deposited.Okay, let's break this down. First, let's define some variables to make it clearer. Let's say the amount Fedya deposited is ( n ) rubles, and the commission rate is ( k % ). The problem says that after applying the commission, he was credited with 847 rubles. So, mathematically, this can be represented as:[n times left(1 - frac{k}{100}right) = 847]This equation means that the amount he deposited minus the commission equals 847 rubles. My goal is to find ( n ), the original amount deposited, given that ( k ) is a natural number less than 30.So, rearranging the equation to solve for ( n ), we get:[n = frac{847}{1 - frac{k}{100}} = frac{847 times 100}{100 - k}]This simplifies to:[n = frac{84700}{100 - k}]Now, since both ( n ) and ( k ) are integers, ( 100 - k ) must be a divisor of 84700. So, I need to find a divisor of 84700 that is less than 100 (because ( k ) is less than 30, so ( 100 - k ) is greater than 70) and results in ( n ) being an integer.Let's factorize 84700 to find its divisors. Breaking it down:[84700 = 847 times 100 = 7 times 121 times 100 = 7 times 11^2 times 2^2 times 5^2]So, the prime factors are 2, 5, 7, and 11. Now, I need to find all the divisors of 84700 that are between 70 and 100 because ( 100 - k ) must be in that range.Let's list the possible divisors:- 70: Is 70 a divisor of 84700? Let's check: 84700 ÷ 70 = 1210. Yes, it is.- 77: 84700 ÷ 77 = 1100. Yes, 77 is a divisor.- 88: 84700 ÷ 88 = 962.5. Not an integer, so 88 is not a divisor.- 98: 84700 ÷ 98 = 864.2857... Not an integer, so 98 is not a divisor.So, the possible values for ( 100 - k ) are 70 and 77. Let's check each one:1. If ( 100 - k = 70 ), then ( k = 30 ). But the problem states that the commission is less than 30%, so ( k ) must be less than 30. Therefore, ( k = 30 ) is not acceptable.2. If ( 100 - k = 77 ), then ( k = 23 ). This is acceptable because 23 is less than 30.So, ( 100 - k = 77 ) is the valid divisor. Plugging this back into the equation for ( n ):[n = frac{84700}{77} = 1100]Therefore, Fedya deposited 1100 rubles.Wait, let me double-check to make sure I didn't make a mistake. If he deposited 1100 rubles and the commission is 23%, then the amount credited should be:[1100 times (1 - 0.23) = 1100 times 0.77 = 847]Yes, that checks out. So, the calculations seem correct.I think I've covered all the necessary steps. I defined the variables, set up the equation, factorized the number, found the appropriate divisor, and verified the solution. It seems solid.

answer：Okay, so I have this inequality to solve: (-3x^2 + 8x + 1 < 0). Hmm, quadratic inequalities can be a bit tricky, but let me see if I can figure this out step by step.First, I remember that to solve a quadratic inequality, it's helpful to find the roots of the corresponding quadratic equation. So, I'll start by setting the quadratic expression equal to zero:[-3x^2 + 8x + 1 = 0]This is a quadratic equation in the standard form (ax^2 + bx + c = 0), where (a = -3), (b = 8), and (c = 1). To find the roots, I can use the quadratic formula:[x = frac{-b pm sqrt{b^2 - 4ac}}{2a}]Plugging in the values:[x = frac{-8 pm sqrt{8^2 - 4(-3)(1)}}{2(-3)} = frac{-8 pm sqrt{64 + 12}}{-6} = frac{-8 pm sqrt{76}}{-6}]Wait, (sqrt{76}) can be simplified. Since (76 = 4 times 19), (sqrt{76} = 2sqrt{19}). So, substituting back:[x = frac{-8 pm 2sqrt{19}}{-6}]I can factor out a 2 from the numerator:[x = frac{2(-4 pm sqrt{19})}{-6} = frac{-4 pm sqrt{19}}{-3}]Multiplying numerator and denominator by -1 to make it look nicer:[x = frac{4 mp sqrt{19}}{3}]So, the roots are:[x = frac{4 + sqrt{19}}{3} quad text{and} quad x = frac{4 - sqrt{19}}{3}]Let me approximate these values to get a better sense. (sqrt{19}) is approximately 4.3589.So,[x approx frac{4 + 4.3589}{3} = frac{8.3589}{3} approx 2.7863][x approx frac{4 - 4.3589}{3} = frac{-0.3589}{3} approx -0.1196]So, the roots are approximately at (x approx -0.1196) and (x approx 2.7863).Now, since the quadratic opens downward (because the coefficient of (x^2) is negative), the graph of the quadratic is a downward-opening parabola. This means that the quadratic expression will be below zero (negative) outside the interval between the roots.Wait, let me think about that again. If the parabola opens downward, the expression is positive between the roots and negative outside. So, since we're looking for where (-3x^2 + 8x + 1 < 0), that should be where the parabola is below the x-axis, which is outside the interval between the roots.But let me double-check that. If I pick a test point between the roots, say (x = 0), and plug it into the expression:[-3(0)^2 + 8(0) + 1 = 1 > 0]So, between the roots, the expression is positive. Therefore, outside the roots, it's negative. So, the solution should be (x < frac{4 - sqrt{19}}{3}) or (x > frac{4 + sqrt{19}}{3}).But wait, the original inequality is (-3x^2 + 8x + 1 < 0), so we're looking for where the expression is negative. Therefore, the solution is indeed (x < frac{4 - sqrt{19}}{3}) or (x > frac{4 + sqrt{19}}{3}).But let me write the exact values without approximating:The roots are (x = frac{4 pm sqrt{19}}{3}). So, the intervals where the expression is negative are:[x < frac{4 - sqrt{19}}{3} quad text{and} quad x > frac{4 + sqrt{19}}{3}]Therefore, the solution to the inequality is:[x in left(-infty, frac{4 - sqrt{19}}{3}right) cup left(frac{4 + sqrt{19}}{3}, inftyright)]But just to make sure I didn't make a mistake, let me test another point in each interval.First interval: (x < frac{4 - sqrt{19}}{3}). Let's pick (x = -1):[-3(-1)^2 + 8(-1) + 1 = -3(1) - 8 + 1 = -3 - 8 + 1 = -10 < 0]Good, it satisfies the inequality.Second interval: (x > frac{4 + sqrt{19}}{3}). Let's pick (x = 3):[-3(3)^2 + 8(3) + 1 = -27 + 24 + 1 = -2 < 0]Wait, that's negative, which is correct because we're looking for where the expression is less than zero. So, that works.But wait, when I picked (x = 0), which is between the roots, the expression was positive, which is why the solution is outside the interval between the roots.So, I think I did it right. The solution is all real numbers less than (frac{4 - sqrt{19}}{3}) and greater than (frac{4 + sqrt{19}}{3}).Just to recap:1. Found the roots using the quadratic formula.2. Determined the direction the parabola opens (downward because (a = -3 < 0)).3. Tested intervals to confirm where the expression is negative.4. Confirmed with test points.So, I'm confident that the solution is correct.

answer：First, I need to determine the combined area of the three rugs. I know that when the rugs are overlapped, they cover a total floor area of 140 square meters. Out of this, 22 square meters are covered by exactly two layers of rug, and 19 square meters are covered by three layers of rug.To find the area covered by exactly one layer of rug, I subtract the areas covered by two and three layers from the total floor area:[text{Area covered by one layer} = 140 - 22 - 19 = 99 text{ square meters}]Next, I calculate the total area of the rugs by considering the contribution of each layer:[text{Total area of the rugs} = text{Area covered by one layer} + 2 times text{Area covered by two layers} + 3 times text{Area covered by three layers}][text{Total area of the rugs} = 99 + 2 times 22 + 3 times 19 = 99 + 44 + 57 = 200 text{ square meters}]Therefore, the combined area of the three rugs is (boxed{200}) square meters.

answer：Okay, so I have this problem about a boat and a stream. The boat's speed in still water is 16 km per hour, and the stream's speed is 2 km per hour. A man rows to a place and then comes back, and the total time he takes is 891.4285714285714 hours. I need to find the distance to the place he rows to.First, I should understand what's happening here. When the man is going downstream, the stream's speed will add to his rowing speed, making him go faster. Conversely, when he's going upstream, the stream's speed will subtract from his rowing speed, making him slower. So, his effective speed downstream is 16 + 2 = 18 km/h, and upstream is 16 - 2 = 14 km/h.Let me denote the distance to the place as D kilometers. So, the time taken to go downstream would be D divided by 18 km/h, and the time taken to come back upstream would be D divided by 14 km/h. The total time is the sum of these two times, which is given as approximately 891.4285714285714 hours.So, I can write the equation:D/18 + D/14 = 891.4285714285714To solve for D, I need to combine these two fractions. The denominators are 18 and 14, so the least common multiple (LCM) of 18 and 14 is 126. Wait, is that right? Let me check:18 factors into 2 * 3^2, and 14 factors into 2 * 7. So, the LCM would be 2 * 3^2 * 7 = 126. Okay, so the common denominator is 126.So, rewriting the equation:(7D)/126 + (9D)/126 = 891.4285714285714Adding the numerators:(7D + 9D)/126 = 891.428571428571416D/126 = 891.4285714285714Now, to solve for D, I can multiply both sides by 126:16D = 891.4285714285714 * 126Let me calculate 891.4285714285714 * 126. Hmm, 891.4285714285714 is actually 891 and 3/7, because 0.4285714285714 is approximately 3/7. So, 891 3/7 * 126.First, let's convert 891 3/7 to an improper fraction:891 * 7 = 62376237 + 3 = 6240So, 6240/7 * 126 = (6240 * 126)/7Simplify 126/7 = 18So, 6240 * 18 = ?Let's calculate 6240 * 10 = 62,4006240 * 8 = 49,920Adding them together: 62,400 + 49,920 = 112,320So, 16D = 112,320Now, divide both sides by 16:D = 112,320 / 16Let's do the division:112,320 ÷ 1616 * 7,000 = 112,000So, 112,320 - 112,000 = 320320 ÷ 16 = 20So, D = 7,000 + 20 = 7,020 kmWait, that seems really far. Is that possible? Rowing for almost 900 hours to go 7,020 km? That would mean he's rowing at an average speed of about 7.8 km/h, which seems slow, but considering the stream's effect, maybe it's possible.Let me double-check my calculations.Starting from the equation:D/18 + D/14 = 891.4285714285714I found the LCM of 18 and 14 to be 126, so:(7D + 9D)/126 = 891.428571428571416D/126 = 891.4285714285714Multiplying both sides by 126:16D = 891.4285714285714 * 126I converted 891.4285714285714 to 6240/7, which is correct because 891 * 7 = 6237, plus 3 is 6240.Then, 6240/7 * 126 = (6240 * 126)/7 = 6240 * 18 = 112,320Then, 112,320 ÷ 16 = 7,020Yes, the calculations seem correct. So, the distance is 7,020 km.But wait, 7,020 km is an extremely long distance for rowing. Maybe I made a mistake in interpreting the total time.Let me check the total time given: 891.4285714285714 hours. That's approximately 891.4286 hours. To get a sense, 891 hours is about 37 days. Rowing for 37 days straight to cover 7,020 km seems unrealistic. Maybe the units are different? Or perhaps I misread the problem.Wait, the problem says the total time is 891.4285714285714 hours. That's correct. So, unless the man is rowing non-stop for over a month, which is improbable, maybe the distance is indeed 7,020 km.Alternatively, perhaps I made a mistake in calculating the LCM or the fractions.Let me try another approach. Instead of using LCM, I'll use decimal equivalents.D/18 + D/14 = 891.4285714285714Calculate D/18 + D/14:Let me find D*(1/18 + 1/14) = 891.4285714285714Compute 1/18 + 1/14:1/18 ≈ 0.05555555561/14 ≈ 0.0714285714Adding them together: 0.0555555556 + 0.0714285714 ≈ 0.126984127So, D * 0.126984127 ≈ 891.4285714285714Therefore, D ≈ 891.4285714285714 / 0.126984127 ≈ ?Let me compute that:891.4285714285714 ÷ 0.126984127First, approximate 0.126984127 ≈ 0.127So, 891.4285714285714 ÷ 0.127 ≈ ?Calculate 891.4285714285714 ÷ 0.127Multiply numerator and denominator by 1000 to eliminate decimals:891,428.5714285714 ÷ 127 ≈ ?Now, let's divide 891,428.5714285714 by 127.127 * 7,000 = 889,000Subtract: 891,428.5714285714 - 889,000 = 2,428.5714285714Now, 127 * 19 = 2,413Subtract: 2,428.5714285714 - 2,413 = 15.5714285714So, total is approximately 7,000 + 19 = 7,019, with a remainder of about 15.57So, approximately 7,019.123 kmWhich is close to 7,020 km, considering the rounding.So, my initial calculation was correct. The distance is approximately 7,020 km.But again, that seems extremely long. Maybe the units are different? Or perhaps the time is in minutes instead of hours? Let me check the problem statement again.It says the total time taken is 891.4285714285714 hours. So, it's definitely hours. So, unless the man is rowing at a superhuman pace, the distance being 7,020 km is correct.Alternatively, maybe I misread the boat's speed. It says 16 kmph in still water, and the stream is 2 kmph. So, downstream 18 kmph, upstream 14 kmph. That seems correct.Another way to think about it: the average speed for the round trip is total distance divided by total time. Total distance is 2D, total time is 891.4285714285714 hours.So, average speed = 2D / 891.4285714285714But average speed for round trip with different speeds is the harmonic mean of the two speeds.So, harmonic mean = 2 * (18 * 14) / (18 + 14) = 2 * 252 / 32 = 504 / 32 = 15.75 km/hSo, average speed is 15.75 km/hThen, total distance is average speed * total time = 15.75 * 891.4285714285714Let me compute that:15.75 * 891.4285714285714First, 15 * 891.4285714285714 = ?15 * 800 = 12,00015 * 91.4285714285714 ≈ 15 * 91.4286 ≈ 1,371.428571428571So, total ≈ 12,000 + 1,371.428571428571 ≈ 13,371.428571428571Now, 0.75 * 891.4285714285714 ≈ ?0.75 * 800 = 6000.75 * 91.4285714285714 ≈ 68.57142857142857So, total ≈ 600 + 68.57142857142857 ≈ 668.5714285714286Adding to the previous total:13,371.428571428571 + 668.5714285714286 ≈ 14,040 kmSo, total distance is 14,040 km, which means one way is 7,020 km. So, that confirms it.Therefore, despite the distance being extremely long, the calculations are consistent. So, the distance to the place he rows to is 7,020 km.

answer：Okay, so I have this sequence problem here. It says that there's a sequence of positive terms {a_n} where a₁ = 2, a₂ = 1, and it satisfies the equation (a_n)/(a_{n+1}) + (a_n)/(a_{n-1}) = 2. I need to find the value of a₁₂. Hmm, let's try to figure this out step by step.First, let me write down what I know. The sequence starts with a₁ = 2 and a₂ = 1. The recurrence relation is given by (a_n)/(a_{n+1}) + (a_n)/(a_{n-1}) = 2. That seems a bit complicated, but maybe I can manipulate it to find a pattern or a simpler relation.Let me try to rewrite the equation. If I factor out a_n from both terms on the left side, I get a_n*(1/a_{n+1} + 1/a_{n-1}) = 2. Then, dividing both sides by a_n, I get 1/a_{n+1} + 1/a_{n-1} = 2/a_n. Hmm, that looks interesting. So, 1/a_{n+1} + 1/a_{n-1} = 2/a_n.Wait a minute, this seems like a linear recurrence relation. If I let b_n = 1/a_n, then the equation becomes b_{n+1} + b_{n-1} = 2b_n. That's a simpler form. So, the sequence {b_n} satisfies b_{n+1} + b_{n-1} = 2b_n. Now, what kind of recurrence is this? Let me think. If I rearrange the terms, it becomes b_{n+1} - 2b_n + b_{n-1} = 0. That looks like a second-order linear homogeneous recurrence relation with constant coefficients. The characteristic equation for this would be r² - 2r + 1 = 0. Let's solve that.The characteristic equation is r² - 2r + 1 = 0. Using the quadratic formula, r = [2 ± sqrt(4 - 4)] / 2 = [2 ± 0]/2 = 1. So, we have a repeated root, r = 1. That means the general solution for the recurrence relation is b_n = (A + Bn)(1)^n = A + Bn. So, b_n is a linear function of n.Great, so b_n = A + Bn. Now, I need to find the constants A and B using the initial conditions. But wait, I have a₁ = 2 and a₂ = 1. Since b_n = 1/a_n, that means b₁ = 1/2 and b₂ = 1/1 = 1.So, let's plug in n = 1 and n = 2 into the general solution:For n = 1: b₁ = A + B(1) = A + B = 1/2.For n = 2: b₂ = A + B(2) = A + 2B = 1.Now, I have a system of two equations:1. A + B = 1/22. A + 2B = 1Let me subtract the first equation from the second to eliminate A:(A + 2B) - (A + B) = 1 - 1/2Which simplifies to B = 1/2.Now, plugging B = 1/2 back into the first equation:A + 1/2 = 1/2So, A = 0.Therefore, the general term for b_n is b_n = 0 + (1/2)n = n/2.Since b_n = 1/a_n, that means 1/a_n = n/2, so a_n = 2/n.Wow, so the nth term of the sequence is just 2 divided by n. Let me check if this makes sense with the initial terms.For n = 1: a₁ = 2/1 = 2, which matches.For n = 2: a₂ = 2/2 = 1, which also matches.Good, that seems consistent. Let me test the recurrence relation with n = 2 to see if it holds.Given a₁ = 2, a₂ = 1, and a₃ = 2/3.Compute (a₂)/(a₃) + (a₂)/(a₁) = (1)/(2/3) + (1)/(2) = 3/2 + 1/2 = 2. Perfect, it satisfies the equation.Similarly, let's check for n = 3. a₃ = 2/3, a₄ = 2/4 = 1/2, a₂ = 1.Compute (a₃)/(a₄) + (a₃)/(a₂) = (2/3)/(1/2) + (2/3)/(1) = (4/3) + (2/3) = 6/3 = 2. It works again.Alright, so the formula a_n = 2/n seems to hold. Therefore, to find a₁₂, I just plug in n = 12.a₁₂ = 2/12 = 1/6.So, the value of a₁₂ is 1/6.**Final Answer**The value of ( a_{12} ) is boxed{dfrac{1}{6}}.

answer：Okay, so I need to figure out the monotonically decreasing interval for the function ( y = sinleft(frac{pi}{3} - 2xright) ). Hmm, let's start by recalling what it means for a function to be monotonically decreasing. If a function is decreasing on an interval, that means as ( x ) increases, ( y ) decreases. So, the slope of the function is negative in that interval.I remember that the sine function has a specific shape with increasing and decreasing intervals. The standard sine function ( y = sin(x) ) increases from ( -frac{pi}{2} ) to ( frac{pi}{2} ) and decreases from ( frac{pi}{2} ) to ( frac{3pi}{2} ), and this pattern repeats every ( 2pi ). But in this case, the function is a bit different because of the ( -2x ) inside the sine function and the ( frac{pi}{3} ) shift.Let me try to rewrite the function to make it easier to analyze. The function is ( y = sinleft(frac{pi}{3} - 2xright) ). I can factor out the negative sign to make it look more familiar: ( y = sinleft(-2x + frac{pi}{3}right) ). Since sine is an odd function, ( sin(-theta) = -sin(theta) ), so this can be rewritten as ( y = -sinleft(2x - frac{pi}{3}right) ).Okay, so now the function is ( y = -sinleft(2x - frac{pi}{3}right) ). This might make it easier to analyze because I can think of it as a sine function that's been reflected over the x-axis and shifted horizontally.Since the function is ( -sin(2x - frac{pi}{3}) ), its behavior will be opposite to that of ( sin(2x - frac{pi}{3}) ). So, where ( sin(2x - frac{pi}{3}) ) is increasing, ( y ) will be decreasing, and vice versa.Let's first find the intervals where ( sin(2x - frac{pi}{3}) ) is increasing. The standard sine function ( sin(u) ) is increasing when ( u ) is between ( -frac{pi}{2} + 2pi k ) and ( frac{pi}{2} + 2pi k ) for any integer ( k ).So, for ( sin(2x - frac{pi}{3}) ) to be increasing, the argument ( 2x - frac{pi}{3} ) must lie in the interval ( [-frac{pi}{2} + 2pi k, frac{pi}{2} + 2pi k] ).Let me write that down:[-frac{pi}{2} + 2pi k leq 2x - frac{pi}{3} leq frac{pi}{2} + 2pi k]Now, I need to solve this inequality for ( x ).First, add ( frac{pi}{3} ) to all parts of the inequality:[-frac{pi}{2} + frac{pi}{3} + 2pi k leq 2x leq frac{pi}{2} + frac{pi}{3} + 2pi k]Let me compute ( -frac{pi}{2} + frac{pi}{3} ):[-frac{pi}{2} + frac{pi}{3} = -frac{3pi}{6} + frac{2pi}{6} = -frac{pi}{6}]Similarly, ( frac{pi}{2} + frac{pi}{3} ):[frac{pi}{2} + frac{pi}{3} = frac{3pi}{6} + frac{2pi}{6} = frac{5pi}{6}]So, the inequality becomes:[-frac{pi}{6} + 2pi k leq 2x leq frac{5pi}{6} + 2pi k]Now, divide all parts by 2 to solve for ( x ):[-frac{pi}{12} + pi k leq x leq frac{5pi}{12} + pi k]So, this tells me that ( sin(2x - frac{pi}{3}) ) is increasing on the intervals ( left[ -frac{pi}{12} + pi k, frac{5pi}{12} + pi k right] ) for any integer ( k ).But remember, our original function is ( y = -sin(2x - frac{pi}{3}) ). Since it's the negative of the sine function, wherever ( sin(2x - frac{pi}{3}) ) is increasing, ( y ) will be decreasing.Therefore, the intervals where ( y = sinleft(frac{pi}{3} - 2xright) ) is monotonically decreasing are the same intervals where ( sin(2x - frac{pi}{3}) ) is increasing, which we found to be ( left[ -frac{pi}{12} + pi k, frac{5pi}{12} + pi k right] ) for any integer ( k ).Let me just double-check my steps to make sure I didn't make a mistake.1. I started with ( y = sinleft(frac{pi}{3} - 2xright) ).2. I rewrote it as ( y = -sinleft(2x - frac{pi}{3}right) ) using the identity ( sin(-theta) = -sin(theta) ).3. I recognized that the function is the negative of a sine function, so its increasing/decreasing behavior is opposite.4. I found the intervals where ( sin(2x - frac{pi}{3}) ) is increasing by solving the inequality ( -frac{pi}{2} + 2pi k leq 2x - frac{pi}{3} leq frac{pi}{2} + 2pi k ).5. I solved for ( x ) and found the intervals ( left[ -frac{pi}{12} + pi k, frac{5pi}{12} + pi k right] ).6. Since ( y ) is the negative of the sine function, these intervals correspond to where ( y ) is decreasing.Everything seems to check out. I think I've got the correct intervals where the function is monotonically decreasing.**Final Answer**The monotonically decreasing interval of the function is boxed{left[ kpi - frac{pi}{12}, kpi + frac{5pi}{12} right]} for any integer ( k ).