💡First, I need to determine the total distance the train must travel to completely cross the bridge. This is the sum of the train's length and the bridge's length.Next, I'll convert the train's speed from kilometers per hour to meters per second to ensure the units are consistent for calculating time.Finally, I'll use the formula Time = Distance / Speed to find out how many seconds it takes for the train to cross the bridge.

💡First, we are given the equation ( R = gS - 3 ). When ( S = 5 ), ( R = 17 ). We need to find the value of ( R ) when ( S = 12 ).**Step 1: Find the value of ( g )**Substitute ( S = 5 ) and ( R = 17 ) into the equation:[17 = g cdot 5 - 3][17 + 3 = 5g][20 = 5g][g = frac{20}{5} = 4]**Step 2: Calculate ( R ) when ( S = 12 )**Now that we know ( g = 4 ), substitute ( S = 12 ) into the equation:[R = 4 cdot 12 - 3][R = 48 - 3][R = 45]Therefore, when ( S = 12 ), ( R ) is 45.

💡Alright, so I have this problem here where I'm given that the solution set of the inequality ( x^2 - ax - b < 0 ) is ( (2, 3) ). I need to determine the solution set of another inequality: ( bx^2 - ax - 1 > 0 ). Hmm, okay, let's break this down step by step.First, I remember that when a quadratic inequality like ( x^2 - ax - b < 0 ) has a solution set between two numbers, say ( (2, 3) ), it means that the quadratic equation ( x^2 - ax - b = 0 ) has roots at ( x = 2 ) and ( x = 3 ). So, the quadratic can be factored as ( (x - 2)(x - 3) ). Let me write that down:( x^2 - ax - b = (x - 2)(x - 3) )Now, if I expand the right side, I can compare coefficients to find the values of ( a ) and ( b ). Let's do that:( (x - 2)(x - 3) = x^2 - 3x - 2x + 6 = x^2 - 5x + 6 )So, comparing this to the original quadratic ( x^2 - ax - b ), I can set up the following equations:1. The coefficient of ( x ) gives me ( -a = -5 ), so ( a = 5 ).2. The constant term gives me ( -b = 6 ), so ( b = -6 ).Alright, so now I know that ( a = 5 ) and ( b = -6 ). The next step is to substitute these values into the second inequality ( bx^2 - ax - 1 > 0 ). Let's do that:( (-6)x^2 - 5x - 1 > 0 )Hmm, that looks a bit messy. Maybe I can rearrange it or factor it to make it easier to solve. Let's rewrite the inequality:( -6x^2 - 5x - 1 > 0 )I notice that all the coefficients are negative or zero, so perhaps I can multiply both sides by -1 to make the quadratic coefficient positive. But wait, I remember that when I multiply or divide an inequality by a negative number, the inequality sign flips. So, let's do that:Multiplying both sides by -1:( 6x^2 + 5x + 1 < 0 )Okay, now I have a quadratic inequality ( 6x^2 + 5x + 1 < 0 ). To solve this, I need to find the roots of the quadratic equation ( 6x^2 + 5x + 1 = 0 ) and then determine where the quadratic is negative.Let's find the roots using the quadratic formula:( x = frac{-b pm sqrt{b^2 - 4ac}}{2a} )Here, ( a = 6 ), ( b = 5 ), and ( c = 1 ). Plugging these values in:( x = frac{-5 pm sqrt{5^2 - 4 cdot 6 cdot 1}}{2 cdot 6} )( x = frac{-5 pm sqrt{25 - 24}}{12} )( x = frac{-5 pm sqrt{1}}{12} )( x = frac{-5 pm 1}{12} )So, the two roots are:1. ( x = frac{-5 + 1}{12} = frac{-4}{12} = -frac{1}{3} )2. ( x = frac{-5 - 1}{12} = frac{-6}{12} = -frac{1}{2} )Alright, so the roots are ( x = -frac{1}{3} ) and ( x = -frac{1}{2} ). Since the quadratic opens upwards (because the coefficient of ( x^2 ) is positive), the quadratic will be below zero (negative) between its two roots. So, the solution to ( 6x^2 + 5x + 1 < 0 ) is:( -frac{1}{2} < x < -frac{1}{3} )But wait, let me double-check that. Since the quadratic opens upwards, it will be positive outside the roots and negative between them. So, yes, between ( -frac{1}{2} ) and ( -frac{1}{3} ), the quadratic is negative. Therefore, the inequality ( 6x^2 + 5x + 1 < 0 ) holds true for ( x ) in ( (-frac{1}{2}, -frac{1}{3}) ).But remember, this was after I multiplied the original inequality by -1, which flipped the inequality sign. So, the original inequality ( -6x^2 - 5x - 1 > 0 ) is equivalent to ( 6x^2 + 5x + 1 < 0 ), and we've found that the solution is ( (-frac{1}{2}, -frac{1}{3}) ).Just to make sure I didn't make any mistakes, let me test a value within the interval ( (-frac{1}{2}, -frac{1}{3}) ), say ( x = -frac{2}{5} ), which is approximately -0.4. Plugging into ( 6x^2 + 5x + 1 ):( 6(-frac{2}{5})^2 + 5(-frac{2}{5}) + 1 = 6(frac{4}{25}) - 2 + 1 = frac{24}{25} - 1 = -frac{1}{25} ), which is indeed less than 0. Good.Now, let's test a value outside the interval, say ( x = 0 ):( 6(0)^2 + 5(0) + 1 = 1 ), which is greater than 0, so the inequality ( 6x^2 + 5x + 1 < 0 ) does not hold here, which is consistent.Similarly, testing ( x = -1 ):( 6(-1)^2 + 5(-1) + 1 = 6 - 5 + 1 = 2 ), which is also greater than 0. So, again, consistent.Therefore, I'm confident that the solution set for the inequality ( bx^2 - ax - 1 > 0 ) is ( (-frac{1}{2}, -frac{1}{3}) ).**Final Answer**The solution set of the inequality ( bx^2 - ax - 1 > 0 ) is boxed{left( -frac{1}{2}, -frac{1}{3} right)}.

💡Alright, so I have this problem where for every natural number ( a ), I need to find the number of natural numbers ( b ) such that ( a + b ) divides ( ab ). Then, among all ( a ) less than or equal to 1983, I have to find the maximum value of this count, which is denoted as ( M(a) ).First, let me try to understand the condition ( a + b mid ab ). This means that when I divide ( ab ) by ( a + b ), there's no remainder. So, ( ab ) is a multiple of ( a + b ). Maybe I can rewrite this condition in a different way to make it easier to handle.Let me consider the equation ( a + b mid ab ). If ( a + b ) divides ( ab ), then it must also divide any linear combination of ( a ) and ( b ). For example, ( a(a + b) - ab = a^2 ). So, ( a + b ) divides ( a^2 ). Similarly, ( a + b ) would also divide ( b^2 ). That's interesting because it connects the problem to the divisors of ( a^2 ).So, if ( a + b ) is a divisor of ( a^2 ), then ( a + b ) must be one of the divisors of ( a^2 ). Let me denote ( d = a + b ). Then, ( d ) is a divisor of ( a^2 ), and ( b = d - a ). Since ( b ) has to be a natural number, ( d ) must be greater than ( a ) because ( b ) must be positive.Therefore, the number of such ( b ) is equal to the number of divisors of ( a^2 ) that are greater than ( a ). But how many divisors does ( a^2 ) have? If I can find the number of divisors of ( a^2 ), I can then find how many of them are greater than ( a ).The number of divisors of a number is determined by its prime factorization. If ( a ) has the prime factorization ( a = p_1^{k_1} p_2^{k_2} cdots p_n^{k_n} ), then ( a^2 = p_1^{2k_1} p_2^{2k_2} cdots p_n^{2k_n} ). The number of divisors of ( a^2 ) is ( (2k_1 + 1)(2k_2 + 1) cdots (2k_n + 1) ).Now, since the divisors of ( a^2 ) come in pairs that multiply to ( a^2 ), each divisor less than ( a ) pairs with a divisor greater than ( a ). The only exception is when ( a ) is a perfect square, in which case ( a ) itself is a divisor. But since we're only interested in divisors greater than ( a ), we can ignore ( a ) itself.Therefore, the number of divisors greater than ( a ) is exactly half of the total number of divisors minus one (to exclude ( a ) itself). So, ( M(a) = frac{tau(a^2) - 1}{2} ), where ( tau(a^2) ) is the number of divisors of ( a^2 ).To maximize ( M(a) ), I need to maximize ( tau(a^2) ) for ( a leq 1983 ). Since ( tau(a^2) ) depends on the exponents in the prime factorization of ( a ), the more prime factors ( a ) has and the higher the exponents, the more divisors ( a^2 ) will have.I should consider different cases based on the number of prime factors of ( a ):1. **Single Prime Factor**: If ( a = p^k ), then ( a^2 = p^{2k} ), and ( tau(a^2) = 2k + 1 ). So, ( M(a) = frac{2k + 1 - 1}{2} = k ). The maximum ( k ) for ( a leq 1983 ) would be when ( p ) is the smallest prime, which is 2. The largest power of 2 less than 1983 is ( 2^{10} = 1024 ), so ( k = 10 ). Thus, ( M(a) = 10 ).2. **Two Prime Factors**: If ( a = p_1^{k_1} p_2^{k_2} ), then ( tau(a^2) = (2k_1 + 1)(2k_2 + 1) ). So, ( M(a) = frac{(2k_1 + 1)(2k_2 + 1) - 1}{2} ). To maximize this, I should choose primes with exponents that balance the product. For example, ( a = 2^6 cdot 3^3 = 1728 ). Then, ( tau(a^2) = (2*6 + 1)(2*3 + 1) = 13 * 7 = 91 ). So, ( M(a) = frac{91 - 1}{2} = 45 ).3. **Three Prime Factors**: If ( a = p_1^{k_1} p_2^{k_2} p_3^{k_3} ), then ( tau(a^2) = (2k_1 + 1)(2k_2 + 1)(2k_3 + 1) ). So, ( M(a) = frac{(2k_1 + 1)(2k_2 + 1)(2k_3 + 1) - 1}{2} ). An example is ( a = 2^5 cdot 3^2 cdot 5 = 1440 ). Then, ( tau(a^2) = (2*5 + 1)(2*2 + 1)(2*1 + 1) = 11 * 5 * 3 = 165 ). So, ( M(a) = frac{165 - 1}{2} = 82 ).4. **Four Prime Factors**: If ( a = p_1^{k_1} p_2^{k_2} p_3^{k_3} p_4^{k_4} ), then ( tau(a^2) = (2k_1 + 1)(2k_2 + 1)(2k_3 + 1)(2k_4 + 1) ). So, ( M(a) = frac{(2k_1 + 1)(2k_2 + 1)(2k_3 + 1)(2k_4 + 1) - 1}{2} ). An example is ( a = 2^4 cdot 3 cdot 5 cdot 7 = 1680 ). Then, ( tau(a^2) = (2*4 + 1)(2*1 + 1)(2*1 + 1)(2*1 + 1) = 9 * 3 * 3 * 3 = 243 ). So, ( M(a) = frac{243 - 1}{2} = 121 ).Comparing all these cases, the maximum ( M(a) ) occurs when ( a ) has four prime factors, specifically ( a = 1680 ), giving ( M(a) = 121 ).**Final Answer**The maximum value of ( M(a) ) for ( a leq 1983 ) is (boxed{121}).

💡Okay, so I have this problem where points A and B are moving on a circle M, which is given by the equation (x - 2)^2 + y^2 = 1. The distance between A and B is √2, and I need to find the range of |OA + OB|, where O is the origin. The options are given as A, B, C, D with different intervals.First, I should recall that the circle M has its center at (2, 0) and a radius of 1 because the equation is in the standard form (x - h)^2 + (y - k)^2 = r^2. So, h is 2, k is 0, and r is 1.Now, points A and B are on this circle, and the distance between them is √2. I remember that the length of a chord in a circle relates to the radius and the distance from the center to the chord. The formula for the length of a chord is 2√(r^2 - d^2), where d is the distance from the center to the chord.Given that |AB| = √2, I can set up the equation: √2 = 2√(1 - d^2). Let me solve for d.Divide both sides by 2: √2 / 2 = √(1 - d^2). Square both sides: ( (√2)/2 )^2 = 1 - d^2. That simplifies to (2/4) = 1 - d^2, so 1/2 = 1 - d^2. Therefore, d^2 = 1 - 1/2 = 1/2, so d = √(1/2) = √2 / 2.So, the distance from the center M to the chord AB is √2 / 2. That means the midpoint N of AB is at a distance of √2 / 2 from M.Now, I need to find |OA + OB|. I remember that in vectors, OA + OB can be expressed as 2ON, where N is the midpoint of AB. So, |OA + OB| = 2|ON|.Therefore, I need to find the range of |ON|, and then multiply it by 2 to get the range of |OA + OB|.Since N is the midpoint of AB, and AB is a chord of the circle M with length √2, N lies on a circle centered at M with radius √2 / 2. So, the locus of N is a circle with center at (2, 0) and radius √2 / 2.Now, I need to find the minimum and maximum distances from the origin O to any point N on this circle. The origin is at (0, 0), and the center of the circle where N lies is at (2, 0). The distance between O and M is 2 units.The minimum distance from O to N would be the distance from O to M minus the radius of the circle where N lies. Similarly, the maximum distance would be the distance from O to M plus the radius.So, minimum |ON| = |OM| - radius = 2 - (√2 / 2). Similarly, maximum |ON| = 2 + (√2 / 2).But wait, the options are given in terms of √2, not √2 / 2. Let me express these in terms of √2.2 - √2 / 2 can be written as 2 - (√2 / 2) = (4 - √2) / 2. Wait, that's not helpful. Alternatively, maybe I can rationalize or express it differently.Wait, actually, when I multiply |ON| by 2 to get |OA + OB|, let's see:Minimum |OA + OB| = 2 * (2 - √2 / 2) = 4 - √2.Maximum |OA + OB| = 2 * (2 + √2 / 2) = 4 + √2.So, the range of |OA + OB| is [4 - √2, 4 + √2].Looking back at the options, option C is [4 - √2, 4 + √2], which matches my result.I think that makes sense. Let me just verify.We know that N is moving on a circle centered at (2, 0) with radius √2 / 2. The origin is at (0, 0). The distance from O to M is 2. The closest point on the circle of N to O would be along the line connecting O and M, subtracting the radius, and the farthest would be adding the radius.So, |ON| ranges from 2 - √2 / 2 to 2 + √2 / 2. Multiplying by 2 gives 4 - √2 and 4 + √2. So, yes, the range is [4 - √2, 4 + √2].Therefore, the correct answer should be option C.

💡Okay, so I have this problem about an arithmetic progression. The sum of n terms is 180, the common difference is 3, and the first term is an integer. I need to find how many possible values n can take, given that n is greater than 1. The options are from A to E, with C being 4. Hmm, let me think through this step by step.First, I remember the formula for the sum of the first n terms of an arithmetic progression (AP). It's S_n = n/2 [2a + (n-1)d], where a is the first term and d is the common difference. In this case, S_n is 180, and d is 3. So I can plug those values into the formula.Let me write that out:180 = (n/2) [2a + (n - 1)*3]Okay, so I can simplify this equation to solve for a or n. Maybe I'll try solving for a first because the problem mentions that the first term is an integer, so that might help me figure out possible n values.Let me multiply both sides by 2 to get rid of the denominator:360 = n [2a + 3(n - 1)]Now, distribute the 3 inside the brackets:360 = n [2a + 3n - 3]Let me expand that:360 = 2an + 3n^2 - 3nHmm, so I have 360 = 3n^2 + 2an - 3n. Maybe I can rearrange this to solve for a.Let me subtract 3n^2 and add 3n to both sides:360 - 3n^2 + 3n = 2anThen, divide both sides by 2n:a = (360 - 3n^2 + 3n) / (2n)Simplify the numerator:a = (360 + 3n - 3n^2) / (2n)I can factor out a 3 in the numerator:a = 3(120 + n - n^2) / (2n)Hmm, but I'm not sure if that helps. Maybe I can factor it differently or see if I can write it in a way that makes it easier to see when a is an integer.Let me write it as:a = (360 - 3n^2 + 3n) / (2n)I can split the fraction into separate terms:a = 360/(2n) - 3n^2/(2n) + 3n/(2n)Simplify each term:a = 180/n - (3n)/2 + 3/2So, a = 180/n - (3n)/2 + 3/2Hmm, since a has to be an integer, the entire expression on the right must be an integer. Let me see if I can write this differently or find a way to ensure that 180/n is an integer because 180/n is a term here.Wait, 180/n must be an integer because the other terms involve fractions with denominator 2, but since a is an integer, the entire expression must result in an integer. So, 180/n must be a number that, when combined with the other terms, results in an integer.Let me think: 180/n must be a rational number, but since n is an integer greater than 1, 180/n will be an integer only if n is a divisor of 180. So, n must be a divisor of 180.Therefore, I can list all the positive divisors of 180 and then check which ones satisfy the condition that a is an integer.First, let me list the divisors of 180. To do that, I can factorize 180.180 = 2^2 * 3^2 * 5^1So, the number of divisors is (2+1)(2+1)(1+1) = 3*3*2 = 18. So, there are 18 positive divisors.Let me list them:1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180.But since n > 1, we can exclude 1. So, possible n values are 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180.That's 17 possible n values. Now, I need to check which of these n values make a an integer.Given that a = 180/n - (3n)/2 + 3/2, let's see.Alternatively, maybe I can write a as:a = (360 - 3n^2 + 3n) / (2n)Which can be written as:a = (360 + 3n - 3n^2) / (2n)Let me factor out a 3 in the numerator:a = 3(120 + n - n^2) / (2n)Hmm, not sure if that helps. Maybe I can rearrange the terms:a = ( -3n^2 + 3n + 360 ) / (2n )Let me factor out a -3 from the first two terms:a = ( -3(n^2 - n) + 360 ) / (2n )Hmm, not sure. Maybe I can write it as:a = (-3n^2 + 3n + 360) / (2n) = (-3n^2)/(2n) + 3n/(2n) + 360/(2n) = (-3n/2) + 3/2 + 180/nWhich is the same as before: a = 180/n - (3n)/2 + 3/2So, to have a integer, 180/n must be a number such that when we subtract (3n)/2 and add 3/2, the result is integer.Let me think about this expression:a = 180/n - (3n)/2 + 3/2Let me combine the constants:a = 180/n - (3n - 3)/2So, a = 180/n - (3(n - 1))/2Since a must be integer, 180/n must be a number such that when we subtract (3(n - 1))/2, the result is integer.Given that 180/n is either integer or a fraction. But since n divides 180, 180/n is integer. So, 180/n is integer, and (3(n - 1))/2 must also result in a number such that when subtracted from an integer, the result is integer.Therefore, (3(n - 1))/2 must be a number with denominator 2, but since we're subtracting it from an integer, the result will be integer only if (3(n - 1))/2 is integer or half-integer.Wait, but 3(n - 1)/2 must be such that when subtracted from an integer, the result is integer. So, 3(n - 1)/2 must be either integer or half-integer. But since a is integer, the entire expression must be integer, so 3(n - 1)/2 must be integer because 180/n is integer.Therefore, 3(n - 1)/2 must be integer. So, 3(n - 1) must be divisible by 2, meaning that (n - 1) must be even because 3 is odd, so for 3(n - 1) to be even, (n - 1) must be even.Therefore, n - 1 is even, so n is odd.So, n must be an odd divisor of 180 greater than 1.Looking back at the list of divisors of 180: 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180.The odd divisors greater than 1 are: 3, 5, 9, 15, 45.Wait, let me check: 3 is odd, 5 is odd, 9 is odd, 15 is odd, 45 is odd. So, n can be 3, 5, 9, 15, 45.But wait, let me verify if these n values actually result in a being integer.Let me test n = 3:a = 180/3 - (3*3)/2 + 3/2 = 60 - 9/2 + 3/2 = 60 - 6/2 = 60 - 3 = 57. So, a = 57, which is integer.n = 5:a = 180/5 - (3*5)/2 + 3/2 = 36 - 15/2 + 3/2 = 36 - 12/2 = 36 - 6 = 30. So, a = 30, integer.n = 9:a = 180/9 - (3*9)/2 + 3/2 = 20 - 27/2 + 3/2 = 20 - 24/2 = 20 - 12 = 8. So, a = 8, integer.n = 15:a = 180/15 - (3*15)/2 + 3/2 = 12 - 45/2 + 3/2 = 12 - 42/2 = 12 - 21 = -9. So, a = -9, which is integer.n = 45:a = 180/45 - (3*45)/2 + 3/2 = 4 - 135/2 + 3/2 = 4 - 132/2 = 4 - 66 = -62. So, a = -62, integer.So, all these n values result in integer a. So, n can be 3, 5, 9, 15, 45. That's 5 values.Wait, but the options only go up to E) 6, and the answer choices are A)2, B)3, C)4, D)5, E)6.So, according to this, the number of possible values for n is 5, which is option D.But wait, in the initial problem statement, it's mentioned that n > 1, so n can be 3,5,9,15,45, which are 5 values. So, the answer should be D)5.But wait, in the initial thought process, I thought it was 4, but after checking, it's 5. Maybe I made a mistake earlier.Wait, let me double-check. The divisors of 180 are 2,3,4,5,6,9,10,12,15,18,20,30,36,45,60,90,180. The odd ones greater than 1 are 3,5,9,15,45. So, 5 values.But wait, let me check n=5:a = 30, which is integer.n=3: a=57, integer.n=9: a=8, integer.n=15: a=-9, integer.n=45: a=-62, integer.So, all 5 values give integer a. So, the number of possible n is 5, which is option D.But wait, in the initial problem, the options are A)2, B)3, C)4, D)5, E)6. So, D is 5, which seems correct.But I'm confused because in the initial thought process, I thought it was 4, but after checking, it's 5. Maybe I missed something.Wait, let me check if n=1 is allowed. The problem says n>1, so n=1 is excluded. So, n=3,5,9,15,45 are all valid, giving 5 values.Therefore, the correct answer should be D)5.But wait, let me check if n=5 is valid. Let me calculate the sum for n=5, a=30, d=3.Sum = 5/2 [2*30 + (5-1)*3] = 5/2 [60 + 12] = 5/2 *72 = 5*36 = 180. Correct.Similarly, n=3: a=57, sum=3/2 [2*57 + 2*3] = 3/2 [114 +6] = 3/2*120=180. Correct.n=9: a=8, sum=9/2 [2*8 +8*3]=9/2 [16+24]=9/2*40=9*20=180. Correct.n=15: a=-9, sum=15/2 [2*(-9) +14*3]=15/2 [-18 +42]=15/2*24=15*12=180. Correct.n=45: a=-62, sum=45/2 [2*(-62) +44*3]=45/2 [-124 +132]=45/2*8=45*4=180. Correct.So, all 5 values are valid. Therefore, the number of possible values for n is 5, which is option D.But wait, in the initial problem, the options are A)2, B)3, C)4, D)5, E)6. So, D is 5, which is correct.But in the initial thought process, I thought it was 4, but after checking, it's 5. So, the correct answer is D)5.