💡First, I recognize that the absolute value of ( c ) being 1 means ( c ) can be either 1 or -1.Next, since ( b ) and ( c ) are opposite numbers, their sum must be zero. This gives me two possible values for ( b ): if ( c = 1 ), then ( b = -1 ); and if ( c = -1 ), then ( b = 1 ).Finally, knowing that the sum of ( a ) and ( b ) is 2, I substitute the two possible values of ( b ) into the equation ( a + b = 2 ). - If ( b = -1 ), then ( a - 1 = 2 ), which means ( a = 3 ).- If ( b = 1 ), then ( a + 1 = 2 ), which means ( a = 1 ).Therefore, the possible values for ( a ) are 3 or 1.

💡First, I need to determine the area of the square. The square has sides of length 20, so its area is calculated as 20 squared, which equals 400.Next, I calculate the area of the circle. The circle has a radius of 10, so its area is π multiplied by 10 squared, resulting in 100π.Since the circle is centered at the center of the square and its radius is exactly half the length of the square's side, the entire circle is contained within the square. This means that the area of the circle does not add any new region outside the square.Therefore, the area of the union of the square and the circle is simply the area of the square, as the circle's area is entirely within the square.

💡First, I need to understand what the numbers (22_b) and (514_b) represent in base (b). In base (b), the number (22_b) is equal to (2b + 2) in decimal, and (514_b) is equal to (5b^2 + 1b + 4) in decimal.Next, I set up the equation based on the given information that the square of (22_b) is (514_b):[(2b + 2)^2 = 5b^2 + b + 4]Expanding the left side of the equation:[4b^2 + 8b + 4 = 5b^2 + b + 4]Subtracting (5b^2 + b + 4) from both sides to bring all terms to one side:[4b^2 + 8b + 4 - 5b^2 - b - 4 = 0]Simplifying the equation:[-b^2 + 7b = 0]Factoring out the common factor:[b(b - 7) = 0]Setting each factor equal to zero gives the possible values of (b):1. (b = 0)2. (b = 7)Since a base cannot be zero, the valid solution is (b = 7).

💡Okay, so I have this math problem here: I need to evaluate the sum of several terms involving exponents of 1 and -1. The expression is: -1^{2010} + (-1)^{2011} + 1^{2012} - 1^{2013} + (-1)^{2014}.Hmm, let's break this down step by step. I remember that exponents can sometimes be tricky, especially when dealing with negative signs. I think the key here is to carefully evaluate each term individually before adding them all together.First, let's look at the first term: -1^{2010}. Wait, is that -(1^{2010}) or (-1)^{2010}? I think it's the former because the negative sign is not inside the parentheses. So, -1^{2010} should be interpreted as -(1^{2010}). Since any number to any power is still 1, this simplifies to -1.Next, the second term is (-1)^{2011}. Here, the negative sign is inside the parentheses, so it's -1 raised to the 2011th power. I remember that when you raise -1 to an odd power, the result is -1. So, (-1)^{2011} is just -1.Moving on to the third term: 1^{2012}. This one seems straightforward because 1 raised to any power is always 1. So, this term is simply 1.The fourth term is -1^{2013}. Similar to the first term, I think this is interpreted as -(1^{2013}). Again, 1 raised to any power is 1, so this simplifies to -1.Finally, the fifth term is (-1)^{2014}. This is -1 raised to an even power. I recall that when you raise -1 to an even power, the result is 1. So, (-1)^{2014} is 1.Now, let's substitute all these simplified values back into the original expression:-1 + (-1) + 1 - 1 + 1.Let me compute this step by step:1. Start with -1.2. Add (-1): -1 + (-1) = -2.3. Add 1: -2 + 1 = -1.4. Subtract 1: -1 - 1 = -2.5. Add 1: -2 + 1 = -1.Wait, that doesn't seem right. Let me double-check my calculations. Maybe I made a mistake in the order or the signs.Starting over:1. First term: -1.2. Second term: -1. So, -1 + (-1) = -2.3. Third term: 1. So, -2 + 1 = -1.4. Fourth term: -1. So, -1 - 1 = -2.5. Fifth term: 1. So, -2 + 1 = -1.Hmm, I'm getting -1 as the final result, but I have a feeling that might not be correct. Let me check each term again to make sure I simplified them correctly.1. -1^{2010}: As I thought, this is -(1^{2010}) = -1.2. (-1)^{2011}: This is indeed -1 because 2011 is odd.3. 1^{2012}: Definitely 1.4. -1^{2013}: Again, -(1^{2013}) = -1.5. (-1)^{2014}: This should be 1 because 2014 is even.So, substituting back: -1 + (-1) + 1 - 1 + 1.Let me add them in a different order to see if that helps:Combine the negative terms first: -1 + (-1) - 1 = -3.Then add the positive terms: 1 + 1 = 2.Now, add those results together: -3 + 2 = -1.Wait, that still gives me -1. But I have a feeling the answer might be 0. Maybe I miscounted the number of terms or their signs.Let me write them out again:1. -1 (from -1^{2010})2. -1 (from (-1)^{2011})3. 1 (from 1^{2012})4. -1 (from -1^{2013})5. 1 (from (-1)^{2014})So, adding them up: -1 -1 +1 -1 +1.Let me group them differently:(-1 -1) + (1 -1) + 1 = (-2) + (0) + 1 = -2 + 1 = -1.Still getting -1. Maybe I'm missing something. Let me check the exponents again.Wait, -1^{2010}: Is that (-1)^{2010} or -(1^{2010})? If it's (-1)^{2010}, then it would be 1 because 2010 is even. But if it's -(1^{2010}), it's -1. The way it's written, without parentheses, it's -(1^{2010}), so -1.Similarly, -1^{2013} is -(1^{2013}) = -1.So, my initial interpretation was correct. Therefore, the sum should be -1.But wait, let me recount:-1 (first term) + (-1) (second term) = -2.Then +1 (third term) = -1.Then -1 (fourth term) = -2.Then +1 (fifth term) = -1.Yes, that's consistent. So, the final answer should be -1.But I'm still unsure because sometimes these problems are designed to cancel out to 0. Maybe I made a mistake in interpreting the first term.Let me consider if -1^{2010} is actually (-1)^{2010}. If that's the case, then it would be 1, because (-1) to an even power is 1.So, if I reinterpret the first term as (-1)^{2010} = 1, then the sum becomes:1 + (-1) + 1 -1 +1.Let's compute that:1 -1 = 0.0 +1 =1.1 -1 =0.0 +1=1.So, that would give me 1.But now I'm confused because the way the term is written is -1^{2010}, which typically means -(1^{2010}) = -1, not (-1)^{2010}.However, sometimes people might interpret it differently. Maybe the problem intended for -1^{2010} to mean (-1)^{2010}.If that's the case, then the sum would be:1 + (-1) +1 -1 +1 =1.But I'm not sure. The standard interpretation is that exponentiation has higher precedence than the negative sign, so -1^{2010} is -(1^{2010}) = -1.Therefore, sticking with my original interpretation, the sum is -1.Wait, but let me check the problem again: it's written as -1^{2010} + (-1)^{2011} + 1^{2012} - 1^{2013} + (-1)^{2014}.So, the first term is -1^{2010}, which is -(1^{2010}) = -1.The second term is (-1)^{2011} = -1.Third term: 1^{2012} =1.Fourth term: -1^{2013} = -(1^{2013}) = -1.Fifth term: (-1)^{2014} =1.So, substituting:-1 + (-1) +1 -1 +1.Let me add them sequentially:Start with 0.Add -1: total is -1.Add -1: total is -2.Add 1: total is -1.Add -1: total is -2.Add 1: total is -1.So, the final result is -1.But wait, maybe I should consider the order of operations more carefully. Let me write each term with its sign:Term 1: -1^{2010} = -1.Term 2: + (-1)^{2011} = -1.Term 3: +1^{2012} = +1.Term 4: -1^{2013} = -1.Term 5: + (-1)^{2014} = +1.So, the expression is:-1 + (-1) +1 -1 +1.Let me group the positive and negative terms:Positive terms: 1 +1 =2.Negative terms: -1 -1 -1 =-3.Total: 2 + (-3) = -1.Yes, that confirms it. The sum is -1.But I'm still a bit uncertain because sometimes these problems are designed to have a neat answer like 0 or 1. Maybe I made a mistake in the signs.Let me double-check each term:1. -1^{2010}: As per order of operations, exponentiation comes before the negative sign, so it's -(1^{2010}) = -1.2. (-1)^{2011}: Since the exponent is odd, it's -1.3. 1^{2012}: 1.4. -1^{2013}: Again, -(1^{2013}) = -1.5. (-1)^{2014}: Even exponent, so 1.So, the terms are: -1, -1, 1, -1, 1.Adding them up: -1 -1 +1 -1 +1.Let me add the first two: -1 -1 = -2.Then add the next two: -2 +1 = -1; -1 -1 = -2.Then add the last term: -2 +1 = -1.Yes, it's consistent. So, the final answer is -1.But wait, in the initial problem, the user wrote the same expression twice, and in the second instance, they included a dollar sign at the end. Maybe that's just a typo.In any case, based on my calculations, the sum is -1.However, I just realized that in the original problem, the fourth term is written as "- 1^{2013}", which is -1^{2013}. As I interpreted earlier, this is -(1^{2013}) = -1. But if it were (-1)^{2013}, it would also be -1, so it doesn't change the result.Therefore, my conclusion remains that the sum is -1.But wait, let me check one more time:-1 (from -1^{2010}) +-1 (from (-1)^{2011}) +1 (from 1^{2012}) -1 (from -1^{2013}) +1 (from (-1)^{2014}).So, substituting:-1 -1 +1 -1 +1.Let me add them in pairs:(-1 -1) = -2,(1 -1) = 0,Then +1.So, -2 +0 +1 = -1.Yes, that's correct.I think I've double-checked enough. The answer is -1.

💡Okay, so I have this problem where I'm given that the tangent of alpha is 4 times the square root of 3, and the cosine of (beta minus alpha) is 13 over 14. Also, both beta and alpha are between 0 and pi over 2, with beta being less than alpha. I need to find the value of cosine alpha and then find the value of beta.Alright, starting with part (1): finding cos(alpha). I know that tan(alpha) is given as 4√3. Since tan is sine over cosine, maybe I can use that to find cos(alpha). Let me recall the identity that relates tangent and cosine. I remember that 1 plus tan squared alpha equals sec squared alpha. So, if I can find sec(alpha), then I can find cos(alpha) because sec is the reciprocal of cos.So, let's write that down:1 + tan²(alpha) = sec²(alpha)Given that tan(alpha) is 4√3, so tan²(alpha) is (4√3)², which is 16 * 3, so 48. Therefore,1 + 48 = sec²(alpha)So, 49 = sec²(alpha)Taking the square root of both sides, sec(alpha) is 7, since alpha is in the first quadrant, so sec(alpha) is positive. Therefore, cos(alpha) is 1 over sec(alpha), which is 1/7.Okay, so that gives me cos(alpha) as 1/7. That seems straightforward.Now, moving on to part (2): finding beta. I know that cos(beta - alpha) is 13/14. Hmm, so I have the cosine of the difference between beta and alpha. Since beta is less than alpha, beta - alpha will be negative, right? So, the angle (beta - alpha) is negative, but cosine is an even function, so cos(beta - alpha) is the same as cos(alpha - beta). So, maybe I can think of it as cos(alpha - beta) = 13/14.But I need to find beta. Maybe I can use some trigonometric identities here. Let me think. If I can express beta in terms of alpha and some other angle, perhaps I can find it.I remember that cos(beta - alpha) can be expanded using the cosine of difference identity:cos(beta - alpha) = cos(beta)cos(alpha) + sin(beta)sin(alpha)But I don't know cos(beta) or sin(beta), so that might not be directly helpful. Alternatively, maybe I can find sin(beta - alpha) and then use the sine of difference identity.Wait, but I know cos(beta - alpha) is 13/14. Since beta - alpha is negative, as beta < alpha, the angle is in the fourth quadrant. So, cosine is positive, which matches 13/14 being positive. Sine of a negative angle would be negative, so sin(beta - alpha) would be negative. Maybe I can find sin(beta - alpha) using the Pythagorean identity.Let me try that. So, if cos(theta) = 13/14, then sin(theta) is sqrt(1 - (13/14)^2). Let me compute that:(13/14)^2 is 169/196. So, 1 - 169/196 is (196 - 169)/196 = 27/196. Therefore, sin(theta) is sqrt(27/196) which is (3√3)/14. But since theta is beta - alpha, which is negative, sin(theta) is negative. So, sin(beta - alpha) is -3√3/14.Okay, so now I have both cos(beta - alpha) and sin(beta - alpha). Maybe I can use these to find beta.Alternatively, perhaps I can express beta as alpha - theta, where theta is a positive angle such that cos(theta) = 13/14 and sin(theta) = 3√3/14. Then, beta = alpha - theta.But I need to find beta, so maybe I can find theta first and then subtract it from alpha.Wait, but I don't know alpha yet, except for its cosine. I know cos(alpha) is 1/7, so I can find sin(alpha) as well. Let me compute sin(alpha):Since cos(alpha) = 1/7, sin(alpha) = sqrt(1 - (1/7)^2) = sqrt(48/49) = (4√3)/7.So, sin(alpha) is 4√3/7.Now, if I consider beta = alpha - theta, then I can write beta as alpha - theta, so theta = alpha - beta. But I know that cos(theta) = 13/14 and sin(theta) = 3√3/14.Alternatively, maybe I can use the sine and cosine of beta in terms of alpha and theta.Wait, perhaps it's better to use the angle addition formula. Let me think. If I can write beta as alpha - theta, then:cos(beta) = cos(alpha - theta) = cos(alpha)cos(theta) + sin(alpha)sin(theta)Similarly, sin(beta) = sin(alpha - theta) = sin(alpha)cos(theta) - cos(alpha)sin(theta)But I don't know cos(beta) or sin(beta), so that might not help directly. Alternatively, maybe I can find beta by using the fact that beta = alpha - theta, and then find theta.But I need to find theta such that cos(theta) = 13/14 and sin(theta) = 3√3/14. So, theta is the angle whose cosine is 13/14 and sine is 3√3/14. Let me compute theta.Wait, but maybe I can find theta by taking the arccos of 13/14. Let me compute that. But I don't have a calculator here, but maybe I can recognize the angle.Wait, 13/14 is close to 1, so theta is a small angle. Let me see, 13/14 is approximately 0.9286. The arccos of that would be around 21 degrees or so, since cos(30 degrees) is about 0.866, and cos(0) is 1, so 0.9286 is somewhere around 21 degrees. But I don't know the exact value. Maybe it's a standard angle.Wait, let me check if 3√3/14 is a standard sine value. 3√3 is approximately 5.196, so 5.196/14 is approximately 0.371. So, sin(theta) is approximately 0.371, which is about 21.7 degrees. Hmm, so theta is approximately 21.7 degrees, which is pi/8 or something? Wait, pi/8 is 22.5 degrees, so maybe it's close to pi/8.But perhaps it's better to express theta in terms of inverse cosine or inverse sine. But maybe I can find theta in terms of known angles.Alternatively, perhaps I can use the fact that tan(theta) = sin(theta)/cos(theta) = (3√3/14)/(13/14) = 3√3/13. So, tan(theta) is 3√3/13. Hmm, that doesn't seem like a standard angle. Maybe it's better to leave it as arctan(3√3/13).But wait, maybe I can find beta in terms of alpha and theta. Since beta = alpha - theta, and I know alpha from tan(alpha) = 4√3, so alpha is arctan(4√3). Let me compute arctan(4√3). Hmm, 4√3 is approximately 6.928, which is a large value, so alpha is close to pi/2. Let me see, tan(pi/3) is √3 ≈ 1.732, tan(pi/4) is 1, tan(pi/6) is 1/√3 ≈ 0.577. So, 4√3 is much larger than tan(pi/3), so alpha is greater than pi/3 but less than pi/2.Wait, but maybe I can find alpha in terms of known angles. Let me think, tan(alpha) = 4√3. Is there a known angle with tangent 4√3? Hmm, not that I can recall. So, perhaps I need to keep it as arctan(4√3).But then, beta = alpha - theta, where theta is arccos(13/14). So, beta would be arctan(4√3) - arccos(13/14). But that seems complicated, and I don't think that's the expected answer. Maybe there's a better way.Wait, perhaps I can use the fact that I know cos(alpha) and sin(alpha), and I know cos(beta - alpha) and sin(beta - alpha), and use the angle addition formula to find cos(beta) or sin(beta).Let me try that. So, beta = (beta - alpha) + alpha. Therefore, cos(beta) = cos((beta - alpha) + alpha) = cos(beta - alpha)cos(alpha) - sin(beta - alpha)sin(alpha).I have all these values:cos(beta - alpha) = 13/14cos(alpha) = 1/7sin(beta - alpha) = -3√3/14sin(alpha) = 4√3/7So, plugging these into the formula:cos(beta) = (13/14)(1/7) - (-3√3/14)(4√3/7)Let me compute each term:First term: (13/14)(1/7) = 13/(14*7) = 13/98Second term: (-3√3/14)(4√3/7) = (-3√3 * 4√3)/(14*7) = (-12*3)/98 = (-36)/98But since it's minus this term, it becomes +36/98.So, cos(beta) = 13/98 + 36/98 = (13 + 36)/98 = 49/98 = 1/2.So, cos(beta) = 1/2. Therefore, beta is the angle whose cosine is 1/2. Since beta is between 0 and pi/2, beta must be pi/3, because cos(pi/3) = 1/2.So, beta is pi/3.Wait, let me double-check the calculations to make sure I didn't make any mistakes.First, cos(beta) = (13/14)(1/7) - (-3√3/14)(4√3/7)Compute (13/14)(1/7): 13/(14*7) = 13/98Compute (-3√3/14)(4√3/7): (-3*4)*(√3*√3)/(14*7) = (-12)*(3)/98 = (-36)/98But since it's minus this term, it becomes +36/98.So, 13/98 + 36/98 = 49/98 = 1/2. Yes, that's correct.Therefore, cos(beta) = 1/2, so beta is pi/3.So, the answers are:(1) cos(alpha) = 1/7(2) beta = pi/3

💡Alright, so I've got this problem here about a quiz where each participant has a unique score. The score for each person is given by the formula ( n + 2 - 2k ), where ( n ) is a constant and ( k ) is their rank. The total score of all participants is 2009, and I need to find the smallest possible value of ( n ).Hmm, okay. Let me break this down. First, the score depends on the rank ( k ), and it's linear in ( k ). So, the higher the rank (which I assume means the better the performance), the lower the score? Wait, no, because if ( k ) increases, the score decreases. So, actually, the lower the rank number, the higher the score. That makes sense because in quizzes, usually, rank 1 is the highest scorer.So, the score for rank 1 would be ( n + 2 - 2(1) = n ). For rank 2, it would be ( n + 2 - 2(2) = n - 2 ). For rank 3, ( n + 2 - 2(3) = n - 4 ), and so on. So, each subsequent rank has a score that's 2 less than the previous one. That forms an arithmetic sequence where the first term is ( n ) and the common difference is -2.Now, the total score is the sum of this arithmetic sequence. Let me recall the formula for the sum of an arithmetic series. The sum ( S ) of the first ( m ) terms of an arithmetic sequence is given by:[S = frac{m}{2} times (2a + (m - 1)d)]where ( a ) is the first term, ( d ) is the common difference, and ( m ) is the number of terms.In this case, the first term ( a ) is ( n ), the common difference ( d ) is -2, and the number of terms ( m ) is the number of participants. So, plugging these into the formula:[S = frac{m}{2} times (2n + (m - 1)(-2))]Simplifying that:[S = frac{m}{2} times (2n - 2(m - 1)) = frac{m}{2} times (2n - 2m + 2) = m(n - m + 1)]So, the total score ( S ) is ( m(n - m + 1) ). We know that this equals 2009:[m(n - m + 1) = 2009]So, solving for ( n ):[n = frac{2009}{m} + m - 1]Okay, so ( n ) is expressed in terms of ( m ). Now, since ( m ) is the number of participants, it has to be a positive integer. Also, ( n ) must be an integer because the scores are defined as ( n + 2 - 2k ), which suggests that ( n ) is an integer constant.So, ( m ) must be a divisor of 2009. Let me factorize 2009 to find its divisors. 2009 divided by 7 is 287, and 287 divided by 7 is 41. So, 2009 is ( 7 times 7 times 41 ). Therefore, the divisors of 2009 are 1, 7, 49, 41, 287, and 2009.Now, I need to find the smallest possible value of ( n ). Since ( n = frac{2009}{m} + m - 1 ), I can compute ( n ) for each divisor ( m ) and see which one gives the smallest ( n ).Let me list them out:1. For ( m = 1 ): [ n = frac{2009}{1} + 1 - 1 = 2009 + 0 = 2009 ] 2. For ( m = 7 ): [ n = frac{2009}{7} + 7 - 1 = 287 + 6 = 293 ] 3. For ( m = 41 ): [ n = frac{2009}{41} + 41 - 1 = 49 + 40 = 89 ] 4. For ( m = 49 ): [ n = frac{2009}{49} + 49 - 1 = 41 + 48 = 88 ] 5. For ( m = 287 ): [ n = frac{2009}{287} + 287 - 1 = 7 + 286 = 292 ] 6. For ( m = 2009 ): [ n = frac{2009}{2009} + 2009 - 1 = 1 + 2008 = 2008 ]So, the values of ( n ) for each ( m ) are 2009, 293, 89, 88, 292, and 2008. The smallest among these is 88, which occurs when ( m = 49 ).Wait a second, but I need to make sure that when ( m = 49 ), the scores are all positive integers. Because if ( n = 88 ), then the score for rank 49 would be:[n + 2 - 2k = 88 + 2 - 2(49) = 90 - 98 = -8]Hmm, that's negative. But scores can't be negative in a quiz, right? So, that might be a problem. Did I miss something?Let me check the problem statement again. It says "no two people had the same score," but it doesn't explicitly say that scores have to be positive. Hmm, but in real quizzes, scores are usually non-negative. Maybe I should consider that the scores must be positive integers.If that's the case, then the score for the last participant (rank ( m )) must be at least 1. So:[n + 2 - 2m geq 1][n geq 2m - 1]So, combining this with our earlier equation ( n = frac{2009}{m} + m - 1 ), we have:[frac{2009}{m} + m - 1 geq 2m - 1][frac{2009}{m} geq m][2009 geq m^2][m leq sqrt{2009} approx 44.82]So, ( m ) must be less than or equal to 44. But from our earlier list of divisors, the possible ( m ) values are 1, 7, 41, 49, 287, 2009. The only ( m ) less than or equal to 44 is 1 and 7 and 41.Wait, 41 is less than 44.82, so that's okay. 49 is greater than 44.82, so it violates the condition that the last score is positive. Therefore, ( m = 49 ) is invalid because it would result in a negative score for the last participant.So, the valid ( m ) values are 1, 7, and 41.Let me recalculate ( n ) for these:1. For ( m = 1 ): [ n = 2009 ] But with only one participant, the score is 2009, which is valid.2. For ( m = 7 ): [ n = 293 ] Let's check the last score: [ n + 2 - 2(7) = 293 + 2 - 14 = 281 ] That's positive, so it's valid.3. For ( m = 41 ): [ n = 89 ] Checking the last score: [ 89 + 2 - 2(41) = 91 - 82 = 9 ] Positive, so that's valid.So, now, the possible ( n ) values are 2009, 293, and 89. The smallest is 89.Wait, but earlier I thought ( m = 49 ) gives ( n = 88 ), which is smaller, but it's invalid because the last score is negative. So, 89 is the smallest valid ( n ).But just to be thorough, let me check if there are any other divisors between 41 and 49 that might satisfy the condition. The next divisor after 41 is 49, which we already saw is too big. There are no other divisors in between because 2009's factors are only 1, 7, 41, 49, 287, 2009.Therefore, the smallest possible value of ( n ) is 89.**Final Answer**The smallest possible value of ( n ) is boxed{89}.