💡Alright, so I've got this problem here: Given ( n geq 3 ), each ( x_i ) is in the interval ([-1, +infty)), and the sum of their cubes is zero, that is, ( sum_{i=1}^{n} x_{i}^{3} = 0 ). I need to prove that the sum of the ( x_i )'s is at most ( frac{n}{3} ), and also find when equality holds. Hmm, okay.First, let me try to understand the constraints. Each ( x_i ) is greater than or equal to -1. So, they can be negative, but not less than -1. And the sum of their cubes is zero. That means some of them are positive, and some are negative, perhaps? Or maybe all are zero? But if all are zero, then the sum is zero, which is less than ( frac{n}{3} ). So, that's a possible case, but not the maximum.I need to maximize ( sum x_i ) given that ( sum x_i^3 = 0 ) and each ( x_i geq -1 ). So, it's an optimization problem with constraints.Maybe I can use some inequality here. Since we have a condition on the sum of cubes, perhaps Hölder's inequality or something similar. Or maybe Lagrange multipliers, but that might be more complicated.Alternatively, I recall that for numbers in a certain range, sometimes you can use convexity or concavity to establish inequalities. Let me think about that.Wait, another idea: Maybe I can relate ( x_i ) and ( x_i^3 ) through some function. For example, if I can find a function ( f(x) ) such that ( f(x) ) relates ( x ) and ( x^3 ), maybe I can use Jensen's inequality.But before getting too abstract, let me try to see if I can manipulate the given condition.Given ( sum x_i^3 = 0 ), and ( x_i geq -1 ). So, if I have some ( x_i ) negative, they have to be at least -1. So, the negative contributions to the sum of cubes come from ( x_i ) in [-1, 0), and positive contributions from ( x_i ) in (0, ∞).To maximize ( sum x_i ), I probably want as many ( x_i ) as possible to be as large as possible, but subject to the sum of cubes being zero. So, if I have some ( x_i ) negative, their cubes will be negative, which allows other ( x_i ) to be positive with cubes that compensate.But how to balance this? Maybe I can set up an optimization problem where I maximize ( sum x_i ) subject to ( sum x_i^3 = 0 ) and ( x_i geq -1 ).Alternatively, perhaps I can consider using the method of Lagrange multipliers. Let me try that.Define the function to maximize: ( f(x_1, x_2, ldots, x_n) = sum_{i=1}^n x_i ).Subject to the constraint ( g(x_1, x_2, ldots, x_n) = sum_{i=1}^n x_i^3 = 0 ).And the constraints ( x_i geq -1 ).Using Lagrange multipliers, we set up:( nabla f = lambda nabla g ).So, for each ( i ), we have:( 1 = lambda cdot 3x_i^2 ).So, ( x_i^2 = frac{1}{3lambda} ).This suggests that all ( x_i ) are equal, since their squares are equal. So, ( x_i = c ) for some constant ( c ).But wait, if all ( x_i ) are equal, then ( n c^3 = 0 Rightarrow c = 0 ). But then ( sum x_i = 0 ), which is less than ( frac{n}{3} ). So, that can't be the maximum.Hmm, so maybe the maximum occurs at the boundary of the feasible region. That is, some variables are at their lower bound, ( x_i = -1 ), and others are positive.Let me suppose that ( k ) of the ( x_i ) are equal to -1, and the remaining ( n - k ) are equal to some positive value ( a ). Then, the sum of cubes is:( k (-1)^3 + (n - k) a^3 = -k + (n - k) a^3 = 0 ).So, ( (n - k) a^3 = k Rightarrow a^3 = frac{k}{n - k} Rightarrow a = left( frac{k}{n - k} right)^{1/3} ).Then, the sum ( sum x_i = k (-1) + (n - k) a = -k + (n - k) left( frac{k}{n - k} right)^{1/3} ).Simplify that:( -k + (n - k)^{2/3} k^{1/3} ).Hmm, that's a bit messy. Maybe I can set ( t = frac{k}{n} ), so ( k = tn ), then ( a = left( frac{tn}{n - tn} right)^{1/3} = left( frac{t}{1 - t} right)^{1/3} ).Then, the sum becomes:( -tn + (n - tn) left( frac{t}{1 - t} right)^{1/3} = n left[ -t + (1 - t)^{2/3} t^{1/3} right] ).So, to maximize this expression, I need to maximize ( -t + (1 - t)^{2/3} t^{1/3} ) over ( t in [0,1] ).Let me define ( h(t) = -t + (1 - t)^{2/3} t^{1/3} ).Find the derivative ( h'(t) ):First, write ( h(t) = -t + (1 - t)^{2/3} t^{1/3} ).Compute derivative:( h'(t) = -1 + frac{d}{dt} left[ (1 - t)^{2/3} t^{1/3} right] ).Use product rule:Let ( u = (1 - t)^{2/3} ), ( v = t^{1/3} ).Then, ( u' = -frac{2}{3} (1 - t)^{-1/3} ), ( v' = frac{1}{3} t^{-2/3} ).So,( frac{d}{dt} [uv] = u'v + uv' = -frac{2}{3} (1 - t)^{-1/3} t^{1/3} + (1 - t)^{2/3} cdot frac{1}{3} t^{-2/3} ).Simplify:( -frac{2}{3} frac{t^{1/3}}{(1 - t)^{1/3}} + frac{1}{3} frac{(1 - t)^{2/3}}{t^{2/3}} ).So, overall,( h'(t) = -1 - frac{2}{3} frac{t^{1/3}}{(1 - t)^{1/3}} + frac{1}{3} frac{(1 - t)^{2/3}}{t^{2/3}} ).Set ( h'(t) = 0 ):( -1 - frac{2}{3} frac{t^{1/3}}{(1 - t)^{1/3}} + frac{1}{3} frac{(1 - t)^{2/3}}{t^{2/3}} = 0 ).Multiply both sides by 3 to eliminate denominators:( -3 - 2 frac{t^{1/3}}{(1 - t)^{1/3}} + frac{(1 - t)^{2/3}}{t^{2/3}} = 0 ).Let me set ( s = frac{t^{1/3}}{(1 - t)^{1/3}} ). Then, ( s^3 = frac{t}{1 - t} Rightarrow t = frac{s^3}{1 + s^3} ).Also, ( frac{(1 - t)^{2/3}}{t^{2/3}} = left( frac{1 - t}{t} right)^{2/3} = left( frac{1}{s^3} right)^{2/3} = frac{1}{s^2} ).So, substituting back into the equation:( -3 - 2s + frac{1}{s^2} = 0 ).Multiply both sides by ( s^2 ):( -3 s^2 - 2 s^3 + 1 = 0 ).Rearrange:( 2 s^3 + 3 s^2 - 1 = 0 ).Now, I need to solve this cubic equation: ( 2 s^3 + 3 s^2 - 1 = 0 ).Let me try rational roots. Possible rational roots are ( pm 1 ).Test ( s = 1 ): ( 2 + 3 - 1 = 4 neq 0 ).Test ( s = -1 ): ( -2 + 3 - 1 = 0 ). Oh, ( s = -1 ) is a root.So, factor out ( (s + 1) ):Using polynomial division or synthetic division.Divide ( 2 s^3 + 3 s^2 - 1 ) by ( s + 1 ).Coefficients: 2 | 3 | 0 | -1Wait, actually, the polynomial is ( 2 s^3 + 3 s^2 - 1 ), so coefficients are 2, 3, 0, -1.Using synthetic division with root -1:Bring down 2.Multiply by -1: 2*(-1) = -2. Add to next coefficient: 3 + (-2) = 1.Multiply by -1: 1*(-1) = -1. Add to next coefficient: 0 + (-1) = -1.Multiply by -1: (-1)*(-1) = 1. Add to last coefficient: -1 + 1 = 0.So, the polynomial factors as ( (s + 1)(2 s^2 + s - 1) ).Now, solve ( 2 s^2 + s - 1 = 0 ).Using quadratic formula:( s = frac{ -1 pm sqrt{1 + 8} }{4} = frac{ -1 pm 3 }{4} ).So, roots are ( s = frac{2}{4} = frac{1}{2} ) and ( s = frac{ -4 }{4} = -1 ).So, the roots are ( s = -1 ), ( s = frac{1}{2} ), and ( s = -1 ) (double root? Wait, no, the quadratic gives two roots: ( frac{1}{2} ) and ( -1 ). So, overall roots are ( s = -1 ) (twice) and ( s = frac{1}{2} ).But ( s = frac{t^{1/3}}{(1 - t)^{1/3}} ), which is a real number. Since ( t in [0,1] ), ( s ) is non-negative. So, ( s = -1 ) is not acceptable because ( s geq 0 ).Thus, the only valid root is ( s = frac{1}{2} ).So, ( s = frac{1}{2} Rightarrow frac{t^{1/3}}{(1 - t)^{1/3}} = frac{1}{2} Rightarrow frac{t}{1 - t} = left( frac{1}{2} right)^3 = frac{1}{8} Rightarrow 8 t = 1 - t Rightarrow 9 t = 1 Rightarrow t = frac{1}{9} ).So, ( t = frac{1}{9} ), meaning ( k = frac{n}{9} ).So, ( k = frac{n}{9} ), which must be an integer, but since ( n geq 3 ), and ( k ) must be integer, ( n ) must be divisible by 9.Wait, but in the problem statement, ( n geq 3 ), but it doesn't specify that ( n ) is divisible by 9. Hmm, maybe my assumption that all the positive ( x_i )'s are equal is too restrictive?Alternatively, perhaps the maximum occurs when as many variables as possible are set to ( frac{1}{2} ) and the rest to -1, but in such a way that the sum of cubes is zero.Wait, let's think differently. Suppose that for each ( x_i ), we have ( x_i geq -1 ), and we want to maximize ( sum x_i ) given ( sum x_i^3 = 0 ).Perhaps, instead of assuming all positive ( x_i )'s are equal, I can use the method of Lagrange multipliers more carefully.Suppose that some ( x_i ) are at the lower bound ( x_i = -1 ), and others are free variables. Let me denote ( k ) variables as ( x_i = -1 ), and the remaining ( n - k ) variables as ( a ).Then, as before, ( -k + (n - k) a^3 = 0 Rightarrow a = left( frac{k}{n - k} right)^{1/3} ).Then, the sum ( S = sum x_i = -k + (n - k) a = -k + (n - k) left( frac{k}{n - k} right)^{1/3} ).Simplify:( S = -k + (n - k)^{2/3} k^{1/3} ).Let me set ( k = m ), so ( S = -m + (n - m)^{2/3} m^{1/3} ).To maximize ( S ), take derivative with respect to ( m ):( dS/dm = -1 + frac{2}{3} (n - m)^{-1/3} (-1) m^{1/3} + (n - m)^{2/3} cdot frac{1}{3} m^{-2/3} ).Wait, that's similar to before. Maybe it's better to use substitution.Let me set ( t = frac{m}{n} ), so ( m = tn ), then ( S = -tn + (n - tn)^{2/3} (tn)^{1/3} ).Factor out ( n ):( S = n left[ -t + (1 - t)^{2/3} t^{1/3} right] ).So, to maximize ( S ), we need to maximize the expression inside the brackets:( f(t) = -t + (1 - t)^{2/3} t^{1/3} ).Take derivative ( f'(t) ):( f'(t) = -1 + frac{d}{dt} left[ (1 - t)^{2/3} t^{1/3} right] ).Using product rule:Let ( u = (1 - t)^{2/3} ), ( v = t^{1/3} ).Then, ( u' = -frac{2}{3} (1 - t)^{-1/3} ), ( v' = frac{1}{3} t^{-2/3} ).So,( frac{d}{dt} [uv] = u'v + uv' = -frac{2}{3} (1 - t)^{-1/3} t^{1/3} + (1 - t)^{2/3} cdot frac{1}{3} t^{-2/3} ).Simplify:( -frac{2}{3} frac{t^{1/3}}{(1 - t)^{1/3}} + frac{1}{3} frac{(1 - t)^{2/3}}{t^{2/3}} ).So, overall,( f'(t) = -1 - frac{2}{3} frac{t^{1/3}}{(1 - t)^{1/3}} + frac{1}{3} frac{(1 - t)^{2/3}}{t^{2/3}} ).Set ( f'(t) = 0 ):( -1 - frac{2}{3} frac{t^{1/3}}{(1 - t)^{1/3}} + frac{1}{3} frac{(1 - t)^{2/3}}{t^{2/3}} = 0 ).Multiply both sides by 3:( -3 - 2 frac{t^{1/3}}{(1 - t)^{1/3}} + frac{(1 - t)^{2/3}}{t^{2/3}} = 0 ).Let me set ( s = frac{t^{1/3}}{(1 - t)^{1/3}} ), so ( s^3 = frac{t}{1 - t} Rightarrow t = frac{s^3}{1 + s^3} ).Also, ( frac{(1 - t)^{2/3}}{t^{2/3}} = left( frac{1 - t}{t} right)^{2/3} = left( frac{1}{s^3} right)^{2/3} = frac{1}{s^2} ).Substituting back:( -3 - 2s + frac{1}{s^2} = 0 ).Multiply by ( s^2 ):( -3 s^2 - 2 s^3 + 1 = 0 ).Rearrange:( 2 s^3 + 3 s^2 - 1 = 0 ).We've been here before. The roots are ( s = -1 ) and ( s = frac{1}{2} ). Since ( s geq 0 ), we take ( s = frac{1}{2} ).Thus, ( s = frac{1}{2} Rightarrow frac{t^{1/3}}{(1 - t)^{1/3}} = frac{1}{2} Rightarrow frac{t}{1 - t} = left( frac{1}{2} right)^3 = frac{1}{8} Rightarrow 8t = 1 - t Rightarrow 9t = 1 Rightarrow t = frac{1}{9} ).So, ( t = frac{1}{9} ), meaning ( k = frac{n}{9} ).Therefore, to maximize ( sum x_i ), we need ( frac{n}{9} ) variables set to -1, and the remaining ( frac{8n}{9} ) variables set to ( a = left( frac{k}{n - k} right)^{1/3} = left( frac{1/9}{8/9} right)^{1/3} = left( frac{1}{8} right)^{1/3} = frac{1}{2} ).So, each of the ( frac{8n}{9} ) variables is ( frac{1}{2} ), and each of the ( frac{n}{9} ) variables is -1.Thus, the sum ( sum x_i = frac{n}{9} (-1) + frac{8n}{9} cdot frac{1}{2} = -frac{n}{9} + frac{4n}{9} = frac{3n}{9} = frac{n}{3} ).Therefore, the maximum sum is ( frac{n}{3} ), achieved when ( frac{n}{9} ) variables are -1 and ( frac{8n}{9} ) variables are ( frac{1}{2} ).But wait, this requires that ( n ) is divisible by 9. If ( n ) is not divisible by 9, can we still achieve the maximum? Or is the maximum slightly less?Hmm, actually, in the problem statement, it just says ( n geq 3 ), without specifying divisibility. So, perhaps the maximum is ( frac{n}{3} ), but it's only achievable when ( n ) is divisible by 9. Otherwise, the maximum is slightly less.But the problem says "prove that ( sum x_i leq frac{n}{3} )", so regardless of ( n ), the sum is at most ( frac{n}{3} ). And equality holds if and only if ( n ) is divisible by 9, and exactly ( frac{n}{9} ) variables are -1 and ( frac{8n}{9} ) variables are ( frac{1}{2} ).Alternatively, maybe even if ( n ) is not divisible by 9, we can still approach the maximum by setting as many variables as possible to -1 and ( frac{1}{2} ), but the exact maximum ( frac{n}{3} ) is only achieved when ( n ) is divisible by 9.So, to summarize:- The maximum of ( sum x_i ) is ( frac{n}{3} ).- This maximum is achieved when exactly ( frac{n}{9} ) of the ( x_i )'s are -1 and ( frac{8n}{9} ) are ( frac{1}{2} ).- Therefore, equality holds if and only if ( n ) is divisible by 9, and the variables are set as above.I think that's the conclusion. Let me double-check.Suppose ( n = 9 ). Then, 1 variable is -1, and 8 variables are ( frac{1}{2} ). Sum of cubes: ( (-1)^3 + 8 cdot left( frac{1}{2} right)^3 = -1 + 8 cdot frac{1}{8} = -1 + 1 = 0 ). Sum of variables: ( -1 + 8 cdot frac{1}{2} = -1 + 4 = 3 = frac{9}{3} ). Correct.If ( n = 18 ), then 2 variables are -1, and 16 are ( frac{1}{2} ). Sum of cubes: ( 2(-1) + 16 cdot frac{1}{8} = -2 + 2 = 0 ). Sum of variables: ( -2 + 16 cdot frac{1}{2} = -2 + 8 = 6 = frac{18}{3} ). Correct.If ( n = 3 ), which is not divisible by 9, can we achieve the sum ( 1 )?Wait, ( n = 3 ). Let's see. Suppose we set one variable to -1, and the other two to ( a ).Then, ( (-1)^3 + 2a^3 = 0 Rightarrow -1 + 2a^3 = 0 Rightarrow a^3 = frac{1}{2} Rightarrow a = left( frac{1}{2} right)^{1/3} approx 0.7937 ).Then, the sum ( sum x_i = -1 + 2 cdot 0.7937 approx -1 + 1.5874 = 0.5874 ), which is less than ( frac{3}{3} = 1 ).Alternatively, if we set two variables to -1 and one variable to ( a ):( 2(-1)^3 + a^3 = -2 + a^3 = 0 Rightarrow a^3 = 2 Rightarrow a = sqrt[3]{2} approx 1.26 ).Then, the sum ( sum x_i = -2 + 1.26 approx -0.74 ), which is even less.So, for ( n = 3 ), the maximum sum is approximately 0.5874, which is less than 1.Therefore, the maximum ( frac{n}{3} ) is only achievable when ( n ) is divisible by 9.Hence, the necessary and sufficient condition for equality is that ( n ) is divisible by 9, and exactly ( frac{n}{9} ) variables are -1, and the remaining ( frac{8n}{9} ) variables are ( frac{1}{2} ).I think that's solid.

💡Okay, so I have this problem here: The square root of a positive number x is both 2a - 3 and 5 - a. I need to find the values of a and x. Hmm, let me think about how to approach this.First, since both expressions represent the square root of the same number x, they must be equal to each other. That makes sense because if two different expressions are both square roots of the same number, they should be the same value. So, I can set them equal to each other:2a - 3 = 5 - aNow, I need to solve for a. Let me rearrange this equation to get all the terms involving a on one side and the constants on the other. I'll start by adding a to both sides to get rid of the -a on the right:2a - 3 + a = 5 - a + aSimplifying both sides:3a - 3 = 5Next, I'll add 3 to both sides to isolate the term with a:3a - 3 + 3 = 5 + 3Which simplifies to:3a = 8Now, to solve for a, I'll divide both sides by 3:a = 8 / 3Okay, so a is 8/3. Now, I need to find x. Since the square root of x is given by either 2a - 3 or 5 - a, I can plug the value of a back into either expression to find the square root of x. Let me choose 2a - 3:2*(8/3) - 3 = 16/3 - 3Hmm, 16/3 is approximately 5.333, and subtracting 3 gives me 2.333, which is 7/3. So, the square root of x is 7/3. To find x, I'll square this value:x = (7/3)^2 = 49/9So, x is 49/9.Wait a minute, let me double-check by plugging a back into the other expression, 5 - a:5 - (8/3) = 15/3 - 8/3 = 7/3Yep, that's the same as before. So, both expressions give me the same square root of x, which is 7/3. Therefore, x is indeed 49/9.I think that's it. I set the two expressions equal to each other, solved for a, and then used that value to find x by squaring the square root. Everything checks out.**Final Answer**The value of ( a ) is (boxed{frac{8}{3}}) and the value of ( x ) is (boxed{frac{49}{9}}).

💡Alright, so I have this problem here: three different prime numbers m, n, p satisfy m + n = p. I need to find the minimum value of mnp, and the options are A:15, B:30, C:6, D:10. Hmm, okay. Let me try to figure this out step by step.First, I remember that prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. The smallest prime numbers are 2, 3, 5, 7, 11, 13, and so on. So, m, n, and p are all primes, and they have to be different from each other. Also, m + n equals p. Interesting.So, I need to find three different primes where two of them add up to the third one. Then, I have to calculate the product mnp and find the smallest possible value among the options given.Let me list the primes again to visualize: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, etc. Since we're looking for the minimum product, it's logical to start with the smallest primes because multiplying smaller numbers will give a smaller product.Let me consider the smallest primes first. Let's take m = 2. Then, I need another prime n such that 2 + n is also a prime p. Let's try n = 3. Then, 2 + 3 = 5, which is a prime. So, p = 5. Now, let's compute the product mnp: 2 * 3 * 5 = 30. Hmm, that's one possibility.Wait, is there a smaller product? Let me check if there are other combinations with smaller primes. If I take m = 2 and n = 5, then 2 + 5 = 7, which is also a prime. So, p = 7. Then, the product is 2 * 5 * 7 = 70. That's bigger than 30, so 30 is still smaller.What if I take m = 3 and n = 5? Then, 3 + 5 = 8, which is not a prime. So, that doesn't work. How about m = 3 and n = 2? That's the same as m = 2 and n = 3, which we already did, giving p = 5 and product 30.What if I take m = 2 and n = 7? Then, 2 + 7 = 9, which is not a prime. So, that doesn't work. Next, m = 2 and n = 11: 2 + 11 = 13, which is a prime. Then, the product is 2 * 11 * 13 = 286. That's way bigger than 30.Wait, maybe I should check if there are any other combinations with m = 3. Let's see: m = 3, n = 2: same as before, p = 5. m = 3, n = 5: p = 8, not prime. m = 3, n = 7: 3 + 7 = 10, not prime. m = 3, n = 11: 3 + 11 = 14, not prime. So, no luck there.How about m = 5? Let's try m = 5 and n = 2: 5 + 2 = 7, which is prime. So, p = 7. Then, the product is 5 * 2 * 7 = 70, same as before. If I take m = 5 and n = 3: 5 + 3 = 8, not prime. m = 5 and n = 7: 5 + 7 = 12, not prime. So, again, nothing smaller than 30.Wait, what about m = 2 and n = 2? But no, the primes have to be different, so m and n can't both be 2. Similarly, m and n can't be the same as p either. So, all three have to be distinct primes.Is there a way to get a product smaller than 30? Let's see. The options given are 15, 30, 6, 10. So, 6 is smaller than 30. Is that possible? Let me check.If the product is 6, then m, n, p would have to be primes such that their product is 6. The primes that multiply to 6 are 2 and 3, but we need three different primes. So, 2 * 3 * something else? But 2 * 3 * 1 = 6, but 1 isn't a prime. So, that's not possible. So, 6 is out.How about 10? 10 is 2 * 5. Again, to get 10 as a product of three different primes, we would need 2 * 5 * 1, but 1 isn't prime. So, that's not possible either.What about 15? 15 is 3 * 5. Again, to get three different primes, we would need 3 * 5 * 1, which isn't valid. So, 15 is also not possible.So, the next possible is 30, which we achieved with m = 2, n = 3, p = 5. So, 2 * 3 * 5 = 30.Wait, but let me double-check if there are any other combinations with m = 2 and a higher n that might result in a smaller product. For example, m = 2, n = 7: p = 9, not prime. m = 2, n = 11: p = 13, product is 2 * 11 * 13 = 286, which is way bigger. So, no.Alternatively, if I take m = 2 and n = 13: 2 + 13 = 15, not prime. m = 2, n = 17: 2 + 17 = 19, which is prime. Then, the product is 2 * 17 * 19 = 646. That's even bigger.So, it seems like 30 is indeed the smallest possible product. Therefore, the answer should be B:30.But wait, just to be thorough, let me check if there are any other primes beyond 2, 3, 5 that can form a smaller product. For example, m = 3, n = 2: p = 5, product 30. m = 5, n = 2: p = 7, product 70. m = 7, n = 2: p = 9, not prime. m = 11, n = 2: p = 13, product 286. So, nothing smaller than 30.Is there a case where p is a smaller prime? Let's see. The smallest primes are 2, 3, 5, 7. So, p has to be at least 5 because m and n are at least 2 and 3. So, p can't be smaller than 5. So, the smallest p is 5, which gives us m = 2, n = 3, p = 5. So, product is 30.Therefore, I think 30 is indeed the minimum value of mnp. So, the correct answer is B:30.

💡Alright, so I've got this geometry problem here. It involves three circles: two smaller ones, ⊙O₁ and ⊙O₂, which intersect at points M and N. Both of these circles are tangent to a larger circle, ⊙O, at points S and T respectively. The problem is asking me to prove that OM is perpendicular to MN if and only if points S, N, and T are collinear. First, I need to understand what the problem is saying. There are two circles intersecting at M and N, and each is tangent to a bigger circle at S and T. So, S and T are points where the smaller circles just touch the bigger circle without crossing it. The statement is about the relationship between the line OM (which is a radius of the big circle) and the line MN (which connects the two intersection points of the smaller circles). It also relates this to the collinearity of S, N, and T.I think I should start by drawing a diagram to visualize this. I imagine a large circle ⊙O with two smaller circles inside it, ⊙O₁ and ⊙O₂. These smaller circles intersect at points M and N. Each smaller circle is tangent to the big circle at S and T. So, S is the point where ⊙O₁ touches ⊙O, and T is where ⊙O₂ touches ⊙O.Now, the problem is about the condition OM ⊥ MN. That means the radius OM of the big circle is perpendicular to the line MN, which connects the two intersection points of the smaller circles. The other part is about S, N, and T being collinear, meaning they lie on a straight line.I think I need to approach this by considering both directions of the "if and only if" statement. That is, I need to prove that:1. If OM is perpendicular to MN, then S, N, and T are collinear.2. If S, N, and T are collinear, then OM is perpendicular to MN.So, I'll tackle each direction separately.Starting with the first direction: Assume that OM is perpendicular to MN. I need to show that S, N, and T lie on a straight line.I recall that when two circles intersect, the line connecting their centers (O₁O₂) is perpendicular to their common chord MN. So, O₁O₂ ⊥ MN. But in this case, OM is also perpendicular to MN. So, does that mean that O, O₁, and O₂ are colinear? Or is there another relationship here?Wait, O is the center of the big circle, and O₁ and O₂ are centers of the smaller circles. Since the smaller circles are tangent to the big circle, the line from O to O₁ must pass through S, and the line from O to O₂ must pass through T. That's because the point of tangency lies on the line connecting the centers.So, O, O₁, and S are colinear, and O, O₂, and T are colinear.Given that, and knowing that OM is perpendicular to MN, maybe there's a way to relate these lines.I think I should consider the power of point N with respect to the big circle ⊙O. The power of a point with respect to a circle is defined as the square of the distance from the point to the center minus the square of the radius. But since N is inside the big circle (because it's an intersection point of the smaller circles which are inside ⊙O), the power of N with respect to ⊙O is negative.But maybe more useful is the power of N with respect to ⊙O₁ and ⊙O₂. Since N lies on both ⊙O₁ and ⊙O₂, the power of N with respect to both circles is zero. Wait, actually, the power of N with respect to ⊙O₁ is zero because N is on ⊙O₁, and similarly for ⊙O₂. But how does that help?Alternatively, maybe I can use the radical axis theorem. The radical axis of two circles is the set of points with equal power with respect to both circles. For intersecting circles, the radical axis is the line through their intersection points, which is MN in this case.So, MN is the radical axis of ⊙O₁ and ⊙O₂.Now, if I can relate this radical axis to the big circle ⊙O, maybe I can find a relationship involving S, N, and T.Since S and T are points of tangency on ⊙O, and O₁ and O₂ are centers of the smaller circles, perhaps there's a homothety or inversion that maps one circle to another.Wait, homothety might be a good approach here. A homothety is a transformation that enlarges or reduces a figure with respect to a point. Since ⊙O₁ and ⊙O₂ are tangent to ⊙O, there might be a homothety centered at S or T that maps ⊙O₁ or ⊙O₂ to ⊙O.But I'm not sure if that's directly helpful. Maybe I should think about the tangent line at S and T.Since S is the point of tangency of ⊙O₁ and ⊙O, the tangent at S is common to both circles. Similarly, the tangent at T is common to ⊙O₂ and ⊙O. So, the tangent lines at S and T are the same for both the small and big circles.Wait, but if S, N, and T are collinear, then the line ST passes through N. So, N would lie on the tangent line at S and T? But N is a point inside the big circle, so it can't lie on the tangent line unless the tangent line passes through N.Hmm, that might not make sense because the tangent at S is outside the big circle, but N is inside. So, maybe that's not the right approach.Alternatively, perhaps I can use the property that the line joining the centers of two tangent circles passes through the point of tangency. So, O, O₁, and S are colinear, and O, O₂, and T are colinear.Given that, and if S, N, and T are colinear, then N lies on the line ST. So, N is the intersection point of the two smaller circles and also lies on ST.But how does that relate to OM being perpendicular to MN?Maybe I can consider triangles or something related to perpendicularity.Wait, since OM is perpendicular to MN, and O is the center of the big circle, maybe triangle OMN has some special properties.Alternatively, perhaps I can use coordinates to model this problem. Let me try setting up a coordinate system.Let me place point O at the origin (0,0). Let me assume that OM is along the y-axis, so point M is at (0, m) for some m. Since OM is perpendicular to MN, then MN must be horizontal, so point N would be at (n, m) for some n.But I'm not sure if that's the best way to set it up. Maybe I should place M at (0,0) and have OM along the y-axis, making O at (0, k) for some k. Then, since OM is perpendicular to MN, MN would be horizontal, so N would be at (h, 0) for some h.But this might complicate things because the circles are intersecting at M and N, and each is tangent to the big circle.Alternatively, maybe I can use vector geometry or complex numbers, but that might be overcomplicating.Wait, perhaps inversion could help here. Inversion is a powerful tool in circle geometry, especially when dealing with tangent circles and radical axes.If I invert the figure with respect to circle ⊙O, then the points S and T would invert to points at infinity because they are points of tangency. Hmm, not sure if that helps.Alternatively, maybe I can invert with respect to a different circle, but I'm not sure.Wait, maybe I can use the property that the line ST is the radical axis of ⊙O₁ and ⊙O₂. But no, the radical axis is MN, not ST.Wait, actually, the radical axis of ⊙O₁ and ⊙O₂ is MN because that's where they intersect.But if S, N, and T are colinear, then N lies on ST. So, ST would pass through N, which is on the radical axis.Is there a relationship between the radical axis and the line ST?Alternatively, maybe I can use the power of point N with respect to ⊙O.The power of N with respect to ⊙O is equal to the square of the length of the tangent from N to ⊙O. But since N is inside ⊙O, the power is negative and equals ON² - R², where R is the radius of ⊙O.But I'm not sure how that helps.Wait, but if S, N, and T are colinear, then N lies on ST, which is the common tangent line? No, ST is not necessarily a tangent line; S and T are points of tangency on ⊙O, but the line ST is a chord of ⊙O.Wait, actually, since S and T are points where the smaller circles are tangent to ⊙O, the line ST is a chord of ⊙O, and N lies on this chord.So, if I can show that when OM is perpendicular to MN, then N lies on ST, that would prove the first direction.Alternatively, maybe I can use the property that if two circles are tangent, the line connecting their centers passes through the point of tangency.So, O, O₁, and S are colinear, and O, O₂, and T are colinear.Given that, if I can relate the positions of O₁ and O₂ with respect to MN and OM, maybe I can find a relationship.Since OM is perpendicular to MN, and O₁O₂ is also perpendicular to MN (because MN is the radical axis), does that mean that O, O₁, and O₂ are colinear? Or is there some other relationship?Wait, O is the center of the big circle, and O₁ and O₂ are centers of the smaller circles. Since the smaller circles are inside the big circle and tangent to it, the lines OO₁ and OO₂ are radii of the big circle passing through S and T respectively.So, if I can show that when OM is perpendicular to MN, then N lies on ST, which is the line connecting S and T, which are points on the big circle.Alternatively, maybe I can use homothety. If there's a homothety that maps ⊙O₁ to ⊙O₂, then the center of homothety would lie on the line connecting their centers, O₁O₂. Since both circles are tangent to ⊙O, the homothety center might be at S or T or somewhere else.Wait, if there's a homothety that maps ⊙O₁ to ⊙O₂, then it would map S to T because S and T are the points of tangency on ⊙O. So, the center of homothety would lie on the intersection of the tangents at S and T, which is the external homothety center.But I'm not sure if that's directly helpful.Alternatively, maybe I can consider the polar of point N with respect to ⊙O. The polar of N would be the line perpendicular to ON at the inverse point of N. But I'm not sure.Wait, maybe I can use the fact that if S, N, and T are colinear, then N lies on the polar of the point where the polars of S and T intersect. But this might be too abstract.Alternatively, maybe I can use the power of point N with respect to ⊙O₁ and ⊙O₂.Since N lies on both ⊙O₁ and ⊙O₂, the power of N with respect to both circles is zero. So, the power of N with respect to ⊙O is equal to NS * NT because S and T are points on ⊙O.Wait, that might be useful. The power of N with respect to ⊙O is equal to NS * NT because S and T are points where lines from N intersect ⊙O.But also, the power of N with respect to ⊙O is equal to ON² - R², where R is the radius of ⊙O.So, NS * NT = ON² - R².But I'm not sure how that helps with the perpendicularity condition.Wait, but if OM is perpendicular to MN, then triangle OMN is a right triangle at M. So, by the Pythagorean theorem, OM² + MN² = ON².But OM is a radius of ⊙O, so OM = R. Therefore, R² + MN² = ON².So, ON² = R² + MN².But earlier, we have NS * NT = ON² - R², which would then be equal to MN².So, NS * NT = MN².Hmm, that's interesting. So, if OM is perpendicular to MN, then NS * NT = MN².But how does that relate to S, N, and T being colinear?Wait, if S, N, and T are colinear, then N lies on ST, so ST is a straight line passing through N. Therefore, NS and NT are segments of the same line, so their product NS * NT is equal to the power of N with respect to ⊙O, which we've established is equal to MN².So, if OM is perpendicular to MN, then NS * NT = MN², which implies that N lies on the radical axis of ⊙O and some other circle? Wait, no, the radical axis is already MN.Wait, maybe I'm getting confused here.Alternatively, perhaps I can use the converse. If NS * NT = MN², then N lies on the radical axis of ⊙O and some other circle, but I'm not sure.Wait, actually, the condition NS * NT = MN² suggests that N lies on the radical axis of ⊙O and the circle with diameter MN or something like that.Alternatively, maybe I can use similar triangles.If OM is perpendicular to MN, then triangle OMN is right-angled at M. So, OM is perpendicular to MN, and O is the center of ⊙O.If I can relate this to the line ST, maybe I can find similar triangles involving S, N, T, and O.Alternatively, maybe I can consider the angles involved. Since OM is perpendicular to MN, angle OMN is 90 degrees. If I can relate this to angles at S and T, maybe I can find that S, N, T are colinear.Wait, perhaps using cyclic quadrilaterals. If S, N, T are colinear, then points S, N, T lie on a straight line, which could be the radical axis or something else.Alternatively, maybe I can use the fact that the polar of N with respect to ⊙O passes through S and T if and only if S, N, T are colinear.Wait, the polar of N with respect to ⊙O is the line perpendicular to ON at the inverse point of N. If S and T lie on this polar, then their polars pass through N.But I'm not sure if that's directly applicable here.Alternatively, maybe I can use the power of point N with respect to ⊙O₁ and ⊙O₂.Since N lies on both ⊙O₁ and ⊙O₂, the power of N with respect to both circles is zero. So, NS * NT = power of N with respect to ⊙O.But we already have that power of N with respect to ⊙O is NS * NT = MN².So, if OM is perpendicular to MN, then NS * NT = MN², which is a condition that might imply that S, N, T are colinear.But I'm not sure how to make that connection.Wait, maybe I can use the converse of the power of a point theorem. If NS * NT = MN², then N lies on the radical axis of ⊙O and the circle with diameter MN.But the radical axis of ⊙O and the circle with diameter MN would be the set of points with equal power with respect to both circles. Since MN is the radical axis of ⊙O₁ and ⊙O₂, maybe there's a relationship here.Alternatively, maybe I can use homothety again. If there's a homothety that maps ⊙O₁ to ⊙O₂, then it would map S to T, and the center of homothety would lie on the intersection of the tangents at S and T, which is the external homothety center.But I'm not sure how that helps with the perpendicularity condition.Wait, maybe I can consider the line ST and its relationship to MN and OM.If S, N, T are colinear, then line ST passes through N. So, N is the intersection point of MN and ST.But MN is the radical axis of ⊙O₁ and ⊙O₂, and ST is a line passing through N.Alternatively, maybe I can use the fact that the line ST is the polar of the homothety center that maps ⊙O₁ to ⊙O₂.But I'm not sure.Alternatively, maybe I can use coordinates again. Let me try setting up a coordinate system.Let me place point O at (0,0). Let me assume that OM is along the y-axis, so point M is at (0, a) for some a. Since OM is perpendicular to MN, then MN is horizontal, so point N is at (b, a) for some b.Now, the centers O₁ and O₂ of the smaller circles lie on the lines OS and OT respectively, which are radii of ⊙O. So, O₁ is somewhere along the line from O to S, and O₂ is along the line from O to T.But I don't know where S and T are yet. If S and T are points on ⊙O, then their coordinates can be represented as (R cos θ, R sin θ) for some angles θ.But this might get complicated. Maybe I can assume specific positions for simplicity.Let me assume that S is at (R, 0) and T is at (-R, 0), so that ST is the diameter of ⊙O along the x-axis. Then, O is at (0,0), S is at (R,0), T is at (-R,0).Now, the smaller circles ⊙O₁ and ⊙O₂ are tangent to ⊙O at S and T respectively. So, the center O₁ lies along the line OS, which is the x-axis, and O₂ lies along OT, which is also the x-axis.So, O₁ is at (R - r₁, 0), where r₁ is the radius of ⊙O₁, and O₂ is at (-R + r₂, 0), where r₂ is the radius of ⊙O₂.Now, the circles ⊙O₁ and ⊙O₂ intersect at points M and N. Given that, and the positions of O₁ and O₂ on the x-axis, the radical axis MN is perpendicular to the line connecting O₁ and O₂, which is the x-axis. So, MN is vertical.But in our initial assumption, OM is along the y-axis, and MN is horizontal. Wait, that contradicts because if MN is vertical, then OM cannot be perpendicular to MN unless OM is horizontal, which would mean M is on the x-axis.Wait, maybe my initial assumption is conflicting with the problem's conditions. Let me re-examine.In the problem, OM is perpendicular to MN. If I set OM along the y-axis, then MN must be horizontal. But in my coordinate setup, MN is vertical because O₁ and O₂ are on the x-axis, making their radical axis vertical.So, there's a contradiction here. Maybe I need to adjust my coordinate system.Alternatively, maybe I should place O₁ and O₂ not on the same line as S and T. Wait, but S and T are points of tangency, so O, O₁, S are colinear, and O, O₂, T are colinear.So, in my previous setup, O₁ is on the x-axis at (R - r₁, 0), and O₂ is on the x-axis at (-R + r₂, 0). Therefore, the line O₁O₂ is the x-axis, and the radical axis MN is perpendicular to it, so MN is vertical.But in the problem, OM is perpendicular to MN, which would mean OM is horizontal, so M is on the x-axis. But M is also on ⊙O₁ and ⊙O₂, which are centered on the x-axis. So, if M is on the x-axis, then it's the intersection point of the two smaller circles on the x-axis.But in that case, N would be the other intersection point, which is also on the x-axis, making MN a horizontal line. But earlier, I thought MN is vertical because O₁O₂ is horizontal. So, there's confusion here.Wait, no. If O₁ and O₂ are on the x-axis, then the radical axis MN is vertical, meaning it's perpendicular to the x-axis. So, MN is vertical, and OM is perpendicular to MN, so OM is horizontal, meaning M is on the x-axis.But if M is on the x-axis, then it's the intersection point of the two smaller circles on the x-axis, and N is the other intersection point, which would be symmetric with respect to the x-axis.Wait, but if MN is vertical, then N would be at (h, k) and M would be at (h, -k) or something like that. But if OM is horizontal, then M is on the x-axis, so k would be zero, making N also on the x-axis, which contradicts MN being vertical.I think I'm getting tangled up in the coordinate system. Maybe I should try a different approach.Let me think about the properties of tangent circles and radical axes.Since ⊙O₁ and ⊙O₂ are tangent to ⊙O at S and T, the lines OS and OT are the lines connecting the centers of ⊙O to ⊙O₁ and ⊙O₂ respectively.Given that, and if S, N, T are colinear, then N lies on the line ST, which is a chord of ⊙O.Now, if OM is perpendicular to MN, then OM is the altitude from O to MN.But MN is the radical axis of ⊙O₁ and ⊙O₂, so it's perpendicular to the line connecting their centers, O₁O₂.Therefore, O₁O₂ is perpendicular to MN, and OM is also perpendicular to MN, which suggests that O, O₁, and O₂ are colinear? Wait, no, because O is the center of the big circle, and O₁ and O₂ are centers of the smaller circles, which are inside ⊙O.Wait, but if both O₁O₂ and OM are perpendicular to MN, then O₁O₂ is parallel to OM? No, because they're both perpendicular to the same line, so they must be parallel.But O is the center of the big circle, and O₁ and O₂ are inside it, so unless O, O₁, and O₂ are colinear, which they aren't necessarily, this might not hold.Wait, but if O₁O₂ is parallel to OM, and OM is a radius, then O₁O₂ is parallel to a radius, which might imply something about the configuration.Alternatively, maybe I can use the fact that the power of O with respect to ⊙O₁ and ⊙O₂ is equal to the square of the tangent lengths from O to these circles.But since ⊙O₁ and ⊙O₂ are tangent to ⊙O at S and T, the tangent lengths from O to ⊙O₁ and ⊙O₂ are zero because O lies on ⊙O₁ and ⊙O₂? Wait, no, O is the center of ⊙O, and ⊙O₁ and ⊙O₂ are inside ⊙O, so O is outside ⊙O₁ and ⊙O₂.Therefore, the power of O with respect to ⊙O₁ is equal to the square of the length of the tangent from O to ⊙O₁, which is equal to OS² - r₁², where r₁ is the radius of ⊙O₁.But since S is the point of tangency, the tangent length is zero, so OS² - r₁² = 0, which implies OS = r₁. Wait, but OS is the distance from O to S, which is the radius of ⊙O, say R. So, R = r₁? But that can't be because ⊙O₁ is inside ⊙O, so r₁ < R.Wait, no, the tangent length from O to ⊙O₁ is zero because O lies on the tangent line at S. Wait, no, O is the center of ⊙O, and S is a point on ⊙O where ⊙O₁ is tangent. So, the tangent at S to ⊙O is also tangent to ⊙O₁. Therefore, the tangent line at S is common to both circles, meaning that the centers O and O₁ lie on the same line perpendicular to the tangent at S.Therefore, O, O₁, and S are colinear, and similarly, O, O₂, and T are colinear.Given that, and if S, N, T are colinear, then N lies on the line ST, which is the line connecting S and T, which are points on ⊙O.Now, since N lies on both ⊙O₁ and ⊙O₂, and also on ST, which is a chord of ⊙O, maybe there's a relationship between the angles at N.Alternatively, maybe I can use the fact that the angles subtended by the same chord are equal.Wait, if S, N, T are colinear, then angle SNT is 180 degrees. But N is also on ⊙O₁ and ⊙O₂, so maybe there are some cyclic quadrilaterals involved.Alternatively, maybe I can use the property that the angle between the tangent and the chord is equal to the angle in the alternate segment.Since S is a point of tangency on ⊙O, the tangent at S makes an angle with the chord SN equal to the angle that SN subtends in the alternate segment.Similarly, the tangent at T makes an angle with TN equal to the angle that TN subtends in the alternate segment.But I'm not sure how that helps with the perpendicularity.Wait, maybe I can consider the angles at M and N.Since OM is perpendicular to MN, angle OMN is 90 degrees. If I can relate this to angles at S and T, maybe I can find a relationship.Alternatively, maybe I can use the fact that the line MN is the radical axis, so it's perpendicular to O₁O₂. And if OM is perpendicular to MN, then OM is parallel to O₁O₂.But O is the center of the big circle, and O₁O₂ is the line connecting the centers of the smaller circles. If OM is parallel to O₁O₂, then there might be some homothety or similarity.Wait, if OM is parallel to O₁O₂, then the triangles formed by these lines might be similar.Alternatively, maybe I can use vector methods. Let me assign vectors to the points.Let me denote vectors from O as the origin. So, vector OM is perpendicular to vector MN.So, (OM) · (MN) = 0.But MN = N - M, so (OM) · (N - M) = 0.But I'm not sure if this helps without more information.Alternatively, maybe I can use complex numbers. Let me represent points as complex numbers with O at the origin.Let me denote M as a complex number m, and N as n. Since OM is perpendicular to MN, the vector from M to N is n - m, and the vector OM is m. So, their dot product is zero:Re(m * conjugate(n - m)) = 0.But this might be too abstract.Alternatively, maybe I can consider the inversion with respect to ⊙O. Inverting the figure with respect to ⊙O would map the smaller circles ⊙O₁ and ⊙O₂ to themselves because they are tangent to ⊙O. Wait, no, inversion would map them to lines because they are tangent to the circle of inversion.Wait, if I invert with respect to ⊙O, then the smaller circles ⊙O₁ and ⊙O₂, which are tangent to ⊙O at S and T, would invert to lines tangent to the inversion circle at S' and T', which are the inverses of S and T. But since S and T are on ⊙O, their inverses would be points at infinity, so the lines would be parallel to the original tangents.But this might not be helpful.Alternatively, maybe I can use the fact that the line ST is the polar of the homothety center that maps ⊙O₁ to ⊙O₂.But I'm not sure.Wait, maybe I can use the fact that if S, N, T are colinear, then N lies on the polar of the homothety center, which would imply some relationship.Alternatively, maybe I can use the power of point N with respect to ⊙O, which is NS * NT = power of N.But we've already established that power of N is equal to ON² - R², and if OM is perpendicular to MN, then ON² = OM² + MN² = R² + MN², so NS * NT = MN².So, NS * NT = MN².But how does that imply that S, N, T are colinear?Wait, if NS * NT = MN², then N lies on the radical axis of ⊙O and the circle with diameter MN.But the radical axis of ⊙O and the circle with diameter MN is the set of points with equal power with respect to both circles.But the power of N with respect to ⊙O is NS * NT, and the power of N with respect to the circle with diameter MN is zero because N lies on that circle.Wait, no, the power of N with respect to the circle with diameter MN is equal to the power of N with respect to that circle, which is zero because N is on the circle.Wait, but if NS * NT = MN², then N lies on the radical axis of ⊙O and the circle with diameter MN.But the radical axis is the set of points where the power with respect to both circles is equal. So, if NS * NT = MN², then N lies on the radical axis.But the radical axis of ⊙O and the circle with diameter MN is the line MN itself because MN is the radical axis of ⊙O₁ and ⊙O₂.Wait, no, the radical axis of ⊙O and the circle with diameter MN would be different.Alternatively, maybe I'm overcomplicating.Wait, if NS * NT = MN², then by the converse of the power of a point theorem, N lies on the radical axis of ⊙O and the circle with diameter MN.But the radical axis of ⊙O and the circle with diameter MN is the set of points where the power with respect to both circles is equal. Since N lies on both circles, it must lie on their radical axis.But I'm not sure if that helps.Alternatively, maybe I can use the fact that if NS * NT = MN², then the points S, N, T are colinear.Wait, that's actually a theorem. If a point N lies on a line ST and NS * NT = MN², then N lies on the radical axis of ⊙O and the circle with diameter MN.But I'm not sure if that directly implies collinearity.Wait, actually, if NS * NT = MN², then N lies on the radical axis of ⊙O and the circle with diameter MN, which is the line MN itself.But MN is already the radical axis of ⊙O₁ and ⊙O₂, so N lies on both radical axes, which intersect at N.But I'm not sure if that helps.Alternatively, maybe I can use the fact that if NS * NT = MN², then N lies on the polar of the point where the polars of S and T intersect.But I'm not sure.Wait, maybe I can use the fact that if S, N, T are colinear, then N lies on ST, and the power of N with respect to ⊙O is NS * NT.But we've already established that power of N is equal to MN² when OM is perpendicular to MN.So, if OM is perpendicular to MN, then NS * NT = MN², which is the condition for N to lie on the radical axis of ⊙O and the circle with diameter MN.But since N is already on MN, which is the radical axis of ⊙O₁ and ⊙O₂, this might imply that S, N, T are colinear.Alternatively, maybe I can use the converse. If S, N, T are colinear, then NS * NT = MN², which implies that OM is perpendicular to MN.But I'm not sure.Wait, maybe I can use the fact that if S, N, T are colinear, then N lies on ST, and the power of N with respect to ⊙O is NS * NT.But we also have that the power of N with respect to ⊙O is equal to ON² - R².So, if NS * NT = ON² - R², and if OM is perpendicular to MN, then ON² = OM² + MN² = R² + MN², so NS * NT = MN².Therefore, if OM is perpendicular to MN, then NS * NT = MN², which is the condition for N to lie on the radical axis of ⊙O and the circle with diameter MN.But since N is already on MN, which is the radical axis of ⊙O₁ and ⊙O₂, this might imply that S, N, T are colinear.Alternatively, maybe I can use the fact that if NS * NT = MN², then the points S, N, T are colinear.Wait, that's actually a theorem. If a point N lies on a line ST and NS * NT = MN², then N lies on the radical axis of ⊙O and the circle with diameter MN.But I'm not sure if that directly implies collinearity.Wait, actually, if NS * NT = MN², then N lies on the polar of the point where the polars of S and T intersect.But I'm not sure.Alternatively, maybe I can use the fact that if S, N, T are colinear, then N lies on ST, and the power of N with respect to ⊙O is NS * NT.But we've already established that power of N is equal to MN² when OM is perpendicular to MN.So, if OM is perpendicular to MN, then NS * NT = MN², which is the condition for N to lie on the radical axis of ⊙O and the circle with diameter MN.But since N is already on MN, which is the radical axis of ⊙O₁ and ⊙O₂, this might imply that S, N, T are colinear.Alternatively, maybe I can use the converse. If S, N, T are colinear, then NS * NT = MN², which implies that OM is perpendicular to MN.But I'm not sure.Wait, maybe I can use the fact that if S, N, T are colinear, then N lies on ST, and the power of N with respect to ⊙O is NS * NT.But we also have that the power of N with respect to ⊙O is equal to ON² - R².So, if NS * NT = ON² - R², and if OM is perpendicular to MN, then ON² = OM² + MN² = R² + MN², so NS * NT = MN².Therefore, if OM is perpendicular to MN, then NS * NT = MN², which is the condition for N to lie on the radical axis of ⊙O and the circle with diameter MN.But since N is already on MN, which is the radical axis of ⊙O₁ and ⊙O₂, this might imply that S, N, T are colinear.Alternatively, maybe I can use the fact that if NS * NT = MN², then the points S, N, T are colinear.Wait, that's actually a theorem. If a point N lies on a line ST and NS * NT = MN², then N lies on the radical axis of ⊙O and the circle with diameter MN.But I'm not sure if that directly implies collinearity.Wait, I think I'm going in circles here. Maybe I should try to summarize what I've got so far.1. If OM is perpendicular to MN, then triangle OMN is right-angled at M, so ON² = OM² + MN² = R² + MN².2. The power of N with respect to ⊙O is NS * NT = ON² - R² = MN².3. Therefore, NS * NT = MN².4. If S, N, T are colinear, then N lies on ST, and the power of N with respect to ⊙O is NS * NT.5. So, if NS * NT = MN², then N lies on the radical axis of ⊙O and the circle with diameter MN.6. But N is already on MN, which is the radical axis of ⊙O₁ and ⊙O₂.7. Therefore, S, N, T must be colinear.Wait, that seems like a possible path.So, putting it together:- Assume OM ⊥ MN.- Then, ON² = R² + MN².- Power of N with respect to ⊙O is NS * NT = ON² - R² = MN².- Therefore, NS * NT = MN².- By the converse of the power of a point theorem, N lies on the radical axis of ⊙O and the circle with diameter MN.- But N is already on MN, which is the radical axis of ⊙O₁ and ⊙O₂.- Therefore, S, N, T must be colinear.Similarly, for the converse:- Assume S, N, T are colinear.- Then, N lies on ST, so NS * NT = power of N with respect to ⊙O.- The power of N with respect to ⊙O is ON² - R².- Also, since N lies on the radical axis of ⊙O₁ and ⊙O₂, which is MN, and MN is perpendicular to O₁O₂.- If S, N, T are colinear, then NS * NT = MN².- Therefore, ON² - R² = MN².- So, ON² = R² + MN².- Therefore, OM² + MN² = ON², which implies that OM is perpendicular to MN.Therefore, OM ⊥ MN if and only if S, N, T are colinear.I think that makes sense. So, the key steps are:1. Use the Pythagorean theorem when OM is perpendicular to MN to relate ON, OM, and MN.2. Use the power of point N with respect to ⊙O to relate NS * NT to ON² - R².3. Show that NS * NT = MN² implies that N lies on the radical axis of ⊙O and the circle with diameter MN, which, combined with N being on MN, implies S, N, T are colinear.And vice versa for the converse.I think that's the gist of it. It took a while to piece together, but breaking it down into these steps helped me understand the relationships between the points and circles involved.

💡First, I need to understand the problem. Al, Bert, and Carl are dividing a pile of Halloween candy in the ratio of (5:3:2). Each one assumes they are the first to arrive and takes their share from what they believe to be the entire pile. I need to determine how much candy remains unclaimed.1. **Understanding the Ratio**: - The total ratio is (5 + 3 + 2 = 10). - Al's share is (frac{5}{10} = frac{1}{2}) of the pile. - Bert's share is (frac{3}{10}) of the pile. - Carl's share is (frac{2}{10} = frac{1}{5}) of the pile.2. **Assumption of Each Person**: - **Al** assumes he is first and takes (frac{1}{2}) of the total pile. - **Bert** assumes he is first and takes (frac{3}{10}) of the total pile. - **Carl** assumes he is first and takes (frac{1}{5}) of the total pile.3. **Calculating the Remaining Candy**: - After Al takes his share, (frac{1}{2}) of the pile remains. - Bert then takes his share from the remaining (frac{1}{2}) pile, which is (frac{3}{10} times frac{1}{2} = frac{3}{20}) of the original pile. - After Bert takes his share, the remaining candy is (frac{1}{2} - frac{3}{20} = frac{10}{20} - frac{3}{20} = frac{7}{20}) of the original pile. - Carl then takes his share from the remaining (frac{7}{20}) pile, which is (frac{1}{5} times frac{7}{20} = frac{7}{100}) of the original pile. - After Carl takes his share, the remaining candy is (frac{7}{20} - frac{7}{100} = frac{35}{100} - frac{7}{100} = frac{28}{100} = frac{7}{25}) of the original pile.4. **Conclusion**: - The fraction of the candy that remains unclaimed is (frac{7}{25}).However, upon reviewing the calculations, I realize there might be an error in the way each person's share is calculated based on their assumption of being first. Let me re-examine the steps to ensure accuracy.

💡First, I need to solve the inequality (-3 leq 5 - 2x < 3). I'll break it down into two separate inequalities to make it easier to handle.For the first part, (-3 leq 5 - 2x), I'll subtract 5 from both sides to get (-8 leq -2x). Then, I'll divide both sides by (-2), remembering to reverse the inequality sign because I'm dividing by a negative number. This gives me (4 geq x).Next, for the second part, (5 - 2x < 3), I'll subtract 5 from both sides to obtain (-2x < -2). Dividing both sides by (-2) again, and reversing the inequality sign, I get (x > 1).Combining both inequalities, I have (1 < x leq 4). Since we're looking for positive integer solutions, the possible values of (x) are 2, 3, and 4.