💡First, I recognize that Joan currently has 7 blue balloons after losing 2.To find out how many balloons Joan had initially, I need to reverse the loss by adding the 2 lost balloons back to the current number.So, I add 7 (current balloons) to 2 (lost balloons), which equals 9.Therefore, Joan initially had 9 blue balloons.

💡Okay, so I need to figure out how many positive integers less than or equal to 10,000 are simultaneously 7-safe, 11-safe, and 13-safe. Hmm, let me start by understanding what a p-safe number is. From the problem statement, a number n is p-safe if it differs in absolute value by more than 2 from all multiples of p. So, for example, if p is 10, the 10-safe numbers are those that are not within 2 units of any multiple of 10. That means numbers like 3,4,5,6,7,13,14,15,16,17,23,... are 10-safe because they are more than 2 away from 10, 20, 30, etc.So, applying this to 7, 11, and 13, I need numbers that are more than 2 away from any multiple of 7, 11, or 13. That means for each modulus, the number n must not be congruent to 0, 1, 2, p-1, or p-2 modulo p. So, for each prime p, the number n must lie in the range [3, p-3] modulo p.Let me break this down for each prime:1. For 7-safe numbers: n mod 7 should not be 0,1,2,5,6. So the allowed residues are 3 and 4. That gives 2 possible residues.2. For 11-safe numbers: n mod 11 should not be 0,1,2,9,10. So the allowed residues are 3,4,5,6,7,8. That gives 6 possible residues.3. For 13-safe numbers: n mod 13 should not be 0,1,2,11,12. So the allowed residues are 3,4,5,6,7,8,9,10. That gives 8 possible residues.So, for each modulus, we have a certain number of allowed residues. Now, since 7, 11, and 13 are all primes and pairwise coprime, we can use the Chinese Remainder Theorem (CRT) to find the number of solutions modulo 7*11*13 = 1001.The total number of allowed combinations is the product of the number of allowed residues for each modulus. So, that's 2 (for 7) * 6 (for 11) * 8 (for 13) = 96. So, there are 96 residues modulo 1001 that satisfy all three conditions.Now, since we're looking for numbers up to 10,000, we can figure out how many complete blocks of 1001 fit into 10,000. Let's compute 10,000 divided by 1001.10,000 ÷ 1001 ≈ 9.99, so there are 9 complete blocks of 1001 in 10,000. Each block contributes 96 numbers, so 9 * 96 = 864.But wait, 9 blocks would cover up to 9*1001 = 9009. So, we have numbers from 1 to 9009 covered by 9 blocks, each contributing 96 numbers. So, 9*96 = 864 numbers.Now, we have the remaining numbers from 9010 to 10,000. How many numbers is that? 10,000 - 9009 = 991 numbers.Now, we need to check how many of these 991 numbers are 7-safe, 11-safe, and 13-safe. Since the pattern repeats every 1001 numbers, the number of valid numbers in the remaining 991 numbers is the same as the number of valid numbers in the first 991 numbers of the next block.So, we can compute how many valid numbers are in the first 991 numbers of the 10th block.But how do we compute that? Well, since each block has 96 valid numbers, and the distribution is uniform, the number of valid numbers in the first 991 numbers can be approximated by (991 / 1001) * 96.But let me think if that's accurate. Actually, since the valid numbers are spread out uniformly, the number of valid numbers in any interval of length N is roughly (N / 1001) * 96. But since we're dealing with integers, we might need to adjust for the exact count.Alternatively, we can compute how many full cycles of 1001 fit into 991, but since 991 is less than 1001, it's just a partial cycle. So, the number of valid numbers in the first 991 numbers is equal to the number of valid numbers in the first 991 numbers of the modulus cycle.To compute this, we can note that each valid number occurs every 1001 numbers, so in 991 numbers, the number of valid numbers is floor(991 / 1001) * 96 + count of valid numbers in the first (991 mod 1001) numbers.But since 991 < 1001, it's just the count of valid numbers in the first 991 numbers. So, we need to find how many numbers from 1 to 991 are 7-safe, 11-safe, and 13-safe.But how do we compute that? Well, since the valid numbers are uniformly distributed, the count would be approximately (991 / 1001) * 96 ≈ 0.99 * 96 ≈ 95.04. But since we can't have a fraction of a number, we need to check exactly.Alternatively, since the total number of valid numbers up to 1001 is 96, the number of valid numbers up to 991 would be 96 minus the number of valid numbers in the last 1001 - 991 = 10 numbers.So, we need to find how many valid numbers are in the last 10 numbers of the 1001 cycle, i.e., numbers 992 to 1001.But wait, actually, since we're considering numbers from 1 to 991, it's the first 991 numbers. So, the number of valid numbers in the first 991 numbers is equal to the total number of valid numbers in 1001 numbers minus the number of valid numbers in the last 10 numbers (992 to 1001).So, if we can find how many valid numbers are in 992 to 1001, we can subtract that from 96 to get the count for 1 to 991.But how do we find the number of valid numbers in 992 to 1001?Well, each number in 992 to 1001 corresponds to a residue modulo 1001. So, we can check each number from 992 to 1001 and see if it's 7-safe, 11-safe, and 13-safe.But that seems tedious. Maybe there's a smarter way.Alternatively, since the valid numbers are uniformly distributed, the number of valid numbers in any interval of length N is roughly (N / 1001) * 96. So, for N=10, it would be roughly (10 / 1001)*96 ≈ 0.96. So, approximately 1 number.But to be precise, let's compute it.First, let's list the numbers from 992 to 1001:992, 993, 994, 995, 996, 997, 998, 999, 1000, 1001.Now, we need to check each of these numbers to see if they are 7-safe, 11-safe, and 13-safe.But since we're dealing with residues modulo 7, 11, and 13, we can compute their residues and check if they fall into the allowed ranges.Let's compute n mod 7, n mod 11, and n mod 13 for each n from 992 to 1001.But this is a bit time-consuming, but let's try.First, let's compute 992 mod 7, 11, 13.Compute 992 ÷ 7: 7*141=987, so 992-987=5. So, 992 mod7=5.Similarly, 992 ÷11: 11*90=990, so 992-990=2. So, 992 mod11=2.992 ÷13: 13*76=988, so 992-988=4. So, 992 mod13=4.Now, check if these residues are allowed.For 7-safe: allowed residues are 3,4. So, 5 is not allowed. Therefore, 992 is not 7-safe.Next, 993:993 mod7: 993-987=6, so 6 mod7=6. Not allowed (since allowed are 3,4). So, 993 is not 7-safe.994:994 mod7: 994-987=7, which is 0 mod7. Not allowed.995:995 mod7: 995-987=8, 8 mod7=1. Not allowed.996:996 mod7: 996-987=9, 9 mod7=2. Not allowed.997:997 mod7: 997-987=10, 10 mod7=3. Allowed for 7-safe.Now, check 997 mod11: 997-990=7. 7 is allowed (since allowed residues for 11 are 3,4,5,6,7,8). So, 7 is allowed.997 mod13: 997-988=9. 9 is allowed (since allowed residues for 13 are 3-10). So, 9 is allowed.Therefore, 997 is 7-safe, 11-safe, and 13-safe. So, 997 is a valid number.Next, 998:998 mod7: 998-987=11, 11 mod7=4. Allowed for 7-safe.998 mod11: 998-990=8. Allowed.998 mod13: 998-988=10. Allowed.So, 998 is valid.999:999 mod7: 999-987=12, 12 mod7=5. Not allowed.1000:1000 mod7: 1000-987=13, 13 mod7=6. Not allowed.1001:1001 mod7=0, mod11=0, mod13=0. All not allowed.So, in the range 992-1001, only 997 and 998 are valid. So, 2 valid numbers.Therefore, the number of valid numbers in the first 991 numbers is 96 (total in 1001) minus 2 (in the last 10 numbers) = 94.Wait, but actually, the first 991 numbers would include numbers from 1 to 991, which is 991 numbers. The total valid numbers in 1-1001 is 96. So, the number of valid numbers in 1-991 is 96 minus the number of valid numbers in 992-1001, which is 2. So, 96 - 2 = 94.Therefore, in the first 991 numbers, there are 94 valid numbers.So, going back, we have 9 complete blocks contributing 9*96=864 numbers, and the remaining 991 numbers contributing 94 numbers. So, total valid numbers up to 10,000 would be 864 + 94 = 958.Wait, but let me double-check. Because 9 blocks cover up to 9009, and then we have 991 numbers left (9010-10000). But in those 991 numbers, how many are valid?Earlier, I thought that the number of valid numbers in 9010-10000 is the same as in 1-991, which is 94. So, total valid numbers would be 9*96 + 94 = 864 + 94 = 958.But let me think again. When we have 10,000 numbers, and each block of 1001 has 96 valid numbers, then the number of valid numbers up to N is floor(N / 1001) * 96 + count_valid(1, N mod 1001).So, N=10,000.10,000 ÷ 1001 = 9 with a remainder of 991. So, floor(10,000 / 1001)=9, and remainder=991.So, total valid numbers=9*96 + count_valid(1,991).We computed count_valid(1,991)=94.So, total=9*96 +94=864+94=958.Therefore, the answer should be 958.But wait, let me make sure that in the last partial block, we correctly counted the valid numbers. Earlier, I found that in the last 10 numbers (992-1001), there are 2 valid numbers. So, in the first 991 numbers, there are 96-2=94 valid numbers.But actually, when considering the numbers from 9010 to 10000, which is 991 numbers, these correspond to the numbers 9010 to 10000. But in terms of residues modulo 1001, 9010 mod1001=9010-9*1001=9010-9009=1. So, 9010 corresponds to 1 mod1001, and 10000 corresponds to 10000 mod1001=10000-9*1001=10000-9009=991.So, the numbers from 9010 to 10000 correspond to residues 1 to 991 modulo1001. Therefore, the number of valid numbers in this range is the same as the number of valid numbers in 1-991, which we found to be 94.Therefore, total valid numbers=9*96 +94=958.So, the final answer is 958.

💡Alright, so I have this problem where I need to find the minimum surface area of a cone that's inscribed around a sphere with a unit radius. Hmm, okay. Let me try to visualize this. There's a sphere of radius 1, and the cone is touching the sphere in some way. I think the cone is circumscribed around the sphere, meaning the sphere is tangent to the cone's side and possibly sits inside the cone.First, I need to recall some basic properties of cones and spheres. A right circular cone has a height (m), a base radius (r), and a slant height (a). The surface area of a cone is given by the formula πr(r + a), where r is the base radius and a is the slant height. So, my goal is to minimize this surface area.Since the cone is inscribed around the sphere, the sphere must be tangent to the cone's lateral surface and also tangent to the base of the cone. Wait, is it tangent to the base? Or is it just tangent to the lateral surface? I think it's tangent to the lateral surface and sits inside the cone, touching the base as well. So, the sphere touches the cone along a circle and also touches the base of the cone.Let me draw a cross-sectional diagram in my mind. The sphere is inside the cone, touching the cone's side and the base. The sphere has radius 1, so the distance from the center of the sphere to the base of the cone is 1. The height of the cone is m, so the distance from the apex of the cone to the base is m. Therefore, the distance from the apex to the center of the sphere is m - 1.Now, considering the cross-section, which is a triangle (the cone) with a circle (the sphere) inside it. The sphere touches the sides of the triangle and the base. The triangle has a height of m and a base radius of r. The sphere has radius 1, so the center of the sphere is 1 unit above the base.I think I can use similar triangles here. The triangle formed by the apex of the cone, the center of the sphere, and the point where the sphere is tangent to the cone's side should be similar to the triangle formed by the apex, the base, and the edge of the base.Let me denote the apex of the cone as point F, the center of the sphere as point O, and the point where the sphere is tangent to the cone's side as point C. Then, triangle FOC is similar to triangle FAB, where A is the base center and B is the edge of the base.So, the ratio of corresponding sides should be equal. The height of triangle FOC is m - 1, and the height of triangle FAB is m. The radius of the sphere is 1, which corresponds to the radius of the cone's base, r. Wait, no, actually, the radius of the sphere is 1, but the radius of the cone's base is r. So, the ratio of the radii should be equal to the ratio of the heights.Therefore, (m - 1)/m = 1/r. Is that correct? Let me see. If triangle FOC is similar to triangle FAB, then the ratio of their corresponding sides should be equal. So, the ratio of the heights is (m - 1)/m, and the ratio of the radii is 1/r. So, yes, (m - 1)/m = 1/r.From this, I can solve for r in terms of m: r = m/(m - 1). Okay, that's one relationship between r and m.Now, I also know that the slant height of the cone, a, can be found using the Pythagorean theorem: a = sqrt(r^2 + m^2). But from the similar triangles, I also have another expression for a. Since the sphere is tangent to the cone's side, the distance from the center of the sphere to the cone's side is equal to the sphere's radius, which is 1.Wait, maybe I can use the formula for the distance from a point to a line in the cross-sectional view. The cone's side is a line in this cross-section, and the center of the sphere is at a distance of 1 from this line.Let me set up a coordinate system. Let me place the apex of the cone at the origin (0,0), and the base of the cone is at (0, m). The center of the sphere is at (0, m - 1). The cone's side is a line from (0,0) to (r, m). So, the equation of this line is y = (m/r)x.The distance from the center of the sphere (0, m - 1) to this line should be equal to 1. The formula for the distance from a point (x0, y0) to the line ax + by + c = 0 is |ax0 + by0 + c| / sqrt(a^2 + b^2).First, let me write the equation of the cone's side in standard form. y = (m/r)x can be rewritten as (m/r)x - y = 0. So, a = m/r, b = -1, c = 0.The distance from (0, m - 1) to this line is |(m/r)(0) + (-1)(m - 1) + 0| / sqrt((m/r)^2 + (-1)^2) = | - (m - 1) | / sqrt((m^2)/r^2 + 1) = (m - 1) / sqrt((m^2)/r^2 + 1).This distance is equal to the sphere's radius, which is 1. So,(m - 1) / sqrt((m^2)/r^2 + 1) = 1Multiply both sides by the denominator:m - 1 = sqrt((m^2)/r^2 + 1)Square both sides:(m - 1)^2 = (m^2)/r^2 + 1Expand the left side:m^2 - 2m + 1 = (m^2)/r^2 + 1Subtract 1 from both sides:m^2 - 2m = (m^2)/r^2Multiply both sides by r^2:m^2 r^2 - 2m r^2 = m^2Bring all terms to one side:m^2 r^2 - 2m r^2 - m^2 = 0Factor out m:m (m r^2 - 2 r^2 - m) = 0Since m ≠ 0, we have:m r^2 - 2 r^2 - m = 0Let me rearrange:m r^2 - m = 2 r^2Factor m:m (r^2 - 1) = 2 r^2So,m = (2 r^2) / (r^2 - 1)Okay, so now I have m in terms of r. Earlier, I had r = m / (m - 1). Let me check if these are consistent.From r = m / (m - 1), we can solve for m:r (m - 1) = mr m - r = mr m - m = rm (r - 1) = rm = r / (r - 1)Wait, that's different from m = (2 r^2)/(r^2 - 1). Hmm, that's confusing. Maybe I made a mistake somewhere.Wait, let's go back. From the similar triangles, I had (m - 1)/m = 1/r, which gives r = m / (m - 1). But from the distance formula, I got m = (2 r^2)/(r^2 - 1). These two expressions for m in terms of r should be consistent. Let me set them equal:From similar triangles: m = r (m - 1)From distance formula: m = (2 r^2)/(r^2 - 1)So,r (m - 1) = (2 r^2)/(r^2 - 1)Multiply both sides by (r^2 - 1):r (m - 1)(r^2 - 1) = 2 r^2Divide both sides by r (assuming r ≠ 0):(m - 1)(r^2 - 1) = 2 rBut from similar triangles, m = r (m - 1). So, m = r m - rBring terms with m to one side:m - r m = - rm (1 - r) = - rm = (- r)/(1 - r) = r / (r - 1)So, m = r / (r - 1). Let me substitute this into the previous equation:(m - 1)(r^2 - 1) = 2 rSubstitute m = r / (r - 1):(r / (r - 1) - 1)(r^2 - 1) = 2 rSimplify the first term:(r / (r - 1) - 1) = (r - (r - 1)) / (r - 1) = (r - r + 1)/(r - 1) = 1 / (r - 1)So,(1 / (r - 1))(r^2 - 1) = 2 rSimplify (r^2 - 1):(r^2 - 1) = (r - 1)(r + 1)So,(1 / (r - 1))(r - 1)(r + 1) = 2 rCancel (r - 1):(r + 1) = 2 rSo,r + 1 = 2 rSubtract r:1 = rWait, that can't be right because if r = 1, then from m = r / (r - 1), m would be undefined (division by zero). So, something's wrong here.Hmm, maybe I made a mistake in setting up the similar triangles. Let me double-check.I said that triangle FOC is similar to triangle FAB. Let me confirm the points. F is the apex, O is the center of the sphere, C is the point of tangency on the cone's side. A is the center of the base, and B is the edge of the base.So, triangle FOC has sides FO = m - 1, OC = 1 (radius of sphere), and FC is the slant height from apex to the point of tangency.Triangle FAB has sides FA = m, AB = r, and FB is the slant height of the cone.Wait, maybe I got the similar triangles wrong. Maybe triangle FOC is similar to triangle FAB, but I need to make sure about the corresponding sides.Alternatively, perhaps triangle FOC is similar to triangle FBC, where C is the point of tangency.Wait, maybe I need to think differently. Let me consider the similar triangles more carefully.In the cross-sectional view, the cone is a triangle with apex F, base AB, and height FA = m. The sphere is inscribed, touching the base at the center A and touching the side FB at point C.So, the center of the sphere O is located along the height FA, at a distance of 1 from the base AB. So, FO = m - 1.Now, the point C is where the sphere is tangent to the side FB. The radius of the sphere at the point of tangency is perpendicular to FB. So, the line OC is perpendicular to FB.Therefore, triangle FOC is a right triangle with right angle at C.So, triangle FOC has sides FO = m - 1, OC = 1, and FC is the hypotenuse.But triangle FAB is also a right triangle with sides FA = m, AB = r, and FB is the hypotenuse.Wait, so triangle FOC is similar to triangle FAB because both are right triangles and they share the angle at F.Therefore, the ratios of corresponding sides should be equal.So,FO / FA = OC / AB = FC / FBWhich gives:(m - 1)/m = 1/r = FC / FBSo, from (m - 1)/m = 1/r, we get r = m / (m - 1), which is consistent with what I had before.But earlier, when I tried to equate this with the expression from the distance formula, I ended up with r = 1, which is a problem.Wait, maybe I made a mistake in the distance formula approach. Let me go back.I set up the equation of the cone's side as y = (m/r)x, and the center of the sphere is at (0, m - 1). The distance from this point to the line y = (m/r)x is equal to 1.The distance formula gave me:(m - 1) / sqrt((m^2)/r^2 + 1) = 1Which led to:(m - 1)^2 = (m^2)/r^2 + 1Expanding:m^2 - 2m + 1 = m^2 / r^2 + 1Subtracting 1:m^2 - 2m = m^2 / r^2Multiplying both sides by r^2:m^2 r^2 - 2m r^2 = m^2Bringing all terms to one side:m^2 r^2 - 2m r^2 - m^2 = 0Factoring:m (m r^2 - 2 r^2 - m) = 0So, m ≠ 0, so:m r^2 - 2 r^2 - m = 0Rearranging:m (r^2 - 1) = 2 r^2Thus,m = (2 r^2) / (r^2 - 1)Okay, so that's correct.But from the similar triangles, we have r = m / (m - 1). Let me substitute m from this into the expression for m.From r = m / (m - 1), we can solve for m:r (m - 1) = mr m - r = mr m - m = rm (r - 1) = rm = r / (r - 1)So, m = r / (r - 1). Now, substitute this into m = (2 r^2)/(r^2 - 1):r / (r - 1) = (2 r^2)/(r^2 - 1)Multiply both sides by (r - 1)(r^2 - 1):r (r^2 - 1) = 2 r^2 (r - 1)Simplify left side:r^3 - rRight side:2 r^3 - 2 r^2Bring all terms to left:r^3 - r - 2 r^3 + 2 r^2 = 0Combine like terms:- r^3 + 2 r^2 - r = 0Factor out -r:- r (r^2 - 2 r + 1) = 0So,- r (r - 1)^2 = 0Thus, solutions are r = 0 or r = 1. But r = 0 doesn't make sense for a cone, and r = 1 would make m = r / (r - 1) undefined. So, this suggests that there's no solution, which contradicts the problem statement.Hmm, that's a problem. Maybe I made a wrong assumption somewhere.Wait, perhaps the similar triangles approach is not correct. Maybe triangle FOC is not similar to triangle FAB. Let me think again.In the cross-sectional view, the cone is a triangle with apex F, base AB, and height FA = m. The sphere is inscribed, touching the base AB at its center A and touching the side FB at point C.The center of the sphere O is located along FA, at a distance of 1 from A, so FO = m - 1.The radius of the sphere is 1, so OC = 1, and OC is perpendicular to FB because the radius at the point of tangency is perpendicular to the tangent line.Therefore, triangle FOC is a right triangle with legs FO = m - 1 and OC = 1, and hypotenuse FC.Similarly, triangle FAB is a right triangle with legs FA = m and AB = r, and hypotenuse FB.Since both triangles share the angle at F, they are similar.Therefore, the ratio of corresponding sides should be equal:FO / FA = OC / AB = FC / FBSo,(m - 1)/m = 1/r = FC / FBFrom (m - 1)/m = 1/r, we get r = m / (m - 1)From FC / FB = 1/r, and FC = sqrt(FO^2 + OC^2) = sqrt((m - 1)^2 + 1), and FB = sqrt(m^2 + r^2)So,sqrt((m - 1)^2 + 1) / sqrt(m^2 + r^2) = 1/rSquare both sides:((m - 1)^2 + 1) / (m^2 + r^2) = 1/r^2Multiply both sides by (m^2 + r^2):(m - 1)^2 + 1 = (m^2 + r^2)/r^2Simplify left side:m^2 - 2m + 1 + 1 = m^2 - 2m + 2Right side:(m^2)/r^2 + 1So,m^2 - 2m + 2 = (m^2)/r^2 + 1Subtract 1:m^2 - 2m + 1 = (m^2)/r^2But from similar triangles, we have r = m / (m - 1), so r^2 = m^2 / (m - 1)^2Substitute into the equation:m^2 - 2m + 1 = (m^2) / (m^2 / (m - 1)^2) = (m^2) * (m - 1)^2 / m^2 = (m - 1)^2So,m^2 - 2m + 1 = (m - 1)^2Which is an identity, so it doesn't give new information. Therefore, the only relationship we have is r = m / (m - 1)So, going back, we have m = (2 r^2)/(r^2 - 1) from the distance formula, and r = m / (m - 1) from similar triangles.Let me substitute m from the similar triangles into the distance formula expression.From r = m / (m - 1), m = r / (r - 1)Substitute into m = (2 r^2)/(r^2 - 1):r / (r - 1) = (2 r^2)/(r^2 - 1)Multiply both sides by (r - 1)(r^2 - 1):r (r^2 - 1) = 2 r^2 (r - 1)Simplify left side:r^3 - rRight side:2 r^3 - 2 r^2Bring all terms to left:r^3 - r - 2 r^3 + 2 r^2 = 0Combine like terms:- r^3 + 2 r^2 - r = 0Factor out -r:- r (r^2 - 2 r + 1) = 0So,- r (r - 1)^2 = 0Solutions are r = 0 or r = 1. But r = 0 is invalid, and r = 1 makes m undefined. So, this suggests that there's no solution, which can't be right.Wait, maybe I made a mistake in the distance formula approach. Let me double-check.I set up the equation of the cone's side as y = (m/r)x, and the center of the sphere is at (0, m - 1). The distance from this point to the line y = (m/r)x is equal to 1.The distance formula is |(m/r)(0) - 1*(m - 1) + 0| / sqrt((m/r)^2 + (-1)^2) = | - (m - 1) | / sqrt((m^2)/r^2 + 1) = (m - 1) / sqrt((m^2)/r^2 + 1)Set equal to 1:(m - 1) / sqrt((m^2)/r^2 + 1) = 1Square both sides:(m - 1)^2 = (m^2)/r^2 + 1Expand left side:m^2 - 2m + 1 = m^2 / r^2 + 1Subtract 1:m^2 - 2m = m^2 / r^2Multiply both sides by r^2:m^2 r^2 - 2m r^2 = m^2Bring all terms to one side:m^2 r^2 - 2m r^2 - m^2 = 0Factor:m (m r^2 - 2 r^2 - m) = 0So, m ≠ 0, so:m r^2 - 2 r^2 - m = 0Rearrange:m (r^2 - 1) = 2 r^2Thus,m = (2 r^2)/(r^2 - 1)Okay, that's correct.From similar triangles, we have r = m / (m - 1). Let me solve for m in terms of r:r = m / (m - 1)Multiply both sides by (m - 1):r (m - 1) = mExpand:r m - r = mBring terms with m to one side:r m - m = rFactor m:m (r - 1) = rThus,m = r / (r - 1)So, m = r / (r - 1). Now, substitute this into m = (2 r^2)/(r^2 - 1):r / (r - 1) = (2 r^2)/(r^2 - 1)Multiply both sides by (r - 1)(r^2 - 1):r (r^2 - 1) = 2 r^2 (r - 1)Simplify left side:r^3 - rRight side:2 r^3 - 2 r^2Bring all terms to left:r^3 - r - 2 r^3 + 2 r^2 = 0Combine like terms:- r^3 + 2 r^2 - r = 0Factor out -r:- r (r^2 - 2 r + 1) = 0So,- r (r - 1)^2 = 0Solutions are r = 0 or r = 1. But r = 0 is invalid, and r = 1 makes m undefined. So, this suggests that there's no solution, which is a contradiction.Wait, maybe I made a wrong assumption about the similar triangles. Maybe triangle FOC is not similar to triangle FAB. Let me think again.Alternatively, perhaps the similar triangles are different. Maybe triangle FOC is similar to triangle FBC, where C is the point of tangency.Let me try that. Triangle FOC has sides FO = m - 1, OC = 1, and FC is the hypotenuse.Triangle FBC has sides FB = sqrt(m^2 + r^2), BC = r, and FC is the same as in triangle FOC.Wait, but triangle FBC is not a right triangle unless C is the midpoint, which it's not. So, maybe that's not the right approach.Alternatively, perhaps I should use the fact that the sphere is tangent to the cone's side and the base, and use calculus to minimize the surface area.Let me try that approach.The surface area of the cone is A = π r (r + a), where a is the slant height, a = sqrt(r^2 + m^2)From the earlier relationship, m = (2 r^2)/(r^2 - 1)So, a = sqrt(r^2 + (2 r^2/(r^2 - 1))^2)Let me compute a:a = sqrt(r^2 + (4 r^4)/(r^2 - 1)^2)To combine the terms, let me write r^2 as r^2 (r^2 - 1)^2 / (r^2 - 1)^2:a = sqrt( [r^2 (r^2 - 1)^2 + 4 r^4] / (r^2 - 1)^2 )Expand the numerator:r^2 (r^4 - 2 r^2 + 1) + 4 r^4 = r^6 - 2 r^4 + r^2 + 4 r^4 = r^6 + 2 r^4 + r^2So,a = sqrt( (r^6 + 2 r^4 + r^2) / (r^2 - 1)^2 ) = sqrt( r^2 (r^4 + 2 r^2 + 1) / (r^2 - 1)^2 )Factor numerator inside the square root:r^4 + 2 r^2 + 1 = (r^2 + 1)^2So,a = sqrt( r^2 (r^2 + 1)^2 / (r^2 - 1)^2 ) = (r (r^2 + 1)) / (r^2 - 1)Therefore, a = r (r^2 + 1)/(r^2 - 1)Now, the surface area A = π r (r + a) = π r (r + r (r^2 + 1)/(r^2 - 1)) = π r [ r + r (r^2 + 1)/(r^2 - 1) ]Factor out r:A = π r^2 [ 1 + (r^2 + 1)/(r^2 - 1) ]Combine the terms inside the brackets:1 + (r^2 + 1)/(r^2 - 1) = [ (r^2 - 1) + (r^2 + 1) ] / (r^2 - 1) = (2 r^2) / (r^2 - 1)So,A = π r^2 * (2 r^2)/(r^2 - 1) = 2 π r^4 / (r^2 - 1)Now, I need to minimize A with respect to r, where r > 1 (since the sphere has radius 1 and the cone must enclose it).So, A(r) = 2 π r^4 / (r^2 - 1)To find the minimum, take the derivative of A with respect to r, set it to zero, and solve for r.Let me compute dA/dr:Let me write A(r) = 2 π r^4 (r^2 - 1)^(-1)Using the quotient rule:dA/dr = 2 π [ (4 r^3)(r^2 - 1) - r^4 (2 r) ] / (r^2 - 1)^2Simplify numerator:4 r^3 (r^2 - 1) - 2 r^5 = 4 r^5 - 4 r^3 - 2 r^5 = 2 r^5 - 4 r^3Factor numerator:2 r^3 (r^2 - 2)So,dA/dr = 2 π [ 2 r^3 (r^2 - 2) ] / (r^2 - 1)^2 = 4 π r^3 (r^2 - 2) / (r^2 - 1)^2Set derivative equal to zero:4 π r^3 (r^2 - 2) / (r^2 - 1)^2 = 0Since 4 π ≠ 0 and denominator ≠ 0 for r > 1, set numerator equal to zero:r^3 (r^2 - 2) = 0Solutions are r = 0 or r = sqrt(2). Since r > 1, r = sqrt(2)So, critical point at r = sqrt(2). Now, check if this is a minimum.Compute second derivative or check sign changes of first derivative.For r slightly less than sqrt(2), say r = 1.4 (since sqrt(2) ≈ 1.414), plug into dA/dr:r = 1.4, r^2 ≈ 1.96 < 2, so (r^2 - 2) < 0, thus dA/dr < 0For r slightly more than sqrt(2), say r = 1.5, r^2 = 2.25 > 2, so (r^2 - 2) > 0, thus dA/dr > 0Therefore, the function A(r) has a minimum at r = sqrt(2)Now, compute the minimum surface area A at r = sqrt(2)A = 2 π r^4 / (r^2 - 1)Substitute r = sqrt(2):r^2 = 2, r^4 = 4So,A = 2 π * 4 / (2 - 1) = 8 π / 1 = 8 πTherefore, the minimum surface area is 8 πSo, the minimum surface area of the cone is 8π.

💡Okay, so I have this problem: "The equation of the directrix of the parabola y = ax² is y = 1. Find the value of a." Hmm, I remember that parabolas have something called a directrix, which is a line that helps define the shape of the parabola. But I'm a bit fuzzy on the exact details, so I need to recall how the directrix relates to the equation of a parabola.First, I think the standard form of a parabola that opens upward or downward is y = (1/(4p))x², where p is the distance from the vertex to the focus or the directrix. Wait, is that right? Let me double-check. I think it's actually x² = 4py for a parabola that opens upward if p is positive, and downward if p is negative. So, in this case, our equation is y = ax², which can be rewritten as x² = (1/a)y. Comparing this to the standard form x² = 4py, it seems like 4p = 1/a, so p = 1/(4a).Now, the directrix of a parabola in the form x² = 4py is y = -p. Since the directrix is given as y = 1, that means -p = 1. So, p = -1. Wait, that seems a bit confusing. If p is negative, does that mean the parabola opens downward? Let me think. Yes, if p is negative, the parabola opens downward, which makes sense because the directrix is above the vertex.So, if p = -1, then from earlier, p = 1/(4a). So, substituting p = -1 into that equation gives -1 = 1/(4a). To solve for a, I can multiply both sides by 4a, which gives -4a = 1. Then, dividing both sides by -4, I get a = -1/4.Wait, let me make sure I didn't make any mistakes. Starting from the standard form x² = 4py, and our equation is x² = (1/a)y. So, 4p = 1/a, which means p = 1/(4a). The directrix is y = -p, and we're told the directrix is y = 1. So, -p = 1, which means p = -1. Plugging that back into p = 1/(4a), we get -1 = 1/(4a). Solving for a, multiply both sides by 4a: -4a = 1, so a = -1/4. Yeah, that seems consistent.I think I got it. The value of a is -1/4.

💡First, calculate the total number of games in the season by adding the games already played to the remaining games: 45 + 55 = 100 games.Next, determine the number of games needed to win 75% of the season: 75% of 100 games is 75 games.The team has already won 35 games, so subtract this from the total needed wins: 75 - 35 = 40 games.Therefore, the team must win 40 of the remaining 55 games to achieve a 75% win rate for the season.

💡First, I will denote the two numbers as ( x ) and ( y ).We are given two pieces of information:1. The sum of the two numbers is 8: ( x + y = 8 ).2. The product of the two numbers is 28: ( xy = 28 ).Our goal is to find the sum of their squares, ( x^2 + y^2 ).To find ( x^2 + y^2 ), I can use the algebraic identity:[(x + y)^2 = x^2 + 2xy + y^2]Substituting the known values into this identity:[8^2 = x^2 + 2(28) + y^2][64 = x^2 + 56 + y^2]Next, I will isolate ( x^2 + y^2 ) by subtracting 56 from both sides:[x^2 + y^2 = 64 - 56][x^2 + y^2 = 8]Therefore, the sum of the squares of the two numbers is 8.