💡Okay, so I have this problem where two farmers agree that pigs are worth 400 each and goats are worth 250 each. When one farmer owes the other money, they settle the debt using pigs or goats, and if needed, they can receive change in the form of goats or pigs. I need to find the smallest positive debt that can be resolved in this way.Hmm, let me try to understand this. So, if Farmer A owes Farmer B some money, Farmer A can pay using pigs or goats, and if the amount isn't exact, Farmer B can give back change using pigs or goats. So, essentially, they are exchanging pigs and goats to make up the exact amount of the debt.I think this is similar to problems where you have to find the smallest amount that can be made using coins of certain denominations. In those problems, the smallest amount is usually the greatest common divisor (GCD) of the denominations. Maybe this is a similar concept here with pigs and goats instead of coins.Let me recall, the GCD of two numbers is the smallest positive integer that can be expressed as a linear combination of those numbers with integer coefficients. So, if I can express the debt as a combination of pigs and goats, then the smallest positive debt would be the GCD of 400 and 250.Wait, let me check that. If I can write the debt D as 400p + 250g, where p and g are integers (they can be positive or negative, representing giving or receiving change), then D must be a multiple of the GCD of 400 and 250.So, first, I should find the GCD of 400 and 250.Let me calculate that. The prime factors of 400 are 2^4 * 5^2, and the prime factors of 250 are 2 * 5^3. The GCD is the product of the smallest powers of the common prime factors, which is 2^1 * 5^2 = 2 * 25 = 50.Okay, so the GCD is 50. That means the smallest positive debt that can be resolved using pigs and goats is 50.But wait, let me make sure I'm not missing something. The problem says that the debt is paid in pigs or goats, with change received as necessary. So, does that mean that the debt must be a multiple of 50? Because 50 is the GCD, any multiple of 50 can be expressed as a combination of pigs and goats.But the question is asking for the smallest positive debt. So, 50 is the smallest positive amount that can be expressed as a combination of 400 and 250 with integer coefficients. Therefore, the smallest positive debt is 50.Let me think of an example to verify this. Suppose the debt is 50. How can this be settled?If Farmer A owes Farmer B 50, Farmer A can give Farmer B 2 goats, which is 2 * 250 = 500, and Farmer B would give back 1 pig, which is 1 * 400 = 400. So, the net amount is 500 - 400 = 50. That works.Alternatively, Farmer A could give Farmer B 1 pig and receive 1 goat as change. That would be 400 - 250 = 150. But that's not 50. Wait, so maybe that's not the way.Wait, no, if Farmer A gives 2 goats, which is 500, and Farmer B gives back 1 pig, which is 400, the difference is 50, which settles the debt. So, that works.Alternatively, could there be a smaller positive debt? Let's say 25. Is that possible? If I try to express 25 as a combination of 400 and 250, I would need integers p and g such that 400p + 250g = 25. But 400 and 250 are both multiples of 50, so their combination would also be a multiple of 50. Therefore, 25 cannot be expressed as such a combination. So, 50 is indeed the smallest positive debt.Another way to think about it is that since both pigs and goats are worth multiples of 50, any combination of them will also be a multiple of 50. Therefore, the smallest positive multiple is 50.So, I think I'm confident now that the smallest positive debt that can be resolved in this way is 50.

💡Okay, so I have this problem where I need to find the acute angles α and β that satisfy the system of equations:2 sin 2β = 3 sin 2α tg β = 3 tg αHmm, let me think about how to approach this. I remember that sin 2θ is equal to 2 sin θ cos θ, so maybe I can rewrite the first equation using that identity. Let me try that.So, 2 sin 2β becomes 2 * 2 sin β cos β, which is 4 sin β cos β. Similarly, 3 sin 2α becomes 3 * 2 sin α cos α, which is 6 sin α cos α. So the first equation becomes:4 sin β cos β = 6 sin α cos αI can simplify this by dividing both sides by 2:2 sin β cos β = 3 sin α cos αHmm, that's still a bit complicated. Maybe I can use the second equation to express one variable in terms of the other. The second equation is tg β = 3 tg α. I know that tg θ is sin θ / cos θ, so:sin β / cos β = 3 (sin α / cos α)Let me denote this as equation (2):sin β / cos β = 3 sin α / cos αMaybe I can solve this for sin β or cos β and substitute back into the first equation. Let's see.From equation (2), I can write sin β = 3 sin α cos β / cos α. Let me substitute this into the first equation.So, the first equation is 2 sin β cos β = 3 sin α cos α. Substituting sin β:2 * (3 sin α cos β / cos α) * cos β = 3 sin α cos αSimplify the left side:2 * 3 sin α cos β * cos β / cos α = 6 sin α cos² β / cos αSo, 6 sin α cos² β / cos α = 3 sin α cos αI can cancel sin α from both sides, assuming sin α ≠ 0, which it isn't because α is an acute angle.So, 6 cos² β / cos α = 3 cos αMultiply both sides by cos α:6 cos² β = 3 cos² αDivide both sides by 3:2 cos² β = cos² αSo, cos² α = 2 cos² βThat's interesting. Let me keep that in mind.Now, from equation (2), I have tg β = 3 tg α. Let me express this as:sin β / cos β = 3 sin α / cos αLet me denote this as equation (2) again.I can write this as:sin β = 3 sin α cos β / cos αBut I also know from the first equation that 2 sin β cos β = 3 sin α cos α. Let me see if I can find a relationship between sin β and sin α.Alternatively, maybe I can express cos β in terms of cos α or vice versa.From cos² α = 2 cos² β, I can write cos α = sqrt(2) cos β, but since α and β are acute angles, cos α and cos β are positive, so I can take the positive square root.So, cos α = sqrt(2) cos βLet me substitute this into equation (2):sin β / cos β = 3 sin α / (sqrt(2) cos β)Simplify the right side:3 sin α / (sqrt(2) cos β) = (3 / sqrt(2)) (sin α / cos β)So, equation (2) becomes:sin β / cos β = (3 / sqrt(2)) (sin α / cos β)Multiply both sides by cos β:sin β = (3 / sqrt(2)) sin αSo, sin β = (3 / sqrt(2)) sin αNow, I have two equations:1. cos² α = 2 cos² β2. sin β = (3 / sqrt(2)) sin αLet me try to express everything in terms of sin α or cos α.From equation 2, sin β = (3 / sqrt(2)) sin αI know that sin² β + cos² β = 1, so let's square both sides of equation 2:sin² β = (9 / 2) sin² αSo, cos² β = 1 - (9 / 2) sin² αFrom equation 1, cos² α = 2 cos² β, so:cos² α = 2 [1 - (9 / 2) sin² α]Simplify:cos² α = 2 - 9 sin² αBut I also know that cos² α = 1 - sin² α, so:1 - sin² α = 2 - 9 sin² αBring all terms to one side:1 - sin² α - 2 + 9 sin² α = 0Simplify:-1 + 8 sin² α = 0So, 8 sin² α = 1sin² α = 1/8sin α = 1/(2 sqrt(2)) or sin α = sqrt(2)/4Since α is acute, sin α is positive, so sin α = sqrt(2)/4Now, let's find cos α:cos² α = 1 - sin² α = 1 - (2/16) = 1 - 1/8 = 7/8So, cos α = sqrt(7/8) = (sqrt(14))/4Now, from equation 2, sin β = (3 / sqrt(2)) sin α = (3 / sqrt(2)) * (sqrt(2)/4) = 3/4So, sin β = 3/4Therefore, cos β = sqrt(1 - sin² β) = sqrt(1 - 9/16) = sqrt(7/16) = sqrt(7)/4Now, let's find α and β.α = arcsin(sqrt(2)/4) ≈ 20.7 degreesβ = arcsin(3/4) ≈ 48.6 degreesLet me check if these satisfy the original equations.First equation: 2 sin 2β = 3 sin 2αCalculate sin 2β = 2 sin β cos β = 2*(3/4)*(sqrt(7)/4) = (6 sqrt(7))/16 = (3 sqrt(7))/8So, 2 sin 2β = 2*(3 sqrt(7)/8) = (3 sqrt(7))/4Now, sin 2α = 2 sin α cos α = 2*(sqrt(2)/4)*(sqrt(14)/4) = (2 sqrt(28))/16 = (2*2 sqrt(7))/16 = (4 sqrt(7))/16 = sqrt(7)/4So, 3 sin 2α = 3*(sqrt(7)/4) = (3 sqrt(7))/4Yes, both sides are equal.Second equation: tg β = 3 tg αtg α = sin α / cos α = (sqrt(2)/4) / (sqrt(14)/4) = sqrt(2)/sqrt(14) = 1/sqrt(7)tg β = sin β / cos β = (3/4) / (sqrt(7)/4) = 3/sqrt(7)So, 3 tg α = 3*(1/sqrt(7)) = 3/sqrt(7), which equals tg β.Both equations are satisfied.So, the acute angles are approximately α ≈ 20.7 degrees and β ≈ 48.6 degrees.

💡To determine the number of digits ( y ) in ( s(99) ), we need to consider the number of digits contributed by each perfect square from ( 1^2 ) to ( 99^2 ).1. **Single-digit perfect squares (1 to 9):** - These are ( 1^2 ) to ( 9^2 ), which are all single-digit numbers. - There are 9 such numbers, contributing 9 digits.2. **Two-digit perfect squares (10 to 99):** - These are ( 10^2 ) to ( 99^2 ), which are two-digit numbers. - There are ( 99 - 10 + 1 = 90 ) such numbers. - Each contributes 2 digits, so they contribute ( 2 times 90 = 180 ) digits.3. **Total number of digits:** - Sum of digits from single-digit and two-digit perfect squares: ( 9 + 180 = 189 ).Therefore, the total number of digits ( y ) in ( s(99) ) is 189.

💡First, I need to determine the initial number of pineapples in the store.After selling 48 pineapples, there were some pineapples left. Out of these remaining pineapples, 9 were rotten and thrown away, leaving 29 fresh pineapples.To find out how many pineapples were left after selling, I add the fresh pineapples to the rotten ones: 29 + 9 = 38 pineapples.Finally, to find the initial number of pineapples, I add the pineapples sold to the remaining pineapples: 38 + 48 = 86 pineapples.

💡First, I need to determine the value of ( a ) such that ( f(a) = 2 ) for the given piecewise function.The function ( f(x) ) is defined differently for ( x < 0 ) and ( x > 0 ). I'll consider each case separately.**Case 1: ( a < 0 )**[f(a) = -frac{2}{a} = 2]Solving for ( a ):[-frac{2}{a} = 2 implies -2 = 2a implies a = -1]**Case 2: ( a > 0 )**[f(a) = log_{2}a = 2]Solving for ( a ):[log_{2}a = 2 implies a = 2^{2} = 4]Therefore, the real number ( a ) can be either (-1) or (4).

💡Okay, so I have this problem about an ellipse where the major axis is twice as long as the minor axis, and I need to find the eccentricity. Hmm, let me think about what I know regarding ellipses.First, I remember that an ellipse has two axes: the major axis and the minor axis. The major axis is the longer one, and the minor axis is the shorter one. The lengths of these axes are related to something called the semi-major axis and semi-minor axis, which are half the lengths of the major and minor axes, respectively. So, if the major axis is twice the minor axis, that means the semi-major axis is also twice the semi-minor axis.Let me denote the semi-major axis as 'a' and the semi-minor axis as 'b'. According to the problem, the major axis is twice the minor axis. Since the major axis is 2a and the minor axis is 2b, the relationship can be written as:2a = 2 * (2b)Wait, that seems a bit confusing. Let me clarify. If the major axis is twice the minor axis, then:Major axis = 2 * Minor axisBut the major axis is 2a and the minor axis is 2b, so:2a = 2 * (2b)Simplifying that, we get:2a = 4bDivide both sides by 2:a = 2bOkay, that makes sense. So the semi-major axis 'a' is twice the semi-minor axis 'b'.Now, I need to find the eccentricity of the ellipse. I recall that the eccentricity 'e' of an ellipse measures how "stretched" it is. For an ellipse, the eccentricity is always between 0 and 1. A circle, which is a special case of an ellipse, has an eccentricity of 0.The formula for eccentricity is:e = c / aWhere 'c' is the distance from the center of the ellipse to one of its foci. I also remember that there's a relationship between 'a', 'b', and 'c' in an ellipse, which is:c² = a² - b²So, if I can find 'c', I can then find 'e' by dividing 'c' by 'a'.Given that a = 2b, I can substitute that into the equation for c²:c² = (2b)² - b²c² = 4b² - b²c² = 3b²Taking the square root of both sides to solve for 'c':c = √(3b²)c = b√3Now, I can plug this back into the formula for eccentricity:e = c / ae = (b√3) / (2b)The 'b' terms cancel out:e = √3 / 2So, the eccentricity of the ellipse is √3 divided by 2.Wait, let me double-check my steps to make sure I didn't make a mistake. Starting from the relationship between the major and minor axes, I correctly set a = 2b. Then, using the formula c² = a² - b², I substituted a = 2b and correctly simplified to c = b√3. Plugging that into e = c / a gives √3 / 2, which seems right.I also remember that for an ellipse where a = 2b, the eccentricity should be a common value, and √3 / 2 is approximately 0.866, which is less than 1, so it makes sense for an ellipse.Therefore, I'm confident that the eccentricity is √3 / 2.