🧠:<think>Alright, so I've got these three math problems to work through. They all seem related to numbers not divisible by 2 or 3 and their representations in specific forms. Let me try to tackle them one by one.Starting with the first problem: "Prove that any number not divisible by 2 or 3 can be represented in the form (6n + 1) or (6n + 5), where (n) is a natural number or zero." Hmm, okay. I remember that when dealing with divisibility by 2 and 3, it's often useful to consider numbers modulo 6 because 6 is the least common multiple of 2 and 3. So maybe I can use modular arithmetic here.Let me think. If I take any integer and divide it by 6, the possible remainders are 0, 1, 2, 3, 4, or 5. So any integer can be written as (6n + r) where (r) is one of those remainders. Now, if a number is not divisible by 2 or 3, that means it can't have 2 or 3 as factors. So let's see which of these remainders correspond to numbers not divisible by 2 or 3.- If (r = 0), then the number is (6n), which is divisible by both 2 and 3.- If (r = 1), the number is (6n + 1). Let's check: 6n is divisible by 2 and 3, so adding 1 makes it not divisible by 2 or 3.- If (r = 2), the number is (6n + 2), which is divisible by 2.- If (r = 3), the number is (6n + 3), which is divisible by 3.- If (r = 4), the number is (6n + 4), which is divisible by 2.- If (r = 5), the number is (6n + 5). Similar to (6n + 1), since 6n is divisible by 2 and 3, adding 5 (which is one less than 6) makes it not divisible by 2 or 3.So, out of all the possible remainders, only (r = 1) and (r = 5) result in numbers not divisible by 2 or 3. Therefore, any number not divisible by 2 or 3 must be of the form (6n + 1) or (6n + 5). That seems to make sense. I think I've got the first part down.Moving on to the second problem: "Prove that the product of two numbers of the form (6n + 1) or (6n + 5) is a number of the form (6n + 1)." Okay, so I need to multiply two numbers that are either (6n + 1) or (6n + 5) and show that the result is always (6m + 1) for some integer (m).Let me consider different cases:1. Both numbers are of the form (6n + 1): [ (6a + 1)(6b + 1) = 36ab + 6a + 6b + 1 = 6(6ab + a + b) + 1 ] This simplifies to (6m + 1) where (m = 6ab + a + b).2. Both numbers are of the form (6n + 5): [ (6a + 5)(6b + 5) = 36ab + 30a + 30b + 25 = 36ab + 30a + 30b + 24 + 1 = 6(6ab + 5a + 5b + 4) + 1 ] Again, this is (6m + 1) where (m = 6ab + 5a + 5b + 4).3. One number is (6n + 1) and the other is (6n + 5): [ (6a + 1)(6b + 5) = 36ab + 30a + 6b + 5 = 6(6ab + 5a + b) + 5 ] This is (6m + 5), not (6m + 1). Wait, but the problem says the product should be (6n + 1). Did I make a mistake?Hmm, maybe I misread the problem. Let me check again. It says, "the product of two numbers of the form (6n + 1) or (6n + 5) is a number of the form (6n + 1)." But in the third case, I got (6m + 5). That contradicts the statement. Maybe the problem is only considering two numbers of the same form? Or perhaps there's a different approach.Wait, maybe the problem is assuming that both numbers are of the same form, either both (6n + 1) or both (6n + 5). If that's the case, then my first two cases show that the product is indeed (6m + 1). But if the two numbers can be of different forms, then the product can be either (6m + 1) or (6m + 5). Maybe the problem is implicitly assuming that both numbers are of the same form. I'll proceed with that assumption for now.Finally, the third problem: "Prove that the product of a number of the form (6n + 1) and a number of the form (6n + 5) is a number of the form (6n + 5)." Okay, so this is the case I stumbled upon in the second problem. Let me re-examine it.Taking (x = 6a + 1) and (y = 6b + 5):[x times y = (6a + 1)(6b + 5) = 36ab + 30a + 6b + 5 = 6(6ab + 5a + b) + 5]This simplifies to (6m + 5) where (m = 6ab + 5a + b). So indeed, the product is of the form (6n + 5).Wait a minute, so in the second problem, if both numbers are of the same form, the product is (6n + 1), but if they're of different forms, the product is (6n + 5). So maybe the second problem needs to specify that both numbers are of the same form. Otherwise, the statement isn't entirely accurate.But assuming that the second problem is considering two numbers of the same form, then my earlier reasoning holds. So, to summarize:1. Any number not divisible by 2 or 3 can be written as (6n + 1) or (6n + 5).2. The product of two numbers of the same form ((6n + 1) or (6n + 5)) results in a number of the form (6n + 1).3. The product of a number of the form (6n + 1) and a number of the form (6n + 5) results in a number of the form (6n + 5).I think that covers all the cases. I just need to make sure my algebraic manipulations are correct and that I haven't missed any possible remainders or forms. Let me double-check my calculations.For the first problem, I considered all possible remainders modulo 6 and correctly identified which ones correspond to numbers not divisible by 2 or 3. That seems solid.In the second problem, when multiplying two numbers of the same form, I correctly expanded the products and factored out 6 to show the result is of the form (6m + 1). The third case, where the forms are different, correctly results in (6m + 5).For the third problem, I specifically addressed the case where one number is (6n + 1) and the other is (6n + 5), showing the product is (6m + 5). That aligns with the problem statement.Overall, I think I've covered all the necessary steps and verified my reasoning. I just need to present this in a clear, step-by-step manner.</think>

🧠:<think>Okay, so I have two math problems to solve here. Let me take them one at a time. Starting with the first one: Calculate ((π - 2023)^0 - 3tan30° + |1 - sqrt{3}|). Hmm, that looks like a mix of exponents, trigonometry, and absolute values. I need to remember the rules for each of these operations.First, ((π - 2023)^0). I remember that any number raised to the power of 0 is 1, right? So, regardless of what (π - 2023) is, as long as it's not zero, this term should be 1. Since (π) is approximately 3.14 and 2023 is obviously much larger, (π - 2023) is definitely not zero. So, this term simplifies to 1.Next, (3tan30°). I need to recall the value of (tan30°). I think (tan30°) is one of those special angles. Let me visualize the 30-60-90 triangle. The sides are in the ratio 1:√3:2, where 1 is opposite the 30° angle, √3 is opposite the 60° angle, and 2 is the hypotenuse. So, (tan30°) is opposite over adjacent, which is (1/√3). Therefore, (3tan30° = 3 times (1/√3)). Wait, that simplifies to (3/√3). But I think it's better to rationalize the denominator. Multiply numerator and denominator by √3: ((3√3)/(√3 times √3) = (3√3)/3 = √3). So, (3tan30° = √3).Now, the last term is (|1 - sqrt{3}|). Absolute value means the distance from zero, so it's always positive. Since (sqrt{3}) is approximately 1.732, which is greater than 1, (1 - sqrt{3}) is negative. Therefore, the absolute value will make it positive: (sqrt{3} - 1).Putting it all together: ((π - 2023)^0 - 3tan30° + |1 - sqrt{3}|) becomes (1 - √3 + (√3 - 1)). Let me simplify that step by step.First, (1 - √3 + √3 - 1). The (√3) and (-√3) cancel each other out, and the 1 and (-1) also cancel each other. So, everything cancels, and I'm left with 0. That seems too simple, but I think that's correct.Okay, moving on to the second problem: Simplify (left(frac{2x + 1}{x - 1} - 1right) ÷ frac{2x + x^2}{x^2 - 2x + 1}). Hmm, this looks like a complex fraction. I need to simplify both the numerator and the denominator separately before dividing them.Starting with the numerator: (frac{2x + 1}{x - 1} - 1). I can rewrite 1 as (frac{x - 1}{x - 1}) to have a common denominator. So, it becomes (frac{2x + 1}{x - 1} - frac{x - 1}{x - 1}). Now, subtract the numerators: ((2x + 1) - (x - 1)) over the common denominator (x - 1).Simplifying the numerator: (2x + 1 - x + 1 = x + 2). So, the numerator simplifies to (frac{x + 2}{x - 1}).Now, the denominator of the complex fraction is (frac{2x + x^2}{x^2 - 2x + 1}). Let me factor both the numerator and the denominator here. The numerator is (2x + x^2), which can be factored as (x(x + 2)). The denominator is (x^2 - 2x + 1), which is a perfect square: ((x - 1)^2). So, the denominator of the complex fraction becomes (frac{x(x + 2)}{(x - 1)^2}).Now, the entire expression is (frac{x + 2}{x - 1} ÷ frac{x(x + 2)}{(x - 1)^2}). Dividing by a fraction is the same as multiplying by its reciprocal. So, it becomes (frac{x + 2}{x - 1} times frac{(x - 1)^2}{x(x + 2)}).Looking at this, I can cancel out common factors. The (x + 2) in the numerator of the first fraction cancels with the (x + 2) in the denominator of the second fraction. Similarly, one ((x - 1)) in the denominator of the first fraction cancels with one ((x - 1)) in the numerator of the second fraction. What's left is (frac{x - 1}{x}).So, the simplified form is (frac{x - 1}{x}). That seems reasonable. Let me double-check my steps to make sure I didn't make any mistakes.First, in the numerator, I correctly combined the fractions by finding a common denominator and subtracting. Then, I factored both the numerator and denominator of the complex fraction correctly. Finally, I multiplied by the reciprocal and canceled out the common terms. Everything seems to check out.I think I've got both problems solved correctly. The first one simplifies to 0, and the second one simplifies to (frac{x - 1}{x}).</think>

🧠:<think>Okay, so I have this geometry problem here. It says that point P is the center of the circle circumscribed around triangle DEF. That means P is the circumcenter of triangle DEF, right? So, PD, PE, and PF are all radii of the circumscribed circle, which means they are equal in length. That's good to know because it tells me that triangles DPE, EPF, and so on are all isosceles.The problem gives me two angles: angle DPE is 110 degrees, and angle EPF is 150 degrees. I need to find the measure of angle EDF. Hmm, angle EDF is one of the angles of triangle DEF, specifically at vertex D. So, I need to relate the given angles at point P to the angles of triangle DEF.Let me try to visualize this. Since P is the circumcenter, it's equidistant from all three vertices D, E, and F. So, PD = PE = PF. That means triangles DPE, EPF, and FPD are all isosceles. In triangle DPE, sides PD and PE are equal, so the base angles at D and E are equal. Similarly, in triangle EPF, sides PE and PF are equal, so the base angles at E and F are equal.Given angle DPE is 110 degrees. Since triangle DPE is isosceles with PD = PE, the base angles at D and E are equal. Let me call each of these angles x. Then, the sum of the angles in triangle DPE is 110 + x + x = 180. So, 2x = 70, which means x = 35 degrees. Therefore, angles PDE and PED are each 35 degrees.Similarly, angle EPF is 150 degrees. Triangle EPF is isosceles with PE = PF, so the base angles at E and F are equal. Let me call each of these angles y. Then, the sum of the angles in triangle EPF is 150 + y + y = 180. So, 2y = 30, which means y = 15 degrees. Therefore, angles PEF and PFE are each 15 degrees.Now, I need to find angle EDF. Let me think about triangle DEF. The angles at D, E, and F are what I need to find, but specifically, angle EDF is at D. How can I relate this to the angles I just found?I remember that in a triangle, the measure of an angle is half the measure of the central angle that subtends the same arc. That is, the inscribed angle theorem. So, angle EDF is an inscribed angle that subtends arc EF. The central angle subtending arc EF is angle EPF, which is 150 degrees. Therefore, angle EDF should be half of 150 degrees, which is 75 degrees.Wait, hold on. Let me make sure I'm applying the theorem correctly. The inscribed angle theorem says that an inscribed angle is half the measure of the central angle that subtends the same arc. So, if angle EDF subtends arc EF, and the central angle for arc EF is angle EPF, which is 150 degrees, then angle EDF should indeed be 75 degrees.But wait, earlier I found angles PDE and PED as 35 degrees each. How does that fit into this? Let me see. In triangle DEF, the angles at D, E, and F are related to the angles at P. Maybe I can use the fact that the sum of the angles in triangle DEF is 180 degrees.Let me denote angle EDF as A, angle DEF as B, and angle DFE as C. Then, A + B + C = 180 degrees. I need to find A.From the earlier calculations, I have angles PDE = 35 degrees and PEF = 15 degrees. How do these relate to angles B and C?Wait, angle PDE is part of angle EDF. Specifically, angle EDF is composed of angles PDE and PDF. But I don't have angle PDF yet. Alternatively, maybe I should consider the entire triangle DEF and the relationships between the angles at P.Alternatively, perhaps I can use the fact that the sum of the central angles around point P should be 360 degrees. So, angle DPE is 110 degrees, angle EPF is 150 degrees, and the remaining angle, which would be angle FPD, can be found by subtracting the sum of the given angles from 360 degrees.So, angle DPE + angle EPF + angle FPD = 360 degrees. Plugging in the known values: 110 + 150 + angle FPD = 360. So, angle FPD = 360 - 260 = 100 degrees.Now, angle FPD is 100 degrees. Since triangle FPD is isosceles with PF = PD, the base angles at F and D are equal. Let me call each of these angles z. Then, the sum of the angles in triangle FPD is 100 + z + z = 180. So, 2z = 80, which means z = 40 degrees. Therefore, angles PFD and PDF are each 40 degrees.Now, going back to triangle DEF, let's see. Angle EDF is composed of angles PDF and PDE? Wait, no. Let me think. In triangle DEF, angle at D is EDF, which is formed by sides ED and FD. The angles at D in triangles PDE and PDF are PDE and PDF, which are 35 degrees and 40 degrees, respectively.Wait, is angle EDF equal to the sum of angles PDE and PDF? Let me visualize this. Point P is the circumcenter, so it's inside triangle DEF. So, from point D, lines are drawn to P, E, and F. So, angle EDF is the angle between ED and FD. The angles PDE and PDF are angles between PD and ED, and PD and FD, respectively.Therefore, angle EDF is actually the sum of angles PDE and PDF. So, angle EDF = angle PDE + angle PDF = 35 degrees + 40 degrees = 75 degrees.Wait, but earlier I thought angle EDF was half of angle EPF, which was 150 degrees, giving 75 degrees. So, both methods give me the same answer, 75 degrees. That seems consistent.But hold on, in my initial thought process, I considered angle EDF as an inscribed angle subtending arc EF, which is 150 degrees, so half of that is 75 degrees. That matches with the other method where I added angles PDE and PDF.Therefore, angle EDF is 75 degrees.But wait, let me double-check. The central angle for arc EF is 150 degrees, so the inscribed angle EDF should be half of that, which is 75 degrees. That seems correct.Alternatively, looking at triangle DEF, if I can find the other angles, maybe I can confirm. Let's see, angle DEF is at E. How can I find that?Angle DEF is composed of angles PED and PEF. From earlier, angle PED is 35 degrees, and angle PEF is 15 degrees. So, angle DEF = angle PED + angle PEF = 35 + 15 = 50 degrees.Similarly, angle DFE is at F. It is composed of angles PFE and PFD. From earlier, angle PFE is 15 degrees, and angle PFD is 40 degrees. So, angle DFE = angle PFE + angle PFD = 15 + 40 = 55 degrees.Now, let's check if the sum of angles in triangle DEF is 180 degrees. So, angle EDF = 75, angle DEF = 50, angle DFE = 55. Adding them up: 75 + 50 + 55 = 180. Perfect, that adds up.Therefore, angle EDF is 75 degrees.Wait, but in the initial problem statement, the user wrote:"Since PD = PE = PF, triangles DPF, EPD, and DFE are all isosceles.1. Calculate angle PDF and angle PEF:angle PEF = 180 - 150 = 30 degrees, and angle PDF = 180 - 110 = 70 degrees.2. Combine these to find angle EDF:angle EDF = angle PED = angle PDF + angle PEF = 70 + 30 = 100 degrees.Since angle PDE = 100 degrees (exact measure of angle PFE due to the symmetry of the isosceles triangle DFE), the Central Angle Theorem helps conclude:angle EDF = 1/2 angle EPD = 1/2 * 100 = 50 degrees."Wait, that seems conflicting with my conclusion of 75 degrees. So, perhaps I made a mistake somewhere.Let me go through their steps.They say triangles DPF, EPD, and DFE are all isosceles. Hmm, DPF is isosceles because PD = PF, EPD is isosceles because PE = PD, and DFE is isosceles? Wait, DFE is triangle DEF, which may not necessarily be isosceles unless DE = DF.But in our case, unless specified, triangle DEF is just any triangle, so it's not necessarily isosceles. So, maybe that's an incorrect assumption.Then, they calculate angle PEF as 180 - 150 = 30 degrees. Wait, angle EPF is 150 degrees, so in triangle EPF, which is isosceles with PE = PF, the base angles are equal. So, angle PEF = angle PFE = (180 - 150)/2 = 15 degrees each. So, their calculation of angle PEF as 30 degrees is incorrect. It should be 15 degrees.Similarly, they calculate angle PDF as 180 - 110 = 70 degrees. But angle DPE is 110 degrees, so in triangle DPE, which is isosceles with PD = PE, the base angles are equal. So, angle PDE = angle PED = (180 - 110)/2 = 35 degrees each. So, their calculation of angle PDF as 70 degrees is incorrect; it should be 35 degrees.Therefore, their step 1 is wrong because they incorrectly calculated angles PEF and PDF.Then, in step 2, they add angle PDF (70 degrees) and angle PEF (30 degrees) to get angle EDF as 100 degrees. But since their initial angles were wrong, this result is incorrect.Then, they say angle PDE = 100 degrees, which is the exact measure of angle PFE due to symmetry. Wait, angle PFE is 15 degrees, so that doesn't make sense. So, their reasoning here is flawed.Then, they apply the Central Angle Theorem, saying angle EDF = 1/2 angle EPD = 1/2 * 100 = 50 degrees. But angle EPD is 110 degrees, not 100 degrees. So, that's another mistake.So, their entire solution is incorrect because of wrong angle calculations and incorrect assumptions about the triangles being isosceles where they aren't, and misapplying the Central Angle Theorem.Therefore, going back to my own solution, I think 75 degrees is the correct answer. Let me just recapitulate:1. Given P is the circumcenter, so PD = PE = PF.2. Given angle DPE = 110 degrees. In triangle DPE, which is isosceles, the base angles are (180 - 110)/2 = 35 degrees each. So, angle PDE = angle PED = 35 degrees.3. Given angle EPF = 150 degrees. In triangle EPF, which is isosceles, the base angles are (180 - 150)/2 = 15 degrees each. So, angle PEF = angle PFE = 15 degrees.4. The remaining central angle at P is angle FPD. Since the total around point P is 360 degrees, angle FPD = 360 - 110 - 150 = 100 degrees.5. In triangle FPD, which is isosceles with PF = PD, the base angles are (180 - 100)/2 = 40 degrees each. So, angle PFD = angle PDF = 40 degrees.6. Now, in triangle DEF, angle EDF is composed of angles PDE and PDF. So, angle EDF = angle PDE + angle PDF = 35 + 40 = 75 degrees.7. Alternatively, using the inscribed angle theorem, angle EDF subtends arc EF, which has a central angle of 150 degrees. Therefore, angle EDF = 1/2 * 150 = 75 degrees.Both methods confirm that angle EDF is 75 degrees.I think that's solid. The initial incorrect solution made mistakes in calculating the base angles and misapplied theorems, but my step-by-step approach seems correct.</think>

🧠:<think>Alright, so I have this problem: I need to find how many complex numbers ( z ) with ( |z| < 20 ) satisfy the equation ( e^z = frac{z - 2}{z + 2} ). Hmm, okay, let's break this down step by step.First, I remember that for complex numbers, the modulus (or absolute value) of a quotient is the quotient of the moduli. So, ( left| frac{z - 2}{z + 2} right| = frac{|z - 2|}{|z + 2|} ). On the other hand, the modulus of ( e^z ) is ( e^{text{Re}(z)} ) because ( e^z = e^{x + yi} = e^x (cos y + i sin y) ), so its modulus is ( e^x ).So, if ( e^z = frac{z - 2}{z + 2} ), then their moduli must be equal. That gives me ( e^x = frac{|z - 2|}{|z + 2|} ). Interesting. Now, I need to figure out what this tells me about ( z ).Let me denote ( z = x + yi ), where ( x ) and ( y ) are real numbers. Then, ( |z - 2| = sqrt{(x - 2)^2 + y^2} ) and ( |z + 2| = sqrt{(x + 2)^2 + y^2} ). So, the equation becomes:( e^x = frac{sqrt{(x - 2)^2 + y^2}}{sqrt{(x + 2)^2 + y^2}} )Hmm, that looks a bit complicated. Maybe I can square both sides to get rid of the square roots:( e^{2x} = frac{(x - 2)^2 + y^2}{(x + 2)^2 + y^2} )Let me rearrange this:( e^{2x} left( (x + 2)^2 + y^2 right) = (x - 2)^2 + y^2 )Expanding both sides:Left side: ( e^{2x} (x^2 + 4x + 4 + y^2) )Right side: ( x^2 - 4x + 4 + y^2 )So, bringing everything to one side:( e^{2x} (x^2 + 4x + 4 + y^2) - (x^2 - 4x + 4 + y^2) = 0 )Let me factor out ( (x^2 + y^2 + 4) ) from both terms:Wait, actually, let me see:( e^{2x} (x^2 + 4x + 4 + y^2) - (x^2 - 4x + 4 + y^2) = 0 )Let me group the terms:( (e^{2x} - 1)(x^2 + y^2 + 4) + 4x e^{2x} + 4x = 0 )Hmm, that seems a bit messy. Maybe there's another approach.Wait, earlier I thought about the modulus, but perhaps I should also consider the argument of both sides. Since ( e^z ) has an argument equal to ( y ) (the imaginary part of ( z )), and ( frac{z - 2}{z + 2} ) has an argument equal to the difference of the arguments of ( z - 2 ) and ( z + 2 ).So, if ( e^z = frac{z - 2}{z + 2} ), then their arguments must be equal modulo ( 2pi ). So:( y equiv arg(z - 2) - arg(z + 2) mod 2pi )Hmm, that might be useful. Let me think about this.Let me denote ( theta = arg(z - 2) - arg(z + 2) ). Then, ( y = theta + 2pi k ) for some integer ( k ).But I'm not sure how to proceed with that. Maybe I should consider specific cases or look for symmetries.Wait, another thought: if ( z ) is purely imaginary, then ( x = 0 ). Let me check if that's possible.If ( x = 0 ), then ( z = yi ), so ( e^{z} = e^{yi} = cos y + i sin y ). On the other hand, ( frac{z - 2}{z + 2} = frac{yi - 2}{yi + 2} ). Let me compute this:Multiply numerator and denominator by the conjugate of the denominator:( frac{(yi - 2)(-yi + 2)}{(yi + 2)(-yi + 2)} = frac{(-y^2 i^2 + 2yi + 2yi - 4)}{y^2 + 4} = frac{(y^2 - 4) + 4yi}{y^2 + 4} )So, ( frac{z - 2}{z + 2} = frac{y^2 - 4}{y^2 + 4} + i frac{4y}{y^2 + 4} )Therefore, if ( z = yi ), the equation becomes:( cos y + i sin y = frac{y^2 - 4}{y^2 + 4} + i frac{4y}{y^2 + 4} )So, equating real and imaginary parts:( cos y = frac{y^2 - 4}{y^2 + 4} )( sin y = frac{4y}{y^2 + 4} )Hmm, interesting. So, we have two equations:1. ( cos y = frac{y^2 - 4}{y^2 + 4} )2. ( sin y = frac{4y}{y^2 + 4} )Let me square both equations and add them:( cos^2 y + sin^2 y = left( frac{y^2 - 4}{y^2 + 4} right)^2 + left( frac{4y}{y^2 + 4} right)^2 )But ( cos^2 y + sin^2 y = 1 ), so:( 1 = frac{(y^2 - 4)^2 + 16y^2}{(y^2 + 4)^2} )Simplify the numerator:( (y^2 - 4)^2 + 16y^2 = y^4 - 8y^2 + 16 + 16y^2 = y^4 + 8y^2 + 16 = (y^2 + 4)^2 )So, ( 1 = frac{(y^2 + 4)^2}{(y^2 + 4)^2} = 1 ). So, this equation is always satisfied. That means that the two equations for ( cos y ) and ( sin y ) are consistent, but they don't give us new information. So, we need to solve them together.Let me consider the ratio of the two equations:( frac{sin y}{cos y} = frac{frac{4y}{y^2 + 4}}{frac{y^2 - 4}{y^2 + 4}} = frac{4y}{y^2 - 4} )So, ( tan y = frac{4y}{y^2 - 4} )This is a transcendental equation, which likely doesn't have a closed-form solution, but we can analyze it to find the number of solutions.Let me define ( f(y) = tan y - frac{4y}{y^2 - 4} ). We need to find the number of roots of ( f(y) = 0 ) with ( |yi| < 20 ), which translates to ( |y| < 20 ).So, ( y in (-20, 20) ).Let me analyze the behavior of ( f(y) ).First, note that ( tan y ) has vertical asymptotes at ( y = frac{pi}{2} + kpi ), where ( k ) is an integer. Similarly, ( frac{4y}{y^2 - 4} ) has vertical asymptotes at ( y = pm 2 ).So, we need to consider the intervals between these asymptotes and see how many times ( f(y) ) crosses zero in each interval.Let me list the critical points:- Vertical asymptotes of ( tan y ): ( y = frac{pi}{2} + kpi )- Vertical asymptotes of ( frac{4y}{y^2 - 4} ): ( y = pm 2 )So, the intervals to consider are:1. ( (-20, -frac{3pi}{2}) )2. ( (-frac{3pi}{2}, -2) )3. ( (-2, -frac{pi}{2}) )4. ( (-frac{pi}{2}, frac{pi}{2}) )5. ( (frac{pi}{2}, 2) )6. ( (2, frac{3pi}{2}) )7. ( (frac{3pi}{2}, 20) )Wait, but actually, ( frac{3pi}{2} ) is approximately 4.712, which is less than 20, so we have more intervals beyond that.But let me think: since ( tan y ) has a period of ( pi ), and ( frac{4y}{y^2 - 4} ) is an odd function, perhaps the behavior is symmetric in some way.But let's focus on positive ( y ) first and then consider negative ( y ) due to symmetry.So, for ( y > 0 ):- The function ( tan y ) increases from 0 to ( +infty ) as ( y ) approaches ( frac{pi}{2} ) from below, then jumps to ( -infty ) and increases again to 0 at ( y = pi ), and so on.- The function ( frac{4y}{y^2 - 4} ) is positive for ( y > 2 ) and negative for ( 0 < y < 2 ). It has a vertical asymptote at ( y = 2 ).So, let's analyze the intervals:1. ( (0, 2) ): - ( tan y ) is positive and increasing from 0 to ( +infty ). - ( frac{4y}{y^2 - 4} ) is negative because ( y^2 - 4 < 0 ). - So, ( f(y) = tan y - frac{4y}{y^2 - 4} ) is positive minus negative, which is positive. So, ( f(y) > 0 ) in this interval. Therefore, no zeros here.2. ( (2, frac{pi}{2}) ): - ( tan y ) is positive and increasing from ( +infty ) to ( +infty ) (but actually, it approaches ( +infty ) as ( y ) approaches ( frac{pi}{2} )). - ( frac{4y}{y^2 - 4} ) is positive because ( y > 2 ), and as ( y ) increases, this function decreases from ( +infty ) to 0. - So, ( f(y) = tan y - frac{4y}{y^2 - 4} ) starts at ( -infty ) (since ( tan y ) approaches ( +infty ) but ( frac{4y}{y^2 - 4} ) also approaches ( +infty ), but which one grows faster? Let me check the limit as ( y to 2^+ ): ( lim_{y to 2^+} tan y = tan 2 approx -2.185 ) (wait, no, ( tan 2 ) is actually negative because 2 radians is in the second quadrant where tangent is negative. Wait, no, 2 radians is approximately 114.6 degrees, which is in the second quadrant where tangent is negative. So, ( tan y ) approaches ( -infty ) as ( y ) approaches ( frac{pi}{2} ) from below, but wait, ( frac{pi}{2} ) is approximately 1.5708, which is less than 2. So, actually, as ( y ) approaches ( frac{pi}{2} ) from below, ( tan y ) approaches ( +infty ). But since ( y ) is approaching 2 from above, which is beyond ( frac{pi}{2} ), so actually, in the interval ( (2, frac{pi}{2}) ), ( y ) is greater than ( frac{pi}{2} ), but wait, ( frac{pi}{2} ) is approximately 1.5708, which is less than 2. So, actually, ( y ) in ( (2, frac{pi}{2}) ) is not possible because ( frac{pi}{2} < 2 ). Wait, that can't be. So, actually, ( frac{pi}{2} ) is approximately 1.5708, so ( 2 > frac{pi}{2} ). Therefore, the interval ( (2, frac{pi}{2}) ) is empty because ( 2 > frac{pi}{2} ). So, actually, the interval after ( y = 2 ) is ( (2, frac{3pi}{2}) ), but wait, ( frac{3pi}{2} ) is approximately 4.712, which is greater than 2.Wait, I think I made a mistake in the intervals. Let me correct that.Since ( frac{pi}{2} approx 1.5708 ), ( pi approx 3.1416 ), ( frac{3pi}{2} approx 4.712 ), and so on. So, the intervals for ( y > 0 ) are:1. ( (0, frac{pi}{2}) )2. ( (frac{pi}{2}, 2) )3. ( (2, frac{3pi}{2}) )4. ( (frac{3pi}{2}, pi) ) Wait, no, ( pi ) is approximately 3.1416, which is less than ( frac{3pi}{2} approx 4.712 ). So, actually, after ( frac{pi}{2} ), the next asymptote is at ( y = 2 ), then ( y = frac{3pi}{2} ), and so on.Wait, perhaps it's better to list the intervals based on the asymptotes:- Vertical asymptotes of ( tan y ): ( y = frac{pi}{2} + kpi )- Vertical asymptotes of ( frac{4y}{y^2 - 4} ): ( y = pm 2 )So, for ( y > 0 ), the critical points are at ( y = 2 ) and ( y = frac{pi}{2} approx 1.5708 ), ( y = frac{3pi}{2} approx 4.712 ), ( y = frac{5pi}{2} approx 7.854 ), etc.So, the intervals for ( y > 0 ) are:1. ( (0, frac{pi}{2}) )2. ( (frac{pi}{2}, 2) )3. ( (2, frac{3pi}{2}) )4. ( (frac{3pi}{2}, frac{5pi}{2}) )5. ( (frac{5pi}{2}, frac{7pi}{2}) )6. ( (frac{7pi}{2}, 20) )Similarly, for ( y < 0 ), the intervals are symmetric.Now, let's analyze each interval for ( y > 0 ):1. ( (0, frac{pi}{2}) ): - ( tan y ) is positive and increasing from 0 to ( +infty ). - ( frac{4y}{y^2 - 4} ) is negative because ( y^2 - 4 < 0 ). - So, ( f(y) = tan y - frac{4y}{y^2 - 4} ) is positive minus negative, which is positive. So, ( f(y) > 0 ) in this interval. Therefore, no zeros here.2. ( (frac{pi}{2}, 2) ): - ( tan y ) is negative because ( y ) is in the second quadrant (between ( frac{pi}{2} ) and ( pi )), but wait, ( 2 ) is approximately 114.6 degrees, which is still in the second quadrant where tangent is negative. - Wait, no, ( tan y ) is negative in ( (frac{pi}{2}, pi) ), but ( 2 ) is less than ( pi approx 3.1416 ). So, in ( (frac{pi}{2}, 2) ), ( tan y ) is negative. - ( frac{4y}{y^2 - 4} ) is negative because ( y^2 - 4 < 0 ) (since ( y < 2 )). - So, ( f(y) = tan y - frac{4y}{y^2 - 4} ) is negative minus negative. Let's see the behavior: - As ( y ) approaches ( frac{pi}{2} ) from above, ( tan y ) approaches ( -infty ), and ( frac{4y}{y^2 - 4} ) approaches ( frac{4 cdot frac{pi}{2}}{(frac{pi}{2})^2 - 4} approx frac{6.283}{2.467 - 4} approx frac{6.283}{-1.533} approx -4.10 ). So, ( f(y) ) approaches ( -infty - (-4.10) = -infty ). - As ( y ) approaches 2 from below, ( tan y ) approaches ( tan 2 approx -2.185 ), and ( frac{4y}{y^2 - 4} ) approaches ( -infty ) because the denominator approaches 0 from the negative side. So, ( f(y) ) approaches ( -2.185 - (-infty) = +infty ). - Therefore, ( f(y) ) goes from ( -infty ) to ( +infty ) in this interval, crossing zero exactly once. So, one solution in ( (frac{pi}{2}, 2) ).3. ( (2, frac{3pi}{2}) ): - ( tan y ) is positive in ( (2, pi) ) and negative in ( (pi, frac{3pi}{2}) ). - Wait, ( frac{3pi}{2} approx 4.712 ), so ( y ) ranges from 2 to approximately 4.712. - Let's break this interval into two parts: ( (2, pi) ) and ( (pi, frac{3pi}{2}) ). - In ( (2, pi) ): - ( tan y ) is positive because ( y ) is in the first quadrant (wait, no, ( pi approx 3.1416 ), so ( 2 ) is approximately 114.6 degrees, which is in the second quadrant where tangent is negative. Wait, no, ( 2 ) radians is approximately 114.6 degrees, which is in the second quadrant where tangent is negative. So, in ( (2, pi) ), ( tan y ) is negative. - ( frac{4y}{y^2 - 4} ) is positive because ( y > 2 ), so ( y^2 - 4 > 0 ). - So, ( f(y) = tan y - frac{4y}{y^2 - 4} ) is negative minus positive, which is negative. So, ( f(y) < 0 ) in this sub-interval. - In ( (pi, frac{3pi}{2}) ): - ( tan y ) is positive because ( y ) is in the third quadrant where tangent is positive. - ( frac{4y}{y^2 - 4} ) is positive because ( y > 2 ). - So, ( f(y) = tan y - frac{4y}{y^2 - 4} ) is positive minus positive. Let's see the behavior: - As ( y ) approaches ( pi ) from above, ( tan y ) approaches 0 from the positive side, and ( frac{4y}{y^2 - 4} ) approaches ( frac{4pi}{pi^2 - 4} approx frac{12.566}{9.8696 - 4} approx frac{12.566}{5.8696} approx 2.14 ). So, ( f(y) ) approaches ( 0 - 2.14 = -2.14 ). - As ( y ) approaches ( frac{3pi}{2} ) from below, ( tan y ) approaches ( +infty ), and ( frac{4y}{y^2 - 4} ) approaches ( frac{4 cdot frac{3pi}{2}}{(frac{3pi}{2})^2 - 4} approx frac{18.849}{22.207 - 4} approx frac{18.849}{18.207} approx 1.035 ). So, ( f(y) ) approaches ( +infty - 1.035 = +infty ). - Therefore, ( f(y) ) goes from ( -2.14 ) to ( +infty ), crossing zero exactly once in this sub-interval. - So, in total, in ( (2, frac{3pi}{2}) ), there is one solution.4. ( (frac{3pi}{2}, frac{5pi}{2}) ): - ( tan y ) is negative in ( (frac{3pi}{2}, 2pi) ) and positive in ( (2pi, frac{5pi}{2}) ). - Wait, ( frac{5pi}{2} approx 7.854 ), so ( y ) ranges from approximately 4.712 to 7.854. - Let's break this into two parts: ( (frac{3pi}{2}, 2pi) ) and ( (2pi, frac{5pi}{2}) ). - In ( (frac{3pi}{2}, 2pi) ): - ( tan y ) is positive because ( y ) is in the fourth quadrant where tangent is negative? Wait, no, in the fourth quadrant, tangent is negative. Wait, ( frac{3pi}{2} ) is 270 degrees, and ( 2pi ) is 360 degrees. So, in ( (frac{3pi}{2}, 2pi) ), ( y ) is in the fourth quadrant where tangent is negative. - ( frac{4y}{y^2 - 4} ) is positive because ( y > 2 ). - So, ( f(y) = tan y - frac{4y}{y^2 - 4} ) is negative minus positive, which is negative. So, ( f(y) < 0 ) in this sub-interval. - In ( (2pi, frac{5pi}{2}) ): - ( tan y ) is positive because ( y ) is in the first quadrant (wait, ( 2pi ) is 360 degrees, so ( y ) is just above 360 degrees, which is equivalent to 0 degrees, so tangent is 0. As ( y ) increases beyond ( 2pi ), it's in the first quadrant where tangent is positive. - ( frac{4y}{y^2 - 4} ) is positive because ( y > 2 ). - So, ( f(y) = tan y - frac{4y}{y^2 - 4} ) is positive minus positive. Let's see the behavior: - As ( y ) approaches ( 2pi ) from above, ( tan y ) approaches 0 from the positive side, and ( frac{4y}{y^2 - 4} ) approaches ( frac{8pi}{(2pi)^2 - 4} approx frac{25.133}{39.478 - 4} approx frac{25.133}{35.478} approx 0.708 ). So, ( f(y) ) approaches ( 0 - 0.708 = -0.708 ). - As ( y ) approaches ( frac{5pi}{2} ) from below, ( tan y ) approaches ( +infty ), and ( frac{4y}{y^2 - 4} ) approaches ( frac{4 cdot frac{5pi}{2}}{(frac{5pi}{2})^2 - 4} approx frac{31.416}{61.685 - 4} approx frac{31.416}{57.685} approx 0.544 ). So, ( f(y) ) approaches ( +infty - 0.544 = +infty ). - Therefore, ( f(y) ) goes from ( -0.708 ) to ( +infty ), crossing zero exactly once in this sub-interval. - So, in total, in ( (frac{3pi}{2}, frac{5pi}{2}) ), there is one solution.5. ( (frac{5pi}{2}, frac{7pi}{2}) ): - ( frac{5pi}{2} approx 7.854 ), ( frac{7pi}{2} approx 11.0 ). - Let's break this into two parts: ( (frac{5pi}{2}, 3pi) ) and ( (3pi, frac{7pi}{2}) ). - In ( (frac{5pi}{2}, 3pi) ): - ( tan y ) is negative because ( y ) is in the third quadrant where tangent is positive? Wait, ( frac{5pi}{2} ) is 450 degrees, which is equivalent to 90 degrees, so ( y ) is just above 450 degrees, which is equivalent to 90 degrees, so tangent is undefined. Wait, actually, ( frac{5pi}{2} ) is equivalent to ( frac{pi}{2} ), so ( y ) is just above ( frac{pi}{2} ), which is in the second quadrant where tangent is negative. - ( frac{4y}{y^2 - 4} ) is positive because ( y > 2 ). - So, ( f(y) = tan y - frac{4y}{y^2 - 4} ) is negative minus positive, which is negative. So, ( f(y) < 0 ) in this sub-interval. - In ( (3pi, frac{7pi}{2}) ): - ( tan y ) is positive because ( y ) is in the first quadrant (wait, ( 3pi ) is 540 degrees, which is equivalent to 180 degrees, so ( y ) is just above 540 degrees, which is equivalent to 180 degrees, so tangent is 0. As ( y ) increases beyond ( 3pi ), it's in the third quadrant where tangent is positive. - ( frac{4y}{y^2 - 4} ) is positive because ( y > 2 ). - So, ( f(y) = tan y - frac{4y}{y^2 - 4} ) is positive minus positive. Let's see the behavior: - As ( y ) approaches ( 3pi ) from above, ( tan y ) approaches 0 from the positive side, and ( frac{4y}{y^2 - 4} ) approaches ( frac{12pi}{(3pi)^2 - 4} approx frac{37.699}{88.826 - 4} approx frac{37.699}{84.826} approx 0.444 ). So, ( f(y) ) approaches ( 0 - 0.444 = -0.444 ). - As ( y ) approaches ( frac{7pi}{2} ) from below, ( tan y ) approaches ( +infty ), and ( frac{4y}{y^2 - 4} ) approaches ( frac{4 cdot frac{7pi}{2}}{(frac{7pi}{2})^2 - 4} approx frac{43.982}{246.740 - 4} approx frac{43.982}{242.740} approx 0.181 ). So, ( f(y) ) approaches ( +infty - 0.181 = +infty ). - Therefore, ( f(y) ) goes from ( -0.444 ) to ( +infty ), crossing zero exactly once in this sub-interval. - So, in total, in ( (frac{5pi}{2}, frac{7pi}{2}) ), there is one solution.6. ( (frac{7pi}{2}, 20) ): - ( frac{7pi}{2} approx 11.0 ), so ( y ) ranges from approximately 11.0 to 20. - Let's break this into intervals based on ( pi ) periods: - ( (frac{7pi}{2}, 4pi) approx (11.0, 12.566) ) - ( (4pi, frac{9pi}{2}) approx (12.566, 14.137) ) - ( (frac{9pi}{2}, 5pi) approx (14.137, 15.708) ) - ( (5pi, frac{11pi}{2}) approx (15.708, 17.279) ) - ( (frac{11pi}{2}, 6pi) approx (17.279, 18.849) ) - ( (6pi, 20) approx (18.849, 20) ) - However, since ( 20 ) is less than ( 7pi approx 21.991 ), the last interval is ( (6pi, 20) ). - Let's analyze each sub-interval: - In each interval ( (kpi, (k+1)pi) ), where ( k ) is an integer, ( tan y ) alternates between positive and negative. - However, since ( y > 2 ), ( frac{4y}{y^2 - 4} ) is always positive. - So, in each interval where ( tan y ) is positive, ( f(y) = tan y - frac{4y}{y^2 - 4} ) could cross zero, and in intervals where ( tan y ) is negative, ( f(y) ) is negative minus positive, which is negative, so no zeros. - Let's check the first sub-interval ( (frac{7pi}{2}, 4pi) approx (11.0, 12.566) ): - ( tan y ) is positive because ( y ) is in the first quadrant (since ( frac{7pi}{2} = 3pi + frac{pi}{2} ), so ( y ) is just above ( 3pi + frac{pi}{2} ), which is in the fourth quadrant where tangent is negative? Wait, no, ( frac{7pi}{2} ) is equivalent to ( frac{3pi}{2} ), so ( y ) is just above ( frac{3pi}{2} ), which is in the fourth quadrant where tangent is negative. - Wait, I'm getting confused. Let me think differently. Since ( tan y ) has a period of ( pi ), the behavior in each interval ( (kpi, (k+1)pi) ) is similar, just shifted by ( kpi ). - So, in each interval where ( tan y ) is positive, we can have a zero crossing. - Let's consider the interval ( (2npi, (2n+1)pi) ) where ( tan y ) is positive in the first half and negative in the second half, but actually, ( tan y ) is positive in ( (2npi, (2n+1)pi/2) ) and negative in ( ((2n+1)pi/2, (2n+1)pi) ). - However, since ( y ) is large, the function ( frac{4y}{y^2 - 4} ) behaves like ( frac{4}{y} ), which decreases as ( y ) increases. - So, in each interval where ( tan y ) is positive, ( f(y) = tan y - frac{4y}{y^2 - 4} ) will cross zero once because ( tan y ) goes from 0 to ( +infty ) and ( frac{4y}{y^2 - 4} ) is decreasing from a positive value to a smaller positive value. - Similarly, in intervals where ( tan y ) is negative, ( f(y) ) is negative minus positive, which is negative, so no zeros. - So, in each interval ( (2npi, (2n+1)pi) ), there is one solution. - Now, let's count how many such intervals are within ( (frac{7pi}{2}, 20) ): - ( frac{7pi}{2} approx 11.0 ) - ( 20 ) is approximately ( 6.366pi ) (since ( 20 / pi approx 6.366 )) - So, the intervals are: - ( (3pi, frac{7pi}{2}) approx (9.425, 11.0) ) – but this is already covered in the previous interval. - ( (4pi, frac{9pi}{2}) approx (12.566, 14.137) ) - ( (5pi, frac{11pi}{2}) approx (15.708, 17.279) ) - ( (6pi, 20) approx (18.849, 20) ) - Wait, actually, starting from ( frac{7pi}{2} approx 11.0 ), the next interval where ( tan y ) is positive is ( (4pi, frac{9pi}{2}) approx (12.566, 14.137) ), then ( (6pi, frac{13pi}{2}) ), but ( 20 ) is less than ( 7pi approx 21.991 ), so the last interval is ( (6pi, 20) approx (18.849, 20) ). - So, the intervals where ( tan y ) is positive and within ( (frac{7pi}{2}, 20) ) are: - ( (4pi, frac{9pi}{2}) approx (12.566, 14.137) ) - ( (6pi, 20) approx (18.849, 20) ) - Wait, actually, ( (5pi, frac{11pi}{2}) approx (15.708, 17.279) ) is also within ( (frac{7pi}{2}, 20) ), but in this interval, ( tan y ) is positive in ( (5pi, frac{11pi}{2}) ) because ( 5pi approx 15.708 ) and ( frac{11pi}{2} approx 17.279 ), which is in the first half of the interval where ( tan y ) is positive. - Similarly, ( (6pi, 20) approx (18.849, 20) ) is in the first half of the interval ( (6pi, frac{13pi}{2}) approx (18.849, 20.42) ), so ( tan y ) is positive here. - Therefore, in ( (frac{7pi}{2}, 20) ), there are three intervals where ( tan y ) is positive: - ( (4pi, frac{9pi}{2}) ) - ( (5pi, frac{11pi}{2}) ) - ( (6pi, 20) ) - Each of these intervals will have one solution. - So, in total, in ( (frac{7pi}{2}, 20) ), there are three solutions.Now, let's summarize the number of solutions for ( y > 0 ):- ( (frac{pi}{2}, 2) ): 1 solution- ( (2, frac{3pi}{2}) ): 1 solution- ( (frac{3pi}{2}, frac{5pi}{2}) ): 1 solution- ( (frac{5pi}{2}, frac{7pi}{2}) ): 1 solution- ( (frac{7pi}{2}, 20) ): 3 solutionsWait, that adds up to 1 + 1 + 1 + 1 + 3 = 7 solutions for ( y > 0 ). But wait, earlier I thought each interval ( (2npi, (2n+1)pi) ) would have one solution, but in reality, it's more nuanced because of the asymptotes at ( y = 2 ) and ( y = frac{pi}{2} ).Wait, perhaps I overcomplicated it. Let me try a different approach. Since ( f(y) = tan y - frac{4y}{y^2 - 4} ), and considering the periodicity of ( tan y ), which has a period of ( pi ), we can expect that in each interval ( (npi - frac{pi}{2}, npi + frac{pi}{2}) ), there might be a solution.But given the asymptotes at ( y = pm 2 ), it's better to consider the intervals between the asymptotes and count the number of times ( f(y) ) crosses zero.Alternatively, perhaps using the argument principle or considering the function ( e^z - frac{z - 2}{z + 2} ) and counting the number of zeros inside ( |z| < 20 ).But that might be more advanced. Alternatively, since we've already considered that ( z ) must be purely imaginary, and we've reduced the problem to solving ( tan y = frac{4y}{y^2 - 4} ), and we've analyzed the intervals, perhaps the total number of solutions is 8, considering both positive and negative ( y ).Wait, in the positive ( y ) side, I found 7 solutions, but that seems inconsistent. Let me recount:From the earlier analysis:- ( (frac{pi}{2}, 2) ): 1- ( (2, frac{3pi}{2}) ): 1- ( (frac{3pi}{2}, frac{5pi}{2}) ): 1- ( (frac{5pi}{2}, frac{7pi}{2}) ): 1- ( (frac{7pi}{2}, 20) ): 3Wait, that's 7 intervals, but actually, each interval ( (2npi, (2n+1)pi) ) for ( n geq 1 ) will have one solution, and considering the asymptotes, perhaps the number is 8.Wait, perhaps it's better to note that for each ( n ), there's a solution in ( (2npi, (2n+1)pi) ) and another in ( (-(2n+1)pi, -2npi) ), leading to 8 solutions in total within ( |y| < 20 ).But I'm not entirely sure. Maybe I should consider that for each positive interval, there's a corresponding negative interval, so the total number of solutions is double the number of positive solutions.Given that ( |y| < 20 ), and considering the periodicity, the number of solutions is approximately ( frac{20}{pi} times 2 approx 12.73 ), but since we have asymptotes and the function behavior, it's likely 8 solutions.Wait, actually, considering the function ( f(y) = tan y - frac{4y}{y^2 - 4} ), and the fact that it crosses zero once in each interval ( (2npi, (2n+1)pi) ) for ( n ) such that ( 2npi < 20 ), and similarly for negative ( y ).Calculating the number of such intervals:- For ( y > 0 ): - ( n = 0 ): ( (0, pi) ) – but we saw no solution here - ( n = 1 ): ( (2pi, 3pi) approx (6.283, 9.425) ) – 1 solution - ( n = 2 ): ( (4pi, 5pi) approx (12.566, 15.708) ) – 1 solution - ( n = 3 ): ( (6pi, 7pi) approx (18.849, 21.991) ) – but ( 20 < 7pi ), so only up to 20, so 1 solution - So, total for ( y > 0 ): 3 solutions- For ( y < 0 ): Similarly, 3 solutions- But earlier, I found solutions in ( (frac{pi}{2}, 2) ), ( (2, frac{3pi}{2}) ), etc., which are not covered by the ( n ) intervals above. So, perhaps the total is 8 solutions.Alternatively, perhaps the number of solutions is 8 because for each ( n ) from -3 to 3, excluding 0, we have one solution in each interval ( (2npi, (2n+1)pi) ) and ( ((2n-1)pi, 2npi) ), but I'm not sure.Wait, perhaps a better way is to note that the function ( f(y) = tan y - frac{4y}{y^2 - 4} ) has a zero in each interval ( (2npi, (2n+1)pi) ) for integer ( n ), except near the asymptotes at ( y = pm 2 ).Given that ( |y| < 20 ), the number of such intervals is approximately ( frac{20}{pi} approx 6.366 ), so about 6 intervals on each side, leading to 12 solutions, but considering the asymptotes and the behavior near ( y = pm 2 ), it's likely 8 solutions.But I'm not entirely confident. Maybe I should look for a pattern or use a graphing approach.Alternatively, considering that each time ( y ) increases by ( pi ), the function ( tan y ) completes a period, and ( frac{4y}{y^2 - 4} ) decreases, so in each period, there's one solution.Given that ( y ) ranges from ( -20 ) to ( 20 ), the number of periods is ( frac{40}{pi} approx 12.73 ), so about 12 solutions, but considering the asymptotes and the behavior near ( y = pm 2 ), it's likely 8 solutions.Wait, perhaps the correct answer is 8 solutions, considering the symmetry and the number of intervals where the function crosses zero.So, after all this analysis, I think the number of complex numbers ( z ) with ( |z| < 20 ) satisfying ( e^z = frac{z - 2}{z + 2} ) is 8.</think>