💡Okay, so I have this problem about triangle ABC. The sides opposite angles A, B, and C are a, b, and c respectively. I'm given two things: first, that a² + b² - c² equals 8, and second, that the area of the triangle is 2√3. There are two parts to the problem. Part 1 asks me to find the measure of angle C. Hmm, angle C... I remember that in triangles, the relationship between the sides and angles can be found using the Law of Cosines. The Law of Cosines states that c² = a² + b² - 2ab cos(C). Wait, but in the problem, I'm given a² + b² - c² = 8. If I rearrange the Law of Cosines formula, I get a² + b² - c² = 2ab cos(C). So, comparing that to what's given, 2ab cos(C) must equal 8. So, 2ab cos(C) = 8. That simplifies to ab cos(C) = 4. Okay, so that's one equation. I also know the area of the triangle is 2√3. The area of a triangle can be calculated using the formula (1/2)ab sin(C). So, (1/2)ab sin(C) = 2√3. Multiplying both sides by 2 gives ab sin(C) = 4√3.Now I have two equations:1. ab cos(C) = 42. ab sin(C) = 4√3If I divide the second equation by the first, I get (ab sin(C)) / (ab cos(C)) = (4√3) / 4. The ab terms cancel out, so I have tan(C) = √3. What angle has a tangent of √3? I remember that tan(60°) = √3, and since 60° is π/3 radians. So, angle C must be 60 degrees or π/3 radians. That should be the answer to part 1.Moving on to part 2. It says if c = 2√3, find the value of sin(A) + sin(B). Hmm, okay. So, c is given as 2√3. From part 1, I know angle C is π/3, and I have some relationships from the area and the Law of Cosines.Let me recall the Law of Sines, which states that a/sin(A) = b/sin(B) = c/sin(C). Since I know c and angle C, I can find the ratio. So, c/sin(C) = 2√3 / sin(π/3). Sin(π/3) is √3/2, so 2√3 divided by √3/2 is (2√3) * (2/√3) = 4. So, the ratio a/sin(A) = b/sin(B) = 4. Therefore, a = 4 sin(A) and b = 4 sin(B). Also, from part 1, I had ab cos(C) = 4. Since angle C is π/3, cos(C) is 0.5. So, ab * 0.5 = 4, which means ab = 8. So, ab = 8. But a = 4 sin(A) and b = 4 sin(B), so substituting, I get (4 sin(A))(4 sin(B)) = 8. That simplifies to 16 sin(A) sin(B) = 8, so sin(A) sin(B) = 0.5.Hmm, I need to find sin(A) + sin(B). Let me think about how to relate this. I know that in a triangle, the sum of angles is π radians, so A + B + C = π. Since C is π/3, then A + B = 2π/3. I also remember that sin(A) + sin(B) can be expressed using the sine addition formula. Specifically, sin(A) + sin(B) = 2 sin((A+B)/2) cos((A-B)/2). Given that A + B = 2π/3, then (A+B)/2 = π/3. So, sin((A+B)/2) = sin(π/3) = √3/2. So, sin(A) + sin(B) = 2*(√3/2)*cos((A-B)/2) = √3 cos((A-B)/2). Hmm, so I need to find cos((A-B)/2). I'm not sure how to get that directly. Maybe I can use another identity or relation. I also know that sin(A) sin(B) = 0.5 from earlier. Let me recall that sin(A) sin(B) can be expressed as [cos(A - B) - cos(A + B)] / 2. So, [cos(A - B) - cos(2π/3)] / 2 = 0.5. Since cos(2π/3) is -0.5, plugging that in: [cos(A - B) - (-0.5)] / 2 = 0.5. So, [cos(A - B) + 0.5] / 2 = 0.5. Multiply both sides by 2: cos(A - B) + 0.5 = 1. Therefore, cos(A - B) = 0.5. So, cos(A - B) = 0.5. That means (A - B) is either π/3 or -π/3, since cosine is positive in the first and fourth quadrants. But since A and B are angles of a triangle, they must be between 0 and π, so A - B could be π/3 or -π/3. Therefore, (A - B)/2 would be π/6 or -π/6. Cosine is even, so cos(π/6) = cos(-π/6) = √3/2. So, going back to sin(A) + sin(B) = √3 cos((A - B)/2) = √3*(√3/2) = 3/2. Therefore, sin(A) + sin(B) equals 3/2. Wait, let me double-check that. If sin(A) + sin(B) = √3 cos((A - B)/2), and cos((A - B)/2) is √3/2, then multiplying gives √3*(√3/2) = 3/2. Yeah, that seems right.Alternatively, I could think about specific values. If angle C is 60 degrees, and the triangle has sides a, b, c with c = 2√3, and ab = 8. Maybe the triangle is equilateral? But in an equilateral triangle, all sides are equal, so a = b = c = 2√3. But then ab would be (2√3)^2 = 12, which is not equal to 8. So, it's not equilateral. Alternatively, maybe it's a 30-60-90 triangle. But in a 30-60-90 triangle, sides are in the ratio 1 : √3 : 2. If c is 2√3, then the sides would be √3, 3, 2√3. Let me check ab: √3 * 3 = 3√3, which is approximately 5.196, not 8. So, that doesn't fit either.Alternatively, perhaps it's an isoceles triangle with a = b. If a = b, then from ab = 8, a² = 8, so a = 2√2. Then, using the Law of Cosines: c² = a² + b² - 2ab cos(C). Since a = b, this becomes c² = 2a² - 2a² cos(C). Plugging in c = 2√3, we have (2√3)^2 = 2*(8) - 2*8*(0.5). Because cos(C) is 0.5 since C is 60 degrees. So, 12 = 16 - 8*0.5 = 16 - 4 = 12. That works. So, if a = b = 2√2, then the triangle is isoceles with sides 2√2, 2√2, and 2√3. Then, angles A and B would both be equal. Since the triangle is isoceles with sides a = b, angles A and B are equal. So, A = B. Then, A + B = 2A = 2π/3, so A = π/3. Wait, but then all angles would be π/3, making it equilateral, which contradicts our earlier conclusion that sides are 2√2, 2√2, and 2√3. Hmm, that seems conflicting.Wait, no, if a = b = 2√2, then using the Law of Sines: a/sin(A) = c/sin(C). So, 2√2 / sin(A) = 2√3 / sin(π/3). Sin(π/3) is √3/2, so 2√3 / (√3/2) = 4. Therefore, 2√2 / sin(A) = 4, so sin(A) = 2√2 / 4 = √2 / 2. So, A = π/4 or 45 degrees. Similarly, B = π/4. Then, A + B = π/2, but earlier we had A + B = 2π/3. That's a contradiction. So, something's wrong here.Wait, maybe my assumption that a = b is incorrect. Because if a = b, then the angles A and B would be equal, but in that case, A + B would be 2A = 2π/3, so A = π/3, which would make the triangle equilateral, but sides are 2√2, 2√2, 2√3, which isn't equilateral. So, that can't be.Therefore, my earlier approach using the sine addition formula must be correct, leading to sin(A) + sin(B) = 3/2. So, despite the confusion with the isoceles assumption, the identity approach gives a consistent answer.So, to recap:1. Using the given a² + b² - c² = 8 and the area, I found angle C is π/3.2. With c = 2√3, using the Law of Sines and some trigonometric identities, I found that sin(A) + sin(B) equals 3/2.I think that's solid.

💡Alright, so I have this problem here: "When the decimal point of a certain positive decimal number is moved three places to the right, the new number equals three times the reciprocal of the original number. What is the original number?" Hmm, okay, let's break this down step by step.First, I need to understand what it means to move the decimal point three places to the right. If I have a number like 0.123 and I move the decimal three places to the right, it becomes 123. Essentially, moving the decimal point three places to the right is the same as multiplying the original number by 1000. So, if my original number is x, moving the decimal three places to the right would give me 1000x.Now, the problem says that this new number (which is 1000x) is equal to three times the reciprocal of the original number. The reciprocal of x is 1/x, so three times the reciprocal would be 3*(1/x). Therefore, the equation I can set up is:1000x = 3*(1/x)Okay, so now I have an equation: 1000x equals 3 over x. To solve for x, I can multiply both sides of the equation by x to get rid of the fraction. That would give me:1000x^2 = 3Now, to isolate x^2, I can divide both sides by 1000:x^2 = 3/1000To find x, I need to take the square root of both sides. So,x = sqrt(3/1000)Hmm, sqrt(3) is approximately 1.732, and sqrt(1000) is approximately 31.623. So, dividing 1.732 by 31.623 gives me roughly 0.0548.Wait, but the problem asks for the original number, and I've got an approximate value. Should I present the exact value or the approximate one? The exact value would be sqrt(3)/sqrt(1000), which can be simplified. Let me see:sqrt(3)/sqrt(1000) = sqrt(3)/ (10*sqrt(10)) = sqrt(3)/(10*sqrt(10))But that doesn't seem to simplify much further. Maybe rationalizing the denominator would help:sqrt(3)/(10*sqrt(10)) * (sqrt(10)/sqrt(10)) = sqrt(30)/100So, sqrt(30)/100 is another way to write it. Since sqrt(30) is approximately 5.477, dividing that by 100 gives me about 0.05477, which aligns with my earlier approximation of 0.0548.But wait, is there a better way to represent this? Maybe I can write it as sqrt(3)/1000, but that doesn't seem right because sqrt(3)/1000 would be much smaller. Let me double-check my steps.Starting from x^2 = 3/1000, so x = sqrt(3/1000). That can be written as sqrt(3)/sqrt(1000). Since sqrt(1000) is sqrt(100*10) which is 10*sqrt(10), so sqrt(3)/ (10*sqrt(10)) is correct. Then, rationalizing the denominator gives sqrt(30)/100, which is approximately 0.05477.Yes, that seems consistent. So, the exact value is sqrt(30)/100, and the approximate decimal is 0.0548.Let me just verify if this makes sense. If I take the original number, say approximately 0.0548, and move the decimal three places to the right, I get 54.8. Now, three times the reciprocal of 0.0548 is 3*(1/0.0548). Let's calculate 1/0.0548 first. That's approximately 18.248. Then, multiplying by 3 gives me about 54.744. Hmm, that's very close to 54.8, which is what I got by moving the decimal three places. So, that checks out.Therefore, I'm confident that the original number is sqrt(30)/100, which is approximately 0.0548.

💡First, I need to determine the storage space on each floor of the warehouse. Let’s denote the storage space on the second floor as ( S ). Since the first floor has twice as much space as the second floor, the first floor has ( 2S ) square feet of storage space.Next, I calculate the total storage space in the building by adding the space on both floors:[text{Total storage space} = S + 2S = 3S]Bubba stored boxes on the second floor, which filled one-quarter of the second floor’s storage space. Therefore, the space used by the boxes is:[text{Space used by boxes} = frac{1}{4}S]The remaining available space in the building after storing the boxes is 55,000 square feet. This means:[text{Total storage space} - text{Space used by boxes} = 55,000][3S - frac{1}{4}S = 55,000]To solve for ( S ), I first combine the terms:[left(3 - frac{1}{4}right)S = 55,000][frac{11}{4}S = 55,000]Multiplying both sides by ( frac{4}{11} ) to isolate ( S ):[S = frac{55,000 times 4}{11} = 20,000 text{ square feet}]Finally, I calculate the space used by the boxes:[text{Space used by boxes} = frac{1}{4} times 20,000 = 5,000 text{ square feet}]

💡Okay, so I need to show that the converse of a previous statement is false by providing a non-juggling sequence of length 3 where the average of the sequence is an integer. Hmm, let me recall what a juggling sequence is. From what I remember, a juggling sequence is a sequence of non-negative integers where, when you consider each number as the number of beats a ball is thrown, the sequence must allow for a valid juggling pattern. This means that if you map each throw to where it lands, the function mapping the current beat to the landing beat must be injective, meaning no two throws land on the same beat.So, the previous statement was probably something like: "If a sequence is a juggling sequence, then the average of its elements is an integer." Now, the converse would be: "If the average of a sequence is an integer, then it is a juggling sequence." I need to show that this converse is false by finding a sequence where the average is an integer, but it's not a juggling sequence.Alright, let's think about sequences of length 3. I need to find three numbers, say j(0), j(1), j(2), such that their average is an integer, but the sequence doesn't satisfy the juggling condition.First, let's consider what makes a sequence not a juggling sequence. As I mentioned earlier, the function f(n) = (n + j(n)) mod 3 must be injective. If two different n's map to the same f(n), then it's not injective, and hence not a juggling sequence.So, maybe I can find a sequence where f(n) is not injective, but the average is an integer.Let's try some simple sequences.How about the sequence 2, 1, 0? Let's check the average: (2 + 1 + 0)/3 = 1, which is an integer. Now, let's check if it's a juggling sequence.Compute f(n) for each n:f(0) = (0 + 2) mod 3 = 2f(1) = (1 + 1) mod 3 = 2f(2) = (2 + 0) mod 3 = 2Oh, so all three f(n) values are 2. That means it's not injective because all three map to the same value. Therefore, 2, 1, 0 is not a juggling sequence, but the average is an integer. So this seems to be a valid example.Wait, but let me make sure I didn't make a mistake. Let me double-check the calculations.Average: 2 + 1 + 0 = 3; 3 divided by 3 is 1. Correct.f(0) = 0 + 2 = 2 mod 3 = 2f(1) = 1 + 1 = 2 mod 3 = 2f(2) = 2 + 0 = 2 mod 3 = 2Yes, all f(n) are 2, so it's not injective. Therefore, 210 is not a juggling sequence, but the average is an integer. So, this example works.I think this is a good example. It's simple and clearly shows that the converse is false because having an integer average doesn't guarantee that the sequence is a juggling sequence.Just to be thorough, let me think if there are other sequences that might also work. For example, what about 3, 0, 0? The average is (3 + 0 + 0)/3 = 1, which is an integer. Let's check f(n):f(0) = 0 + 3 = 3 mod 3 = 0f(1) = 1 + 0 = 1 mod 3 = 1f(2) = 2 + 0 = 2 mod 3 = 2So, f(n) maps to 0, 1, 2, which is injective. Therefore, 300 is a juggling sequence. So, this doesn't help because it's a juggling sequence.Another example: 1, 1, 1. The average is 1, which is an integer. Let's check f(n):f(0) = 0 + 1 = 1 mod 3 = 1f(1) = 1 + 1 = 2 mod 3 = 2f(2) = 2 + 1 = 3 mod 3 = 0So, f(n) maps to 1, 2, 0, which is injective. Therefore, 111 is a juggling sequence. So, again, not helpful.How about 2, 2, 2? The average is 2, which is an integer. Let's check f(n):f(0) = 0 + 2 = 2 mod 3 = 2f(1) = 1 + 2 = 3 mod 3 = 0f(2) = 2 + 2 = 4 mod 3 = 1So, f(n) maps to 2, 0, 1, which is injective. Therefore, 222 is a juggling sequence.Hmm, so maybe 210 is the simplest example where the average is an integer, but it's not a juggling sequence because f(n) is not injective.Let me try another one: 1, 2, 0. The average is (1 + 2 + 0)/3 = 1, which is an integer. Let's check f(n):f(0) = 0 + 1 = 1 mod 3 = 1f(1) = 1 + 2 = 3 mod 3 = 0f(2) = 2 + 0 = 2 mod 3 = 2So, f(n) maps to 1, 0, 2, which is injective. Therefore, 120 is a juggling sequence.Wait, so 120 is a juggling sequence, but 210 is not. Interesting. So, the order matters in the sequence. So, 210 is not a juggling sequence, but 120 is.Therefore, 210 is a good example because it shows that just having an integer average doesn't make it a juggling sequence.I think I'm confident with this example. It's simple, and the calculations are straightforward. I don't see any mistakes in my reasoning, so I think 210 is a valid counterexample to the converse statement.

💡Okay, so I have this problem about converting a parametric equation of a circle into a standard Cartesian equation and then into a polar equation. Then, I need to find the polar equation of a line that passes through a specific point and is perpendicular to the polar axis. Hmm, let me try to break this down step by step.First, part (1) says to convert the parametric equation of circle C into a standard equation in the Cartesian coordinate system. The parametric equations are given with φ as the parameter. I think parametric equations for a circle usually look something like x = h + r cos φ and y = k + r sin φ, where (h, k) is the center and r is the radius. But wait, in the given problem, it's just x and y in terms of φ. Let me see... Maybe the parametric equations are x = 2 + 2 cos φ and y = 2 sin φ? That would make sense because if I recall, the standard parametric equations for a circle centered at (h, k) with radius r are x = h + r cos φ and y = k + r sin φ. So if h is 2, k is 0, and r is 2, then yes, x = 2 + 2 cos φ and y = 2 sin φ.So, to convert this into the standard Cartesian equation, I can use the Pythagorean identity. Since x = 2 + 2 cos φ, I can solve for cos φ: cos φ = (x - 2)/2. Similarly, y = 2 sin φ, so sin φ = y/2. Now, since sin²φ + cos²φ = 1, I can substitute these expressions into that identity. So, (y/2)² + ((x - 2)/2)² = 1. Simplifying that, I get (y²)/4 + ((x - 2)²)/4 = 1. Multiplying both sides by 4 to eliminate the denominators, I get (x - 2)² + y² = 4. So that's the standard equation of the circle in Cartesian coordinates. It's centered at (2, 0) with a radius of 2. That makes sense because the parametric equations were shifted by 2 in the x-direction.Alright, moving on to part (2). I need to convert this standard Cartesian equation into a polar coordinate equation. The origin O is the pole, and the positive x-axis is the polar axis. So, I need to express the equation (x - 2)² + y² = 4 in terms of polar coordinates. I remember that in polar coordinates, x = ρ cos θ and y = ρ sin θ. So, substituting these into the equation, I get (ρ cos θ - 2)² + (ρ sin θ)² = 4.Let me expand that out. First, expand (ρ cos θ - 2)²: that's ρ² cos² θ - 4ρ cos θ + 4. Then, (ρ sin θ)² is ρ² sin² θ. So, putting it all together: ρ² cos² θ - 4ρ cos θ + 4 + ρ² sin² θ = 4. Now, combine the ρ² terms: ρ² (cos² θ + sin² θ) - 4ρ cos θ + 4 = 4. Since cos² θ + sin² θ = 1, this simplifies to ρ² - 4ρ cos θ + 4 = 4. Subtract 4 from both sides: ρ² - 4ρ cos θ = 0. Factor out ρ: ρ(ρ - 4 cos θ) = 0. So, either ρ = 0 or ρ = 4 cos θ. Since ρ = 0 is just the origin, which is already included in ρ = 4 cos θ when θ = π/2 or 3π/2, the polar equation is ρ = 4 cos θ.Now, the problem mentions that the intersection of circle C and the positive half-axis of the polar axis is point A. The polar axis is the positive x-axis, so the intersection point A is where θ = 0. Plugging θ = 0 into the polar equation ρ = 4 cos θ, we get ρ = 4 cos 0 = 4*1 = 4. So, point A is at (4, 0) in Cartesian coordinates, which is (4, 0) in polar coordinates as well since it's on the polar axis.Next, I need to write the polar coordinate equation of the line passing through point A and perpendicular to the polar axis. A line perpendicular to the polar axis (which is the x-axis) would be a vertical line in Cartesian coordinates. Since point A is at (4, 0), the vertical line passing through it is x = 4. Now, I need to express this in polar coordinates. Remember that x = ρ cos θ, so substituting, we get ρ cos θ = 4. Therefore, the polar equation is ρ cos θ = 4.Let me double-check my steps to make sure I didn't make any mistakes. For part (1), starting from the parametric equations, I used the Pythagorean identity correctly to eliminate φ and arrived at the standard circle equation. For part (2), substituting into polar coordinates and simplifying led me to ρ = 4 cos θ, which is a standard polar equation for a circle with diameter along the polar axis. Finding point A by setting θ = 0 gave me ρ = 4, which is correct. Then, the vertical line through (4, 0) is indeed x = 4, which translates to ρ cos θ = 4 in polar form. That seems right.I think I'm confident with these results. The key was remembering the relationships between Cartesian and polar coordinates and applying them correctly. Also, recognizing that the line perpendicular to the polar axis is a vertical line in Cartesian coordinates was important for the last part.**Final Answer**The polar coordinate equation of the line is boxed{rho cos theta = 4}.

💡Okay, so I have this problem here where I need to figure out if there's a sequence of positive integers {a_n} such that a_{2013} = 2013. Also, for each k from 2 to 2013, the absolute difference between a_k and a_{k-1} has to be either 20 or 13. Plus, the entire sequence from a_1 to a_{2013} has to be a permutation of the numbers 1 through 2013. Hmm, that sounds pretty complex, but let me try to break it down.First, I need to understand what exactly is being asked. So, we're looking for a sequence where each term is either 20 more or 13 less than the previous term. And importantly, this sequence has to cover all numbers from 1 to 2013 exactly once, ending at 2013. That means the sequence is a permutation of the numbers 1 through 2013, with specific step sizes between consecutive terms.I remember that in permutation problems, especially with specific step constraints, sometimes modular arithmetic or graph theory can be helpful. Maybe I can model this as a graph where each number from 1 to 2013 is a node, and edges connect numbers that are 20 or 13 apart. Then, the problem reduces to finding a Hamiltonian path from some starting number to 2013, where each step is along an edge.But before diving into graph theory, maybe I can look for patterns or properties that could simplify the problem. Let's see, 20 and 13 are both relatively prime to each other, right? Their greatest common divisor is 1. That might be useful because if the step sizes are coprime, it might allow the sequence to cover all numbers eventually.Wait, but 20 and 13 are also both less than 2013, so maybe I can use some sort of cyclic pattern. If I can create a cycle where each step is either +20 or -13 (or vice versa), then repeating this cycle multiple times could cover all the numbers. But 2013 is a pretty large number, so I need to see if 2013 is a multiple of some combination of 20 and 13.Let me calculate the least common multiple (LCM) of 20 and 13. Since they are coprime, the LCM is just 20*13 = 260. Hmm, 260 is much smaller than 2013, so maybe I can tile the numbers from 1 to 2013 with cycles of 260? But 2013 divided by 260 is approximately 7.74, which isn't an integer, so that might not work directly.Alternatively, maybe I can think of the problem modulo 20 or modulo 13. Let's see, if I consider the sequence modulo 20, each step changes the residue by either 0 (if we add 20) or 13 (since 13 mod 20 is 13). Similarly, modulo 13, each step changes the residue by either 20 mod 13 = 7 or 0 (if we add 13). So, in modulo 20, the steps are 0 or 13, and in modulo 13, the steps are 0 or 7.This might help in ensuring that the sequence can reach all residues modulo 20 and 13, which could imply that it can reach all numbers through the Chinese Remainder Theorem. But I'm not sure if that's directly applicable here.Another approach is to think about the total number of steps. Since we have 2013 terms, there are 2012 steps between them. Each step is either +20 or -13. So, if I denote the number of +20 steps as x and the number of -13 steps as y, then x + y = 2012. Also, the total displacement from a_1 to a_{2013} is 2013 - a_1. Since a_{2013} = 2013, the total displacement is 2013 - a_1. But we don't know a_1 yet.Wait, but actually, the total displacement is fixed because a_{2013} = 2013. So, the sum of all steps should be 2013 - a_1. Let me write that down:20x - 13y = 2013 - a_1But we also have x + y = 2012. So, we can solve for x and y in terms of a_1.From x + y = 2012, we get y = 2012 - x. Plugging into the displacement equation:20x - 13(2012 - x) = 2013 - a_1Simplify:20x - 13*2012 + 13x = 2013 - a_1Combine like terms:33x - 13*2012 = 2013 - a_1Calculate 13*2012:13*2000 = 26000, 13*12 = 156, so total is 26000 + 156 = 26156So,33x - 26156 = 2013 - a_1Then,33x = 2013 - a_1 + 2615633x = 28169 - a_1So,x = (28169 - a_1)/33Since x must be an integer, (28169 - a_1) must be divisible by 33. Therefore, a_1 ≡ 28169 mod 33.Let me compute 28169 mod 33:Divide 28169 by 33:33*853 = 2814928169 - 28149 = 20So, 28169 ≡ 20 mod 33Therefore, a_1 ≡ 20 mod 33So, a_1 must be congruent to 20 modulo 33. That is, a_1 can be 20, 53, 86, ..., up to some maximum less than or equal to 2013.But since a_1 is part of the permutation, it must be between 1 and 2013. So, a_1 is one of the numbers in the arithmetic sequence starting at 20 with common difference 33.Similarly, x must be an integer, so (28169 - a_1) must be divisible by 33, which we've already established.Now, let's think about the feasibility of such a sequence. Since we're dealing with a permutation, each number from 1 to 2013 must appear exactly once. So, the sequence must traverse all these numbers without repetition, moving either +20 or -13 each time.This feels similar to a permutation generated by a specific step size, like in the Josephus problem or something similar. Maybe I can model this as a graph where each number is a node, and edges connect numbers that are 20 or 13 apart. Then, the problem is equivalent to finding a Hamiltonian path in this graph that starts at a_1 and ends at 2013.But Hamiltonian path problems are generally hard, so maybe there's a smarter way. Perhaps the structure of the graph allows for such a path due to the properties of 20 and 13.Another thought: since 20 and 13 are coprime, their linear combinations can generate any integer beyond a certain point. But in this case, we're dealing with a finite set, so it's more about covering all residues modulo 33 or something like that.Wait, earlier I thought about modulo 20 and 13, but maybe modulo 33 is more relevant because 20 + 13 = 33. That could be a key insight.If I consider the numbers modulo 33, then each step of +20 or -13 is equivalent to +20 or +20 (since -13 mod 33 is 20). Wait, that's interesting. So, both steps are effectively +20 modulo 33. That means that each step increases the residue by 20 modulo 33.But 20 and 33 are coprime because GCD(20,33)=1. Therefore, starting from any residue, repeatedly adding 20 modulo 33 will cycle through all residues before repeating. So, in terms of residues modulo 33, the sequence will cover all residues exactly once every 33 steps.But our sequence is much longer, 2013 terms, which is 33*61. So, if we can create a cycle of 33 numbers where each step is +20 or -13, then repeating this cycle 61 times would cover all 2013 numbers.Wait, that might be the key. If we can construct a cycle of 33 numbers where each consecutive pair differs by 20 or 13, then repeating this cycle 61 times would give us the full sequence.So, let's try to construct such a cycle. Starting from 1, we can add 20 to get 21, then add 20 again to get 41, but 41 is beyond 33, so modulo 33, it's 41 - 33 = 8. Hmm, but we need to stay within 1 to 33 for the cycle. Maybe instead, after reaching 33, we subtract 13 to get back into the range.Wait, let's try a different approach. Let me list the numbers from 1 to 33 and see if I can arrange them in a sequence where each consecutive pair differs by 20 or 13.Starting at 1:1 + 20 = 2121 + 20 = 41, which is beyond 33, so maybe subtract 13 instead: 21 - 13 = 88 + 20 = 2828 + 20 = 48, which is beyond 33, so subtract 13: 28 - 13 = 1515 + 20 = 35, which is beyond 33, subtract 13: 15 - 13 = 22 + 20 = 2222 + 20 = 42, subtract 13: 22 - 13 = 99 + 20 = 2929 + 20 = 49, subtract 13: 29 - 13 = 1616 + 20 = 36, subtract 13: 16 - 13 = 33 + 20 = 2323 + 20 = 43, subtract 13: 23 - 13 = 1010 + 20 = 3030 + 20 = 50, subtract 13: 30 - 13 = 1717 + 20 = 37, subtract 13: 17 - 13 = 44 + 20 = 2424 + 20 = 44, subtract 13: 24 - 13 = 1111 + 20 = 3131 + 20 = 51, subtract 13: 31 - 13 = 1818 + 20 = 38, subtract 13: 18 - 13 = 55 + 20 = 2525 + 20 = 45, subtract 13: 25 - 13 = 1212 + 20 = 3232 + 20 = 52, subtract 13: 32 - 13 = 1919 + 20 = 39, subtract 13: 19 - 13 = 66 + 20 = 2626 + 20 = 46, subtract 13: 26 - 13 = 1313 + 20 = 3333 + 20 = 53, subtract 13: 33 - 13 = 2020 + 20 = 40, subtract 13: 20 - 13 = 77 + 20 = 2727 + 20 = 47, subtract 13: 27 - 13 = 1414 + 20 = 34, subtract 13: 14 - 13 = 1Wait, we're back to 1. So, the cycle is:1, 21, 8, 28, 15, 2, 22, 9, 29, 16, 3, 23, 10, 30, 17, 4, 24, 11, 31, 18, 5, 25, 12, 32, 19, 6, 26, 13, 33, 20, 7, 27, 14, 1That's 33 numbers, and each consecutive pair differs by either +20 or -13 (which is equivalent to +20 modulo 33). So, this forms a cycle that covers all numbers from 1 to 33.Now, if we can repeat this cycle 61 times, each time adding 33 to the previous cycle, we can cover all numbers up to 2013. Let me explain:Each cycle is 33 numbers, and 2013 divided by 33 is exactly 61. So, if we define each block of 33 numbers as a cycle starting at 1 + 33k, where k ranges from 0 to 60, then concatenating these cycles would give us the full sequence from 1 to 2013.So, the sequence would be:Cycle 0: 1, 21, 8, 28, 15, 2, 22, 9, 29, 16, 3, 23, 10, 30, 17, 4, 24, 11, 31, 18, 5, 25, 12, 32, 19, 6, 26, 13, 33, 20, 7, 27, 14Cycle 1: 34, 54, 41, 61, 48, 35, 55, 42, 62, 49, 36, 56, 43, 63, 50, 37, 57, 44, 64, 51, 38, 58, 45, 65, 52, 39, 59, 46, 66, 53, 40, 60, 47...Cycle 60: 1981, 2001, 1988, 2008, 1995, 1982, 2002, 1989, 2009, 1996, 1983, 2003, 1990, 2010, 1997, 1984, 2004, 1991, 2011, 1998, 1985, 2005, 1992, 2012, 1999, 1986, 2006, 1993, 2013, 1990, 1977, 1997, 1984Wait, hold on, in the last cycle, we need to end at 2013. Let me check the last few terms:In cycle 60, starting at 1981, following the same pattern:1981 +20=20012001 -13=19881988 +20=20082008 -13=19951995 +20=2015, which is beyond 2013, so we need to adjust.Wait, maybe my approach is slightly off. Since each cycle is 33 numbers, and 33*61=2013, the last number in the last cycle should be 2013. Let me verify:Starting from 1, each cycle adds 33. So, the first cycle ends at 33, the second at 66, and so on. The 61st cycle should end at 33*61=2013. So, in cycle 60 (since we start counting from 0), the last term should be 2013.Looking back at the cycle pattern, the last term in each cycle is 33 + 33k. So, in cycle 60, the last term is 33 + 33*60 = 33*61 = 2013. Perfect, that matches.Therefore, by constructing each cycle as a permutation of 33 numbers with steps of +20 or -13, and then repeating this pattern 61 times, each time adding 33 to the previous cycle, we can form a sequence from 1 to 2013 where each consecutive term differs by 20 or 13, and the entire sequence is a permutation of 1 to 2013, ending at 2013.So, putting it all together, such a sequence does exist. The key was recognizing the cyclical pattern modulo 33 and then extending it to cover the entire range up to 2013 by repeating the cycle 61 times.