answer：Okay, so I have this problem about arithmetic and geometric progressions. Let me try to understand it step by step. First, the problem says that {a_n} is an arithmetic progression. That means each term increases by a constant difference. So, if I denote the first term as a₁ and the common difference as d, then the nth term is a₁ + (n-1)d. Got that.Then, {b_n} is a geometric progression with a common ratio of 2. So, each term is twice the previous one. If the first term is b₁, then the nth term is b₁ * 2^(n-1). That makes sense.Now, the problem gives me some equalities: a₂ - b₂ = a₃ - b₃ = b₄ - a₄. Hmm, okay. So, the difference between the second term of the arithmetic progression and the second term of the geometric progression is equal to the difference between the third terms, and that's also equal to the difference between the fourth term of the geometric progression and the fourth term of the arithmetic progression.Part (1) asks me to prove that a₁ = b₁. So, the first terms of both sequences are equal. Interesting. Let me try to write down the expressions for these differences.First, let's write down the terms:For {a_n}:- a₂ = a₁ + d- a₃ = a₁ + 2d- a₄ = a₁ + 3dFor {b_n}:- b₂ = b₁ * 2^(2-1) = 2b₁- b₃ = b₁ * 2^(3-1) = 4b₁- b₄ = b₁ * 2^(4-1) = 8b₁Now, let's compute the differences given:1. a₂ - b₂ = (a₁ + d) - 2b₁2. a₃ - b₃ = (a₁ + 2d) - 4b₁3. b₄ - a₄ = 8b₁ - (a₁ + 3d)According to the problem, these three expressions are equal. So, I can set them equal to each other:(a₁ + d) - 2b₁ = (a₁ + 2d) - 4b₁and(a₁ + 2d) - 4b₁ = 8b₁ - (a₁ + 3d)Let me solve the first equation first:(a₁ + d) - 2b₁ = (a₁ + 2d) - 4b₁Simplify both sides:Left side: a₁ + d - 2b₁Right side: a₁ + 2d - 4b₁Subtract left side from both sides:0 = (a₁ + 2d - 4b₁) - (a₁ + d - 2b₁)0 = a₁ + 2d - 4b₁ - a₁ - d + 2b₁0 = d - 2b₁So, d = 2b₁. Okay, so the common difference of the arithmetic progression is twice the first term of the geometric progression.Now, let's use the second equality:(a₁ + 2d) - 4b₁ = 8b₁ - (a₁ + 3d)Substitute d = 2b₁ into this equation:Left side: (a₁ + 2*(2b₁)) - 4b₁ = (a₁ + 4b₁) - 4b₁ = a₁Right side: 8b₁ - (a₁ + 3*(2b₁)) = 8b₁ - (a₁ + 6b₁) = 8b₁ - a₁ - 6b₁ = (8b₁ - 6b₁) - a₁ = 2b₁ - a₁So, we have:a₁ = 2b₁ - a₁Bring a₁ to the left side:a₁ + a₁ = 2b₁2a₁ = 2b₁Divide both sides by 2:a₁ = b₁So, that proves part (1). The first terms of both sequences are equal.Now, moving on to part (2). It asks for the number of elements in the set {k | b_k = a_m + a₁, 1 ≤ m ≤ 500}. So, we need to find all integers k such that the k-th term of the geometric progression equals the sum of the m-th term of the arithmetic progression and the first term a₁, where m ranges from 1 to 500.Let me write down the expressions for b_k and a_m:b_k = b₁ * 2^(k-1)a_m = a₁ + (m - 1)dBut from part (1), we know that a₁ = b₁ and d = 2b₁. So, let's substitute these into the expressions.First, since a₁ = b₁, let's denote a₁ = b₁ = c for simplicity. Then, d = 2c.So, the expressions become:b_k = c * 2^(k-1)a_m = c + (m - 1)*2c = c + 2c(m - 1) = c(1 + 2(m - 1)) = c(2m - 1)Now, the equation we need is:b_k = a_m + a₁c * 2^(k-1) = c(2m - 1) + cSimplify the right side:c(2m - 1) + c = c(2m - 1 + 1) = c(2m)So, the equation becomes:c * 2^(k-1) = c * 2mSince c is non-zero (as it's the first term of a geometric progression with ratio 2, which can't be zero), we can divide both sides by c:2^(k-1) = 2mDivide both sides by 2:2^(k-2) = mSo, m = 2^(k-2)But m must be an integer between 1 and 500, inclusive. So, we need to find all integers k such that m = 2^(k-2) is an integer between 1 and 500.Let's find the possible values of k:Since m = 2^(k-2) must be ≥ 1, 2^(k-2) ≥ 1 ⇒ k - 2 ≥ 0 ⇒ k ≥ 2.Also, m = 2^(k-2) ≤ 500 ⇒ 2^(k-2) ≤ 500.Let's find the maximum k such that 2^(k-2) ≤ 500.Compute powers of 2:2^0 = 12^1 = 22^2 = 42^3 = 82^4 = 162^5 = 322^6 = 642^7 = 1282^8 = 2562^9 = 512Wait, 2^9 = 512, which is greater than 500. So, the maximum exponent where 2^(k-2) ≤ 500 is when k-2 = 8 ⇒ k = 10.So, k can be from 2 to 10 inclusive.Let me list the possible k and corresponding m:k=2: m=2^(2-2)=2^0=1k=3: m=2^(3-2)=2^1=2k=4: m=2^(4-2)=2^2=4k=5: m=2^(5-2)=2^3=8k=6: m=2^(6-2)=2^4=16k=7: m=2^(7-2)=2^5=32k=8: m=2^(8-2)=2^6=64k=9: m=2^(9-2)=2^7=128k=10: m=2^(10-2)=2^8=256Wait, but m=256 is still ≤500, but for k=11, m=2^(11-2)=512, which is >500, so k=11 is invalid.So, k can be 2,3,4,5,6,7,8,9,10. That's 9 values.Wait, but let me check if m=256 is indeed ≤500. Yes, 256 is less than 500. So, k=10 is valid.But wait, when k=10, m=256, which is within the range. So, the number of elements is from k=2 to k=10, which is 9 elements.Wait, but let me double-check. For k=2, m=1; k=3, m=2; k=4, m=4; k=5, m=8; k=6, m=16; k=7, m=32; k=8, m=64; k=9, m=128; k=10, m=256.Yes, that's 9 values. So, the set {k | b_k = a_m + a₁, 1 ≤ m ≤ 500} has 9 elements.Wait, but let me think again. Is there any possibility that for some k, m could be beyond 500? For example, if k=11, m=512, which is beyond 500, so it's excluded. So, k can't be 11. So, the maximum k is 10.Therefore, the number of elements is 9.I think that's it. So, the answer for part (2) is 9.

answer：Okay, so I need to find the sum of all even positive divisors of 140. Hmm, let me think about how to approach this. I remember that to find divisors, it's helpful to start with the prime factorization of the number. So, first, I should factorize 140 into its prime components.Alright, 140. Let's see, 140 divided by 2 is 70. Then, 70 divided by 2 is 35. Okay, so I have two 2s so far. Now, 35 divided by 5 is 7, and 7 is a prime number. So, putting it all together, the prime factors of 140 are 2 squared, 5, and 7. So, 140 = 2² × 5¹ × 7¹.Now, I need to find all the even positive divisors. Wait, what makes a divisor even? Well, a number is even if it's divisible by 2. So, any divisor that includes at least one factor of 2 will be even. That means to find the even divisors, I can consider the prime factorization but ensure that the exponent of 2 is at least 1.So, the prime factorization is 2² × 5¹ × 7¹. For the even divisors, the exponent of 2 can be 1 or 2 (since it can't be 0, as that would make it odd). The exponents for 5 and 7 can be 0 or 1, as usual.To find the sum of the divisors, I can use the formula for the sum of divisors, which is the product of the sums of each prime power. But since we're only considering even divisors, we'll adjust the exponents accordingly.So, for the prime 2, the exponent can be 1 or 2. That gives us (2¹ + 2²). For the prime 5, the exponent can be 0 or 1, so (1 + 5). Similarly, for the prime 7, the exponent can be 0 or 1, so (1 + 7).Wait, but I think I need to factor out the 2 first. Let me think again. If I factor out a 2 from each even divisor, then the remaining part will be a divisor of 140 divided by 2, which is 70. So, the sum of the even divisors of 140 is equal to 2 times the sum of all divisors of 70.Okay, that makes sense. So, first, let's find the sum of all divisors of 70. The prime factorization of 70 is 2 × 5 × 7, so using the sum of divisors formula, it's (1 + 2) × (1 + 5) × (1 + 7). Calculating that: 3 × 6 × 8. Let me compute that step by step.3 × 6 is 18, and 18 × 8 is 144. So, the sum of all divisors of 70 is 144. Therefore, the sum of all even divisors of 140 is 2 × 144, which is 288.Wait, let me double-check that. If I list out all the even divisors of 140, they should be 2, 4, 10, 14, 20, 28, 70, and 140. Let me add those up: 2 + 4 is 6, plus 10 is 16, plus 14 is 30, plus 20 is 50, plus 28 is 78, plus 70 is 148, plus 140 is 288. Yep, that matches. So, the sum is indeed 288.I think that's it. I started by factorizing 140, realized that even divisors must include at least one 2, so I adjusted the exponents accordingly. Then, I used the sum of divisors formula on the adjusted number, which was 70, and multiplied by 2 to get the final sum. Listing out the divisors and adding them up confirmed the result. So, I'm confident that the sum of all even positive divisors of 140 is 288.

answer：Okay, so I have this problem here about a line intersecting the y-axis and a direct proportion function. Let me try to understand what it's asking step by step.First, the line given is y = x + 3. I know that the y-intercept of a line is where it crosses the y-axis, which happens when x = 0. So, for part ①, I need to find the coordinates of point A where the line intersects the y-axis. That should be straightforward. I just plug in x = 0 into the equation y = x + 3. Let me do that:y = 0 + 3 = 3So, point A is at (0, 3). That seems simple enough.Moving on to part ②, it says the line intersects the graph of the direct proportion function y = kx at point B (-1, m). I need to find the value of m. Since point B is on both the line y = x + 3 and the direct proportion function y = kx, I can use the coordinates of B to find m.The coordinates of B are (-1, m). So, plugging x = -1 into the line equation y = x + 3:y = (-1) + 3 = 2So, m should be 2. That gives me point B as (-1, 2). Okay, that makes sense.Now, part ③ asks for the equation of the direct proportion function. A direct proportion function is of the form y = kx, where k is the constant of proportionality. Since point B (-1, 2) lies on this function, I can use its coordinates to find k.Plugging x = -1 and y = 2 into y = kx:2 = k*(-1)Solving for k:k = 2 / (-1) = -2So, the equation of the direct proportion function is y = -2x. Got it.Finally, part ④ asks for the area of triangle AOB, where O is the origin. Points A, O, and B form a triangle. To find the area, I can use the coordinates of these points.Point O is (0, 0), point A is (0, 3), and point B is (-1, 2). I can plot these points to visualize the triangle. Point O is the origin, point A is straight up on the y-axis, and point B is in the second quadrant.The area of a triangle can be found using the formula:Area = (1/2) * base * heightIn this case, I can consider OA as the base. OA is along the y-axis from (0, 0) to (0, 3), so its length is 3 units. The height would be the horizontal distance from the y-axis to point B, which is the absolute value of the x-coordinate of B, so | -1 | = 1 unit.Wait, but actually, since point B is at (-1, 2), the base could also be considered as the distance from O to B, but I think using OA as the base and the horizontal distance as the height is simpler.Alternatively, I can use the coordinates to calculate the area using the shoelace formula. Let me try that to verify.The shoelace formula for three points (x1, y1), (x2, y2), (x3, y3) is:Area = (1/2) |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|Plugging in the points O(0, 0), A(0, 3), and B(-1, 2):Area = (1/2) |0*(3 - 2) + 0*(2 - 0) + (-1)*(0 - 3)|Simplify:= (1/2) |0 + 0 + (-1)*(-3)|= (1/2) |0 + 0 + 3|= (1/2) * 3 = 1.5Wait, that's different from my initial thought. Hmm, maybe I made a mistake in choosing the base and height.Let me think again. If I consider OA as the base, which is 3 units, then the height would be the perpendicular distance from point B to the y-axis, which is indeed 1 unit (the absolute value of the x-coordinate of B). So, area should be (1/2)*3*1 = 1.5.But earlier, using the shoelace formula, I also got 1.5. So, why did I initially think it was 3? Maybe I confused the base and height.Alternatively, perhaps I should consider the vectors or the determinant method.Another way is to note that the area can be calculated using the determinant formula for the area of a triangle given by three points:Area = (1/2) | (x_A * y_B - x_B * y_A) |Since point O is the origin, the formula simplifies to that.Plugging in the coordinates:x_A = 0, y_A = 3x_B = -1, y_B = 2Area = (1/2) | (0*2 - (-1)*3) | = (1/2) |0 + 3| = (1/2)*3 = 1.5So, it's consistent. Therefore, the area is 1.5, which is 3/2.Wait, but in the initial problem, the user's solution had the area as 3. Maybe I need to double-check.Alternatively, perhaps the base is OA (3 units) and the height is the x-coordinate of B, which is 1 unit. So, area is (1/2)*3*1 = 1.5.Alternatively, if I consider the base as OB, which is the distance from O to B, which is sqrt((-1)^2 + 2^2) = sqrt(1 + 4) = sqrt(5). Then, the height would be the distance from A to the line OB. That seems more complicated.Alternatively, maybe I should use the formula for the area of a triangle with coordinates.Yes, the shoelace formula is reliable. So, according to that, the area is 1.5.But in the initial problem, the user's solution had the area as 3. Maybe I made a mistake.Wait, let's recast the points:O(0,0), A(0,3), B(-1,2)Plotting these, O is at the origin, A is straight up on the y-axis, and B is to the left of the y-axis at (-1,2). So, the triangle is formed by these three points.Using the shoelace formula:List the coordinates in order: O(0,0), A(0,3), B(-1,2), and back to O(0,0).Compute the sum of x_i y_{i+1}:(0*3) + (0*2) + (-1*0) = 0 + 0 + 0 = 0Compute the sum of y_i x_{i+1}:(0*0) + (3*(-1)) + (2*0) = 0 + (-3) + 0 = -3Take the absolute difference: |0 - (-3)| = |3| = 3Area = (1/2)*3 = 1.5Wait, so that's consistent with my earlier calculation. So, the area is 1.5, which is 3/2.But in the initial problem, the user's solution had the area as 3. Maybe there was a miscalculation there.Alternatively, perhaps the user considered the base as OA (3 units) and the height as the y-coordinate of B, which is 2 units, giving area (1/2)*3*2 = 3. But that would be incorrect because the height should be the perpendicular distance from B to OA, which is the y-axis. The perpendicular distance from B(-1,2) to the y-axis is indeed 1 unit, not 2.Therefore, the correct area should be 1.5, not 3.Wait, but let me think again. If I consider vectors OA and OB, the area is half the magnitude of the cross product.Vector OA is (0,3), vector OB is (-1,2).Cross product in 2D is scalar: OA_x * OB_y - OA_y * OB_x = 0*2 - 3*(-1) = 0 + 3 = 3Area is (1/2)*|3| = 1.5Yes, so that confirms it. The area is 1.5.Therefore, the initial solution might have had an error in calculating the area.So, to summarize:① Point A is at (0,3)② m = 2③ The direct proportion function is y = -2x④ The area of triangle AOB is 1.5, which is 3/2.But in the initial problem, the user's solution had the area as 3. Maybe they considered the base as OA (3) and the height as the y-coordinate of B (2), which is incorrect because the height should be the horizontal distance from B to OA, which is 1.Alternatively, perhaps they used a different method.But according to the correct methods, the area should be 1.5.So, I think the correct answers are:① (0,3)② 2③ y = -2x④ 1.5 or 3/2But since the problem might expect an integer, maybe it's 3. But according to calculations, it's 1.5.Wait, perhaps I made a mistake in the shoelace formula.Let me try again:Coordinates: O(0,0), A(0,3), B(-1,2)Shoelace formula:Multiply and sum diagonally down to the right:(0*3) + (0*2) + (-1*0) = 0 + 0 + 0 = 0Multiply and sum diagonally up to the right:(0*0) + (3*(-1)) + (2*0) = 0 + (-3) + 0 = -3Subtract the two sums: 0 - (-3) = 3Take absolute value and multiply by 1/2: (1/2)*3 = 1.5Yes, that's correct.Alternatively, using the formula for the area of a triangle with vertices at (x1,y1), (x2,y2), (x3,y3):Area = |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))/2|Plugging in:x1=0, y1=0; x2=0, y2=3; x3=-1, y3=2Area = |0*(3 - 2) + 0*(2 - 0) + (-1)*(0 - 3)| / 2= |0 + 0 + 3| / 2 = 3/2 = 1.5Yes, so the area is indeed 1.5.Therefore, the initial solution had an error in calculating the area as 3 instead of 1.5.So, my final answers are:① (0,3)② 2③ y = -2x④ 1.5 or 3/2But since the problem might expect an integer, maybe it's better to write it as 3/2.Alternatively, perhaps the user intended the area to be 3, but according to calculations, it's 1.5.I think I should go with 1.5 as the correct area.

answer：Alright, so I need to find the product of the numerator and the denominator when the repeating decimal 0.overline{012} is expressed as a fraction in its simplest form. Hmm, okay, let's break this down step by step.First, I remember that repeating decimals can be converted into fractions by using some algebra. The general method involves setting the decimal equal to a variable, multiplying both sides by a power of 10 to shift the decimal point so that the repeating part lines up, and then subtracting the original equation to eliminate the repeating part. Let me try that here.Let me denote x as the repeating decimal:x = 0.overline{012}So, x = 0.012012012... and so on. Now, I need to figure out what power of 10 to multiply x by so that the decimal parts align. Since the repeating sequence "012" has three digits, I should multiply by 10^3, which is 1000.Multiplying both sides by 1000:1000x = 12.overline{012}So now, 1000x = 12.012012012... Now, if I subtract the original x from this equation, the repeating parts should cancel out:1000x - x = 12.overline{012} - 0.overline{012}This simplifies to:999x = 12So, solving for x:x = 12 / 999Okay, so now I have the fraction 12/999. But I need to make sure this is in its simplest form. To do that, I need to find the greatest common divisor (GCD) of 12 and 999 and divide both numerator and denominator by it.Let's find the GCD of 12 and 999. First, I can list the factors of 12:1, 2, 3, 4, 6, 12And the factors of 999:1, 3, 9, 27, 37, 111, 333, 999Looking for the common factors, I see that both 12 and 999 are divisible by 3. Let me check if 3 is the greatest common divisor.Dividing 12 by 3 gives 4, and dividing 999 by 3 gives 333. So, the simplified fraction is 4/333.Now, the problem asks for the product of the numerator and the denominator in this reduced fraction. The numerator is 4, and the denominator is 333. So, multiplying them together:4 * 333 = ?Let me calculate that:4 * 300 = 12004 * 33 = 132Adding them together: 1200 + 132 = 1332Therefore, the product of the numerator and the denominator is 1332.Wait, let me double-check my steps to make sure I didn't make any mistakes.1. I set x = 0.overline{012}.2. Multiplied both sides by 1000 to get 1000x = 12.overline{012}.3. Subtracted x from 1000x to get 999x = 12.4. Solved for x to get 12/999.5. Simplified the fraction by dividing numerator and denominator by their GCD, which is 3, resulting in 4/333.6. Calculated the product of 4 and 333 to get 1332.Everything seems to check out. I don't think I made any calculation errors. The key steps were recognizing the repeating decimal, setting up the equation correctly, and simplifying the fraction properly.Just to be thorough, let me verify that 4/333 is indeed equal to 0.overline{012}.Dividing 4 by 333:333 goes into 4 zero times. Add a decimal point and a zero, making it 40. 333 goes into 40 zero times. Add another zero, making it 400. 333 goes into 400 once (333), subtracting gives 67. Bring down another zero, making it 670. 333 goes into 670 twice (666), subtracting gives 4. Bring down another zero, making it 40. Now we're back to where we started.So, the decimal repeats as 0.012012..., which is 0.overline{012}. That confirms that 4/333 is correct.Therefore, the product of the numerator and denominator is indeed 1332.

answer：Okay, so I have this inequality to solve: (frac{x^2 - 16}{x + 4} < 0). Hmm, let me think about how to approach this. I remember that when dealing with inequalities involving rational expressions, I need to consider the critical points where the expression can change its sign, which are the zeros of the numerator and the undefined points of the denominator.First, I should factor the numerator to see if I can simplify the expression. The numerator is (x^2 - 16), which looks like a difference of squares. I recall that (a^2 - b^2 = (a - b)(a + b)), so applying that here, I get:[x^2 - 16 = (x - 4)(x + 4)]So now, the inequality becomes:[frac{(x - 4)(x + 4)}{x + 4} < 0]Wait a minute, I notice that ((x + 4)) is present in both the numerator and the denominator. Can I cancel them out? Well, yes, but I have to remember that (x + 4) cannot be zero because division by zero is undefined. So, (x + 4 neq 0) implies (x neq -4). I'll keep that in mind.After canceling the ((x + 4)) terms, the inequality simplifies to:[x - 4 < 0]Which simplifies further to:[x < 4]But hold on, I need to consider the restriction I noted earlier, which is (x neq -4). So, the solution (x < 4) includes all real numbers less than 4, but I have to exclude (x = -4) because the original expression is undefined there.Therefore, the solution set is all real numbers less than 4, excluding -4. In interval notation, this is written as:[(-infty, -4) cup (-4, 4)]Let me double-check to make sure I didn't make any mistakes. I factored the numerator correctly, canceled out the common term, and considered the restriction on (x). Testing a value in each interval should confirm the solution.For example, let's test (x = -5), which is in ((-infty, -4)):[frac{(-5)^2 - 16}{-5 + 4} = frac{25 - 16}{-1} = frac{9}{-1} = -9 < 0]So, that works.Next, test (x = 0), which is in ((-4, 4)):[frac{0^2 - 16}{0 + 4} = frac{-16}{4} = -4 < 0]That also works.Finally, test (x = 5), which is greater than 4:[frac{5^2 - 16}{5 + 4} = frac{25 - 16}{9} = frac{9}{9} = 1 > 0]Which doesn't satisfy the inequality, as expected.I think I've covered all the necessary steps and checked my work. The solution seems solid.

answer：Okay, so I have this problem about a technology company with two teams, Team A and Team B, working on solving a technical challenge. The company set up a reward system where the team that successfully solves the problem gets a reward. The probabilities given are that Team A has a 2/3 chance of solving it, and Team B has a 3/4 chance. Part (1) asks for the probability distribution of ξ, which is the number of teams rewarded at the end, and also the expected value Eξ. Alright, so first, I need to figure out all the possible values ξ can take. Since there are two teams, ξ can be 0, 1, or 2. That makes sense because either none, one, or both teams could solve the problem.To find the probability distribution, I need to calculate P(ξ=0), P(ξ=1), and P(ξ=2). Starting with P(ξ=0), which is the probability that neither team solves the problem. Since the events are independent, I can multiply the probabilities of each team not solving it. So, for Team A, the probability of not solving is 1 - 2/3 = 1/3. For Team B, it's 1 - 3/4 = 1/4. Multiplying these together: (1/3) * (1/4) = 1/12. So, P(ξ=0) = 1/12.Next, P(ξ=1) is the probability that exactly one team solves the problem. This can happen in two ways: either Team A solves it and Team B doesn't, or Team B solves it and Team A doesn't. Calculating each scenario:- Team A solves, Team B doesn't: (2/3) * (1/4) = 2/12 = 1/6.- Team B solves, Team A doesn't: (1/3) * (3/4) = 3/12 = 1/4.Adding these together: 1/6 + 1/4. To add these, I need a common denominator, which is 12. So, 1/6 is 2/12 and 1/4 is 3/12. Adding them gives 5/12. So, P(ξ=1) = 5/12.Finally, P(ξ=2) is the probability that both teams solve the problem. Since the events are independent, multiply the probabilities: (2/3) * (3/4) = 6/12 = 1/2. So, P(ξ=2) = 1/2.Let me just double-check these probabilities to make sure they sum up to 1: 1/12 + 5/12 + 6/12 = 12/12 = 1. Perfect, that checks out.Now, for the expected value Eξ. The expected value is calculated as the sum of each outcome multiplied by its probability. So:Eξ = 0 * P(ξ=0) + 1 * P(ξ=1) + 2 * P(ξ=2)= 0 * (1/12) + 1 * (5/12) + 2 * (1/2)= 0 + 5/12 + 1= 5/12 + 12/12= 17/12So, Eξ is 17/12.Moving on to part (2). Let η be the square of the difference between the number of rewarded teams and the number of unrewarded teams. So, first, I need to figure out what η can be.Since there are two teams, the number of rewarded teams can be 0, 1, or 2. Therefore, the number of unrewarded teams would be 2, 1, or 0 respectively. The difference between the number of rewarded and unrewarded teams would be:- If ξ=0: 0 - 2 = -2. The square of that is (-2)^2 = 4.- If ξ=1: 1 - 1 = 0. The square is 0^2 = 0.- If ξ=2: 2 - 0 = 2. The square is 2^2 = 4.So, η can be either 0 or 4. Now, event C is defined as "the function f(x) = |η - 7/2|^x decreases monotonically in its domain." I need to find the probability of event C.First, let's understand what it means for f(x) to decrease monotonically. A function f(x) is monotonically decreasing if, as x increases, f(x) decreases. For exponential functions like |η - 7/2|^x, the behavior depends on the base of the exponent.The general form is b^x. If b > 1, then b^x is increasing. If 0 < b < 1, then b^x is decreasing. If b = 1, it's constant. If b < 0, it's not defined for all real x, but since η is either 0 or 4, let's compute |η - 7/2|.Compute |η - 7/2| for η=0 and η=4:- If η=0: |0 - 7/2| = 7/2 = 3.5- If η=4: |4 - 7/2| = |4 - 3.5| = 0.5So, the base of the exponent is either 3.5 or 0.5.Now, for f(x) = (3.5)^x, since 3.5 > 1, this function is increasing, not decreasing. For f(x) = (0.5)^x, since 0.5 < 1, this function is decreasing.Therefore, event C occurs only when η=4 because that's when the base is 0.5, making the function monotonically decreasing.So, I need to find P(η=4). From earlier, η=4 occurs when ξ=0 or ξ=2. Wait, hold on. Earlier, I saw that η=4 when ξ=0 or ξ=2. But looking back, when ξ=0, η=4, and when ξ=2, η=4 as well. So, η=4 corresponds to both ξ=0 and ξ=2.But wait, no. Let me clarify:Wait, η is the square of the difference between the number of rewarded teams and the number of unrewarded teams. So, if ξ=0, then the difference is 0 - 2 = -2, square is 4. If ξ=1, difference is 1 - 1 = 0, square is 0. If ξ=2, difference is 2 - 0 = 2, square is 4.So, η=4 occurs when ξ=0 or ξ=2. Therefore, P(η=4) = P(ξ=0) + P(ξ=2) = 1/12 + 1/2 = 1/12 + 6/12 = 7/12.Therefore, the probability of event C is 7/12.Wait, but let me double-check. So, event C is when f(x) decreases monotonically, which happens only when η=4, as we saw. So, P(C) = P(η=4) = 7/12.Yes, that seems correct.So, summarizing:(1) The probability distribution of ξ is:- P(ξ=0) = 1/12- P(ξ=1) = 5/12- P(ξ=2) = 1/2And the expected value Eξ = 17/12.(2) The probability of event C is 7/12.I think that's all. I don't see any mistakes in my reasoning, but let me just go through it once more.For part (1), calculating the probabilities for each ξ value by considering the independent events of each team solving or not solving. Then, summing up the probabilities to ensure they add up to 1, which they do.For part (2), understanding that η is based on the difference squared, leading to η being 0 or 4. Then, analyzing the function f(x) to see when it's decreasing, which only happens when η=4. Calculating the probability of η=4 by adding the probabilities of ξ=0 and ξ=2, which gives 7/12.Yep, that all makes sense. I think I'm confident with these answers.