answer:Given the problem: We need to find a sequence of 20 numbers, such that the sum of any three consecutive numbers is positive, and the total sum of all 20 numbers is negative. One possible approach is to look for a periodic sequence. Let's consider a sequence of the form: [ a, a, b, a, a, b, a, a, b, a, a, b, a, a, b, a, a, b, a, a ] This structure repeats every 3 numbers and ensures the sum of any three consecutive numbers can be expressed as (2a + b). For this sequence to satisfy the given conditions, we need to ensure: 1. The sum of any three consecutive numbers is positive: [ 2a + b > 0 ] 2. The total sum of all 20 numbers is negative: [ 14a + 6b < 0 ] Let's solve these inequalities step-by-step: 1. **Condition for the sum of three consecutive numbers**: [ 2a + b > 0 implies b > -2a ] 2. **Condition for the total sum of 20 numbers**: [ 14a + 6b < 0 implies 7a + 3b < 0 implies 3b < -7a implies b < -frac{7}{3}a ] Combining these two inequalities, we get: [ -2a < b < -frac{7}{3}a ] We need to find suitable values of (a) and (b) such that these inequalities hold. Let's take (a = -3) (a negative value to help make the total sum negative) and find (b): For (a = -3): [ -2a = -2(-3) = 6 ] [ -frac{7}{3}a = -frac{7}{3}(-3) = 7 ] Thus, (b) should satisfy: [ 6 < b < 7 ] Choosing a value of (b) in this interval, we can take (b = 6.5). Hence, the sequence can be: [ -3, -3, 6.5, -3, -3, 6.5, -3, -3, 6.5, -3, -3, 6.5, -3, -3, 6.5, -3, -3, 6.5, -3, -3 ] **Verification**: 1. **Sum of any three consecutive numbers**: [ -3 + (-3) + 6.5 = 0.5 > 0 ] 2. **Total sum of all 20 numbers**: [ 14(-3) + 6(6.5) = -42 + 39 = -3 ] [ -3 < 0 ] Both conditions are satisfied. Conclusion: [ boxed{-3, -3, 6.5, -3, -3, 6.5, -3, -3, 6.5, -3, -3, 6.5, -3, -3, 6.5, -3, -3, 6.5, -3, -3} ]

answer:First, let's calculate the total cost before the discount and tax: Cost of basketball cards = 2 packs * 3/pack = 6 Cost of baseball cards = 5 decks * 4/deck = 20 Total cost before discount = 6 + 20 = 26 Now, let's apply the 10% discount: Discount amount = 10% of 26 = 0.10 * 26 = 2.60 Total cost after discount = 26 - 2.60 = 23.40 Next, we apply the 7% sales tax: Sales tax amount = 7% of 23.40 = 0.07 * 23.40 = 1.638 Since we usually round to the nearest cent, the sales tax amount is 1.64. Total cost after tax = 23.40 + 1.64 = 25.04 Finally, let's calculate the change from the 50 bill: Change = 50 - 25.04 = 24.96 Olivia received boxed{24.96} in change.

answer:To solve this problem, we need to utilize properties of circles, spheres, and geometry in three dimensions. Let's break down the solution step-by-step. 1. **Understand the Definition and Given Conditions**: - We have a sphere of radius 1. - Several arcs of great circles are located on this sphere. A great circle is defined as the intersection of the sphere with a plane that passes through its center. - The total length of these arcs is less than (2pi). 2. **Visual Representation**: - Consider the sphere with its center at the origin (O). - Each great circle intersects the sphere along a circumference that is equivalent to a circle on a flat plane with a radius of 1. - The sum of the lengths of these arcs collectively must be less than (2pi), which means the total "arc length" sum is less than the circumference of the sphere. 3. **Geometric Interpretation**: - Let these arcs be represented by the intersections of the sphere with the planes (Pi_i) (where (i) is the index of each plane). - Each intersection (Pi_i) with the sphere forms a great circle on the surface of the sphere. 4. **Total Arc Length Consideration**: - Given the total length of these arcs on the surface of the sphere sums to less than (2pi), it implies the angles sub-tended by these arcs at the center of the sphere collectively have a measure less than (2pi). This follows because the length of an arc of a great circle (= theta) (where (theta) is the angle subtended at the center). 5. **Existence of a Non-Intersecting Plane**: - Imagine collecting all the planes (Pi_i) together. The total angular measure around the sphere summed from these arcs is less than (2pi). - This means these planes only slice through a portion of the sphere's total (4pi) steradians (the surface area measure of the sphere). 6. **Finding the Desired Plane**: - A key insight is that if you imagine choosing a direction orthogonal to one of these planes, then rotating about this direction will span an arc-length angular measure of (2pi), covering the entire circle perpendicular to any small circle arcs. - Because the collective measure of the arcs (think of them as slices) sums to less than (2pi), there must remain some direction through which a plane passing through the center of the sphere does not intersect any of these arcs. 7. **Choose a Perpendicular Plane**: - Among all planes passing through (O), there will always be at least one such plane that does not intersect these arcs because the total span of arcs is insufficient to cover (2pi). 8. **Conclusion**: - Thus, we can always find at least one plane through (O) that avoids all the arcs. Such a plane does not intersect any of the great circle arcs. [ therefore boxed{text{There is a plane passing through the center of the sphere that does not intersect any of the arcs.}} ]

answer:# Problem Solution: Part 1: Given the original price of Xiaohong's purchase at Supermarket A is 400 yuan, we calculate the amount spent as follows: - For the first 300 yuan, a 10% discount is applied, so the cost is 300 times 0.9 = 270 yuan. - For the remaining 100 yuan (400 - 300), a 30% discount is applied, so the cost is 100 times 0.7 = 70 yuan. Therefore, the total amount spent at Supermarket A is 270 + 70 = 340 yuan. Hence, the amount spent by Xiaohong at Supermarket A is boxed{340 text{yuan}}. Part 2: For Supermarket A, the functional relationship of y_{1} with respect to x is determined as follows: - When x leq 300, the discount is 10%, so y_{1} = 0.9x. - When x > 300, the first 300 yuan gets a 10% discount, and the remaining amount gets a 30% discount. Thus, y_{1} = 0.9 times 300 + 0.7(x - 300) = 270 + 0.7x - 210 = 0.7x + 60. Therefore, the functional relationship for Supermarket A is: y_{1} = begin{cases} 0.9x & (0 leq x leq 300) 0.7x + 60 & (x > 300) end{cases} For Supermarket B, the functional relationship of y_{2} with respect to x is determined as follows: - When x leq 100, there is no discount, so y_{2} = x. - When x > 100, only the part exceeding 100 yuan gets a 20% discount, so y_{2} = 100 + 0.8(x - 100) = 0.8x + 20. Therefore, the functional relationship for Supermarket B is: y_{2} = begin{cases} x & (0 leq x leq 100) 0.8x + 20 & (x > 100) end{cases} Part 3: Given the original price of Xiaogang's purchase is 450 yuan, we calculate the amount spent at both supermarkets: - At Supermarket A: y_{1} = 0.7 times 450 + 60 = 315 + 60 = 375 yuan. - At Supermarket B: y_{2} = 0.8 times 450 + 20 = 360 + 20 = 380 yuan. Since 375 < 380, it is more cost-effective for Xiaogang to shop at Supermarket A. Therefore, for an original price of 450 yuan, shopping at Supermarket A is more cost-effective, and the answer is boxed{text{Supermarket A}}.