answer:Given a convex 1000-gon contains 1000 vertices, with an additional 500 points inside, none of which are collinear. We need to find the number of triangles obtained by triangulating this 1000-gon such that all 1500 points are vertices of the triangles, and no other points are involved. **Step-by-Step Solution:** 1. **Sum of Interior Angles of the Polygon:** The sum of the interior angles of a convex n-gon is given by: [ S = (n-2) times 180^circ ] For a 1000-gon: [ S = (1000-2) times 180^circ = 998 times 180^circ ] 2. **Interior Angles at Extra Points:** The additional 500 points will create internal angles that sum up to 360^circ at each point: [ 500 times 360^circ ] 3. **Total Angle Sum for Triangles:** Triangulation divides the polygon into triangles. Each triangle has interior angles summing to 180^circ. If there are T triangles, their combined angle sum should match the total from both steps 1 and 2: [ text{Total Sum} = 998 times 180^circ + 500 times 360^circ ] 4. **Simplify the Calculation:** Simplifying the sum, note that each 360^circ for the extra points can be considered as 2 full triangles' angles: [ text{Sum of angles from triangles} = (998 + 2 times 500) times 180^circ ] 5. **Determine the Number of Triangles:** The given total angle sum directly translates to the number of triangles: [ T = 998 + 2 times 500 ] [ T = 998 + 1000 ] [ T = 1998 ] **Conclusion:** The number of triangles obtained through the triangulation is: [ boxed{1998} ]

answer:According to the problem, we know that the second term of the binomial expansion is binom{5}{1} cdot ax = 10x, thus a = 2. The third term is binom{5}{2} cdot (ax)^2 = bx^2, which means b = 40. Therefore, b = boxed{40}.

answer:To solve this problem, let's analyze the configuration of the cubes after the wooden block is cut into sixteen 1-inch cubes. The block, initially 4 inches long, 4 inches wide, and 1 inch high, is painted red on all six sides. When it is cut into 1-inch cubes, the distribution of red faces among these cubes can be categorized based on their positions within the original block: 1. **Corner Cubes**: There are 4 corner cubes in the block. Each corner cube has 3 faces exposed and painted red. Since they are at the corners, they have 3 sides painted. 2. **Edge Cubes**: Along the edges, excluding the corners, there are 8 cubes. Each of these cubes has 2 faces exposed and painted red. They are positioned along the edges but not at the corners, resulting in 2 sides painted. 3. **Face Cubes**: On each face of the block, excluding the edges, there are 4 central cubes (but since the block is only 1 inch high, these are essentially the same as the edge cubes in this context). These cubes have only 1 face painted red. They are at the center of each face. 4. **Inner Cubes**: Finally, the cubes that are completely inside (if the block were larger) would have no faces painted, but in this case, since the block is only 1 inch high, there are no completely inner cubes. Given the problem's interest in cubes with an even number of red faces, we focus on the corner cubes and the central cubes on the faces. However, since the block is only 1 inch high, all cubes are either on an edge or a corner, and there are no "central" cubes in the traditional sense. The correction to the initial analysis is as follows: - **Corner Cubes**: 4 corner cubes, each with 3 red faces, which was incorrectly identified. The correct observation is that each corner cube has 3 faces exposed and painted red, which is an odd number and not what we're looking for. - **Edge Cubes (excluding corners)**: 8 cubes with 2 red faces, which is an even number and matches our criteria. - **Central Cubes on the Faces**: In the context of this problem, these would be the cubes not on the edge or corner, but given the dimensions, all cubes are either corner or edge cubes. Therefore, the correct analysis for cubes with an even number of red faces includes only the edge cubes (excluding corners), which are 8 in number. The corner cubes, despite the initial incorrect categorization, do not meet the criteria for an even number of red faces. Hence, the number of cubes with an even number of red faces is boxed{8}.

answer:To find the length of the desk, we can use the formula for the perimeter of a rectangle, which is: Perimeter = 2 * (length + width) Given that the perimeter is 46 cm and the width is 9 cm, we can plug these values into the formula: 46 = 2 * (length + 9) Now, we can solve for the length: 46 = 2 * length + 2 * 9 46 = 2 * length + 18 46 - 18 = 2 * length 28 = 2 * length Now, divide both sides by 2 to find the length: 28 / 2 = length 14 = length So, the length of the desk is boxed{14} centimeters.