answer:- Use the identity ((a+b)^2 = a^2 + 2ab + b^2). - Let ( a = 100 ) and ( b = 5 ), so ( 105 = 100 + 5 ). - Calculate each term separately: - ( 100^2 = 10000 ) - ( 2 times 100 times 5 = 1000 ) - ( 5^2 = 25 ) - Sum these values to find ( 105^2 ): [ 10000 + 1000 + 25 = 11025 ] - Therefore, ( 105^2 = boxed{11025} ).

answer:1. **Identify the total number of points and the position of P:** The square now contains 121 grid points in an 11 x 11 grid. The center point, P, is located at the center grid point, which is at coordinates (6,6) given 1-based indexing. 2. **Determine the number of points excluding P:** Excluding point P, there are 121 - 1 = 120 points. 3. **Identify lines of symmetry through P:** Since P is exactly at the center of the square, lines through P extending symmetrically to the edges of the square will include the vertical center line, horizontal center line, and the two main diagonals. 4. **Count the number of points along each line of symmetry:** Each symmetry line through P intersects: - 5 grid points on each side of P along the horizontal and vertical lines (excluding P). - 5 grid points on each side of P along each diagonal. 5. **Calculate the total number of symmetric points:** For each of the 4 symmetry lines, there are 10 symmetric points per line (5 on each side of P and excluding P itself). Therefore, total symmetric points = 4 lines × 10 points/line = 40 points. 6. **Calculate the probability:** The probability that a randomly chosen point Q (from the 120 points excluding P) is on a line of symmetry through P is: [ text{Probability} = frac{40}{120} = frac{1}{3} ] Thus, the probability that the line PQ is a line of symmetry for the square is frac{1{3}}. The final answer is boxed{textbf{(C)} frac{1}{3}}

answer:From the divisibility condition, we have [p(x)]^3 - x^2 = 0 at x = 2, x = -2, and x = 7. Therefore, we can set: p(2) = 2, p(-2) = -2, p(7) = 3. Assuming p(x) = ax^2 + bx + c, the system of equations becomes: [ begin{align*} 4a + 2b + c &= 2, 4a - 2b + c &= -2, 49a + 7b + c &= 3. end{align*} ] Solving this, from the first two equations, subtract the second from the first to obtain: [ 4b = 4 implies b = 1. ] Adding the first two equations gives us: [ 8a + 2c = 0 implies 4a + c = 0. ] Substitute into the third equation: [ 49a + 7 + c = 3 implies 49a + 7 = 3 - c implies 49a + 7 = 3 - 4a implies 53a = -4 implies a = -frac{4}{53}. ] Then from 4a + c = 0, we find: [ c = -4a = frac{16}{53}. ] The polynomial is: [ p(x) = -frac{4}{53}x^2 + x + frac{16}{53}. ] Evaluating at x = 12: [ p(12) = -frac{4}{53}(12^2) + 12 + frac{16}{53} = -frac{4}{53}(144) + 12 + frac{16}{53} = -frac{576}{53} + 12 = -frac{576}{53} + frac{636}{53} = frac{60}{53}. ] Thus, p(12) = boxed{frac{60}{53}}.

answer:To solve this problem, we must consider the entire set of triangles that can be formed by cutting the square paper along specific points within the paper. Let's break the solution down step by step. 1. **Define the Set M**: - The set M includes the four vertices of the square and the n additional points given inside the square. Thus, M contains n + 4 points. 2. **Criteria for Triangles**: - Each triangle must have vertices that are elements of M. - No point from M should be inside any triangle except for its vertices. 3. **Angles and Properties**: - Each point inside the square affects the triangles that can be formed heading out from this point. - The sum of the interior angles around any given point in the square is 360^circ. For the vertices of the square, the sum of the angles for the triangles emerging from each vertex is 90^circ. 4. **Triangle Angle Sum Considerations**: - Each triangle has an internal angle sum of 180^circ. - Let’s calculate the total internal angle sum of all the triangles formed: We know that: [ n times 360^circ + 4 times 90^circ = 2n times 180^circ + 2 times 180^circ ] Simplifying this, [ n times 360^circ + 360^circ = (2n + 2) times 180^circ ] This relation confirms that the total number of triangles formed is: [ 2n + 2 ] 5. **Cutting Process**: - Each triangle has 3 sides. - The sum of the lengths of all sides of all the triangles combined is (2n + 2) times 3 = 6n + 6. - However, the square's 4 edges do not need cutting and do not contribute to new cuts. 6. **Number of Cuts Needed**: - The total sides from step 5 include the 4 sides of the original square which should be subtracted. - Each cut forms a common edge between pairs of triangles. Thus, the total number of new cuts is: [ frac{6n + 6 - 4}{2} = 3n + 1 ] **Conclusion**: - The number of triangles formed is: [ boxed{2n + 2} ] - The number of cuts required is: [ boxed{3n + 1} ] Through this structured approach, we successfully determined the number of triangles that can be cut from the square and the number of cuts needed.