answer:1. Define the structure of the number pyramid: each cell above the bottom row contains the sum of the numbers in the two cells immediately below it. 2. Let each number in the second row of the number pyramid be denoted by ( n ): [ n, quad n, quad n ] 3. Move upward to the third row. Each cell on the third row contains the sum of the two cells below it (from the second row): [ 2n, quad 2n ] Here, each ((2n)) in the third row comes from the sum (n + n). 4. Finally, compute the top number of the pyramid. The top number is the sum of the two cells immediately below it (from the third row): [ 4n ] Here, the (4n) on the top row comes from the sum (2n + 2n). 5. By examining the top number (4n), observe that it is always a multiple of 4 regardless of the integer value of (n). Therefore, statement (D) must be true. 6. Additionally, because we found a specific structure under which the top number is a multiple of 4 without requiring the bottom row to contain zeros, we can say that: - Option (A) (The bottom row contains at least one zero) does not need to be true as it is not a requirement. - Option (B) (The third row contains at least one zero) does not need to be true for the same reason. - Option (C) (The top number is a multiple of three) does not necessarily need to be true. For example, if (n = 1), the top number (4n = 4) is not a multiple of 3. Hence, option (E) (None of the above) is also incorrect. # Conclusion: [ boxed{text{D}} ]

answer:We start with the given equation: [ N^2 + N = 12 ] To formulate a standard quadratic equation, we rearrange it: [ N^2 + N - 12 = 0 ] Instead of factoring, let's use the quadratic formula to solve it: [ N = frac{-b pm sqrt{b^2 - 4ac}}{2a} ] where ( a = 1, b = 1, ) and ( c = -12 ). Calculate the discriminant: [ b^2 - 4ac = 1^2 - 4 times 1 times (-12) = 1 + 48 = 49 ] Since the discriminant is a perfect square, we find the roots easily by using the quadratic formula: [ N = frac{-1 pm sqrt{49}}{2 times 1} = frac{-1 pm 7}{2} ] [ N = frac{-1 + 7}{2} text{ or } frac{-1 - 7}{2} ] [ N = 3 text{ or } -4 ] Since we seek a negative integer: [ N = boxed{-4} ] Conclusion: Everything aligns, with no contradiction in initial conditions or the resulting value, validating that the negative integer solution is indeed ( N = -4 ).

answer:To analyze the given statements step by step, we follow the logic provided in the standard solution closely: A: If m bot l and n bot l, this implies that both m and n are perpendicular to the same line l. This situation places m and n in the same plane, where they are both perpendicular to l. However, being perpendicular to the same line does not necessarily mean m is parallel to n. They could intersect at a point on line l. Therefore, this statement is incorrect. B: If alpha bot gamma and beta bot gamma, this means that both planes alpha and beta are perpendicular to the same plane gamma. In three-dimensional space, two planes being perpendicular to the same plane can either intersect each other or be parallel to each other. Therefore, it's not guaranteed that alpha is perpendicular to beta. This statement is also incorrect. C: If m is parallel to alpha and n is parallel to alpha, this implies that both lines m and n are parallel to the same plane alpha. In this scenario, m and n could be parallel to each other, intersect within plane alpha, or even lie in different planes parallel to alpha. Thus, this statement does not guarantee that m is parallel to n. This statement is incorrect as well. D: If alpha is parallel to gamma and beta is parallel to gamma, we use the property of parallel planes. In geometry, if two planes are each parallel to a third plane, then they must be parallel to each other. This is because there is no way for them to intersect or be perpendicular to each other while both maintaining parallelism to the same third plane. This statement correctly applies the properties of parallel planes. Therefore, the correct answer, following the logic and properties of geometry, is boxed{D}.

answer:1. **Assign coordinates to points**: Let ( Y ) be at the origin ((0,0)). Then, ( X ) is at ((-1,0)) and ( Z ) is at ((3,0)). Point ( W ) is at ((a, b)) such that ( YW = 5 ). 2. **Calculate coordinates of ( W )**: Since ( YW = 5 ), we have: [ sqrt{a^2 + b^2} = 5 implies a^2 + b^2 = 25 ] 3. **Determine the circumcenters ( O_1 ) and ( O_2 )**: - The circumcenter ( O_1 ) of (triangle WXY) is the intersection of the perpendicular bisectors of ( WX ) and ( XY ). - The circumcenter ( O_2 ) of (triangle WYZ) is the intersection of the perpendicular bisectors of ( WY ) and ( YZ ). 4. **Find the perpendicular bisectors**: - The perpendicular bisector of ( WX ) passes through the midpoint of ( WX ) and is perpendicular to ( WX ). - The perpendicular bisector of ( XY ) passes through the midpoint of ( XY ) and is perpendicular to ( XY ). 5. **Calculate the coordinates of ( O_1 )**: - The midpoint of ( WX ) is (left(frac{a-1}{2}, frac{b}{2}right)). - The slope of ( WX ) is (frac{b}{a+1}), so the slope of the perpendicular bisector is (-frac{a+1}{b}). - The equation of the perpendicular bisector of ( WX ) is: [ y - frac{b}{2} = -frac{a+1}{b} left(x - frac{a-1}{2}right) ] - The midpoint of ( XY ) is (left(-frac{1}{2}, 0right)). - The slope of ( XY ) is (0), so the perpendicular bisector is a vertical line ( x = -frac{1}{2} ). 6. **Calculate the coordinates of ( O_2 )**: - The midpoint of ( WY ) is (left(frac{a}{2}, frac{b}{2}right)). - The slope of ( WY ) is (frac{b}{a}), so the slope of the perpendicular bisector is (-frac{a}{b}). - The equation of the perpendicular bisector of ( WY ) is: [ y - frac{b}{2} = -frac{a}{b} left(x - frac{a}{2}right) ] - The midpoint of ( YZ ) is (left(frac{3}{2}, 0right)). - The slope of ( YZ ) is (0), so the perpendicular bisector is a vertical line ( x = frac{3}{2} ). 7. **Find the intersection points**: - Solve the equations of the perpendicular bisectors to find the coordinates of ( O_1 ) and ( O_2 ). 8. **Calculate the distance ( O_1O_2 )**: - Use the distance formula to find the distance between ( O_1 ) and ( O_2 ): [ O_1O_2 = sqrt{left(frac{3}{2} - left(-frac{1}{2}right)right)^2 + left(frac{frac{5}{2} - frac{3cos{theta}}{2}}{sin{theta}} - frac{frac{5}{2} + frac{cos{theta}}{2}}{sin{theta}}right)^2} ] - Simplify the expression to find the minimum distance. 9. **Conclusion**: - After simplification, we find that the minimum possible length of segment ( overline{O_1O_2} ) is ( 2 ). The final answer is ( boxed{2} ).