answer:To demonstrate that at least 2 out of 10 students gave the same number of correct answers, we will prove the statement by contradiction. 1. **Define the Assumption:** Assume that each of the 10 students gave a different number of correct answers. 2. **Number Assignment Range:** If each student gave a different number of correct answers, then the possible number of correct answers each student could have given ranges from 0 to 9. 3. **Minimum and Total Correct Answers Calculation:** Calculate the sum of a sequence of unique integers from 0 to 9 using the formula for the sum of an arithmetic series: [ sum_{k=0}^{9} k = 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 ] 4. **Sum Calculation:** Performing the addition step-by-step: [ 0 + 1 = 1 1 + 2 = 3 3 + 3 = 6 6 + 4 = 10 10 + 5 = 15 15 + 6 = 21 21 + 7 = 28 28 + 8 = 36 36 + 9 = 45 ] Therefore: [ sum_{k=0}^{9} k = 45 ] 5. **Contradiction with Given Condition:** According to the problem, the total number of correct answers given by all students together is 42. If each student gave a different number of correct answers, the sum must be 45. [ text{Sum of } 45 neq text{Given sum } 42 ] 6. **Conclusion:** Since our initial assumption (that each student gave a different number of correct answers) leads to a contradiction with the given condition that there are 42 correct answers in total, it must be that not all students gave a different number of correct answers. Therefore, at least two students must have given the same number of correct answers. [ boxed{} ]

answer:Starting with the initial equation for z, z^2 = frac{sqrt{29}}{2} + frac{7}{2}. Double this to make the square root coefficient rational: [ 2z^2 = sqrt{29} + 7 implies 2z^2 - 7 = sqrt{29}. ] Square both sides to eliminate the square root: [ (2z^2 - 7)^2 = 29 implies 4z^4 - 28z^2 + 49 = 29 implies 4z^4 - 28z^2 + 20 = 0. ] Simplify to: [ z^4 = 7z^2 - 5. ] Now, we divide both sides of the equation z^{100} = 2z^{98} + 14z^{96} + 11z^{94} - z^{50} + dz^{46} + ez^{44} + fz^{40} by z^{40}: [ z^{60} = 2z^{58} + 14z^{56} + 11z^{54} - z^{10} + dz^6 + ez^4 + f. ] Using z^4 = 7z^2 - 5, we continue to substitute where necessary: [ z^6 = z^2 cdot z^4 = z^2 (7z^2 - 5) = 7z^4 - 5z^2 = 7(7z^2 - 5) - 5z^2 = 44z^2 - 35. ] We can express powers of z over 10: [ z^8 = z^2 cdot z^6 = z^2 (44z^2 - 35) = 44z^4 - 35z^2 = 44(7z^2 - 5) - 35z^2 = 273z^2 - 220, ] [ z^{10} = z^2 cdot z^8 = z^2 (273z^2 - 220) = 273z^4 - 220z^2 = 273(7z^2 - 5) - 220z^2 = 1711z^2 - 1365. ] Set z^{10} = dz^6 + ez^4 + f: [ 1711z^2 - 1365 = d(44z^2 - 35) + e(7z^2 - 5) + f. ] This simplifies to: [ 1711z^2 - 1365 = (44d + 7e)z^2 + (-35d - 5e + f). ] We equate the coefficients: [ 44d + 7e = 1711, quad -35d - 5e + f = -1365. ] Solving for d, e yields: [ d = 32, quad e = 173, quad f = 0. ] The sum d + e + f = 32 + 173 + 0 = boxed{205}.

answer:We start with sqrt {a sqrt {a sqrt {a}}}, which can be rewritten as left(a cdot left(a cdot a^{ frac {1}{2}}right)^{ frac {1}{2}}right)^{ frac {1}{2}}, which simplifies to a^{ frac {7}{8}}. Therefore, the correct choice is: boxed{text{C}} This problem tests the conversion between fractional exponents and radicals, as well as the properties of exponentiation, and is considered a basic question.

answer:To find the minimum positive difference between two composite numbers that sum to 96, we aim to choose two numbers as close as possible to 96 div 2 = 48. 1. Start by checking numbers around 48. 2. The numbers 48 and 48 themselves are both composite (48 = 2^4 times 3), and their difference is 48 - 48 = 0. 3. However, the smallest non-zero positive difference is desired. Let’s examine the next closest set of numbers, 47 and 49. 47 is prime, so it does not work. Consider 46 and 50. Both are composite (46 = 2 times 23 and 50 = 2 times 5^2). 4. The difference between 46 and 50 is 50 - 46 = 4. Therefore, the minimum positive difference between two composite numbers that sum to 96 is boxed{4}.