answer:We need to find all real-valued functions ( f ) on the positive reals which satisfy the functional equation: [ f(x + y) = f(x^2 + y^2) ] for all positive ( x ) and ( y ). 1. **Introduce Variables**: Let ( x + y = k ) and ( x^2 + y^2 = h ). 2. **Relate ( xy ) to ( k ) and ( h )**: Since ( x ) and ( y ) are the roots of the quadratic polynomial: [ z^2 - kz + xy = 0, ] by the Vieta's formulas, we know: [ x + y = k quad text{and} quad xy = c, ] where ( c ) is some constant to be determined. 3. **Express ( xy ) in terms of ( k ) and ( h )**: By squaring ( x + y = k ): [ (x + y)^2 = x^2 + 2xy + y^2. ] Therefore, [ k^2 = x^2 + 2xy + y^2 = h + 2xy. ] Solving for ( xy ), we get: [ xy = frac{k^2 - h}{2}. ] 4. **Determine Conditions for Positive Roots ( x ) and ( y )**: For ( x ) and ( y ) to be real and positive: - The discriminant of the quadratic must be non-negative: [ k^2 - 4 cdot left(frac{k^2 - h}{2}right) geq 0. ] Simplifying this: [ k^2 - 2(k^2 - h) geq 0 Rightarrow k^2 - 2k^2 + 2h geq 0 Rightarrow 2h - k^2 geq 0 Rightarrow 2h geq k^2 Rightarrow h geq frac{k^2}{2}. ] - Both ( x ) and ( y ) must be positive: [ h < k^2. ] Hence, if ( k > 0 ) and: [ frac{k^2}{2} leq h < k^2, ] we have: [ f(k) = f(h). ] 5. **Intervals and Overlapping Coverage**: Consider intervals ( A_n ) such that ( k = (1.2)^n ): [ A_n = left[ frac{(1.2)^{2n}}{2}, (1.2)^{2n} right). ] These intervals clearly overlap and cover the positive reals: [ ldots, A_{-2}, A_{-1}, A_0, A_1, A_2, ldots. ] 6. **Conclusion**: Since the intervals ( A_n ) overlap and cover all positive reals, and within each interval ( A_n ), the function ( f ) must be constant, it follows that ( f ) is constant on the entire range of positive reals. Thus, the only real-valued functions that satisfy the given functional equation are constant functions. [boxed{f(x) = c text{ for some constant } c text{ and all positive } x.}]

answer:To solve this problem, we aim to find two prime numbers ( p ) and ( q ) such that ( p + q = 140 ), ( p neq q ), ( p < q ), and ( p < 50 ). We need to maximize ( q - p ). 1. **Identify the condition**: We need ( p ) and ( q ) to be prime numbers, ( p < q ), and ( p < 50 ). 2. **Check feasible primes under 50**: - **Subtract the smallest primes less than 50 from 140 and check if the result is prime**: - ( 140 - 3 = 137 ) (prime) - ( 140 - 7 = 133 ) (not prime) - ( 140 - 11 = 129 ) (not prime) - ( 140 - 17 = 123 ) (not prime) - ( 140 - 19 = 121 ) (not prime) - ( 140 - 23 = 117 ) (not prime) - ( 140 - 29 = 111 ) (not prime) - ( 140 - 31 = 109 ) (prime) - ( 140 - 37 = 103 ) (prime) - ( 140 - 41 = 99 ) (not prime) - ( 140 - 43 = 97 ) (prime) - ( 140 - 47 = 93 ) (not prime) 3. **Calculate the differences**: - For pairs ( (3, 137) ), ( (31, 109) ), ( (37, 103) ), and ( (43, 97) ), the differences are ( 134 ), ( 78 ), ( 66 ), and ( 54 ) respectively. 4. **Largest difference**: The largest difference under these conditions is ( 137 - 3 = 134 ). Thus, the largest possible difference between two prime numbers that sum to 140, with one of the primes less than 50, is ( 134 ). The final answer is boxed{D}

answer:To find the value of the question mark, we need to solve for it in the equation: √? / 19 = 4 First, we can isolate the square root by multiplying both sides of the equation by 19: √? = 4 * 19 Now, calculate the right side: √? = 76 Next, we need to square both sides to remove the square root: (√?)^2 = 76^2 ? = 76 * 76 Now, calculate the square of 76: ? = 5776 So the value of the question mark is boxed{5776} .

answer:1. **Calculate the current average score**: Ben's current test scores are 95, 85, 75, 65, and 90. The average of these scores is: [ text{Average} = frac{95 + 85 + 75 + 65 + 90}{5} = frac{410}{5} = 82 ] 2. **Determine the desired average score**: Ben wants to raise his average by 4 points. Therefore, the target average score after his next test is: [ 82 + 4 = 86 ] 3. **Calculate the total score required to achieve the desired average**: After the next test, Ben will have taken 6 tests. To achieve an average of 86, the total score for all 6 tests must be: [ 86 times 6 = 516 ] 4. **Determine the score needed on the next test**: The sum of his current scores is 410. To find out the score needed on his next test to reach a total of 516, we calculate: [ text{Required score} = 516 - 410 = 106 ] Thus, the minimum score Ben needs on his next test to achieve his goal is 106. The final answer is boxed{textbf{(C)} 106}