answer:Solution: 1. Let the number of people in the company be ( n ), and suppose each person sent ( k ) letters. Since each letter is received by all members except the sender, each letter was sent to ( n - 1 ) people. 2. The total number of letters received by all members of the company is given as 440. 3. The total number of letters sent by one person is ( k ). Thus, the total number of letters received by the rest of the ( n - 1 ) people from one person is ( k(n-1) ). 4. Since there are ( n ) people each sending ( k ) letters, the total number of letters received is: [ k(n-1) times n = 440 ] Simplifying this gives us: [ kn(n-1) = 440 ] 5. The number 440 factors into prime multiples as follows: [ 440 = 2^3 times 5 times 11 ] 6. Next, we need to find ( n ) such that ( n(n-1) ) divides 440. Furthermore, ( n ) and ( n-1 ) must be two consecutive factors of 440. Given: [ n(n-1) < 440 ] 7. Performing a trial of possible values satisfying the constraints: - Try for ( n = 11 ): [ n(n-1) = 11 times 10 = 110 quad (Not , 440) ] - Try for ( n = 22 ): [ n(n-1) = 22 times 21 = 462 quad (Not , 440) ] 8. Now trying decomposition for each value within ( 2^3 times 5 times 11 ): - Try ( n times (n-1) ) directly: - For small values considered, other combinations, no viable ( n ) where both ( n(n-1) < 22 ): Thus, feasible options understanding breakdown calculated gives us ( n = 11 times 2, 5 times 2 ) feasibly ( n = 1,2 20. # Conclusion: The probable values for ( n ) given factors confirming and scenarios depict ( n = 2 , 5, text { or } 11 and we consolidate it through boxed as: [ boxed{2, 5 text { or } 11} ]

answer:To solve for the range of the real number m where the function f(x)={x}^{2}-frac{1}{2}x+m has a "value-preserving" interval, we analyze the behavior of the function and apply the given conditions. First, we note that the function is monotonically decreasing on the interval (-∞,frac{1}{4}] and monotonically increasing on [frac{1}{4},+∞). This is determined by analyzing the derivative of f(x), which changes sign at x=frac{1}{4}. # Case 1: [a,b]⊆[frac{1}{4},+∞) - For f(x) to have a "value-preserving" interval, it must intersect the line y=x at two points within the interval. This gives us f(x)=x or g(x)={x}^{2}-frac{3}{2}x+m=0. - The discriminant Delta of g(x) must be greater than 0 for two distinct roots: Delta = left(frac{3}{2}right)^2 - 4cdot1cdot m = frac{9}{4} - 4m > 0 Rightarrow m < frac{9}{16}. - Evaluating g(x) at x=frac{1}{4}, we get gleft(frac{1}{4}right)=left(frac{1}{4}right)^{2}-frac{3}{2}left(frac{1}{4}right)+m=frac{1}{16}-frac{3}{8}+m geq 0 Rightarrow m geq frac{5}{16}. Combining these conditions, we find that frac{5}{16} leq m < frac{9}{16}. # Case 2: [a,b]⊆(-∞,frac{1}{4}] - Given f(a)=b and f(b)=a, subtracting these equations leads to a+b+frac{1}{2}=0. This implies a relationship between a and b. - Substituting a+b=-frac{1}{2} into the equation for f(x) and rearranging, we define a new function h(x)={x}^{2}+frac{1}{2}x+m+frac{1}{2}=0 and analyze it similarly. - The discriminant of h(x) must be greater than 0 for two distinct roots: Delta = -4m-frac{7}{4} > 0 Rightarrow m < -frac{7}{16}. - Evaluating h(x) at x=frac{1}{4}, we get hleft(frac{1}{4}right)=left(frac{1}{4}right)^{2}+frac{1}{2}left(frac{1}{4}right)+m+frac{1}{2} geq 0 Rightarrow m geq -frac{11}{16}. Combining these conditions, we find that -frac{11}{16} leq m < -frac{7}{16}. # Conclusion The function f(x)={x}^{2}-frac{1}{2}x+m has a "value-preserving" interval if m is in the range [frac{5}{16},frac{9}{16}) cup [-frac{11}{16},-frac{7}{16}). Therefore, the answer is: boxed{[frac{5}{16},frac{9}{16}) cup [-frac{11}{16},-frac{7}{16})}.

answer:Expanding the given equation, we have [a^4 + 2a^2b^2 + b^4 - (a^2 + b^2)c^2 = 3a^2b^2,] which simplifies to [a^4 + b^4 - c^2(a^2 + b^2) = a^2b^2.] Let's rearrange it into a more useful form for applying the cosine law: [a^4 + b^4 - a^2b^2 = c^2(a^2 + b^2).] By the Law of Cosines again, we have [c^2 = a^2 + b^2 - 2abcos C,] plugging the value from our rearranged equation: [a^4 + b^4 - a^2b^2 = (a^2 + b^2)(a^2 + b^2 - 2abcos C).] Simplify to find cos C: [cos C = frac{a^4 + b^4 - a^2b^2}{2ab(a^2 + b^2)}.] If we assume (a = b), then [cos C = frac{2a^4 - a^4}{2a^2 cdot 2a^2} = frac{a^4}{4a^4} = frac{1}{4},] hence, [C = cos^{-1}left(frac{1}{4}right) = boxed{75.5^circ} text{ (approximately)}.]

answer:If Big Joe is 8 feet tall, and he is one foot taller than Ben, then Ben is 8 - 1 = 7 feet tall. Since Ben is one foot taller than Larry, Larry is 7 - 1 = 6 feet tall. Frank is one foot shorter than Larry, so Frank is 6 - 1 = 5 feet tall. Pepe is half a foot shorter than Frank, so Pepe is 5 - 0.5 = 4.5 feet tall. To convert 4.5 feet to feet and inches, we know that 1 foot is equal to 12 inches. So, 0.5 feet is equal to 0.5 * 12 inches. 0.5 * 12 inches = 6 inches. Therefore, Pepe is boxed{4} feet 6 inches tall.