answer:To solve this problem, we need to find the smallest and largest integers ( n ) such that ( 400 < n^2 < 1600 ). 1. **Finding the smallest integer ( n )**: We look for the smallest ( n ) where ( n^2 > 400 ). Since ( 20^2 = 400 ), the next integer is ( 21 ). Checking, ( 21^2 = 441 ) and indeed ( 441 > 400 ). So, the smallest ( n ) is 21. 2. **Finding the largest integer ( n )**: Next, we look for the largest ( n ) where ( n^2 < 1600 ). Since ( 40^2 = 1600 ), the previous integer is ( 39 ). Checking, ( 39^2 = 1521 ) and ( 1521 < 1600 ). So, the largest ( n ) is 39. 3. **Counting the integers**: The integers ( n ) satisfying ( 400 < n^2 < 1600 ) are those from 21 to 39 inclusive. The count is ( 39 - 21 + 1 = 19 ). The number of positive integers ( n ) satisfying ( 400 < n^2 < 1600 ) is ( boxed{19} ).

answer:Since (m^2-3m-4) + (m^2-5m-6)i is imaginary, it follows that m^2-5m-6 neq 0. Solving this yields m neq -1 and m neq 6, Therefore, the answer is: m neq -1 and m neq 6. By the definition of an imaginary number, we get m^2-5m-6 neq 0, which can be solved accordingly. This question tests the basic concept of complex numbers and is considered a basic problem. Accurately understanding the related concepts is key to solving the problem. Thus, the real number m satisfies boxed{m neq -1 text{ and } m neq 6}.

answer:# Problem: In ( triangle ABC ), prove: [ frac{b^{2} cos A}{a} + frac{c^{2} cos B}{b} + frac{a^{2} cos C}{c} = frac{a^{4} + b^{4} + c^{4}}{2abc}. ] 1. **Use the Law of Cosines**: Recall that in any triangle, the Law of Cosines states: [ cos A = frac{b^2 + c^2 - a^2}{2bc}, ] [ cos B = frac{a^2 + c^2 - b^2}{2ac}, ] [ cos C = frac{a^2 + b^2 - c^2}{2ab}. ] 2. **Substitute the Cosine Formulas**: The left-hand side of the equation given is: [ frac{b^2 cos A}{a} + frac{c^2 cos B}{b} + frac{a^2 cos C}{c}. ] Substitute the expressions for (cos A), (cos B), and (cos C) from the Law of Cosines into this: [ frac{b^2 cos A}{a} = frac{b^2}{a} cdot frac{b^2 + c^2 - a^2}{2bc}, ] [ frac{c^2 cos B}{b} = frac{c^2}{b} cdot frac{a^2 + c^2 - b^2}{2ac}, ] [ frac{a^2 cos C}{c} = frac{a^2}{c} cdot frac{a^2 + b^2 - c^2}{2ab}. ] 3. **Simplify Each Term**: Simplify each term separately: [ frac{b^2}{a} cdot frac{b^2 + c^2 - a^2}{2bc} = frac{b^2(b^2 + c^2 - a^2)}{2abc}, ] [ frac{c^2}{b} cdot frac{a^2 + c^2 - b^2}{2ac} = frac{c^2(a^2 + c^2 - b^2)}{2abc}, ] [ frac{a^2}{c} cdot frac{a^2 + b^2 - c^2}{2ab} = frac{a^2(a^2 + b^2 - c^2)}{2abc}. ] 4. **Combine the Results**: The left-hand side now becomes: [ frac{b^2(b^2 + c^2 - a^2)}{2abc} + frac{c^2(a^2 + c^2 - b^2)}{2abc} + frac{a^2(a^2 + b^2 - c^2)}{2abc}. ] 5. **Combine the Fractions**: Since all terms have the common denominator (2abc), you can combine them: [ frac{b^2(b^2 + c^2 - a^2) + c^2(a^2 + c^2 - b^2) + a^2(a^2 + b^2 - c^2)}{2abc}. ] 6. **Expand the Numerator**: Simplify the expression in the numerator: [ = frac{b^4 + b^2 c^2 - a^2 b^2 + c^4 + c^2 a^2 - b^2 c^2 + a^4 + a^2 b^2 - a^2 c^2}{2abc}. ] 7. **Combine Like Terms**: Combine like terms in the numerator: [ = frac{a^4 + b^4 + c^4 + 2 a^2 b^2 + 2 c^2 a^2 + 2 b^2 c^2 - a^2b^2 - c^2b^2 - a^2c^2}{2abc}. ] 8. **Simplified Form**: Since the mixed terms (- a^2 b^2), (- c^2 b^2), and (- a^2 c^2) balance out, the final simplified form is: [ = frac{a^4 + b^4 + c^4}{2abc}. ] # Conclusion: Therefore: [ frac{b^{2} cos A}{a} + frac{c^{2} cos B}{b} + frac{a^{2} cos C}{c} = frac{a^{4} + b^{4} + c^{4}}{2abc}. ] blacksquare

answer:Let's denote the amount invested by a as x. Since the profits are distributed according to the amount invested and the time period for which the money was invested, we can say that the ratio of their investments is equal to the ratio of their profits. Given that b invested 11000 and received 2200 as his share, we can write the following ratio for b: Profit/Investment = 2200/11000 Similarly, for a, we have: Profit/Investment = 1400/x Since the time period is the same for both a and b (8 months), we can equate the ratios: 2200/11000 = 1400/x Now, we can solve for x: 2200 * x = 1400 * 11000 x = (1400 * 11000) / 2200 x = 1400 * 5 x = 7000 Therefore, a invested boxed{7000} .