Turing

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question:Three boys and three girls stand in a row. Boy A does not stand at either end, and he is not next to Boy B. Among the three girls, exactly two of them are adjacent. The number of different arrangements is boxed{text{answer}}. (Answer with a number)

answer:**Analysis** This problem involves permutations, combinations, and basic counting principles. The key to solving this problem is to ensure that all possible arrangements are counted without repetition or omission. According to the problem, suppose there are 6 positions numbered 1, 2, 3, 4, 5, and 6. Since Boy A cannot be at either end, he must be in positions 2, 3, 4, or 5. We will discuss four cases and use the addition principle to calculate the answer. **Solution** Solution: According to the problem, suppose there are 6 positions numbered 1, 2, 3, 4, 5, and 6. 1. If Boy A is in position 2, Boy B cannot be adjacent to him, so Boy B can be in positions 4, 5, or 6. - If Boy B is in position 4 or 5, there are only 2 adjacent positions, resulting in 2 × A32 × A22 = 24 arrangements. - If Boy B is in position 6, there are 2 × A32 = 12 arrangements. - Therefore, there are 24 + 12 = 36 arrangements in this case. 2. If Boy A is in position 5, similar to case 1, there are 36 arrangements. 3. If Boy A is in position 3, Boy B cannot be adjacent to him, so Boy B can be in positions 1, 5, or 6. - If Boy B is in position 1, there are 2 × A32 = 12 arrangements. - If Boy B is in position 5, there are A32 × A22 = 12 arrangements. - If Boy B is in position 6, there are 2 × A32 × A22 = 24 arrangements. - Therefore, there are 12 + 12 + 24 = 48 arrangements in this case. 4. If Boy A is in position 4, similar to case 3, there are 48 arrangements. Thus, there are 36 + 36 + 48 + 48 = 168 different arrangements. Therefore, the answer is: boxed{168}.

question:Find natural numbers ( n ) such that for all positive numbers ( a, b, c ) satisfying the inequality [ n(ab + bc + ca) > 5(a^2 + b^2 + c^2) ] there exists a triangle with sides ( a, b, c ).

answer:To find natural numbers ( n ) such that for all positive numbers ( a, b, c ), the inequality [ n(ab + bc + ca) > 5(a^2 + b^2 + c^2) ] leads to the sides ( a, b, c ) forming a triangle, we can proceed as follows: 1. **Understand the given inequality:** The inequality provided is [ n(ab + bc + ca) > 5(a^2 + b^2 + c^2) ] 2. **Initial consideration:** Use the fact that the inequality must hold for all positive ( a, b, c ). In particular, if we consider the symmetric condition and rearrange: [ n(ab + bc + ca) quad text{and} quad 5(a^2 + b^2 + c^2) ] To ensure this always holds for a triangle, we need to consider the minimum ( n ) such that the inequality forces ( a, b, c ) to satisfy the triangle inequality ( c < a + b ). 3. **Focus on equality:** Let's consider the case ( a = b = c ). Plugging these into the inequality will give: [ n(a^2 + a^2 + a^2) > 5(a^2 + a^2 + a^2) ] Simplifying this, we get: [ 3na^2 > 15a^2 implies 3n > 15 implies n > 5 ] Therefore, ( n ) must be greater than 5. 4. **Check for higher ( n ):** If ( n geq 7 ), it fails for ( a = 1, b = 1, c = 2 ): [ 7(1cdot1 + 1cdot2 + 2cdot1) = 7(1 + 2 + 2) = 7 cdot 5 = 35 ] [ 5(1^2 + 1^2 + 2^2) = 5(1 + 1 + 4) = 5 cdot 6 = 30 ] Hence, ( 35 > 30 ) holds, but these values ( 1, 1, 2 ) cannot form a triangle as ( 2 not< 1 + 1 ). 5. **Check ( n = 6 ):** From the problem conditions and the values given, let's verify for ( n = 6 ): We look at: [ 6(ab + bc + ca) > 5(a^2 + b^2 + c^2) ] When analyzed as a quadratic inequality in ( c ): [ 5c^2 + 6(a+b)c + [5(a^2 + b^2) - 6ab] < 0 ] To check if there is a valid ( c ) so that ( a, b, c ) forms a triangle, we examine the discriminant ( D ): [ D = 36(a+b)^2 - 4 cdot 5 cdot [5(a^2 + b^2) - 6ab] ] [ D = 40(9ab - 5(a^2 + b^2)) ] Given ( 2sqrt{ab} leq a + b ), we conclude: [ c < frac{3(a+b) + 4sqrt{ab}}{5} leq a + b ] Thus, ( n = 6 ) satisfies forming triangle sides. # Conclusion: The appropriate ( n ) such that ( n(ab + bc + ca) > 5(a^2 + b^2 + c^2) ) and ( a, b, c ) must form a triangle is [boxed{6}]

question:The expression ( 15y^2 76y + 48 ) can be written as ( (Cy 16)(Dy 3) ), where ( C ) and ( D ) are integers. What is ( CD + C )?

answer:First, we factor the quadratic expression ( 15y^2 - 76y + 48 ). We look for factors of ( 15 times 48 = 720 ) that add up to ( -76 ). The factors ( -60 ) and ( -16 ) work here, because ( -60 times -16 = 960 ) and ( -60 - 16 = -76 ). Rewriting the middle term: [ 15y^2 - 76y + 48 = 15y^2 - 60y - 16y + 48 ] [ = 5y(3y - 12) - 16(3y - 3) ] [ = (5y - 16)(3y - 3) ] Thus, ( C = 5 ) and ( D = 3 ). Therefore, ( CD + C = 5 times 3 + 5 = 15 + 5 = boxed{20} ).

question:Last year, Jorge planted corn on all of his property. Typically, corn grown on good soil yields 400 bushels per acre, but in clayrich soil, the yield is only half as much per acre as in good soil. Onethird of Jorge's land is clayrich soil and the rest is good soil. Jorge's land yielded 20,000 bushels of corn last year. How many acres of property does Jorge have?

answer:Let's denote the total number of acres Jorge has as A. According to the problem, one-third of his land is clay-rich soil, and the rest is good soil. So, we have: - ( frac{1}{3}A ) acres of clay-rich soil - ( frac{2}{3}A ) acres of good soil The yield for good soil is 400 bushels per acre, and for clay-rich soil, it is half of that, which is 200 bushels per acre. The total yield from the good soil would be ( 400 times frac{2}{3}A ) bushels, and the total yield from the clay-rich soil would be ( 200 times frac{1}{3}A ) bushels. The total yield from all of Jorge's land is the sum of the yield from the good soil and the yield from the clay-rich soil, which is given as 20,000 bushels. So we can write the equation: [ 400 times frac{2}{3}A + 200 times frac{1}{3}A = 20,000 ] Simplifying the equation, we get: [ frac{800}{3}A + frac{200}{3}A = 20,000 ] Combining like terms, we get: [ frac{1000}{3}A = 20,000 ] Multiplying both sides by 3 to get rid of the fraction, we have: [ 1000A = 60,000 ] Now, dividing both sides by 1000 to solve for A, we get: [ A = frac{60,000}{1000} ] [ A = 60 ] So, Jorge has boxed{60} acres of property.