🧠:<think>First, I need to calculate the total distance driven by Jane, Mike, and Carl. Each of them drives 100 miles, so the total distance is:[100 text{ miles} times 3 = 300 text{ miles}]Next, I'll determine the amount of gasoline each person uses for their 100-mile drive.- Jane's Gasoline Usage: [ frac{100 text{ miles}}{30 text{ miles per gallon}} = frac{10}{3} text{ gallons} ]- Mike's Gasoline Usage: [ frac{100 text{ miles}}{15 text{ miles per gallon}} = frac{20}{3} text{ gallons} ]- Carl's Gasoline Usage: [ frac{100 text{ miles}}{20 text{ miles per gallon}} = 5 text{ gallons} ]Now, I'll sum up the total gasoline used by all three:[frac{10}{3} text{ gallons} + frac{20}{3} text{ gallons} + 5 text{ gallons} = frac{30}{3} text{ gallons} + 5 text{ gallons} = 10 text{ gallons} + 5 text{ gallons} = 15 text{ gallons}]Finally, to find the combined miles per gallon (mpg), I'll divide the total distance by the total gasoline used:[frac{300 text{ miles}}{15 text{ gallons}} = 20 text{ miles per gallon}]</think>

🧠:<think>First, simplify the exponent in the term (3^{2+4+6}):[2 + 4 + 6 = 12]Thus, the expression becomes:[3^{12} - (3^2 + 3^4 + 3^6) + (3^2 cdot 3^4 cdot 3^6)]Next, calculate each power of 3:[3^{12}, quad 3^2 = 9, quad 3^4 = 81, quad 3^6 = 729]Then, evaluate the product of powers:[3^2 cdot 3^4 cdot 3^6 = 3^{2+4+6} = 3^{12}]Substitute these values back into the expression:[3^{12} - (9 + 81 + 729) + 3^{12}]Calculate the sum inside the parentheses:[9 + 81 + 729 = 819]Now substitute and simplify:[3^{12} + 3^{12} - 819]Recognizing that (3^{12} = 531441), the expression simplifies to:[531441 + 531441 - 819 = 1062063 - 819 = 1062242]The value of the expression is (boxed{textbf{(B)} 1062242}).</think>

🧠:<think>First, we determine the total number of games each player participated in:- Liam won 6 games and lost 3 games, so he played a total of (6 + 3 = 9) games.- Noah won 4 games and lost 4 games, so he played a total of (4 + 4 = 8) games.- Since Olivia competes an equal number of times with each opponent, her total number of games must be 8 as well (from Noah's games).Next, we set up the total game environment:- Each game involves two players, so each game is counted twice when summing individual games.- Let (x) be the number of games Olivia won. Thus, Olivia played (x + 4) games. Since everyone plays the same number of games, we set this equal to Noah’s and Liam's: [ x + 4 = 9 text{ (from Liam's games)} ]Solving for Olivia's games:[x + 4 = 9][x = 9 - 4 = 5]Finally, we conclude:- Each contestant meets others three times each.- Olivia’s total score is (1) point per win times the number of games she won, calculated as (5) games. [ 5 text{ points} ]The final answer is (boxed{textbf{(C)} 5}).</think>

🧠:<think>To solve the problem, we need to determine the value of ( Q ) for the polynomial ( z^5 - 15z^4 + Pz^3 + Qz^2 + Rz + 64 ) where all roots are positive integers.1. Identify the Roots: - The sum of the roots is given by the coefficient of ( z^4 ), which is 15. - The product of the roots is given by the constant term, which is 64. - We need to find five positive integers that sum to 15 and multiply to 64. A suitable set of roots is ( 8, 4, 1, 1, 1 ).2. Calculate the Third Symmetric Sum: - The third symmetric sum ( s_3 ) is the sum of all products of the roots taken three at a time. - The products are calculated as follows: - Three 1s: ( 1 times 1 times 1 = 1 ) - Two 1s and an 8: ( 8 times 1 times 1 = 8 ) - Two 1s and a 4: ( 4 times 1 times 1 = 4 ) - One 8, one 4, and one 1: ( 8 times 4 times 1 = 32 ) - Summing these products: ( 1 + 8 + 4 + 32 = 45 ).3. Determine ( Q ): - The value of ( Q ) is the negation of the third symmetric sum, so ( Q = -45 ).Thus, the value of ( Q ) is ( boxed{D} ).</think>