answer:Let's prove the inequality (a^{4} + b^{4} + c^{4} geqslant abc(a + b + c)) for positive real numbers (a, b,) and (c). 1. **Starting Point:** Consider the following pairwise inequalities involving squares: [ a^{4} + b^{4} geqslant 2a^{2}b^{2}, quad b^{4} + c^{4} geqslant 2b^{2}c^{2}, quad a^{4} + c^{4} geqslant 2a^{2}c^{2}. ] These inequalities follow directly from the Arithmetic Mean-Geometric Mean Inequality (AM-GM Inequality), which states that for any non-negative (x) and (y), [ frac{x + y}{2} geqslant sqrt{xy} quad text{or equivalently} quad x + y geqslant 2sqrt{xy}. ] Applying (x = a^4) and (y = b^4) gives us the first inequality, and similarly for the others. 2. **Summing the Inequalities:** Add the three inequalities: [ (a^{4} + b^{4}) + (b^{4} + c^{4}) + (a^{4} + c^{4}) geqslant 2a^{2}b^{2} + 2b^{2}c^{2} + 2a^{2}c^{2}. ] Simplifying the left-hand side, we get: [ 2(a^{4} + b^{4} + c^{4}) geqslant 2a^{2}b^{2} + 2b^{2}c^{2} + 2a^{2}c^{2}. ] 3. **Rewriting the Summed Result:** Divide both sides of the inequality by 2: [ a^{4} + b^{4} + c^{4} geqslant a^{2}b^{2} + b^{2}c^{2} + a^{2}c^{2}. ] 4. **Applying Constants and Rearranging:** Consider the expressions (b^{2} + c^{2} geqslant 2bc), (a^{2} + c^{2} geqslant 2ac), and (a^{2} + b^{2} geqslant 2ab), which again follow from the AM-GM Inequality: [ begin{aligned} a^2(b^2 + c^2) &geqslant a^2 cdot 2bc = 2a^2bc, b^2(a^2 + c^2) &geqslant b^2 cdot 2ac = 2b^2ac, c^2(a^2 + b^2) &geqslant c^2 cdot 2ab = 2c^2ab. end{aligned} ] 5. **Strengthening the Inequality:** Substitute these inequalities back into the original inequality obtained from summation: [ a^{2}b^{2} + b^{2}c^{2} + a^{2}c^{2} geqslant ab(a^2 + c^2) + bc(b^2 + a^2) + ac(c^2 + b^2). ] Since (a^{2}b^{2} + b^{2}c^{2} + a^{2}c^{2}) is on the left-hand side, we relate: [ a^{2}b^{2} + b^{2}c^{2} + a^{2}c^{2} geqslant 2a^2bc + 2b^2ac + 2c^2ab = 2(a^2bc + b^2ac + c^2ab) ] 6. **Final Inequality Result:** We obtain: [ 2(a^{4} + b^{4} + c^{4}) geqslant 2(a^{2}bc + b^{2}ac + c^{2}ab). ] Dividing both sides by 2: [ a^{4} + b^{4} + c^{4} geqslant a^{2}bc + b^{2}ac + c^{2}ab. ] And rearrange to achieve the desired inequality: [ a^{4} + b^{4} + c^{4} geqslant abc(a + b + c). ] # Conclusion: [ boxed{a^{4} + b^{4} + c^{4} geqslant abc(a + b + c)} ]

answer:First, let's find out how many cars have valid tickets. Since 75% of the cars have valid tickets, we can calculate this number by taking 75% of the total number of cars in the parking lot. Number of cars with valid tickets = 75% of 300 = 0.75 * 300 = 225 cars Now, we know that 30 people tried to park without paying, which means they don't have valid tickets. Since these 30 cars are part of the total 300 cars, we can subtract them from the number of cars with valid tickets to find out how many cars have permanent parking passes. Number of cars with permanent parking passes = Number of cars with valid tickets - Number of cars without valid tickets = 225 - 30 = 195 cars Now, to find the fraction of cars with valid tickets that have permanent parking passes, we divide the number of cars with permanent parking passes by the number of cars with valid tickets. Fraction of cars with permanent parking passes = Number of cars with permanent parking passes / Number of cars with valid tickets = 195 / 225 = 13 / 15 Therefore, the fraction of the cars with valid tickets that have permanent parking passes is boxed{13/15} .

answer:Let's denote the number of pages of reading homework Rachel had as R. According to the information given, Rachel had 3 more pages of math homework than reading homework. Since we know she had 5 pages of math homework, we can write the following equation: Math homework = Reading homework + 3 Substituting the known value of math homework (5 pages), we get: 5 = R + 3 To find the value of R (the number of pages of reading homework), we need to subtract 3 from both sides of the equation: 5 - 3 = R R = 2 Therefore, Rachel had boxed{2} pages of reading homework.

answer:Given that a > b, we want to determine the relationship between -2-a and -2-b. Starting with the given inequality: a > b Multiplying both sides by -1 (remember, multiplying by a negative number reverses the inequality): -a < -b Now, adding -2 to both sides (adding the same number to both sides does not change the direction of the inequality): -2 + (-a) < -2 + (-b) Simplifying both sides gives us: -2-a < -2-b Therefore, the correct answer is boxed{<}. This step-by-step solution adheres to the basic properties of inequalities, demonstrating how multiplying by a negative number reverses the inequality and adding the same number to both sides does not change the direction of the inequality.