answer:Lorraine's children ate 25% of the 16 brownies, which is 0.25 * 16 = 4 brownies. After the children ate some, there were 16 - 4 = 12 brownies left. The family then ate 50% of the remaining 12 brownies, which is 0.50 * 12 = 6 brownies. After the family ate, there were 12 - 6 = 6 brownies left. Lorraine ate 1 more brownie after everyone went to bed, so there were 6 - 1 = boxed{5} brownies left over.

answer:Given that the polynomial equation is [x^3 + bx + c = 0,] and one of its roots is 5-sqrt{2}, we know that the coefficients of the polynomial, b and c, are rational numbers. This implies that if a root has an irrational part, its radical conjugate must also be a root to ensure the coefficients remain rational. Therefore, since 5-sqrt{2} is a root, its radical conjugate, 5+sqrt{2}, must also be a root. By Vieta's formulas, we know that the sum of the roots of a cubic polynomial x^3 + bx + c = 0 is 0 (since the coefficient of x^2 is 0 in this case). Let's denote the roots of the polynomial as r_1, r_2, and r_3. We know that r_1 = 5-sqrt{2} and r_2 = 5+sqrt{2}. Thus, we have: [r_1 + r_2 + r_3 = 0.] Substituting the known roots into the equation gives: [(5-sqrt{2}) + (5+sqrt{2}) + r_3 = 0.] Simplifying the left side: [10 + r_3 = 0.] Solving for r_3: [r_3 = -10.] Therefore, the integer root of the polynomial is boxed{-10}.

answer:1. **Interpret the historical context**: The problem mentions that after the death of A. S. Pushkin in 1837 during a duel, a financial allowance known as a 'pension' was granted to his wife and daughters. This allowance is described as a lifelong provision. 2. **Modern equivalent analysis**: The task is to determine which modern type of pension this historical financial support corresponds to. In contemporary terms, pensions are typically categorized based on the circumstance of eligibility. 3. **Identify the types of pensions**: - Retirement pension: Typically granted upon reaching a certain age after fulfilling specific work criteria. - Disability pension: Provided to individuals who are unable to work due to a disability. - Survivor's pension: This is paid to family members following the death of the main provider of the family, intended to support those who were financially dependent on the deceased. 4. **Apply context to identify pension type**: Pushkin's wife and daughters were provided with a pension following his death. This was intended to offer financial support to his dependents who lost their main provider. 5. **Determine the answer**: The pension provided to Pushkin’s family fits the definition of a survivor's or dependent's pension as it was a financial provision given to family members following the death of the main income earner. # Conclusion: The pension allocated to Alexander Pushkin’s relatives after his death is equivalent to a modern-day survivor's pension. boxed{text{survivor's pension}}

answer:To find the discount on the backpacks, we first need to calculate the total cost of the backpacks without the monogramming fee. Since monogramming each backpack costs 12.00 and Grandma wants to order 5 backpacks, the total cost for monogramming is: 5 backpacks * 12.00/backpack = 60.00 Now, we know that the total cost including the monogramming is 140.00. To find the cost of the backpacks before the monogramming, we subtract the monogramming cost from the total cost: 140.00 (total cost) - 60.00 (monogramming cost) = 80.00 (cost of backpacks without monogramming) If the backpacks cost 80.00 after the discount, and the original monogramming cost of 60.00 is added back in, the original price before the discount would have been: 80.00 (discounted backpacks) + 60.00 (monogramming) = 140.00 (original price before discount) Since the original price before the discount is the same as the total cost after the discount, it appears there is no discount applied to the backpacks. The total cost of boxed{140.00} includes both the backpacks and the monogramming fee.