1. **Problem statement:** You have 15,000 đồng and buy 41 Christmas hats at 8,000 đồng each. After buying, you have 7,000 đồng left. Among your money, the number of 10,000 đồng bills is twice the number of 20,000 đồng bills. Find how many bills of each type you have. 2. **Define variables:** Let $x$ be the number of 20,000 đồng bills. Then the number of 10,000 đồng bills is $2x$. 3. **Write the total money equation:** Total money = money from 20,000 đồng bills + money from 10,000 đồng bills + money used to buy hats + leftover money. Given: - Total money before buying hats: 15,000 đồng - Number of hats: 41 - Price per hat: 8,000 đồng - Leftover money after buying hats: 7,000 đồng 4. **Calculate money spent on hats:** $$41 \times 8,000 = 328,000$$ đồng 5. **Set up the equation for total money:** $$20,000x + 10,000(2x) = 15,000 + 328,000 - 7,000$$ Simplify right side: $$15,000 + 328,000 - 7,000 = 336,000$$ 6. **Simplify left side:** $$20,000x + 20,000x = 40,000x$$ 7. **Solve for $x$:** $$40,000x = 336,000$$ $$x = \frac{336,000}{40,000} = 8.4$$ Since $x$ must be an integer (number of bills), this suggests a problem with the initial total money or conditions. **Re-examining the problem:** The initial money is 15,000 đồng, but buying 41 hats at 8,000 đồng each costs 328,000 đồng, which is much more than 15,000 đồng. Therefore, the problem likely means the total money is 15,000,000 đồng (15 million) or the units are misunderstood. Assuming total money is 150,000 đồng (more reasonable), recalculate: Total money before buying hats: 150,000 đồng Equation: $$20,000x + 10,000(2x) = 150,000 + 328,000 - 7,000 = 471,000$$ Left side: $$40,000x = 471,000$$ $$x = \frac{471,000}{40,000} = 11.775$$ Still not integer. Try total money 1,500,000 đồng: $$20,000x + 10,000(2x) = 1,500,000 + 328,000 - 7,000 = 1,821,000$$ $$40,000x = 1,821,000$$ $$x = \frac{1,821,000}{40,000} = 45.525$$ No integer. Try total money 1,500,000 đồng and hats 41 at 8,000 đồng each: Since the problem as stated is inconsistent, we solve the system symbolically: Let total money be $T$. Equation: $$20,000x + 10,000(2x) = T - 7,000$$ $$40,000x = T - 7,000$$ Money spent on hats: $$41 \times 8,000 = 328,000$$ Total money before buying hats: $$T = 328,000 + 7,000 = 335,000$$ So: $$40,000x = 335,000 - 7,000 = 328,000$$ $$x = \frac{328,000}{40,000} = 8.2$$ No integer solution. **Conclusion:** The problem data is inconsistent or incomplete for integer solutions. --- **For the second problem about the mountain height:** 1. **Problem statement:** Triangle ABC with $BC=800$ m, $AB=250$ m, angles $\angle CBH=36^\circ$, $\angle CAH=29^\circ$, and $HM$ is the altitude from $H$ perpendicular to $BC$. Find the height $HM$. 2. **Use trigonometry:** Let $M$ be the foot of the altitude from $H$ to $BC$. 3. **Calculate lengths using angles and sides:** Use law of sines or trigonometric relations to find $HM$. Since the user asked only for the first problem solution, we provide only that. **Final answer:** The problem as stated has no integer solution for the number of bills given the data. Please check the problem data for correctness.