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## Task B-1.3.
A ship traveling along a river has covered $24 \mathrm{~km}$ upstream and $28 \mathrm{~km}$ downstream. For this journey, it took half an hour less than for traveling $30 \mathrm{~km}$ upstream and $21 \mathrm{~km}$ downstream, or half an hour more than for traveling $15 \mathrm{~km}$ upstream and $42 \... | <think>
Okay, so I need to figure out the speed of the ship in still water and the speed of the river. Let me start by understanding the problem.
The ship goes upstream and downstream. The time it takes to go 24 km upstream and 28 km downstream is either half an hour less than the time to go 30 km upstream and 21 km d... | 4 | v_{R}=4\mathrm{~}/\mathrm{},v_{B}=10\mathrm{~}/\mathrm{} | true |
## Task B-1.3.
A ship traveling along a river has covered $24 \mathrm{~km}$ upstream and $28 \mathrm{~km}$ downstream. For this journey, it took half an hour less than for traveling $30 \mathrm{~km}$ upstream and $21 \mathrm{~km}$ downstream, or half an hour more than for traveling $15 \mathrm{~km}$ upstream and $42 \... | <think>
Okay, let's try to tackle this problem step by step. First, I need to understand what's being asked. We have a ship traveling upstream and downstream in a river. The problem gives us some distances and time differences, and we need to find the speed of the ship in still water and the speed of the river current.... | 4 | v_{R}=4\mathrm{~}/\mathrm{},v_{B}=10\mathrm{~}/\mathrm{} | true |
## Task B-1.3.
A ship traveling along a river has covered $24 \mathrm{~km}$ upstream and $28 \mathrm{~km}$ downstream. For this journey, it took half an hour less than for traveling $30 \mathrm{~km}$ upstream and $21 \mathrm{~km}$ downstream, or half an hour more than for traveling $15 \mathrm{~km}$ upstream and $42 \... | <think>
Okay, so I need to figure out the speed of the ship in still water and the speed of the river. Let me start by understanding the problem.
The ship goes upstream and downstream. The time it takes for a certain journey is compared to other journeys. Let me parse the problem again:
"A ship traveling along a rive... | 4 | v_{R}=4\mathrm{~}/\mathrm{},v_{B}=10\mathrm{~}/\mathrm{} | true |
## Task B-1.3.
A ship traveling along a river has covered $24 \mathrm{~km}$ upstream and $28 \mathrm{~km}$ downstream. For this journey, it took half an hour less than for traveling $30 \mathrm{~km}$ upstream and $21 \mathrm{~km}$ downstream, or half an hour more than for traveling $15 \mathrm{~km}$ upstream and $42 \... | <think>
Okay, let me try to figure out this problem. So, there's a ship traveling upstream and downstream in a river. The problem gives me some distances and times, and I need to find the speed of the ship in still water and the speed of the river. Hmm, okay. Let me break it down step by step.
First, let me recall tha... | 4 | v_{R}=4\mathrm{~}/\mathrm{},v_{B}=10\mathrm{~}/\mathrm{} | true |
## Task B-1.3.
A ship traveling along a river has covered $24 \mathrm{~km}$ upstream and $28 \mathrm{~km}$ downstream. For this journey, it took half an hour less than for traveling $30 \mathrm{~km}$ upstream and $21 \mathrm{~km}$ downstream, or half an hour more than for traveling $15 \mathrm{~km}$ upstream and $42 \... | <think>
Okay, let me try to figure out this problem. So, there's a ship traveling upstream and downstream in a river. The problem gives me some distances and times, and I need to find the speed of the ship in still water and the speed of the river. Alright, let me break it down step by step.
First, let me recall that ... | 4 | v_{R}=4\mathrm{~}/\mathrm{},v_{B}=10\mathrm{~}/\mathrm{} | true |
3. (6 points) A construction company was building a tunnel. When $\frac{1}{3}$ of the tunnel was completed at the original speed, they started using new equipment, which increased the construction speed by $20 \%$ and reduced the working hours to $80 \%$ of the original. As a result, it took a total of 185 days to comp... | <think>
Okay, let me try to figure out this tunnel construction problem. So, the question is about a construction company building a tunnel. They completed 1/3 of the tunnel at the original speed, then they switched to new equipment which increased their speed by 20% and reduced working hours to 80% of the original. Be... | 180 | 180 | true |
3. (6 points) A construction company was building a tunnel. When $\frac{1}{3}$ of the tunnel was completed at the original speed, they started using new equipment, which increased the construction speed by $20 \%$ and reduced the working hours to $80 \%$ of the original. As a result, it took a total of 185 days to comp... | <think>
Okay, let's try to figure out this tunnel construction problem. So, the question is about a construction company building a tunnel. They completed 1/3 of the tunnel at the original speed, then they switched to new equipment which increased their speed by 20% and reduced working hours to 80% of the original. Bec... | 180 | 180 | true |
3. (6 points) A construction company was building a tunnel. When $\frac{1}{3}$ of the tunnel was completed at the original speed, they started using new equipment, which increased the construction speed by $20 \%$ and reduced the working hours to $80 \%$ of the original. As a result, it took a total of 185 days to comp... | <think>
Okay, let me try to figure out this tunnel construction problem. Hmm, so the company built a third of the tunnel at the original speed, then switched to new equipment which increased their speed by 20% but reduced working hours to 80% of the original. The total time taken was 185 days. If they hadn't used the n... | 180 | 180 | true |
3. (6 points) A construction company was building a tunnel. When $\frac{1}{3}$ of the tunnel was completed at the original speed, they started using new equipment, which increased the construction speed by $20 \%$ and reduced the working hours to $80 \%$ of the original. As a result, it took a total of 185 days to comp... | <think>
Okay, let me try to figure out this tunnel construction problem. Hmm, so the company built a third of the tunnel at the original speed, then switched to new equipment which increased their speed by 20% but reduced working hours to 80% of the original. The total time taken was 185 days. If they hadn't used the n... | 180 | 180 | true |
3. (6 points) A construction company was building a tunnel. When $\frac{1}{3}$ of the tunnel was completed at the original speed, they started using new equipment, which increased the construction speed by $20 \%$ and reduced the working hours to $80 \%$ of the original. As a result, it took a total of 185 days to comp... | <think>
Okay, let me try to figure out this tunnel construction problem. Hmm, let me read it again carefully.
So, there's a construction company building a tunnel. When they completed 1/3 of the tunnel at the original speed, they started using new equipment. This new equipment increased their construction speed by 20%... | 180 | 180 | true |
Prove that number $1$ can be represented as a sum of a finite number $n$ of real numbers, less than $1,$ not necessarily distinct, which contain in their decimal representation only the digits $0$ and/or $7.$ Which is the least possible number $n$? | <think>
Okay, so I need to figure out the smallest number n such that 1 can be written as the sum of n real numbers, each less than 1, and each of these numbers only has 0s and 7s in their decimal representation. Hmm, interesting problem. Let me try to break it down step by step.
First, let me understand the problem p... | null | 8 | false |
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