problem stringclasses 28
values | prefix stringlengths 1.99k 109k | answer stringclasses 28
values | progress stringlengths 5 9 | terminate_reward float64 0 1 | prefix_length int64 724 32.5k | avg_terminate_response_length float64 379 3.01k |
|---|---|---|---|---|---|---|
Find the sum of all integer bases $b>9$ for which $17_b$ is a divisor of $97_b.$ | <think>
Okay, so I need to find all integer bases b greater than 9 where the number 17_b is a divisor of 97_b. Then, sum all those bases. Let me start by understanding what the problem is asking.
First, let me recall that in base b, the number 17_b represents 1*b^1 + 7*b^0 = b + 7. Similarly, 97_b is 9*b^1 + 7*b^0 = 9... | 70 | 6 / 28 | 1 | 1,465 | 616.875 |
Find the sum of all integer bases $b>9$ for which $17_b$ is a divisor of $97_b.$ | <think>
Okay, so I need to find all integer bases b greater than 9 where the number 17_b is a divisor of 97_b. Then, sum all those bases. Hmm, let me start by understanding what this means.
First, in base b, the number 17_b represents 1*b^1 + 7*b^0 = b + 7. Similarly, 97_b is 9*b^1 + 7*b^0 = 9b + 7. The problem states... | 70 | 12 / 30 | 1 | 1,833 | 578.25 |
Find the sum of all integer bases $b>9$ for which $17_b$ is a divisor of $97_b.$ | <think>
Okay, so I need to find all integer bases b greater than 9 where the number 17_b is a divisor of 97_b. Then, sum all those bases. Let me start by understanding what the problem is asking.
First, I know that in base b, the number 17_b represents 1*b^1 + 7*b^0, which is b + 7. Similarly, 97_b is 9*b + 7. So, the... | 70 | 8 / 18 | 1 | 1,230 | 580.25 |
Find the sum of all integer bases $b>9$ for which $17_b$ is a divisor of $97_b.$ | <think>
Okay, so I need to find all integer bases b greater than 9 where the number 17_b is a divisor of 97_b. Then, sum all those bases. Let me start by understanding what the problem is asking.
First, let me recall that in base b, the number 17_b represents 1*b^1 + 7*b^0 = b + 7. Similarly, 97_b is 9*b^1 + 7*b^0 = 9... | 70 | 7 / 21 | 1 | 1,455 | 650.5 |
Find the sum of all integer bases $b>9$ for which $17_b$ is a divisor of $97_b.$ | <think>
Okay, so I need to find all integer bases b greater than 9 where the number 17_b is a divisor of 97_b. Then, sum all those bases. Let me start by understanding what the problem is asking.
First, let me recall that in base b, the number 17_b represents 1*b^1 + 7*b^0 = b + 7. Similarly, 97_b is 9*b + 7. The prob... | 70 | 6 / 26 | 1 | 1,068 | 571.375 |
Find the sum of all integer bases $b>9$ for which $17_b$ is a divisor of $97_b.$ | <think>
Okay, so I need to find all integer bases b greater than 9 where the number 17 in base b divides 97 in base b. Then, sum all those bases. Hmm, let me think about how to approach this.
First, I remember that numbers in different bases can be converted to decimal (base 10) to make the divisibility check easier. ... | 70 | 4 / 15 | 1 | 1,081 | 557.0625 |
Find the sum of all integer bases $b>9$ for which $17_b$ is a divisor of $97_b.$ | <think>
Okay, so I need to find all integer bases b greater than 9 where the number 17_b is a divisor of 97_b. Then, sum all those bases. Let me start by understanding what the problem is asking.
First, let me recall that in base b, the number 17_b represents 1*b^1 + 7*b^0 = b + 7. Similarly, 97_b is 9*b^1 + 7*b^0 = 9... | 70 | 6 / 23 | 1 | 1,128 | 655.5625 |
Find the sum of all integer bases $b>9$ for which $17_b$ is a divisor of $97_b.$ | <think>
Okay, so I need to find all integer bases b greater than 9 where the number 17_b is a divisor of 97_b. Then, sum all those bases. Hmm, let me think about how to approach this.
First, I remember that numbers in different bases can be converted to decimal (base 10) to make the divisibility easier to check. So, l... | 70 | 6 / 34 | 1 | 991 | 572 |
Find the sum of all integer bases $b>9$ for which $17_b$ is a divisor of $97_b.$ | <think>
Okay, so I need to find all integer bases b greater than 9 where the number 17_b is a divisor of 97_b. Then sum all those bases. Let me start by understanding what the problem is asking.
First, bases in numeral systems. In base b, the number 17_b means 1*b^1 + 7*b^0, which is b + 7. Similarly, 97_b is 9*b + 7.... | 70 | 3 / 10 | 1 | 724 | 547.375 |
Find the sum of all integer bases $b>9$ for which $17_b$ is a divisor of $97_b.$ | <think>
Okay, so I need to find all integer bases b greater than 9 where the number 17_b is a divisor of 97_b. Then, sum all those bases. Hmm, let me start by understanding what the problem is asking.
First, I remember that in any base b, the number 17_b represents 1*b^1 + 7*b^0, which is b + 7. Similarly, 97_b is 9*b... | 70 | 7 / 16 | 1 | 1,329 | 596.25 |
Find the sum of all integer bases $b>9$ for which $17_b$ is a divisor of $97_b.$ | <think>
Okay, so I need to find all integer bases b greater than 9 where the number 17_b is a divisor of 97_b. Then, sum all those bases. Hmm, let me start by understanding what the problem is asking.
First, I know that in base b, the number 17_b represents 1*b^1 + 7*b^0 = b + 7. Similarly, 97_b is 9*b^1 + 7*b^0 = 9b ... | 70 | 8 / 22 | 1 | 1,123 | 660 |
Find the sum of all integer bases $b>9$ for which $17_b$ is a divisor of $97_b.$ | <think>
Okay, so I need to find all integer bases b greater than 9 where the number 17_b is a divisor of 97_b. Then, sum all those bases. Hmm, let me start by understanding what the problem is asking.
First, let me recall that in base b, the number 17_b represents 1*b^1 + 7*b^0, which is b + 7. Similarly, 97_b is 9*b ... | 70 | 6 / 13 | 1 | 1,283 | 743.8125 |
Find the sum of all integer bases $b>9$ for which $17_b$ is a divisor of $97_b.$ | <think>
Okay, so I need to find all integer bases b greater than 9 where the number 17_b is a divisor of 97_b. Then, sum all those bases. Hmm, let me start by understanding what 17_b and 97_b mean in base b.
First, in any base b, a number like 17_b represents 1*b^1 + 7*b^0. Similarly, 97_b is 9*b^1 + 7*b^0. So, conver... | 70 | 4 / 18 | 1 | 786 | 536.375 |
Find the sum of all integer bases $b>9$ for which $17_b$ is a divisor of $97_b.$ | <think>
Okay, so I need to find all integer bases b greater than 9 where the number 17_b is a divisor of 97_b. Then, sum all those bases. Let me start by understanding what the problem is asking.
First, let me recall that in base b, the number 17_b represents 1*b^1 + 7*b^0 = b + 7. Similarly, 97_b is 9*b^1 + 7*b^0 = 9... | 70 | 5 / 14 | 1 | 827 | 577 |
Find the sum of all integer bases $b>9$ for which $17_b$ is a divisor of $97_b.$ | <think>
Okay, so I need to find all integer bases b greater than 9 where the number 17_b is a divisor of 97_b. Then, sum all those bases. Hmm, let me start by understanding what the problem is asking.
First, numbers in different bases. For example, 17_b means 1*b + 7 in decimal, right? Similarly, 97_b is 9*b + 7. So, ... | 70 | 10 / 31 | 1 | 1,723 | 804.4375 |
Find the sum of all integer bases $b>9$ for which $17_b$ is a divisor of $97_b.$ | <think>
Okay, so I need to find all integer bases b greater than 9 where the number 17_b is a divisor of 97_b. Then, sum all those bases. Let me start by understanding what this means.
First, in base b, the number 17_b represents 1*b^1 + 7*b^0 = b + 7. Similarly, 97_b is 9*b^1 + 7*b^0 = 9b + 7. The problem states that... | 70 | 8 / 19 | 0.875 | 1,467 | 850.75 |
In $\triangle ABC$ points $D$ and $E$ lie on $\overline{AB}$ so that $AD < AE < AB$, while points $F$ and $G$ lie on $\overline{AC}$ so that $AF < AG < AC$. Suppose $AD = 4$, $DE = 16$, $EB = 8$, $AF = 13$, $FG = 52$, and $GC = 26$. Let $M$ be the reflection of $D$ through $F$, and let $N$ be the reflection of $G$ thro... | <think>
Okay, so I need to find the area of this heptagon AFNBCEM in triangle ABC. The problem gives me some lengths on the sides AB and AC, with points D, E on AB and F, G on AC. There are reflections involved, and the area of quadrilateral DEGF is given as 288.
First, let me try to visualize the triangle. Let me sk... | 588 | 23 / 44 | 1 | 5,400 | 1,129.625 |
In $\triangle ABC$ points $D$ and $E$ lie on $\overline{AB}$ so that $AD < AE < AB$, while points $F$ and $G$ lie on $\overline{AC}$ so that $AF < AG < AC$. Suppose $AD = 4$, $DE = 16$, $EB = 8$, $AF = 13$, $FG = 52$, and $GC = 26$. Let $M$ be the reflection of $D$ through $F$, and let $N$ be the reflection of $G$ thro... | <think>
Okay, so I need to find the area of this heptagon AFNBCEM in triangle ABC. The problem gives me some lengths on the sides AB and AC, and some reflections of points. Let me try to visualize this first.
First, let me sketch triangle ABC. Points D and E are on AB with AD = 4, DE = 16, and EB = 8. So AB is divided... | 588 | 52 / 105 | 1 | 7,619 | 1,059.9375 |
In $\triangle ABC$ points $D$ and $E$ lie on $\overline{AB}$ so that $AD < AE < AB$, while points $F$ and $G$ lie on $\overline{AC}$ so that $AF < AG < AC$. Suppose $AD = 4$, $DE = 16$, $EB = 8$, $AF = 13$, $FG = 52$, and $GC = 26$. Let $M$ be the reflection of $D$ through $F$, and let $N$ be the reflection of $G$ thro... | <think>
Okay, let's try to tackle this geometry problem. It seems a bit complex with all these points and reflections, but maybe if I break it down step by step, it will make more sense.
First, let me try to visualize triangle ABC. Points D and E are on AB, with AD = 4, DE = 16, and EB = 8. So AB is divided into three... | 588 | 27 / 51 | 0.875 | 6,241 | 975.4375 |
In $\triangle ABC$ points $D$ and $E$ lie on $\overline{AB}$ so that $AD < AE < AB$, while points $F$ and $G$ lie on $\overline{AC}$ so that $AF < AG < AC$. Suppose $AD = 4$, $DE = 16$, $EB = 8$, $AF = 13$, $FG = 52$, and $GC = 26$. Let $M$ be the reflection of $D$ through $F$, and let $N$ be the reflection of $G$ thro... | <think>
Okay, let me try to tackle this geometry problem. It seems a bit complex with all these points and reflections, but I'll take it step by step.
First, let me parse the problem again to make sure I understand all the given information.
We have triangle ABC. Points D and E are on AB such that AD < AE < AB. Simil... | 588 | 60 / 94 | 1 | 7,239 | 1,010.4375 |
In $\triangle ABC$ points $D$ and $E$ lie on $\overline{AB}$ so that $AD < AE < AB$, while points $F$ and $G$ lie on $\overline{AC}$ so that $AF < AG < AC$. Suppose $AD = 4$, $DE = 16$, $EB = 8$, $AF = 13$, $FG = 52$, and $GC = 26$. Let $M$ be the reflection of $D$ through $F$, and let $N$ be the reflection of $G$ thro... | <think>
Okay, so I need to find the area of this heptagon AFNBCEM in triangle ABC. The problem gives me some information about points D, E, F, G on the sides AB and AC, with specific lengths. There are reflections involved, M and N, and the area of quadrilateral DEGF is 288. Let me try to visualize the triangle first.
... | 588 | 31 / 57 | 1 | 4,927 | 1,144.125 |
In $\triangle ABC$ points $D$ and $E$ lie on $\overline{AB}$ so that $AD < AE < AB$, while points $F$ and $G$ lie on $\overline{AC}$ so that $AF < AG < AC$. Suppose $AD = 4$, $DE = 16$, $EB = 8$, $AF = 13$, $FG = 52$, and $GC = 26$. Let $M$ be the reflection of $D$ through $F$, and let $N$ be the reflection of $G$ thro... | <think>
Okay, let's try to tackle this geometry problem. It seems a bit complex with all these points and reflections, but maybe if I break it down step by step, it will make more sense.
First, let me parse the given information:
We have triangle ABC. Points D and E are on AB with AD < AE < AB. Points F and G are on ... | 588 | 49 / 67 | 1 | 8,534 | 1,112.125 |
In $\triangle ABC$ points $D$ and $E$ lie on $\overline{AB}$ so that $AD < AE < AB$, while points $F$ and $G$ lie on $\overline{AC}$ so that $AF < AG < AC$. Suppose $AD = 4$, $DE = 16$, $EB = 8$, $AF = 13$, $FG = 52$, and $GC = 26$. Let $M$ be the reflection of $D$ through $F$, and let $N$ be the reflection of $G$ thro... | <think>
Okay, so I need to find the area of this heptagon AFNBCEM in triangle ABC. The problem gives me some lengths on sides AB and AC, and some areas related to quadrilateral DEGF. Let me try to visualize the triangle first.
First, triangle ABC. Points D and E are on AB with AD = 4, DE = 16, and EB = 8. So AB is AD ... | 588 | 33 / 65 | 1 | 7,254 | 1,027.9375 |
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