data_source stringclasses 1
value | prompt listlengths 2 2 | ability stringclasses 1
value | extra_info dict | response stringlengths 1.24k 66.4k | has_thinking bool 2
classes | score float64 0 1 |
|---|---|---|---|---|---|---|
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given that Henry writes the quotient of $ ab $ divided by $ N+1 $ for each pair $ (a, b) $ with $ 1 \leq a, b \leq N $. We are to find all positive integers $ N $ such that the **sum** of all these numbers is:
$$
\frac{N^3 - N^2 + 2}{4}
$$
---
### Step 1: Understand the problem
We are to sum the values of:
... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given that for a positive integer $ N $, Henry writes the quotient of $ ab $ divided by $ N+1 $ for each integer pair $ (a, b) $ where $ 1 \le a, b \le N $. We are to find all such $ N $ for which the **sum of all these $ N^2 $ numbers** is:
$$
\frac{N^3 - N^2 + 2}{4}
$$
Let’s break this down step-by-step.
--... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given that Henry writes the **quotient** of $ ab $ divided by $ N+1 $ for each integer pair $(a,b)$ such that $ 1 \le a,b \le N $. He writes this number on the board for $ N^2 $ such pairs.
We are to find all **positive integers** $ N $ such that the **sum** of all these numbers is:
$$
\frac{N^3 - N^2 + 2}{4}
... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given that for a positive integer $ N $, Henry writes the quotient $ \frac{ab}{N+1} $ on the board for every pair $ (a, b) $ such that $ 1 \le a, b \le N $. We are to find all such $ N $ for which the **sum of these $ N^2 $ numbers** is equal to:
$$
\frac{N^3 - N^2 + 2}{4}
$$
---
### Step 1: Understand the ex... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given a problem involving a sum of quotients, and we are to find all positive integers $ N $ such that the total sum of these quotients equals a specific expression.
---
### Problem Restatement
For a given positive integer $ N $, Henry writes the **quotient of** $ ab $ **divided by** $ N+1 $ on the board for ... | false | 1 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given a problem involving the sum of numbers written on a board by Henry. Let's break it down step by step.
---
### **Understanding the Problem:**
Henry writes the quotient of $ ab $ divided by $ N+1 $ for each integer pair $ (a, b) $ such that $ 1 \le a, b \le N $.
So, for each pair $ (a, b) $, he writes:
... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given a problem involving summation over all integer pairs $(a, b)$ with $1 \le a, b \le N$ of the quotient $\frac{ab}{N+1}$. We are to find all positive integers $N$ such that the **sum of these quotients** equals $\frac{N^3 - N^2 + 2}{4}$.
---
### Step 1: Understand the problem
Henry writes the number $\fra... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given a problem where Henry writes the quotient of $ ab $ divided by $ N+1 $ for each integer pair $ (a, b) $ such that $ 1 \leq a, b \leq N $. Then, we're told that the **sum** of all these numbers he wrote is:
$$
\frac{N^3 - N^2 + 2}{4}
$$
We are to **find all positive integers $ N $** for which this is true... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given a problem where Henry writes the quotient of $ ab $ divided by $ N+1 $ for each integer pair $ (a, b) $ with $ 1 \leq a, b \leq N $, and we are told that the **sum** of all these numbers is:
$$
\frac{N^3 - N^2 + 2}{4}
$$
We are to **find all positive integers $ N $** such that this condition is satisfied... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given that Henry writes the quotient of $ ab $ divided by $ N+1 $ for each integer pair $ (a, b) $ with $ 1 \le a, b \le N $. So, for each pair, he writes:
$$
\frac{ab}{N+1}
$$
We are to find all positive integers $ N $ such that the **sum** of these $ N^2 $ numbers is:
$$
\frac{N^3 - N^2 + 2}{4}
$$
---
###... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given the following problem:
> For a given positive integer $ N $, Henry writes the quotient of $ ab $ divided by $ N+1 $ on the board for each integer pair $ (a, b) $ where $ 1 \le a, b \le N $. We are to find all values of $ N $ such that the **sum of the $ N^2 $ numbers** Henry wrote is:
$$
\frac{N^3 - N^2 +... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given a problem involving a sum over all pairs $(a, b)$ with $1 \le a, b \le N$, and the sum of the numbers written on the board is:
$$
\sum_{a=1}^N \sum_{b=1}^N \frac{ab}{N+1} = \frac{N^3 - N^2 + 2}{4}
$$
We are to **find all positive integers $N$** for which this equation holds.
---
### Step 1: Simplify th... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given a problem involving a function of two variables, $ a $ and $ b $, where $ 1 \le a, b \le N $, and for each pair, Henry writes the value of $ \frac{ab}{N+1} $. We are to find all positive integers $ N $ such that the **sum** of all these values is equal to:
$$
\frac{N^3 - N^2 + 2}{4}
$$
---
### Step 1: U... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given a problem involving a sum of quotients, and we need to find all positive integers $ N $ such that the sum of the numbers written on the board is equal to:
$$
\frac{N^3 - N^2 + 2}{4}
$$
---
### Step 1: Understanding the Setup
Henry writes the **quotient** of $ ab $ divided by $ N+1 $ for each pair $ (a,... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given that Henry writes the **quotient** of $ ab $ divided by $ N+1 $ for each pair $ (a, b) $ where $ 1 \le a, b \le N $. Then, the **sum of all these numbers** is given to be:
$$
\frac{N^3 - N^2 + 2}{4}
$$
We are to find all **positive integers** $ N $ for which this holds.
---
### Step 1: Understanding th... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given a problem involving a sum over all pairs of integers $(a, b)$ with $1 \leq a, b \leq N$. Henry writes the quotient $\frac{ab}{N+1}$ for each such pair, and we are told that the **sum** of all these numbers is:
$$
\frac{N^3 - N^2 + 2}{4}
$$
We are to **find all positive integers $N$** for which this is tr... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given that Henry writes the quotient of $ ab $ divided by $ N+1 $ for each pair $ (a, b) $ with $ 1 \le a, b \le N $, and we are to find all positive integers $ N $ such that the **sum of all these quotients** is equal to:
$$
\frac{N^3 - N^2 + 2}{4}
$$
---
### Step 1: Understand the expression
For each pair ... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given a problem where Henry writes the quotient of $ ab $ divided by $ N+1 $ on the board for each integer pair $ (a, b) $ with $ 1 \le a, b \le N $. We are to find all positive integers $ N $ such that the **sum of all these numbers** is equal to:
$$
\frac{N^3 - N^2 + 2}{4}
$$
---
### Step 1: Understanding t... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given that Henry writes the quotient of $ ab $ divided by $ N+1 $ for each integer pair $ (a, b) $ with $ 1 \le a, b \le N $. That is, for each such pair, he writes:
$$
\frac{ab}{N+1}
$$
So, the total sum of all these numbers is:
$$
\sum_{a=1}^{N} \sum_{b=1}^{N} \frac{ab}{N+1} = \frac{1}{N+1} \sum_{a=1}^{N} \... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given that Henry writes the quotient of $ ab $ divided by $ N+1 $ on the board for each integer pair $ (a, b) $ with $ 1 \leq a, b \leq N $. The total number of such pairs is $ N^2 $, and the sum of all these $ N^2 $ numbers is given as:
$$
\sum_{a=1}^{N} \sum_{b=1}^{N} \frac{ab}{N+1} = \frac{N^3 - N^2 + 2}{4}
... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given that Henry writes the quotient of $ ab $ divided by $ N+1 $ for each pair $ (a, b) $ with $ 1 \leq a, b \leq N $. We are to find all positive integers $ N $ for which the sum of all these numbers is equal to:
$$
\frac{N^3 - N^2 + 2}{4}
$$
---
### Step 1: Understand the expression
Henry writes the value:... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given that Henry writes the quotient of $ ab $ divided by $ N+1 $ for each pair $ (a, b) $ with $ 1 \le a, b \le N $. The problem asks us to find all positive integers $ N $ such that the **sum** of all these numbers is:
$$
\frac{N^3 - N^2 + 2}{4}
$$
---
### Step 1: Understand the structure of the problem
He... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given a problem where for a given positive integer $ N $, Henry writes the value of
$$
\frac{ab}{N+1}
$$
on the board for every integer pair $ (a,b) $ such that $ 1 \le a, b \le N $. We are to find all positive integers $ N $ such that the **sum of all these numbers** is equal to
$$
\frac{N^3 - N^2 + 2}{4}.
$... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given a problem involving a sum over all integer pairs $(a, b)$ where $1 \le a, b \le N$. For each pair, Henry writes the **quotient** of $ab$ divided by $N+1$ on the board. That is, for each pair $(a, b)$, the number written is:
$$
\frac{ab}{N+1}
$$
We are told that the **sum** of all these $N^2$ numbers is:
... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given that Henry writes the quotient $\frac{ab}{N+1}$ for each integer pair $(a,b)$ with $1 \le a, b \le N$, and we are to find all $N$ such that the **sum** of all these numbers is:
$$
\frac{N^3 - N^2 + 2}{4}
$$
---
### **Step 1: Understanding the Sum**
Henry writes the value $\frac{ab}{N+1}$ for each pair ... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given that for each pair of integers $(a, b)$ with $1 \le a, b \le N$, Henry writes the value $\frac{ab}{N+1}$ on the board. We are to find all positive integers $N$ such that the **sum of all these numbers** is equal to:
$$
\frac{N^3 - N^2 + 2}{4}
$$
---
### Step 1: Understand the Total Sum
We are summing o... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given a problem involving a function of two integers $ a $ and $ b $, where both range from 1 to $ N $, and for each such pair, Henry writes the quotient of $ ab $ divided by $ N+1 $, i.e.,
$$
\frac{ab}{N+1}
$$
We are to find all positive integers $ N $ for which the **sum of all such numbers** written on the ... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given that Henry writes the quotient of $ ab $ divided by $ N+1 $ for each integer pair $ (a, b) $ with $ 1 \le a, b \le N $. That is, for each such pair, he writes:
$$
\frac{ab}{N+1}
$$
We are to find all positive integers $ N $ such that the **sum of all these numbers** is equal to:
$$
\frac{N^3 - N^2 + 2}{... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given the problem:
> For a given positive integer $ N $, Henry writes the quotient of $ ab $ divided by $ N+1 $ on the board for each integer pair $ (a, b) $ where $ 1 \leq a, b \leq N $. Find all $ N $ such that the sum of the $ N^2 $ numbers Henry wrote on the board is $ \frac{N^3 - N^2 + 2}{4} $.
---
### S... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given a problem involving the sum of certain expressions over all pairs $(a, b)$ where $1 \le a, b \le N$. Specifically, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each such pair. We are to find all positive integers $N$ such that the **sum of all the values written on the board** equals... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given that Henry writes the quotient of $ ab $ divided by $ N+1 $ for each pair $ (a, b) $ with $ 1 \le a, b \le N $. We are to find all positive integers $ N $ such that the **sum** of all these numbers is equal to:
$$
\frac{N^3 - N^2 + 2}{4}
$$
---
### Step 1: Understand the setup
For each pair $ (a, b) $ ... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given a problem involving a sum of quotients for all integer pairs $(a, b)$ with $1 \le a, b \le N$. The goal is to find all positive integers $N$ for which the sum of the quotients $\left\lfloor \frac{ab}{N+1} \right\rfloor$ is equal to $\frac{N^3 - N^2 + 2}{4}$.
---
### Step 1: Understanding the Problem
Hen... | false | 1 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given that Henry writes the quotient of $ ab $ divided by $ N+1 $ on the board for each integer pair $ (a,b) $ with $ 1 \le a,b \le N $. We are to find all positive integers $ N $ such that the **sum** of all these numbers is:
$$
\frac{N^3 - N^2 + 2}{4}
$$
---
### Step 1: Understanding the structure of the pr... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given that for each integer pair $(a, b)$ with $1 \le a, b \le N$, Henry writes the value of $\frac{ab}{N+1}$ on the board. We are to find all positive integers $N$ such that the **sum** of all such values is:
$$
\frac{N^3 - N^2 + 2}{4}.
$$
---
### Step 1: Understand the sum
Henry writes the number $\frac{ab... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given that Henry writes the quotient of $ ab $ divided by $ N+1 $ for each pair $ (a, b) $ with $ 1 \le a, b \le N $, and we are to find all positive integers $ N $ such that the **sum of these $ N^2 $ numbers** is equal to:
$$
\frac{N^3 - N^2 + 2}{4}
$$
---
### Step 1: Understand the expression Henry writes
... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given that for a positive integer $ N $, Henry writes the quotient $ \left\lfloor \frac{ab}{N+1} \right\rfloor $ on the board for each pair $ (a,b) $ with $ 1 \le a, b \le N $. Then the **sum** of all these numbers is equal to:
$$
\frac{N^3 - N^2 + 2}{4}
$$
We are to find all such $ N $.
---
### Step 1: Unde... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given a problem where Henry writes the quotient of $ ab $ divided by $ N+1 $ for each pair $ (a, b) $ with $ 1 \le a, b \le N $. We are to find all positive integers $ N $ such that the **sum of all these $ N^2 $ numbers** equals:
$$
\frac{N^3 - N^2 + 2}{4}
$$
---
### Step 1: Understand the expression Henry w... | false | 0 |
Hwilner/imo-answerbench | [
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \boxed{}."
},
{
"role": "user",
"content": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find a... | math | {
"index": 0,
"split": "train",
"solution": "N/A",
"question": "For a given positive integer $N$, Henry writes the quotient of $ab$ divided by $N+1$ on the board for each integer pair $(a,b)$ where $1\\le a,b\\le N$. Find all $N$ such that the sum of the $N^2$ numbers Henry wrote on the board is $\\frac{N^3-N^2... | We are given a problem involving the sum of certain quotients written on a board by Henry. Let's carefully break it down step by step.
---
### **Understanding the Problem**
Henry writes the **quotient** of $ ab $ divided by $ N + 1 $ for each pair of integers $ (a, b) $ such that $ 1 \le a, b \le N $. That is, for e... | false | 0 |
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