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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 $\frac{N^3-N^2+2}{4}$.
| 3 | imo-bench | Let $S$ be the sum of the $N^2$ numbers Henry wrote on the board. The numbers are the quotients of $ab$ divided by $N+1$ for $1 \le a,b \le N$. The sum is given by:
$$S = \sum_{a=1}^N \sum_{b=1}^N \left\lfloor \frac{ab}{N+1} \right\rfloor$$
We are given the condition $S = \frac{N^3-N^2+2}{4}$.
Let $M = N+1$. The sum c... | gemini | We are given a problem involving a sum over integer pairs $(a, b)$ with $1 \le a, b \le N$, and we are to find all positive integers $N$ such that the sum of the values
$$
\frac{ab}{N+1}
$$
over all such pairs equals
$$
\frac{N^3 - N^2 + 2}{4}.
$$
---
### Step 1: Understand the sum
Henry writes $\frac{ab}{N+1}$ f... | 0 | {
"content": "The problem says Henry writes the *quotient* of $ab$ divided by $N+1$. In mathematics, when we say 'quotient' in this context, it means the integer part — that is, $\\left\\lfloor \\frac{ab}{N+1} \\right\\rfloor$. So instead of summing $\\frac{ab}{N+1}$, we must sum $\\left\\lfloor \\frac{ab}{N+1} \\rig... | We are given a problem involving a sum over integer pairs $(a, b)$ with $1 \le a, b \le N$, and we are to find all positive integers $N$ such that the sum of the values
The problem says Henry writes the *quotient* of $ab$ divided by $N+1$. In mathematics, when we say 'quotient' in this context, it means the integer pa... | # Detailed Verification Log
- **Substep 1:** "We are given a problem involving a sum over integer pairs $(a, b)$ with $1 \le a, b \le N$, and we are to find all positive integers $N$ such that the sum of the values"
✅ Correct — sets up the problem correctly.
- **Substep 2:** "$\frac{ab}{N+1}$"
✅ Correct — ide... | <|im_start|>user
You are an expert mathematician teaching a Math Olympiad class. You will be given a problem and a high-level oracle solution to the problem. Your task is to solve the problem step-by-step guided by the high-level oracle solution.
# Problem #
For a given positive integer $N$, Henry writes the quotient ... | We are given a problem involving a sum over integer pairs $(a, b)$ with $1 \le a, b \le N$, and we are to find all positive integers $N$ such that the sum of the values
The problem says Henry writes the *quotient* of $ab$ divided by $N+1$. In mathematics, when we say 'quotient' in this context, it means the integer pa... | We are given a problem involving a sum of floor functions over integer pairs $(a, b)$, and we are to **find all positive integers $N$** such that the sum of the quotients $\left\lfloor \frac{ab}{N+1} \right\rfloor$ for $1 \le a, b \le N$ equals:
$$
\frac{N^3 - N^2 + 2}{4}
$$
We will solve this step-by-step, guided by... |
"Find all functions $g:\\mathbb{R}\\rightarrow\\mathbb{R}$ which is not a linear function and satisf(...TRUNCATED) | $g(x)=2x^{3}+c, g(x)=-2x^{3}+c$ | imo-bench | "Let $P(x,y,z)$ be the assertion\n\\[4g\\left(x^{2}y+y^{2}z+z^{2}x\\right)-(g(y)-g(x))(g(z)-g(y))(g((...TRUNCATED) | gemini | "We are given a functional equation involving a function $ g: \\mathbb{R} \\to \\mathbb{R} $, which (...TRUNCATED) | 1 | {"content":"Try a function of the form $ g(x) = 2x^3 + b $ for some constant $ b $. Plug it into the(...TRUNCATED) | "We are given a functional equation involving a function $ g: \\mathbb{R} \\to \\mathbb{R} $, which (...TRUNCATED) | "# Detailed Verification Log\n\n- **Substep 1:** \"We are given a functional equation involving a fu(...TRUNCATED) | "<|im_start|>user\nYou are an expert mathematician teaching a Math Olympiad class. You will be given(...TRUNCATED) | "We are given a functional equation involving a function $ g: \\mathbb{R} \\to \\mathbb{R} $, which (...TRUNCATED) | "We are given the functional equation:\n\n$$\n4g\\left(x^{2}y+y^{2}z+z^{2}x\\right)-(g(y)-g(x))(g(z)(...TRUNCATED) |
"Let $u \\ge 2$ be a given positive integer. Find the smallest real number $C$ such that for all rea(...TRUNCATED) | $2^{u-2}$ | imo-bench | "Let the given inequality be\n$$ \\frac{t^{2^u}+1}{2} \\le (C(t-1)^2+t)^{2^{u-1}} $$\nLet $n = 2^{u-(...TRUNCATED) | gemini | "We are given a positive integer $ u \\geq 2 $, and we are to find the **smallest real number** $ C (...TRUNCATED) | 5 | {"content":"After verifying that both sides are equal at $ t = 1 $, consider the behavior near $ t =(...TRUNCATED) | "We are given a positive integer $ u \\geq 2 $, and we are to find the **smallest real number** $ C (...TRUNCATED) | "# Detailed Verification Log\n\n- **Substep 1–3:** The student correctly identifies the problem an(...TRUNCATED) | "<|im_start|>user\nYou are an expert mathematician teaching a Math Olympiad class. You will be given(...TRUNCATED) | "We are given a positive integer $ u \\geq 2 $, and we are to find the **smallest real number** $ C (...TRUNCATED) | "We are given a problem involving a real inequality for a fixed integer $ u \\ge 2 $, and we are to (...TRUNCATED) |
"Let $P$ be a function from the set of integers to itself such that for all integers $h, m$,\n\\[P^{(...TRUNCATED) | $P(x)=-1, P(x)=x+1$ | imo-bench | "Let the given functional equation be\n\\[P^{h^2 + m^2}(h+m-1) = mP(m-1) + hP(h-1) + (h+m-1) \\quad (...TRUNCATED) | gemini | "We are given a functional equation involving a function $ P: \\mathbb{Z} \\to \\mathbb{Z} $, define(...TRUNCATED) | 7 | {"content":"Try a constant function $ P(x) = c $ for all $ x \\in \\mathbb{Z} $. Substitute into the(...TRUNCATED) | "We are given a functional equation involving a function $ P: \\mathbb{Z} \\to \\mathbb{Z} $, define(...TRUNCATED) | "# Detailed Verification Log\n\n- **Substep 1:** Correct. The student correctly identifies the funct(...TRUNCATED) | "<|im_start|>user\nYou are an expert mathematician teaching a Math Olympiad class. You will be given(...TRUNCATED) | "We are given a functional equation involving a function $ P: \\mathbb{Z} \\to \\mathbb{Z} $, define(...TRUNCATED) | "We are given a functional equation involving a function $ P: \\mathbb{Z} \\to \\mathbb{Z} $, define(...TRUNCATED) |
"Let $a_1, a_2, \\ldots, a_{2026}$ satisfy $3^i < a_i < 3^{i+1}$ for each $i$. For each $1\\le i\\le(...TRUNCATED) | 1012 | imo-bench | "Let $A_i$ be the given expression for $1 \\le i \\le 2025$. The definition of $A_i$ involves the va(...TRUNCATED) | gemini | "We are given a sequence $ a_1, a_2, \\ldots, a_{2026} $ such that for each $ i $,\n\n$$\n3^i < a_i (...TRUNCATED) | 11 | {"content":"Try computing $ A_3 $ (i.e., for $ i = 3 $, so $ x \\leq 4 $) with the same values $ a_1(...TRUNCATED) | "We are given a sequence $ a_1, a_2, \\ldots, a_{2026} $ such that for each $ i $,\n\n$$\n3^i < a_i (...TRUNCATED) | "# Detailed Verification Log\n\n- **Substep 1–2:** Correct. The problem is clearly stated with the(...TRUNCATED) | "<|im_start|>user\nYou are an expert mathematician teaching a Math Olympiad class. You will be given(...TRUNCATED) | "We are given a sequence $ a_1, a_2, \\ldots, a_{2026} $ such that for each $ i $,\n\n$$\n3^i < a_i (...TRUNCATED) | "We are given a sequence $ a_1, a_2, \\ldots, a_{2026} $ such that \n$$\n3^i < a_i < 3^{i+1} \\quad(...TRUNCATED) |
"Let $x_0, x_1, \\ldots$ be a sequence of real numbers such that $x_0 = 0$, $x_1 = 1$, and for each (...TRUNCATED) | $-\frac{2023}{2024^2}$ | imo-bench | "Let the given recurrence relation be $x_k = \\frac{1}{t_k} \\sum_{i=1}^{t_k} x_{k-i}$ for some inte(...TRUNCATED) | gemini | "We are given a sequence $ x_0, x_1, x_2, \\ldots $ of real numbers defined by:\n\n- $ x_0 = 0 $\n- (...TRUNCATED) | 14 | {"content":"Consider the possibility that $ x_{2025} $ can be **larger** than $ x_{2024} $, even tho(...TRUNCATED) | "We are given a sequence $ x_0, x_1, x_2, \\ldots $ of real numbers defined by:\n\n- $ x_0 = 0 $\n- (...TRUNCATED) | "# Detailed Verification Log\n\n- **Substep 1:** Correct. The problem is correctly stated with initi(...TRUNCATED) | "<|im_start|>user\nYou are an expert mathematician teaching a Math Olympiad class. You will be given(...TRUNCATED) | "We are given a sequence $ x_0, x_1, x_2, \\ldots $ of real numbers defined by:\n\n- $ x_0 = 0 $\n- (...TRUNCATED) | "We are given a sequence $ x_0, x_1, x_2, \\ldots $ of real numbers defined by:\n\n- $ x_0 = 0 $\n- (...TRUNCATED) |
"Find the maximal value of\n\n\\[\nS=\\sqrt[3]{\\frac{x}{y+13}}+\\sqrt[3]{\\frac{y}{z+13}}+\\sqrt[3](...TRUNCATED) | $2\sqrt[3]{\frac{196}{13}}$ | imo-bench | "Let the function be\n\\[S=\\sqrt[3]{\\frac{x}{y+13}}+\\sqrt[3]{\\frac{y}{z+13}}+\\sqrt[3]{\\frac{z}(...TRUNCATED) | gemini | "We are given the expression:\n\n$$\nS = \\sqrt[3]{\\frac{x}{y+13}} + \\sqrt[3]{\\frac{y}{z+13}} + \(...TRUNCATED) | 18 | {"content":"Before trying to maximize $ T = \\sum \\frac{x}{y+13} $, consider that the actual expres(...TRUNCATED) | "We are given the expression:\n\n$$\nS = \\sqrt[3]{\\frac{x}{y+13}} + \\sqrt[3]{\\frac{y}{z+13}} + \(...TRUNCATED) | "# Detailed Verification Log\n\n- **Substep 1:** Correct. The problem is clearly stated with the exp(...TRUNCATED) | "<|im_start|>user\nYou are an expert mathematician teaching a Math Olympiad class. You will be given(...TRUNCATED) | "We are given the expression:\n\n$$\nS = \\sqrt[3]{\\frac{x}{y+13}} + \\sqrt[3]{\\frac{y}{z+13}} + \(...TRUNCATED) | "We are given the expression:\n\n$$\nS = \\sqrt[3]{\\frac{x}{y+13}} + \\sqrt[3]{\\frac{y}{z+13}} + \(...TRUNCATED) |
"Determine all functions $Q$ from the real numbers to itself such that $Q(0)\\neq0$ and\n\n$$\\frac{(...TRUNCATED) | $Q(x)=-2, Q(x)=2x-2$ | imo-bench | "Let $P(m,n)$ be the assertion $\\frac{1}{2}Q(m+n)^{2}-Q(m)Q(n)=\\max\\left\\{ Q\\left(m^{2}+n^{2}\\(...TRUNCATED) | gemini | "We are given a functional equation involving a function $ Q: \\mathbb{R} \\to \\mathbb{R} $, with t(...TRUNCATED) | 25 | {"content":"Try to find a non-constant solution by assuming a linear form. Suppose $ Q(x) = 2x - 2 $(...TRUNCATED) | "We are given a functional equation involving a function $ Q: \\mathbb{R} \\to \\mathbb{R} $, with t(...TRUNCATED) | "# Detailed Verification Log\n\n- **Substep 1:** Correct. The problem is clearly stated.\n- **Subste(...TRUNCATED) | "<|im_start|>user\nYou are an expert mathematician teaching a Math Olympiad class. You will be given(...TRUNCATED) | "We are given a functional equation involving a function $ Q: \\mathbb{R} \\to \\mathbb{R} $, with t(...TRUNCATED) | "We are given a functional equation involving a function $ Q: \\mathbb{R} \\to \\mathbb{R} $, with t(...TRUNCATED) |
"Given a positive integer $a$, find the maximum possible value of\n$$ \\sum_{1 \\le m < n \\le 2a} ((...TRUNCATED) | $2025^2 a(a-1)$ | imo-bench | "Let $S$ be the expression to be maximized:\n$$ S = \\sum_{1 \\le m < n \\le 2a} (n-m-a)p_mp_n $$\nT(...TRUNCATED) | gemini | "We are given a positive integer $ a $, and we are to **maximize** the sum:\n\n$$\n\\sum_{1 \\le m <(...TRUNCATED) | 28 | {"content":"After identifying that symmetric assignments (like setting $ p_i = x $ for $ i \\le a $,(...TRUNCATED) | "We are given a positive integer $ a $, and we are to **maximize** the sum:\n\n$$\n\\sum_{1 \\le m <(...TRUNCATED) | "# Detailed Verification Log\n\n- **Substep 1:** Correct. The problem is clearly stated: maximize a (...TRUNCATED) | "<|im_start|>user\nYou are an expert mathematician teaching a Math Olympiad class. You will be given(...TRUNCATED) | "We are given a positive integer $ a $, and we are to **maximize** the sum:\n\n$$\n\\sum_{1 \\le m <(...TRUNCATED) | "We are given a positive integer $ a $, and we are to **maximize** the expression:\n\n$$\n\\sum_{1 \(...TRUNCATED) |
"Suppose that $g:\\mathbb{Z}\\to O$, where $O$ is the set of odd integers, satisfies\n$$g(a + g(a) +(...TRUNCATED) | 8109 | imo-bench | "Let the given functional equation be denoted by $P(a,b)$:\n$$g(a + g(a) + b)-g(a+b) = g(a-b)-g(a-g((...TRUNCATED) | gemini | "We are given a functional equation involving a function $ g: \\mathbb{Z} \\to O $, where $ O $ is t(...TRUNCATED) | 32 | {"content":"Try to define sequences based on the residue of n modulo 3. Let u_k = g(3k), v_k = g(3k+(...TRUNCATED) | "We are given a functional equation involving a function $ g: \\mathbb{Z} \\to O $, where $ O $ is t(...TRUNCATED) | "# Detailed Verification Log\n\n- **Substep 1:** Correct. The problem is correctly stated with the f(...TRUNCATED) | "<|im_start|>user\nYou are an expert mathematician teaching a Math Olympiad class. You will be given(...TRUNCATED) | "We are given a functional equation involving a function $ g: \\mathbb{Z} \\to O $, where $ O $ is t(...TRUNCATED) | "We are given a functional equation involving a function $ g: \\mathbb{Z} \\to O $, where $ O $ is t(...TRUNCATED) |
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