problem stringlengths 33 9.01k | answer stringlengths 0 417 | nohint_responses sequencelengths 8 8 | hint_responses sequencelengths 8 8 | nohint_correct sequencelengths 8 8 | hint_correct sequencelengths 8 8 | source stringclasses 7
values | difficulty float64 1 9.5 | model stringclasses 1
value | all_hints stringlengths 8 4.6k |
|---|---|---|---|---|---|---|---|---|---|
Let $ a, b \in \mathbb{N}$ with $ 1 \leq a \leq b,$ and $ M \equal{} \left[\frac {a \plus{} b}{2} \right].$ Define a function $ f: \mathbb{Z} \mapsto \mathbb{Z}$ by
\[ f(n) \equal{} \begin{cases} n \plus{} a, & \text{if } n \leq M, \\
n \minus{} b, & \text{if } n >M. \end{cases}
\]
Let $ f^1(n) \equal{} f(n),$ $ f_{i ... | \frac {a + b}{\gcd(a,b)} | [
"Let's analyze the problem step-by-step.\n\n---\n\n**Given:**\n\n- \\(a, b \\in \\mathbb{N}\\) with \\(1 \\leq a \\leq b\\).\n- \\(M = \\left\\lfloor \\frac{a+b}{2} \\right\\rfloor\\).\n- Function \\(f: \\mathbb{Z} \\to \\mathbb{Z}\\) defined by\n \\[\n f(n) = \\begin{cases}\n n + a, & \\text{if } n \\leq M, \\\... | [
"Let's analyze the problem step by step.\n\n---\n\n**Given:**\n\n- \\( a, b \\in \\mathbb{N} \\) with \\( 1 \\leq a \\leq b \\).\n- \\( M = \\left[\\frac{a+b}{2}\\right] \\) (integer part).\n- The function \\( f: \\mathbb{Z} \\to \\mathbb{Z} \\) defined as:\n \\[\n f(n) = \\begin{cases}\n n + a, & \\text{if } n ... | [
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<note>
<description>Iterative piecewise linear processes can be viewed as sequences of uniform “up” moves of size A and “down” moves of size B. After k steps, the net change equals (number of ups)·A – (number of downs)·B. Finding when the process returns to its start amounts to making this net change zero... |
We are given an infinite deck of cards, each with a real number on it. For every real number $x$, there is exactly one card in the deck that has $x$ written on it. Now two players draw disjoint sets $A$ and $B$ of $100$ cards each from this deck. We would like to define a rule that declares one of them a winner. This r... | 100 | [
"Let's analyze the problem step-by-step.\n\n---\n\n### Problem restatement:\n- There is an infinite deck of cards with one card for every real number.\n- Two players each pick a **disjoint** set of 100 cards, \\(A\\) and \\(B\\).\n- We want to define a **rule** to decide the winner between \\(A\\) and \\(B\\), sati... | [
"Let's carefully analyze the problem and the provided notes step-by-step.\n\n---\n\n### Problem Restatement\n\n- We have an infinite deck with one card for every real number.\n- Two players pick **disjoint** sets \\( A \\) and \\( B \\), each of size 100.\n- We want a rule to declare a winner between \\( A \\) and ... | [
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] | imo_shortlist | 9 | gpt-4.1-mini | <notes>
<note>
<description>Whenever two sorted lists of equal size are compared by relative order, one can encode their interleaving as a binary sequence (or “shuffle”) of length 2n with exactly n zeros and n ones. A zero at position k means the k-th smallest card belongs to list A; a one means it belongs to B.... |
"Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \\cdots$ defined by \\[ (...TRUNCATED) | M > 1 | ["Given a positive integer \\( M \\), consider the sequence \\(\\{a_k\\}\\) defined by:\n\\[\na_0 = (...TRUNCATED) | ["Given the sequence defined by:\n\\[\na_0 = M + \\frac{1}{2}, \\quad M \\in \\mathbb{Z}^+,\n\\]\nan(...TRUNCATED) | [
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"Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: the(...TRUNCATED) | n=3 | ["**Problem:**\n\nFind the smallest positive integer \\( n \\) such that there exist infinitely many(...TRUNCATED) | ["**Problem restatement:**\n\nFind the smallest positive integer \\( n \\) such that there exist inf(...TRUNCATED) | [
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] | imo_shortlist | 9.5 | gpt-4.1-mini | "<notes>\n <note>\n <description>For systems of rational equations, compare the number of indepe(...TRUNCATED) |
"For every $ n\\in\\mathbb{N}$ let $ d(n)$ denote the number of (positive) divisors of $ n$. Find al(...TRUNCATED) | f(n) = \prod_{i=1}^k p_i^{p_i^{\alpha_i} - 1} | ["Let's analyze the problem carefully.\n\n---\n\n**Problem:**\n\nFor every \\( n \\in \\mathbb{N} \\(...TRUNCATED) | ["**Problem recap:**\n\nWe want all functions \\( f: \\mathbb{N} \\to \\mathbb{N} \\) such that\n\n1(...TRUNCATED) | [
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false,
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false,
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] | imo_shortlist | 9 | gpt-4.1-mini | "<notes>\n <note>\n <description>Any positive integer’s total number of divisors is computed d(...TRUNCATED) |
"Five identical empty buckets of $2$-liter capacity stand at the vertices of a regular pentagon. Cin(...TRUNCATED) | \text{No} | ["**Problem restatement:**\n\n- There are 5 identical buckets arranged at the vertices of a regular (...TRUNCATED) | ["**Problem restatement:**\n\n- There are 5 identical buckets arranged at the vertices of a regular (...TRUNCATED) | [
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"$A$ and $B$ play a game, given an integer $N$, $A$ writes down $1$ first, then every player sees th(...TRUNCATED) | N = \text{the sum of distinct odd powers of }2 | ["Let's analyze the game carefully.\n\n---\n\n**Game Description:**\n\n- There is a fixed integer \\(...TRUNCATED) | ["**Problem restatement:**\n\nPlayers \\( A \\) and \\( B \\) play a game starting with the integer (...TRUNCATED) | [
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] | imo_shortlist | 7 | gpt-4.1-mini | "<notes>\n <note>\n <description>Identify and classify terminal states as losing positions (P-po(...TRUNCATED) |
"Find all pairs of functions $ f : \\mathbb R \\to \\mathbb R$, $g : \\mathbb R \\to \\mathbb R$ suc(...TRUNCATED) | $f(x) = t(x-t)/(t+1), g(x) = t(x-t) \text{ for any real t not equal to -1.}$ | ["Given the functional equation for all real \\( x, y \\):\n\\[\nf(x + g(y)) = x f(y) - y f(x) + g(x(...TRUNCATED) | ["Given the functional equation for all real \\( x,y \\):\n\\[\nf(x + g(y)) = x f(y) - y f(x) + g(x)(...TRUNCATED) | [
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] | imo_shortlist | 6.25 | gpt-4.1-mini | "<notes>\n <note>\n <description>Use strategic substitution of special constants (often zeros or(...TRUNCATED) |
"The following operation is allowed on a finite graph: Choose an arbitrary cycle of length 4 (if the(...TRUNCATED) | n | ["**Problem restatement:**\n\nStart with the complete graph \\( K_n \\) on \\( n \\geq 4 \\) vertice(...TRUNCATED) | ["**Problem Restatement:**\n\nStart with the complete graph \\( K_n \\) on \\( n \\geq 4 \\) vertice(...TRUNCATED) | [
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"Let $ABC$ be an acute triangle. Let $DAC,EAB$, and $FBC$ be isosceles triangles exterior to $ABC$,(...TRUNCATED) | 4 | ["**Problem restatement:** \nGiven an acute triangle \\(ABC\\), construct exterior isosceles triang(...TRUNCATED) | ["**Problem restatement:**\n\nLet \\(ABC\\) be an acute triangle. Construct three isosceles triangle(...TRUNCATED) | [
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