challenge_id stringclasses 487
values | source stringclasses 1
value | split stringclasses 2
values | theorem_name stringclasses 487
values | challenge_lean stringclasses 487
values | prover_id stringclasses 1
value | attempt int64 0 39 | stop_reason stringclasses 1
value | trace stringlengths 1.17k 47.5k | grade stringclasses 2
values | grade_detail stringclasses 3
values | curation_status stringclasses 4
values | n_valid int64 0 37 | n_invalid int64 0 39 | interval_low float64 0 0.84 | interval_high float64 0.08 1 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
aimeII_2001_p3 | minif2f | validation | aimeII_2001_p3 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 0 | eos | ### Detailed Proof and Analysis
First, we need to understand the problem. We have a sequence of integers `x(n)` defined for all natural numbers `n`, with initial values:
- `x(1) = 211`,
- `x(2) = 375`,
- `x(3) = 420`,
- `x(4) = 523`,
and for `n ≥ 5`, it satisfies the recurrence relation:
`x(n) = x(n - 1) - x(n - 2) + ... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aimeII_2001_p3 | minif2f | validation | aimeII_2001_p3 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 1 | eos | ### Detailed Proof and Analysis
First, we need to determine the values of `x 5`, `x 6`, ..., `x 975` using the recurrence relation:
\[ x_n = x_{n-1} - x_{n-2} + x_{n-3} - x_{n-4} \]
for \( n \geq 5 \). The initial conditions are:
1. \( x_1 = 211 \)
2. \( x_2 = 375 \)
3. \( x_3 = 420 \)
4. \( x_4 = 523 \)
#### Step 1:... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aimeII_2001_p3 | minif2f | validation | aimeII_2001_p3 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 2 | eos | ### Detailed Proof and Analysis
First, we observe that the problem involves a recurrence relation for a sequence of integers indexed by natural numbers. The recurrence is not straightforward, as it's non-linear and depends on the previous four terms. The initial terms are given, and we need to compute \(x_{531} + x_{7... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aimeII_2020_p6 | minif2f | validation | aimeII_2020_p6 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 0 | eos | ### Detailed Proof and Analysis
First, we need to understand the problem. We have a sequence \( t_n \) defined by:
- \( t_1 = 20 \)
- \( t_2 = 21 \)
- For \( n \geq 3 \), \( t_n = \frac{5 \cdot t_{n-1} + 1}{25 \cdot t_{n-2}} \)
We are to compute \( \text{den}(t_{2020}) + \text{num}(t_{2020}) \) where:
1. \( \text{den... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aimeII_2020_p6 | minif2f | validation | aimeII_2020_p6 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 1 | eos | ### Detailed Proof and Analysis
First, let's understand the problem and the sequence `t(n)`:
1. **Initial Terms**:
- `t(1) = 20`
- `t(2) = 21`
2. **Recurrence Relation for `n ≥ 3`**:
- `t(n) = (5 * t(n - 1) + 1) / (25 * t(n - 2))`
Our goal is to compute `t(2020)` and show that its numerator and denominator... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aimeII_2020_p6 | minif2f | validation | aimeII_2020_p6 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 2 | eos | ### Detailed Proof and Analysis
First, we observe the sequence defined by:
- \( t_1 = 20 \),
- \( t_2 = 21 \),
- For \( n \geq 3 \), \( t_n = \frac{5 \cdot t_{n-1} + 1}{25 \cdot t_{n-2}} \).
We need to compute the denominator and numerator of \( t_{2020} \) (in reduced form) and show that \( \text{den} + \text{num} =... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aimeI_2000_p7 | minif2f | validation | aimeI_2000_p7 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 0 | eos | ### Detailed Proof and Analysis
First, we need to understand the problem and the given conditions. We have three positive real numbers \( x, y, z \) such that:
1. \( x \cdot y \cdot z = 1 \),
2. \( x + \frac{1}{z} = 5 \),
3. \( y + \frac{1}{x} = 29 \),
4. \( z + \frac{1}{y} = m \) (where \( m \) is a rational number w... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aimeI_2000_p7 | minif2f | validation | aimeI_2000_p7 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 1 | eos | ### Detailed Proof and Analysis
First, let's carefully restate and understand the problem.
**Problem Analysis:**
We have positive real numbers \( x, y, z \) (this is given by \( 0 < x, 0 < y, 0 < z \)) such that:
1. \( x \cdot y \cdot z = 1 \),
2. \( x + \frac{1}{z} = 5 \),
3. \( y + \frac{1}{x} = 29 \),
4. \( z + \f... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aimeI_2000_p7 | minif2f | validation | aimeI_2000_p7 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 2 | eos | ### Detailed Proof and Analysis
#### Understanding the Problem
We have three positive real numbers \( x, y, z \) such that:
1. \( x \cdot y \cdot z = 1 \),
2. \( x + \frac{1}{z} = 5 \),
3. \( y + \frac{1}{x} = 29 \),
4. \( z + \frac{1}{y} = m \), where \( m \) is a positive rational number,
and we need to show that \(... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1983_p1 | minif2f | test | aime_1983_p1 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 0 | eos | ### Detailed Proof and Analysis
First, we need to understand the problem and the given conditions. We have four positive integers `x, y, z, w` with `x, y, z > 1` and `w ≥ 0`. The logarithms of `w` with respect to `x`, `y`, and `x * y * z` are given as:
1. `log w / log x = 24`,
2. `log w / log y = 40`, and
3. `log w / ... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1983_p1 | minif2f | test | aime_1983_p1 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 1 | eos | ### Detailed Proof and Analysis
First, let's recall the properties of logarithms:
1. If \( a, b > 1 \) and \( w > 0 \), then \( \log_a w \) and \( \log_b w \) are related by:
\[
\log_a w = \frac{\log_e w}{\log_e a}, \quad \log_b w = \frac{\log_e w}{\log_e b}
\]
where \( \log_e \) is the natural logarithm.
... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1983_p1 | minif2f | test | aime_1983_p1 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 2 | eos | ### Detailed Proof and Analysis
#### Understanding the Problem
We have natural numbers \( x, y, z > 1 \) and a non-negative real number \( w \geq 0 \). The conditions involving logarithms are:
1. \( \frac{\ln w}{\ln x} = 24 \)
2. \( \frac{\ln w}{\ln y} = 40 \)
3. \( \frac{\ln w}{\ln (x \cdot y \cdot z)} = 12 \)
We ne... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1983_p2 | minif2f | test | aime_1983_p2 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 0 | eos | ### Detailed Proof and Analysis
First, we need to understand the problem. We have real numbers \( x \) and \( p \), where \( 0 < p < 15 \), \( p \leq x \leq 15 \), and a function \( f \) defined by \( f(x) = |x - p| + |x - 15| + |x - p - 15| \). We need to prove that \( f(x) \geq 15 \), i.e., \( |x - p| + |x - 15| + |... | valid | comparator accepted: statement unmodified, kernel-checked | single_class | 3 | 0 | 0.804828 | 1 |
aime_1983_p2 | minif2f | test | aime_1983_p2 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 1 | eos | ### Detailed Proof and Analysis
First, let's understand the problem and the goal:
**Given:**
1. \( 0 < p < 15 \)
2. \( p \leq x \leq 15 \)
3. \( f(x) = |x - p| + |x - 15| + |x - p - 15| \)
**Prove:** \( f(x) \geq 15 \).
#### Step 1: Understand the expression for \( f(x) \)
Since \( f(x) = |x - p| + |x - 15| + |x - ... | valid | comparator accepted: statement unmodified, kernel-checked | single_class | 3 | 0 | 0.804828 | 1 |
aime_1983_p2 | minif2f | test | aime_1983_p2 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 2 | eos | ### Detailed Proof and Analysis
**Problem Analysis:**
We need to prove that \( f(x) \geq 15 \) under the given conditions:
1. \( 0 < p < 15 \),
2. \( p \leq x \leq 15 \),
3. \( f(x) = |x - p| + |x - 15| + |x - p - 15| \).
**Key Observations:**
1. The absolute value expressions \( |x - p| \), \( |x - 15| \), and \( |x... | valid | comparator accepted: statement unmodified, kernel-checked | single_class | 3 | 0 | 0.804828 | 1 |
aime_1983_p3 | minif2f | test | aime_1983_p3 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 0 | eos | ### Detailed Proof and Analysis
First, we need to understand the problem. We have a function `f : ℝ → ℝ` defined as:
\[ f(x) = x^2 + 18x + 30 - 2\sqrt{x^2 + 18x + 45} \]
We are given that the preimage of `0` under `f`, i.e., `f⁻¹(0)`, is a finite set (this is `h₁ : Fintype (f⁻¹' {0})`). We need to show that the produc... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1983_p3 | minif2f | test | aime_1983_p3 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 1 | eos | ### Detailed Proof and Analysis
First, let's understand the problem and the function involved.
#### Problem Analysis
We are given a function `f : ℝ → ℝ` defined by:
\[ f(x) = x^2 + (18x + 30) - 2\sqrt{x^2 + (18x + 45)} \]
and we need to find the product of all real numbers `x` such that `f(x) = 0` (i.e., all `x` in t... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1983_p3 | minif2f | test | aime_1983_p3 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 2 | eos | ### Detailed Proof and Analysis
#### Understanding the Problem
We have a function `f : ℝ → ℝ` defined by:
\[ f(x) = x^2 + (18x + 30) - 2 \sqrt{x^2 + (18x + 45)} \]
We need to show that the product of all `x` such that `f(x) = 0` is `20`.
#### Simplifying the Function `f(x)`
First, let's simplify the expression inside... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1983_p9 | minif2f | validation | aime_1983_p9 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 0 | eos | ### Detailed Proof and Analysis
First, we need to prove that for all \( x \in (0, \pi) \),
\[
12 \leq \frac{9x^2 \sin^2 x + 4}{x \sin x}.
\]
This is equivalent to proving that
\[
12 x \sin x \leq 9x^2 \sin^2 x + 4,
\]
assuming \( x \sin x > 0 \) (since \( x \in (0, \pi) \) and \( \sin x > 0 \) because \( x \in (0, \pi... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1983_p9 | minif2f | validation | aime_1983_p9 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 1 | eos | ### Detailed Proof and Analysis
First, we need to prove that for all real numbers \( x \) in the open interval \( (0, \pi) \), the inequality \( 12 \leq \frac{9x^2 \sin^2 x + 4}{x \sin x} \) holds.
#### Step 1: Simplify the Inequality
The inequality can be rewritten as:
\[ 12x \sin x \leq 9x^2 \sin^2 x + 4 \]
This is... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1983_p9 | minif2f | validation | aime_1983_p9 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 2 | eos | ### Detailed Proof and Analysis
#### 1. Understanding the Problem
We need to prove that for all \( x \in (0, \pi) \), the inequality:
\[ 12 \leq \frac{9x^2 \sin^2 x + 4}{x \sin x} \]
holds.
#### 2. Rewriting the Inequality
First, multiply both sides by the denominator (since \( x \sin x > 0 \) in \( (0, \pi) \)):
\[... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1984_p1 | minif2f | test | aime_1984_p1 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 0 | eos | ### Detailed Proof and Analysis
First, we need to understand the problem. We have a sequence `u : ℕ → ℚ` with the property that `u(n + 1) = u(n) + 1` for all `n` (i.e., the sequence is an arithmetic progression with a common difference of 1). Additionally, the sum of the first 98 terms of `u` (from `n = 1` to `n = 98`... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1984_p1 | minif2f | test | aime_1984_p1 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 1 | eos | ### Detailed Proof and Analysis
First, let's understand the problem and the given conditions.
1. **Problem Statement**:
- We have a sequence `u : ℕ → ℚ` where each term satisfies `u(n + 1) = u(n) + 1` for all `n`. This means that the sequence is an arithmetic progression with a common difference of `1`.
- The s... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1984_p1 | minif2f | test | aime_1984_p1 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 2 | eos | ### Detailed Proof and Analysis
#### Understanding the Problem
We have a sequence `u : ℕ → ℚ` defined by a recurrence relation:
1. For all `n ∈ ℕ`, `u(n + 1) = u(n) + 1`.
This is an arithmetic sequence where the difference between consecutive terms is `1`.
We are given:
2. `∑_{k=0}^{97} u(k + 1) = 137`.
But notice t... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1984_p15 | minif2f | validation | aime_1984_p15 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 0 | eos | ### Detailed Proof and Analysis
First, we need to understand the problem. We have four real numbers `x`, `y`, `z`, `w` and four equations, each involving a sum of four fractions. The denominators in each fraction are differences of squares, and the numerators are squares of the variables.
The goal is to prove that `x... | valid | comparator accepted: statement unmodified, kernel-checked | single_class | 3 | 0 | 0.804828 | 1 |
aime_1984_p15 | minif2f | validation | aime_1984_p15 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 1 | eos | ### Detailed Proof and Analysis
First, let's simplify the denominators in each equation:
1. **First equation:**
- `2² - 1 = 4 - 1 = 3`
- `2² - 3² = 4 - 9 = -5`
- `2² - 5² = 4 - 25 = -21`
- `2² - 7² = 4 - 49 = -45`
- The equation becomes: `x² / 3 + y² / (-5) + z² / (-21) + w² / (-45) = 1`.
However, ... | valid | comparator accepted: statement unmodified, kernel-checked | single_class | 3 | 0 | 0.804828 | 1 |
aime_1984_p15 | minif2f | validation | aime_1984_p15 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 2 | eos | ### Detailed Proof and Analysis
First, let's carefully interpret and simplify the given problem. The problem involves four real numbers \(x, y, z, w\) and four equations. Each equation is of the form:
\[
\frac{x^2}{a^2 - 1} + \frac{y^2}{a^2 - 3^2} + \frac{z^2}{a^2 - 5^2} + \frac{w^2}{a^2 - 7^2} = 1,
\]
for \(a = 2, 4,... | valid | comparator accepted: statement unmodified, kernel-checked | single_class | 3 | 0 | 0.804828 | 1 |
aime_1984_p5 | minif2f | validation | aime_1984_p5 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 0 | eos | ### Detailed Proof and Analysis
First, recall the definition of `logb`:
\[ \text{logb}_b x = \frac{\ln x}{\ln b}. \]
Thus, the given equations can be rewritten as:
1. \[ \frac{\ln a}{\ln 8} + \frac{\ln (b^2)}{\ln 4} = 5 \]
2. \[ \frac{\ln b}{\ln 8} + \frac{\ln (a^2)}{\ln 4} = 7 \]
Simplify the denominators using log... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1984_p5 | minif2f | validation | aime_1984_p5 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 1 | eos | ### Detailed Proof and Analysis
First, recall that `Real.logb b x` is defined as `Real.log x / Real.log b`. The given equations can be rewritten as:
1. `Real.log a / Real.log 8 + Real.log (b²) / Real.log 4 = 5`
2. `Real.log b / Real.log 8 + Real.log (a²) / Real.log 4 = 7`
We can simplify the denominators `Real.log 8... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1984_p5 | minif2f | validation | aime_1984_p5 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 2 | eos | ### Detailed Proof and Analysis
#### Understanding the Problem
We have two equations involving logarithms with different bases:
1. `log₈ a + log₄ (b²) = 5`
2. `log₈ b + log₄ (a²) = 7`
We need to prove that `a * b = 512`, where `a` and `b` are positive integers (or natural numbers).
#### Preliminary Notes
1. **Logari... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1984_p7 | minif2f | test | aime_1984_p7 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 0 | eos | ### Detailed Proof and Analysis
First, we need to understand the problem and the given conditions:
1. **Definitions**:
- For all integers `n ≥ 1000`, `f(n) = n - 3`.
- For all integers `n < 1000`, `f(n) = f(f(n + 5))` (a recursive condition).
2. **Goal**: Prove that `f(84) = 997`.
#### Observations:
- The rec... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1984_p7 | minif2f | test | aime_1984_p7 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 1 | eos | ### Detailed Proof and Analysis
First, let's understand the problem and the given conditions:
1. **Function Definition for \( n \geq 1000 \)**:
- For all integers \( n \) such that \( n \geq 1000 \), \( f(n) = n - 3 \).
2. **Recurrence Relation for \( n < 1000 \)**:
- For all integers \( n \) such that \( n < ... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1984_p7 | minif2f | test | aime_1984_p7 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 2 | eos | ### Detailed Proof and Analysis
#### Understanding the Problem
We have a function `f : ℤ → ℤ` defined for integers `n`:
1. For `n ≥ 1000`, `f(n) = n - 3`.
2. For `n < 1000`, the function satisfies the recursive-like condition `f(n) = f(f(n + 5))` (assuming the recursion terminates after finitely many steps).
We are to... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1987_p5 | minif2f | test | aime_1987_p5 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 0 | eos | ### Detailed Proof and Analysis
First, we need to solve for `y² + 3 * (x² * y²) = 30 * x² + 517` with the goal of showing that `3 * (x² * y²) = 588`.
1. **Rearrange the equation**:
\[
y^2 + 3x^2 y^2 = 30x^2 + 517 \\
3x^2 y^2 + y^2 - 30x^2 - 517 = 0
\]
Alternatively, we can factor the left-hand side:
... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1987_p5 | minif2f | test | aime_1987_p5 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 1 | eos | ### Detailed Proof and Analysis
First, let's recall the problem and the goal:
Given integers \( x \) and \( y \) such that \( y^2 + 3x^2 y^2 = 30x^2 + 517 \), we need to prove that \( 3x^2 y^2 = 588 \).
#### Observations:
1. Notice that \( y^2 (1 + 3x^2) = 30x^2 + 517 \). This can be rewritten as \( y^2 (1 + 3x^2) =... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1987_p5 | minif2f | test | aime_1987_p5 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 2 | eos | ### Detailed Proof and Analysis
#### 1. Understand the Problem
We are given two integers `x` and `y` satisfying the equation:
\[ y^2 + 3x^2 y^2 = 30x^2 + 517. \]
We need to prove that `3x^2 y^2 = 588`.
#### 2. Rewrite the Equation
First, factor the left-hand side (LHS):
\[ LHS = y^2 + 3x^2 y^2 = y^2 (1 + 3x^2) = 30x^... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1987_p8 | minif2f | validation | aime_1987_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 0 | eos | ### Detailed Proof and Analysis
First, we need to understand the problem. We are to prove that `112` is the greatest element in the set `{n : ℕ | 0 < n ∧ ∃! k : ℕ, (8 : ℝ) / 15 < n / (n + k) ∧ (n : ℝ) / (n + k) < 7 / 13}`. In other words:
1. `112` is in the set.
2. If `n` is in the set, then `n ≤ 112`.
#### Step 1: P... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1987_p8 | minif2f | validation | aime_1987_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 1 | eos | ### Detailed Proof and Analysis
First, we need to understand the problem and rephrase it correctly. The set in question is:
\[ S = \{ n \in \mathbb{N} \mid 0 < n \land \exists! k \in \mathbb{N}, \frac{8}{15} < \frac{n}{n + k} \land \frac{n}{n + k} < \frac{7}{13} \} \]
We are to show that \( 112 \) is the greatest elem... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1987_p8 | minif2f | validation | aime_1987_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 2 | eos | ### Detailed Proof and Analysis
#### Understanding the Problem
We need to prove that `112` is the greatest element in the set:
\[ S = \{ n \in \mathbb{N} \mid 0 < n \land \exists! k \in \mathbb{N}, \frac{8}{15} < \frac{n}{n + k} \land \frac{n}{n + k} < \frac{7}{13} \}. \]
In other words, `112` is the largest natural n... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1988_p3 | minif2f | validation | aime_1988_p3 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 0 | eos | ### Detailed Proof and Analysis
First, recall the definition of `logb`:
\[ \text{logb}_b x = \frac{\ln x}{\ln b}. \]
Thus, the equation can be rewritten as:
\[ \log_2 (\log_8 x) = \log_8 (\log_2 x). \]
We must solve this for \(x > 0\).
#### Step 1: Simplify the Logarithms
The base \(8\) in the logarithms can be re... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1988_p3 | minif2f | validation | aime_1988_p3 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 1 | eos | ### Detailed Proof and Analysis
First, recall that `Real.logb b x` is defined as `Real.log x / Real.log b`. The given equation is:
\[ \log_2 (\log_8 x) = \log_8 (\log_2 x) \]
#### Key Observations and Steps:
1. **Simplify Logarithmic Bases**:
- We can use the change of base formula to rewrite everything in terms... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1988_p3 | minif2f | validation | aime_1988_p3 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 2 | eos | ### Detailed Proof and Analysis
#### Understanding the Problem
We need to prove that if \( x > 0 \) and \(\log_2 (\log_8 x) = \log_8 (\log_2 x)\), then \((\log_2 x)^2 = 27\).
#### Rewriting the Logarithms
First, recall the identity \(\log_b a = \frac{\ln a}{\ln b}\). Thus, we can rewrite all the logarithms in terms o... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1988_p4 | minif2f | validation | aime_1988_p4 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 0 | eos | ### Detailed Proof and Analysis
First, we need to understand the problem and the given conditions.
**Problem Statement:**
For a natural number `n` and a sequence `a : ℕ → ℝ`, such that for every `n`, `|a n| < 1`, and the sum of the absolute values of the first `n` terms is `19 + |∑ k in Finset.range n, a k|`, prove t... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1988_p4 | minif2f | validation | aime_1988_p4 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 1 | eos | ### Detailed Proof and Analysis
First, let's carefully restate and understand the problem.
**Problem Analysis:**
We have a sequence of real numbers `a_k` (indexed by `k ∈ ℕ`) such that for every `n ∈ ℕ`, `|a_n| < 1`. We are given that:
\[ \sum_{k=0}^{n-1} |a_k| = 19 + \left| \sum_{k=0}^{n-1} a_k \right|. \]
Here, the... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1988_p4 | minif2f | validation | aime_1988_p4 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 2 | eos | ### Detailed Proof and Analysis
#### Understanding the Problem
We have a sequence of real numbers `a : ℕ → ℝ` such that for every `n ∈ ℕ`, `|a n| < 1`. We are given that the sum of the absolute values of the first `n` terms of the sequence is equal to `19 + |sum of the first `n` terms of the sequence|`, i.e.,
`∑ k i... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1988_p8 | minif2f | test | aime_1988_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 0 | eos | ### Detailed Proof and Analysis
First, we need to understand the problem. We have a function `f : ℕ → ℕ → ℝ` with the following properties:
1. For any positive integer `x`, `f x x = x` (this includes `f 0 0` but is vacuously applied since `x` is a positive integer).
2. For any positive integers `x` and `y`, `f x y = f... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1988_p8 | minif2f | test | aime_1988_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 1 | eos | ### Detailed Proof and Analysis
First, let's carefully analyze the problem and understand the hypotheses and the goal.
**Problem Analysis:**
We have a function `f : ℕ → ℕ → ℝ` with the following properties:
1. For all `x > 0`, `f(x, x) = x` (this is a very strong condition, but we can exploit it).
2. For all `x, y > ... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1988_p8 | minif2f | test | aime_1988_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 2 | eos | ### Detailed Proof and Analysis
#### Understanding the Problem
We have a function `f : ℕ → ℕ → ℝ` with the following properties:
1. For all positive integers `x`, `f(x, x) = x`.
2. For all positive integers `x, y`, `f(x, y) = f(y, x)`.
3. For all positive integers `x, y`, `(x + y) * f(x, y) = y * f(x, x + y)`.
We nee... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 0 | eos | ### Detailed Proof and Analysis
First, we have the following system of linear equations:
1. \( a + 4b + 9c + 16d + 25e + 36f + 49g = 1 \)
2. \( 4a + 9b + 16c + 25d + 36e + 49f + 64g = 12 \)
3. \( 9a + 16b + 25c + 36d + 49e + 64f + 81g = 123 \)
We need to find the value of \( 16a + 25b + 36c + 49d + 64e + 81f + 100g \)... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 1 | eos | ### Detailed Proof and Analysis
First, let's understand the problem and set up the variables correctly. We have a system of linear equations:
1. \( a + 4b + 9c + 16d + 25e + 36f + 49g = 1 \)
2. \( 4a + 9b + 16c + 25d + 36e + 49f + 64g = 12 \)
3. \( 9a + 16b + 25c + 36d + 49e + 64f + 81g = 123 \)
We need to find \( 1... | invalid | lean did not elaborate cleanly (status=error) | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 10 | eos | ### Detailed Proof and Analysis
First, let's understand the problem:
We have a system of linear equations:
1. \( a + 4b + 9c + 16d + 25e + 36f + 49g = 1 \)
2. \( 4a + 9b + 16c + 25d + 36e + 49f + 64g = 12 \)
3. \( 9a + 16b + 25c + 36d + 49e + 64f + 81g = 123 \)
We need to find the value of \( 16a + 25b + 36c + 49d + 6... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 11 | eos | ### Detailed Proof and Analysis
First, let's observe the system of equations:
1. \( a + 4b + 9c + 16d + 25e + 36f + 49g = 1 \)
2. \( 4a + 9b + 16c + 25d + 36e + 49f + 64g = 12 \)
3. \( 9a + 16b + 25c + 36d + 49e + 64f + 81g = 123 \)
We need to find the value of \( 16a + 25b + 36c + 49d + 64e + 81f + 100g \).
#### St... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 12 | eos | ### Detailed Proof and Analysis
First, we need to find a relationship between the given equations to find `16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g`. The key observation is that the coefficients in each equation can be seen as a quadratic sequence. Specifically, they are squares of consecutive int... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 13 | eos | ### Detailed Proof and Analysis
**Problem Analysis:**
We have a system of linear equations with real variables `a`, `b`, `c`, `d`, `e`, `f`, `g`:
1. `a + 4b + 9c + 16d + 25e + 36f + 49g = 1` (`h₀`)
2. `4a + 9b + 16c + 25d + 36e + 49f + 64g = 12` (`h₁`)
3. `9a + 16b + 25c + 36d + 49e + 64f + 81g = 123` (`h₂`)
We need ... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 14 | eos | ### Detailed Proof and Analysis
We are given a system of linear equations in real numbers \(a, b, c, d, e, f, g\):
1. \(a + 4b + 9c + 16d + 25e + 36f + 49g = 1\),
2. \(4a + 9b + 16c + 25d + 36e + 49f + 64g = 12\),
3. \(9a + 16b + 25c + 36d + 49e + 64f + 81g = 123\),
and we need to find the value of the linear combin... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 15 | eos | ### Detailed Proof
**Problem Analysis:**
We are given three linear equations in seven variables `a`, `b`, `c`, `d`, `e`, `f`, `g` with integer coefficients. We need to find the value of `16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g` under the constraints:
1. `a + 4 * b + 9 * c + 16 * d + 25 * e + 36 *... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 16 | eos | ### Detailed Proof and Analysis
First, we are given three linear equations in seven variables `a`, `b`, `c`, `d`, `e`, `f`, `g`:
1. \( a + 4b + 9c + 16d + 25e + 36f + 49g = 1 \) (1)
2. \( 4a + 9b + 16c + 25d + 36e + 49f + 64g = 12 \) (2)
3. \( 9a + 16b + 25c + 36d + 49e + 64f + 81g = 123 \) (3)
We need to find th... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 17 | eos | ### Detailed Proof and Analysis
First, we are given three linear equations in real numbers \( a, b, c, d, e, f, g \):
1. \( a + 4b + 9c + 16d + 25e + 36f + 49g = 1 \) (1)
2. \( 4a + 9b + 16c + 25d + 36e + 49f + 64g = 12 \) (2)
3. \( 9a + 16b + 25c + 36d + 49e + 64f + 81g = 123 \) (3)
We need to prove that:
\[ 16a... | invalid | lean did not elaborate cleanly (status=error) | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 18 | eos | ### Detailed Proof
**Problem Analysis:**
We have a system of linear equations in real numbers:
1. \( a + 4b + 9c + 16d + 25e + 36f + 49g = 1 \)
2. \( 4a + 9b + 16c + 25d + 36e + 49f + 64g = 12 \)
3. \( 9a + 16b + 25c + 36d + 49e + 64f + 81g = 123 \)
We need to find the value of \( 16a + 25b + 36c + 49d + 64e + 81f + ... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 19 | eos | ### Detailed Proof
#### Problem Analysis
We are given a system of linear equations:
1. \( a + 4b + 9c + 16d + 25e + 36f + 49g = 1 \)
2. \( 4a + 9b + 16c + 25d + 36e + 49f + 64g = 12 \)
3. \( 9a + 16b + 25c + 36d + 49e + 64f + 81g = 123 \)
We need to find the value of \( 16a + 25b + 36c + 49d + 64e + 81f + 100g ... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 2 | eos | ### Detailed Proof and Analysis
We have a system of linear equations:
1. \( a + 4b + 9c + 16d + 25e + 36f + 49g = 1 \) (Equation 1)
2. \( 4a + 9b + 16c + 25d + 36e + 49f + 64g = 12 \) (Equation 2)
3. \( 9a + 16b + 25c + 36d + 49e + 64f + 81g = 123 \) (Equation 3)
We need to find the value of:
\[ 16a + 25b + 36c + 49... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 20 | eos | ### Detailed Proof and Analysis
First, observe the pattern in the coefficients:
1. For each variable, the coefficient is \( k^2 \) where \( k \) is its position (starting from 1).
2. The variables \( a, b, c, d, e, f, g \) correspond to the positions \( k = 1, 2, 3, 4, 5, 6, 7 \).
3. The given equations are of the for... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 21 | eos | ### Detailed Proof
**Problem Analysis:**
We have a system of linear equations with real coefficients. Our goal is to find the value of \( 16a + 25b + 36c + 49d + 64e + 81f + 100g \) given the three equations:
1. \( a + 4b + 9c + 16d + 25e + 36f + 49g = 1 \) (Equation 1),
2. \( 4a + 9b + 16c + 25d + 36e + 49f + 64g = 1... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 22 | eos | ### Detailed Proof and Analysis
First, let's understand the problem. We have a system of linear equations in the real numbers \(a, b, c, d, e, f, g\):
1. \( a + 4b + 9c + 16d + 25e + 36f + 49g = 1 \)
2. \( 4a + 9b + 16c + 25d + 36e + 49f + 64g = 12 \)
3. \( 9a + 16b + 25c + 36d + 49e + 64f + 81g = 123 \)
We need to ... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 23 | eos | ### Detailed Proof
We are given three linear equations in the variables \(a, b, c, d, e, f, g\):
1. \(a + 4b + 9c + 16d + 25e + 36f + 49g = 1\),
2. \(4a + 9b + 16c + 25d + 36e + 49f + 64g = 12\),
3. \(9a + 16b + 25c + 36d + 49e + 64f + 81g = 123\).
We need to find the value of \(16a + 25b + 36c + 49d + 64e + 81f + 10... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 24 | eos | ### Detailed Proof and Analysis
**Problem:** Given real numbers `a, b, c, d, e, f, g` satisfying the three equations:
1. `a + 4b + 9c + 16d + 25e + 36f + 49g = 1`,
2. `4a + 9b + 16c + 25d + 36e + 49f + 64g = 12`,
3. `9a + 16b + 25c + 36d + 49e + 64f + 81g = 123`,
prove that:
`16a + 25b + 36c + 49d + 64e + 81f + 100g ... | invalid | lean did not elaborate cleanly (status=error) | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 25 | eos | ### Detailed Proof and Analysis
First, let's understand the problem:
We have three linear equations in seven variables:
1. \( a + 4b + 9c + 16d + 25e + 36f + 49g = 1 \) (1)
2. \( 4a + 9b + 16c + 25d + 36e + 49f + 64g = 12 \) (2)
3. \( 9a + 16b + 25c + 36d + 49e + 64f + 81g = 123 \) (3)
We need to find the value of \(... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 26 | eos | ### Detailed Proof and Analysis
First, let's carefully analyze the given system of linear equations:
1. \( a + 4b + 9c + 16d + 25e + 36f + 49g = 1 \) (1)
2. \( 4a + 9b + 16c + 25d + 36e + 49f + 64g = 12 \) (2)
3. \( 9a + 16b + 25c + 36d + 49e + 64f + 81g = 123 \) (3)
4. The goal is to find \( 16a + 25b + 36c + 49d... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 27 | eos | ### Detailed Proof and Analysis
**Problem Analysis:**
We have a system of three linear equations in seven variables \(a, b, c, d, e, f, g\):
1. \( a + 4b + 9c + 16d + 25e + 36f + 49g = 1 \) (1)
2. \( 4a + 9b + 16c + 25d + 36e + 49f + 64g = 12 \) (2)
3. \( 9a + 16b + 25c + 36d + 49e + 64f + 81g = 123 \) (3)
We need... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 28 | eos | ### Detailed Proof and Analysis
#### Problem Analysis
We are given a system of linear equations in the variables \(a, b, c, d, e, f, g\):
1. \(a + 4b + 9c + 16d + 25e + 36f + 49g = 1\),
2. \(4a + 9b + 16c + 25d + 36e + 49f + 64g = 12\),
3. \(9a + 16b + 25c + 36d + 49e + 64f + 81g = 123\).
We need to prove that:
\[ 16... | invalid | lean did not elaborate cleanly (status=error) | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 29 | eos | ### Detailed Proof and Analysis
First, let's understand the problem. We have three linear equations in real numbers, each representing a weighted sum of \( a, b, c, d, e, f, g \) with coefficients that are perfect squares:
1. \( a + 4b + 9c + 16d + 25e + 36f + 49g = 1 \) (h₀)
2. \( 4a + 9b + 16c + 25d + 36e + 49f + ... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 3 | eos | ### Detailed Proof and Analysis
We have a system of linear equations:
1. \( a + 4b + 9c + 16d + 25e + 36f + 49g = 1 \)
2. \( 4a + 9b + 16c + 25d + 36e + 49f + 64g = 12 \)
3. \( 9a + 16b + 25c + 36d + 49e + 64f + 81g = 123 \)
We need to find the value of \( 16a + 25b + 36c + 49d + 64e + 81f + 100g \).
First, ob... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 30 | eos | ### Detailed Proof and Analysis
First, we are given a system of three linear equations in real numbers:
1. \( a + 4b + 9c + 16d + 25e + 36f + 49g = 1 \),
2. \( 4a + 9b + 16c + 25d + 36e + 49f + 64g = 12 \),
3. \( 9a + 16b + 25c + 36d + 49e + 64f + 81g = 123 \).
We need to find \( 16a + 25b + 36c + 49d + 64e + 81f + 1... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 31 | eos | ### Detailed Proof and Analysis
We are given three linear equations with real variables `a, b, c, d, e, f, g`:
1. `a + 4b + 9c + 16d + 25e + 36f + 49g = 1` (Equation 1)
2. `4a + 9b + 16c + 25d + 36e + 49f + 64g = 12` (Equation 2)
3. `9a + 16b + 25c + 36d + 49e + 64f + 81g = 123` (Equation 3)
We need to find the value... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 32 | eos | ### Detailed Proof and Analysis
First, let's recall the problem:
We have real numbers \( a, b, c, d, e, f, g \) satisfying the following three equations:
1. \( a + 4b + 9c + 16d + 25e + 36f + 49g = 1 \),
2. \( 4a + 9b + 16c + 25d + 36e + 49f + 64g = 12 \),
3. \( 9a + 16b + 25c + 36d + 49e + 64f + 81g = 123 \).
We nee... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 33 | eos | ### Detailed Proof and Analysis
We are given a system of linear equations in variables \(a, b, c, d, e, f, g\):
1. \(a + 4b + 9c + 16d + 25e + 36f + 49g = 1\)
2. \(4a + 9b + 16c + 25d + 36e + 49f + 64g = 12\)
3. \(9a + 16b + 25c + 36d + 49e + 64f + 81g = 123\)
We need to find the value of \(16a + 25b + 36c + 49d + 64... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 34 | eos | ### Detailed Proof and Analysis
We are given three linear equations with seven variables \(a, b, c, d, e, f, g\):
1. \(a + 4b + 9c + 16d + 25e + 36f + 49g = 1\)
2. \(4a + 9b + 16c + 25d + 36e + 49f + 64g = 12\)
3. \(9a + 16b + 25c + 36d + 49e + 64f + 81g = 123\)
We need to find the value of:
\[ 16a + 25b + 36c + 49d ... | invalid | lean did not elaborate cleanly (status=error) | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 35 | eos | ### Detailed Proof
**Problem Analysis:**
We have a system of three linear equations in seven variables:
1. \( a + 4b + 9c + 16d + 25e + 36f + 49g = 1 \) (1)
2. \( 4a + 9b + 16c + 25d + 36e + 49f + 64g = 12 \) (2)
3. \( 9a + 16b + 25c + 36d + 49e + 64f + 81g = 123 \) (3)
We need to find the value of \( 16a ... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 36 | eos | ### Detailed Proof and Analysis
We are given a system of linear equations in the variables \(a, b, c, d, e, f, g\) with real coefficients:
1. \( a + 4b + 9c + 16d + 25e + 36f + 49g = 1 \) (Equation 1)
2. \( 4a + 9b + 16c + 25d + 36e + 49f + 64g = 12 \) (Equation 2)
3. \( 9a + 16b + 25c + 36d + 49e + 64f + 81g = 123 ... | invalid | lean did not elaborate cleanly (status=error) | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 37 | eos | ### Detailed Proof
First, we are given a system of linear equations in real numbers:
1. \( a + 4b + 9c + 16d + 25e + 36f + 49g = 1 \)
2. \( 4a + 9b + 16c + 25d + 36e + 49f + 64g = 12 \)
3. \( 9a + 16b + 25c + 36d + 49e + 64f + 81g = 123 \)
We need to find the value of:
\[ 16a + 25b + 36c + 49d + 64e + 81f + 100g \]
... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 38 | eos | ### Detailed Proof and Analysis
First, let's understand the problem. We have three linear equations in seven variables (`a`, `b`, `c`, `d`, `e`, `f`, `g`):
1. `a + 4b + 9c + 16d + 25e + 36f + 49g = 1` (Equation `h₀`)
2. `4a + 9b + 16c + 25d + 36e + 49f + 64g = 12` (Equation `h₁`)
3. `9a + 16b + 25c + 36d + 49e + 64f +... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 39 | eos | ### Detailed Proof and Analysis
We are given a system of three linear equations in seven variables:
1. \( a + 4b + 9c + 16d + 25e + 36f + 49g = 1 \) (1)
2. \( 4a + 9b + 16c + 25d + 36e + 49f + 64g = 12 \) (2)
3. \( 9a + 16b + 25c + 36d + 49e + 64f + 81g = 123 \) (3)
We need to find the value of \( 16a + 25b + 36c ... | invalid | lean did not elaborate cleanly (status=error) | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 4 | eos | ### Detailed Proof and Analysis
We have the following system of linear equations:
1. \( a + 4b + 9c + 16d + 25e + 36f + 49g = 1 \) (1)
2. \( 4a + 9b + 16c + 25d + 36e + 49f + 64g = 12 \) (2)
3. \( 9a + 16b + 25c + 36d + 49e + 64f + 81g = 123 \) (3)
We need to find:
\( 16a + 25b + 36c + 49d + 64e + 81f + 100g = ... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 5 | eos | ### Detailed Proof and Analysis
First, we are given three linear equations in real numbers \(a, b, c, d, e, f, g\):
1. \(a + 4b + 9c + 16d + 25e + 36f + 49g = 1\) (1)
2. \(4a + 9b + 16c + 25d + 36e + 49f + 64g = 12\) (2)
3. \(9a + 16b + 25c + 36d + 49e + 64f + 81g = 123\) (3)
We need to find the value of \(16a +... | invalid | lean did not elaborate cleanly (status=error) | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 6 | eos | ### Detailed Proof
**Problem Analysis:**
We have a system of linear equations in real variables \( a, b, c, d, e, f, g \):
1. \( a + 4b + 9c + 16d + 25e + 36f + 49g = 1 \)
2. \( 4a + 9b + 16c + 25d + 36e + 49f + 64g = 12 \)
3. \( 9a + 16b + 25c + 36d + 49e + 64f + 81g = 123 \)
We are to find the value of:
\[ 16a + 25... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 7 | eos | ### Detailed Proof
**Problem Analysis:**
We are given three linear equations in seven variables `a, b, c, d, e, f, g`:
1. `a + 4b + 9c + 16d + 25e + 36f + 49g = 1` (Equation 1)
2. `4a + 9b + 16c + 25d + 36e + 49f + 64g = 12` (Equation 2)
3. `9a + 16b + 25c + 36d + 49e + 64f + 81g = 123` (Equation 3)
We need to find t... | invalid | lean did not elaborate cleanly (status=error) | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 8 | eos | ### Detailed Proof and Analysis
First, let's interpret the problem and understand what we are given and what we need to prove. We have three linear equations in the real variables \(a, b, c, d, e, f, g\):
1. \(a + 4b + 9c + 16d + 25e + 36f + 49g = 1\)
2. \(4a + 9b + 16c + 25d + 36e + 49f + 64g = 12\)
3. \(9a + 16b + ... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1989_p8 | minif2f | test | aime_1989_p8 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 9 | eos | ### Detailed Proof
**Problem Analysis:**
We have three linear equations in seven real variables:
1. \( a + 4b + 9c + 16d + 25e + 36f + 49g = 1 \) (Equation 1)
2. \( 4a + 9b + 16c + 25d + 36e + 49f + 64g = 12 \) (Equation 2)
3. \( 9a + 16b + 25c + 36d + 49e + 64f + 81g = 123 \) (Equation 3)
We need to find the value o... | valid | comparator accepted: statement unmodified, kernel-checked | needs_more_info | 31 | 9 | 0.658945 | 0.872676 |
aime_1990_p15 | minif2f | test | aime_1990_p15 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 0 | eos | ### Detailed Proof and Analysis
First, we have the following system of equations:
1. \( a x + b y = 3 \)
2. \( a x^2 + b y^2 = 7 \)
3. \( a x^3 + b y^3 = 16 \)
4. \( a x^4 + b y^4 = 42 \)
We are to find \( a x^5 + b y^5 \).
#### Observations:
1. The equations resemble a linear recurrence.
2. We can treat \( S_k = a ... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1990_p15 | minif2f | test | aime_1990_p15 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 1 | eos | ### Detailed Proof and Analysis
First, we need to find a recurrence relation for the expressions involving powers of \(x\) and \(y\). We start by noting that the given conditions are:
1. \(ax + by = 3\)
2. \(ax^2 + by^2 = 7\)
3. \(ax^3 + by^3 = 16\)
4. \(ax^4 + by^4 = 42\)
We observe that:
\[ ax^2 + by^2 = (ax + by)x... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1990_p15 | minif2f | test | aime_1990_p15 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 2 | eos | ### Detailed Proof and Analysis
We are given real numbers \( a, b, x, y \) and four equations:
1. \( a x + b y = 3 \)
2. \( a x^2 + b y^2 = 7 \)
3. \( a x^3 + b y^3 = 16 \)
4. \( a x^4 + b y^4 = 42 \)
We need to find \( a x^5 + b y^5 \).
#### Observations:
1. The four equations resemble recurrence relations, and the... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1990_p2 | minif2f | validation | aime_1990_p2 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 0 | eos | ### Detailed Proof and Analysis
First, we need to simplify the expression:
\[ (52 + 6 \sqrt{43})^{3/2} - (52 - 6 \sqrt{43})^{3/2} \]
#### Key Observations:
1. The exponents are fractional, but it turns out that this expression can be simplified using identities for cube roots of conjugates. However, Lean's notation `... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1990_p2 | minif2f | validation | aime_1990_p2 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 1 | eos | ### Detailed Proof and Analysis
First, let's recall the problem and set up some notation to simplify it. We need to compute:
\[
(52 + 6 \sqrt{43})^{3/2} - (52 - 6 \sqrt{43})^{3/2}.
\]
However, Lean interprets `(3 / 2)` as integer division, which is `1`. This means the expression is simplified to:
\[
(52 + 6 \sqrt{43})... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1990_p2 | minif2f | validation | aime_1990_p2 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 2 | eos | ### Detailed Proof and Analysis
First, recall that:
1. For real numbers \( a > 0 \) and \( b > 0 \), the expression \( a^b \) is well-defined and real-valued when \( a \geq 0 \) and \( b \) is a positive real number. However, the Lean 4 code interprets \( x^{(3 / 2)} \) as `(x : ℝ) ^ (3 / 2 : ℕ)`, which is a bit unusu... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1990_p4 | minif2f | test | aime_1990_p4 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 0 | eos | ### Detailed Proof and Analysis
First, we need to solve for `x` in the equation:
\[ \frac{1}{x^2 - 10x - 29} + \frac{1}{x^2 - 10x - 45} - \frac{2}{x^2 - 10x - 69} = 0. \]
This problem involves quadratic polynomials in the denominators, and we need to ensure that the denominators are non-zero to avoid division by zero... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1990_p4 | minif2f | test | aime_1990_p4 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 1 | eos | ### Detailed Proof and Analysis
First, let's recall the problem and set up the notation. We have a real number `x > 0` such that the following conditions are satisfied for the expressions `A = x² - 10x - 29`, `B = x² - 10x - 45`, and `C = x² - 10x - 69`:
1. `A ≠ 0`,
2. `B ≠ 0`,
3. `C ≠ 0`, and
4. The equation `1/A + 1... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1990_p4 | minif2f | test | aime_1990_p4 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 2 | eos | ### Detailed Proof and Analysis
#### Understanding the Problem
We have a real number \( x > 0 \) and four quadratic expressions in \( x \):
1. \( x^2 - 10x - 29 \neq 0 \),
2. \( x^2 - 10x - 45 \neq 0 \),
3. \( x^2 - 10x - 69 \neq 0 \),
4. The equation involving the reciprocals:
\[
\frac{1}{x^2 - 10x - 29} + \fra... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1991_p1 | minif2f | validation | aime_1991_p1 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 0 | eos | ### Detailed Proof and Analysis
First, we have the following problem:
1. \( x \) and \( y \) are positive integers.
2. \( xy + (x + y) = 71 \).
3. \( x^2 y + x y^2 = 880 \).
4. We need to prove \( x^2 + y^2 = 146 \).
#### Step 1: Simplify \( x^2 y + x y^2 \)
Factor out \( xy \) from \( x^2 y + x y^2 \):
\[ x^2 y + x ... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1991_p1 | minif2f | validation | aime_1991_p1 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 1 | eos | ### Detailed Proof and Analysis
First, we are given:
1. \(x\) and \(y\) are positive integers.
2. \(xy + (x + y) = 71\).
3. \(x^2 y + xy^2 = 880\).
We need to prove that \(x^2 + y^2 = 146\).
#### Step 1: Factor the Second Equation
The second equation can be factored as:
\[xy(x + y) = 880\]
Notice that this resembles... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
aime_1991_p1 | minif2f | validation | aime_1991_p1 | import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Log
import Mathlib.Data.Complex.Exponential
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.ZMod.Defs
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Basic
import Mathlib.Da... | deepseek-ai/DeepSeek-Prover-V2-7B | 2 | eos | ### Detailed Proof and Analysis
#### Understanding the Problem
We have two positive integers \( x \) and \( y \) such that:
1. \( xy + (x + y) = 71 \)
2. \( x^2 y + x y^2 = 880 \)
We need to find \( x^2 + y^2 \).
First, let's simplify the second equation using the fact that \( x^2 y + x y^2 = x y (x + y) \). But we ... | invalid | lean did not elaborate cleanly (status=error) | single_class | 0 | 3 | 0 | 0.195172 |
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