id string | source string | source_commit string | source_id string | source_split string | license string | informal string | header string | formal_statement string | statement string | area string | difficulty string | split string | seed int64 | formal_statement_original string | converted string | source_tags list | informal_redacted bool |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
mathd_algebra_441 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | mathd_algebra_441 | test | MIT | Assuming $x\ne0$, simplify $\frac{12}{x \cdot x} \cdot \frac{x^4}{14x}\cdot \frac{35}{3x}$. Show that it is 10. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem mathd_algebra_441 (x : β) (hβ : x β 0) :
12 / (x * x) * (x ^ 4 / (14 * x)) * (35 / (3 * x)) = 10 := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem mathd_algebra_441 (x : β) (hβ : x β 0) :
12 / (x * x) * (x ^ 4 / (14 * x)) * (35 / (3 * x)) = 10 := by
| algebra | easy | dev | 2,026 | null | null | null | null |
mathd_algebra_209 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | mathd_algebra_209 | test | MIT | Suppose that $h(x)=f^{-1}(x)$. If $h(2)=10$, $h(10)=1$ and $h(1)=2$, what is $f(f(10))$? Show that it is 1. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem mathd_algebra_209 (Ο : Equiv β β) (hβ : Ο.2 2 = 10) (hβ : Ο.2 10 = 1) (hβ : Ο.2 1 = 2) :
Ο.1 (Ο.1 10) = 1 := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem mathd_algebra_209 (Ο : Equiv β β) (hβ : Ο.2 2 = 10) (hβ : Ο.2 10 = 1) (hβ : Ο.2 1 = 2) :
Ο.1 (Ο.1 10) = 1 := by
| algebra | easy | dev | 2,026 | null | null | null | null |
mathd_algebra_69 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | mathd_algebra_69 | valid | MIT | An audience of $450$ people is seated in an auditorium. Each row contains the same number of seats and each seat in the auditorium is occupied. With three fewer seats per row, and five extra rows, the same audience could still be seated, occupying all seats. How many rows does the auditorium have? Show that it is 25. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem mathd_algebra_69 (rows seats : β) (hβ : rows * seats = 450)
(hβ : (rows + 5) * (seats - 3) = 450) : rows = 25 := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem mathd_algebra_69 (rows seats : β) (hβ : rows * seats = 450)
(hβ : (rows + 5) * (seats - 3) = 450) : rows = 25 := by
| algebra | easy | dev | 2,026 | null | null | null | null |
mathd_algebra_206 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | mathd_algebra_206 | valid | MIT | The polynomial $p(x) = x^2+ax+b$ has distinct roots $2a$ and $b$. Find $a+b$. Show that it is -1. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem mathd_algebra_206 (a b : β) (f : β β β) (hβ : β x, f x = x ^ 2 + a * x + b) (hβ : 2 * a β b)
(hβ : f (2 * a) = 0) (hβ : f b = 0) : a + b = -1 := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem mathd_algebra_206 (a b : β) (f : β β β) (hβ : β x, f x = x ^ 2 + a * x + b) (hβ : 2 * a β b)
(hβ : f (2 * a) = 0) (hβ : f b = 0) : a + b = -1 := by
| algebra | easy | test | 2,026 | null | null | null | null |
mathd_algebra_455 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | mathd_algebra_455 | valid | MIT | At each basketball practice last week, Jenny made twice as many free throws as she made at the previous practice. At her fifth practice she made 48 free throws. How many free throws did she make at the first practice? Show that it is 3. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem mathd_algebra_455 (x : β) (hβ : 2 * (2 * (2 * (2 * x))) = 48) : x = 3 := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem mathd_algebra_455 (x : β) (hβ : 2 * (2 * (2 * (2 * x))) = 48) : x = 3 := by
| algebra | easy | test | 2,026 | null | null | null | null |
mathd_algebra_234 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | mathd_algebra_234 | valid | MIT | What is the sixth term in the geometric sequence $\frac{27}{125}, \frac{9}{25}, \frac{3}{5},\ldots$? Express your answer as a common fraction. Show that it is \frac{25}{9}. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem mathd_algebra_234 (d : β) (hβ : 27 / 125 * d = 9 / 25) : 3 / 5 * d ^ 3 = 25 / 9 := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem mathd_algebra_234 (d : β) (hβ : 27 / 125 * d = 9 / 25) : 3 / 5 * d ^ 3 = 25 / 9 := by
| algebra | easy | test | 2,026 | null | null | null | null |
mathd_algebra_160 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | mathd_algebra_160 | test | MIT | For each plumbing repair job, Mr. Wrench charges $N$ dollars for coming out to the house plus $x$ dollars per hour that he works at the house. He charged $\$97$ for a one-hour repair job and $\$265$ for a five-hour repair job. What is his charge for a two-hour repair job? Show that it is \$ 139. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem mathd_algebra_160 (n x : β) (hβ : n + x = 97) (hβ : n + 5 * x = 265) : n + 2 * x = 139 := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem mathd_algebra_160 (n x : β) (hβ : n + x = 97) (hβ : n + 5 * x = 265) : n + 2 * x = 139 := by
| algebra | easy | test | 2,026 | null | null | null | null |
mathd_algebra_288 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | mathd_algebra_288 | test | MIT | A point $(x,y)$ on the coordinate plane with both coordinates negative is a distance of 6 units from the $x$-axis. It is a distance of 15 units from the point $(8,3)$. It is a distance $\sqrt{n}$ from the origin. What is $n$? Show that it is 52. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem mathd_algebra_288 (x y : β) (n : NNReal) (hβ : x < 0 β§ y < 0) (hβ : abs y = 6)
(hβ : Real.sqrt ((x - 8) ^ 2 + (y - 3) ^ 2) = 15)
(hβ : Real.sqrt (x ^ 2 + y ^ 2) = Real.sqrt n) : n = 52 := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem mathd_algebra_288 (x y : β) (n : NNReal) (hβ : x < 0 β§ y < 0) (hβ : abs y = 6)
(hβ : Real.sqrt ((x - 8) ^ 2 + (y - 3) ^ 2) = 15)
(hβ : Real.sqrt (x ^ 2 + y ^ 2) = Real.sqrt n) : n = 52 := by
| algebra | easy | test | 2,026 | null | null | null | null |
mathd_numbertheory_353 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | mathd_numbertheory_353 | test | MIT | Let $S = 2010 + 2011 + \cdots + 4018$. Compute the residue of $S$, modulo 2009. Show that it is 0. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem mathd_numbertheory_353 (s : β) (hβ : s = β k β Finset.Icc 2010 4018, k) : s % 2009 = 0 := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem mathd_numbertheory_353 (s : β) (hβ : s = β k β Finset.Icc 2010 4018, k) : s % 2009 = 0 := by
| number_theory | easy | dev | 2,026 | theorem mathd_numbertheory_353 (s : β) (hβ : s = β k in Finset.Icc 2010 4018, k) : s % 2009 = 0 := by
| big_operator_in_to_mem | null | null |
mathd_numbertheory_232 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | mathd_numbertheory_232 | valid | MIT | Compute $(3^{-1}+5^{-1})^{-1}\pmod{31}$. Express your answer as an integer from $0$ to $30$, inclusive. Show that it is 29. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem mathd_numbertheory_232 (x y z : ZMod 31) (hβ : x = 3β»ΒΉ) (hβ : y = 5β»ΒΉ)
(hβ : z = (x + y)β»ΒΉ) : z = 29 := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem mathd_numbertheory_232 (x y z : ZMod 31) (hβ : x = 3β»ΒΉ) (hβ : y = 5β»ΒΉ)
(hβ : z = (x + y)β»ΒΉ) : z = 29 := by
| number_theory | easy | dev | 2,026 | null | null | null | null |
mathd_numbertheory_430 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | mathd_numbertheory_430 | test | MIT | If $A$, $B$, and $C$ represent three distinct digits from 1 to 9 and they satisfy the following equations, what is the value of the sum $A+B+C$? (In the equation below, $AA$ represents a two-digit number both of whose digits are $A$.) $$A+B=C$$$$AA-B=2\times C$$$$C\times B=AA+A$$ Show that it is 8. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem mathd_numbertheory_430 (a b c : β) (hβ : 1 β€ a β§ a β€ 9) (hβ : 1 β€ b β§ b β€ 9)
(hβ : 1 β€ c β§ c β€ 9) (hβ : a β b) (hβ : a β c) (hβ
: b β c) (hβ : a + b = c)
(hβ : 10 * a + a - b = 2 * c) (hβ : c * b = 10 * a + a + a) : a + b + c = 8 := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem mathd_numbertheory_430 (a b c : β) (hβ : 1 β€ a β§ a β€ 9) (hβ : 1 β€ b β§ b β€ 9)
(hβ : 1 β€ c β§ c β€ 9) (hβ : a β b) (hβ : a β c) (hβ
: b β c) (hβ : a + b = c)
(hβ : 10 * a + a - b = 2 * c) (hβ : c * b = 10 * a +... | number_theory | easy | test | 2,026 | null | null | null | null |
mathd_numbertheory_328 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | mathd_numbertheory_328 | test | MIT | What is the remainder when 5 to the 999,999th power is divided by 7? Show that it is 6. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem mathd_numbertheory_328 : 5 ^ 999999 % 7 = 6 := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem mathd_numbertheory_328 : 5 ^ 999999 % 7 = 6 := by
| number_theory | easy | test | 2,026 | null | null | null | null |
mathd_numbertheory_64 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | mathd_numbertheory_64 | valid | MIT | What is the smallest positive integer that satisfies the congruence $30x \equiv 42 \pmod{47}$? Show that it is 39. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem mathd_numbertheory_64 : IsLeast { x : β | 30 * x β‘ 42 [MOD 47] } 39 := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem mathd_numbertheory_64 : IsLeast { x : β | 30 * x β‘ 42 [MOD 47] } 39 := by
| number_theory | easy | test | 2,026 | null | null | null | null |
mathd_numbertheory_321 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | mathd_numbertheory_321 | test | MIT | Notice that \[35\cdot40=1400.\]Find some integer $n$ with $0\leq n<1399$ such that $n$ is the multiplicative inverse to 160 modulo 1399. Show that it is 1058. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem mathd_numbertheory_321 (n : ZMod 1399) (hβ : n = 160β»ΒΉ) : n = 1058 := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem mathd_numbertheory_321 (n : ZMod 1399) (hβ : n = 160β»ΒΉ) : n = 1058 := by
| number_theory | easy | test | 2,026 | null | null | null | null |
mathd_numbertheory_345 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | mathd_numbertheory_345 | test | MIT | What is the remainder when $2000+2001+2002+2003+2004+2005+2006$ is divided by $7$? Show that it is 0. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem mathd_numbertheory_345 : (2000 + 2001 + 2002 + 2003 + 2004 + 2005 + 2006) % 7 = 0 := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem mathd_numbertheory_345 : (2000 + 2001 + 2002 + 2003 + 2004 + 2005 + 2006) % 7 = 0 := by
| number_theory | easy | test | 2,026 | null | null | null | null |
amc12a_2015_p10 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | amc12a_2015_p10 | valid | MIT | Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=80$. What is $x$?
$ \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 26$ Show that it is \textbf{(E)}\ 26. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem amc12a_2015_p10 (x y : β€) (hβ : 0 < y) (hβ : y < x) (hβ : x + y + x * y = 80) : x = 26 := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem amc12a_2015_p10 (x y : β€) (hβ : 0 < y) (hβ : y < x) (hβ : x + y + x * y = 80) : x = 26 := by
| algebra | medium | dev | 2,026 | null | null | null | null |
aime_1991_p1 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | aime_1991_p1 | valid | MIT | Find $x^2+y^2_{}$ if $x_{}^{}$ and $y_{}^{}$ are positive integers such that
$\begin{align*}
xy+x+y&=71, \\
x^2y+xy^2&=880.
\end{align*}$ Show that it is 146. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem aime_1991_p1 (x y : β) (hβ : 0 < x β§ 0 < y) (hβ : x * y + (x + y) = 71)
(hβ : x ^ 2 * y + x * y ^ 2 = 880) : x ^ 2 + y ^ 2 = 146 := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem aime_1991_p1 (x y : β) (hβ : 0 < x β§ 0 < y) (hβ : x * y + (x + y) = 71)
(hβ : x ^ 2 * y + x * y ^ 2 = 880) : x ^ 2 + y ^ 2 = 146 := by
| algebra | medium | dev | 2,026 | null | null | null | null |
amc12_2001_p9 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | amc12_2001_p9 | valid | MIT | Let $f$ be a function satisfying $f(xy) = \frac{f(x)}y$ for all positive real numbers $x$ and $y$. If $f(500) =3$, what is the value of $f(600)$?
$(\mathrm{A})\ 1 \qquad (\mathrm{B})\ 2 \qquad (\mathrm{C})\ \frac52 \qquad (\mathrm{D})\ 3 \qquad (\mathrm{E})\ \frac{18}5$ Show that it is \textbf{C } \frac52. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem amc12_2001_p9 (f : β β β) (hβ : β x > 0, β y > 0, f (x * y) = f x / y) (hβ : f 500 = 3) :
f 600 = 5 / 2 := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem amc12_2001_p9 (f : β β β) (hβ : β x > 0, β y > 0, f (x * y) = f x / y) (hβ : f 500 = 3) :
f 600 = 5 / 2 := by
| algebra | medium | test | 2,026 | null | null | null | null |
amc12_2000_p20 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | amc12_2000_p20 | test | MIT | If $x,y,$ and $z$ are positive numbers satisfying
$x + \frac{1}{y} = 4,\qquad y + \frac{1}{z} = 1, \qquad \text{and} \qquad z + \frac{1}{x} = \frac{7}{3}$
Then what is the value of $xyz$ ?
$\text {(A)}\ \frac{2}{3} \qquad \text {(B)}\ 1 \qquad \text {(C)}\ \frac{4}{3} \qquad \text {(D)}\ 2 \qquad \text {(E)}\ \frac{... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem amc12_2000_p20 (x y z : β) (hβ : 0 < x β§ 0 < y β§ 0 < z) (hβ : x + 1 / y = 4)
(hβ : y + 1 / z = 1) (hβ : z + 1 / x = 7 / 3) : x * y * z = 1 := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem amc12_2000_p20 (x y z : β) (hβ : 0 < x β§ 0 < y β§ 0 < z) (hβ : x + 1 / y = 4)
(hβ : y + 1 / z = 1) (hβ : z + 1 / x = 7 / 3) : x * y * z = 1 := by
| algebra | medium | test | 2,026 | null | null | null | null |
amc12a_2008_p25 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | amc12a_2008_p25 | test | MIT | A sequence $(a_1,b_1)$, $(a_2,b_2)$, $(a_3,b_3)$, $\ldots$ of points in the coordinate plane satisfies
$(a_{n + 1}, b_{n + 1}) = (\sqrt {3}a_n - b_n, \sqrt {3}b_n + a_n)$ for $n = 1,2,3,\ldots$.
Suppose that $(a_{100},b_{100}) = (2,4)$. What is $a_1 + b_1$?
$\mathrm{(A)}\ -\frac{1}{2^{97}}\qquad\mathrm{(B)}\ -\fra... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem amc12a_2008_p25 (a b : β β β) (hβ : β n, a (n + 1) = Real.sqrt 3 * a n - b n)
(hβ : β n, b (n + 1) = Real.sqrt 3 * b n + a n) (hβ : a 100 = 2) (hβ : b 100 = 4) :
a 1 + b 1 = 1 / 2 ^ 98 := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem amc12a_2008_p25 (a b : β β β) (hβ : β n, a (n + 1) = Real.sqrt 3 * a n - b n)
(hβ : β n, b (n + 1) = Real.sqrt 3 * b n + a n) (hβ : a 100 = 2) (hβ : b 100 = 4) :
a 1 + b 1 = 1 / 2 ^ 98 := by
| algebra | medium | test | 2,026 | null | null | null | null |
algebra_absapbon1pabsapbleqsumabsaon1pabsa | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | algebra_absapbon1pabsapbleqsumabsaon1pabsa | test | MIT | Show that for any real numbers $a$ and $b$, $\frac{|a+b|}{1+|a+b|}\leq \frac{|a|}{1+|a|}+\frac{|b|}{1+|b|}$. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem algebra_absapbon1pabsapbleqsumabsaon1pabsa (a b : β) :
abs (a + b) / (1 + abs (a + b)) β€ abs a / (1 + abs a) + abs b / (1 + abs b) := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem algebra_absapbon1pabsapbleqsumabsaon1pabsa (a b : β) :
abs (a + b) / (1 + abs (a + b)) β€ abs a / (1 + abs a) + abs b / (1 + abs b) := by
| algebra | medium | test | 2,026 | null | null | null | null |
amc12b_2003_p6 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | amc12b_2003_p6 | valid | MIT | The second and fourth terms of a geometric sequence are $2$ and $6$. Which of the following is a possible first term?
$\textbf{(A) } -\sqrt{3} \qquad\textbf{(B) } -\frac{2\sqrt{3}}{3} \qquad\textbf{(C) } -\frac{\sqrt{3}}{3} \qquad\textbf{(D) } \sqrt{3} \qquad\textbf{(E) } 3$ Show that it is \textbf{(B)}\ -\frac{2\sqr... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem amc12b_2003_p6 (a r : β) (u : β β β) (hβ : β k, u k = a * r ^ k) (hβ : u 1 = 2)
(hβ : u 3 = 6) : u 0 = 2 / Real.sqrt 3 β¨ u 0 = -(2 / Real.sqrt 3) := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem amc12b_2003_p6 (a r : β) (u : β β β) (hβ : β k, u k = a * r ^ k) (hβ : u 1 = 2)
(hβ : u 3 = 6) : u 0 = 2 / Real.sqrt 3 β¨ u 0 = -(2 / Real.sqrt 3) := by
| algebra | medium | test | 2,026 | null | null | null | null |
amc12b_2004_p3 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | amc12b_2004_p3 | valid | MIT | If $x$ and $y$ are positive integers for which $2^x3^y=1296$, what is the value of $x+y$?
$(\mathrm {A})\ 8 \qquad (\mathrm {B})\ 9 \qquad (\mathrm {C})\ 10 \qquad (\mathrm {D})\ 11 \qquad (\mathrm {E})\ 12$ Show that it is 8. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem amc12b_2004_p3 (x y : β) (hβ : 2 ^ x * 3 ^ y = 1296) : x + y = 8 := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem amc12b_2004_p3 (x y : β) (hβ : 2 ^ x * 3 ^ y = 1296) : x + y = 8 := by
| number_theory | medium | dev | 2,026 | null | null | null | null |
induction_divisibility_3div2tooddnp1 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | induction_divisibility_3div2tooddnp1 | valid | MIT | For a natural number $n$, show that $3 \mid (2^{2n+1}+1)$. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem induction_divisibility_3div2tooddnp1 (n : β) : 3 β£ 2 ^ (2 * n + 1) + 1 := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem induction_divisibility_3div2tooddnp1 (n : β) : 3 β£ 2 ^ (2 * n + 1) + 1 := by
| number_theory | medium | dev | 2,026 | null | null | null | null |
amc12b_2002_p3 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | amc12b_2002_p3 | valid | MIT | For how many positive integers $n$ is $n^2 - 3n + 2$ a [[prime]] number?
$\mathrm{(A)}\ \text{none}
\qquad\mathrm{(B)}\ \text{one}
\qquad\mathrm{(C)}\ \text{two}
\qquad\mathrm{(D)}\ \text{more\ than\ two,\ but\ finitely\ many}
\qquad\mathrm{(E)}\ \text{infinitely\ many}$ Show that it is \mathrm{(B)}\ \text{one}. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem amc12b_2002_p3 (S : Finset β)
(hβ : β n : β, n β S β 0 < n β§ Nat.Prime (n ^ 2 + 2 - 3 * n)) :
S.card = 1 := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem amc12b_2002_p3 (S : Finset β)
(hβ : β n : β, n β S β 0 < n β§ Nat.Prime (n ^ 2 + 2 - 3 * n)) :
S.card = 1 := by
| number_theory | medium | test | 2,026 | null | null | null | null |
amc12a_2008_p15 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | amc12a_2008_p15 | valid | MIT | Let $k={2008}^{2}+{2}^{2008}$. What is the units digit of $k^2+2^k$?
$\mathrm{(A)}\ 0\qquad\mathrm{(B)}\ 2\qquad\mathrm{(C)}\ 4\qquad\mathrm{(D)}\ 6\qquad\mathrm{(E)}\ 8$ Show that it is D. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem amc12a_2008_p15 (k : β) (hβ : k = 2008 ^ 2 + 2 ^ 2008) : (k ^ 2 + 2 ^ k) % 10 = 6 := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem amc12a_2008_p15 (k : β) (hβ : k = 2008 ^ 2 + 2 ^ 2008) : (k ^ 2 + 2 ^ k) % 10 = 6 := by
| number_theory | medium | test | 2,026 | null | null | null | null |
numbertheory_notequiv2i2jasqbsqdiv8 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | numbertheory_notequiv2i2jasqbsqdiv8 | test | MIT | Let $a$ and $b$ be two integer numbers. Show that the following statement is false: $a$ and $b$ are both even iff $8 \mid a^2 + b^2$. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem numbertheory_notequiv2i2jasqbsqdiv8 :
Β¬β a b : β€, (β i j, a = 2 * i β§ b = 2 * j) β β k, a ^ 2 + b ^ 2 = 8 * k := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem numbertheory_notequiv2i2jasqbsqdiv8 :
Β¬β a b : β€, (β i j, a = 2 * i β§ b = 2 * j) β β k, a ^ 2 + b ^ 2 = 8 * k := by
| number_theory | medium | test | 2,026 | null | null | null | null |
numbertheory_3pow2pownm1mod2pownp3eq2pownp2 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | numbertheory_3pow2pownm1mod2pownp3eq2pownp2 | test | MIT | Show that for positive integer $n$, the remainder of $3^{2^n} - 1 \equiv 2^{n+2} \mod 2^{n+3}$. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem numbertheory_3pow2pownm1mod2pownp3eq2pownp2 (n : β) (hβ : 0 < n) :
(3 ^ 2 ^ n - 1) % 2 ^ (n + 3) = 2 ^ (n + 2) := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem numbertheory_3pow2pownm1mod2pownp3eq2pownp2 (n : β) (hβ : 0 < n) :
(3 ^ 2 ^ n - 1) % 2 ^ (n + 3) = 2 ^ (n + 2) := by
| number_theory | medium | test | 2,026 | null | null | null | null |
numbertheory_nckeqnm1ckpnm1ckm1 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | numbertheory_nckeqnm1ckpnm1ckm1 | valid | MIT | Show that for positive integers $n$ and $k$ with $k \leq n$, we have
$\binom{n}{k} = \binom{n-1}{k} + \binom{n-1}{k-1}$. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem numbertheory_nckeqnm1ckpnm1ckm1 (n k : β) (hβ : 0 < n β§ 0 < k) (hβ : k β€ n) :
Nat.choose n k = Nat.choose (n - 1) k + Nat.choose (n - 1) (k - 1) := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem numbertheory_nckeqnm1ckpnm1ckm1 (n k : β) (hβ : 0 < n β§ 0 < k) (hβ : k β€ n) :
Nat.choose n k = Nat.choose (n - 1) k + Nat.choose (n - 1) (k - 1) := by
| combinatorics | medium | dev | 2,026 | null | null | null | null |
numbertheory_sumkmulnckeqnmul2pownm1 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | numbertheory_sumkmulnckeqnmul2pownm1 | valid | MIT | Show that for positive integers $n$ and $k$, if $k \leq n$, then $\sum_{k=1}^n (k*C_n^k) = n * 2^{n-1}$. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem numbertheory_sumkmulnckeqnmul2pownm1 (n : β) (hβ : 0 < n) :
(β k β Finset.Icc 1 n, k * Nat.choose n k) = n * 2 ^ (n - 1) := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem numbertheory_sumkmulnckeqnmul2pownm1 (n : β) (hβ : 0 < n) :
(β k β Finset.Icc 1 n, k * Nat.choose n k) = n * 2 ^ (n - 1) := by
| combinatorics | medium | test | 2,026 | theorem numbertheory_sumkmulnckeqnmul2pownm1 (n : β) (hβ : 0 < n) :
(β k in Finset.Icc 1 n, k * Nat.choose n k) = n * 2 ^ (n - 1) := by
| big_operator_in_to_mem | null | null |
imo_1987_p4 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | imo_1987_p4 | valid | MIT | Prove that there is no function $f $ from the set of non-negative integers into itself such that $f(f(n)) = n + 1987 $ for every $n $. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem imo_1987_p4 (f : β β β) : β n, f (f n) β n + 1987 := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem imo_1987_p4 (f : β β β) : β n, f (f n) β n + 1987 := by
| algebra | hard | dev | 2,026 | null | null | null | null |
imo_2006_p3 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | imo_2006_p3 | valid | MIT | Show that for any real numbers $a$, $b$, and $c$, we have $(ab(a^2 - b^2)) + (bc(b^2 - c^2)) + (ca(c^2 - a^2)) \leq \frac{9\sqrt{2}}{32}(a^2 + b^2 + c^2)^2$. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem imo_2006_p3 (a b c : β) :
a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2) β€
9 * Real.sqrt 2 / 32 * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2 := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem imo_2006_p3 (a b c : β) :
a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2) β€
9 * Real.sqrt 2 / 32 * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2 := by
| algebra | hard | dev | 2,026 | null | null | null | null |
imo_1965_p2 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | imo_1965_p2 | test | MIT | Consider the system of equations
$a_{11}x_1 + a_{12}x_2 + a_{13}x_3 = 0$
$a_{21}x_1 + a_{22}x_2 + a_{23}x_3 = 0$
$a_{31}x_1 + a_{32}x_2 + a_{33}x_3 = 0$
with unknowns $x_1$, $x_2$, $x_3$. The coefficients satisfy the conditions:
(a) $a_{11}$, $a_{22}$, $a_{33}$ are positive numbers;
(b) the remaining coefficients are... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem imo_1965_p2 (x y z : β) (a : β β β) (hβ : 0 < a 0 β§ 0 < a 4 β§ 0 < a 8)
(hβ : a 1 < 0 β§ a 2 < 0) (hβ : a 3 < 0 β§ a 5 < 0) (hβ : a 6 < 0 β§ a 7 < 0)
(hβ : 0 < a 0 + a 1 + a 2) (hβ
: 0 < a 3 + a 4 + a 5) (hβ : 0 < a 6 + a 7 + a 8)
(hβ : a 0 * x + a 1 * y + a 2 * z = 0) (hβ : a 3 * x + a 4 * y + a 5 * z ... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem imo_1965_p2 (x y z : β) (a : β β β) (hβ : 0 < a 0 β§ 0 < a 4 β§ 0 < a 8)
(hβ : a 1 < 0 β§ a 2 < 0) (hβ : a 3 < 0 β§ a 5 < 0) (hβ : a 6 < 0 β§ a 7 < 0)
(hβ : 0 < a 0 + a 1 + a 2) (hβ
: 0 < a 3 + a 4 + a 5) (hβ : ... | algebra | hard | test | 2,026 | null | null | null | null |
imo_1993_p5 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | imo_1993_p5 | valid | MIT | Let $\mathbb{N} = \{1,2,3, \ldots\}$. Determine if there exists a strictly increasing function $f: \mathbb{N} \mapsto \mathbb{N}$ with the following properties:
(i) $f(1) = 2$;
(ii) $f(f(n)) = f(n) + n, (n \in \mathbb{N})$. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem imo_1993_p5 : β f : β β β, f 1 = 2 β§ β n, f (f n) = f n + n β§ β n, f n < f (n + 1) := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem imo_1993_p5 : β f : β β β, f 1 = 2 β§ β n, f (f n) = f n + n β§ β n, f n < f (n + 1) := by
| algebra | hard | test | 2,026 | null | null | null | null |
imo_1978_p5 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | imo_1978_p5 | valid | MIT | Let $f$ be an injective function from ${1,2,3,\ldots}$ in itself. Prove that for any $n$ we have: $\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.$ | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem imo_1978_p5 (n : β) (a : β β β) (hβ : Function.Injective a) (hβ : a 0 = 0) (hβ : 0 < n) :
(β k β Finset.Icc 1 n, (1 : β) / k) β€ β k β Finset.Icc 1 n, (a k : β) / k ^ 2 := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem imo_1978_p5 (n : β) (a : β β β) (hβ : Function.Injective a) (hβ : a 0 = 0) (hβ : 0 < n) :
(β k β Finset.Icc 1 n, (1 : β) / k) β€ β k β Finset.Icc 1 n, (a k : β) / k ^ 2 := by
| algebra | hard | test | 2,026 | theorem imo_1978_p5 (n : β) (a : β β β) (hβ : Function.Injective a) (hβ : a 0 = 0) (hβ : 0 < n) :
(β k in Finset.Icc 1 n, (1 : β) / k) β€ β k in Finset.Icc 1 n, (a k : β) / k ^ 2 := by
| big_operator_in_to_mem | null | null |
imo_1974_p3 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | imo_1974_p3 | test | MIT | Prove that the number $\sum^n_{k=0}\binom{2n+1}{2k+1}2^{3k}$ is not divisible by $5$ for any integer $n\ge0.$ | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem imo_1974_p3 (n : β) :
Β¬5 β£ β k β Finset.range (n + 1), Nat.choose (2 * n + 1) (2 * k + 1) * 2 ^ (3 * k) := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem imo_1974_p3 (n : β) :
Β¬5 β£ β k β Finset.range (n + 1), Nat.choose (2 * n + 1) (2 * k + 1) * 2 ^ (3 * k) := by
| number_theory | hard | dev | 2,026 | theorem imo_1974_p3 (n : β) :
Β¬5 β£ β k in Finset.range (n + 1), Nat.choose (2 * n + 1) (2 * k + 1) * 2 ^ (3 * k) := by
| big_operator_in_to_mem | null | null |
imo_1987_p6 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | imo_1987_p6 | valid | MIT | Let $n $ be an integer greater than or equal to 2. Prove that if $k^2 + k + n $ is prime for all integers $k $ such that $ 0 \leq k \leq \sqrt{n/3} $, then $k^2 + k + n $ is prime for all integers $k $ such that $0 \leq k \leq n - 2 $. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem imo_1987_p6 (p : β) (f : β β β) (hβ : β x, f x = x ^ 2 + x + p)
(hβ : β k : β, k β€ Nat.floor (Real.sqrt (p / 3)) β Nat.Prime (f k)) :
β i β€ p - 2, Nat.Prime (f i) := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem imo_1987_p6 (p : β) (f : β β β) (hβ : β x, f x = x ^ 2 + x + p)
(hβ : β k : β, k β€ Nat.floor (Real.sqrt (p / 3)) β Nat.Prime (f k)) :
β i β€ p - 2, Nat.Prime (f i) := by
| number_theory | hard | test | 2,026 | null | null | null | null |
imo_1979_p1 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | imo_1979_p1 | valid | MIT | If $p$ and $q$ are natural numbers so that$ \frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+ \ldots -\frac{1}{1318}+\frac{1}{1319}, $prove that $p$ is divisible with $1979$. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem imo_1979_p1 (p q : β) (hβ : 0 < q)
(hβ : (β k β Finset.Icc (1 : β) 1319, (-1) ^ (k + 1) * ((1 : β) / k)) = p / q) : 1979 β£ p := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem imo_1979_p1 (p q : β) (hβ : 0 < q)
(hβ : (β k β Finset.Icc (1 : β) 1319, (-1) ^ (k + 1) * ((1 : β) / k)) = p / q) : 1979 β£ p := by
| number_theory | hard | test | 2,026 | theorem imo_1979_p1 (p q : β) (hβ : 0 < q)
(hβ : (β k in Finset.Icc (1 : β) 1319, (-1) ^ (k + 1) * ((1 : β) / k)) = p / q) : 1979 β£ p := by
| big_operator_in_to_mem | null | null |
imo_1984_p6 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | imo_1984_p6 | test | MIT | Let $a,b,c,d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^k$ and $b+c=2^m$ for some integers $k$ and $m$, then $a=1$. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem imo_1984_p6 (a b c d k m : β) (hβ : 0 < a β§ 0 < b β§ 0 < c β§ 0 < d)
(hβ : Odd a β§ Odd b β§ Odd c β§ Odd d) (hβ : a < b β§ b < c β§ c < d) (hβ : a * d = b * c)
(hβ : a + d = 2 ^ k) (hβ
: b + c = 2 ^ m) : a = 1 := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem imo_1984_p6 (a b c d k m : β) (hβ : 0 < a β§ 0 < b β§ 0 < c β§ 0 < d)
(hβ : Odd a β§ Odd b β§ Odd c β§ Odd d) (hβ : a < b β§ b < c β§ c < d) (hβ : a * d = b * c)
(hβ : a + d = 2 ^ k) (hβ
: b + c = 2 ^ m) : a = 1 :=... | number_theory | hard | test | 2,026 | null | null | null | null |
imo_1964_p1_2 | miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl) | 2c4ba9119eef74d0d611f494261b2c5bae98c69a | imo_1964_p1_2 | valid | MIT | Show that for any natural number $n$, $7$ does not divide $2^n + 1$. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
| theorem imo_1964_p1_2 (n : β) : Β¬7 β£ 2 ^ n + 1 := by
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem imo_1964_p1_2 (n : β) : Β¬7 β£ 2 ^ n + 1 := by
| number_theory | hard | test | 2,026 | null | null | null | null |
putnam_1966_b4 | PutnamBench (lean4/src, informal/putnam.json) | a635813f5edf297d928f9bd5ef8324a26bdde09c | putnam_1966_b4 | null | Apache-2.0 (Lean statement); informal text available with permission from the MAA | [Informal statement not redistributed: PutnamBench states it is available with permission from the MAA. Problem: Putnam 1966 B4; source: PutnamBench informal/putnam.json, commit a635813f.] | import Mathlib
open Topology Filter
| theorem putnam_1966_b4
(m n : β)
(S : Finset β)
(hS : (β i β S, i > 0) β§ S.card = m * n + 1)
: β T β S, (T.card = m + 1 β§ β j β T, β i β T, i β j β Β¬(j β£ i)) β¨ (T.card = n + 1 β§ β i β T, β j β T, j < i β j β£ i) := by
| import Mathlib
open Topology Filter
theorem putnam_1966_b4
(m n : β)
(S : Finset β)
(hS : (β i β S, i > 0) β§ S.card = m * n + 1)
: β T β S, (T.card = m + 1 β§ β j β T, β i β T, i β j β Β¬(j β£ i)) β¨ (T.card = n + 1 β§ β i β T, β j β T, j < i β j β£ i) := by
| combinatorics | hard | dev | 2,026 | null | null | [
"number_theory",
"combinatorics"
] | true |
putnam_1965_a4 | PutnamBench (lean4/src, informal/putnam.json) | a635813f5edf297d928f9bd5ef8324a26bdde09c | putnam_1965_a4 | null | Apache-2.0 (Lean statement); informal text available with permission from the MAA | [Informal statement not redistributed: PutnamBench states it is available with permission from the MAA. Problem: Putnam 1965 A4; source: PutnamBench informal/putnam.json, commit a635813f.] | import Mathlib
open EuclideanGeometry Topology Filter Complex
| theorem putnam_1965_a4
{G B : Type*}
[Fintype G] [Nonempty G]
[Fintype B] [Nonempty B]
(dances : G β B β Prop)
(h : (Β¬β b : B, β g : G, dances g b) β§ β g : G, β b : B, dances g b)
: β g h : G, β b c : B, dances g b β§ dances h c β§ Β¬dances h b β§ Β¬dances g c := by
| import Mathlib
open EuclideanGeometry Topology Filter Complex
theorem putnam_1965_a4
{G B : Type*}
[Fintype G] [Nonempty G]
[Fintype B] [Nonempty B]
(dances : G β B β Prop)
(h : (Β¬β b : B, β g : G, dances g b) β§ β g : G, β b : B, dances g b)
: β g h : G, β b c : B, dances g b β§ dances h c β§ Β¬dances h b β§ Β¬dances g c ... | combinatorics | hard | dev | 2,026 | null | null | [
"combinatorics"
] | true |
putnam_1964_b2 | PutnamBench (lean4/src, informal/putnam.json) | a635813f5edf297d928f9bd5ef8324a26bdde09c | putnam_1964_b2 | null | Apache-2.0 (Lean statement); informal text available with permission from the MAA | [Informal statement not redistributed: PutnamBench states it is available with permission from the MAA. Problem: Putnam 1964 B2; source: PutnamBench informal/putnam.json, commit a635813f.] | import Mathlib
open Set Function Filter Topology
| theorem putnam_1964_b2
(S : Type*) [Fintype S] [Nonempty S]
(P : Finset (Set S))
(hPP : β T β P, β U β P, T β© U β β
)
(hPS : Β¬β T : Set S, T β P β§ (β U β P, T β© U β β
))
: (P.card = 2 ^ (Fintype.card S - 1)) := by
| import Mathlib
open Set Function Filter Topology
theorem putnam_1964_b2
(S : Type*) [Fintype S] [Nonempty S]
(P : Finset (Set S))
(hPP : β T β P, β U β P, T β© U β β
)
(hPS : Β¬β T : Set S, T β P β§ (β U β P, T β© U β β
))
: (P.card = 2 ^ (Fintype.card S - 1)) := by
| combinatorics | hard | test | 2,026 | null | null | [
"set_theory",
"combinatorics"
] | true |
putnam_1995_b1 | PutnamBench (lean4/src, informal/putnam.json) | a635813f5edf297d928f9bd5ef8324a26bdde09c | putnam_1995_b1 | null | Apache-2.0 (Lean statement); informal text available with permission from the MAA | [Informal statement not redistributed: PutnamBench states it is available with permission from the MAA. Problem: Putnam 1995 B1; source: PutnamBench informal/putnam.json, commit a635813f.] | import Mathlib
open Filter Topology Real Nat
| theorem putnam_1995_b1
(Pi : Finpartition (Finset.range 9) β (Finset.range 9) β β)
(Pi_def : β partition k, Pi partition k = (Exists.choose (Finpartition.exists_mem partition k.2)).card) :
β Pt1 Pt2,
β x y : Finset.range 9,
x β y β§
Pi Pt1 x = Pi Pt1 y β§
Pi Pt2 x = Pi Pt2 y := by
| import Mathlib
open Filter Topology Real Nat
theorem putnam_1995_b1
(Pi : Finpartition (Finset.range 9) β (Finset.range 9) β β)
(Pi_def : β partition k, Pi partition k = (Exists.choose (Finpartition.exists_mem partition k.2)).card) :
β Pt1 Pt2,
β x y : Finset.range 9,
x β y β§
Pi Pt1 x = Pi Pt1 y... | combinatorics | hard | test | 2,026 | null | null | [
"combinatorics"
] | true |
putnam_1965_b2 | PutnamBench (lean4/src, informal/putnam.json) | a635813f5edf297d928f9bd5ef8324a26bdde09c | putnam_1965_b2 | null | Apache-2.0 (Lean statement); informal text available with permission from the MAA | [Informal statement not redistributed: PutnamBench states it is available with permission from the MAA. Problem: Putnam 1965 B2; source: PutnamBench informal/putnam.json, commit a635813f.] | import Mathlib
open EuclideanGeometry Topology Filter Complex
| theorem putnam_1965_b2
(n : β)
(hn : n > 1)
(won : Fin n β Fin n β Bool)
(hirrefl : β i : Fin n, won i i = false)
(hantisymm : β i j : Fin n, i β j β won i j = Β¬won j i)
(w l : Fin n β β€)
(hw : w = fun r : Fin n => β j : Fin n, (if won r j then 1 else 0))
(hl : l = fun r : Fin n => n - 1 - w r)
: β r : Fin n, (w r)^2 =... | import Mathlib
open EuclideanGeometry Topology Filter Complex
theorem putnam_1965_b2
(n : β)
(hn : n > 1)
(won : Fin n β Fin n β Bool)
(hirrefl : β i : Fin n, won i i = false)
(hantisymm : β i j : Fin n, i β j β won i j = Β¬won j i)
(w l : Fin n β β€)
(hw : w = fun r : Fin n => β j : Fin n, (if won r j then 1 else 0))
... | combinatorics | hard | test | 2,026 | null | null | [
"combinatorics"
] | true |
Formal-MAS evaluation set
45 competition problems stated in Lean 4 (Mathlib), used to evaluate the Formal-MAS multi-agent prover. The set was fixed before any tuning on it. Code: https://huggingface.co/NicolasT08/formal-mas
Files
data/eval.jsonl: one problem per line:id,source,source_id,source_commit,source_split,license,informal,header,formal_statement,statement,area,difficulty,split(dev/test),seed; some records also keepformal_statement_originalandconverted(see "Allowed edits").data/curation/:criteria.yaml(difficulty, area, grid, seed, labels),discarded.json(every discarded problem and why),fidelity_review.yaml(informal-to-formal review, notes, verification limits, post-test review),build_info.json(Lean back-end of the build).docs/results_explained.md: the results in prose, table by table.docs/results/: the study's tables (results.md,tables.tex), figure SR(k), method notes, the Baseline-B review sheet and the scripts that regenerate them from the run logs (the run logs themselves are not published).
The split column is the study's own dev / test partition (15 / 30), not a Hub split: the
file is exposed as one table; filter on split.
Sources and licenses
| Source | Commit | License |
|---|---|---|
miniF2F, Lean 4 port in DeepSeek-Prover-V1.5 (datasets/minif2f.jsonl) |
2c4ba911 |
MIT |
PutnamBench (lean4/src, informal/putnam.json) |
a635813f |
Apache-2.0 (Lean statements) |
Each record keeps its source, original id, source commit and license (license column).
The set has no single license; each part keeps the license of its source (see LICENSE.md):
- miniF2F problems (40 records; Lean statements and informal text): MIT, as in the
DeepSeek-Prover-V1.5 Lean 4 port (
LICENSE-CODE). - PutnamBench Lean statements (5 records): Apache-2.0.
- PutnamBench informal statements: not redistributed. PutnamBench states that they are available
with permission from the MAA, so for the 5
putnam_*recordsinformalholds a reference to the problem instead (informal_redacted: true); the Lean statements (Apache-2.0) are included.
Grid, difficulty and split
Difficulty is decided only by the competition of origin: easy = mathd_*; medium = AMC 12, AIME
and the textbook families algebra_*, numbertheory_*, induction_*; hard = IMO, IMO Shortlist,
Putnam. Area labels come from reading the informal text (strict combinatorics; geometry,
trigonometry and real analysis out of scope). PutnamBench is used only for hard Γ combinatorics,
where miniF2F has no pool.
| algebra | number_theory | combinatorics | total | |
|---|---|---|---|---|
| easy | 8 (3 dev) | 7 (2 dev) | 0 | 15 |
| medium | 7 (2 dev) | 6 (2 dev) | 2 (1 dev) | 15 |
| hard | 5 (2 dev) | 5 (1 dev) | 5 (2 dev) | 15 |
| total | 20 | 18 | 7 | 45 |
Target 5 per cell; miniF2F has no strict combinatorics at easy level and only 2 at medium, so the
missing 8 were reassigned within the same difficulty. Within a cell, problems are taken in the order
of sha256("2026:<id>"); a problem that does not compile with sorry is replaced by the next one.
dev slots (15) are allocated per cell by largest remainder. No system result informed selection.
dev was used for all tuning; test (30) only for the final runs.
Allowed edits
β x in s/β x in sββ x β s/β x β sby script (original kept informal_statement_original, record marked"converted": "big_operator_in_to_mem").- PutnamBench's
:= sorrybecomes:= by, the format of all records.
Fidelity review
Every statement was compared with its informal text by hand (data/curation/fidelity_review.yaml):
the review establishes that statements match the informal problem, not that they are true. After
the test runs, every solved problem with a fidelity note was reviewed again by hand before being
counted (post_test_review); no reported number changed.
Contamination
All problems are public (miniF2F, PutnamBench and their competitions), so they may be in the training data of the models evaluated; results measure the system on public problems, not on unseen ones. Lean checks every accepted proof, so contamination can make a proof easier to find but cannot make a wrong proof pass.
Results (test split, n = 30)
Full system (C5) 24/30 verified; without the Critic (C4) 14/30; LLM-only Lean proof (C1) 20/30;
without retrieval (C3) 20/30. See docs/results/results.md.
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