id stringlengths 20 28 | math_problem stringlengths 88 843 | solution stringlengths 1 421 ⌀ | formal_statement stringlengths 187 675 | formal_statement_easy stringlengths 188 667 | is_hard bool 2
classes |
|---|---|---|---|---|---|
fimo_2008_algebra_p3_2 | Let $S \subseteq \mathbb{R}$ be a set of real numbers. We say that a pair $(f, g)$ of functions from $S$ into $S$ is a Spanish Couple on $S$, if they satisfy the following conditions:
(i) Both functions are strictly increasing, i.e. $f(x)<f(y)$ and $g(x)<g(y)$ for all $x, y \in S$ with $x<y$
(ii) The inequality $f(g(... | YES | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2008_algebra_p3_2_solution : Prop := sorry
-- true
theorem fimo_2008_algebra_p3_2
(S : Set ℝ)
(hS : S = {x | ∃ a b : PNat, x = (a : ℝ) - 1 / (b : ℝ)}) :
(∃ f g : S → S, StrictMono f ∧ Strict... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2008_algebra_p3_2_solution : Prop := true
theorem fimo_2008_algebra_p3_2
(S : Set ℝ)
(hS : S = {x | ∃ a b : PNat, x = (a : ℝ) - 1 / (b : ℝ)}) :
(∃ f g : S → S, StrictMono f ∧ StrictMono g ∧ ... | true |
fimo_2008_algebra_p3_1 | Let $S \subseteq \mathbb{R}$ be a set of real numbers. We say that a pair $(f, g)$ of functions from $S$ into $S$ is a Spanish Couple on $S$, if they satisfy the following conditions:
(i) Both functions are strictly increasing, i.e. $f(x)<f(y)$ and $g(x)<g(y)$ for all $x, y \in S$ with $x<y$
(ii) The inequality $f(g(... | NO | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2008_algebra_p3_1_solution : Prop := sorry
-- false
theorem fimo_2008_algebra_p3_1 :
(∃ f g : PNat → PNat, StrictMono f ∧ StrictMono g ∧ ∀ x, f (g (g x)) < g (f x)) ↔ fimo_2008_algebra_p3_1_solu... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2008_algebra_p3_1_solution : Prop := false
theorem fimo_2008_algebra_p3_1 :
(∃ f g : PNat → PNat, StrictMono f ∧ StrictMono g ∧ ∀ x, f (g (g x)) < g (f x)) ↔ fimo_2008_algebra_p3_1_solution := b... | true |
fimo_2016_number_theory_p2 | Let $\tau(n)$ be the number of positive divisors of $n$. Let $\tau_{1}(n)$ be the number of positive divisors of $n$ which have remainders 1 when divided by 3 . Find all possible integral values of the fraction $\frac{\tau(10 n)}{\tau_{1}(10 n)}$ | all composite numbers together with 2 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2016_number_theory_p2_solution : Set ℤ := sorry
-- Nat.cast '' {k : ℕ | k = 2 ∨ Composite k}
theorem fimo_2016_number_theory_p2
(τ : ℕ → ℕ)
(hτ : τ = fun n => (Nat.divisors n).card)
(τ₁ : ℕ... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2016_number_theory_p2_solution : Set ℤ := Nat.cast '' {k : ℕ | k = 2 ∨ Composite k}
theorem fimo_2016_number_theory_p2
(τ : ℕ → ℕ)
(hτ : τ = fun n => (Nat.divisors n).card)
(τ₁ : ℕ → ℕ)
(... | true |
fimo_2016_algebra_p5_1 | Prove that for every positive integer $n$, there exists a fraction $\frac{a}{b}$ where $a$ and $b$ are integers satisfying $0<b \leqslant \sqrt{n}+1$ and $\sqrt{n} \leqslant \frac{a}{b} \leqslant \sqrt{n+1}$. | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2016_algebra_p5_1
(n : ℕ)
(hn : 0 < n) :
∃ (a b : ℤ), 0 < b ∧ (b : ℝ) ≤ Real.sqrt (n : ℝ) + 1 ∧ Real.sqrt (n : ℝ) ≤ (a : ℝ) / (b : ℝ) ∧ (a : ℝ) / (b : ℝ) ≤ Real.sqrt ((n : ℝ) + 1) := by sorry
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2016_algebra_p5_1
(n : ℕ)
(hn : 0 < n) :
∃ (a b : ℤ), 0 < b ∧ (b : ℝ) ≤ Real.sqrt (n : ℝ) + 1 ∧ Real.sqrt (n : ℝ) ≤ (a : ℝ) / (b : ℝ) ∧ (a : ℝ) / (b : ℝ) ≤ Real.sqrt ((n : ℝ) + 1) := by sorry
| false |
fimo_2009_algebra_p7 | Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all real $x, y$ the identity
$$
f(x f(x+y))=f(y f(x))+x^{2} .
$$ | f(x)=x and f(x)=-x | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2009_algebra_p7_solution : Set (ℝ → ℝ) := sorry
-- { (fun x => x), (fun x => -x) }
theorem fimo_2009_algebra_p7 :
{f : ℝ → ℝ | ∀ x y, f (x * f (x + y)) = f (y * f x) + x ^ 2} = fimo_2009_algebra... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2009_algebra_p7_solution : Set (ℝ → ℝ) := { (fun x => x), (fun x => -x) }
theorem fimo_2009_algebra_p7 :
{f : ℝ → ℝ | ∀ x y, f (x * f (x + y)) = f (y * f x) + x ^ 2} = fimo_2009_algebra_p7_solut... | true |
fimo_2010_number_theory_p1_2 | Find the least positive integer $n$ for which there exists a set $\left\{s_{1}, s_{2}, \ldots, s_{n}\right\}$ consisting of $n$ distinct positive integers such that
$$
\left(1-\frac{1}{s_{1}}\right)\left(1-\frac{1}{s_{2}}\right) \ldots\left(1-\frac{1}{s_{n}}\right)=\frac{42}{2010} .
$$ | n=48 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Finset
noncomputable abbrev fimo_2010_number_theory_p1_2_solution : ℕ := sorry
-- 48
theorem fimo_2010_number_theory_p1_2 :
sInf { n : ℕ | n > 0 ∧ ∃ S : Finset ℕ, S.card = n ∧ (∀ s ∈ S, 0 < s) ∧ ∏ s in S, (1 - (1 : ℚ) /... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Finset
noncomputable abbrev fimo_2010_number_theory_p1_2_solution : ℕ := 48
theorem fimo_2010_number_theory_p1_2 :
sInf { n : ℕ | n > 0 ∧ ∃ S : Finset ℕ, S.card = n ∧ (∀ s ∈ S, 0 < s) ∧ ∏ s in S, (1 - (1 : ℚ) / s) = 42 ... | true |
fimo_2010_number_theory_p1_1 | Find the least positive integer $n$ for which there exists a set $\left\{s_{1}, s_{2}, \ldots, s_{n}\right\}$ consisting of $n$ distinct positive integers such that
$$
\left(1-\frac{1}{s_{1}}\right)\left(1-\frac{1}{s_{2}}\right) \ldots\left(1-\frac{1}{s_{n}}\right)=\frac{51}{2010} .
$$ | n=39 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Finset
noncomputable abbrev fimo_2010_number_theory_p1_1_solution : ℕ := sorry
-- 39
theorem fimo_2010_number_theory_p1_1 :
sInf { n : ℕ | n > 0 ∧ ∃ S : Finset ℕ, S.card = n ∧ (∀ s ∈ S, 0 < s) ∧ ∏ s in S, (1 - (1 : ℚ) /... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Finset
noncomputable abbrev fimo_2010_number_theory_p1_1_solution : ℕ := 39
theorem fimo_2010_number_theory_p1_1 :
sInf { n : ℕ | n > 0 ∧ ∃ S : Finset ℕ, S.card = n ∧ (∀ s ∈ S, 0 < s) ∧ ∏ s in S, (1 - (1 : ℚ) / s) = 51 ... | true |
fimo_2018_algebra_p7 | Find the maximal value of
$$
S=\sqrt[3]{\frac{a}{b+7}}+\sqrt[3]{\frac{b}{c+7}}+\sqrt[3]{\frac{c}{d+7}}+\sqrt[3]{\frac{d}{a+7}}
$$
where $a, b, c, d$ are nonnegative real numbers which satisfy $a+b+c+d=100$. | \frac{8}{\sqrt[3]{7}} | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2018_algebra_p7_solution : ℝ := sorry
-- 8 / (7 ^ (1/3 : ℝ))
theorem fimo_2018_algebra_p7 :
sSup {s : ℝ | ∃ a b c d : ℝ, 0 ≤ a ∧ 0 ≤ b ∧ 0 ≤ c ∧ 0 ≤ d ∧ a + b + c + d = 100 ∧ s = (a / (b + 7)) ... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2018_algebra_p7_solution : ℝ := 8 / (7 ^ (1/3 : ℝ))
theorem fimo_2018_algebra_p7 :
sSup {s : ℝ | ∃ a b c d : ℝ, 0 ≤ a ∧ 0 ≤ b ∧ 0 ≤ c ∧ 0 ≤ d ∧ a + b + c + d = 100 ∧ s = (a / (b + 7)) ^ (1/3 : ... | true |
fimo_2016_number_theory_p3 | Define $P(n)=n^{2}+n+1$. For any positive integers $a$ and $b$, the set
$$
\{P(a), P(a+1), P(a+2), \ldots, P(a+b)\}
$$
is said to be fragrant if none of its elements is relatively prime to the product of the other elements. Determine the smallest size of a fragrant set. | 6 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2016_number_theory_p3_solution : ℕ := sorry
-- 6
theorem fimo_2016_number_theory_p3 :
sInf {k : ℕ | k ≥ 2 ∧ ∃ a : ℕ, a > 0 ∧
(∀ i ∈ Finset.Icc a (a + k - 1),
∃ j ∈ Finset.Icc a (a + k... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2016_number_theory_p3_solution : ℕ := 6
theorem fimo_2016_number_theory_p3 :
sInf {k : ℕ | k ≥ 2 ∧ ∃ a : ℕ, a > 0 ∧
(∀ i ∈ Finset.Icc a (a + k - 1),
∃ j ∈ Finset.Icc a (a + k - 1), i ... | true |
fimo_2015_algebra_p2 | Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ with the property that
$$
f(x-f(y))=f(f(x))-f(y)-1
$$
holds for all $x, y \in \mathbb{Z}$. | there are two such functions, namely the constant function $x \mapsto-1$ and the successor function $x \mapsto x+1$. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2015_algebra_p2_solution : Set (ℤ → ℤ) := sorry
-- { (fun _ => -1), (fun x => x + 1) }
theorem fimo_2015_algebra_p2 :
{ f : ℤ → ℤ | ∀ x y : ℤ, f (x - f y) = f (f x) - f y - 1 } = fimo_2015_alge... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2015_algebra_p2_solution : Set (ℤ → ℤ) := { (fun _ => -1), (fun x => x + 1) }
theorem fimo_2015_algebra_p2 :
{ f : ℤ → ℤ | ∀ x y : ℤ, f (x - f y) = f (f x) - f y - 1 } = fimo_2015_algebra_p2_so... | true |
fimo_2016_algebra_p3 | Find all integers $n \geqslant 3$ with the following property: for all real numbers $a_{1}, a_{2}, \ldots, a_{n}$ and $b_{1}, b_{2}, \ldots, b_{n}$ satisfying $\left|a_{k}\right|+\left|b_{k}\right|=1$ for $1 \leqslant k \leqslant n$, there exist $x_{1}, x_{2}, \ldots, x_{n}$, each of which is either $-1$ or 1 , such th... | n can be any odd integer greater than or equal to 3 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2016_algebra_p3_solution : Set ℕ := sorry
-- { n | 3 ≤ n ∧ Odd n }
theorem fimo_2016_algebra_p3 :
{ n : ℕ | 3 ≤ n ∧
(∀ (a b : Fin n → ℝ),
(∀ k, abs (a k) + abs (b k) = 1) →
... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2016_algebra_p3_solution : Set ℕ := { n | 3 ≤ n ∧ Odd n }
theorem fimo_2016_algebra_p3 :
{ n : ℕ | 3 ≤ n ∧
(∀ (a b : Fin n → ℝ),
(∀ k, abs (a k) + abs (b k) = 1) →
(∃ x : ... | true |
fimo_2009_algebra_p3 | Determine all functions $f$ from the set of positive integers into the set of positive integers such that for all $x$ and $y$ there exists a non degenerated triangle with sides of lengths
$$
x, \quad f(y) \text { and } f(y+f(x)-1) .
$$ | the identity function $f(x)=x$ | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Function
abbrev fimo_2009_algebra_p3_solution : Set (ℕ+ → ℕ+) := sorry
-- { id }
theorem fimo_2009_algebra_p3 :
{ f : ℕ+ → ℕ+ | (∀ x y,
x + f y > f (y + f x - 1) ∧
x + f (y + f x - 1) > f y ∧
f y + f (y... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Function
abbrev fimo_2009_algebra_p3_solution : Set (ℕ+ → ℕ+) := { id }
theorem fimo_2009_algebra_p3 :
{ f : ℕ+ → ℕ+ | (∀ x y,
x + f y > f (y + f x - 1) ∧
x + f (y + f x - 1) > f y ∧
f y + f (y + f x - ... | true |
fimo_2016_algebra_p8 | Determine the largest real number $a$ such that for all $n \geqslant 1$ and for all real numbers $x_{0}, x_{1}, \ldots, x_{n}$ satisfying $0=x_{0}<x_{1}<x_{2}<\cdots<x_{n}$, we have
$$
\frac{1}{x_{1}-x_{0}}+\frac{1}{x_{2}-x_{1}}+\cdots+\frac{1}{x_{n}-x_{n-1}} \geqslant a\left(\frac{2}{x_{1}}+\frac{3}{x_{2}}+\cdots+\fr... | \frac{4}{9} | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2016_algebra_p8_solution : ℝ := sorry
-- 4 / 9
theorem fimo_2016_algebra_p8 :
sSup { a : ℝ |
∀ (n : ℕ) (hn : 1 ≤ n)
(x : Fin (n + 1) → ℝ),
x 0 = 0 →
(∀ i : Fin n, x (i.cas... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2016_algebra_p8_solution : ℝ := 4 / 9
theorem fimo_2016_algebra_p8 :
sSup { a : ℝ |
∀ (n : ℕ) (hn : 1 ≤ n)
(x : Fin (n + 1) → ℝ),
x 0 = 0 →
(∀ i : Fin n, x (i.castSucc) < ... | true |
fimo_2012_number_theory_p4_2 | An integer $a$ is called friendly if the equation $\left(m^{2}+n\right)\left(n^{2}+m\right)=a(m-n)^{3}$ has a solution over the positive integers.
Decide whether $a=2$ is friendly. | a=2 is not friendly | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
def friendly (a : ℕ) : Prop :=
∃ m n : ℕ, 0 < m ∧ 0 < n ∧ ((m ^ 2 + n) * (n ^ 2 + m) : ℕ) = a * (m - n) ^ 3
noncomputable abbrev fimo_2012_number_theory_p4_2_solution : Bool := sorry
-- false
theorem fimo_2012_number_t... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
def friendly (a : ℕ) : Prop :=
∃ m n : ℕ, 0 < m ∧ 0 < n ∧ ((m ^ 2 + n) * (n ^ 2 + m) : ℕ) = a * (m - n) ^ 3
noncomputable abbrev fimo_2012_number_theory_p4_2_solution : Bool := false
theorem fimo_2012_number_theory_p4_... | true |
fimo_2012_algebra_p1 | Find all the functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that
$$
f(a)^{2}+f(b)^{2}+f(c)^{2}=2 f(a) f(b)+2 f(b) f(c)+2 f(c) f(a)
$$
for all integers $a, b, c$ satisfying $a+b+c=0$. | the only possible solutions of the functional equation are the constant function $f_{1}(x)=0$ and the following functions:
$$f_{2}(x)=k x^{2} \quad f_{3}(x)=\left\{\begin{array}{ccc}0 & x \text { even } \\k & x \text { odd }\end{array}\right. \quad f_{4}(x)=\left\{\begin{array}{ccc}0 & x \equiv 0 & (\bmod 4) \\k & x \... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Int
noncomputable abbrev fimo_2012_algebra_p1_solution : Set (ℤ → ℤ) := sorry
-- {f | (f = fun _ => 0) ∨
-- (∃ k : ℤ, k ≠ 0 ∧ f = fun x => k * x ^ 2) ∨
-- (∃ k : ℤ, k ≠ 0 ∧ f = fun x => if Even x then 0 else k)... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Int
noncomputable abbrev fimo_2012_algebra_p1_solution : Set (ℤ → ℤ) := {f | (f = fun _ => 0) ∨
theorem fimo_2012_algebra_p1 :
{f : ℤ → ℤ | ∀ a b c : ℤ, a + b + c = 0 → f a ^ 2 + f b ^ 2 + f c ^ 2 = 2 * f a * f b + 2 * ... | true |
fimo_2016_algebra_p2 | Find the smallest real constant $C$ such that for any positive real numbers $a_{1}, a_{2}, a_{3}, a_{4}$ and $a_{5}$ (not necessarily distinct), one can always choose distinct subscripts $i, j, k$ and $l$ such that
$$
\left|\frac{a_{i}}{a_{j}}-\frac{a_{k}}{a_{l}}\right| \leqslant C .
$$ | \frac{1}{2} | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2016_algebra_p2_solution : ℝ := sorry
-- 1/2
theorem fimo_2016_algebra_p2 :
sInf { C : ℝ | 0 ≤ C ∧
∀ a : Fin 5 → ℝ, (∀ i, 0 < a i) →
∃ i j k l : Fin 5, i ≠ j ∧ i ≠ k ∧ i ≠ l ∧ j ≠ k ∧ j... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2016_algebra_p2_solution : ℝ := 1/2
theorem fimo_2016_algebra_p2 :
sInf { C : ℝ | 0 ≤ C ∧
∀ a : Fin 5 → ℝ, (∀ i, 0 < a i) →
∃ i j k l : Fin 5, i ≠ j ∧ i ≠ k ∧ i ≠ l ∧ j ≠ k ∧ j ≠ l ∧ k ... | true |
fimo_2020_algebra_p3 | Suppose that $a, b, c, d$ are positive real numbers satisfying $(a+c)(b+d)=a c+b d$. Find the smallest possible value of
$$
S=\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}
$$ | 8 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2020_algebra_p3_solution : ℝ := sorry
-- 8
theorem fimo_2020_algebra_p3 :
sInf {S | ∃ (a b c d : ℝ), 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d ∧ (a + c) * (b + d) = a * c + b * d ∧ S = a / b + b / c + c / ... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2020_algebra_p3_solution : ℝ := 8
theorem fimo_2020_algebra_p3 :
sInf {S | ∃ (a b c d : ℝ), 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d ∧ (a + c) * (b + d) = a * c + b * d ∧ S = a / b + b / c + c / d + d / a... | true |
fimo_2017_number_theory_p8 | Let $p$ be an odd prime number and $\mathbb{Z}_{>0}$ be the set of positive integers. Suppose that a function $f: \mathbb{Z}_{>0} \times \mathbb{Z}_{>0} \rightarrow\{0,1\}$ satisfies the following properties:
\begin{itemize}
\item $f(1,1)=0$
\item $f(a, b)+f(b, a)=1$ for any pair of relatively prime positive inte... | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2017_number_theory_p8
(p : ℕ)
(hp_prime : p.Prime)
(hp_odd : Odd p)
(f : ℕ+ → ℕ+ → ℕ)
(hf_codomain : ∀ a b : ℕ+, f a b = 0 ∨ f a b = 1)
(hf1 : f 1 1 = 0)
(hf2 : ∀ a b : ℕ+, Nat.Coprime a b → ¬ (a... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2017_number_theory_p8
(p : ℕ)
(hp_prime : p.Prime)
(hp_odd : Odd p)
(f : ℕ+ → ℕ+ → ℕ)
(hf_codomain : ∀ a b : ℕ+, f a b = 0 ∨ f a b = 1)
(hf1 : f 1 1 = 0)
(hf2 : ∀ a b : ℕ+, Nat.Coprime a b → ¬ (a... | false |
fimo_2013_number_theory_p6 | Determine all functions $f: \mathbb{Q} \longrightarrow \mathbb{Z}$ satisfying
$$
f\left(\frac{f(x)+a}{b}\right)=f\left(\frac{x+a}{b}\right)
$$
for all $x \in \mathbb{Q}, a \in \mathbb{Z}$, and $b \in \mathbb{Z}_{>0}$. (Here, $\mathbb{Z}_{>0}$ denotes the set of positive integers.) | there are three kinds of such functions, which are: all constant functions, the floor function, and the ceiling function. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2013_number_theory_p6_solution : Set (ℚ → ℤ) := sorry
-- Set.range (fun c : ℤ => Function.const ℚ c) ∪ {Int.floor, Int.ceil}
theorem fimo_2013_number_theory_p6 :
{ f : ℚ → ℤ | ∀ (x : ℚ) (a : ℤ)... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2013_number_theory_p6_solution : Set (ℚ → ℤ) := Set.range (fun c : ℤ => Function.const ℚ c) ∪ {Int.floor, Int.ceil}
theorem fimo_2013_number_theory_p6 :
{ f : ℚ → ℤ | ∀ (x : ℚ) (a : ℤ) (b : ℤ),... | true |
fimo_2020_algebra_p1_2 | For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$,
$$
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x
$$ | b_{n}=N / 2 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2020_algebra_p1_2_solution : ℕ → ℝ := sorry
-- fun N => (N : ℝ) / 2
theorem fimo_2020_algebra_p1_2
(N : ℕ)
(hN : 0 < N) :
sInf {b | ∀ (x : ℝ), ((x ^ (2 * N) + 1) / 2) ^ (1 / (N : ℝ)) ≤ b * ... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2020_algebra_p1_2_solution : ℕ → ℝ := fun N => (N : ℝ) / 2
theorem fimo_2020_algebra_p1_2
(N : ℕ)
(hN : 0 < N) :
sInf {b | ∀ (x : ℝ), ((x ^ (2 * N) + 1) / 2) ^ (1 / (N : ℝ)) ≤ b * (x - 1) ^... | true |
fimo_2010_number_theory_p5 | Let $\mathbb{N}$ be the set of all positive integers. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that the number $(f(m)+n)(m+f(n))$ is a square for all $m, n \in \mathbb{N}$. | all functions of the form $f(n)=n+c$, where $c in \mathbb{N} \cup\{0\}$ | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2010_number_theory_p5_solution : Set (ℕ → ℕ) := sorry
-- { f | ∃ c, ∀ n > 0, f n = n + c }
theorem fimo_2010_number_theory_p5 :
{ f : ℕ → ℕ | (∀ n, n > 0 → f n > 0) ∧ (∀ m n, m > 0 → n > 0 → Is... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2010_number_theory_p5_solution : Set (ℕ → ℕ) := { f | ∃ c, ∀ n > 0, f n = n + c }
theorem fimo_2010_number_theory_p5 :
{ f : ℕ → ℕ | (∀ n, n > 0 → f n > 0) ∧ (∀ m n, m > 0 → n > 0 → IsSquare ((... | true |
fimo_2009_number_theory_p4 | Find all positive integers $n$ such that there exists a sequence of positive integers $a_{1}, a_{2}, \ldots, a_{n}$ satisfying
$$
a_{k+1}=\frac{a_{k}^{2}+1}{a_{k-1}+1}-1
$$
for every $k$ with $2 \leq k \leq n-1$. | n=1,2,3,4 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2009_number_theory_p4_solution : Set ℕ := sorry
-- {1, 2, 3, 4}
theorem fimo_2009_number_theory_p4 :
{ n : ℕ | n > 0 ∧
(∃ a : ℕ → ℕ,
(∀ i ∈ Finset.Icc 1 n, a i > 0) ∧
(∀ k ∈ Fins... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2009_number_theory_p4_solution : Set ℕ := {1, 2, 3, 4}
theorem fimo_2009_number_theory_p4 :
{ n : ℕ | n > 0 ∧
(∃ a : ℕ → ℕ,
(∀ i ∈ Finset.Icc 1 n, a i > 0) ∧
(∀ k ∈ Finset.Icc 2 ... | true |
fimo_2016_number_theory_p5 | Let $a$ be a positive integer which is not a square number. Denote by $A$ the set of all positive integers $k$ such that
$$
k=\frac{x^{2}-a}{x^{2}-y^{2}}
$$
for some integers $x$ and $y$ with $x>\sqrt{a}$. Denote by $B$ the set of all positive integers $k$ such that (1) is satisfied for some integers $x$ and $y$ with... | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2016_number_theory_p5
(a : ℕ)
(ha_pos : 0 < a)
(ha_not_sq : ¬ IsSquare a)
(A : Set ℕ)
(hA : A = { k : ℕ | 0 < k ∧ ∃ (x y : ℤ), (x : ℝ) > Real.sqrt (a : ℝ) ∧ (k : ℤ) * (x^2 - y^2) = x^2 - (a : ℤ) })
... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2016_number_theory_p5
(a : ℕ)
(ha_pos : 0 < a)
(ha_not_sq : ¬ IsSquare a)
(A : Set ℕ)
(hA : A = { k : ℕ | 0 < k ∧ ∃ (x y : ℤ), (x : ℝ) > Real.sqrt (a : ℝ) ∧ (k : ℤ) * (x^2 - y^2) = x^2 - (a : ℤ) })
... | false |
fimo_2018_number_theory_p6 | Let $f:\{1,2,3, \ldots\} \rightarrow\{2,3, \ldots\}$ be a function such that $f(m+n) \mid f(m)+f(n)$ for all pairs $m, n$ of positive integers. Prove that there exists a positive integer $c>1$ which divides all values of $f$. | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2018_number_theory_p6
(f : ℕ+ → ℕ)
(hf_codomain : ∀ n, f n ≥ 2)
(hf_property : ∀ m n, f (m + n) ∣ f m + f n) :
∃ c > 1, ∀ n, c ∣ f n := by
sorry
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2018_number_theory_p6
(f : ℕ+ → ℕ)
(hf_codomain : ∀ n, f n ≥ 2)
(hf_property : ∀ m n, f (m + n) ∣ f m + f n) :
∃ c > 1, ∀ n, c ∣ f n := by
sorry
| false |
fimo_2020_number_theory_p7 | Let $\mathcal{S}$ be a set consisting of $n \geqslant 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_{1}, a_{2}, \ldots, a_{n}$ so that $a_{i}$ does not divide $a_{i-1}+a_{i+1}$ for all $i=2,3, \ldots, n-1$. | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2020_number_theory_p7
(S : Finset ℕ)
(hn : 3 ≤ S.card)
(h_pos : ∀ x ∈ S, 0 < x)
(h_sum : ∀ {x y z : ℕ}, x ∈ S → y ∈ S → z ∈ S → x ≠ y → z ≠ x + y) :
∃ a : List ℕ,
a.Nodup ∧
a.toFinset = S ∧
... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2020_number_theory_p7
(S : Finset ℕ)
(hn : 3 ≤ S.card)
(h_pos : ∀ x ∈ S, 0 < x)
(h_sum : ∀ {x y z : ℕ}, x ∈ S → y ∈ S → z ∈ S → x ≠ y → z ≠ x + y) :
∃ a : List ℕ,
a.Nodup ∧
a.toFinset = S ∧
... | false |
fimo_2007_algebra_p2 | Consider those functions $f: \mathbb{N} \rightarrow \mathbb{N}$ which satisfy the condition
$$
f(m+n) \geq f(m)+f(f(n))-1
$$
for all $m, n \in \mathbb{N}$. Find all possible values of $f(2007)$.
$(\mathbb{N}$ denotes the set of all positive integers.) | 1, 2, .., 2008 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2007_algebra_p2_solution : Set ℕ := sorry
-- Set.Icc 1 2008
theorem fimo_2007_algebra_p2 :
{ (f 2007 : ℕ) | (f : ℕ+ → ℕ+) (h_cond : ∀ m n, f (m + n) + 1 ≥ f m + f (f n)) } = fimo_2007_algebra_p... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2007_algebra_p2_solution : Set ℕ := Set.Icc 1 2008
theorem fimo_2007_algebra_p2 :
{ (f 2007 : ℕ) | (f : ℕ+ → ℕ+) (h_cond : ∀ m n, f (m + n) + 1 ≥ f m + f (f n)) } = fimo_2007_algebra_p2_solutio... | true |
fimo_2018_algebra_p1 | Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f: \mathbb{Q}>0 \rightarrow \mathbb{Q}>0$ satisfying
$$
f\left(x^{2} f(y)^{2}\right)=f(x)^{2} f(y)
$$
for all $x, y \in \mathbb{Q}>0$. | $f(x)=1$ for all $x \in \mathbb{Q}>0$ | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
abbrev PosRat := { r : Rat // 0 < r }
noncomputable abbrev fimo_2018_algebra_p1_solution : Set (PosRat → PosRat) := sorry
-- { fun (_ : PosRat) => 1 }
theorem fimo_2018_algebra_p1 :
{ f : (PosRat → PosRat) | ∀ (x y : P... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
abbrev PosRat := { r : Rat // 0 < r }
noncomputable abbrev fimo_2018_algebra_p1_solution : Set (PosRat → PosRat) := { fun (_ : PosRat) => 1 }
theorem fimo_2018_algebra_p1 :
{ f : (PosRat → PosRat) | ∀ (x y : PosRat), f... | true |
fimo_2016_algebra_p5_2 | Prove that there are infinitely many positive integers $n$ such that there is no fraction $\frac{a}{b}$ where $a$ and $b$ are integers satisfying $0<b \leqslant \sqrt{n}$ and $\sqrt{n} \leqslant \frac{a}{b} \leqslant \sqrt{n+1}$. | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2016_algebra_p5_2 :
Set.Infinite { n : ℕ | n > 0 ∧ ¬∃ (a b : ℤ), 0 < b ∧ (b : ℝ) ≤ Real.sqrt (n : ℝ) ∧ Real.sqrt (n : ℝ) ≤ (a : ℝ) / (b : ℝ) ∧ (a : ℝ) / (b : ℝ) ≤ Real.sqrt ((n : ℝ) + 1) } := by sorry
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2016_algebra_p5_2 :
Set.Infinite { n : ℕ | n > 0 ∧ ¬∃ (a b : ℤ), 0 < b ∧ (b : ℝ) ≤ Real.sqrt (n : ℝ) ∧ Real.sqrt (n : ℝ) ≤ (a : ℝ) / (b : ℝ) ∧ (a : ℝ) / (b : ℝ) ≤ Real.sqrt ((n : ℝ) + 1) } := by sorry
| false |
fimo_2018_algebra_p2 | Find all positive integers $n \geqslant 3$ for which there exist real numbers $a_{1}, a_{2}, \ldots, a_{n}$, $a_{n+1}=a_{1}, a_{n+2}=a_{2}$ such that
$$
a_{i} a_{i+1}+1=a_{i+2}
$$
for all $i=1,2, \ldots, n$. | n can be any multiple of 3 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology
abbrev fimo_2018_algebra_p2_solution : Set ℕ := sorry
-- {n | n ≥ 3 ∧ 3 ∣ n}
theorem fimo_2018_algebra_p2 :
{n : ℕ | n ≥ 3 ∧ (∃ a : (ZMod n) → ℝ, ∀ i : ZMod n, a i * a (i + 1) + 1 = a (i + 2))} = fimo_2018_algebra_p2_soluti... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology
abbrev fimo_2018_algebra_p2_solution : Set ℕ := {n | n ≥ 3 ∧ 3 ∣ n}
theorem fimo_2018_algebra_p2 :
{n : ℕ | n ≥ 3 ∧ (∃ a : (ZMod n) → ℝ, ∀ i : ZMod n, a i * a (i + 1) + 1 = a (i + 2))} = fimo_2018_algebra_p2_solution := by ... | true |
fimo_2012_number_theory_p7 | Find all $n \in \mathbb{N}$ for which there exist nonnegative integers $a_{1}, a_{2}, \ldots, a_{n}$ such that
$$
\frac{1}{2^{a_{1}}}+\frac{1}{2^{a_{2}}}+\cdots+\frac{1}{2^{a_{n}}}=\frac{1}{3^{a_{1}}}+\frac{2}{3^{a_{2}}}+\cdots+\frac{n}{3^{a_{n}}}=1 .
$$ | Such numbers $a_{1}, a_{2}, \ldots, a_{n}$ exist if and only if $n \equiv 1(\bmod 4)$ or $n \equiv 2(\bmod 4)$. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2012_number_theory_p7_solution (n : ℕ) : Prop := sorry
-- n % 4 = 1 ∨ n % 4 = 2
theorem fimo_2012_number_theory_p7
(n : ℕ) :
(∃ (a : Fin n → ℕ),
(∑ i : Fin n, (1 : ℝ) / (2 ^ (a i))) = 1 ∧... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2012_number_theory_p7_solution (n : ℕ) : Prop := n % 4 = 1 ∨ n % 4 = 2
theorem fimo_2012_number_theory_p7
(n : ℕ) :
(∃ (a : Fin n → ℕ),
(∑ i : Fin n, (1 : ℝ) / (2 ^ (a i))) = 1 ∧
(∑ i... | true |
fimo_2007_number_theory_p1 | Find all pairs $(k, n)$ of positive integers for which $7^{k}-3^{n}$ divides $k^{4}+n^{2}$. | (2,4) | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2007_number_theory_p1_solution : Set (ℕ × ℕ) := sorry
-- { (2, 4) }
theorem fimo_2007_number_theory_p1 :
{ p : ℕ × ℕ | 0 < p.1 ∧ 0 < p.2 ∧ ((7^p.1 : ℤ) - (3^p.2 : ℤ)) ∣ (p.1^4 + p.2^2 : ℤ) } = ... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2007_number_theory_p1_solution : Set (ℕ × ℕ) := { (2, 4) }
theorem fimo_2007_number_theory_p1 :
{ p : ℕ × ℕ | 0 < p.1 ∧ 0 < p.2 ∧ ((7^p.1 : ℤ) - (3^p.2 : ℤ)) ∣ (p.1^4 + p.2^2 : ℤ) } = fimo_2007... | true |
fimo_2007_algebra_p3 | Let $n$ be a positive integer, and let $x$ and $y$ be positive real numbers such that $x^{n}+y^{n}=1$. Prove that
$$
\left(\sum_{k=1}^{n} \frac{1+x^{2 k}}{1+x^{4 k}}\right)\left(\sum_{k=1}^{n} \frac{1+y^{2 k}}{1+y^{4 k}}\right)<\frac{1}{(1-x)(1-y)} .
$$ | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat
theorem fimo_2007_algebra_p3
(n : ℕ)
(x y : ℝ)
(hn : n > 0)
(hx : x > 0)
(hy : y > 0)
(h_sum_pow : x^n + y^n = 1) :
(∑ k in Finset.Icc 1 n, (1 + x^(2 * k)) / (1 + x^(4 * k))) *
(∑ k in Finset.Icc 1 n, (1 + y^(2 * k)) / ... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat
theorem fimo_2007_algebra_p3
(n : ℕ)
(x y : ℝ)
(hn : n > 0)
(hx : x > 0)
(hy : y > 0)
(h_sum_pow : x^n + y^n = 1) :
(∑ k in Finset.Icc 1 n, (1 + x^(2 * k)) / (1 + x^(4 * k))) *
(∑ k in Finset.Icc 1 n, (1 + y^(2 * k)) / ... | false |
fimo_2021_algebra_p8 | Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy
$$
(f(a)-f(b))(f(b)-f(c))(f(c)-f(a))=f\left(a b^{2}+b c^{2}+c a^{2}\right)-f\left(a^{2} b+b^{2} c+c^{2} a\right)
$$
for all real numbers $a, b, c$. | $f(x)=\alpha x+\beta$ or $f(x)=\alpha x^{3}+\beta$ where $\alpha \in\{-1,0,1\}$ and $\beta \in \mathbb{R}$ | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2021_algebra_p8_solution : Set (ℝ → ℝ) := sorry
-- {f | ∃ (α β : ℝ), ((α = -1 ∨ α = 0 ∨ α = 1) ∧ f = fun x => α * x + β) ∨ ((α = -1 ∨ α = 0 ∨ α = 1) ∧ f = fun x => α * x^3 + β)}
theorem fimo_2021... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2021_algebra_p8_solution : Set (ℝ → ℝ) := {f | ∃ (α β : ℝ), ((α = -1 ∨ α = 0 ∨ α = 1) ∧ f = fun x => α * x + β) ∨ ((α = -1 ∨ α = 0 ∨ α = 1) ∧ f = fun x => α * x^3 + β)}
theorem fimo_2021_algebra_... | true |
fimo_2008_algebra_p2_1 | Prove the inequality
$$
\frac{x^{2}}{(x-1)^{2}}+\frac{y^{2}}{(y-1)^{2}}+\frac{z^{2}}{(z-1)^{2}} \geq 1
$$
for real numbers $x, y, z \neq 1$ satisfying the condition $x y z=1$. | null | import Mathlib
set_option maxHeartbeats 0
open Real
theorem fimo_2008_algebra_p2_1
(x y z : ℝ)
(h_ne_1 : x ≠ 1 ∧ y ≠ 1 ∧ z ≠ 1)
(h_prod_eq_1 : x * y * z = 1) :
x^2 / (x - 1)^2 + y^2 / (y - 1)^2 + z^2 / (z - 1)^2 ≥ 1 := by sorry
| import Mathlib
set_option maxHeartbeats 0
open Real
theorem fimo_2008_algebra_p2_1
(x y z : ℝ)
(h_ne_1 : x ≠ 1 ∧ y ≠ 1 ∧ z ≠ 1)
(h_prod_eq_1 : x * y * z = 1) :
x^2 / (x - 1)^2 + y^2 / (y - 1)^2 + z^2 / (z - 1)^2 ≥ 1 := by sorry
| false |
fimo_2006_number_theory_p5 | Find all integer solutions of the equation
$$
\frac{x^{7}-1}{x-1}=y^{5}-1 \text {. }
$$ | the given equation has no integer solutions | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Int
noncomputable abbrev fimo_2006_number_theory_p5_solution : Set (ℤ × ℤ) := sorry
-- ∅
theorem fimo_2006_number_theory_p5 :
{ xy : ℤ × ℤ | ∑ i in Finset.range 7, xy.1 ^ i = xy.2 ^ 5 - 1 } = fimo_2006_number_theory_p5_... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Int
noncomputable abbrev fimo_2006_number_theory_p5_solution : Set (ℤ × ℤ) := ∅
theorem fimo_2006_number_theory_p5 :
{ xy : ℤ × ℤ | ∑ i in Finset.range 7, xy.1 ^ i = xy.2 ^ 5 - 1 } = fimo_2006_number_theory_p5_solution ... | true |
fimo_2015_number_theory_p4 | Suppose that $a_{0}, a_{1}, \ldots$ and $b_{0}, b_{1}, \ldots$ are two sequences of positive integers satisfying $a_{0}, b_{0} \geqslant 2$ and
$$
a_{n+1}=\operatorname{gcd}\left(a_{n}, b_{n}\right)+1, \quad b_{n+1}=\operatorname{lcm}\left(a_{n}, b_{n}\right)-1
$$
for all $n \geqslant 0$. Prove that the sequence $\le... | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2015_number_theory_p4
(a b : Nat → Nat)
(h_a0_ge_2 : a 0 ≥ 2)
(h_b0_ge_2 : b 0 ≥ 2)
(h_recurrence_a : ∀ n, a (n + 1) = Nat.gcd (a n) (b n) + 1)
(h_recurrence_b : ∀ n, b (n + 1) = Nat.lcm (a n) (b n) ... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2015_number_theory_p4
(a b : Nat → Nat)
(h_a0_ge_2 : a 0 ≥ 2)
(h_b0_ge_2 : b 0 ≥ 2)
(h_recurrence_a : ∀ n, a (n + 1) = Nat.gcd (a n) (b n) + 1)
(h_recurrence_b : ∀ n, b (n + 1) = Nat.lcm (a n) (b n) ... | false |
fimo_2006_number_theory_p7 | Prove that, for every positive integer $n$, there exists an integer $m$ such that $2^{m}+m$ is divisible by $n$. | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2006_number_theory_p7
(n : ℕ)
(hn : n > 0) :
∃ m : ℕ,
n ∣ (2^m + m) := by sorry
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2006_number_theory_p7
(n : ℕ)
(hn : n > 0) :
∃ m : ℕ,
n ∣ (2^m + m) := by sorry
| false |
fimo_2016_number_theory_p4 | Let $n, m, k$ and $l$ be positive integers with $n \neq 1$ such that $n^{k}+m n^{l}+1$ divides $n^{k+l}-1$. Prove that
\begin{itemize}
\item $m=1$ and $l=2 k ;$ or
\item $l \mid k$ and $m=\frac{n^{k-l}-1}{n^{l}-1}$.
\end{itemize} | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2016_number_theory_p4
(n m k l : ℕ)
(hn_pos : 0 < n)
(hm_pos : 0 < m)
(hk_pos : 0 < k)
(hl_pos : 0 < l)
(hn_ne_1 : n ≠ 1)
(h_div : n^k + m * n^l + 1 ∣ n^(k+l) - 1) :
(m = 1 ∧ l = 2 * k) ∨ (l ∣ ... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2016_number_theory_p4
(n m k l : ℕ)
(hn_pos : 0 < n)
(hm_pos : 0 < m)
(hk_pos : 0 < k)
(hl_pos : 0 < l)
(hn_ne_1 : n ≠ 1)
(h_div : n^k + m * n^l + 1 ∣ n^(k+l) - 1) :
(m = 1 ∧ l = 2 * k) ∨ (l ∣ ... | false |
fimo_2021_number_theory_p7 | Let $a_{1}, a_{2}, a_{3}, \ldots$ be an infinite sequence of positive integers such that $a_{n+2 m}$ divides $a_{n}+a_{n+m}$ for all positive integers $n$ and $m$. Prove that this sequence is eventually periodic, i.e. there exist positive integers $N$ and $d$ such that $a_{n}=a_{n+d}$ for all $n>N$. | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2021_number_theory_p7
(a : ℕ+ → ℕ+)
(h_div : ∀ n m : ℕ+, a (n + 2 * m) ∣ a n + a (n + m)) :
∃ N d : ℕ+, ∀ n : ℕ+, n > N → a n = a (n + d) := by sorry
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2021_number_theory_p7
(a : ℕ+ → ℕ+)
(h_div : ∀ n m : ℕ+, a (n + 2 * m) ∣ a n + a (n + m)) :
∃ N d : ℕ+, ∀ n : ℕ+, n > N → a n = a (n + d) := by sorry
| false |
fimo_2018_number_theory_p5 | Four positive integers $x, y, z$, and $t$ satisfy the relations
$$
x y-z t=x+y=z+t
$$
Is it possible that both $x y$ and $z t$ are perfect squares? | no | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2018_number_theory_p5_solution : Prop := sorry
-- False
theorem fimo_2018_number_theory_p5 :
(∃ (x y z t : ℕ), 0 < x ∧ 0 < y ∧ 0 < z ∧ 0 < t ∧ x * y = z * t + (x + y) ∧ x + y = z + t ∧ IsSquare... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2018_number_theory_p5_solution : Prop := False
theorem fimo_2018_number_theory_p5 :
(∃ (x y z t : ℕ), 0 < x ∧ 0 < y ∧ 0 < z ∧ 0 < t ∧ x * y = z * t + (x + y) ∧ x + y = z + t ∧ IsSquare (x * y) ... | true |
fimo_2014_algebra_p5 | Consider all polynomials $P(x)$ with real coefficients that have the following property: for any two real numbers $x$ and $y$ one has
$$
\left|y^{2}-P(x)\right| \leqslant 2|x| \quad \text { if and only if } \quad\left|x^{2}-P(y)\right| \leqslant 2|y| \text {. }
$$
Determine all possible values of $P(0)$. | the set of possible values of P(0) is (-infinity, 0) U {1} | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Polynomial
noncomputable abbrev fimo_2014_algebra_p5_solution : Set ℝ := sorry
-- Set.Iio 0 ∪ {1}
theorem fimo_2014_algebra_p5 :
{c : ℝ | ∃ P : Polynomial ℝ, (∀ x y : ℝ, (|y^2 - eval x P| ≤ 2 * |x|) ↔ (|x^2 - eval y P| ... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Polynomial
noncomputable abbrev fimo_2014_algebra_p5_solution : Set ℝ := Set.Iio 0 ∪ {1}
theorem fimo_2014_algebra_p5 :
{c : ℝ | ∃ P : Polynomial ℝ, (∀ x y : ℝ, (|y^2 - eval x P| ≤ 2 * |x|) ↔ (|x^2 - eval y P| ≤ 2 * |y|... | true |
fimo_2010_algebra_p1 | Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that the equality
$$
f([x] y)=f(x)[f(y)] .
$$
holds for all $x, y \in \mathbb{R}$. Here, by $[x]$ we denote the greatest integer not exceeding $x$. | $f(x)=$ const $=C$, where $C=0$ or $1 \leq C<2$ | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2010_algebra_p1_solution_set : Set (ℝ → ℝ) := sorry
-- { f | ∃ C, (C = 0 ∨ (1 ≤ C ∧ C < 2)) ∧ (∀ x, f x = C) }
theorem fimo_2010_algebra_p1 :
{ f : ℝ → ℝ | ∀ x y : ℝ, f (Int.floor x * y) = f x ... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2010_algebra_p1_solution_set : Set (ℝ → ℝ) := { f | ∃ C, (C = 0 ∨ (1 ≤ C ∧ C < 2)) ∧ (∀ x, f x = C) }
theorem fimo_2010_algebra_p1 :
{ f : ℝ → ℝ | ∀ x y : ℝ, f (Int.floor x * y) = f x * Int.flo... | true |
fimo_2009_algebra_p5 | Let $f$ be any function that maps the set of real numbers into the set of real numbers. Prove that there exist real numbers $x$ and $y$ such that
$$
f(x-f(y))>y f(x)+x .
$$ | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2009_algebra_p5
(f : ℝ → ℝ) :
∃ x y : ℝ, f (x - f y) > y * f x + x := by sorry
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2009_algebra_p5
(f : ℝ → ℝ) :
∃ x y : ℝ, f (x - f y) > y * f x + x := by sorry
| false |
fimo_2013_algebra_p2 | Prove that in any set of 2000 distinct real numbers there exist two pairs $a>b$ and $c>d$ with $a \neq c$ or $b \neq d$, such that
$$
\left|\frac{a-b}{c-d}-1\right|<\frac{1}{100000} \text {. }
$$ | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2013_algebra_p2
(S : Finset ℝ)
(hS_card : S.card = 2000) :
∃ (a b c d : ℝ),
a ∈ S ∧ b ∈ S ∧ c ∈ S ∧ d ∈ S ∧
a > b ∧ c > d ∧
((a ≠ c) ∨ (b ≠ d)) ∧
abs (((a - b) / (c - d)) - 1) < 1 / 10000... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2013_algebra_p2
(S : Finset ℝ)
(hS_card : S.card = 2000) :
∃ (a b c d : ℝ),
a ∈ S ∧ b ∈ S ∧ c ∈ S ∧ d ∈ S ∧
a > b ∧ c > d ∧
((a ≠ c) ∨ (b ≠ d)) ∧
abs (((a - b) / (c - d)) - 1) < 1 / 10000... | false |
fimo_2019_algebra_p7_1 | Let $\mathbb{Z}$ be the set of integers. We consider functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying
$$
f(f(x+y)+y)=f(f(x)+y)
$$
for all integers $x$ and $y$. For such a function, we say that an integer $v$ is $f$-rare if the set
$$
X_{v}=\{x \in \mathbb{Z}: f(x)=v\}
$$
is finite and nonempty.
Prove th... | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set
theorem fimo_2019_algebra_p7_1 :
∃ (f : ℤ → ℤ),
(∀ x y : ℤ, f (f (x + y) + y) = f (f x + y)) ∧
(∃ v : ℤ, (Set.Finite {x : ℤ | f x = v}) ∧ (Set.Nonempty {x : ℤ | f x = v})) := by sorry
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set
theorem fimo_2019_algebra_p7_1 :
∃ (f : ℤ → ℤ),
(∀ x y : ℤ, f (f (x + y) + y) = f (f x + y)) ∧
(∃ v : ℤ, (Set.Finite {x : ℤ | f x = v}) ∧ (Set.Nonempty {x : ℤ | f x = v})) := by sorry
| false |
fimo_2007_number_theory_p4 | For every integer $k \geq 2$, prove that $2^{3 k}$ divides the number
$$
\left(\begin{array}{c}
2^{k+1} \\
2^{k}
\end{array}\right)-\left(\begin{array}{c}
2^{k} \\
2^{k-1}
\end{array}\right)
$$
but $2^{3 k+1}$ does not. | null | import Mathlib
open Nat
theorem fimo_2007_number_theory_p4
(k : ℕ)
(hk : k ≥ 2) :
(2^(3*k) ∣ (choose (2^(k+1)) (2^k) - choose (2^k) (2^(k-1)))) ∧
¬ (2^(3*k+1) ∣ (choose (2^(k+1)) (2^k) - choose (2^k) (2^(k-1)))) := by sorry
| import Mathlib
open Nat
theorem fimo_2007_number_theory_p4
(k : ℕ)
(hk : k ≥ 2) :
(2^(3*k) ∣ (choose (2^(k+1)) (2^k) - choose (2^k) (2^(k-1)))) ∧
¬ (2^(3*k+1) ∣ (choose (2^(k+1)) (2^k) - choose (2^k) (2^(k-1)))) := by sorry
| false |
fimo_2019_number_theory_p2 | Find all triples $(a, b, c)$ of positive integers such that $a^{3}+b^{3}+c^{3}=(a b c)^{2}$. | $(1,2,3)$ and its permutations | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2019_number_theory_p2_solution : Finset (ℕ × ℕ × ℕ) := sorry
-- { (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1) }
theorem fimo_2019_number_theory_p2 :
{ (a, b, c) : ℕ × ℕ × ℕ | 0 < a ∧ 0... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2019_number_theory_p2_solution : Finset (ℕ × ℕ × ℕ) := { (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1) }
theorem fimo_2019_number_theory_p2 :
{ (a, b, c) : ℕ × ℕ × ℕ | 0 < a ∧ 0 < b ∧ 0 ... | true |
fimo_2010_algebra_p5 | Denote by $\mathbb{Q}^{+}$the set of all positive rational numbers. Determine all functions $f: \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}$ which satisfy the following equation for all $x, y \in \mathbb{Q}^{+}$:
$$
f\left(f(x)^{2} y\right)=x^{3} f(x y) .
$$ | $f(x)=\frac{1}{x}$ | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2010_algebra_p5_solution : Set ({q : ℚ // 0 < q} → {q : ℚ // 0 < q}) := sorry
-- {fun x => x⁻¹}
theorem fimo_2010_algebra_p5 :
{f : {q : ℚ // 0 < q} → {q : ℚ // 0 < q} | ∀ (x y : {q : ℚ // 0 < ... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2010_algebra_p5_solution : Set ({q : ℚ // 0 < q} → {q : ℚ // 0 < q}) := {fun x => x⁻¹}
theorem fimo_2010_algebra_p5 :
{f : {q : ℚ // 0 < q} → {q : ℚ // 0 < q} | ∀ (x y : {q : ℚ // 0 < q}), f (f... | true |
fimo_2015_algebra_p5 | Let $2 \mathbb{Z}+1$ denote the set of odd integers. Find all functions $f: \mathbb{Z} \rightarrow 2 \mathbb{Z}+1$ satisfying
$$
f(x+f(x)+y)+f(x-f(x)-y)=f(x+y)+f(x-y)
$$
for every $x, y \in \mathbb{Z}$. | fix an odd positive integer $d$, an integer $k$, and odd integers $\ell_{0}, \ell_{1}, \ldots, \ell_{d-1}$. Then the function defined as
$$f(m d+i)=2 k m d+\ell_{i} d \quad(m \in \mathbb{Z}, \quad i=0,1, \ldots, d-1)$$
satisfies the problem requirements, and these are all such functions. | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2015_algebra_p5_solution : Set (ℤ → ℤ) := sorry
-- { f | ∃ (d : ℤ) (hd_pos : 0 < d) (hd_odd : Odd d) (k : ℤ) (l : ℤ → ℤ) (_ : ∀ i, 0 ≤ i ∧ i < d → Odd (l i)), ∀ n, f n = (2 * k * (n / d) + l (n % ... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2015_algebra_p5_solution : Set (ℤ → ℤ) := { f | ∃ (d : ℤ) (hd_pos : 0 < d) (hd_odd : Odd d) (k : ℤ) (l : ℤ → ℤ) (_ : ∀ i, 0 ≤ i ∧ i < d → Odd (l i)), ∀ n, f n = (2 * k * (n / d) + l (n % d)) * d }... | true |
fimo_2007_number_theory_p2 | Let $b, n>1$ be integers. Suppose that for each $k>1$ there exists an integer $a_{k}$ such that $b-a_{k}^{n}$ is divisible by $k$. Prove that $b=A^{n}$ for some integer $A$. | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2007_number_theory_p2
(b : ℤ) (n : ℕ)
(hb : b > 1)
(hn : n > 1)
(h_divisibility : ∀ k : ℕ, k > 1 → ∃ a_k : ℤ, (k : ℤ) ∣ (b - a_k^n)) :
∃ A : ℤ, b = A^n := by
sorry
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2007_number_theory_p2
(b : ℤ) (n : ℕ)
(hb : b > 1)
(hn : n > 1)
(h_divisibility : ∀ k : ℕ, k > 1 → ∃ a_k : ℤ, (k : ℤ) ∣ (b - a_k^n)) :
∃ A : ℤ, b = A^n := by
sorry
| false |
fimo_2006_algebra_p1 | A sequence of real numbers $a_{0}, a_{1}, a_{2}, \ldots$ is defined by the formula
$$
a_{i+1}=\left\lfloor a_{i}\right\rfloor \cdot\left\langle a_{i}\right\rangle \quad \text { for } \quad i \geq 0 ;
$$
here $a_{0}$ is an arbitrary real number, $\left\lfloor a_{i}\right\rfloor$ denotes the greatest integer not exceed... | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2006_algebra_p1
(a : ℕ → ℝ)
(h₀ : ∀ i, a (i + 1) = ⌊a i⌋ * (a i - ⌊a i⌋)) :
∃ n : ℕ, ∀ i ≥ n, a i = a (i + 2) := by sorry
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2006_algebra_p1
(a : ℕ → ℝ)
(h₀ : ∀ i, a (i + 1) = ⌊a i⌋ * (a i - ⌊a i⌋)) :
∃ n : ℕ, ∀ i ≥ n, a i = a (i + 2) := by sorry
| false |
fimo_2017_algebra_p6 | Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f(f(x) f(y))+f(x+y)=f(x y)
$$
for all $x, y \in \mathbb{R}$. | there are 3 solutions:
$$x \mapsto 0 \quad \text { or } \quad x \mapsto x-1 \quad \text { or } \quad x \mapsto 1-x \quad(x \in \mathbb{R}).$$ | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2017_algebra_p6_solution : Set (ℝ → ℝ) := sorry
-- { (fun x => 0), (fun x => x - 1), (fun x => 1 - x) }
theorem fimo_2017_algebra_p6 :
{ f : ℝ → ℝ | ∀ x y : ℝ, f (f x * f y) + f (x + y) = f (x ... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2017_algebra_p6_solution : Set (ℝ → ℝ) := { (fun x => 0), (fun x => x - 1), (fun x => 1 - x) }
theorem fimo_2017_algebra_p6 :
{ f : ℝ → ℝ | ∀ x y : ℝ, f (f x * f y) + f (x + y) = f (x * y) } = ... | true |
fimo_2013_number_theory_p1 | Let $\mathbb{Z}_{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that
$$
m^{2}+f(n) \mid m f(m)+n
$$
for all positive integers $m$ and $n$. | f(n)=n | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Int
abbrev fimo_2013_number_theory_p1_solution : Set (ℕ+ → ℕ+) := sorry
-- {id}
theorem fimo_2013_number_theory_p1 :
{f : ℕ+ → ℕ+ | ∀ m n : ℕ+, (m^2 + f n) ∣ (m * f m + n)} = fimo_2013_number_theory_p1_solution := by so... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Int
abbrev fimo_2013_number_theory_p1_solution : Set (ℕ+ → ℕ+) := {id}
theorem fimo_2013_number_theory_p1 :
{f : ℕ+ → ℕ+ | ∀ m n : ℕ+, (m^2 + f n) ∣ (m * f m + n)} = fimo_2013_number_theory_p1_solution := by sorry
| true |
fimo_2018_algebra_p4 | Let $a_{0}, a_{1}, a_{2}, \ldots$ be a sequence of real numbers such that $a_{0}=0, a_{1}=1$, and for every $n \geqslant 2$ there exists $1 \leqslant k \leqslant n$ satisfying
$$
a_{n}=\frac{a_{n-1}+\cdots+a_{n-k}}{k}
$$
Find the maximal possible value of $a_{2018}-a_{2017}$. | \frac{2016}{2017^{2}} | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2018_algebra_p4_solution : ℝ := sorry
-- 2016 / (2017^2)
theorem fimo_2018_algebra_p4 :
sSup ((fun (a : ℕ → ℝ) => a 2018 - a 2017) ''
{ a | a 0 = 0 ∧ a 1 = 1 ∧
∀ n ≥ 2, ∃ k ∈ Finset.I... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2018_algebra_p4_solution : ℝ := 2016 / (2017^2)
theorem fimo_2018_algebra_p4 :
sSup ((fun (a : ℕ → ℝ) => a 2018 - a 2017) ''
{ a | a 0 = 0 ∧ a 1 = 1 ∧
∀ n ≥ 2, ∃ k ∈ Finset.Icc 1 n, a... | true |
fimo_2020_algebra_p8 | Let $\mathbb{R}^{+}$be the set of positive real numbers. Determine all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that, for all positive real numbers $x$ and $y$,
$$
f(x+f(x y))+y=f(x) f(y)+1
$$ | f(x)=x+1 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2020_algebra_p8_solution : Set ({r : ℝ // 0 < r} → {r : ℝ // 0 < r}) := sorry
-- {fun x => x + 1}
theorem fimo_2020_algebra_p8 :
{f : {r : ℝ // 0 < r} → {r : ℝ // 0 < r} | (∀ x y, f (x + f (x *... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2020_algebra_p8_solution : Set ({r : ℝ // 0 < r} → {r : ℝ // 0 < r}) := {fun x => x + 1}
theorem fimo_2020_algebra_p8 :
{f : {r : ℝ // 0 < r} → {r : ℝ // 0 < r} | (∀ x y, f (x + f (x * y)) + y ... | true |
fimo_2014_number_theory_p7 | Let $c \geqslant 1$ be an integer. Define a sequence of positive integers by $a_{1}=c$ and
$$
a_{n+1}=a_{n}^{3}-4 c \cdot a_{n}^{2}+5 c^{2} \cdot a_{n}+c
$$
for all $n \geqslant 1$. Prove that for each integer $n \geqslant 2$ there exists a prime number $p$ dividing $a_{n}$ but none of the numbers $a_{1}, \ldots, a_{... | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2014_number_theory_p7
(c : ℤ)
(hc : 1 ≤ c)
(a : ℕ → ℤ)
(ha₁ : a 1 = c)
(ha_rec : ∀ n ≥ 1, a (n + 1) = (a n)^3 - 4 * c * (a n)^2 + 5 * c^2 * (a n) + c)
(ha_pos : ∀ n ≥ 1, 0 < a n) :
∀ n ≥ 2, ∃ p, ... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2014_number_theory_p7
(c : ℤ)
(hc : 1 ≤ c)
(a : ℕ → ℤ)
(ha₁ : a 1 = c)
(ha_rec : ∀ n ≥ 1, a (n + 1) = (a n)^3 - 4 * c * (a n)^2 + 5 * c^2 * (a n) + c)
(ha_pos : ∀ n ≥ 1, 0 < a n) :
∀ n ≥ 2, ∃ p, ... | false |
fimo_2017_algebra_p2 | Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard:
\begin{itemize}
\item In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) numbers o... | -2,0,2 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2017_algebra_p2_solution : Set ℝ := sorry
-- {-2, 0, 2}
theorem fimo_2017_algebra_p2
(line1 : Set ℝ → Set ℝ)
(hline1 : ∀ S, line1 S = { x | ∃ a ∈ S, ∃ b ∈ S, x = a - b })
(line2 : ℝ → Set ℝ... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2017_algebra_p2_solution : Set ℝ := {-2, 0, 2}
theorem fimo_2017_algebra_p2
(line1 : Set ℝ → Set ℝ)
(hline1 : ∀ S, line1 S = { x | ∃ a ∈ S, ∃ b ∈ S, x = a - b })
(line2 : ℝ → Set ℝ → Set ℝ)... | true |
fimo_2011_algebra_p3 | Determine all pairs $(f, g)$ of functions from the set of real numbers to itself that satisfy
$$
g(f(x+y))=f(x)+(2 x+y) g(y)
$$
for all real numbers $x$ and $y$. | either both $f$ and $g$ vanish identically, or there exists a real number $C$ such that $f(x)=x^{2}+C$ and $g(x)=x$ for all real numbers $x$ | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2011_algebra_p3_solution_set : Set ((ℝ → ℝ) × (ℝ → ℝ)) := sorry
-- { p : (ℝ → ℝ) × (ℝ → ℝ) | (p.1 = (fun _ => 0) ∧ p.2 = (fun _ => 0)) ∨ (∃ C : ℝ, p.1 = (fun x => x^2 + C) ∧ p.2 = (fun x => x)) }
... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2011_algebra_p3_solution_set : Set ((ℝ → ℝ) × (ℝ → ℝ)) := { p : (ℝ → ℝ) × (ℝ → ℝ) | (p.1 = (fun _ => 0) ∧ p.2 = (fun _ => 0)) ∨ (∃ C : ℝ, p.1 = (fun x => x^2 + C) ∧ p.2 = (fun x => x)) }
theorem ... | true |
fimo_2012_algebra_p5 | Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the conditions
$$
f(1+x y)-f(x+y)=f(x) f(y) \text { for all } x, y \in \mathbb{R}
$$
and $f(-1) \neq 0$ | the function $f(x)=x-1, x \in \mathbb{R}$ | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2012_algebra_p5_solution : Set (ℝ → ℝ) := sorry
-- {fun x => x - 1}
theorem fimo_2012_algebra_p5 :
{f : ℝ → ℝ | (∀ x y : ℝ, f (1 + x * y) - f (x + y) = f x * f y) ∧ f (-1) ≠ 0} = fimo_2012_alge... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2012_algebra_p5_solution : Set (ℝ → ℝ) := {fun x => x - 1}
theorem fimo_2012_algebra_p5 :
{f : ℝ → ℝ | (∀ x y : ℝ, f (1 + x * y) - f (x + y) = f x * f y) ∧ f (-1) ≠ 0} = fimo_2012_algebra_p5_so... | true |
fimo_2018_algebra_p5 | Determine all functions $f:(0, \infty) \rightarrow \mathbb{R}$ satisfying
$$
\left(x+\frac{1}{x}\right) f(y)=f(x y)+f\left(\frac{y}{x}\right)
$$
for all $x, y>0$. | $f(x)=C_{1} x+ rac{C_{2}}{x}$ with arbitrary constants $C_{1}$ and $C_{2}$ | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2018_algebra_p5_solution : Set ({x : ℝ // 0 < x} → ℝ) := sorry
-- { f | ∃ C₁ C₂, f = fun x => C₁ * x.val + C₂ / x.val }
theorem fimo_2018_algebra_p5 :
{ f : {x : ℝ // 0 < x} → ℝ | ∀ (x y : {x :... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2018_algebra_p5_solution : Set ({x : ℝ // 0 < x} → ℝ) := { f | ∃ C₁ C₂, f = fun x => C₁ * x.val + C₂ / x.val }
theorem fimo_2018_algebra_p5 :
{ f : {x : ℝ // 0 < x} → ℝ | ∀ (x y : {x : ℝ // 0 <... | true |
fimo_2006_algebra_p5 | Let $a, b, c$ be the sides of a triangle. Prove that
$$
\frac{\sqrt{b+c-a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}}+\frac{\sqrt{c+a-b}}{\sqrt{c}+\sqrt{a}-\sqrt{b}}+\frac{\sqrt{a+b-c}}{\sqrt{a}+\sqrt{b}-\sqrt{c}} \leq 3
$$ | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2006_algebra_p5
(a b c : ℝ)
(h_pos : a > 0 ∧ b > 0 ∧ c > 0)
(h_triangle : a + b > c ∧ a + c > b ∧ b + c > a) :
(sqrt (b + c - a) / (sqrt b + sqrt c - sqrt a)) +
(sqrt (c + a - b) / (sqrt c + sqrt a -... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2006_algebra_p5
(a b c : ℝ)
(h_pos : a > 0 ∧ b > 0 ∧ c > 0)
(h_triangle : a + b > c ∧ a + c > b ∧ b + c > a) :
(sqrt (b + c - a) / (sqrt b + sqrt c - sqrt a)) +
(sqrt (c + a - b) / (sqrt c + sqrt a -... | false |
fimo_2010_algebra_p8 | Given six positive numbers $a, b, c, d, e, f$ such that $a<b<c<d<e<f$. Let $a+c+e=S$ and $b+d+f=T$. Prove that
$$
2 S T>\sqrt{3(S+T)(S(b d+b f+d f)+T(a c+a e+c e))} .
$$ | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2010_algebra_p8
(a b c d e f S T : ℝ)
(h_pos : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d ∧ 0 < e ∧ 0 < f)
(h_order : a < b ∧ b < c ∧ c < d ∧ d < e ∧ e < f)
(hS : S = a + c + e)
(hT : T = b + d + f) :
2 * S * T... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2010_algebra_p8
(a b c d e f S T : ℝ)
(h_pos : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d ∧ 0 < e ∧ 0 < f)
(h_order : a < b ∧ b < c ∧ c < d ∧ d < e ∧ e < f)
(hS : S = a + c + e)
(hT : T = b + d + f) :
2 * S * T... | false |
fimo_2017_algebra_p8 | Assume that a function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfies the following condition:
For every $x, y \in \mathbb{R}$ such that $(f(x)+y)(f(y)+x)>0$, we have $f(x)+y=f(y)+x$.
Prove that $f(x)+y \leqslant f(y)+x$ whenever $x>y$. | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2017_algebra_p8
(f : ℝ → ℝ)
(h_cond : ∀ x y : ℝ, (f x + y) * (f y + x) > 0 → f x + y = f y + x) :
∀ x y : ℝ, x > y → f x + y ≤ f y + x := by sorry
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2017_algebra_p8
(f : ℝ → ℝ)
(h_cond : ∀ x y : ℝ, (f x + y) * (f y + x) > 0 → f x + y = f y + x) :
∀ x y : ℝ, x > y → f x + y ≤ f y + x := by sorry
| false |
fimo_2008_algebra_p7_1 | Prove that for any four positive real numbers $a, b, c, d$ the inequality
$$
\frac{(a-b)(a-c)}{a+b+c}+\frac{(b-c)(b-d)}{b+c+d}+\frac{(c-d)(c-a)}{c+d+a}+\frac{(d-a)(d-b)}{d+a+b} \geq 0
$$
holds. | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2008_algebra_p7_1
(a b c d : ℝ)
(ha : 0 < a)
(hb : 0 < b)
(hc : 0 < c)
(hd : 0 < d) :
((a - b) * (a - c) / (a + b + c)) +
((b - c) * (b - d) / (b + c + d)) +
((c - d) * (c - a) / (c + d + a)) +... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2008_algebra_p7_1
(a b c d : ℝ)
(ha : 0 < a)
(hb : 0 < b)
(hc : 0 < c)
(hd : 0 < d) :
((a - b) * (a - c) / (a + b + c)) +
((b - c) * (b - d) / (b + c + d)) +
((c - d) * (c - a) / (c + d + a)) +... | false |
fimo_2008_algebra_p7_2 | Prove that for any four positive real numbers $a, b, c, d$ the inequality
$$
\frac{(a-b)(a-c)}{a+b+c}+\frac{(b-c)(b-d)}{b+c+d}+\frac{(c-d)(c-a)}{c+d+a}+\frac{(d-a)(d-b)}{d+a+b} \geq 0
$$
holds. Determine all cases of equality. | a=c and simultaneously b=d | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
abbrev fimo_2008_algebra_p7_2_equality (a b c d : ℝ) : Prop := sorry
-- a = c ∧ b = d
theorem fimo_2008_algebra_p7_2
(a b c d : ℝ)
(h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d)
(ineq_lhs : ℝ)
(h_ineq_lhs : ineq_lhs = (a - ... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
abbrev fimo_2008_algebra_p7_2_equality (a b c d : ℝ) : Prop := a = c ∧ b = d
theorem fimo_2008_algebra_p7_2
(a b c d : ℝ)
(h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d)
(ineq_lhs : ℝ)
(h_ineq_lhs : ineq_lhs = (a - b) * (a -... | true |
fimo_2007_algebra_p4 | Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$such that
$$
f(x+f(y))=f(x+y)+f(y)
$$
for all $x, y \in \mathbb{R}^{+}$. (Symbol $\mathbb{R}^{+}$denotes the set of all positive real numbers.) | $f(x)=2 x$ | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2007_algebra_p4_solution : {x : ℝ // 0 < x} → {x : ℝ // 0 < x} := sorry
-- fun x => ⟨2 * x.val, by { apply mul_pos; norm_num; exact x.property }⟩
theorem fimo_2007_algebra_p4
(f : {x : ℝ // 0 <... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2007_algebra_p4_solution : {x : ℝ // 0 < x} → {x : ℝ // 0 < x} := fun x => ⟨2 * x.val, by { apply mul_pos; norm_num; exact x.property }⟩
theorem fimo_2007_algebra_p4
(f : {x : ℝ // 0 < x} → {x ... | true |
fimo_2017_algebra_p7 | Let $a_{0}, a_{1}, a_{2}, \ldots$ be a sequence of integers and $b_{0}, b_{1}, b_{2}, \ldots$ be a sequence of positive integers such that $a_{0}=0, a_{1}=1$, and
$$
a_{n+1}=\left\{\begin{array}{ll}
a_{n} b_{n}+a_{n-1}, & \text { if } b_{n-1}=1 \\
a_{n} b_{n}-a_{n-1}, & \text { if } b_{n-1}>1
\end{array} \quad \text {... | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2017_algebra_p7
(a : ℕ → ℤ)
(b : ℕ → ℕ+)
(h_a0 : a 0 = 0)
(h_a1 : a 1 = 1)
(h_recurrence : ∀ n : ℕ, n ≥ 1 →
a (n + 1) = if b (n - 1) = 1 then a n * (b n : ℤ) + a (n - 1)
else a n ... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2017_algebra_p7
(a : ℕ → ℤ)
(b : ℕ → ℕ+)
(h_a0 : a 0 = 0)
(h_a1 : a 1 = 1)
(h_recurrence : ∀ n : ℕ, n ≥ 1 →
a (n + 1) = if b (n - 1) = 1 then a n * (b n : ℤ) + a (n - 1)
else a n ... | false |
fimo_2012_algebra_p6 | Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function, and let $f^{m}$ be $f$ applied $m$ times. Suppose that for every $n \in \mathbb{N}$ there exists a $k \in \mathbb{N}$ such that $f^{2 k}(n)=n+k$, and let $k_{n}$ be the smallest such $k$. Prove that the sequence $k_{1}, k_{2}, \ldots$ is unbounded. | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2012_algebra_p6
(f : ℕ → ℕ)
(h_cond : ∀ n : ℕ, ∃ k : ℕ, f^[2*k] n = n + k) :
(∀ M : ℕ, ∃ n : ℕ, Nat.find (h_cond n) > M) := by
sorry
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2012_algebra_p6
(f : ℕ → ℕ)
(h_cond : ∀ n : ℕ, ∃ k : ℕ, f^[2*k] n = n + k) :
(∀ M : ℕ, ∃ n : ℕ, Nat.find (h_cond n) > M) := by
sorry
| false |
fimo_2008_algebra_p6 | Let $f: \mathbb{R} \rightarrow \mathbb{N}$ be a function which satisfies
$$
f\left(x+\frac{1}{f(y)}\right)=f\left(y+\frac{1}{f(x)}\right) \quad \text { for all } x, y \in \mathbb{R}
$$
Prove that there is a positive integer which is not a value of $f$. | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat PNat
theorem fimo_2008_algebra_p6
(f : ℝ → ℕ+)
(h_feq : ∀ x y : ℝ, f (x + (1 : ℝ) / (f y : ℝ)) = f (y + (1 : ℝ) / (f x : ℝ))) :
∃ n : ℕ+, n ∉ Set.range f := by sorry
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat PNat
theorem fimo_2008_algebra_p6
(f : ℝ → ℕ+)
(h_feq : ∀ x y : ℝ, f (x + (1 : ℝ) / (f y : ℝ)) = f (y + (1 : ℝ) / (f x : ℝ))) :
∃ n : ℕ+, n ∉ Set.range f := by sorry
| false |
fimo_2013_algebra_p3 | Let $\mathbb{Q}_{>0}$ be the set of positive rational numbers. Let $f: \mathbb{Q}_{>0} \rightarrow \mathbb{R}$ be a function satisfying the conditions
$$
\begin{aligned}
& f(x) f(y) \geqslant f(x y), \\
& f(x+y) \geqslant f(x)+f(y)
\end{aligned}
$$
for all $x, y \in \mathbb{Q}_{>0}$. Given that $f(a)=a$ for some rati... | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2013_algebra_p3
(f : {q : ℚ // 0 < q} → ℝ)
(f_mul : ∀ x y : {q : ℚ // 0 < q}, f x * f y ≥ f (x * y))
(f_add : ∀ x y : {q : ℚ // 0 < q}, f (x + y) ≥ f x + f y)
(a : {q : ℚ // 0 < q})
(ha_gt_1 : a > 1)... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2013_algebra_p3
(f : {q : ℚ // 0 < q} → ℝ)
(f_mul : ∀ x y : {q : ℚ // 0 < q}, f x * f y ≥ f (x * y))
(f_add : ∀ x y : {q : ℚ // 0 < q}, f (x + y) ≥ f x + f y)
(a : {q : ℚ // 0 < q})
(ha_gt_1 : a > 1)... | false |
fimo_2014_number_theory_p1 | Let $n \geqslant 2$ be an integer, and let $A_{n}$ be the set
$$
A_{n}=\left\{2^{n}-2^{k} \mid k \in \mathbb{Z}, 0 \leqslant k<n\right\} .
$$
Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_{n}$. | $(n-2) 2^{n}+1$ | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2014_number_theory_p1_solution (n : ℕ) : ℕ := sorry
-- (n - 2) * 2^n + 1
theorem fimo_2014_number_theory_p1
(n : ℕ)
(h₀ : n ≥ 2)
(An : Set ℕ)
(hAn : An = (fun k : Fin n => 2^n - 2^(k : ℕ)... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2014_number_theory_p1_solution (n : ℕ) : ℕ := (n - 2) * 2^n + 1
theorem fimo_2014_number_theory_p1
(n : ℕ)
(h₀ : n ≥ 2)
(An : Set ℕ)
(hAn : An = (fun k : Fin n => 2^n - 2^(k : ℕ)) '' Set.... | true |
fimo_2009_number_theory_p6 | Let $k$ be a positive integer. Show that if there exists a sequence $a_{0}, a_{1}, \ldots$ of integers satisfying the condition
$$
a_{n}=\frac{a_{n-1}+n^{k}}{n} \text { for all } n \geq 1,
$$
then $k-2$ is divisible by 3 . | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Int
theorem fimo_2009_number_theory_p6
(k : ℕ)
(hk : k > 0)
(h_exists_sequence : ∃ a : ℕ → ℤ, ∀ n : ℕ, n ≥ 1 → (n : ℤ) * (a n) = (a (n-1)) + ((n : ℤ) ^ k)) :
(3 : ℤ) ∣ ((k : ℤ) - 2) := by sorry
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Int
theorem fimo_2009_number_theory_p6
(k : ℕ)
(hk : k > 0)
(h_exists_sequence : ∃ a : ℕ → ℤ, ∀ n : ℕ, n ≥ 1 → (n : ℤ) * (a n) = (a (n-1)) + ((n : ℤ) ^ k)) :
(3 : ℤ) ∣ ((k : ℤ) - 2) := by sorry
| false |
fimo_2008_algebra_p5 | Let $a, b, c, d$ be positive real numbers such that
$$
a b c d=1 \quad \text { and } \quad a+b+c+d>\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}
$$
Prove that
$$
a+b+c+d<\frac{b}{a}+\frac{c}{b}+\frac{d}{c}+\frac{a}{d} .
$$ | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2008_algebra_p5
(a b c d : ℝ)
(h_pos : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d)
(h_prod_eq_1 : a * b * c * d = 1)
(h_ineq_1 : a + b + c + d > a / b + b / c + c / d + d / a) :
a + b + c + d < b / a + c / b + d ... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2008_algebra_p5
(a b c d : ℝ)
(h_pos : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d)
(h_prod_eq_1 : a * b * c * d = 1)
(h_ineq_1 : a + b + c + d > a / b + b / c + c / d + d / a) :
a + b + c + d < b / a + c / b + d ... | false |
fimo_2011_algebra_p2 | Determine all sequences $\left(x_{1}, x_{2}, \ldots, x_{2011}\right)$ of positive integers such that for every positive integer $n$ there is an integer $a$ with
$$
x_{1}^{n}+2 x_{2}^{n}+\cdots+2011 x_{2011}^{n}=a^{n+1}+1 .
$$ | the only sequence that satisfies the condition is
$$ (x_{1}, \ldots, x_{2011})=(1, k, \ldots, k) \quad \text { with } k=2+3+\cdots+2011=2023065 $$ | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2011_algebra_p2_solution : Set (Fin 2011 → ℕ) := sorry
-- { fun i => if i = 0 then 1 else 2023065 }
theorem fimo_2011_algebra_p2 :
{ x : Fin 2011 → ℕ |
(∀ i, x i > 0) ∧
(∀ n > 0, ∃ a : ... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2011_algebra_p2_solution : Set (Fin 2011 → ℕ) := { fun i => if i = 0 then 1 else 2023065 }
theorem fimo_2011_algebra_p2 :
{ x : Fin 2011 → ℕ |
(∀ i, x i > 0) ∧
(∀ n > 0, ∃ a : ℤ,
... | true |
fimo_2018_algebra_p3 | Given any set $S$ of positive integers, show that at least one of the following two assertions holds:
(1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x \in F} 1 / x=\sum_{x \in G} 1 / x$;
(2) There exists a positive rational number $r<1$ such that $\sum_{x \in F} 1 / x \neq r$ for all fini... | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2018_algebra_p3 (S : Set ℕ+) :
(∃ (F G : Finset ℕ+),
F.toSet ⊆ S ∧ G.toSet ⊆ S ∧
F ≠ G ∧
(∑ x in F, (1 : ℝ) / x.val) = (∑ x in G, (1 : ℝ) / x.val))
∨
(∃ (r : ℚ),
0 < r ∧ r < 1 ∧
∀ (F ... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2018_algebra_p3 (S : Set ℕ+) :
(∃ (F G : Finset ℕ+),
F.toSet ⊆ S ∧ G.toSet ⊆ S ∧
F ≠ G ∧
(∑ x in F, (1 : ℝ) / x.val) = (∑ x in G, (1 : ℝ) / x.val))
∨
(∃ (r : ℚ),
0 < r ∧ r < 1 ∧
∀ (F ... | false |
fimo_2018_number_theory_p1 | Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the numbers of divisors of $s n$ and of $s k$ are equal. | all pairs $(n, k)$ such that $n \nmid k$ and $k \nmid n$ | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat PNat
noncomputable abbrev fimo_2018_number_theory_p1_solution (n k : ℕ+) : Prop := sorry
-- ¬ ((n : ℕ) ∣ (k : ℕ)) ∧ ¬ ((k : ℕ) ∣ (n : ℕ))
theorem fimo_2018_number_theory_p1
(n k : ℕ+)
(h_distinct : n ≠ k) :
(∃ s : ℕ... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat PNat
noncomputable abbrev fimo_2018_number_theory_p1_solution (n k : ℕ+) : Prop := ¬ ((n : ℕ) ∣ (k : ℕ)) ∧ ¬ ((k : ℕ) ∣ (n : ℕ))
theorem fimo_2018_number_theory_p1
(n k : ℕ+)
(h_distinct : n ≠ k) :
(∃ s : ℕ+, (Nat.c... | true |
fimo_2014_number_theory_p5 | Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p-1}+y$ and $x+y^{p-1}$ are both powers of $p$. | $(p, x, y) \in\{(3,2,5),(3,5,2)\} \cup\left{\left(2, n, 2^{k}-n\right) \mid 0<n<2^{k}\right}$ | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2014_number_theory_p5_solution : Set (ℕ × ℕ × ℕ) := sorry
-- ({(3, 2, 5), (3, 5, 2)} : Set (ℕ × ℕ × ℕ)) ∪ { (p, x, y) | p = 2 ∧ x > 0 ∧ y > 0 ∧ ∃ k, x + y = 2^k }
theorem fimo_2014_number_theory_... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2014_number_theory_p5_solution : Set (ℕ × ℕ × ℕ) := ({(3, 2, 5), (3, 5, 2)} : Set (ℕ × ℕ × ℕ)) ∪ { (p, x, y) | p = 2 ∧ x > 0 ∧ y > 0 ∧ ∃ k, x + y = 2^k }
theorem fimo_2014_number_theory_p5 :
{ ... | true |
fimo_2006_number_theory_p2 | For $x \in(0,1)$ let $y \in(0,1)$ be the number whose $n$th digit after the decimal point is the $\left(2^{n}\right)$ th digit after the decimal point of $x$. Show that if $x$ is rational then so is $y$. | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2006_number_theory_p2_idiomatic
(x y : ℝ)
(h₀ : 0 < x ∧ x < 1)
(h₁ : 0 < y ∧ y < 1)
(h₂ : ∀ n : ℕ, (Nat.floor (y * 10^n) % 10) = (Nat.floor (x * 10^(2^n)) % 10))
(hx_rational : ∃ q : ℚ, q = x) :
∃ ... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2006_number_theory_p2_idiomatic
(x y : ℝ)
(h₀ : 0 < x ∧ x < 1)
(h₁ : 0 < y ∧ y < 1)
(h₂ : ∀ n : ℕ, (Nat.floor (y * 10^n) % 10) = (Nat.floor (x * 10^(2^n)) % 10))
(hx_rational : ∃ q : ℚ, q = x) :
∃ ... | false |
fimo_2021_number_theory_p1 | Determine all integers $n \geqslant 1$ for which there exists a pair of positive integers $(a, b)$ such that no cube of a prime divides $a^{2}+b+3$ and
$$
\frac{a b+3 b+8}{a^{2}+b+3}=n
$$ | n=2 | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2021_number_theory_p1_solution : Set ℕ := sorry
-- {2}
theorem fimo_2021_number_theory_p1 :
{ n : ℕ | n ≥ 1 ∧ ∃ (a b : ℕ), a > 0 ∧ b > 0 ∧ (∀ p, Nat.Prime p → ¬ p^3 ∣ a^2 + b + 3) ∧ a * b + 3 *... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2021_number_theory_p1_solution : Set ℕ := {2}
theorem fimo_2021_number_theory_p1 :
{ n : ℕ | n ≥ 1 ∧ ∃ (a b : ℕ), a > 0 ∧ b > 0 ∧ (∀ p, Nat.Prime p → ¬ p^3 ∣ a^2 + b + 3) ∧ a * b + 3 * b + 8 = ... | true |
fimo_2019_algebra_p1 | Let $\mathbb{Z}$ be the set of integers. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that, for all integers $a$ and $b$
$$
f(2 a)+2 f(b)=f(f(a+b))
$$ | $f(n)=0$ and $f(n)=2 n+K$ for any constant $K \in \mathbb{Z}$ | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
abbrev fimo_2019_algebra_p1_solution : Set (ℤ → ℤ) := sorry
-- {(fun _ => 0)} ∪ {f | ∃ K : ℤ, f = (fun n => 2 * n + K)}
theorem fimo_2019_algebra_p1 (f : ℤ → ℤ) :
(∀ a b : ℤ, f (2 * a) + 2 * f b = f (f (a + b))) ↔ f ∈ f... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
abbrev fimo_2019_algebra_p1_solution : Set (ℤ → ℤ) := {(fun _ => 0)} ∪ {f | ∃ K : ℤ, f = (fun n => 2 * n + K)}
theorem fimo_2019_algebra_p1 (f : ℤ → ℤ) :
(∀ a b : ℤ, f (2 * a) + 2 * f b = f (f (a + b))) ↔ f ∈ fimo_2019_... | true |
fimo_2017_algebra_p1 | Let $a_{1}, a_{2}, \ldots, a_{n}, k$, and $M$ be positive integers such that
$$
\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}=k \quad \text { and } \quad a_{1} a_{2} \ldots a_{n}=M .
$$
If $M>1$, prove that the polynomial
$$
P(x)=M(x+1)^{k}-\left(x+a_{1}\right)\left(x+a_{2}\right) \cdots\left(x+a_{n}\right)... | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2017_algebra_p1
(n : ℕ)
(a : Fin n → ℕ)
(k : ℕ)
(M : ℕ)
(h_a_pos : ∀ i, 0 < a i)
(h_k_pos : 0 < k)
(h_M_pos : 0 < M)
(h_sum : ∑ i : Fin n, (1 : ℝ) / (a i : ℝ) = (k : ℝ))
(h_prod : ∏ i : Fin n... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2017_algebra_p1
(n : ℕ)
(a : Fin n → ℕ)
(k : ℕ)
(M : ℕ)
(h_a_pos : ∀ i, 0 < a i)
(h_k_pos : 0 < k)
(h_M_pos : 0 < M)
(h_sum : ∑ i : Fin n, (1 : ℝ) / (a i : ℝ) = (k : ℝ))
(h_prod : ∏ i : Fin n... | false |
fimo_2019_number_theory_p5 | Let $a$ be a positive integer. We say that a positive integer $b$ is $a-\operatorname{good}$ if $\left(\begin{array}{c}a n \\ b\end{array}\right)-1$ is divisible by $a n+1$ for all positive integers $n$ with $a n \geqslant b$. Suppose $b$ is a positive integer such that $b$ is $a$-good, but $b+2$ is not $a$-good. Prove... | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
def a_good (a b : ℕ) : Prop :=
∀ n : ℕ, 0 < n ∧ a * n ≥ b → (a * n + 1) ∣ (choose (a * n) b - 1)
theorem fimo_2019_number_theory_p5
(a b : ℕ)
(ha : 0 < a)
(hb : 0 < b)
(h_b_good : a_good a b)
(h_b_plus_2_not_g... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
def a_good (a b : ℕ) : Prop :=
∀ n : ℕ, 0 < n ∧ a * n ≥ b → (a * n + 1) ∣ (choose (a * n) b - 1)
theorem fimo_2019_number_theory_p5
(a b : ℕ)
(ha : 0 < a)
(hb : 0 < b)
(h_b_good : a_good a b)
(h_b_plus_2_not_g... | false |
fimo_2012_number_theory_p1 | Call admissible a set $A$ of integers that has the following property:
$$
\text { If } x, y \in A \text { (possibly } x=y \text { ) then } x^{2}+k x y+y^{2} \in A \text { for every integer } k \text {. }
$$
Determine all pairs $m, n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is... | a pair of integers $m, n$ fulfills the condition if and only if $\operatorname{gcd}(m, n)=1$ | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2012_number_theory_p1_solution : ℤ → ℤ → Prop := sorry
-- fun m n => Int.gcd m n = 1
theorem fimo_2012_number_theory_p1
(m n : ℤ)
(hm : m ≠ 0)
(hn : n ≠ 0) :
(∀ (A : Set ℤ), (∀ x ∈ A, ∀ y... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2012_number_theory_p1_solution : ℤ → ℤ → Prop := fun m n => Int.gcd m n = 1
theorem fimo_2012_number_theory_p1
(m n : ℤ)
(hm : m ≠ 0)
(hn : n ≠ 0) :
(∀ (A : Set ℤ), (∀ x ∈ A, ∀ y ∈ A, ∀ k... | true |
fimo_2013_number_theory_p2 | Prove that for any pair of positive integers $k$ and $n$ there exist $k$ positive integers $m_{1}, m_{2}, \ldots, m_{k}$ such that
$$
1+\frac{2^{k}-1}{n}=\left(1+\frac{1}{m_{1}}\right)\left(1+\frac{1}{m_{2}}\right) \cdots\left(1+\frac{1}{m_{k}}\right) .
$$ | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Int
theorem fimo_2013_number_theory_p2
(k n : ℕ)
(hk : 0 < k)
(hn : 0 < n) :
∃ (m : Fin k → ℕ), (∀ i, 0 < m i) ∧ ((1 : ℚ) + (↑(2^k - 1) : ℚ) / (↑n : ℚ) = ∏ i : Fin k, ((1 : ℚ) + (1 : ℚ) / (↑(m i) : ℚ))) := by sorry... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Int
theorem fimo_2013_number_theory_p2
(k n : ℕ)
(hk : 0 < k)
(hn : 0 < n) :
∃ (m : Fin k → ℕ), (∀ i, 0 < m i) ∧ ((1 : ℚ) + (↑(2^k - 1) : ℚ) / (↑n : ℚ) = ∏ i : Fin k, ((1 : ℚ) + (1 : ℚ) / (↑(m i) : ℚ))) := by sorry... | false |
fimo_2012_number_theory_p6 | Let $x$ and $y$ be positive integers. If $x^{2^{n}}-1$ is divisible by $2^{n} y+1$ for every positive integer $n$, prove that $x=1$. | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2012_number_theory_p6
(x y : ℕ)
(hx_pos : 0 < x)
(hy_pos : 0 < y)
(h_div : ∀ (n : ℕ), 0 < n → (2^n * y + 1) ∣ (x^(2^n) - 1)) :
x = 1 := by sorry
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2012_number_theory_p6
(x y : ℕ)
(hx_pos : 0 < x)
(hy_pos : 0 < y)
(h_div : ∀ (n : ℕ), 0 < n → (2^n * y + 1) ∣ (x^(2^n) - 1)) :
x = 1 := by sorry
| false |
fimo_2008_number_theory_p6 | Prove that there exist infinitely many positive integers $n$ such that $n^{2}+1$ has a prime divisor greater than $2 n+\sqrt{2 n}$. | null | import Mathlib
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2008_number_theory_p6 :
Set.Infinite {n : ℕ | n > 0 ∧ ∃ p : ℕ, Nat.Prime p ∧ p ∣ (n^2 + 1) ∧ (p : ℝ) > 2 * (n : ℝ) + Real.sqrt (2 * (n : ℝ))} := by sorry
| import Mathlib
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2008_number_theory_p6 :
Set.Infinite {n : ℕ | n > 0 ∧ ∃ p : ℕ, Nat.Prime p ∧ p ∣ (n^2 + 1) ∧ (p : ℝ) > 2 * (n : ℝ) + Real.sqrt (2 * (n : ℝ))} := by sorry
| false |
fimo_2008_number_theory_p1 | Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a, b, c$ are integers (not necessarily positive) satisfying the equations
$$
a^{n}+p b=b^{n}+p c=c^{n}+p a,
$$
then $a=b=c$. | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2008_number_theory_p1
(n : ℕ)
(p : ℕ)
(a b c : ℤ)
(hn : n > 0)
(hp : p.Prime)
(h_eq1 : (a : ℤ)^n + p * b = (b : ℤ)^n + p * c)
(h_eq2 : (b : ℤ)^n + p * c = (c : ℤ)^n + p * a) :
a = b ∧ b = c := ... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2008_number_theory_p1
(n : ℕ)
(p : ℕ)
(a b c : ℤ)
(hn : n > 0)
(hp : p.Prime)
(h_eq1 : (a : ℤ)^n + p * b = (b : ℤ)^n + p * c)
(h_eq2 : (b : ℤ)^n + p * c = (c : ℤ)^n + p * a) :
a = b ∧ b = c := ... | false |
fimo_2020_algebra_p4 | Let $a, b, c, d$ be four real numbers such that $a \geqslant b \geqslant c \geqslant d>0$ and $a+b+c+d=1$. Prove that
$$
(a+2 b+3 c+4 d) a^{a} b^{b} c^{c} d^{d}<1
$$ | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2020_algebra_p4
(a b c d : ℝ)
(h₀ : a ≥ b)
(h₁ : b ≥ c)
(h₂ : c ≥ d)
(h₃ : d > 0)
(h₄ : a + b + c + d = 1) :
(a + 2 * b + 3 * c + 4 * d) * (a ^ a) * (b ^ b) * (c ^ c) * (d ^ d) < 1 := by sorry
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2020_algebra_p4
(a b c d : ℝ)
(h₀ : a ≥ b)
(h₁ : b ≥ c)
(h₂ : c ≥ d)
(h₃ : d > 0)
(h₄ : a + b + c + d = 1) :
(a + 2 * b + 3 * c + 4 * d) * (a ^ a) * (b ^ b) * (c ^ c) * (d ^ d) < 1 := by sorry
| false |
fimo_2015_number_theory_p2 | Let $a$ and $b$ be positive integers such that $a ! b$ ! is a multiple of $a !+b !$. Prove that $3 a \geqslant 2 b+2$. | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2015_number_theory_p2
(a b : ℕ)
(ha : 0 < a)
(hb : 0 < b)
(h_div : (a.factorial + b.factorial) ∣ (a.factorial * b.factorial)) :
3 * a ≥ 2 * b + 2 := by sorry
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2015_number_theory_p2
(a b : ℕ)
(ha : 0 < a)
(hb : 0 < b)
(h_div : (a.factorial + b.factorial) ∣ (a.factorial * b.factorial)) :
3 * a ≥ 2 * b + 2 := by sorry
| false |
fimo_2012_number_theory_p2 | Find all triples $(x, y, z)$ of positive integers such that $x \leq y \leq z$ and
$$
x^{3}\left(y^{3}+z^{3}\right)=2012(x y z+2) .
$$ | (2,251,252) | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
abbrev fimo_2012_number_theory_p2_solution : Set (ℕ × ℕ × ℕ) := sorry
-- { (2, 251, 252) }
theorem fimo_2012_number_theory_p2 :
{ t : ℕ × ℕ × ℕ | let (x, y, z) := t; 1 ≤ x ∧ x ≤ y ∧ y ≤ z ∧ x^3 * (y^3 + z^3) = 2012 * (x... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
abbrev fimo_2012_number_theory_p2_solution : Set (ℕ × ℕ × ℕ) := { (2, 251, 252) }
theorem fimo_2012_number_theory_p2 :
{ t : ℕ × ℕ × ℕ | let (x, y, z) := t; 1 ≤ x ∧ x ≤ y ∧ y ≤ z ∧ x^3 * (y^3 + z^3) = 2012 * (x * y * z ... | true |
fimo_2011_algebra_p6 | Let $f$ be a function from the set of real numbers to itself that satisfies
$$
f(x+y) \leq y f(x)+f(f(x))
$$
for all real numbers $x$ and $y$. Prove that $f(x)=0$ for all $x \leq 0$. | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2011_algebra_p6
(f : ℝ → ℝ)
(h_ineq : ∀ x y : ℝ, f (x + y) ≤ y * f x + f (f x)) :
∀ x : ℝ, x ≤ 0 → f x = 0 := by sorry
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2011_algebra_p6
(f : ℝ → ℝ)
(h_ineq : ∀ x y : ℝ, f (x + y) ≤ y * f x + f (f x)) :
∀ x : ℝ, x ≤ 0 → f x = 0 := by sorry
| false |
fimo_2016_number_theory_p6 | Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-m n$ is nonzero and divides $m f(m)+n f(n)$. | $f(n)=n^{2}$ for any $n \in \mathbb{N}$ | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2016_number_theory_p6_solution : Set (PNat → PNat) := sorry
-- {fun n ↦ n^2}
theorem fimo_2016_number_theory_p6 :
{ f : PNat → PNat | ∀ m n : PNat,
(f m : ℤ) + (f n : ℤ) - (m : ℤ) * (n : ℤ)... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2016_number_theory_p6_solution : Set (PNat → PNat) := {fun n ↦ n^2}
theorem fimo_2016_number_theory_p6 :
{ f : PNat → PNat | ∀ m n : PNat,
(f m : ℤ) + (f n : ℤ) - (m : ℤ) * (n : ℤ) ≠ 0 ∧
... | true |
fimo_2007_number_theory_p3 | Let $X$ be a set of 10000 integers, none of them is divisible by 47 . Prove that there exists a 2007-element subset $Y$ of $X$ such that $a-b+c-d+e$ is not divisible by 47 for any $a, b, c, d, e \in Y$. | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Int
theorem fimo_2007_number_theory_p3
(X : Finset ℤ)
(hX_card : X.card = 10000)
(hX_not_div_47 : ∀ x ∈ X, ¬ (47 ∣ x)) :
∃ (Y : Finset ℤ),
Y ⊆ X ∧
Y.card = 2007 ∧
(∀ (a b c d e : ℤ),
a ∈ Y → b ∈ Y... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Int
theorem fimo_2007_number_theory_p3
(X : Finset ℤ)
(hX_card : X.card = 10000)
(hX_not_div_47 : ∀ x ∈ X, ¬ (47 ∣ x)) :
∃ (Y : Finset ℤ),
Y ⊆ X ∧
Y.card = 2007 ∧
(∀ (a b c d e : ℤ),
a ∈ Y → b ∈ Y... | false |
fimo_2008_algebra_p4 | For an integer $m$, denote by $t(m)$ the unique number in $\{1,2,3\}$ such that $m+t(m)$ is a multiple of 3 . A function $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfies $f(-1)=0, f(0)=1, f(1)=-1$ and
$$
f\left(2^{n}+m\right)=f\left(2^{n}-t(m)\right)-f(m) \quad \text { for all integers } m, n \geq 0 \text { with } 2^{... | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
def t (m : ℤ) : ℕ :=
let m_mod_3 := m % 3
if m_mod_3 = 0 then 3
else if m_mod_3 = 1 then 2
else 1
theorem fimo_2008_algebra_p4
(f : ℤ → ℤ)
(hf_neg_1 : f (-1) = 0)
(hf_0 : f 0 = 1)
(hf_1 : f 1 = -1)
(hf_r... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
def t (m : ℤ) : ℕ :=
let m_mod_3 := m % 3
if m_mod_3 = 0 then 3
else if m_mod_3 = 1 then 2
else 1
theorem fimo_2008_algebra_p4
(f : ℤ → ℤ)
(hf_neg_1 : f (-1) = 0)
(hf_0 : f 0 = 1)
(hf_1 : f 1 = -1)
(hf_r... | false |
fimo_2021_algebra_p1 | Let $n$ be an integer, and let $A$ be a subset of $\left\{0,1,2,3, \ldots, 5^{n}\right\}$ consisting of $4 n+2$ numbers. Prove that there exist $a, b, c \in A$ such that $a<b<c$ and $c+2 a>3 b$. | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2021_algebra_p1
(n : ℕ)
(hn : n ≥ 1)
(A : Finset ℕ)
(hA_subset : ∀ x ∈ A, x ≤ 5^n)
(hA_card : A.card = 4 * n + 2) :
∃ a b c, a ∈ A ∧ b ∈ A ∧ c ∈ A ∧ a < b ∧ b < c ∧ c + 2 * a > 3 * b := by sorry
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2021_algebra_p1
(n : ℕ)
(hn : n ≥ 1)
(A : Finset ℕ)
(hA_subset : ∀ x ∈ A, x ≤ 5^n)
(hA_card : A.card = 4 * n + 2) :
∃ a b c, a ∈ A ∧ b ∈ A ∧ c ∈ A ∧ a < b ∧ b < c ∧ c + 2 * a > 3 * b := by sorry
| false |
fimo_2015_algebra_p1 | Suppose that a sequence $a_{1}, a_{2}, \ldots$ of positive real numbers satisfies
$$
a_{k+1} \geqslant \frac{k a_{k}}{a_{k}^{2}+(k-1)}
$$
for every positive integer $k$. Prove that $a_{1}+a_{2}+\cdots+a_{n} \geqslant n$ for every $n \geqslant 2$. | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2015_algebra_p1
(a : ℕ → ℝ)
(h_pos : ∀ k : ℕ, 1 ≤ k → a k > 0)
(h_ineq : ∀ k : ℕ, 1 ≤ k → a (k+1) ≥ (k : ℝ) * a k / (a k ^ 2 + ((k : ℝ) - 1)))
(n : ℕ)
(h_n_ge_2 : 2 ≤ n) :
(∑ i in Finset.Icc 1 n, a... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2015_algebra_p1
(a : ℕ → ℝ)
(h_pos : ∀ k : ℕ, 1 ≤ k → a k > 0)
(h_ineq : ∀ k : ℕ, 1 ≤ k → a (k+1) ≥ (k : ℝ) * a k / (a k ^ 2 + ((k : ℝ) - 1)))
(n : ℕ)
(h_n_ge_2 : 2 ≤ n) :
(∑ i in Finset.Icc 1 n, a... | false |
fimo_2013_algebra_p5 | Let $\mathbb{Z}_{\geqslant 0}$ be the set of all nonnegative integers. Find all the functions $f: \mathbb{Z}_{\geqslant 0} \rightarrow \mathbb{Z}_{\geqslant 0}$ satisfying the relation
$$
f(f(f(n)))=f(n+1)+1
$$
for all $n \in \mathbb{Z}_{\geqslant 0}$. | there are two such functions: $f(n)=n+1$ for all $n \in \mathbb{Z}_{\geqslant 0}$, and
$$
f(n)=\left\{\begin{array}{ll}
n+1, & n \equiv 0(\bmod 4) \text { or } n \equiv 2(\bmod 4), \\
n+5, & n \equiv 1(\bmod 4), \\
n-3, & n \equiv 3(\bmod 4)
\end{array} \quad \text { for all } n \in \mathbb{Z}_{\geqslant 0} .
$$
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2013_algebra_p5_solution : Set (ℕ → ℕ) := sorry
-- { (fun n => n + 1), (fun n => if n % 4 = 0 ∨ n % 4 = 2 then n + 1 else if n % 4 = 1 then n + 5 else n - 3) }
theorem fimo_2013_algebra_p5 :
{f... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
noncomputable abbrev fimo_2013_algebra_p5_solution : Set (ℕ → ℕ) := { (fun n => n + 1), (fun n => if n % 4 = 0 ∨ n % 4 = 2 then n + 1 else if n % 4 = 1 then n + 5 else n - 3) }
theorem fimo_2013_algebra_p5 :
{f : ℕ → ℕ ... | true |
fimo_2021_algebra_p4 | Show that for all real numbers $x_{1}, \ldots, x_{n}$ the following inequality holds:
$$
\sum_{i=1}^{n} \sum_{j=1}^{n} \sqrt{\left|x_{i}-x_{j}\right|} \leqslant \sum_{i=1}^{n} \sum_{j=1}^{n} \sqrt{\left|x_{i}+x_{j}\right|}
$$ | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2021_algebra_p4 (n : ℕ) (x : Fin n → ℝ) :
(∑ i : Fin n, ∑ j : Fin n, sqrt (abs (x i - x j))) ≤ (∑ i : Fin n, ∑ j : Fin n, sqrt (abs (x i + x j))) := by
sorry
| import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem fimo_2021_algebra_p4 (n : ℕ) (x : Fin n → ℝ) :
(∑ i : Fin n, ∑ j : Fin n, sqrt (abs (x i - x j))) ≤ (∑ i : Fin n, ∑ j : Fin n, sqrt (abs (x i + x j))) := by
sorry
| false |
fimo_2020_algebra_p6 | Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that
$$
f^{a^{2}+b^{2}}(a+b)=a f(a)+b f(b) \quad \text { for every } a, b \in \mathbb{Z} .
$$
Here, $f^{n}$ denotes the $n^{\text {th }}$ iteration of $f$, i.e., $f^{0}(x)=x$ and $f^{n+1}(x)=f\left(f^{n}(x)\right)$ for all $n \geqslant 0$. | f(x)=0 for all x \in \mathbb{Z}, or f(x)=x+1 for all x \in \mathbb{Z} | import Mathlib
import Aesop
set_option maxHeartbeats 0
open Function
noncomputable abbrev fimo_2020_algebra_p6_solution : Set (ℤ → ℤ) := sorry
-- { fun _ => 0, fun x => x + 1 }
theorem fimo_2020_algebra_p6 :
{ f : ℤ → ℤ | ∀ a b : ℤ, f^[(a ^ 2 + b ^ 2).toNat] (a + b) = a * f a + b * f b } = fimo_2020_algebra_p6_so... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open Function
noncomputable abbrev fimo_2020_algebra_p6_solution : Set (ℤ → ℤ) := { fun _ => 0, fun x => x + 1 }
theorem fimo_2020_algebra_p6 :
{ f : ℤ → ℤ | ∀ a b : ℤ, f^[(a ^ 2 + b ^ 2).toNat] (a + b) = a * f a + b * f b } = fimo_2020_algebra_p6_solution :=... | true |
fimo_2019_algebra_p2 | Let $u_{1}, u_{2}, \ldots, u_{2019}$ be real numbers satisfying
$$
u_{1}+u_{2}+\cdots+u_{2019}=0 \text { and } u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1 .
$$
Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that
$$
a b \leqslant-\frac{1}{2019} .... | null | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
def N_val : ℕ := 2019
theorem fimo_2019_algebra_p2
(u : Fin N_val → ℝ)
(h_sum : ∑ i : Fin N_val, u i = 0)
(h_sum_sq : ∑ i : Fin N_val, (u i)^2 = 1)
(a : ℝ)
(b : ℝ)
(h_a_def : ∀ i : Fin N_val, a ≤ u i)
(h_a_e... | import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
def N_val : ℕ := 2019
theorem fimo_2019_algebra_p2
(u : Fin N_val → ℝ)
(h_sum : ∑ i : Fin N_val, u i = 0)
(h_sum_sq : ∑ i : Fin N_val, (u i)^2 = 1)
(a : ℝ)
(b : ℝ)
(h_a_def : ∀ i : Fin N_val, a ≤ u i)
(h_a_e... | false |
End of preview. Expand in Data Studio
MiniF2F-Hard & FIMO-Hard
Expert-reannotated Hard Mode variants of the MiniF2F and FIMO theorem-proving benchmarks, released with our paper Discover and Prove: An Open-source Agentic Framework for Hard Mode Automated Theorem Proving in Lean 4 (ACL 2026).
In Hard Mode, the final answer is not embedded in the formal statement:
the system must first discover the answer before constructing a formal proof —
mirroring what a human competitor actually faces. Each solution-style statement
carries two sorry placeholders (one for the answer, one for the proof).
Configs
| Config | Examples | Hard Mode | Lean |
|---|---|---|---|
minif2f_hard |
244 | 197 | Lean 4 |
fimo_hard |
149 | 70 | Lean 4 |
from datasets import load_dataset
minif2f = load_dataset("liuchengwu/discover-and-prove", "minif2f_hard", split="train")
fimo = load_dataset("liuchengwu/discover-and-prove", "fimo_hard", split="train")
Fields
id— problem identifiermath_problem— natural-language problem statementsolution— ground-truth final answer (nullfor proof-style problems)formal_statement— Lean 4 Hard Mode statement (twosorrys for solution-style)formal_statement_easy— Lean 4 Easy Mode statement (FIMO only)is_hard— whether the problem is a Hard Mode (solution-style) problem
Notes
- MiniF2F-Hard contains 197 Hard Mode problems (the corrected version used in the paper's main table and Table 3).
- Three FIMO statements had the final answer leaking into the natural-language problem; the leaked text was removed so the answer must be discovered.
- FIMO was ported from Lean 3 to Lean 4. PutnamBench and CombiBench (used as-is in the paper) are available from their upstream repositories.
License
MIT. Derived from MiniF2F and FIMO (both MIT).
Citation
@inproceedings{liu-etal-2026-discover,
title = "Discover and Prove: An Open-source Agentic Framework for Hard Mode Automated Theorem Proving in Lean 4",
author = "Liu, Chengwu and Yin, Yichun and Yuan, Ye and Xie, Jiaxuan and Li, Botao and Li, Siqi and Shen, Jianhao and Xu, Yan and Shang, Lifeng and Zhang, Ming",
booktitle = "Proceedings of the 64th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)",
year = "2026",
address = "San Diego, California, United States",
publisher = "Association for Computational Linguistics",
url = "https://aclanthology.org/2026.acl-long.3/",
pages = "117--133"
}
- Downloads last month
- 77