## TPTP Problem File: RAL046^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : RAL046^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Inequalities)
% Problem  : International Mathematical Olympiad, 2012, Problem 2
% Version  : [Mat16] axioms : Especial.
% English  : Let n >= 3 be an integer, and let a2, a3, ..., a_n be positive
%            real numbers such that  a2 a3 ... a_n = 1. Prove that (1 + a2)^2
%            (1 + a3)^2 ... (1 + a_n)^n > n^n.

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : IMO-2012-2.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3485 (   0 unit;1199 type;   0 defn)
%            Number of atoms       : 45336 (2209 equality;22712 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39631 ( 104   ~; 233   |;1175   &;35991   @)
%                                         (1095 <=>;1033  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  : 2408 (2408   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1246 (1199   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   : 8059 (  66 sgn;7088   !; 429   ?; 406   ^)
%                                         (8059   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1977 (   6 prd;   9 fun;  23 num;1939 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: RCF+PA; Score: 7; Author: Jumma Kudo;
%            Generated: 2014-10-17
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf(p,conjecture,(
! [V_n: \$int,V_a: 'seq.Seq'] :
( ( ( \$greatereq @ V_n @ 3 )
& ! [V_k: \$int] :
( ( ( \$lesseq @ 2 @ V_k )
& ( \$lesseq @ V_k @ V_n ) )
=> ( \$less @ 0.0 @ ( 'seq.nth-term-of/2' @ V_a @ ( 'seq.index/1' @ V_k ) ) ) )
& ( ( 'seq.prod-from-to/3' @ V_a @ ( 'seq.index/1' @ 2 ) @ ( 'seq.index/1' @ V_n ) )
= 1.0 ) )
=> ( \$greater
@ ( 'seq.prod-from-to/3'
@ ( 'seq.seq/1'
@ ^ [V_k_dot_0: \$int] :
( '^/2' @ ( \$sum @ 1.0 @ ( 'seq.nth-term-of/2' @ V_a @ ( 'seq.index/1' @ V_k_dot_0 ) ) ) @ 2.0 ) )
@ ( 'seq.index/1' @ 2 )
@ ( 'seq.index/1' @ V_n ) )
@ ( \$to_real @ ( 'int.^/2' @ V_n @ V_n ) ) ) ) )).

%------------------------------------------------------------------------------
```