## TPTP Problem File: RAL041^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : RAL041^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Inequalities)
% Problem  : International Mathematical Olympiad, 2004, Problem 4
% Version  : [Mat16] axioms : Especial.
% English  : Let n >= 3 be an integer. Let t1, t2, ..., t_n be positive real
%            numbers such that n^2 + 1 > (t1 + t2 + ... + t_n)(1/t1 + 1/t2 +
%            ... + 1/t_n).  Show that t_i, t_j , t_k are side lengths of a
%            triangle for all i, j, k with 1 <= i < j < k <= n.

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : IMO-2004-4.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3485 (   0 unit;1199 type;   0 defn)
%            Number of atoms       : 45362 (2209 equality;22726 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39657 ( 104   ~; 233   |;1179   &;36012   @)
%                                         (1095 <=>;1034  =>;   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   : 8063 (  66 sgn;7092   !; 429   ?; 406   ^)
%                                         (8063   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1980 (   6 prd;   9 fun;  23 num;1942 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: RCF+PA; Score: 7; Author: Jumma Kudo;
%            Generated: 2014-10-31
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf(p,conjecture,(
! [V_n: \$int,V_t: 'seq.Seq',V_ti: 'seq.Seq'] :
( ( ( 'seq.is-sequence/1' @ V_t )
& ! [V_k_dot_0: \$int] :
( ( ( \$lesseq @ 1 @ V_k_dot_0 )
& ( \$lesseq @ V_k_dot_0 @ V_n ) )
=> ( \$greater @ ( 'seq.nth-term-of/2' @ V_t @ ( 'seq.index/1' @ V_k_dot_0 ) ) @ 0.0 ) )
& ( V_ti
= ( 'seq.seq/1'
@ ^ [V_k: \$int] :
( \$quotient @ 1.0 @ ( 'seq.nth-term-of/2' @ V_t @ ( 'seq.index/1' @ V_k ) ) ) ) )
& ( \$greater @ ( \$sum @ ( '^/2' @ ( \$to_real @ V_n ) @ 2.0 ) @ 1.0 ) @ ( \$product @ ( 'seq.sum-from-to/3' @ V_t @ ( 'seq.index/1' @ 1 ) @ ( 'seq.index/1' @ V_n ) ) @ ( 'seq.sum-from-to/3' @ V_ti @ ( 'seq.index/1' @ 1 ) @ ( 'seq.index/1' @ V_n ) ) ) ) )
=> ! [V_i: \$int,V_j: \$int,V_k_dot_1: \$int] :
( ( ( \$lesseq @ 1 @ V_i )
& ( \$less @ V_i @ V_j )
& ( \$less @ V_j @ V_k_dot_1 )
& ( \$lesseq @ V_k_dot_1 @ V_n ) )
=> ( 'are-triangle-edges/3' @ ( 'seq.nth-term-of/2' @ V_t @ ( 'seq.index/1' @ V_i ) ) @ ( 'seq.nth-term-of/2' @ V_t @ ( 'seq.index/1' @ V_j ) ) @ ( 'seq.nth-term-of/2' @ V_t @ ( 'seq.index/1' @ V_k_dot_1 ) ) ) ) ) )).

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