## TPTP Problem File: RAL029^1.p

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```%------------------------------------------------------------------------------
% File     : RAL029^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Basics of equation/inequality)
% Problem  : International Mathematical Olympiad, 1972, Problem 4
% Version  : [Mat16] axioms : Especial.
% English  : Find all solutions (x_1, x_2, x_3, x_4, x_5) of the system of
%            inequalities
%            (x_1^2 - x_3 x_5)(x_2^2 - x_3 x_5) =< 0
%            (x_2^2 - x_4 x_1)(x_3^2 - x_4 x_1) =< 0
%            (x_3^2 - x_5 x_2)(x_4^2 - x_5 x_2) =< 0
%            (x_4^2 - x_1 x_3)(x_5^2 - x_1 x_3) =< 0
%            (x_5^2 - x_2 x_4)(x_1^2 - x_2 x_4) =< 0
%            where x_1, x_2, x_3, x_4, x_5 are positive real numbers.

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : IMO-1972-4.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3485 (   0 unit;1199 type;   0 defn)
%            Number of atoms       : 45416 (2209 equality;22740 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39712 ( 104   ~; 233   |;1182   &;36067   @)
%                                         (1095 <=>;1031  =>;   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   : 8061 (  66 sgn;7085   !; 434   ?; 406   ^)
%                                         (8061   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1979 (   6 prd;   9 fun;  23 num;1941 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: RCF; Score: 7; Author: Jumma Kudo;
%            Generated: 2014-11-27
%            ^ [V_XL_dot_0: ( 'ListOf' @ \$real )] :
%            ? [V_X1_dot_0: \$real,V_X2_dot_0: \$real,V_X3_dot_0: \$real,V_X4_dot_0: \$real,V_X5_dot_0: \$real] :
%              ( ( V_XL_dot_0
%                = ( 'cons/2' @ \$real @ V_X1_dot_0 @ ( 'cons/2' @ \$real @ V_X2_dot_0 @ ( 'cons/2' @ \$real @ V_X3_dot_0 @ ( 'cons/2' @ \$real @ V_X4_dot_0 @ ( 'cons/2' @ \$real @ V_X5_dot_0 @ ( 'nil/0' @ \$real ) ) ) ) ) ) )
%              & ( V_X1_dot_0 = V_X2_dot_0 )
%              & ( \$less @ 0.0 @ V_X1_dot_0 )
%              & ( \$less @ 0.0 @ V_X2_dot_0 )
%              & ( \$less @ 0.0 @ V_X3_dot_0 )
%              & ( \$less @ 0.0 @ V_X4_dot_0 )
%              & ( \$less @ 0.0 @ V_X5_dot_0 )
%              & ( V_X2_dot_0 = V_X3_dot_0 )
%              & ( V_X3_dot_0 = V_X4_dot_0 )
%              & ( V_X4_dot_0 = V_X5_dot_0 ) ) )
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf(p_qustion,conjecture,
( 'find/1' @ ( 'ListOf' @ \$real )
@ ^ [V_XL: ( 'ListOf' @ \$real )] :
? [V_X1: \$real,V_X2: \$real,V_X3: \$real,V_X4: \$real,V_X5: \$real] :
( ( \$lesseq @ ( \$product @ ( \$difference @ ( '^/2' @ V_X1 @ 2.0 ) @ ( \$product @ V_X3 @ V_X5 ) ) @ ( \$difference @ ( '^/2' @ V_X2 @ 2.0 ) @ ( \$product @ V_X3 @ V_X5 ) ) ) @ 0.0 )
& ( \$lesseq @ ( \$product @ ( \$difference @ ( '^/2' @ V_X2 @ 2.0 ) @ ( \$product @ V_X1 @ V_X4 ) ) @ ( \$difference @ ( '^/2' @ V_X3 @ 2.0 ) @ ( \$product @ V_X4 @ V_X1 ) ) ) @ 0.0 )
& ( \$lesseq @ ( \$product @ ( \$difference @ ( '^/2' @ V_X3 @ 2.0 ) @ ( \$product @ V_X2 @ V_X5 ) ) @ ( \$difference @ ( '^/2' @ V_X4 @ 2.0 ) @ ( \$product @ V_X2 @ V_X5 ) ) ) @ 0.0 )
& ( \$lesseq @ ( \$product @ ( \$difference @ ( '^/2' @ V_X4 @ 2.0 ) @ ( \$product @ V_X1 @ V_X3 ) ) @ ( \$difference @ ( '^/2' @ V_X5 @ 2.0 ) @ ( \$product @ V_X1 @ V_X3 ) ) ) @ 0.0 )
& ( \$lesseq @ ( \$product @ ( \$difference @ ( '^/2' @ V_X5 @ 2.0 ) @ ( \$product @ V_X2 @ V_X4 ) ) @ ( \$difference @ ( '^/2' @ V_X1 @ 2.0 ) @ ( \$product @ V_X2 @ V_X4 ) ) ) @ 0.0 )
& ( \$less @ 0.0 @ V_X1 )
& ( \$less @ 0.0 @ V_X2 )
& ( \$less @ 0.0 @ V_X3 )
& ( \$less @ 0.0 @ V_X4 )
& ( \$less @ 0.0 @ V_X5 )
& ( V_XL
= ( 'cons/2' @ \$real @ V_X1 @ ( 'cons/2' @ \$real @ V_X2 @ ( 'cons/2' @ \$real @ V_X3 @ ( 'cons/2' @ \$real @ V_X4 @ ( 'cons/2' @ \$real @ V_X5 @ ( 'nil/0' @ \$real ) ) ) ) ) ) ) ) )).

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