## TPTP Problem File: RAL025^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : RAL025^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Simultaneous equations)
% Problem  : International Mathematical Olympiad, 1966, Problem 5
% Version  : [Mat16] axioms : Especial.
% English  : Solve the system of equations
%            |a_1 - a_2| x_2 + |a_1 - a_3| x_3 + |a_1 - a_4| x_4 = 1
%            |a_2 - a_1| x_1 + |a_2 - a_3| x_3 + |a_2 - a_4| x_4 = 1
%            |a_3 - a_1| x_1 + |a_3 - a_2| x_2 + |a_3 - a_4| x_4 = 1
%            |a_4 - a_1| x_1 + |a_4 - a_2| x_2 + |a_4 - a_3| x_3 = 1
%            where a_1, a_2, a_3, a_4 are four different 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-1966-5.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3489 (   0 unit;1203 type;   0 defn)
%            Number of atoms       : 45417 (2213 equality;22716 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39705 ( 104   ~; 233   |;1177   &;36065   @)
%                                         (1095 <=>;1031  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  : 2408 (2408   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1250 (1203   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   : 8060 (  66 sgn;7085   !; 433   ?; 406   ^)
%                                         (8060   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1978 (   6 prd;   9 fun;  23 num;1940 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: RCF; Score: 7; Author: Jumma Kudo;
%            Generated: 2014-12-03
%            ^ [V_xs_dot_0: ( 'ListOf' @ \$real )] :
%              ( ( 'pairwise-distinct/1' @ \$real @ ( 'cons/2' @ \$real @ 'A1/0' @ ( 'cons/2' @ \$real @ 'A2/0' @ ( 'cons/2' @ \$real @ 'A3/0' @ ( 'cons/2' @ \$real @ 'A4/0' @ ( 'nil/0' @ \$real ) ) ) ) ) )
%              & ( V_xs_dot_0
%                = ( 'cons/2' @ \$real @ ( \$quotient @ 1.0 @ ( \$difference @ 'A1/0' @ 'A4/0' ) ) @ ( 'cons/2' @ \$real @ 0.0 @ ( 'cons/2' @ \$real @ 0.0 @ ( 'cons/2' @ \$real @ ( \$quotient @ 1.0 @ ( \$difference @ 'A1/0' @ 'A4/0' ) ) @ ( 'nil/0' @ \$real ) ) ) ) ) ) ) )
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf('A1/0_type',type,(
'A1/0': \$real )).

thf('A2/0_type',type,(
'A2/0': \$real )).

thf('A3/0_type',type,(
'A3/0': \$real )).

thf('A4/0_type',type,(
'A4/0': \$real )).

thf(p_qustion,conjecture,
( 'find/1' @ ( 'ListOf' @ \$real )
@ ^ [V_xs: ( 'ListOf' @ \$real )] :
( ( 'pairwise-distinct/1' @ \$real @ ( 'cons/2' @ \$real @ 'A1/0' @ ( 'cons/2' @ \$real @ 'A2/0' @ ( 'cons/2' @ \$real @ 'A3/0' @ ( 'cons/2' @ \$real @ 'A4/0' @ ( 'nil/0' @ \$real ) ) ) ) ) )
& ? [V_X1: \$real,V_X2: \$real,V_X3: \$real,V_X4: \$real] :
( ( V_xs
= ( 'cons/2' @ \$real @ V_X1 @ ( 'cons/2' @ \$real @ V_X2 @ ( 'cons/2' @ \$real @ V_X3 @ ( 'cons/2' @ \$real @ V_X4 @ ( 'nil/0' @ \$real ) ) ) ) ) )
& ( ( \$sum @ ( \$product @ ( 'abs/1' @ ( \$difference @ 'A1/0' @ 'A2/0' ) ) @ V_X2 ) @ ( \$sum @ ( \$product @ ( 'abs/1' @ ( \$difference @ 'A1/0' @ 'A3/0' ) ) @ V_X3 ) @ ( \$product @ ( 'abs/1' @ ( \$difference @ 'A1/0' @ 'A4/0' ) ) @ V_X4 ) ) )
= 1.0 )
& ( ( \$sum @ ( \$product @ ( 'abs/1' @ ( \$difference @ 'A2/0' @ 'A1/0' ) ) @ V_X1 ) @ ( \$sum @ ( \$product @ ( 'abs/1' @ ( \$difference @ 'A2/0' @ 'A3/0' ) ) @ V_X3 ) @ ( \$product @ ( 'abs/1' @ ( \$difference @ 'A2/0' @ 'A4/0' ) ) @ V_X4 ) ) )
= 1.0 )
& ( ( \$sum @ ( \$product @ ( 'abs/1' @ ( \$difference @ 'A3/0' @ 'A2/0' ) ) @ V_X2 ) @ ( \$sum @ ( \$product @ ( 'abs/1' @ ( \$difference @ 'A3/0' @ 'A1/0' ) ) @ V_X1 ) @ ( \$product @ ( 'abs/1' @ ( \$difference @ 'A3/0' @ 'A4/0' ) ) @ V_X4 ) ) )
= 1.0 )
& ( ( \$sum @ ( \$product @ ( 'abs/1' @ ( \$difference @ 'A4/0' @ 'A2/0' ) ) @ V_X2 ) @ ( \$sum @ ( \$product @ ( 'abs/1' @ ( \$difference @ 'A4/0' @ 'A3/0' ) ) @ V_X3 ) @ ( \$product @ ( 'abs/1' @ ( \$difference @ 'A4/0' @ 'A1/0' ) ) @ V_X1 ) ) )
= 1.0 ) ) ) )).

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