## TPTP Problem File: RAL020^1.p

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```%------------------------------------------------------------------------------
% File     : RAL020^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Simultaneous equations)
% Problem  : International Mathematical Olympiad, 1961, Problem 1
% Version  : [Mat16] axioms : Especial.
% English  : Solve the system of equations: x+y+z = a; x^2 +y^2 +z^2 = b^2.
%            xy = z^2 where a and b are constants. Give the conditions that a
%            and b must satisfy so that x, y, z (the solutions of the system)
%            are distinct positive numbers.

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : IMO-1961-1.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3485 (   0 unit;1199 type;   0 defn)
%            Number of atoms       : 45354 (2215 equality;22722 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39641 ( 107   ~; 233   |;1181   &;35994   @)
%                                         (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: 6; Author: Jumma Kudo;
%            Generated: 2014-12-16
%            ^ [V_ab_dot_0: ( 'ListOf' @ \$real )] :
%            ? [V_a_dot_0: \$real,V_b_dot_0: \$real] :
%              ( ( V_ab_dot_0
%                = ( 'cons/2' @ \$real @ V_a_dot_0 @ ( 'cons/2' @ \$real @ V_b_dot_0 @ ( 'nil/0' @ \$real ) ) ) )
%              & ( \$less @ ( 'abs/1' @ V_b_dot_0 ) @ V_a_dot_0 )
%              & ( \$less @ V_a_dot_0 @ ( \$product @ ( 'sqrt/1' @ 3.0 ) @ ( 'abs/1' @ V_b_dot_0 ) ) ) ) )
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf(p_qustion,conjecture,
( 'find/1' @ ( 'ListOf' @ \$real )
@ ^ [V_ab: ( 'ListOf' @ \$real )] :
? [V_a: \$real,V_b: \$real,V_x: \$real,V_y: \$real,V_z: \$real] :
( ( V_ab
= ( 'cons/2' @ \$real @ V_a @ ( 'cons/2' @ \$real @ V_b @ ( 'nil/0' @ \$real ) ) ) )
& ( ( \$sum @ V_x @ ( \$sum @ V_y @ V_z ) )
= V_a )
& ( ( \$sum @ ( '^/2' @ V_x @ 2.0 ) @ ( \$sum @ ( '^/2' @ V_y @ 2.0 ) @ ( '^/2' @ V_z @ 2.0 ) ) )
= ( '^/2' @ V_b @ 2.0 ) )
& ( ( \$product @ V_x @ V_y )
= ( '^/2' @ V_z @ 2.0 ) )
& ( \$greater @ V_x @ 0.0 )
& ( \$greater @ V_y @ 0.0 )
& ( \$greater @ V_z @ 0.0 )
& ( V_x != V_y )
& ( V_y != V_z )
& ( V_x != V_z ) ) )).

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