## TPTP Problem File: RAL033^1.p

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```%------------------------------------------------------------------------------
% File     : RAL033^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Inequalities)
% Problem  : International Mathematical Olympiad, 1984, Problem 1
% Version  : [Mat16] axioms : Especial.
% English  : Prove that 0 =< yz + zx + xy - 2xyz =< 7/27, where x, y and z
%            are non-negative real numbers for which x + y + z = 1.

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : IMO-1984-1.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3485 (   0 unit;1199 type;   0 defn)
%            Number of atoms       : 45354 (2209 equality;22723 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39649 ( 104   ~; 233   |;1176   &;36009   @)
%                                         (1095 <=>;1032  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  : 2408 (2408   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1247 (1199   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   : 8058 (  66 sgn;7088   !; 429   ?; 405   ^)
%                                         (8058   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1978 (   6 prd;   9 fun;  24 num;1939 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: RCF; Score: 7; Author: Jumma Kudo;
%            Generated: 2014-11-20
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf(p,conjecture,(
! [V_x: \$real,V_y: \$real,V_z: \$real] :
( ( ( \$greatereq @ V_x @ 0.0 )
& ( \$greatereq @ V_y @ 0.0 )
& ( \$greatereq @ V_z @ 0.0 )
& ( ( \$sum @ V_x @ ( \$sum @ V_y @ V_z ) )
= 1.0 ) )
=> ( ( \$lesseq @ 0.0 @ ( \$difference @ ( \$sum @ ( \$product @ V_x @ V_y ) @ ( \$sum @ ( \$product @ V_y @ V_z ) @ ( \$product @ V_z @ V_x ) ) ) @ ( \$product @ 2.0 @ ( \$product @ V_x @ ( \$product @ V_y @ V_z ) ) ) ) )
& ( \$lesseq @ ( \$difference @ ( \$sum @ ( \$product @ V_x @ V_y ) @ ( \$sum @ ( \$product @ V_y @ V_z ) @ ( \$product @ V_z @ V_x ) ) ) @ ( \$product @ 2.0 @ ( \$product @ V_x @ ( \$product @ V_y @ V_z ) ) ) ) @ ( \$quotient @ 7.0 @ 27.0 ) ) ) ) )).

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