## TPTP Problem File: RAL028^1.p

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```%------------------------------------------------------------------------------
% File     : RAL028^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Inequalities)
% Problem  : International Mathematical Olympiad, 1969, Problem 6
% Version  : [Mat16] axioms : Especial.
% English  : Prove that for all real numbers x_1, x_2, y_1, y_2, z_1, z_2,
%            with x_1 > 0, x_2 > 0, x_1 y_1 - z_1^2 > 0, x_2 y_2 - z_2^2 > 0,
%            the inequality 8/(x_1+x_2)(y_1+y_2) - (z_1+z_2)^2 =<
%            1/x_1 y_1 - z_1^2 + 1/x_2 y_2 - z_2^2 is satisfied. Give necessary
%            and sufficient conditions for equality.

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : IMO-1969-6.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3485 (   0 unit;1199 type;   0 defn)
%            Number of atoms       : 45353 (2208 equality;22719 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39650 ( 104   ~; 233   |;1175   &;36011   @)
%                                         (1095 <=>;1032  =>;   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;7091   !; 429   ?; 405   ^)
%                                         (8061   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1980 (   6 prd;   9 fun;  23 num;1942 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: RCF; Score: 8; Author: Jumma Kudo;
%            Generated: 2014-12-18
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf(p1,conjecture,(
! [V_x1: \$real,V_x2: \$real,V_y1: \$real,V_y2: \$real,V_z1: \$real,V_z2: \$real] :
( ( ( \$greater @ V_x1 @ 0.0 )
& ( \$greater @ V_x2 @ 0.0 )
& ( \$greater @ ( \$difference @ ( \$product @ V_x1 @ V_y1 ) @ ( '^/2' @ V_z1 @ 2.0 ) ) @ 0.0 )
& ( \$greater @ ( \$difference @ ( \$product @ V_x2 @ V_y2 ) @ ( '^/2' @ V_z2 @ 2.0 ) ) @ 0.0 ) )
=> ( \$lesseq @ ( \$quotient @ 8.0 @ ( \$difference @ ( \$product @ ( \$sum @ V_x1 @ V_x2 ) @ ( \$sum @ V_y1 @ V_y2 ) ) @ ( '^/2' @ ( \$sum @ V_z1 @ V_z2 ) @ 2.0 ) ) ) @ ( \$sum @ ( \$quotient @ 1.0 @ ( \$difference @ ( \$product @ V_x1 @ V_y1 ) @ ( '^/2' @ V_z1 @ 2.0 ) ) ) @ ( \$quotient @ 1.0 @ ( \$difference @ ( \$product @ V_x2 @ V_y2 ) @ ( '^/2' @ V_z2 @ 2.0 ) ) ) ) ) ) )).
%------------------------------------------------------------------------------
```