## TPTP Problem File: RAL042^1.p

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```%------------------------------------------------------------------------------
% File     : RAL042^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Functional equations)
% Problem  : International Mathematical Olympiad, 2008, Problem 4
% Version  : [Mat16] axioms : Especial.
% English  : Find all functions f : (0, infinity) -> (0, infinity) (so, f is
%            a function from the positive real numbers to the positive real
%            numbers) such that (f(w)^2 + f(x)^2) / (f(y^2) + f(z^2)) = (w^2
%            + x^2)/(y^2 + z^2) for all positive real numbers w, x, y, z,
%            satisfying wx = yz.

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : IMO-2008-4.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3485 (   0 unit;1199 type;   0 defn)
%            Number of atoms       : 45365 (2211 equality;22725 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39655 ( 104   ~; 233   |;1178   &;36011   @)
%                                         (1095 <=>;1034  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  : 2410 (2410   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1246 (1199   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   : 8062 (  66 sgn;7091   !; 429   ?; 406   ^)
%                                         (8062   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1980 (   6 prd;   9 fun;  23 num;1942 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: ZF; Score: 7; Author: Jumma Kudo;
%            Generated: 2014-10-22
%            ^ [V_f_dot_0: ( \$real > \$real )] :
%              ( ( V_f_dot_0
%                = ( ^ [V_x_dot_3: \$real] :
%                      ( 'if/3' @ \$real
%                      @ ^ [V___dot_0: 'Unit'] :
%                          ( \$less @ V_x_dot_3 @ 0.0 )
%                      @ 0.0
%                      @ V_x_dot_3 ) ) )
%              | ( V_f_dot_0
%                = ( ^ [V_x_dot_2: \$real] :
%                      ( 'if/3' @ \$real
%                      @ ^ [V__: 'Unit'] :
%                          ( \$less @ V_x_dot_2 @ 0.0 )
%                      @ 0.0
%                      @ ( \$quotient @ 1.0 @ V_x_dot_2 ) ) ) ) ) )
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf(p_qustion,conjecture,
( 'find/1' @ ( \$real > \$real )
@ ^ [V_f: ( \$real > \$real )] :
( ! [V_x_dot_1: \$real] :
( ( \$lesseq @ V_x_dot_1 @ 0.0 )
=> ( ( V_f @ V_x_dot_1 )
= 0.0 ) )
& ! [V_x_dot_0: \$real] :
( ( \$less @ 0.0 @ V_x_dot_0 )
=> ( \$less @ 0.0 @ ( V_f @ V_x_dot_0 ) ) )
& ! [V_w: \$real,V_x: \$real,V_y: \$real,V_z: \$real] :
( ( ( ( \$product @ V_w @ V_x )
= ( \$product @ V_y @ V_z ) )
& ( \$greater @ V_x @ 0.0 )
& ( \$greater @ V_y @ 0.0 )
& ( \$greater @ V_w @ 0.0 )
& ( \$greater @ V_z @ 0.0 ) )
=> ( ( \$quotient @ ( \$sum @ ( '^/2' @ ( V_f @ V_w ) @ 2.0 ) @ ( '^/2' @ ( V_f @ V_x ) @ 2.0 ) ) @ ( \$sum @ ( V_f @ ( '^/2' @ V_y @ 2.0 ) ) @ ( V_f @ ( '^/2' @ V_z @ 2.0 ) ) ) )
= ( \$quotient @ ( \$sum @ ( '^/2' @ V_w @ 2.0 ) @ ( '^/2' @ V_x @ 2.0 ) ) @ ( \$sum @ ( '^/2' @ V_y @ 2.0 ) @ ( '^/2' @ V_z @ 2.0 ) ) ) ) ) ) )).

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