## TPTP Problem File: RAL048^1.p

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```%------------------------------------------------------------------------------
% File     : RAL048^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Functional equations)
% Problem  : International Mathematical Olympiad, 2013, Problem 5
% Version  : [Mat16] axioms : Especial.
% English  : Let Q_{>0} be the set of positive rational numbers.  Let f :
%            Q_{>0} rightarrow R be a function satisfying the following three
%            conditions: (i)   for all x, y in Q_{>0}, we have f(x)f(y) geq
%            f(xy); (ii)  for all x, y in Q_{>0}, we have f(x + y) geq f(x) +
%            f(y); (iii) there exists a rational number a > 1 such that f(a)
%            = a.  Prove that f(x) = x for all x in Q_{>0}.

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : IMO-2013-5.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3485 (   0 unit;1199 type;   0 defn)
%            Number of atoms       : 45350 (2210 equality;22727 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39643 ( 104   ~; 233   |;1180   &;35997   @)
%                                         (1095 <=>;1034  =>;   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   : 8060 (  66 sgn;7089   !; 430   ?; 405   ^)
%                                         (8060   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1978 (   6 prd;   9 fun;  23 num;1940 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: ZF; Score: 7; Author: Yiyang Zhan;
%            Generated: 2014-10-24
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf(p,conjecture,(
! [V_f: 'R2R'] :
( ( ! [V_x: \$real,V_y: \$real] :
( ( ( \$is_rat @ V_x )
& ( \$is_rat @ V_y )
& ( \$less @ 0.0 @ V_x )
& ( \$less @ 0.0 @ V_y ) )
=> ( ( \$greatereq @ ( \$product @ ( 'funapp/2' @ V_f @ V_x ) @ ( 'funapp/2' @ V_f @ V_y ) ) @ ( 'funapp/2' @ V_f @ ( \$product @ V_x @ V_y ) ) )
& ( \$greatereq @ ( 'funapp/2' @ V_f @ ( \$sum @ V_x @ V_y ) ) @ ( \$sum @ ( 'funapp/2' @ V_f @ V_x ) @ ( 'funapp/2' @ V_f @ V_y ) ) ) ) )
& ? [V_a: \$real] :
( ( \$is_rat @ V_a )
& ( \$greater @ V_a @ 1.0 )
& ( ( 'funapp/2' @ V_f @ V_a )
= V_a ) ) )
=> ! [V_x_dot_0: \$real] :
( ( ( \$is_rat @ V_x_dot_0 )
& ( \$less @ 0.0 @ V_x_dot_0 ) )
=> ( ( 'funapp/2' @ V_f @ V_x_dot_0 )
= V_x_dot_0 ) ) ) )).

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