## TPTP Problem File: RAL047^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : RAL047^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra
% Problem  : International Mathematical Olympiad, 2012, Problem 4
% Version  : [Mat16] axioms : Especial.
% English  : Find all functions f : Z -> Z such that, for all integers a, b,
%            c that satisfy a + b + c = 0, the following equality holds:
%            f(a)^2 + f(b)^2 + f(c)^2 = 2 f(a) f(b) + 2 f(b) f(c) + 2 f(c)
%            f(a).  (Here Z denotes the set of integers.)

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : IMO-2012-4.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3485 (   0 unit;1199 type;   0 defn)
%            Number of atoms       : 45342 (2210 equality;22720 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39634 ( 104   ~; 233   |;1172   &;35998   @)
%                                         (1095 <=>;1032  =>;   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   : 8059 (  66 sgn;7088   !; 429   ?; 406   ^)
%                                         (8059   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1977 (   6 prd;   9 fun;  23 num;1939 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: ZF; Score: 7; Author: Jumma Kudo;
%            Generated: 2014-10-17
%            ^ [V_f_dot_0: ( \$int > \$int )] :
%            ? [V_k: \$int] :
%              ( ( V_k != 0 )
%              & ( ( V_f_dot_0
%                  = ( ^ [V_x_dot_2: \$int] : 0 ) )
%                | ( V_f_dot_0
%                  = ( ^ [V_x_dot_1: \$int] :
%                        ( 'if/3' @ \$int
%                        @ ^ [V___dot_2: 'Unit'] :
%                            ( ( \$remainder_f @ V_x_dot_1 @ 2 )
%                            = 0 )
%                        @ 0
%                        @ V_k ) ) )
%                | ( V_f_dot_0
%                  = ( ^ [V_x_dot_0: \$int] :
%                        ( 'if/3' @ \$int
%                        @ ^ [V___dot_1: 'Unit'] :
%                            ( ( \$remainder_f @ V_x_dot_0 @ 4 )
%                            = 0 )
%                        @ 0
%                        @ ( 'if/3' @ \$int
%                          @ ^ [V___dot_0: 'Unit'] :
%                              ( ( \$remainder_f @ V_x_dot_0 @ 4 )
%                              = 1 )
%                          @ V_k
%                          @ ( 'if/3' @ \$int
%                            @ ^ [V__: 'Unit'] :
%                                ( ( \$remainder_f @ V_x_dot_0 @ 4 )
%                                = 2 )
%                            @ ( \$product @ 4 @ V_k )
%                            @ V_k ) ) ) ) )
%                | ( V_f_dot_0
%                  = ( ^ [V_x: \$int] :
%                        ( \$product @ V_k @ ( 'int.^/2' @ V_x @ 2 ) ) ) ) ) ) )
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf(p_qustion,conjecture,
( 'find/1' @ ( \$int > \$int )
@ ^ [V_f: ( \$int > \$int )] :
! [V_a: \$int,V_b: \$int,V_c: \$int] :
( ( ( \$sum @ V_a @ ( \$sum @ V_b @ V_c ) )
= 1 )
=> ( ( \$sum @ ( 'int.^/2' @ ( V_f @ V_a ) @ 2 ) @ ( \$sum @ ( 'int.^/2' @ ( V_f @ V_b ) @ 2 ) @ ( 'int.^/2' @ ( V_f @ V_c ) @ 2 ) ) )
= ( \$sum @ ( \$product @ 2 @ ( \$product @ ( V_f @ V_b ) @ ( V_f @ V_c ) ) ) @ ( \$sum @ ( \$product @ 2 @ ( \$product @ ( V_f @ V_a ) @ ( V_f @ V_b ) ) ) @ ( \$product @ 2 @ ( \$product @ ( V_f @ V_a ) @ ( V_f @ V_c ) ) ) ) ) ) ) )).

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