## TPTP Problem File: RAL044^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : RAL044^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Functional equations)
% Problem  : International Mathematical Olympiad, 2010, Problem 1
% Version  : [Mat16] axioms : Especial.
% English  : Determine all functions f : R -> R such that the equality
%            f(floor(x) y) = f(x) floor(f(y)) holds for all x, y in R. (Here
%            floor(z) denotes the greatest integer less than or equal to z.)

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : IMO-2010-1.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3485 (   0 unit;1199 type;   0 defn)
%            Number of atoms       : 45309 (2209 equality;22706 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39603 ( 104   ~; 233   |;1172   &;35968   @)
%                                         (1095 <=>;1031  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  : 2410 (2410   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1247 (1199   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   : 8058 (  66 sgn;7087   !; 429   ?; 406   ^)
%                                         (8058   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1977 (   6 prd;  10 fun;  23 num;1938 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: ZF; Score: 7; Author: Jumma Kudo;
%            Generated: 2014-10-21
%            ^ [V_f_dot_0: ( \$real > \$real )] :
%            ? [V_a: \$real] :
%              ( ( \$lesseq @ 1.0 @ V_a )
%              & ( \$less @ V_a @ 2.0 )
%              & ( ( V_f_dot_0
%                  = ( ^ [V_x_dot_1: \$real] : 0.0 ) )
%                | ( V_f_dot_0
%                  = ( ^ [V_x_dot_0: \$real] : V_a ) ) ) ) )
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf(p_qustion,conjecture,
( 'find/1' @ ( \$real > \$real )
@ ^ [V_f: ( \$real > \$real )] :
! [V_x: \$real,V_y: \$real] :
( ( V_f @ ( \$product @ ( \$floor @ V_x ) @ V_y ) )
= ( \$product @ ( V_f @ V_x ) @ ( \$floor @ ( V_f @ V_y ) ) ) ) )).

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