## TPTP Problem File: RAL027^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : RAL027^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Functions)
% Problem  : International Mathematical Olympiad, 1968, Problem 5
% Version  : [Mat16] axioms : Especial.
% English  : Let f be a real-valued function defined for all real numbers x
%            such that, for some positive constant a, the equation f(x + a)
%            = 1/2 + sqrt(f(x) - [f(x)]^2) holds for all x. (a) Prove that the
%            function f is periodic (i.e., there exists a positive number b
%            such that f(x + b) = f(x) for all x).  (b) For a = 1, give an
%            example of a non-constant function with the required properties.

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : IMO-1968-5.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3485 (   0 unit;1199 type;   0 defn)
%            Number of atoms       : 45323 (2210 equality;22711 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39616 ( 104   ~; 233   |;1172   &;35980   @)
%                                         (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   : 8060 (  66 sgn;7088   !; 431   ?; 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: Takuya Matsuzaki;
%            Generated: 2015-01-24
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf(p1,conjecture,(
! [V_f: 'R2R'] :
( ? [V_a: \$real] :
! [V_x: \$real] :
( ( 'funapp/2' @ V_f @ ( \$sum @ V_x @ V_a ) )
= ( \$sum @ ( \$quotient @ 1.0 @ 2.0 ) @ ( 'sqrt/1' @ ( \$difference @ ( 'funapp/2' @ V_f @ V_x ) @ ( '^/2' @ ( 'funapp/2' @ V_f @ V_x ) @ 2.0 ) ) ) ) )
=> ? [V_b: \$real] :
! [V_x_dot_0: \$real] :
( ( 'funapp/2' @ V_f @ ( \$sum @ V_x_dot_0 @ V_b ) )
= ( 'funapp/2' @ V_f @ V_x_dot_0 ) ) ) )).
%------------------------------------------------------------------------------
```