## TPTP Problem File: RAL037^1.p

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```%------------------------------------------------------------------------------
% File     : RAL037^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Functions)
% Problem  : International Mathematical Olympiad, 1998, Problem 6
% Version  : [Mat16] axioms : Especial.
% English  : Determine all finite sets S of at least three points in the plane
%            which satisfy the following condition: for any two distinct points
%            A and B in S, the perpendicular bisector of the line segment AB is
%            an axis of symmetry for S.

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : IMO-1998-6.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3485 (   0 unit;1199 type;   0 defn)
%            Number of atoms       : 45334 (2211 equality;22716 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39625 ( 104   ~; 233   |;1176   &;35983   @)
%                                         (1095 <=>;1034  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  : 2409 (2409   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1247 (1199   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   : 8061 (  66 sgn;7089   !; 430   ?; 406   ^)
%                                         (8061   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1980 (   6 prd;   9 fun;  24 num;1941 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: ZF; Score: 7; Author: Jumma Kudo;
%            Generated: 2014-11-06
%            ^ [V_y_dot_0: \$int] : ( V_y_dot_0 = 120 ) )
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf(p_qustion,conjecture,
( 'find/1' @ \$int
@ ^ [V_y: \$int] :
? [V_f: ( \$int > \$int )] :
( ! [V_n_dot_0: \$int] :
( ( \$lesseq @ V_n_dot_0 @ 0 )
=> ( ( V_f @ V_n_dot_0 )
= 0 ) )
& ! [V_n: \$int] :
( ( \$greater @ V_n @ 0 )
=> ( \$greater @ ( V_f @ V_n ) @ 0 ) )
& ! [V_s: \$int,V_t: \$int] :
( ( ( \$less @ 0 @ V_s )
& ( \$less @ 0 @ V_t ) )
=> ( ( V_f @ ( \$product @ ( 'int.^/2' @ V_t @ 2 ) @ ( V_f @ V_s ) ) )
= ( \$product @ V_s @ ( 'int.^/2' @ ( V_f @ V_t ) @ 2 ) ) ) )
& ( V_y
= ( V_f @ 1998 ) ) ) )).

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