## TPTP Problem File: MSC032^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : MSC032^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Miscellaneous
% Problem  : International Mathematical Olympiad, 2003, Problem 1
% Version  : [Mat16] axioms : Especial.
% English  : S is the set {1, 2, 3, ..., 1000000}. Show that for any subset A
%            of S with 101 elements we can find 100 distinct elements xi of
%            S, such that the sets { a + xi | a in A } are all pairwise
%            disjoint.

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : IMO-2003-1.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3485 (   0 unit;1199 type;   0 defn)
%            Number of atoms       : 45339 (2213 equality;22714 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39630 ( 104   ~; 233   |;1178   &;35988   @)
%                                         (1095 <=>;1032  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  : 2408 (2408   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1249 (1199   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   : 8063 (  66 sgn;7086   !; 433   ?; 408   ^)
%                                         (8063   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1981 (   6 prd;   9 fun;  26 num;1940 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: PA(comb); Score: 7; Author: Yiyang Zhan;
%            Generated: 2014-11-19
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf(p1,conjecture,(
? [V_S: ( 'SetOf' @ \$int )] :
( ( V_S
= ( 'set-by-def/1' @ \$int
@ ^ [V_n_dot_0: \$int] :
( ( \$lesseq @ 1 @ V_n_dot_0 )
& ( \$lesseq @ V_n_dot_0 @ 1000000 ) ) ) )
& ! [V_A: ( 'SetOf' @ \$int )] :
( ( ( 'is-subset-of/2' @ \$int @ V_A @ V_S )
& ( ( 'int.cardinality-of/1' @ V_A )
= 101 ) )
=> ? [V_X: ( 'SetOf' @ \$int )] :
( ( 'is-subset-of/2' @ \$int @ V_X @ V_S )
& ( ( 'int.cardinality-of/1' @ V_X )
= 100 )
& ( 'pairwise-disjoint/1' @ ( 'SetOf' @ \$int )
@ ( 'set-by-def/1' @ ( 'SetOf' @ \$int )
@ ^ [V_Ax: ( 'SetOf' @ \$int )] :
? [V_x: \$int] :
( V_Ax
= ( 'set-by-def/1' @ \$int
@ ^ [V_n: \$int] :
? [V_a: \$int] :
( ( 'elem/2' @ \$int @ V_a @ V_A )
& ( V_n
= ( \$sum @ V_x @ V_a ) ) ) ) ) ) ) ) ) ) )).

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