## TPTP Problem File: SEV442^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEV442^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Set Theory (Combinatorics, finite sets)
% Problem  : International Mathematical Olympiad, 1994, Problem 1
% Version  : [Mat16] axioms : Especial.
% English  : Let m and n be positive integers. Let a_1, a_2, ..., a_m be
%            distinct elements of \{1, 2, ... , n\} such that whenever a_i +
%            a_j =< n for some i, j, 1 =< i =< j =< m, there exists k, 1
%            =< k =< m, with a_i + a_j = a_k. Prove that \a_1 + a_2
%            + ... + a_m/m >= (n+1)/2.

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : IMO-1994-1.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3485 (   0 unit;1199 type;   0 defn)
%            Number of atoms       : 45413 (2211 equality;22742 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39706 ( 105   ~; 233   |;1186   &;36052   @)
%                                         (1095 <=>;1035  =>;   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   : 8064 (  66 sgn;7093   !; 430   ?; 405   ^)
%                                         (8064   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1982 (   6 prd;   9 fun;  23 num;1944 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: PA; Score: 7; Author: Jumma Kudo;
%            Generated: 2014-11-11
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf(p,conjecture,(
! [V_n: \$int,V_m: \$int,V_an: ( 'ListOf' @ \$int )] :
( ( ( \$greater @ V_m @ 0 )
& ( \$greater @ V_n @ 0 )
& ( ( 'list-len/1' @ \$int @ V_an )
= V_m )
& ! [V_i_dot_1: \$int] :
( ( ( \$lesseq @ 1 @ V_i_dot_1 )
& ( \$lesseq @ V_i_dot_1 @ V_m ) )
=> ( ( \$lesseq @ 1 @ ( 'nth/2' @ \$int @ ( \$difference @ V_i_dot_1 @ 1 ) @ V_an ) )
& ( \$lesseq @ ( 'nth/2' @ \$int @ ( \$difference @ V_i_dot_1 @ 1 ) @ V_an ) @ V_n ) ) )
& ! [V_i_dot_0: \$int,V_j_dot_0: \$int] :
( ( ( \$lesseq @ 1 @ V_i_dot_0 )
& ( \$less @ V_i_dot_0 @ V_j_dot_0 )
& ( \$lesseq @ V_j_dot_0 @ V_m ) )
=> ( ( 'nth/2' @ \$int @ ( \$difference @ V_i_dot_0 @ 1 ) @ V_an )
!= ( 'nth/2' @ \$int @ ( \$difference @ V_j_dot_0 @ 1 ) @ V_an ) ) )
& ! [V_i: \$int,V_j: \$int] :
( ( ( \$lesseq @ 1 @ V_i )
& ( \$lesseq @ V_i @ V_j )
& ( \$lesseq @ V_j @ V_m )
& ( \$lesseq @ ( \$sum @ ( 'nth/2' @ \$int @ ( \$difference @ V_i @ 1 ) @ V_an ) @ ( 'nth/2' @ \$int @ ( \$difference @ V_j @ 1 ) @ V_an ) ) @ V_n ) )
=> ? [V_k: \$int] :
( ( \$lesseq @ 1 @ V_k )
& ( \$lesseq @ V_k @ V_m )
& ( ( \$sum @ ( 'nth/2' @ \$int @ ( \$difference @ V_i @ 1 ) @ V_an ) @ ( 'nth/2' @ \$int @ ( \$difference @ V_j @ 1 ) @ V_an ) )
= ( 'nth/2' @ \$int @ ( \$difference @ V_k @ 1 ) @ V_an ) ) ) ) )
=> ( \$greatereq @ ( \$quotient @ ( \$to_rat @ ( 'int.sum/1' @ V_an ) ) @ ( \$to_rat @ V_m ) ) @ ( \$quotient @ ( \$to_rat @ ( \$sum @ V_n @ 1 ) ) @ ( \$to_rat @ 2 ) ) ) ) )).

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