## TPTP Problem File: RAL049^1.p

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```%------------------------------------------------------------------------------
% File     : RAL049^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Computation of expressions)
% Problem  : International Mathematical Olympiad, 2014, Problem 1
% Version  : [Mat16] axioms : Especial.
% English  : Let a_0 < a_1 < a_2 < ... be an infinite sequence of positive
%            integers.  Prove that there exists a unique integer n geq 1 such
%            that a_n < a_0 + a_1 + ... + a_n/n leq a_{n+1}.

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : IMO-2014-1.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3485 (   0 unit;1199 type;   0 defn)
%            Number of atoms       : 45388 (2209 equality;22731 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39683 ( 104   ~; 233   |;1178   &;36038   @)
%                                         (1095 <=>;1035  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  : 2409 (2409   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1246 (1199   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   : 8060 (  66 sgn;7089   !; 430   ?; 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: Jumma Kudo;
%            Generated: 2014-10-10
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf(p,conjecture,(
! [V_a: ( \$int > \$int )] :
( ( ! [V_n_dot_0: \$int] :
( ( \$lesseq @ 0 @ V_n_dot_0 )
=> ( \$lesseq @ 1 @ ( V_a @ V_n_dot_0 ) ) )
& ! [V_n: \$int] :
( ( \$lesseq @ 0 @ V_n )
=> ( \$lesseq @ ( V_a @ V_n ) @ ( V_a @ ( \$sum @ V_n @ 1 ) ) ) ) )
=> ? [V_n_dot_1: \$int] :
( ( \$lesseq @ 1 @ V_n_dot_1 )
& ( \$less @ ( \$to_rat @ ( V_a @ V_n_dot_1 ) ) @ ( \$quotient @ ( \$to_rat @ ( 'int.sum/1' @ ( 'map/2' @ \$int @ \$int @ V_a @ ( 'int.iota/2' @ 0 @ V_n_dot_1 ) ) ) ) @ ( \$to_rat @ V_n_dot_1 ) ) )
& ( \$lesseq @ ( \$quotient @ ( \$to_rat @ ( 'int.sum/1' @ ( 'map/2' @ \$int @ \$int @ V_a @ ( 'int.iota/2' @ 0 @ V_n_dot_1 ) ) ) ) @ ( \$to_rat @ V_n_dot_1 ) ) @ ( \$to_rat @ ( V_a @ ( \$sum @ V_n_dot_1 @ 1 ) ) ) )
& ! [V_m: \$int] :
( ( ( \$lesseq @ 1 @ V_m )
& ( \$less @ ( \$to_rat @ ( V_a @ V_m ) ) @ ( \$quotient @ ( \$to_rat @ ( 'int.sum/1' @ ( 'map/2' @ \$int @ \$int @ V_a @ ( 'int.iota/2' @ 0 @ V_m ) ) ) ) @ ( \$to_rat @ V_m ) ) )
& ( \$lesseq @ ( \$quotient @ ( \$to_rat @ ( 'int.sum/1' @ ( 'map/2' @ \$int @ \$int @ V_a @ ( 'int.iota/2' @ 0 @ V_m ) ) ) ) @ ( \$to_rat @ V_m ) ) @ ( \$to_rat @ ( V_a @ ( \$sum @ V_m @ 1 ) ) ) ) )
=> ( V_n_dot_1 = V_m ) ) ) ) )).

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