## TPTP Problem File: MSC033^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : MSC033^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Miscellaneous
% Problem  : International Mathematical Olympiad, 2014, Problem 2
% Version  : [Mat16] axioms : Especial.
% English  : Let n geq 2 be an integer. Consider an n times n chessboard
%            consisting of n^2 unit squares. A configuration of n rooks on
%            this board is peaceful if every row and every column contains
%            exactly one rook. Find the greatest positive integer k such
%            that, for each peaceful configuration of n rooks, there is a k
%            times k square which does not contain a rook on any of its k^2
%            unit squares.

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : IMO-2014-2.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3486 (   0 unit;1200 type;   0 defn)
%            Number of atoms       : 45432 (2211 equality;22744 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39728 ( 107   ~; 233   |;1190   &;36068   @)
%                                         (1095 <=>;1035  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  : 2408 (2408   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1247 (1200   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   : 8065 (  66 sgn;7091   !; 431   ?; 407   ^)
%                                         (8065   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1983 (   6 prd;   9 fun;  23 num;1945 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: PA(comb); Score: 7; Author: Jumma Kudo;
%            Generated: 2014-10-15
%            ^ [V_k_dot_1: \$int] :
%              ( ( \$less @ 'n/0' @ ( 'int.^/2' @ ( \$sum @ V_k_dot_1 @ 1 ) @ 2 ) )
%              & ( \$lesseq @ ( 'int.^/2' @ V_k_dot_1 @ 2 ) @ 'n/0' ) ) )
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf('n/0_type',type,(
'n/0': \$int )).

thf(p_qustion,conjecture,
( 'find/1' @ \$int
@ ^ [V_max_k: \$int] :
( 'int.maximum/2'
@ ( 'set-by-def/1' @ \$int
@ ^ [V_k: \$int] :
! [V_ps: ( 'ListOf' @ ( 'ListOf' @ \$int ) )] :
( ( ( ( 'list-len/1' @ ( 'ListOf' @ \$int ) @ V_ps )
= 'n/0' )
& ! [V_i: \$int] :
( ( ( \$lesseq @ 0 @ V_i )
& ( \$less @ V_i @ 'n/0' ) )
=> ( ( 'int.is-lattice-point/1' @ ( 'nth/2' @ ( 'ListOf' @ \$int ) @ V_i @ V_ps ) )
& ( \$lesseq @ 1 @ ( 'int.x-coord/1' @ ( 'nth/2' @ ( 'ListOf' @ \$int ) @ V_i @ V_ps ) ) )
& ( \$lesseq @ ( 'int.x-coord/1' @ ( 'nth/2' @ ( 'ListOf' @ \$int ) @ V_i @ V_ps ) ) @ 'n/0' )
& ( \$lesseq @ 1 @ ( 'int.y-coord/1' @ ( 'nth/2' @ ( 'ListOf' @ \$int ) @ V_i @ V_ps ) ) )
& ( \$lesseq @ ( 'int.y-coord/1' @ ( 'nth/2' @ ( 'ListOf' @ \$int ) @ V_i @ V_ps ) ) @ 'n/0' ) ) )
& ! [V_k_dot_0: \$int,V_l: \$int] :
( ( ( \$lesseq @ 0 @ V_k_dot_0 )
& ( \$less @ V_k_dot_0 @ 'n/0' )
& ( \$lesseq @ 0 @ V_l )
& ( \$less @ V_l @ 'n/0' ) )
=> ( ( ( 'int.x-coord/1' @ ( 'nth/2' @ ( 'ListOf' @ \$int ) @ V_k_dot_0 @ V_ps ) )
!= ( 'int.x-coord/1' @ ( 'nth/2' @ ( 'ListOf' @ \$int ) @ V_l @ V_ps ) ) )
& ( ( 'int.y-coord/1' @ ( 'nth/2' @ ( 'ListOf' @ \$int ) @ V_k_dot_0 @ V_ps ) )
!= ( 'int.y-coord/1' @ ( 'nth/2' @ ( 'ListOf' @ \$int ) @ V_l @ V_ps ) ) ) ) ) )
=> ? [V_x: \$int,V_y: \$int] :
( ( \$lesseq @ 0 @ V_x )
& ( \$less @ ( \$sum @ V_x @ V_k ) @ 'n/0' )
& ( \$lesseq @ 0 @ V_y )
& ( \$less @ ( \$sum @ V_y @ V_k ) @ 'n/0' )
& ! [V_i_dot_0: \$int,V_j: \$int] :
( ( ( \$lesseq @ V_x @ V_i_dot_0 )
& ( \$less @ V_i_dot_0 @ ( \$sum @ V_x @ V_k ) )
& ( \$lesseq @ V_y @ V_j )
& ( \$less @ V_j @ ( \$sum @ V_y @ V_k ) ) )
=> ~ ( 'member/2' @ ( 'ListOf' @ \$int ) @ ( 'cons/2' @ \$int @ V_i_dot_0 @ ( 'cons/2' @ \$int @ V_j @ ( 'nil/0' @ \$int ) ) ) @ V_ps ) ) ) ) )
@ V_max_k ) )).

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