## TPTP Problem File: RAL032^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : RAL032^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Simultaneous equations)
% Problem  : International Mathematical Olympiad, 1979, Problem 5
% Version  : [Mat16] axioms : Especial.
% English  : Find all real numbers a for which there exist non-negative real
%            numbers x_1, x_2, x_3, x_4, x_5 satisfying the relations
%            sum_{k=1}^{5} k x_k = a, sum_{k=1}^{5} k^3 x_k = a^2,
%            sum_{k=1}^{5} k^5 x_k = a^3.

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : IMO-1979-5.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3485 (   0 unit;1199 type;   0 defn)
%            Number of atoms       : 45374 (2211 equality;22725 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39665 ( 104   ~; 233   |;1179   &;36023   @)
%                                         (1095 <=>;1031  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  : 2408 (2408   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1253 (1199   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   : 8061 (  66 sgn;7085   !; 434   ?; 406   ^)
%                                         (8061   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1987 (   6 prd;   9 fun;  30 num;1942 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: RCF; Score: 7; Author: Yiyang Zhan;
%            Generated: 2014-12-26
%            ^ [V_a_dot_0: \$real] :
%              ( ( V_a_dot_0 = 0.0 )
%              | ( V_a_dot_0 = 1.0 )
%              | ( V_a_dot_0 = 4.0 )
%              | ( V_a_dot_0 = 9.0 )
%              | ( V_a_dot_0 = 16.0 )
%              | ( V_a_dot_0 = 25.0 ) ) )
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf(p_qustion,conjecture,
( 'find/1' @ \$real
@ ^ [V_a: \$real] :
? [V_x1: \$real,V_x2: \$real,V_x3: \$real,V_x4: \$real,V_x5: \$real] :
( ( \$greatereq @ V_x1 @ 0.0 )
& ( \$greatereq @ V_x2 @ 0.0 )
& ( \$greatereq @ V_x3 @ 0.0 )
& ( \$greatereq @ V_x4 @ 0.0 )
& ( \$greatereq @ V_x5 @ 0.0 )
& ( ( \$sum @ V_x1 @ ( \$sum @ ( \$product @ 2.0 @ V_x2 ) @ ( \$sum @ ( \$product @ 3.0 @ V_x3 ) @ ( \$sum @ ( \$product @ 4.0 @ V_x4 ) @ ( \$product @ 5.0 @ V_x5 ) ) ) ) )
= V_a )
& ( ( \$sum @ V_x1 @ ( \$sum @ ( \$product @ 8.0 @ V_x2 ) @ ( \$sum @ ( \$product @ 27.0 @ V_x3 ) @ ( \$sum @ ( \$product @ 64.0 @ V_x4 ) @ ( \$product @ 125.0 @ V_x5 ) ) ) ) )
= ( \$product @ V_a @ V_a ) )
& ( ( \$sum @ V_x1 @ ( \$sum @ ( \$product @ 32.0 @ V_x2 ) @ ( \$sum @ ( \$product @ 243.0 @ V_x3 ) @ ( \$sum @ ( \$product @ 1024.0 @ V_x4 ) @ ( \$product @ 3125.0 @ V_x5 ) ) ) ) )
= ( \$product @ V_a @ ( \$product @ V_a @ V_a ) ) ) ) )).

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