## TPTP Problem File: RAL066^1.p

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```%------------------------------------------------------------------------------
% File     : RAL066^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Numbers and algebraic expressions)
% Problem  : Tohoku University, 2013, Science Course, Problem 1
% Version  : [Mat16] axioms : Especial.
% English  : Let k be a real number. For the cubic equation f(x)= x^3 - k x^2
%            - 1, let {alpha}, {beta}, and {gamma} be the three solutions of
%            the equation f(x)= 0. Assume that g(x) is a cubic equation of
%            which the coefficient of x^3 is 1, and let {alpha} {beta},
%            {beta} {gamma}, and {gamma} {alpha} be the three solutions of
%            the equation g(x)= 0.  (1) Represent g(x) using k.  (2) Find the
%            value of k such that the equations f(x)= 0 and g(x)= 0 have
%            common solutions.

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : Univ-Tohoku-2013-Ri-1.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3486 (   0 unit;1200 type;   0 defn)
%            Number of atoms       : 45362 (2209 equality;22713 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39658 ( 104   ~; 233   |;1174   &;36021   @)
%                                         (1095 <=>;1031  =>;   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   : 8062 (  66 sgn;7085   !; 435   ?; 406   ^)
%                                         (8062   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1980 (   6 prd;   9 fun;  23 num;1942 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: RCF; Author: Takehiro Naito; Generated: 2015-01-08
%            ^ [V_gc_dot_0: ( ''ListOf'' @ \$real )] :
%              ( V_gc_dot_0
%              = ( 'cons/2' @ \$real @ -1.0 @ ( 'cons/2' @ \$real @ 'k/0' @ ( 'cons/2' @ \$real @ 0.0 @ ( 'cons/2' @ \$real @ 1.0 @ ( 'nil/0' @ \$real ) ) ) ) ) ) )
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf('k/0_type',type,(
'k/0': \$real )).

thf(p1_qustion,conjecture,
( 'find/1' @ ( 'ListOf' @ \$real )
@ ( ^ [V_gc: ( 'ListOf' @ \$real )] :
? [V_c0: \$real,V_c1: \$real,V_c2: \$real] :
( ( V_gc
= ( 'cons/2' @ \$real @ V_c0 @ ( 'cons/2' @ \$real @ V_c1 @ ( 'cons/2' @ \$real @ V_c2 @ ( 'cons/2' @ \$real @ 1.0 @ ( 'nil/0' @ \$real ) ) ) ) ) )
& ? [V_alpha: \$real,V_beta: \$real,V_gamma: \$real] :
( ( 'are-solutions-of/2' @ ( 'cons/2' @ \$real @ V_alpha @ ( 'cons/2' @ \$real @ V_beta @ ( 'cons/2' @ \$real @ V_gamma @ ( 'nil/0' @ \$real ) ) ) ) @ ( 'poly-equation/1' @ ( 'cons/2' @ \$real @ ( \$uminus @ 1.0 ) @ ( 'cons/2' @ \$real @ 0.0 @ ( 'cons/2' @ \$real @ ( \$uminus @ 'k/0' ) @ ( 'cons/2' @ \$real @ 1.0 @ ( 'nil/0' @ \$real ) ) ) ) ) ) )
& ( 'are-solutions-of/2' @ ( 'cons/2' @ \$real @ ( \$product @ V_alpha @ V_beta ) @ ( 'cons/2' @ \$real @ ( \$product @ V_beta @ V_gamma ) @ ( 'cons/2' @ \$real @ ( \$product @ V_gamma @ V_alpha ) @ ( 'nil/0' @ \$real ) ) ) ) @ ( 'poly-equation/1' @ V_gc ) ) ) ) ) )).
%------------------------------------------------------------------------------
```