## TPTP Problem File: RAL051^1.p

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```%------------------------------------------------------------------------------
% File     : RAL051^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Complex numbers and complex plane)
% Problem  : Hokkaido University, 2005, Humanities Course, Problem 1
% Version  : [Mat16] axioms : Especial.
% English  : Answer the following questions: (1) Find the range of the value
%            of the real number k such that the quadratic equation of x,
%            x^2-2 k x-3 k^2+1=0 has imaginary solutions.  (2) Find the
%            minimum and maximum values of F(k)=int_0^k(x^2 - 2 k x - 3 k^2 +
%            1)d x in the range of k found in (1).

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : Univ-Hokkaido-2005-Bun-1.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3485 (   0 unit;1199 type;   0 defn)
%            Number of atoms       : 45311 (2208 equality;22701 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39608 ( 104   ~; 233   |;1172   &;35973   @)
%                                         (1095 <=>;1031  =>;   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   : 8056 (  66 sgn;7085   !; 429   ?; 406   ^)
%                                         (8056   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1975 (   6 prd;   9 fun;  23 num;1937 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: RCF; Author: Ukyo Suzuki; Generated: 2014-09-27
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf(p1_qustion,conjecture,
( 'find/1' @ \$real
@ ( ^ [V_k: \$real] :
( 'complex.has-complex-solution/1' @ ( 'complex.quad-equation/3' @ ( \$sum @ ( \$uminus @ ( \$product @ 3.0 @ ( '^/2' @ V_k @ 2.0 ) ) ) @ 1.0 ) @ ( \$uminus @ ( \$product @ 2.0 @ V_k ) ) @ 1.0 ) ) ) )).
%------------------------------------------------------------------------------
```