## TPTP Problem File: RAL057^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : RAL057^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Functions and their graphs)
% Problem  : Kyushu University, 2005, Humanities Course, Problem 1
% Version  : [Mat16] axioms : Especial.
% English  : Let a be a positive real number, and consider the point A(0,
%            a+1/2 a) and the curve C: y = a x^2 ( xge 0). Let P the point on
%            the curve C that is the closest to the point A. Answer the
%            following questions: (1) Find the coordinates of the point P and
%            the length of the line segment AP. (2) Find the area S(a) of the
%            region enclosed by the curve C, the y axis, and the line segment
%            AP. (3) When a > 0, find the minimum value of the area S(a) and
%            the value of a that gives the minimum value.

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

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3486 (   0 unit;1200 type;   0 defn)
%            Number of atoms       : 45348 (2211 equality;22715 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39639 ( 104   ~; 233   |;1179   &;35996   @)
%                                         (1095 <=>;1032  =>;   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   : 8061 (  66 sgn;7086   !; 432   ?; 407   ^)
%                                         (8061   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1976 (   6 prd;   9 fun;  23 num;1938 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: RCF; Author: Tomoya Ishii; Generated: 2014-05-21
%            ^ [V_P_dot_0: '2d.Point'] :
%              ( ( \$less @ 0.0 @ 'a/0' )
%              & ( V_P_dot_0
%                = ( '2d.point/2' @ 1.0 @ 'a/0' ) ) ) )
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf('a/0_type',type,(
'a/0': \$real )).

thf(p1P_qustion,conjecture,
( 'find/1' @ '2d.Point'
@ ( ^ [V_P: '2d.Point'] :
? [V_A: '2d.Point',V_C: '2d.Shape',V_AP: \$real] :
( ( \$less @ 0.0 @ 'a/0' )
& ( V_A
= ( '2d.point/2' @ 0.0 @ ( \$sum @ 'a/0' @ ( \$quotient @ 1.0 @ ( \$product @ 2.0 @ 'a/0' ) ) ) ) )
& ( V_C
= ( '2d.graph-of/1'
@ ( 'fun/1'
@ ( ^ [V_x: \$real] :
( \$product @ 'a/0' @ ( '^/2' @ V_x @ 2.0 ) ) ) ) ) )
& ( '2d.on/2' @ V_P @ V_C )
& ( \$lesseq @ 0.0 @ ( '2d.x-coord/1' @ V_P ) )
& ( V_AP
= ( '2d.length-of/1' @ ( '2d.seg/2' @ V_A @ V_P ) ) )
& ! [V_Q: '2d.Point'] :
( ( ( '2d.on/2' @ V_Q @ V_C )
& ( \$lesseq @ 0.0 @ ( '2d.x-coord/1' @ V_Q ) ) )
=> ( \$lesseq @ ( '2d.length-of/1' @ ( '2d.seg/2' @ V_A @ V_P ) ) @ ( '2d.length-of/1' @ ( '2d.seg/2' @ V_A @ V_Q ) ) ) ) ) ) )).
%------------------------------------------------------------------------------
```