[The Journal of mathematics at home University] the fifth session of the 254th [2013] national college students' mathematical competition [mathematics] questions

1 ($ 15' $) plane $ \ BBR ^ 2 $ the first two circles with a radius of $ r$ $ C_1 $ and $ C_2 $ are tangent to $ p $, scroll the circle $ C_2 $ along the circumference of $ C_1 $ (without moving) for one week. Then, the $ p $ point on $ C_2 $ also moves with $ C_2 $. note the motion track curve $ \ VGA $ is $ p $, called the heart line. set $ C $ to the Circle centered on the initial position (cut point) of $ p $, and its radius is $ r$. Note: $ \ Bex \ GAMMA: \ BBR ^ 2 \ cup \ sed {\ infty} \ To \ BBR ^ 2 \ cup \ sed {\ infty} \ EEx $ is the inverse transformation of the circle $ C $, it maps $ Q \ In \ BBR ^ 2 \ BS \ sed {p} $ to the point on the ray $ PQ $, and $ \ overrightarrow {PQ} \ cdot \ overrightarrow {PQ '} = R ^ 2 $. proof: $ \ gamma (\ VGA) $ is a parabolic.

 

2 ($10 '$) set $ N $ square matrix of order $ B (t) $ and $ n \ times 1 $ matrix $ g (t) $ \ Bex B (t) = (B _ {IJ} (t), \ quad B (t) = \ sex {\ BA {L} B _1 (t) \\\ vdots \ B _n (t) \ EA }, \ EEx $ where $ B _ {IJ} (t) $ and $ B _ I (t) $ are all real-system polynomial about $ T $, $ I, j =, \ cdots, N $. note $ D (t) = \ det B (t) $, $ d_ I (t) $ Replace $ B (t) $ with $ B (t) $ the determining factor of the $ N $ level matrix obtained after column $ I $ in the determinant. if $ D (t) $ has a solid root $ T_0 $, $ B (T_0) x = B (T_0) $ becomes a compatible linear equations about $ x $. proof: $ D (t), D_1 (t), D_2 (t), \ cdots, D_n (t) $ must have a number of times $ \ geq 1 $.

 

3 ($ 15' $) set $ f (x) $ on $ [0, a] $ the upper and second-order continuous micro, $ f' (0) = 1 $, $ f'' (0) \ NEQ 0 $, and $0 <F (x) <X, X \ In (0, a) $. make $ X _ {n + 1} = f (x_n), X_1 \ In (0, a) $.

(1) proof: $ \ sed {x_n} $ converges and calculates its limit;

(2) Do you want to converge $ \ sed {nx_n} $? If it converges, find its limit. If it does not converge, explain the reasons.

 

4 ($ 15' $) set $ A> 1 $, $ f :( 0, + \ infty) \ To (0, + \ infty) $ micro. proof: There is a positive sequence of $ + \ infty $ \ sed {x_n} $, making $ f' (x_n) <F (ax_n), \ n =, \ cdots $.

 

5 ($ 20' $): $ F: [-\ To \ BBR $ is an even function. $ F $ is an addition function on $ [] $; set $ G $ to the Convex Function on $ [-] $, that is, $ \ Bex g (Tx + (1-T) y) \ Leq Tg (x) + (1-T) G (Y), \ quad \ forall \ X, Y \ in [0, 1], \ quad \ forall \ t \ in [0, 1]. \ EEx $ trial certificate: $ \ Bex 2 \ int _ {-1} ^ 1 F (x) g (x) \ RD x \ geq \ int _ {-1} ^ 1 F (x) \ RD x \ cdot \ int _ {-1} ^ 1g (x) \ rd x. \ EEx $

 

6 ($ 25' $) set $ \ BBR ^ {n \ times N }$ to $ N $ all of the first-order matrix, $ E _ {IJ} $ to $ (I, j) $ element $1 $, other elements $ N $ of $0 $, $ I, j = 1, 2, \ cdots, N $. note $ \ vga_r $ indicates all the solid phalanx with the rank of $ r$, $ r =, 2, \ cdots, N $; and make $ \ Phi: \ BBR ^ {n \ times n} \ To \ BBR ^ {n \ times n} $ is a multiplier, that is, $ \ Bex \ PHI (AB) = \ PHI (a) \ cdot \ PHI (B), \ quad \ forall \ A, B \ In \ BBR ^ {n \ times n }. \ EEx $ proof:

(1) For $ \ forall \ A, B \ In \ vga_r $, $ \ rank \ PHI (A) = \ rank \ PHI (B) $.

(2) If $ \ PHI (0) = 0 $ and $ r = 1 $ matrix $ W $ makes $ \ PHI (w) = 0 $, the reversible square matrix $ r$ must exist so that $ \ Bex \ PHI (E _ {IJ}) = Re _ {IJ} r ^ {-1 }, \ quad \ forall \ I, j = 1, 2, \ cdots, N. \ EEx $