2008日本东京大学入学考试数学试题

20081Consider a transformation f which maps the point (x,y )to (3x +y,−2x )in the x −y palne.Denote the image of P as f (P ).Define the lines l 0,l 1,···by f as follows.

(a )l 0:3x +2y =1.

(b )Let l n +1be the line which f (P )draws when the point P moves on l n .

Denote l n as a n x +b n y =1by linear expression.

(1)Express a n +1,b n +1in terms of a n ,b n .

(2)Let D n be the domain by which the inequality a n x +b n y >1determines.Draw the domain of the points which are contained in all of D 0,D 1,D 2,···.2There are many two kinds of black and white cards.When you have k cards in hand,consider the following procedure (A ).

(A )You choose one card from k cards in hand with equal probability 1k and replace the card with different color one.

Answer the following questions.

(1)When you have 4cards with 2white and 2black,first,find the probabilty such that you will first have 4cards in same color after repeating the procedure (A )n times.

(2)When you have 6cards with 3white and 3black,first,find the probabilty such that you will first have 6cards in same color after repeating the procedure (A )n times.3(1)A regular octahedron is placed on a horizontal rest.Draw the plan of top-view for the regular octahedron.

(2)Let G 1,G 2be the barycentres of the two faces of the regular octahedron parallel to each other.Find the volume of the solid by revolving the regular tetrahedron about the line G 1G 2as the axis of rotation.4Given two points P,Q on the parabola y =x 2.Let h be the y coordinate of the segment P Q .

(1)Let L and m be the length and slope of the segment P Q respectively.Express h in terms of L,m .

(2)When L is fixed,find the minimum value of h .5For a positive integer n ,denote 10n −19=n’s 1 111···111by n .For example 1=1,2=11,3=111.

(1)Let m be a non negative integer.Prove that 3m is divisible by 3m and indivisible by 3m +1./This file was downloaded from the AoPS −MathLinks Math Olympiad Resources Page Page 1http://www.mathlinks.ro/

2008

(2)Prove that n is divisible by 27is the necessary and sufficient condition for which n is divisible by 27.6In the x −y plane,find the area of the region bounded by the parameterized curve as follows. x =cos 2t y =t sin t

(0≤t ≤2π)/This file was downloaded from the AoPS −MathLinks Math Olympiad Resources Page Page 2http://www.mathlinks.ro/