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The University of Tokyo 2013 Entrance Examination—Science Course Math

Question 1

For real numbers α and β, point Pn(Xn, Yn) on a plane is determined by the following:

      (X0, Y0) = (1, 0)

      (Xn+1, Yn+1) = (αXn – βYn, βXn + αYn) (n = 0, 1, 2, …)


Find all (α, β) such that the following conditions (i) and (ii) are satisfied.

     (i) P0 = P6

     (ii) P0, P1, P2, P3, P4, and P5 are different from each other.

 

Question 2

f(x) and g(x) are functions of x as follows, where α is a real number and x > 0.

f(x) =  cos x 
 x
g(x) = sin x + ax

Find all α such that the graphs of y = f(x) and y = g(x) intersect at exactly three points where x > 0.

 

Question 3

There are two people, A and B. They have one coin where the probability of getting heads or tails when tossed is ½ each. A is in possession of the coin first. The next steps are repeated.

(i) When A is in possession of the coin, A tosses the coin. If the result is heads, A gets 1 point and A remains in possession of the coin. If the result is tails, no points are given and the coin is handed to B.

(ii) When B is in possession of the coin, B tosses the coin. If the result is heads, B gets 1 point, and B remains in possession of the coin. If the result is tails, no points are given and the coin is handed to A.

The first person to reach 2 points is the winner. For example, suppose after four coin tosses, the results are heads, tails, heads, heads. At this point, A has 1 point and B has 2 points, so B becomes the winner.


(1) After the coin has been tossed n times (including tosses by both A and B), what is the probability p(n) that A is the winner?

(2)   Determine
Σ
n=1
p(n)

Question 4

In ABC, BAC= 90°, = 1, and =√3. Suppose a point P inside ABC satisfies the following:


(1) Determine APB and APC.

(2) Determine , and .


Question 5

Consider proving the following proposition P.

Proposition P:  There exists a natural number (an integer greater than or equal to 1) A that satisfies the following conditions (a) and (b).

(a) A is a product of three consecutive natural numbers.

(b) When A is expressed in decimal notation, the number 1 appears 99 times or more in a row somewhere within the expression.

Answer the following questions.


(1) Suppose y is a natural number. Determine the range of a positive real number x that will satisfy the following inequality.

x3 + 3yx2 < (x+y-1) (x + y) (x + y + 1) < x3 + (3y + 1) x2

(2) Prove proposition P.

 

Question 6

In a coordinate system, square S in the xy-plane is determined by the following inequalities: |x|1, |y|1. The four corners of S are defined as follows: A(-1, 1, 0), B (1, 1, 0), C(1, -1, 0), D(-1, -1, 0). The three-dimensional body formed by rotating square S about line BD is V1. The three-dimensional body formed by rotating square S about line AC is V2.


(1) For a real number t that satisfies 0 t < 1, what is the area of the cross section of V1 with the plane where x = t?

(2) What is the volume of the space shared by both V1 and V2?


(end of the problems)