## Feature Topics

### The University of Tokyo 2013 Entrance Examination—Science Course Math

**Question
1**

For
real numbers α and β, point P_{n}(X_{n}, Y_{n}) on a
plane is determined by the following:

_{0},
Y_{0}) = (1, 0)

_{n+1},
Y_{n+1}) = (αX_{n} – βY_{n}, βX_{n} + αY_{n})
(n = 0, 1, 2, …)

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

(i)
P_{0} = P_{6}

(ii)
P_{0}, P_{1}, P_{2}, P_{3}, P_{4}, and
P_{5} 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*+ a

*x*

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.

*x*^{3} + 3*yx*^{2} < (*x*+*y*-1) (*x* + *y*) (*x* + *y*
+ 1) < *x*^{3} + (3*y* + 1) *x*^{2}

**(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 V_{1}.
The three-dimensional body formed by rotating square S about line AC is V_{2}.

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

**(2)
**What is the volume of the space shared by both V_{1} and V_{2}?

**(end of the problems)**