## Feature Topics

### Nagoya University Entrance Examination (Year 2012) :Mathematics

*Time limit: 150 min.

#### 1. Let *a* be a positive constant and *y* = x^{3} – *a*^{2}*x* be the equation of the curve *C* in the *xy*-plane.

(1) Let *l* be a tangential line of *C* that passes through the point A(*t, t ^{3} – a^{2}t*) on

*C*. Find the area

*S(t)*of a figure bounded by

*l*and

*C*. Assume t ≠ 0.

(2) Let *b* be a real number. Find the number of tangent lines of *C* that pass through the point B(*2a, b*) in the *xy*-plane.

(3) Assume that only two tangent lines of *C*, *l*_{1} and *l*_{2}, pass through the point B(*2a, b*). Neither *l*_{1} nor *l*_{2} passes through the origin (0, 0). The area of the figure bounded by *l*_{1} and *C* is *S*_{1}. The area of the figure bounded by *l*_{2} and *C* is *S*_{2}. Assuming *S*_{1} ≥ *S*_{2}, find the value of *S*_{1}/*S*_{2}.

#### 2. For a positive integer *n*, a function *f*_{n}(*x*) of a real number *x* is defined by:

_{n}

where *f _{0}*(

*x*) =

*xe*

^{x}.

(1) Find *f _{1}*(

*x*).

When , find a definite integral assuming that real numbers *a*, *b*, and *c* are constants.

(3) Find *f _{2n}*(

*x*) for a positive integer

*n*.

#### 3. Let *n* be an integer (*n* ≥ 2). A single number, between 1 to *n*, is written on *n* pieces of card. A different number is written on each card. Randomly select one card from these *n* pieces of card, record the number written on the card, and then return it to the stack of cards. Repeat this trial three times and assign the minimum and the maximum integer values as *X* and *Y*, respectively. Answer the following questions. Note: *j* and *k* are positive integers and *j* + *k* ≤ *n*. *s* is a positive integer and *s* ≤ *n* - 1.

(1) Find the probability that satisfies both conditions: *X* ≥ *j* and *Y* ≤ *j* + *k*.

(2) Find the probability that satisfies both conditions: *X* = *j* and *Y* = *j* + *k*.

(3) Find the probability P(*s*) that satisfies the condition *Y* – *X* = *s*.

(4) When *n* is an even number, find *s* that maximizes *P(s)*.

#### 4. Let *m* and *p* be odd numbers (*m, p* ≥ 3). *m* is not divisible by *p*.

(1) Find the coefficient of the *x*^{2} term in the polynomial expansion of (*x* – 1)^{101}.

(2) Show that (*p* – 1)^{m} + l is divisible by *p*.

(3) Show that (*p* – 1)^{m} + l is not divisible by *p*^{2}.

(4) Let *r* be a positive integer and *s* = 3^{r–1}*m*. Show that 2^{s} + 1 is divisible by 3^{r}.