The Ninth Annual Catonsville Mathematics Competition 2000

Problems

Peter Collins, a heavy smoker, has decided to give up smoking gradually. This year he resolved not to smoke in any month with an 'r' in its name (e.g., April) and also not to smoke in any day of the week with an 'r' in its name (e.g., Friday). On how many days in the year 2000 can he smoke?
```
A)  64              B)  67            C)  71            D)  73
```

In the following long division, each letter denotes a distinct digit from 0 to 9. Which of the 10 digits is missing?

                                    W E B N E
                            _________________
                      W E B ) I N T E R N E T
                                B T U
                              _________
                                  B R R
                                  A R T
                                  ________   
                                  E B U N
                                  E B A I
                                  ___________   
                                        B E T
                                        A R T
                                        _____
                                        E U N

A)  0               B)  4            C)  6              D)  9

What are the last 4 digits of 3²⁰⁰⁰?

A) 0001             B) 0401          C) 4001            D) 4401

It is well-known that the points of intersection of two perpendicular tangents to a parabola lie on its directrix. The points of intersection of two perpendicular normals to the parabola y² = 4px lie on
```
A) the x-axis  y = 0;              B) the parabola y² = p(x - 3p);

C) the parabola y² = p(x - 3p);    D) the circle (x - 4p)² + y² = p².
```
The Fibonacci numbers are defined as follows
F₁ = 1, F₂ = 1 , F_n = F_{n - 1} + F_{n - 2} , n > 2 .
Show that if n is an odd integer,
F₁F_n - F₂F_{n - 1} + F₃F_{n - 2} - . . . - F_{n - 1}F₂ + F_nF₁ =
```
A) 0               B) F_n            C) F_{n + 1}            D) _nC_[n/2].
```
( [x] stands for the greatest integer less than or equal to x. )
If _nC_r stands for the number of ways r objects can be chosen from n different objects,
_nC₀ - _n-1C₁ + _n-2C₂ - . . . =
```
A) 0               B) 1             C) 2 cos((n+1)p)/3  D) (2/Ö3) sin((n+1)p)/3
```
If _nC_r stands for the number of ways r objects can be chosen from n different objects,
_nC₀ + _n-1C₁ + _n-2C₂ + . . . =
```
A) 0               B) F_n            C) F_{n + 1}            D) _nC_[n/2].
```
If P, Q, R are three points on the ellipse

x²/a² + y²/b² = 1,
the maximum value of the area of the triangle PQR is
```
A) pab            B) pab/2         C) 3Ö3ab/4        D) unknown
```
Let A, B, C, D be four points on a plane and P be another point. The sum of squares of distances of P from two of the four points equals the sum of distances from the other two, i.e.,

(PA)² + (PC)² = (PB)² + (PD)²
if and only if ABCD is a
```
A) parallelogram,  B) rectangle,   C) square,        D) rhombus.
```
It is known that men from the village of St. Mary Mead always lie while the women always tell the truth. In the neighboring village of Chipping Cleghorn, it is just the opposite, men tell the truth and women lie. As a result, there is so much animosity between the residents of the two villages that they rarely if ever intermarry. Louis, an anthropologist meets two married couples in a pub on the edge of the two villages and had the following conversation :
- Harry said, "My wife is a liar. And she calls me a liar."
- Rodney said, "My wife is always truthful."
- Jane said, "My husband is not a liar."
- Gladys said, "My husband says I am not a liar."
Who are the married couple from Chipping Cleghorn?
```
A) Harry and Jane  B) Harry and Gladys  C) Rodney And Jane  D) Rodney and Gladys
```

For solutions click here

or call Dr. Nilotpal Ghosh (410) 455-4281
or email : nghosh@ccbc.cc.md.us

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