The Sixth Annual Catonsville Mathematics Competition 1997

Problems

What is the condition that the two parabolas y = x² + ax and x = y² + ay intersect at four distinct points?
```
A) a > 0	B) a < 0	C) |a - 1| > 2	D) |a| < 2
```
An icosahedron is a polyhedron (solid figure with plane faces) with 20 triangular faces. It has 30 edges and 12 vertices. At each vertex, five faces meet, so that each vertex has five "neighbors" joined by an edge. A diagonal is a line segment joining two vertices which is not an edge. What is the total number of diagonals in an icosahedron?
```
A) 36		B) 140		C) 160		D) 200
```
If F_n is the n-th Fibonacci number defined by
F₁ = 1, F₂ = 1 ; F_n = F_{n - 1} + F_{n - 2}, n > 2 ;
which of the following is not always true?
A) If m divides n, then F_m divides F_n
B) F₁² + F₂² + F₃² + . . . + F_n² = F_n F_{n + 1}
C) The integer part of the arithmetic mean of F₂ , F₄ , F₆ , . . . , F_2n is 2^{n -1}.
D) F₁F₄ + F₂F₅ + . . . + F_nF_{n + 3} = [F_{n + 2}]² - 1
The number of ways n couples can dance so that no man dances with his wife is
```
A) (n - 1)!	B) (n - 1)(n - 2)	C) (n - 1)    	
D) none of the above.
```
The product of three positive integers is 36. Even if I tell you the sum of the three numbers you cannot be sure what they are. What is the sum of the numbers?
```
A) 11		B) 13   		C) 14		D) 16
```
A quadrilateral on a plane is called cyclic if all four vertices lie on a circle. If ABCD is a cyclic quadrilateral which of the following is not always true?
A) The sum of opposite angles is 180°.
B) The area of the quadrilateral is [(s - a)(s - b)(s - c)(s - d)]^1/2
where a, b, c, d are the side-lengths and s = (a + b + c + d)/2.
C) (AB)(BC) + (AD)(CD) = 2(h₁ + h₂) R , where h₁ and h₂ are the lengths of the perpendiculars from B and D on the diagonal AC and R is the radius of the circle through the vertices.
D) (AB)(BC) + (AD)(CD) = (AB)(AD) + (BC)(CD).
ABCD is a quadrilateral in the plane. Let R₁, R₂, R₃, and R₄ be the radii of the circles passing through respectively, the points A, B, C, the points B, C, D, the points C, D, A, and the points D, B, A. Consider the statements (refer to problem 6.)
(i) ABCD is cyclic if all four of R₁, R₂, R₃, and R₄ are equal.
(ii) ABCD is cyclic if three of R₁, R₂, R₃, and R₄ are equal.
(iii) ABCD is cyclic if any two of R₁, R₂, R₃, and R₄ are equal.
Which of the following is true?
A) Only (i) ,
B) (i) and (ii) ,
C) (i), (ii), and (iii) ,
D) None of them.
The number of different ways 4 letters can be chosen from the word MISSISSIPPI , where different ordering of the same set of letters is counted as the same, is
```
A) 21		B) 28		C) 33		D) 44
```
If k is a positive integer > 1, let n_k be the largest integer such that k raised to the power n_k does not exceed 1997. What is the value of the sum of all n_k 's for k = 2 to infinity?
```
A) 1998		B) 2064		C) 2596		D) 3994
```

Each letter in the following long division denotes a different digit from 0 to 9. What is U?

			                        A Q N
                                        _____________
				A N N E ) Q U I L T S
					  A N N E
					  _______
					    U I E T
					    S N N L
                                            _________   
					    A I I Q S
					    A E N E S
                                            _________	
						I U C

A) 3		B) 5		C) 7		D) 8

For solutions click here

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

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