The Second Annual Catonsville Mathematics Competition 1993

Problems

a and b are two positive integers such that when a² + b² is divided by a + b, the quotient is 18 and the remainder is 7. How many different ordered pairs of values of a and b are possible?
```
A) 4		B) 6		C) 10		D) 12
```
If n is a positive integer such that 2^{n - 1} is divisible by 7, then n must be
```
A) odd		B) prime	C) divisible by 3	D) triangular
```
The n-th Farey sequence F_n consists of all rational numbers x = p/q , such that
0 Ł x Ł 1 and q Ł n in ascending order.
e.g., F₄ = {0, 1/4, 1/3, 1/2, 2/3, 3/4, 1}
and F₅ = {0, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1}
Find the sum of the 65 numbers in the 14-th Farey sequence F₁₄ ?
```
A) 13.5		B) 21		C) 32.5		D) 35.5
```
Which one of the following numbers cannot be the area of a triangle with perimeter 60?
```
A) 96		B) 144		C) 169  	D) 175 
```
The maximum possible value of the product of positive integers whose sum is 20 is
```
A) 1024		B) 1296		C) 1458		D) 2048
```

What is the sum of the digits of the quotient in the following long division?

			      * * 8 * *
		      -----------------	
		* * * ) * * * * * * * *
			* * * *
			-------
			    * * * *
			      * * *
			    -------
				* * * *
				* * * *
				-------

A) 26		B) 28		C) 34		D) 42

If you have only lots of 5˘ and 29˘ stamps, what is the smallest number of stamps needed to make up the amount $2.44?
```
A) 10		B) 15		C) 20		D) 49
```
A, B, C are 3 points on a plane and a point P moves in such a way that
(AP)² + (BP)² = (CP)² at all times. What is the path of P? You may assume that ĐACB is an acute angle.
```
A) a line	B) a circle	C) an ellipse	D) a right triangle
```
A, B, C are 3 points on a plane and a point P moves in such a way that
(AP)² + (BP)² = 2(CP)² at all times. What is the path of P? You may assume that C is not the midpoint of AB.
```
A) a line	B) a circle	C) an ellipse	D) none of the above
```
A said "B always speaks the truth."
B said "A is lying."
C said "A and B are both lying."
Who is speaking the truth?
```
A) A		B) B		C) C		D) none of them
```
Three players A, B, and C play the following game. Three distinct positive integers p, q, and r are written on three cards, one number on each card. In each round the three cards are dealt among the players and each player collects as many chips from a pile as the number on his or her card. They keep the chips but return the cards which are dealt again. After a few rounds A, B, and C have amassed respectively 31, 26, and 28 chips. How many rounds were played?
```
A) 3		B) 5		C) 17		D) no one knows
```
A rectangular box can be completely filled with 1 inch cube blocks, but when you try to fill it with 2 inch cube blocks there are spaces left in each direction, i.e., along the length, the width, and the height of the box, and you can fill only 64% of its volume. The box can be put inside a 20 inch cube cardboard container. What is the volume of the box in cubic inches?
```
A) 105		B) 125		C) 525		D) 2197
```

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