The Fifth Annual Catonsville Mathematics Competition 1996

SOME DEFINITIONS

• The circum-radius of a triangle is the radius of its circum-circle, the circle that passes through the three vertices. The center is called the circum-center.
• The in-radius of a triangle is the radius of its in-circle, i.e., the largest circle lying inside the triangle. The center is called the in-center.
• The orthocenter of a triangle is the point of concurrence of the three altitudes.
• The pedal triangle of a triangle is the triangle whose vertices are the feet of the perpendiculars from the vertices to the opposite sides.

Problems

1. A polygon has 629 distinct diagonals. What is the number of its sides?

   A) 29		B) 33		C) 37		D) 43

2. a, b are positive integers (a > b). When a² + b² is divided by a + b, the quotient is q and the remainder is r. If q² + r = 1996, what is a?

   A) 44		B) 48		C) 54		D) 62

3. 2¹⁹⁹⁶ is a 601 digit number beginning with 7 and ending in 6. If A is the sum of digits of 2¹⁹⁹⁶ and B is the sum of digits of A, what is the sum of digits of B?

   A) 4		B) 7		C) 13		D) 16

4. The product of the in-radius and the circum-radius of a triangle with side lengths a, b, c is

   A) abc/2(a + b + c)
   B) (a² + b² + c²)/18
   C) (ab + bc + ca)/18
   D) none of the above.

5. A murder was committed by someone. Three suspects, A, B, and C were interrogated. A and B claimed innocence while C confessed to the crime. Assuming only one of them is lying who is the murderer?

   A) A		B) B   		C) C		D) none of them.

6. ABCD is a quadrilateral on a plane and a point P moves on the plane in such a way that the sum of squares of its distances from the four points A, B, C, D remains constant. What is the curve traced out by the point P?

   A) a circle	B) a hyperbola	C) a parabola	D) a line

7. ABCD is a quadrilateral in the plane, all of whose angles are less than 180°.What is the location of the point P such that the sum of the distances of P from the four vertices A, B, C, D is minimum?

   A) the point of intersection of the two diagonals of ABCD,
   B) the center of the smallest circle containing ABCD,
   C) the center of the largest circle inscribed in ABCD,
   D) the center of mass of ABCD.

8. Let DEF be the pedal triangle of the triangle ABC. The ratio of the products of the sidelengths of the triangles DEF and ABC is

   A) 3(abc)^1/3/4(a + b + c)
   B) 1/4
   C) |cos A cos B cos C|
   D) sin(A/2) sin(B/2) sin(C/2)

9. The orthocenter of a triangle is

   A) the centroid	B) the circum-center C) the in-center D) the orthocenter

   of the pedal triangle of the original triangle.

10. If a is the smallest positive integer which leaves 2 as remainder when divided by 3, 1 as remainder when divided by 5, and 3 as remainder when divided by 7, while b is the smallest positive integer which leaves 1 as remainder when divided by 3, 2 as remainder when divided by 5, and 3 as remainder when divided by 7, what is a + b?

    A) 48		B) 52		C) 153		D) 258

Back to Catonsville Mathematics Competition Page

For solutions click here

or call Dr. Nilotpal Ghosh (410) 455-4281
or email : Nilotpal Ghosh