MathDB

8

Part of 1999 Harvard-MIT Mathematics Tournament

Problems(6)

1999 Advanced Topics #8: Largest Possible Circle

Source:

6/21/2012
Let CC be a circle with two diameters intersecting at an angle of 3030 degrees. A circle SS is tangent to both diameters and to CC, and has radius 11. Find the largest possible radius of CC.
1999 Algebra #8: Monic Quartic Polynomial

Source:

6/21/2012
If f(x)f(x) is a monic quartic polynomial such that f(−1)=−1f(-1)=-1, f(2)=−4f(2)=-4, f(−3)=−9f(-3)=-9, and f(4)=−16f(4)=-16, find f(1)f(1).
algebrapolynomial
1999 Calculus #8: Circle Inside a Circle

Source:

6/21/2012
A circle is randomly chosen in a circle of radius 11 in the sense that a point is randomly chosen for its center, then a radius is chosen at random so that the new circle is contained in the original circle. What is the probability that the new circle contains the center of the original circle?
calculusprobability
1999 HMMT Geometry # 8

Source:

3/3/2024
Squares ABKLABKL, BCMNBCMN, CAOPCAOP are drawn externally on the sides of a triangle ABCABC. The line segments KLKL, MNMN, OPOP, when extended, form a triangle A′B′C′A'B'C'. Find the area of A′B′C′A'B'C' if ABCABC is an equilateral triangle of side length 22.
geometry
1999 HMMT Team #8

Source:

3/8/2024
Find all the roots of (x2+3x+2)(x2−7x+12)(x2−2x−1)+24=0(x^2 + 3x + 2)(x^2 - 7x + 12)(x^2- 2x -1) + 24 = 0.
algebra
1999 Oral #8: Dividing a Number of a Certain Form

Source:

10/20/2012
What is the smallest square-free composite number that can divide a number of the form 4242…42±14242\ldots 42\pm 1?