MathDB

3

Part of 2008 Harvard-MIT Mathematics Tournament

Problems(6)

Quadratic Inequality

Source:

3/2/2008
Determine all real numbers a a such that the inequality |x^2 \plus{} 2ax \plus{} 3a|\le2 has exactly one solution in x x.
quadraticsinequalitiesfunctionconicsparabolaanalytic geometryabsolute value
Definite Integral=1/4

Source: HMMT 2008 Calculus Problem 3

3/2/2008
(4) Find all y>1 y > 1 satisfying \int^y_1x\ln x\ dx \equal{} \frac {1}{4}.
calculusintegrationlogarithmscalculus computations
Farmer John's Diverse Animals

Source:

3/2/2008
Farmer John has 5 5 cows, 4 4 pigs, and 7 7 horses. How many ways can he pair up the animals so that every pair consists of animals of different species? (Assume that all animals are distinguishable from each other.)
countingdistinguishability
Segment Outside Triangle

Source:

3/6/2008
Let ABC ABC be a triangle with \angle BAC \equal{} 90^\circ. A circle is tangent to the sides AB AB and AC AC at X X and Y Y respectively, such that the points on the circle diametrically opposite X X and Y Y both lie on the side BC BC. Given that AB \equal{} 6, find the area of the portion of the circle that lies outside the triangle. [asy]import olympiad; import math; import graph;
unitsize(20mm); defaultpen(fontsize(8pt));
pair A = (0,0); pair B = A + right; pair C = A + up;
pair O = (1/3, 1/3);
pair Xprime = (1/3,2/3); pair Yprime = (2/3,1/3);
fill(Arc(O,1/3,0,90)--Xprime--Yprime--cycle,0.7*white);
draw(A--B--C--cycle); draw(Circle(O, 1/3)); draw((0,1/3)--(2/3,1/3)); draw((1/3,0)--(1/3,2/3));
label("AA",A, SW); label("BB",B, down); label("CC",C, left); label("XX",(1/3,0), down); label("YY",(0,1/3), left);[/asy]
geometrysimilar triangles
Dogs, Cats, and Milk

Source:

3/17/2008
There are 5 5 dogs, 4 4 cats, and 7 7 bowls of milk at an animal gathering. Dogs and cats are distinguishable, but all bowls of milk are the same. In how many ways can every dog and cat be paired with either a member of the other species or a bowl of milk such that all the bowls of milk are taken?
countingdistinguishability
Coloring 2x2008 Grid

Source:

3/23/2008
How many ways can you color the squares of a 2×2008 2 \times 2008 grid in 3 colors such that no two squares of the same color share an edge?
LaTeX