MathDB

9

Part of 2008 Harvard-MIT Mathematics Tournament

Problems(5)

Real Root(s)?

Source: HMMT 2008 Algebra Problem 9

3/2/2008
Let S S be the set of points (a,b) (a,b) with 0≤a,b≤1 0\le a,b\le1 such that the equation x^4 \plus{} ax^3 \minus{} bx^2 \plus{} ax \plus{} 1 \equal{} 0 has at least one real root. Determine the area of the graph of S S.
geometryfunctionconicsparabola
Limit of Bunch of Powers of n

Source: HMMT 2008 Calculus Problem 9

3/2/2008
(7) Evaluate the limit \lim_{n\rightarrow\infty} n^{\minus{}\frac{1}{2}\left(1\plus{}\frac{1}{n}\right)} \left(1^1\cdot2^2\cdot\cdots\cdot n^n\right)^{\frac{1}{n^2}}.
limitintegrationHMMTlogarithmscalculusreal analysiscalculus computations
King on Infinite Chessboard

Source: HMMT 2008 Combinatorics Problem 9

3/3/2008
On an infinite chessboard (whose squares are labeled by (x,y) (x, y), where x x and y y range over all integers), a king is placed at (0,0) (0, 0). On each turn, it has probability of 0.1 0.1 of moving to each of the four edge-neighboring squares, and a probability of 0.05 0.05 of moving to each of the four diagonally-neighboring squares, and a probability of 0.4 0.4 of not moving. After 2008 2008 turns, determine the probability that the king is on a square with both coordinates even. An exact answer is required.
probabilityanalytic geometry
Incenter and Square of Ratio

Source: HMMT 2008 Geometry Problem 9

3/9/2008
Let ABC ABC be a triangle, and I I its incenter. Let the incircle of ABC ABC touch side BC BC at D D, and let lines BI BI and CI CI meet the circle with diameter AI AI at points P P and Q Q, respectively. Given BI \equal{} 6, CI \equal{} 5, DI \equal{} 3, determine the value of (DP/DQ)2 \left( DP / DQ \right)^2.
geometryincenterratioangle bisector
Triangle in Base of Cone

Source:

3/23/2008
Consider a circular cone with vertex V V, and let ABC ABC be a triangle inscribed in the base of the cone, such that AB AB is a diameter and AC \equal{} BC. Let L L be a point on BV BV such that the volume of the cone is 4 times the volume of the tetrahedron ABCL ABCL. Find the value of BL/LV BL/LV.
geometry3D geometrytetrahedronpyramidsimilar triangles