MathDB

4

Part of 2008 Harvard-MIT Mathematics Tournament

Problems(6)

Functional Equation

Source:

3/2/2008
The function f f satisfies f(x) \plus{} f(2x \plus{} y) \plus{} 5xy \equal{} f(3x \minus{} y) \plus{} 2x^2 \plus{} 1 for all real numbers x x, y y. Determine the value of f(10) f(10).
function
Log and Quadratic Quotient Limit

Source: HMMT 2008 Calculus Problem 4

3/2/2008
(4) Let a a, b b be constants such that \lim_{x\rightarrow1}\frac {(\ln(2 \minus{} x))^2}{x^2 \plus{} ax \plus{} b} \equal{} 1. Determine the pair (a,b) (a,b).
quadraticslimitlogarithmscalculuscalculus computations
Kermit's Strange Backyard

Source:

3/2/2008
Kermit the frog enjoys hopping around the infinite square grid in his backyard. It takes him 1 1 Joule of energy to hop one step north or one step south, and 1 1 Joule of energy to hop one step east or one step west. He wakes up one morning on the grid with 100 100 Joules of energy, and hops till he falls asleep with 0 0 energy. How many different places could he have gone to sleep?
Congruent Circumcircles (really this time!)

Source:

3/9/2008
In a triangle ABC ABC, take point D D on BC BC such that DB \equal{} 14, DA \equal{} 13, DC \equal{} 4, and the circumcircle of ADB ADB is congruent to the circumcircle of ADC ADC. What is the area of triangle ABC ABC?
geometrycircumcircle
ab-cd

Source:

3/23/2008
Suppose that a,b,c,d a, b, c, d are real numbers satisfying a≥b≥c≥d≥0 a \geq b \geq c \geq d \geq 0, a^2 \plus{} d^2 \equal{} 1, b^2 \plus{} c^2 \equal{} 1, and ac \plus{} bd \equal{} 1/3. Find the value of ab \minus{} cd.
trigonometrycomplex numbers
Real Solutions

Source:

3/23/2008
Find the real solution(s) to the equation (x \plus{} y)^2 \equal{} (x \plus{} 1)(y \minus{} 1).
quadratics