MathDB

4

Part of 1999 Harvard-MIT Mathematics Tournament

Problems(6)

1999 Advanced Topics #4: Reversing Orders by Switching

Source:

6/21/2012
You are given 16 pieces of paper numbered 16,15,…,2,116, 15, \ldots , 2, 1 in that order. You want to put them in the order 1,2,…,15,161, 2, \ldots , 15, 16 switching only two adjacent pieces of paper at a time. What is the minimum number of switches necessary?
1999 Algebra #4: All Possible Values of Ratio

Source:

6/21/2012
Find all possible values of da\dfrac{d}{a} where a2−6ad+8d2=0a^2-6ad+8d^2=0, a≠0a\neq 0.
ratio
1999 Calculus #4: Functional Equation

Source:

6/21/2012
ff is a continuous real-valued function such that f(x+y)=f(x)f(y)f(x+y)=f(x)f(y) for all real xx, yy. If f(2)=5f(2)=5, find f(5)f(5).
calculusfunction
1999 HMMT Geometry # 4

Source:

3/3/2024
A cross-section of a river is a trapezoid with bases 1010 and 1616 and slanted sides of length 55. At this section the water is flowing at π\pi mph. A little ways downstream is a dam where the water flows through 44 identical circular holes at 1616 mph. What is the radius of the holes?
geometry
1999 HMMT Team #4

Source:

3/8/2024
Consider the equation FORTY+TEN+TEN=SIXTYFORTY + TEN + TEN = SIXTY , where each of the ten letters represents a distinct digit from 00 to 99. Find all possible values of SIXTYSIXTY .
number theory
1999 Oral #4: Trigonometrico-Geometric Sequence

Source:

10/20/2012
Evaluate ∑n=0∞cos⁡nθ2n\displaystyle\sum_{n=0}^\infty \dfrac{\cos n\theta}{2^n}, where cos⁡θ=15\cos\theta = \dfrac{1}{5}.
trigonometrygeometric series