Mark
I think my answers are on the same track,
there are a number of ways to look at the problem.
Porthole
There are numerous variations on the main equation to the problem. If we assume (as we may) that the radius of the porthole is 1 unit, then the area, A, swept out by a wiper of length L is given by the equation
A = L2arccos(L/2).
Solving for A to be half the area of the porthole gives two solutions:
L = 21/2, 31/2.
A little straightforward calculus gives a maximum of about 52.45% of the area of the porthole with a wiper of length approximately L = 1.588.
The only other solution(s) with two digit numbers are (20 + 25)2 = 2025, and a generously interpreted (98 + 01)2 = 9801.
To arrive at the answer one can simply check all possible pairs of two digit numbers -- a daunting task by hand, but a simple task when armed with just about any programmable calculator. Alternatively, one can proceed as follow.
We want to solve the equation
(a + b)2 = 100a + b,
where a and b are two digit numbers. Reading the equation modulo 9 and modulo 11 gives the two equations
(a + b)2 = a + b (mod 9) and (a + b)2 = a + b (mod 11),
These equations give you that a + b has remainder 0 or 1 upon division by 9, and 0 or 1 upon division by 11. This restricts a + b to be one of 45, 55, or 99; note that (a + b)2 is a four digit number so a + b is at most 100. Squaring each of these gives the solutions above.
Solutions with integers of more than two digits can be handled in either of the ways above -- brute force, or a little number theory.
A sampling of some of the more interesting oddities:
(4 + 3 + 1 + 649)2 = 431649
(842 + 72 + 4)2 = 842724
(585886298 + 179545801)2 = 585886298179545801
a = 25*10n-2 +,- 5*10(n/2)-1, b = 25*10n-2, n even
(9 + 11 + 25)3 = 91125
(17147 + 18793 + 19616)3 = 171471879319616.
Integers
Let A1,...,A1998 be the numbers selected, and let Sk = A1 + ... + Ak. If any one of the Sk is divisible by 1998, we're done. In the contrary case, all of the Sk have a non-zero remainder when divided by 1998.
By the pigeonhole principle (there are 1998 sums but only 1997 possible remainder -- 1 to 1997) two of the sums must have the same remainder upon division by 1998. The difference of these two sums is divisible by 1998.
Recently I went on a vacation with my wife and her extended family. The dining room staff at the resort said that the 14 of us could not be seated at the same table, so they put us at two tables, of 8 and 6 seats respectively. So I wondered, if everyone wants to sit at least once with everyone else, how many meals do we need to accomplish this
When I say "sit with," I don't mean "sit next to", I just mean sit at the same table.
This leads to a more general question. Given any two numbers M and N, if M+N people are seated at tables of M and N respectively, how many meals does it take
This is not a trick question. At a given meal you have to sit in one seat throughout the meal, the tables do not overlap, etc.)
One night, in a month of the spring season, a certain young lady was lovingly happy with her husband in a big mansion, white as the moon, set in a pleasure garden with trees bent down with flowers and fruits, and resonant with the sweet sounds of parrots, cuckoos and bees which were all intoxicated with the honey of the flowers.
Then, on a love- quarrel arising between husband and wife, her pearl necklace was broken. One third of the pearls were collected by the maid-servant, one sixth fell on the bed - then half of what remained and half of what remained thereafter and again one half of what remained thereafter and so on, six times in all, fell scattered everywhere. 1,161 pearls were still left on the string; how many pearls had there been in the necklace
At 3:00pm, a boat is 12.5 miles due west of a radar station and traveling at 11mph in a direction that is 57.3 degrees south of an east-west line. At what time will the boat be closest to the radar station
Solve (if anyone can, which I doubt) the ff. systems:
a) x+y+2z=-1, 2xy+3z=0, -y+z=2
b) x+y+2z=1, 3x-y+z=-1, -x+3y+4z=-1
c) x-y+z-2w=3, -x+y+z+w=2, -x+2y+2z-w=9, x-y+2z+w=2
