
Exercises and problems in Informatics September 2001 
Sz. 1. We learned from an article of József Bölcsföldi and
Géza Balázs (KöMaL Homepage Dec. 2000) that ``already the Pythagoreans
knew about numbers that form a friendly pairwhich is a pair
(a,b) of natural numbers with both numbers being equal
to the sum of the proper divisors (i.e. including 1 but excluding the
number itself) of the other. Clearly, one number in a friendly pair
has many divisors (typed in bold below), while the other one has only
few of them. Some friendly pairs include (220, 284),
(1184, 1210), (2620, 2924),
(5020, 5564), (6232, 6368),
(10744, 10856), ...'' Write a program which asks for two
natural numbers (N<M<10^{6}), then prints
all the friendly pairs (a,b) with N<a,
b<M. (10 points)
Sz. 2. In most programming languages an ellipse can be
displayed by a single command, however this ellipse must have axes
parallel to the edges of the screen. Write a program that enables us
to draw the ellipse in any position. The parameters are the length of
the axes and the angle between the major axis and the upper edge of
the screen in degrees. (10 points)
Sz. 3. We deposit our savings for N years (1N\(\displaystyle le\)10) according to the following:
1. Throughout N years, at the beginning of every month we place
A Forints into the bank, and lock it up for one year at a rate
of interest X percent (X>0 a real number). 2. When
the lockup is over, at the beginning of the next month we get the
yearly interest, which is put back and locked up again for one year
together with our savings so far. Make a sheet (named PENZ.XLS) which
the numbers N, A and X can be entered into, then
our bank balance is calculated throughout N years. The sheet
should contain exactly N rows, that is unnecessary rows should
not be visible even if N is changed. Example (with N=3,
A=1000, X=10)
1st year:  1000  2000  3000  4000  5000  6000  7000  8000  9000  10000  11000  12000 
2nd year:  13100  14200  15300  16400  17500  18600  19700  20800  21900  23000  24100  25200 
3rd year:  26410  27620  28830  30040  31250  32460  33670  34880  36090  37300  38510  39720  
(10 points)
Send your solutions to the following email address:
Deadline: 13 October 2001
