Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Exercises and problems in Informatics
September 2001

Please read The Conditions of the Problem Solving Competition.

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 pair--which 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<106), 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 (1leN\(\displaystyle le\)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:100020003000400050006000700080009000100001100012000
    2nd year:131001420015300164001750018600197002080021900230002410025200
    3rd year:264102762028830300403125032460336703488036090373003851039720

(10 points)

Send your solutions to the following e-mail address:

Deadline: 13 October 2001