Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Exercises and problems in Informatics
September 2003

Please read The Conditions of the Problem Solving Competition.

I. 55. An integer n>1 is said to be highly composite, if the number of its divisors is greater than those of any positive integers smaller than n.

Write your program (i55.pas, ...) that displays all highly composite numbers up to a given positive integer N (1\(\displaystyle \le\)N \(\displaystyle \le\)1 000 000).
(10 points)

I. 56. Write a program (i56.pas, ...) which draws a given phase of the Moon. Full Moon is represented as a filled circle, while New Moon should be displayed as an empty circle centered at the middle of the screen and with radius of 100 pixels. The parameter of your program should be the diameter of the crescent defined according to the Figures.
(10 points)

20014010010

I. 57. A polynomial P(x) of degree n is given by its coefficients P(x)=a0+a1 x+a2 x2+...+an xn. The derivative of P(x) is the following polynomial of degree n-1

P'(x)=a1+2a2 x+3a3 x2+...+n an xn-1.

Further derivatives of this polynomial can be computed until all of its coefficients become zero.

Prepare your sheet (i57.xls) that produces the derivatives of a polynomial of degree at most 10 as shown in the Figure.
(10 points)

P(x)=1+1x1+1x2+1x3+1x4+1x5+1x6+1x7+1x8
11+2x1+3x2+4x3+5x4+6x5+7x6+8x7+0x8
22+6x1+12x2+20x3+30x4+42x5+56x6+0x7+0x8
36+24x1+60x2+120x3+210x4+336x5+0x6+0x7+0x8
424+120x1+360x2+840x3+1680x4+0x5+0x6+0x7+0x8
5120+720x1+2520x2+6720x3+0x4+0x5+0x6+0x7+0x8
6720+5040x1+20160x2+0x3+0x4+0x5+0x6+0x7+0x8
75040+40320x1+0x2+0x3+0x4+0x5+0x6+0x7+0x8
840320+0x1+0x2+0x3+0x4+0x5+0x6+0x7+0x8

Send your solutions to the following e-mail address:

Deadline: 13 October 2003