English Issue, December 2002  
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Problems in information technology
(3436.)
I. 34. Binomial coefficients can be used to represent natural numbers in the socalled binomial base. For a fixed m (2\(\displaystyle \le\)m \(\displaystyle \le\)50) every natural number n (0\(\displaystyle \le\)n\(\displaystyle \le\)10000) can uniquely be represented as
\(\displaystyle n={a_1\choose1}+{a_2\choose2}+\cdots+ {a_m\choose m}\), where 0\(\displaystyle \le\)a_{1}<a_{2}<...<a_{m}.
Your program (I34.pas, ...) should read the numbers n and m, then display the corresponding sequence a_{1},a_{2},...,a_{m}.
Example. Let n=41, then a_{1}=1, a_{2}=2, a_{3}=4, a_{4}=7, because
\(\displaystyle 41={1\choose1}+{2\choose2}+{4\choose3}+{7\choose4}=1+ 1+4+35. \)
(10 points)
I. 35. We put an ant close beside the base of a cylinderjacket with radius R and height H. In every minute the ant creeps upwards M centimetres. The cylinder is rotated around its axis (which is just the Zaxis) anticlockwise completing T turns per minute. The ant starts from the point (R,0,0), and we are watching it at an angle of ALPHA degree relative to the Yaxis, see Figure 1.
Figure 1  Figure 2 
Write your program (I35.pas, ...) which reads the values of R (1\(\displaystyle \le\)R\(\displaystyle \le\)50), H (1\(\displaystyle \le\)H\(\displaystyle \le\)200), M (1\(\displaystyle \le\)M\(\displaystyle \le\)H), T (1\(\displaystyle \le\)T\(\displaystyle \le\)100) and ALPHA (0\(\displaystyle \le\)ALPHA<90), then displays the axonometric projection to the plane Y=0 of the path of the ant using continuous line on the visible side of the cylinder and dotted line on the back side.
Example. Figure 2 shows the path of the ant with R=50, H=200, M=1, T=40, ALPHA=30.
(10 points)
I. 36. According to the trinomial theorem
\(\displaystyle {(x+y+z)}^n=\sum_{\textstyle{0\le a,b,c\le n\atop a+b+c=n}} {a+b+c\choose a,b,c}x^ay^bz^c. \)
The trinomial coefficients can be computed, for example, by the formula
\(\displaystyle {a+b+c\choose a,b,c}=\frac{(a+b+c)!}{a!b!c!}. \)
However, these factorials can be very large, thus their direct computation is not always feasible. Nevertheless, writing trinomial coefficients as a product of binomial coefficients can settle this problem.
Prepare your sheet (I36.xls) which, if n (n=a+b+c, n\(\displaystyle \le\)20) is entered into a given cell, displays a table of trinomial coefficients, similar to the one below.
The example shows the coefficients when n=5.

(10 points)