Exercises and problems in Informatics 
Please read The Conditions of the Problem Solving Competition.
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 (0n10000) can uniquely be represented as
\(\displaystyle n={a_1\choose1}+{a_2\choose2}+\dots+{a_m\choose m}\), where 0a_{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 the figure.
1. ábra  2. ábra 
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 (1T100) 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, n20) 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)