Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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English Issue, December 2002

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Problems in information technology

I. 34. Binomial coefficients can be used to represent natural numbers in the so-called 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\(\displaystyle \le\)a1<a2<...<am.

Your program (I34.pas, ...) should read the numbers n and m, then display the corresponding sequence a1,a2,...,am.

Example. Let n=41, then a1=1, a2=2, a3=4, a4=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 cylinder-jacket 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 Z-axis) 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 Y-axis, see Figure 1.

Figure 1Figure 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)