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

I. 61. The distribution of primes among the integers is not uniform in the sense that there are smaller or larger gaps between consecutive prime numbers.

For any given H (1$\displaystyle \le$H $\displaystyle \le$10 000) and A, B (2$\displaystyle \le$A<B$\displaystyle \le$10 000 000), your program (i61.pas, ...) should determine the intervals of length H in the interval [A,B] containing the maximum and the minimum number of primes. (10 points)

I. 62. Shearing in physics is the deformation of a rigid body under a load. Most conveniently one can demonstrate this by means of a deck of cards resting on a table and overlapping them on each other: the cards are slid on each other to an extent proportional to their distance from the table (i.e. Cxdistance).

Prepare your animation program (i62.pas, ...) that applies a shear with proportion of C to a rectangle (with lower left corner at the origin, width A (1$\displaystyle \le$A $\displaystyle \le$100) and height B (1$\displaystyle \le$B $\displaystyle \le$100)) meanwhile that its lower left corner translates into point (P,Q) (-200$\displaystyle \le$P,Q$\displaystyle \le$200). (10 points)

I. 63. A bath tube of given volume is to be filled up with the hottest possible water in a given time. There are N watertaps (1$\displaystyle \le$N $\displaystyle \le$10) and every minute the ith tap yields Di (0$\displaystyle \le$Di $\displaystyle \le$100) cubic decimetres of water at Fi (0< Fi < 100) degree Celsius. The order of taps in the sheet is arbitrary.

Prepare your sheet (i63.xls) that computes which taps should be turned on and for how long each in order to get the hottest possible full bath of water in a given time. (10 points)

 Taps: Volume: Time: 6 3000 60 Degree: 34 30 22 18 15 12 cubic decimetres: 20 20 15 15 30 5 Times: 60 60 40 0 0 0