| PROBLEMS FROM THE 3RD STAGE OF THE 10TH LATVIAN OLYMPIAD IN INFORMATICS | |
1. "THE VALUE OF AN EXPRESSION"
The operation # is defined for any two positive integers.
If x and y are positive integers, then
(x#y) = sum_of_digits_in_x_decimal_notation ×
greatest_digit_in_y_decimal_notation +
least_digit_in_y_decimal_notation
For example, (9#30) = 9×3+0 = 27,
but (30#9) = 3×9+9 = 36.
Let us say that this problem's expression
is such an expression which is either the single variable a
(whose value is positive integer), or it can be written in the
form
(this problem's expression # this
problem's expression).
For example, the following expressions are this problem's expressions:
a
(a#a)
((a#a)#a)
(a#((a#a)#((a#a)#a)))
You are to write a program, which for the given value of a
determines what is the least possible number of operations # with
which this problem's expression with given value K
(K - positive integer) can be written.
Input data
From the keyboard two positive integer values are input - the
value of integer variable a (1 £
a £ 999999999) and the value of the
expression K (1 £ K £ 999999999) .
Output data
On the screen the least necessary number of operations # must
be written. If for the given value of a it is impossible
to obtain K as a value of this problem's expression
, then the word 'NEVAR'
must be written
Examples
| Input data | Output data | Comment |
| 718 81 | 3 | The corresponding this problem's expression is
((a#(a#a))#a),
because its value is ((718#(718#718))#718)=((718#129)#718)=(145#718)=81
|
| Input data | Output data | |
| 999 333 | NEVAR |
2. "THE FOLDED SHEET"
| Edges of a rectangular sheet of paper
are marked by letters A,R,Z,D. The length of the
edges Z and D is a , while that of
edges A and R - b centimeters. On the side Z, x centimetrs from the
side R, the point P is dotted, but on the side D, y
centimeters from the side R, the point Q is dotted
(see Figure 1) The sheet is folded so that the folding line coincides
with the segment PQ (see Figure 2). You are to calculate the area covered by the folded
sheet. Input data Integers a (0<a£1000), b (0<b£1000), x (0£x£a), y (0£y£a) are inputed from the keyboard. Output data On the screen the area covered by the folded sheet (in
square centimeters) with at least 6 significant digits
must be written. Examples
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3. "THE STRANGE SEQUENCE"
There is a sequence of positive integers {ai}. For each i
(i>1) ai is the
least possible integer with the following features:
1) ai > ai-1,
2) the sum of ai
digits equals the sum of 4* ai-1
digits.
For the given values of the first sequence member a1 and the index n, you
must find and output the value of the an
.
Input data
The values of integers a1 (0< a1<20)
and n (0<n<10000) are input from the keyboard.
Output data
You must write one integer on the screen - the value of the an . For the testing,
only data where the corresponding an
value does not increase 109 are to be used.
Example
| Input data | Output data | Comment |
| 4 5 | 79 | The first five members of the sequence are : 4,7,19,49,79 |
4. "CUTTING OUT OF FACTORIALS"
A factorial of a positive integer n (which is noted as
n!) is multiplication of all integers from 1 till n:
n!=1×2×3×...×n. What is the least number of factorials
to be cut out of multiplication of the first k factorials
1!×2!×3!×...×k! so that the multiplication of remaining
factorials will be the square of a some integer ?
Input data
From the keyboard the value of k(2 £
k £ 500) is input.
Output data
You must output in the increasing order the values of the
numbers whose factorials have to be be cut out. If there are
several solutions, you must output just one of them.
Examples
| Input data | Output data | Comment | |
| 4 | 2 | 4!×3!×1!=24×6×1=144=122 | |
| Input data | Output data | Comment | |
| 6 | 2 3 | 6!×5!×4!×1!=720×120×24×1=2073600=14402 | |
| or | |||
| 2 4 | 6!×5!×3!×1!=720×120×6×1=518400=7202 |
Figure 1
Figure 2
