CYPRUS MATHEMATICAL SOCIETY
REGIONAL COMPETITION
NOVEMBER 2014
LYCEUM C’
Date: 08/11/2014
Time: 10:00 -12:00
INSTRUCTIONS
1. Solve all the problems by giving full answers.
2. Each problem is marked with 10 points.
3. Write with blue or black ink (Shapes can be drawn with pencil).
4. The use of corrective liquid (Tip-Ex) is not allowed.
5. The use of a calculator is not allowed.
PROBLEMS
PROBLEM 1
Let the function ( )
with
√
Value Theorem are satisfied in the interval
Α)
Β)
√
√
√
√
√
√
and
with
(
and the conditions of the Mean). Prove that
√
PROBLEM 2
( )
Let ( )
,
( )
( )
, and
. Find the polynomial ( ).
PROBLEM 3
Two small circles cross each other and they both touch another
bigger circle as shown in thw figure. The points
lie on the
same line and
. Calculate the area of the
shaded region.
PROBLEM 4
A calculator calculates (approximately) the powers of the number
different values of ) using the sum
(a) Prove that for every
of
(the values of
for
.
the approximation of the calculator is less than the real value
.
(b) For which values of the approximation of the calculator is good (it has less error)? For
the values close to 0 or for bigger values? Justify your answer.