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National and Regional Contests
Albania Contests
Albania Team Selection Test
2011 Albania Team Selection Test
2011 Albania Team Selection Test
Part of
Albania Team Selection Test
Subcontests
(5)
5
1
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The sweeties shop "Olympiad"
The sweeties shop called "Olympiad" sells boxes of
6
,
9
6,9
6
,
9
or
20
20
20
chocolates. Groups of students from a school that is near the shop collect money to buy a chocolate for each student; to make this they buy a box and than give to everybody a chocolate. Like this students can create groups of
15
=
6
+
9
15=6+9
15
=
6
+
9
students,
38
=
2
∗
9
+
20
38=2*9+20
38
=
2
∗
9
+
20
students, etc. The seller has promised to the students that he can satisfy any group of students, and if he will need to open a new box of chocolate for any group (like groups of
4
,
7
4,7
4
,
7
or
10
10
10
students) than he will give all the chocolates for free to this group. Can there be constructed the biggest group that profits free chocolates, and if so, how many students are there in this group?
1
1
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find all the possible x-coordinates of the vertex of parabol
The given parabola
y
=
a
x
2
+
b
x
+
c
y=ax^2+bx+c
y
=
a
x
2
+
b
x
+
c
doesn't intersect the
X
X
X
-axis and passes from the points
A
(
−
2
,
1
)
A(-2,1)
A
(
−
2
,
1
)
and
B
(
2
,
9
)
B(2,9)
B
(
2
,
9
)
. Find all the possible values of the
x
x
x
coordinates of the vertex of this parabola.
4
1
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find all primes p such that 2^p+p^2 is also prime
Find all prime numbers p such that
2
p
+
p
2
2^p+p^2
2
p
+
p
2
is also a prime number.
3
1
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find angle and ratios of angles in acute triangle
In the acute angle triangle
A
B
C
ABC
A
BC
the point
O
O
O
is the center of the circumscribed circle and the lines
O
A
,
O
B
,
O
C
OA,OB,OC
O
A
,
OB
,
OC
intersect sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
respectively in points
M
,
N
,
P
M,N,P
M
,
N
,
P
such that
∠
N
M
P
=
9
0
o
\angle NMP=90^o
∠
NMP
=
9
0
o
.(a) Find the ratios
∠
A
M
N
∠
N
M
C
\frac{\angle AMN}{\angle NMC}
∠
NMC
∠
A
MN
,
∠
A
M
P
∠
P
M
B
\frac{\angle AMP}{\angle PMB}
∠
PMB
∠
A
MP
.(b) If any of the angles of the triangle
A
B
C
ABC
A
BC
is
6
0
o
60^o
6
0
o
, find the two other angles.
2
1
Hide problems
triangles with perimeter equal to area
The area and the perimeter of the triangle with sides
10
,
8
,
6
10,8,6
10
,
8
,
6
are equal. Find all the triangles with integral sides whose area and perimeter are equal.