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Problems
Contests
International Contests
May Olympiad
2008 May Olympiad
2008 May Olympiad
Part of
May Olympiad
Subcontests
(5)
5
2
Hide problems
3 pieces of 4 squares each for 7x7 board
Matthias covered a
7
×
7
7 \times 7
7
×
7
square board, divided into
1
×
1
1 \times 1
1
×
1
squares, with pieces of the following three types without gaps or overlaps, and without going off the board. https://cdn.artofproblemsolving.com/attachments/9/9/8a2e63f723cbdf188f22344054f364f1924d47.gif Each type
1
1
1
piece covers exactly
3
3
3
squares and each type
2
2
2
or type
3
3
3
piece covers exactly
4
4
4
squares. Determine the number of pieces of type
1
1
1
that Matías could have used. (Pieces can be rotated and flipped.)
25 coins on 16x16 board
On a
16
x
16
16 x 16
16
x
16
board,
25
25
25
coins are placed, as in the figure. It is allowed to select
8
8
8
rows and
8
8
8
columns and remove from the board all the coins that are in those
16
16
16
lines. Determine if it is possible to remove all coins from the board. https://cdn.artofproblemsolving.com/attachments/1/5/e2c7379a6f47e2e8b8c9b989b85b96454a38e1.gif If the answer is yes, indicate the
8
8
8
rows and
8
8
8
columns selected, and if no, explain why.
3
2
Hide problems
n such that 1010... 101 is prime
In numbers
1010...101
1010... 101
1010...101
Ones and zeros alternate, if there are
n
n
n
ones, there are
n
−
1
n -1
n
−
1
zeros (
n
≥
2
n \ge 2
n
≥
2
).Determine the values of
n
n
n
for which the number
1010...101
1010... 101
1010...101
, which has
n
n
n
ones, is prime.
1-2008 written on blackboard, replace two numbers with their difference
On a blackboard are written all the integers from
1
1
1
to
2008
2008
2008
inclusive. Two numbers are deleted and their difference is written. For example, if you erase
5
5
5
and
241
241
241
, you write
236
236
236
. This continues, erasing two numbers and writing their difference, until only one number remains. Determine if the number left at the end can be
2008
2008
2008
. What about
2007
2007
2007
? In each case, if the answer is affirmative, indicate a sequence with that final number, and if it is negative, explain why.
2
2
Hide problems
center of a rectangle lies in the segment connecting the projections
Let
A
B
C
D
ABCD
A
BC
D
be a rectangle and
P
P
P
be a point on the side
A
D
AD
A
D
such that
∠
B
P
C
=
9
0
o
\angle BPC = 90^o
∠
BPC
=
9
0
o
. The perpendicular from
A
A
A
on
B
P
BP
BP
cuts
B
P
BP
BP
at
M
M
M
and the perpendicular from
D
D
D
on
C
P
CP
CP
cuts
C
P
CP
CP
in
N
N
N
. Show that the center of the rectangle lies in the
M
N
MN
MN
segment.
grades of exams in olympic school
In the Olympic school the exams are graded with whole numbers, the lowest possible grade is
0
0
0
, and the highest is
10
10
10
. In the arithmetic class the teacher takes two exams. This year he has
15
15
15
students. When one of his students gets less than
3
3
3
on the first exam and more than
7
7
7
on the second exam, he calls him an overachieving student. The teacher, at the end of correcting the exams, averaged the
30
30
30
grades and obtained
8
8
8
. What is the largest number of students who passed this class could have had?
1
2
Hide problems
Sum(powers of 2)
In a blackboard, it's written the following expression
1
−
2
−
2
2
−
2
3
−
2
4
−
2
5
−
2
6
−
2
7
−
2
8
−
2
9
−
2
10
1-2-2^2-2^3-2^4-2^5-2^6-2^7-2^8-2^9-2^{10}
1
−
2
−
2
2
−
2
3
−
2
4
−
2
5
−
2
6
−
2
7
−
2
8
−
2
9
−
2
10
We put parenthesis by different ways and then we calculate the result. For example:
1
−
2
−
(
2
2
−
2
3
)
−
2
4
−
(
2
5
−
2
6
−
2
7
)
−
2
8
−
(
2
9
−
2
10
)
=
403
1-2-\left(2^2-2^3\right)-2^4-\left(2^5-2^6-2^7\right)-2^8-\left( 2^9-2^{10}\right)= 403
1
−
2
−
(
2
2
−
2
3
)
−
2
4
−
(
2
5
−
2
6
−
2
7
)
−
2
8
−
(
2
9
−
2
10
)
=
403
and
1
−
(
2
−
2
2
(
−
2
3
−
2
4
)
−
(
2
5
−
2
6
−
2
7
)
)
−
(
2
8
−
2
9
)
−
2
10
=
−
933
1-\left(2-2^2 \left(-2^3-2^4 \right)-\left(2^5-2^6-2^7\right)\right)- \left(2^8- 2^9 \right)-2^{10}= -933
1
−
(
2
−
2
2
(
−
2
3
−
2
4
)
−
(
2
5
−
2
6
−
2
7
)
)
−
(
2
8
−
2
9
)
−
2
10
=
−
933
How many different results can we obtain?
6-digit multiple of 45
How many different numbers with
6
6
6
digits and multiples of
45
45
45
can be written by adding one digit to the left and one to the right of
2008
2008
2008
?
4
2
Hide problems
What's the minimum....
In the plane we have
16
16
16
lines(not parallel and not concurrents), we have
120
120
120
point(s) of intersections of this lines. Sebastian has to paint this
120
120
120
points such that in each line all the painted points are with colour differents, find the minimum(quantity) of colour(s) that Sebastian needs to paint this points. If we have have
15
15
15
lines(in this situation we have
105
105
105
points), what's the minimum(quantity) of colour(s)?
Medium 2
Let
A
B
F
ABF
A
BF
be a right-angled triangle with
∠
A
F
B
=
90
\angle AFB = 90
∠
A
FB
=
90
, a square
A
B
C
D
ABCD
A
BC
D
is externally to the triangle. If
F
A
=
6
FA = 6
F
A
=
6
,
F
B
=
8
FB = 8
FB
=
8
and
E
E
E
is the circumcenter of the square
A
B
C
D
ABCD
A
BC
D
, determine the value of
E
F
EF
EF