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Problems
Contests
International Contests
May Olympiad
2004 May Olympiad
2004 May Olympiad
Part of
May Olympiad
Subcontests
(5)
5
2
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3 square pieces on 9x9
On a
9
×
9
9\times 9
9
×
9
board, divided into
1
×
1
1\times 1
1
×
1
squares, pieces of the form Each piece covers exactly
3
3
3
squares. (a) Starting from the empty board, what is the maximum number of pieces that can be placed? (b) Starting from the board with
3
3
3
pieces already placed as shown in the diagram below, what is the maximum number of pieces that can be placed? https://cdn.artofproblemsolving.com/attachments/d/4/3bd010828accb2d1811d49eb17fa69662ff60d.gif
2 digits written on 90 cards
There are
90
90
90
cards and two different digits are written on each one:
01
01
01
,
02
02
02
,
03
03
03
,
04
04
04
,
05
05
05
,
06
06
06
,
07
07
07
,
08
08
08
,
09
09
09
,
10
10
10
,
12
12
12
, and so on up to
98
98
98
. A set of cards is correct if it does not contain any cards whose first digit is the same as the second digit of another card in the set. We call the value of a set of cards the sum of the numbers written on each card. For example, the four cards
04
04
04
,
35
35
35
,
78
78
78
and
98
98
98
form a correct set and their value is
215
215
215
, since
04
+
35
+
78
+
98
=
215
04+35+78+98=215
04
+
35
+
78
+
98
=
215
. Find a correct set that has the largest possible value. Explain why it is impossible to achieve a correct set of higher value.
2
2
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n packages of 3 candies to give away at birthday party
Pepito's mother wants to prepare
n
n
n
packages of
3
3
3
candies to give away at the birthday party, and for this she will buy assorted candies of
3
3
3
different flavors. You can buy any number of candies but you can't choose how many of each taste. She wants to put one candy of each flavor in each package, and if this is not possible she will use only candy of one flavor and all the packages will have
3
3
3
candies of that flavor. Determine the least number of candies that must be purchased in order to assemble the n packages. He explains why if he buys fewer candies, he is not sure that he will be able to assemble the packages the way he wants.
5 rectangles inside a 11x11 rectangle
Inside an
11
×
11
11\times 11
11
×
11
square, Pablo drew a rectangle and extending its sides divided the square into
5
5
5
rectangles, as shown in the figure. https://cdn.artofproblemsolving.com/attachments/5/a/7774da7085f283b3aae74fb5ff472572571827.gif Sofía did the same, but she also managed to make the lengths of the sides of the
5
5
5
rectangles be whole numbers between
1
1
1
and
10
10
10
, all different. Show a figure like the one Sofia made.
1
2
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5 positive integers whose product = sum
Julián writes five positive integers, not necessarily different, such that their product is equal to their sum. What could be the numbers that Julian writes?
product of 4 digts equals a no with last digit 7
Javier multiplies four digits, not necessarily different, and obtains a number ending in
7
7
7
. Determine how much the sum of the four digits that Javier multiplies can be worth. Give all the possibilities.
4
2
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diophantine system x y z=4104, x+y+z=77
Find all the natural numbers
x
,
y
,
z
x, y, z
x
,
y
,
z
that satisfy simultaneously
{
x
y
z
=
4104
x
+
y
+
z
=
77
\begin{cases} x y z=4104 \\ x+y+z=77 \end{cases}
{
x
yz
=
4104
x
+
y
+
z
=
77
given double area, find an angle, squares related (May Olympiad 2004 L1)
In a square
A
B
C
D
ABCD
A
BC
D
of diagonals
A
C
AC
A
C
and
B
D
BD
B
D
, we call
O
O
O
at the center of the square. A square
P
Q
R
S
PQRS
PQRS
is constructed with sides parallel to those of
A
B
C
D
ABCD
A
BC
D
with
P
P
P
in segment
A
O
,
Q
AO, Q
A
O
,
Q
in segment
B
O
,
R
BO, R
BO
,
R
in segment
C
O
,
S
CO, S
CO
,
S
in segment
D
O
DO
D
O
. If area of
A
B
C
D
ABCD
A
BC
D
equals two times the area of
P
Q
R
S
PQRS
PQRS
, and
M
M
M
is the midpoint of the
A
B
AB
A
B
side, calculate the measure of the angle
∠
A
M
P
\angle AMP
∠
A
MP
.
3
2
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8x2 pool table, a ball that traveled 29 m starting from the center, no of hits
We have a pool table
8
8
8
meters long and
2
2
2
meters wide with a single ball in the center. We throw the ball in a straight line and, after traveling
29
29
29
meters, it stops at a corner of the table. How many times did the ball hit the edges of the table? Note: When the ball rebounds on the edge of the table, the two angles that form its trajectory with the edge of the table are the same.
changing numbers in 5x5 board, 2004 steps
In each square of a
5
×
5
5\times 5
5
×
5
board is written
1
1
1
or
−
1
-1
−
1
. In each step, the number of each of the
25
25
25
cells is replaced by the result of the multiplication of the numbers of all its neighboring cells. Initially we have the board of the figure. https://cdn.artofproblemsolving.com/attachments/2/d/29b8e5df29526630102ac400a3a2b2f8fee4f7.gif Show how the board looks after
2004
2004
2004
steps.Clarification: Two squares are neighbors if they have a common side.