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Problems
Contests
International Contests
May Olympiad
1996 May Olympiad
1996 May Olympiad
Part of
May Olympiad
Subcontests
(5)
4
2
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angle wanted, square related
Let
A
B
C
D
ABCD
A
BC
D
be a square and let point
F
F
F
be any point on side
B
C
BC
BC
. Let the line perpendicular to
D
F
DF
D
F
, that passes through
B
B
B
, intersect line
D
C
DC
D
C
at
Q
Q
Q
. What is value of
∠
F
Q
C
\angle FQC
∠
FQC
?
numbers on drawing
(a) In this drawing, there are three squares on each side of the square. Place a natural number in each of the boxes so that the sum of the numbers of two adjacent boxes is always odd. https://cdn.artofproblemsolving.com/attachments/e/6/75517b7d49857abd3f8f0430a70ae5b0eb1554.gif(b) In this drawing, there are now four squares on each side of the triangle. Justify why a natural number cannot be placed in each box so that the sum of the numbers in two adjacent boxes is always odd. https://cdn.artofproblemsolving.com/attachments/c/8/061895b9c1cdcb132f7d37087873b7de3fb5f3.gif(c) If you now draw a polygon with
51
51
51
sides and on each side you place
50
50
50
boxes, taking care that there is a box at each vertex. Can you place a natural number in each box so that the sum of the numbers in two adjacent boxes is always odd? Why?
5
2
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moves on 10x10 grid
You have a
10
×
10
10 \times 10
10
×
10
grid. A "move" on the grid consists of moving
7
7
7
squares to the right and
3
3
3
squares down. In case of exiting by a line, it continues at the beginning (left) of the same line and in case of ending a column, it continues at the beginning of the same column (above). Where should we start so that after
1996
1996
1996
moves we end up in a corner?
corrent and incorrect answers
In an electronic game of questions and answers, for each correct answer the player adds
5
5
5
points on the screen, for each incorrect answer
2
2
2
points are subtracted and when the player does not answer, no score is added or subtracted. Each game has
30
30
30
questions. Francisco played
5
5
5
games and in all of them he obtained the same number of points, greater than zero, but the number of correct answers, errors and unanswered questions in each game was different. Give all the possible scores that Francisco could obtain.
3
2
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Natalia and Marcela counting numbers
Natalia and Marcela count
1
1
1
by
1
1
1
starting together at
1
1
1
, but Marcela's speed is triple that of Natalia (when Natalia says her second number, Marcela says the fourth number). When the difference of the numbers that they say in unison is any of the multiples of
29
29
29
, between
500
500
500
and
600
600
600
, Natalia continues counting normally and Marcela begins to count downwards in such a way that, at one point, the two say in unison the same number. What is said number?
2 cylindrical containers with water
A
A
A
and
B
B
B
are two cylindrical containers that contain water. The height of the water at
A
A
A
is
1000
1000
1000
cm and at
B
B
B
,
350
350
350
cm. Using a pump, water is transferred from
A
A
A
to
B
B
B
. It is noted that, in container
A
A
A
, the height of the water decreases
4
4
4
cm per minute and in
B
B
B
it increases
9
9
9
cm per minute. After how much time, since the pump was started, will the heights at
A
A
A
and
B
B
B
be the same?
2
2
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3375 cubes of side 1
Joining
1
5
3
=
3375
15^3 = 3375
1
5
3
=
3375
cubes of
1
1
1
cm
3
^3
3
, bodies with a volume of
3375
3375
3375
cm
3
^3
3
can be built. Indicate how two bodies
A
A
A
and
B
B
B
are constructed with
3375
3375
3375
cubes each and such that the lateral surface of
B
B
B
is
10
10
10
times the lateral surface of
A
A
A
.
3-digit numbers
Considering the three-digit natural numbers, how many of them, when adding two of their digits, are double of their remainder? Justify your answer.
1
2
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minimal sum of areas of triangles insides a rectangle
Let
A
B
C
D
ABCD
A
BC
D
be a rectangle. A line
r
r
r
moves parallel to
A
B
AB
A
B
and intersects diagonal
A
C
AC
A
C
, forming two triangles opposite the vertex, inside the rectangle. Prove that the sum of the areas of these triangles is minimal when
r
r
r
passes through the midpoint of segment
A
D
AD
A
D
.
dividing a right trapezoid in equal areas
A terrain (
A
B
C
D
ABCD
A
BC
D
) has a rectangular trapezoidal shape. The angle in
A
A
A
measures
9
0
o
90^o
9
0
o
.
A
B
AB
A
B
measures
30
30
30
m,
A
D
AD
A
D
measures
20
20
20
m and
D
C
DC
D
C
measures 45 m. This land must be divided into two areas of the same area, drawing a parallel to the
A
D
AD
A
D
side . At what distance from
D
D
D
do we have to draw the parallel? https://1.bp.blogspot.com/-DnyNY3x4XKE/XNYvRUrLVTI/AAAAAAAAKLE/gohd7_S9OeIi-CVUVw-iM63uXE5u-WmGwCK4BGAYYCw/s400/image002.gif