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Problems
Contests
International Contests
May Olympiad
2009 May Olympiad
2009 May Olympiad
Part of
May Olympiad
Subcontests
(5)
1
2
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replacing number a by (1/2a) or (1-a)
Initially, the number
1
1
1
is written on the blackboard. At each step, the number on the blackboard is erased and another is written, which is obtained by applying any of the following operations:Operation A: Multiply the number on the board with
1
2
\frac12
2
1
. Operation B: Subtract the number on the board from
1
1
1
.For example, if the number
3
8
\frac38
8
3
is on the board, it can be replaced by
1
2
3
8
=
3
16
\frac12 \frac38=\frac{3}{16}
2
1
8
3
=
16
3
or by
1
−
3
8
=
5
8
1-\frac38=\frac58
1
−
8
3
=
8
5
.Give a sequence of steps after which the number on the board is
2009
2
20009
\frac{2009}{2^{20009}}
2
20009
2009
.
2-digit numbers that are assigned 8
Each two-digit natural number is assigned a digit as follows: Its digits are multiplied. If the result is a digit, this is the assigned digit. If the result is a two-digit number, these two figures are multiplied, and if the result is a digit, this is the assigned digit. Otherwise, the operation is repeated. For example, the digit assigned to
32
32
32
is
6
6
6
since
3
×
=
6
3 \times = 6
3
×
=
6
; the digit assigned to
93
93
93
is
4
4
4
since
9
×
3
=
27
9 \times 3 = 27
9
×
3
=
27
,
2
×
7
=
14
2 \times 7 = 14
2
×
7
=
14
,
1
×
4
=
4
1 \times 4 = 4
1
×
4
=
4
. Find all the two-digit numbers that are assigned
8
8
8
.
3
2
Hide problems
sum game 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13
In the following sum:
1
+
2
+
3
+
4
+
5
+
6
1 + 2 + 3 + 4 + 5 + 6
1
+
2
+
3
+
4
+
5
+
6
, if we remove the first two “+” signs, we obtain the new sum
123
+
4
+
5
+
6
=
138
123 + 4 + 5 + 6 = 138
123
+
4
+
5
+
6
=
138
. By removing three “
+
+
+
” signs, we can obtain
1
+
23
+
456
=
480
1 + 23 + 456 = 480
1
+
23
+
456
=
480
. Let us now consider the sum
1
+
2
+
3
+
4
+
5
+
6
+
7
+
8
+
9
+
10
+
11
+
12
+
13
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13
1
+
2
+
3
+
4
+
5
+
6
+
7
+
8
+
9
+
10
+
11
+
12
+
13
, in which some “
+
+
+
” signs are to be removed. What are the three smallest multiples of
100
100
100
that we can get in this way?
26 cards with 1 number each , two of 1-13
There are
26
26
26
cards and each one has a number written on it. There are two with
1
1
1
, two with
2
2
2
, two with
3
3
3
, and so on up to two with
12
12
12
and two with
13
13
13
. You have to distribute the
26
26
26
cards in piles so that the following two conditions are met:
∙
\bullet
∙
If two cards have the same number they are in the same pile.
∙
\bullet
∙
No pile contains a card whose number is equal to the sum of the numbers of two cards in that same pile. Determine what is the minimum number of stacks to make. Give an example with the distribution of the cards for that number of stacks and justify why it is impossible to have fewer stacks.Clarification: Two squares are neighbors if they have a common side.
4
2
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red or blue on cells of 5x5
Each square of a
5
×
5
5 \times 5
5
×
5
board is painted red or blue, in such a way that the following condition is fulfilled: “For any two rows and two columns, of the
4
4
4
squares that are in their intersections, there are
4
4
4
,
2
2
2
or
0
0
0
painted red.” How many ways can the board be painted?
areas inside 3 circles (May Olympiad 2009 L1 - Olimpiada de Mayo)
Three circumferences are tangent to each other, as shown in the figure. The region of the outer circle that is not covered by the two inner circles has an area equal to
2
π
2 \pi
2
π
. Determine the length of the
P
Q
PQ
PQ
segment . https://cdn.artofproblemsolving.com/attachments/a/e/65c08c47d4d20a05222a9b6cf65e84a25283b7.png
2
2
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angle chasing with convex ABCD, equailateral ABD, isosceles BCD, <C=90^o
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral such that the triangle
A
B
D
ABD
A
B
D
is equilateral and the triangle
B
C
D
BCD
BC
D
is isosceles, with
∠
C
=
9
0
o
\angle C = 90^o
∠
C
=
9
0
o
. If
E
E
E
is the midpoint of the side
A
D
AD
A
D
, determine the measure of the angle
∠
C
E
D
\angle CED
∠
CE
D
.
p+q^2+r^3=200 diophantine in primes
Find prime numbers
p
,
q
,
r
p , q , r
p
,
q
,
r
such that
p
+
q
2
+
r
3
=
200
p+q^2+r^3=200
p
+
q
2
+
r
3
=
200
. Give all the possibilities.Remember that the number
1
1
1
is not prime.
5
2
Hide problems
Any ideas???
A game of solitaire strats of with
25
25
25
cards. Some are facing up and sum are facing down. In each move a card that's facing up should me choosen, taken away, and turning over the cards next to it (if there are cards next to it). The game is won when you have accomplished to take all the
25
25
25
cards from the table. If you initially start with
n
n
n
cards facing up, find all the values of
n
n
n
such that the game can be won. Explain how to win the game, independently from the initial placement of the cards facing up, justify your answer for why it is impossible to win with other values of
n
n
n
. Two cards are neighboring when one is immediately next to the other, to the left or right.Example: The card marked
A
A
A
has two neighboring cards and the one marked with only a
B
B
B
has only one neighboring card. After taking a card there is a hole left, such that the card marked
C
C
C
has only one neighboring card, and the one marked
D
D
D
does'nt have any.
add walks on grid 55 lines x 45 lines
An ant walks along the lines of a grid made up of
55
55
55
horizontal lines and
45
45
45
vertical lines. You want to paint some sections of lines so that the ant can go from any intersection to any other intersection, walking exclusively along painted sections. If the distance between consecutive lines is
10
10
10
cm, what is the least possible number of centimeters that should be painted? What is the higher value?