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Problems
Contests
International Contests
May Olympiad
2015 May Olympiad
2015 May Olympiad
Part of
May Olympiad
Subcontests
(5)
1
2
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10 euro bills, coins of 1 euro May Olympiad (Olimpiada de Mayo) 2015 L2 P1
Ana and Celia sell various objects and obtain for each object as many euros as objects they sold. The money obtained is made up of some
10
10
10
euro bills and less than
10
10
10
coins of
1
1
1
euro . They decide to distribute the money as follows: Ana takes a
10
10
10
euro bill and then Celia, and so on successively until Ana takes the last
10
10
10
euro note, and Celia takes all the
1
1
1
euro coins . How many euros more than Celia did Ana take? Give all the possibilities.[hide=original wording]Ana y Celia venden varios objetos y obtienen por cada objeto tantos euros como objetos vendieron. El dinero obtenido está constituido por algunos billetes de 10 euros y menos de 10 monedas de 1 euro. Deciden repartir el dinero del siguiente modo: Ana toma un billete de 10 euros y después Celia, y así sucesivamente hasta que Ana toma el último billete de 10 euros, y Celia se lleva todas las monedas de 1 euro. ¿Cuántos euros más que Celia se llevó Ana? Dar todas las posibilidades.
3digit number wanted, 4 hints May Olympiad (Olimpiada de Mayo) 2015 L1 P1
The teacher secretly thought of a three-digit
S
S
S
number. Students
A
,
B
,
C
A, B, C
A
,
B
,
C
and
D
D
D
tried to guess, saying, respectively,
541
541
541
,
837
837
837
,
291
291
291
and
846
846
846
. The teacher told them, “Each of you got it right exactly one digit of
S
S
S
and in the correct position ”. What is the number
S
S
S
?
4
2
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erase 2 numbers with prime sum 1-510 May Olympiad (Olimpiada de Mayo) 2015 L2 P4
The first
510
510
510
positive integers are written on a blackboard:
1
,
2
,
3
,
.
.
.
,
510
1, 2, 3, ..., 510
1
,
2
,
3
,
...
,
510
. An operation consists of of erasing two numbers whose sum is a prime number. What is the maximum number of operations in a row what can be done? Show how it is accomplished and explain why it can be done in no more operations.
no=13 x (sum of digits) May Olympiad (Olimpiada de Mayo) 2015 L1 P4
We say that a number is superstitious when it is equal to
13
13
13
times the sum of its digits . Find all superstitious numbers.
2
2
Hide problems
7x7 being colored
We have a 7x7 board. We want to color some 1x1 squares such that any 3x3 sub-board have more painted 1x1 than no painted 1x1. What is the smallest number of 1x1 that we need to color?
6 indistinguishable coins, 2 false May Olympiad (Olimpiada de Mayo) 2015 L2 P2
6
6
6
indistinguishable coins are given,
4
4
4
are authentic, all of the same weight, and
2
2
2
are false, one is more light than the real ones and the other one, heavier than the real ones. The two false ones together weigh same as two authentic coins. Find two authentic coins using a balance scale twice only by two plates, no weights.Clarification: A two-pan scale only reports if the left pan weighs more, equal or less that right.
3
2
Hide problems
perpendiculare segments inside a regular 9-gon
Let
A
B
C
D
E
F
G
H
I
ABCDEFGHI
A
BC
D
EFG
H
I
be a regular polygon of
9
9
9
sides. The segments
A
E
AE
A
E
and
D
F
DF
D
F
intersect at
P
P
P
. Prove that
P
G
PG
PG
and
A
F
AF
A
F
are perpendicular.
Quadrilateral ABCD
In the quadrilateral
A
B
C
D
ABCD
A
BC
D
, we have
∠
C
\angle C
∠
C
is triple of
∠
A
\angle A
∠
A
, let
P
P
P
be a point in the side
A
B
AB
A
B
such that
∠
D
P
A
=
90
º
\angle DPA = 90º
∠
D
P
A
=
90º
and let
Q
Q
Q
be a point in the segment
D
A
DA
D
A
where
∠
B
Q
A
=
90
º
\angle BQA = 90º
∠
BQ
A
=
90º
the segments
D
P
DP
D
P
and
C
Q
CQ
CQ
intersects in
O
O
O
such that
B
O
=
C
O
=
D
O
BO = CO = DO
BO
=
CO
=
D
O
, find
∠
A
\angle A
∠
A
and
∠
C
\angle C
∠
C
.
5
2
Hide problems
65 points in a plane(combinat)
If you have
65
65
65
points in a plane, we will make the lines that passes by any two points in this plane and we obtain exactly
2015
2015
2015
distinct lines, prove that least
4
4
4
points are collinears!!
friendships among 26 people, May Olympiad (Olimpiada de Mayo) 2015 L1 P5
Twenty-six people gather in a house. Alicia is friends with only one person, Bruno is friends with two people, Carlos is a friend of three, Daniel is four, Elías is five, and so following each person is friend of a person more than the previous person, until reaching Yvonne, the person number twenty-five, who is a friend to everyone. How many people is Zoila a friend of, person number twenty-six?Clarification: If
A
A
A
is a friend of
B
B
B
then
B
B
B
is a friend of
A
A
A
.