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Problems
Contests
National and Regional Contests
Iran Contests
Iran MO (2nd Round)
2014 Iran MO (2nd Round)
2014 Iran MO (2nd Round)
Part of
Iran MO (2nd Round)
Subcontests
(3)
1
2
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Stuff Baskets everyone.
A basket is called "Stuff Basket" if it includes
10
10
10
kilograms of rice and
30
30
30
number of eggs. A market is to distribute
100
100
100
Stuff Baskets. We know that there is totally
1000
1000
1000
kilograms of rice and
3000
3000
3000
number of eggs in the baskets, but some of market's baskets include either more or less amount of rice or eggs. In each step, market workers can select two baskets and move an arbitrary amount of rice or eggs between selected baskets. Starting from an arbitrary situation, what's the minimum number of steps that workers provide
100
100
100
Stuff Baskets?
Equation n^{n^{n}}=m^{m}
Find all positive integers
(
m
,
n
)
(m,n)
(
m
,
n
)
such that
n
n
n
=
m
m
.
n^{n^{n}}=m^{m}.
n
n
n
=
m
m
.
2
2
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Powerful sets: a, b in S, then a^b or b^a in S
A subset
S
S
S
of positive real numbers is called powerful if for any two distinct elements
a
,
b
a, b
a
,
b
of
S
S
S
, at least one of
a
b
a^{b}
a
b
or
b
a
b^{a}
b
a
is also an element of
S
S
S
. a) Give an example of a four elements powerful set. b) Prove that every finite powerful set has at most four elements.
Angles and points on a square
Let
A
B
C
D
ABCD
A
BC
D
be a square. Let
N
,
P
N,P
N
,
P
be two points on sides
A
B
,
A
D
AB, AD
A
B
,
A
D
, respectively such that
N
P
=
N
C
NP=NC
NP
=
NC
, and let
Q
Q
Q
be a point on
A
N
AN
A
N
such that
∠
Q
P
N
=
∠
N
C
B
\angle QPN = \angle NCB
∠
QPN
=
∠
NCB
. Prove that
∠
B
C
Q
=
1
2
∠
A
Q
P
.
\angle BCQ = \dfrac{1}{2} \angle AQP .
∠
BCQ
=
2
1
∠
A
QP
.
3
2
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Old Inequality on Iran NMO
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be three non-negative real numbers such that
x
2
+
y
2
+
z
2
=
2
(
x
y
+
y
z
+
z
x
)
.
x^2+y^2+z^2=2(xy+yz+zx).
x
2
+
y
2
+
z
2
=
2
(
x
y
+
yz
+
z
x
)
.
Prove that
x
+
y
+
z
3
≥
2
x
y
z
3
.
\dfrac{x+y+z}{3} \ge \sqrt[3]{2xyz}.
3
x
+
y
+
z
≥
3
2
x
yz
.
Professionous Riddlous, society of 1+\sqrt{2n} members
Members of "Professionous Riddlous" society have been divided into some groups, and groups are changed in a special way each weekend: In each group, one of the members is specified as the best member, and the best members of all groups separate from their previous group and form a new group. If a group has only one member, that member joins the new group and the previous group will be removed. Suppose that the society has
n
n
n
members at first, and all the members are in one group. Prove that a week will come, after which number of members of each group will be at most
1
+
2
n
1+\sqrt{2n}
1
+
2
n
.