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Contests
National and Regional Contests
Iran Contests
Iran MO (3rd Round)
2015 Iran MO (3rd round)
2015 Iran MO (3rd round)
Part of
Iran MO (3rd Round)
Subcontests
(6)
6
1
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iran inquality
a
1
,
a
2
,
…
,
a
n
>
0
a_1,a_2,\dots ,a_n>0
a
1
,
a
2
,
…
,
a
n
>
0
are positive real numbers such that
∑
i
=
1
n
1
a
i
=
n
\sum_{i=1}^{n} \frac{1}{a_i}=n
∑
i
=
1
n
a
i
1
=
n
prove that:
∑
i
<
j
(
a
i
−
a
j
a
i
+
a
j
)
2
≤
n
2
2
(
1
−
n
∑
i
=
1
n
a
i
)
\sum_{i<j} \left(\frac{a_i-a_j}{a_i+a_j}\right)^2\le\frac{n^2}{2}\left(1-\frac{n}{\sum_{i=1}^{n}a_i}\right)
∑
i
<
j
(
a
i
+
a
j
a
i
−
a
j
)
2
≤
2
n
2
(
1
−
∑
i
=
1
n
a
i
n
)
3
3
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Two variable polynomial
Does there exist an irreducible two variable polynomial
f
(
x
,
y
)
∈
Q
[
x
,
y
]
f(x,y)\in \mathbb{Q}[x,y]
f
(
x
,
y
)
∈
Q
[
x
,
y
]
such that it has only four roots
(
0
,
1
)
,
(
1
,
0
)
,
(
0
,
−
1
)
,
(
−
1
,
0
)
(0,1),(1,0),(0,-1),(-1,0)
(
0
,
1
)
,
(
1
,
0
)
,
(
0
,
−
1
)
,
(
−
1
,
0
)
on the unit circle.
Quadratic Residues
Let
p
>
5
p>5
p
>
5
be a prime number and
A
=
{
b
1
,
b
2
,
…
,
b
p
−
1
2
}
A=\{b_1,b_2,\dots,b_{\frac{p-1}{2}}\}
A
=
{
b
1
,
b
2
,
…
,
b
2
p
−
1
}
be the set of all quadratic residues modulo
p
p
p
, excluding zero. Prove that there doesn't exist any natural
a
,
c
a,c
a
,
c
satisfying
(
a
c
,
p
)
=
1
(ac,p)=1
(
a
c
,
p
)
=
1
such that set
B
=
{
a
b
1
+
c
,
a
b
2
+
c
,
…
,
a
b
p
−
1
2
+
c
}
B=\{ab_1+c,ab_2+c,\dots,ab_{\frac{p-1}{2}}+c\}
B
=
{
a
b
1
+
c
,
a
b
2
+
c
,
…
,
a
b
2
p
−
1
+
c
}
and set
A
A
A
are disjoint modulo
p
p
p
.This problem was proposed by Amir Hossein Pooya.
equal angles
Let
A
B
C
ABC
A
BC
be a triangle. consider an arbitrary point
P
P
P
on the plain of
△
A
B
C
\triangle ABC
△
A
BC
. Let
R
,
Q
R,Q
R
,
Q
be the reflections of
P
P
P
wrt
A
B
,
A
C
AB,AC
A
B
,
A
C
respectively. Let
R
Q
∩
B
C
=
T
RQ\cap BC=T
RQ
∩
BC
=
T
. Prove that
∠
A
P
B
=
∠
A
P
C
\angle APB=\angle APC
∠
A
PB
=
∠
A
PC
if and if only
∠
A
P
T
=
9
0
∘
\angle APT=90^{\circ}
∠
A
PT
=
9
0
∘
.
2
3
Hide problems
some subset
M
0
⊂
N
M_0 \subset \mathbb{N}
M
0
⊂
N
is a non-empty set with a finite number of elements. Ali produces sets
M
1
,
M
2
,
.
.
.
,
M
n
M_1,M_2,...,M_n
M
1
,
M
2
,
...
,
M
n
in the following order: In step
n
n
n
, Ali chooses an element of
M
n
−
1
M_{n-1}
M
n
−
1
like
b
n
b_n
b
n
and defines
M
n
M_n
M
n
as
M
n
=
{
b
n
m
+
1
∣
m
∈
M
n
−
1
}
M_n = \left \{ b_nm+1 \vert m\in M_{n-1} \right \}
M
n
=
{
b
n
m
+
1∣
m
∈
M
n
−
1
}
Prove that at some step Ali reaches a set which no element of it divides another element of it.
No function f,g
Prove that there are no functions
f
,
g
:
R
→
R
f,g:\mathbb{R}\rightarrow \mathbb{R}
f
,
g
:
R
→
R
such that
∀
x
,
y
∈
R
:
\forall x,y\in \mathbb{R}:
∀
x
,
y
∈
R
:
f
(
x
2
+
g
(
y
)
)
−
f
(
x
2
)
+
g
(
y
)
−
g
(
x
)
≤
2
y
f(x^2+g(y)) -f(x^2)+g(y)-g(x) \leq 2y
f
(
x
2
+
g
(
y
))
−
f
(
x
2
)
+
g
(
y
)
−
g
(
x
)
≤
2
y
and
f
(
x
)
≥
x
2
f(x)\geq x^2
f
(
x
)
≥
x
2
. Proposed by Mohammad Ahmadi
Two parallel lines
Let
A
B
C
ABC
A
BC
be a triangle with orthocenter
H
H
H
and circumcenter
O
O
O
. Let
K
K
K
be the midpoint of
A
H
AH
A
H
. point
P
P
P
lies on
A
C
AC
A
C
such that
∠
B
K
P
=
9
0
∘
\angle BKP=90^{\circ}
∠
B
K
P
=
9
0
∘
. Prove that
O
P
∥
B
C
OP\parallel BC
OP
∥
BC
.
5
3
Hide problems
find all Polynomials
Find all polynomials
p
(
x
)
∈
R
[
x
]
p(x)\in\mathbb{R}[x]
p
(
x
)
∈
R
[
x
]
such that for all
x
∈
R
x\in \mathbb{R}
x
∈
R
:
p
(
5
x
)
2
−
3
=
p
(
5
x
2
+
1
)
p(5x)^2-3=p(5x^2+1)
p
(
5
x
)
2
−
3
=
p
(
5
x
2
+
1
)
such that:
a
)
p
(
0
)
≠
0
a) p(0)\neq 0
a
)
p
(
0
)
=
0
b
)
p
(
0
)
=
0
b) p(0)=0
b
)
p
(
0
)
=
0
a prime number more than 30
p
>
30
p>30
p
>
30
is a prime number. Prove that one of the following numbers is in form of
x
2
+
y
2
x^2+y^2
x
2
+
y
2
.
p
+
1
,
2
p
+
1
,
3
p
+
1
,
.
.
.
.
,
(
p
−
3
)
p
+
1
p+1 , 2p+1 , 3p+1 , .... , (p-3)p+1
p
+
1
,
2
p
+
1
,
3
p
+
1
,
....
,
(
p
−
3
)
p
+
1
O coincides the incenter
Let
A
B
C
ABC
A
BC
be a triangle with orthocenter
H
H
H
and circumcenter
O
O
O
. Let
R
R
R
be the radius of circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
. Let
A
′
,
B
′
,
C
′
A',B',C'
A
′
,
B
′
,
C
′
be the points on
A
H
→
,
B
H
→
,
C
H
→
\overrightarrow{AH},\overrightarrow{BH},\overrightarrow{CH}
A
H
,
B
H
,
C
H
respectively such that
A
H
.
A
A
′
=
R
2
,
B
H
.
B
B
′
=
R
2
,
C
H
.
C
C
′
=
R
2
AH.AA'=R^2,BH.BB'=R^2,CH.CC'=R^2
A
H
.
A
A
′
=
R
2
,
B
H
.
B
B
′
=
R
2
,
C
H
.
C
C
′
=
R
2
. Prove that
O
O
O
is incenter of
△
A
′
B
′
C
′
\triangle A'B'C'
△
A
′
B
′
C
′
.
4
3
Hide problems
Constant polynomial
p
(
x
)
∈
C
[
x
]
p(x)\in \mathbb{C}[x]
p
(
x
)
∈
C
[
x
]
is a polynomial such that:
∀
z
∈
C
,
∣
z
∣
=
1
⟹
p
(
z
)
∈
R
\forall z\in \mathbb{C}, |z|=1\Longrightarrow p(z)\in \mathbb{R}
∀
z
∈
C
,
∣
z
∣
=
1
⟹
p
(
z
)
∈
R
Prove that
p
(
x
)
p(x)
p
(
x
)
is constant.
the same prime factors
a
,
b
,
c
,
d
,
k
,
l
a,b,c,d,k,l
a
,
b
,
c
,
d
,
k
,
l
are positive integers such that for every natural number
n
n
n
the set of prime factors of
n
k
+
a
n
+
c
,
n
l
+
b
n
+
d
n^k+a^n+c,n^l+b^n+d
n
k
+
a
n
+
c
,
n
l
+
b
n
+
d
are same. prove that
k
=
l
,
a
=
b
,
c
=
d
k=l,a=b,c=d
k
=
l
,
a
=
b
,
c
=
d
.
Three collinear points
Let
A
B
C
ABC
A
BC
be a triangle with incenter
I
I
I
. Let
K
K
K
be the midpoint of
A
I
AI
A
I
and
B
I
∩
⊙
(
△
A
B
C
)
=
M
,
C
I
∩
⊙
(
△
A
B
C
)
=
N
BI\cap \odot(\triangle ABC)=M,CI\cap \odot(\triangle ABC)=N
B
I
∩
⊙
(
△
A
BC
)
=
M
,
C
I
∩
⊙
(
△
A
BC
)
=
N
. points
P
,
Q
P,Q
P
,
Q
lie on
A
M
,
A
N
AM,AN
A
M
,
A
N
respectively such that
∠
A
B
K
=
∠
P
B
C
,
∠
A
C
K
=
∠
Q
C
B
\angle ABK=\angle PBC,\angle ACK=\angle QCB
∠
A
B
K
=
∠
PBC
,
∠
A
C
K
=
∠
QCB
. Prove that
P
,
Q
,
I
P,Q,I
P
,
Q
,
I
are collinear.
1
3
Hide problems
Inequality Iran 2015
x
,
y
,
z
x,y,z
x
,
y
,
z
are three real numbers inequal to zero satisfying
x
+
y
+
z
=
x
y
z
x+y+z=xyz
x
+
y
+
z
=
x
yz
. Prove that
∑
(
x
2
−
1
x
)
2
≥
4
\sum (\frac{x^2-1}{x})^2 \geq 4
∑
(
x
x
2
−
1
)
2
≥
4
Proposed by Amin Fathpour
primes less than 1394
Prove that there are infinitely natural numbers
n
n
n
such that
n
n
n
can't be written as a sum of two positive integers with prime factors less than
1394
1394
1394
.
intersection of two circles
Let
A
B
C
D
ABCD
A
BC
D
be the trapezoid such that
A
B
∥
C
D
AB\parallel CD
A
B
∥
C
D
. Let
E
E
E
be an arbitrary point on
A
C
AC
A
C
. point
F
F
F
lies on
B
D
BD
B
D
such that
B
E
∥
C
F
BE\parallel CF
BE
∥
CF
. Prove that circumcircles of
△
A
B
F
,
△
B
E
D
\triangle ABF,\triangle BED
△
A
BF
,
△
BE
D
and the line
A
C
AC
A
C
are concurrent.