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Problems
Contests
National and Regional Contests
Iran Contests
Iran MO (3rd Round)
2004 Iran MO (3rd Round)
2004 Iran MO (3rd Round)
Part of
Iran MO (3rd Round)
Subcontests
(29)
30
1
Hide problems
Polynomials that (m,n)=1 implies (p(m),p(n))=1
Find all polynomials
p
∈
Z
[
x
]
p\in\mathbb Z[x]
p
∈
Z
[
x
]
such that (m,n)\equal{}1\Rightarrow (p(m),p(n))\equal{}1
28
1
Hide problems
Find the solutions of p=m^2+n^2, p|m^3+n^3-4
Find all prime numbers
p
p
p
such that
p
=
m
2
+
n
2
p = m^2 + n^2
p
=
m
2
+
n
2
and
p
∣
m
3
+
n
3
−
4
p\mid m^3+n^3-4
p
∣
m
3
+
n
3
−
4
.
27
1
Hide problems
A line intersect all segments
Δ
1
,
…
,
Δ
n
\Delta_1,\ldots,\Delta_n
Δ
1
,
…
,
Δ
n
are
n
n
n
concurrent segments (their lines concur) in the real plane. Prove that if for every three of them there is a line intersecting these three segments, then there is a line that intersects all of the segments.
26
1
Hide problems
All points lie on a hemisphere
Finitely many points are given on the surface of a sphere, such that every four of them lie on the surface of open hemisphere. Prove that all points lie on the surface of an open hemisphere.
25
1
Hide problems
Convex subsets of R^3 every three of which intersect
Finitely many convex subsets of
R
3
\mathbb R^3
R
3
are given, such that every three have non-empty intersection. Prove that there exists a line in
R
3
\mathbb R^3
R
3
that intersects all of these subsets.
24
1
Hide problems
Find angles of triangle ABC
In triangle
A
B
C
ABC
A
BC
, points
M
,
N
M,N
M
,
N
lie on line
A
C
AC
A
C
such that MA\equal{}AB and NB\equal{}NC. Also
K
,
L
K,L
K
,
L
lie on line
B
C
BC
BC
such that KA\equal{}KB and LA\equal{}LC. It is know that KL\equal{}\frac12{BC} and MN\equal{}AC. Find angles of triangle
A
B
C
ABC
A
BC
.
23
1
Hide problems
Partition of family of sets
F
\mathcal F
F
is a family of 3-subsets of set
X
X
X
. Every two distinct elements of
X
X
X
are exactly in
k
k
k
elements of
F
\mathcal F
F
. It is known that there is a partition of
F
\mathcal F
F
to sets
X
1
,
X
2
X_1,X_2
X
1
,
X
2
such that each element of
F
\mathcal F
F
has non-empty intersection with both
X
1
,
X
2
X_1,X_2
X
1
,
X
2
. Prove that
∣
X
∣
≤
4
|X|\leq4
∣
X
∣
≤
4
.
22
1
Hide problems
A representation for two subsets of set
Suppose that
F
\mathcal F
F
is a family of subsets of
X
X
X
.
A
,
B
A,B
A
,
B
are two subsets of
X
X
X
s.t. each element of
F
\mathcal{F}
F
has non-empty intersection with
A
,
B
A, B
A
,
B
. We know that no subset of
X
X
X
with n \minus{} 1 elements has this property. Prove that there is a representation
A
,
B
A,B
A
,
B
in the form A \equal{} \{a_1,\dots,a_n\} and B \equal{} \{b_1,\dots,b_n\}, such that for each
1
≤
i
≤
n
1\leq i\leq n
1
≤
i
≤
n
, there is an element of
F
\mathcal F
F
containing both
a
i
,
b
i
a_i, b_i
a
i
,
b
i
.
21
1
Hide problems
p|a_1^k + ... + a_n^k for infinitely many p
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \ldots, a_n
a
1
,
a
2
,
…
,
a
n
are integers, not all equal. Prove that there exist infinitely many prime numbers
p
p
p
such that for some
k
k
k
p\mid a_1^k \plus{} \dots \plus{} a_n^k.
20
1
Hide problems
0 or 1 is root of p(x)
p
(
x
)
p(x)
p
(
x
)
is a polynomial in
Z
[
x
]
\mathbb{Z}[x]
Z
[
x
]
such that for each
m
,
n
∈
N
m,n\in \mathbb{N}
m
,
n
∈
N
there is an integer
a
a
a
such that
n
∣
p
(
a
m
)
n\mid p(a^m)
n
∣
p
(
a
m
)
. Prove that
0
0
0
or
1
1
1
is a root of
p
(
x
)
p(x)
p
(
x
)
.
19
1
Hide problems
Solutions of p^3=p^2+q^2+r^2 in primes
Find all integer solutions of p^3\equal{}p^2\plus{}q^2\plus{}r^2 where
p
,
q
,
r
p,q,r
p
,
q
,
r
are primes.
18
1
Hide problems
Prime divisors in a set
Prove that for any
n
n
n
, there is a subset
{
a
1
,
…
,
a
n
}
\{a_1,\dots,a_n\}
{
a
1
,
…
,
a
n
}
of
N
\mathbb N
N
such that for each subset
S
S
S
of
{
1
,
…
,
n
}
\{1,\dots,n\}
{
1
,
…
,
n
}
,
∑
i
∈
S
a
i
\sum_{i\in S}a_i
∑
i
∈
S
a
i
has the same set of prime divisors.
17
1
Hide problems
Primitive roots
Let p\equal{}4k\plus{}1 be a prime. Prove that
p
p
p
has at least \frac{\phi(p\minus{}1)}2 primitive roots.
16
1
Hide problems
good
Let
A
B
C
ABC
A
BC
be a triangle . Let point
X
X
X
be in the triangle and
A
X
AX
A
X
intersects
B
C
BC
BC
in
Y
Y
Y
. Draw the perpendiculars
Y
P
,
Y
Q
,
Y
R
,
Y
S
YP,YQ,YR,YS
Y
P
,
Y
Q
,
Y
R
,
Y
S
to lines
C
A
,
C
X
,
B
X
,
B
A
CA,CX,BX,BA
C
A
,
CX
,
BX
,
B
A
respectively. Find the necessary and sufficient condition for
X
X
X
such that
P
Q
R
S
PQRS
PQRS
be cyclic .
15
1
Hide problems
nice [physical geometry]
This problem is easy but nobody solved it. point
A
A
A
moves in a line with speed
v
v
v
and
B
B
B
moves also with speed
v
′
v'
v
′
that at every time the direction of move of
B
B
B
goes from
A
A
A
.We know
v
≥
v
′
v \geq v'
v
≥
v
′
.If we know the point of beginning of path of
A
A
A
, then
B
B
B
must be where at first that
B
B
B
can catch
A
A
A
.
12
1
Hide problems
Hypernumber
N
10
\mathbb{N}_{10}
N
10
is generalization of
N
\mathbb{N}
N
that every hypernumber in
N
10
\mathbb{N}_{10}
N
10
is something like:
.
.
.
a
2
a
1
a
0
‾
\overline{...a_2a_1a_0}
...
a
2
a
1
a
0
with
a
i
∈
0
,
1..9
a_i \in {0,1..9}
a
i
∈
0
,
1..9
(Notice that
.
.
.
000
‾
∈
N
10
\overline {...000} \in \mathbb{N}_{10}
...000
∈
N
10
) Also we easily have
+
,
∗
+,*
+
,
∗
in
N
10
\mathbb{N}_{10}
N
10
. first
k
k
k
number of
a
∗
b
a*b
a
∗
b
= first
k
k
k
nubmer of (first
k
k
k
number of a * first
k
k
k
number of b) first
k
k
k
number of
a
+
b
a+b
a
+
b
= first
k
k
k
nubmer of (first
k
k
k
number of a + first
k
k
k
number of b) Fore example
.
.
.
999
‾
+
.
.
.
0001
‾
=
.
.
.
000
‾
\overline {...999}+ \overline {...0001}= \overline {...000}
...999
+
...0001
=
...000
Prove that every monic polynomial in
N
10
[
x
]
\mathbb{N}_{10}[x]
N
10
[
x
]
with degree
d
d
d
has at most
d
2
d^2
d
2
roots.
7
1
Hide problems
Easy
Suppose
F
F
F
is a polygon with lattice vertices and sides parralell to x-axis and y-axis.Suppose
S
(
F
)
,
P
(
F
)
S(F),P(F)
S
(
F
)
,
P
(
F
)
are area and perimeter of
F
F
F
. Find the smallest k that:
S
(
F
)
≤
k
.
P
(
F
)
2
S(F) \leq k.P(F)^2
S
(
F
)
≤
k
.
P
(
F
)
2
1
1
Hide problems
Golden set
We say
m
∘
n
m \circ n
m
∘
n
for natural m,n
⟺
\Longleftrightarrow
⟺
nth number of binary representation of m is 1 or mth number of binary representation of n is 1. and we say
m
∙
n
m \bullet n
m
∙
n
if and only if
m
,
n
m,n
m
,
n
doesn't have the relation
∘
\circ
∘
We say
A
⊂
N
A \subset \mathbb{N}
A
⊂
N
is golden
⟺
\Longleftrightarrow
⟺
∀
U
,
V
⊂
A
\forall U,V \subset A
∀
U
,
V
⊂
A
that are finite and arenot empty and
U
∩
V
=
∅
U \cap V = \emptyset
U
∩
V
=
∅
,There exist
z
∈
A
z \in A
z
∈
A
that
∀
x
∈
U
,
y
∈
V
\forall x \in U,y \in V
∀
x
∈
U
,
y
∈
V
we have
z
∘
x
,
z
∙
y
z \circ x ,z \bullet y
z
∘
x
,
z
∙
y
Suppose
P
\mathbb{P}
P
is set of prime numbers.Prove if
P
=
P
1
∪
.
.
.
∪
P
k
\mathbb{P}=P_1 \cup ... \cup P_k
P
=
P
1
∪
...
∪
P
k
and
P
i
∩
P
j
=
∅
P_i \cap P_j = \emptyset
P
i
∩
P
j
=
∅
then one of
P
1
,
.
.
.
,
P
k
P_1,...,P_k
P
1
,
...
,
P
k
is golden.
3
1
Hide problems
Beautiful
Suppose
V
=
Z
2
n
V= \mathbb{Z}_2^n
V
=
Z
2
n
and for a vector
x
=
(
x
1
,
.
.
x
n
)
x=(x_1,..x_n)
x
=
(
x
1
,
..
x
n
)
in
V
V
V
and permutation
σ
\sigma
σ
.We have
x
σ
=
(
x
σ
(
1
)
,
.
.
.
,
x
σ
(
n
)
)
x_{\sigma}=(x_{\sigma(1)},...,x_{\sigma(n)})
x
σ
=
(
x
σ
(
1
)
,
...
,
x
σ
(
n
)
)
Suppose
n
=
4
k
+
2
,
4
k
+
3
n=4k+2,4k+3
n
=
4
k
+
2
,
4
k
+
3
and
f
:
V
→
V
f:V \to V
f
:
V
→
V
is injective and if
x
x
x
and
y
y
y
differ in more than
n
/
2
n/2
n
/2
places then
f
(
x
)
f(x)
f
(
x
)
and
f
(
y
)
f(y)
f
(
y
)
differ in more than
n
/
2
n/2
n
/2
places. Prove there exist permutaion
σ
\sigma
σ
and vector
v
v
v
that
f
(
x
)
=
x
σ
+
v
f(x)=x_{\sigma}+v
f
(
x
)
=
x
σ
+
v
8
1
Hide problems
Easy
P
\mathbb{P}
P
is a n-gon with sides
l
1
,
.
.
.
,
l
n
l_1 ,...,l_n
l
1
,
...
,
l
n
and vertices on a circle. Prove that no n-gon with this sides has area more than
P
\mathbb{P}
P
4
1
Hide problems
Easy
We have finite white and finite black points that for each 4 oints there is a line that white points and black points are at different sides of this line.Prove there is a line that all white points and black points are at different side of this line.
13
1
Hide problems
Easy
Suppose
f
f
f
is a polynomial in
Z
[
X
]
\mathbb{Z}[X]
Z
[
X
]
and m is integer .Consider the sequence
a
i
a_i
a
i
like this
a
1
=
m
a_1=m
a
1
=
m
and
a
i
+
1
=
f
(
a
i
)
a_{i+1}=f(a_i)
a
i
+
1
=
f
(
a
i
)
find all polynomials
f
f
f
and alll integers
m
m
m
that for each
i
i
i
:
a
i
∣
a
i
+
1
a_i | a_{i+1}
a
i
∣
a
i
+
1
10
1
Hide problems
Easy [contraction on the euclidean plane]
f
:
R
2
→
R
2
f:\mathbb{R}^2 \to \mathbb{R}^2
f
:
R
2
→
R
2
is injective and surjective. Distance of
X
X
X
and
Y
Y
Y
is not less than distance of
f
(
X
)
f(X)
f
(
X
)
and
f
(
Y
)
f(Y)
f
(
Y
)
. Prove for
A
A
A
in plane:
S
(
A
)
≥
S
(
f
(
A
)
)
S(A) \geq S(f(A))
S
(
A
)
≥
S
(
f
(
A
))
where
S
(
A
)
S(A)
S
(
A
)
is area of
A
A
A
2
1
Hide problems
Beautiful
A
A
A
is a compact convex set in plane. Prove that there exists a point
O
∈
A
O \in A
O
∈
A
, such that for every line
X
X
′
XX'
X
X
′
passing through
O
O
O
, where
X
X
X
and
X
′
X'
X
′
are boundary points of
A
A
A
, then
1
2
≤
O
X
O
X
′
≤
2.
\frac12 \leq \frac {OX}{OX'} \leq 2.
2
1
≤
O
X
′
OX
≤
2.
14
1
Hide problems
easy one
We define
f
:
N
→
N
f: \mathbb{N} \rightarrow \mathbb{N}
f
:
N
→
N
, f(n) \equal{} \sum_{k \equal{} 1}^{n}(k,n). a) Show that if \gcd(m,n)\equal{}1 then we have f(mn)\equal{}f(m)\cdot f(n); b) Show that \sum_{d|n}f(d) \equal{} nd(n).
5
1
Hide problems
easy permutation
assume that k,n are two positive integer
k
≤
n
k\leq n
k
≤
n
count the number of permutation
{
1
,
…
,
n
}
\{\ 1,\dots ,n\}\
{
1
,
…
,
n
}
st for any
1
≤
i
,
j
≤
k
1\leq i,j\leq k
1
≤
i
,
j
≤
k
and any positive integer m we have
f
m
(
i
)
≠
j
f^m(i)\neq j
f
m
(
i
)
=
j
(
f
m
f^m
f
m
meas iterarte function,)
9
1
Hide problems
almost easy [prove < ROS = < BAC]
Let
A
B
C
ABC
A
BC
be a triangle, and
O
O
O
the center of its circumcircle. Let a line through the point
O
O
O
intersect the lines
A
B
AB
A
B
and
A
C
AC
A
C
at the points
M
M
M
and
N
N
N
, respectively. Denote by
S
S
S
and
R
R
R
the midpoints of the segments
B
N
BN
BN
and
C
M
CM
CM
, respectively. Prove that
∡
R
O
S
=
∡
B
A
C
\measuredangle ROS=\measuredangle BAC
∡
ROS
=
∡
B
A
C
.
11
1
Hide problems
nice one
assume that ABC is acute traingle and AA' is median we extend it until it meets circumcircle at A". let
A
P
a
AP_a
A
P
a
be a diameter of the circumcircle. the pependicular from A' to
A
P
a
AP_a
A
P
a
meets the tangent to circumcircle at A" in the point
X
a
X_a
X
a
; we define
X
b
,
X
c
X_b,X_c
X
b
,
X
c
similary . prove that
X
a
,
X
b
,
X
c
X_a,X_b,X_c
X
a
,
X
b
,
X
c
are one a line.
6
1
Hide problems
table
assume that we have a n*n table we fill it with 1,...,n such that each number exists exactly n times prove that there exist a row or column such that at least
n
\sqrt{n}
n
diffrent number are contained.