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Problems
Contests
National and Regional Contests
Iran Contests
Iran MO (3rd Round)
1996 Iran MO (3rd Round)
1996 Iran MO (3rd Round)
Part of
Iran MO (3rd Round)
Subcontests
(5)
1
2
Hide problems
Inequality with 4 variables - \sum \frac{a}{b+2c+3d} ≥ 2/3
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be positive real numbers. Prove that
a
b
+
2
c
+
3
d
+
b
c
+
2
d
+
3
a
+
c
d
+
2
a
+
3
b
+
d
a
+
2
b
+
3
c
≥
2
3
.
\frac{a}{b+2c+3d}+\frac{b}{c+2d+3a}+\frac{c}{d+2a+3b}+\frac{d}{a+2b+3c} \geq \frac{2}{3}.
b
+
2
c
+
3
d
a
+
c
+
2
d
+
3
a
b
+
d
+
2
a
+
3
b
c
+
a
+
2
b
+
3
c
d
≥
3
2
.
On the equation 2^x + 3^y = z^2 (old)
Find all non-negative integer solutions of the equation
2
x
+
3
y
=
z
2
.
2^x + 3^y = z^2 .
2
x
+
3
y
=
z
2
.
6
1
Hide problems
Find all pairs of primes
Find all pairs
(
p
,
q
)
(p,q)
(
p
,
q
)
of prime numbers such that
m
3
p
q
≡
m
(
m
o
d
3
p
q
)
∀
m
∈
Z
.
m^{3pq} \equiv m \pmod{3pq} \qquad \forall m \in \mathbb Z.
m
3
pq
≡
m
(
mod
3
pq
)
∀
m
∈
Z
.
2
3
Hide problems
In a parallelogram show that PA+PB+AD≥PE
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram. Construct the equilateral triangle
D
C
E
DCE
D
CE
on the side
D
C
DC
D
C
and outside of parallelogram. Let
P
P
P
be an arbitrary point in plane of
A
B
C
D
ABCD
A
BC
D
. Show that
P
A
+
P
B
+
A
D
≥
P
E
.
PA+PB+AD \geq PE.
P
A
+
PB
+
A
D
≥
PE
.
PQRS is a square ==> ABCD is a square
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral. Construct the points
P
,
Q
,
R
,
P,Q,R,
P
,
Q
,
R
,
and
S
S
S
on continue of
A
B
,
B
C
,
C
D
,
AB,BC,CD,
A
B
,
BC
,
C
D
,
and
D
A
DA
D
A
, respectively, such that
B
P
=
C
Q
=
D
R
=
A
S
.
BP=CQ=DR=AS.
BP
=
CQ
=
D
R
=
A
S
.
Show that if
P
Q
R
S
PQRS
PQRS
is a square, then
A
B
C
D
ABCD
A
BC
D
is also a square.
Radical axis perpendicular to MK
Consider a semicircle of center
O
O
O
and diameter
A
B
AB
A
B
. A line intersects
A
B
AB
A
B
at
M
M
M
and the semicircle at
C
C
C
and
D
D
D
s.t.
M
C
>
M
D
MC>MD
MC
>
M
D
and
M
B
<
M
A
MB<MA
MB
<
M
A
. The circumcircles od the
A
O
C
AOC
A
OC
and
B
O
D
BOD
BO
D
intersect again at
K
K
K
. Prove that
M
K
⊥
K
O
MK\perp KO
M
K
⊥
K
O
.
3
3
Hide problems
Find all sets of real numbers
Find all sets of real numbers
{
a
1
,
a
2
,
…
,
1375
}
\{a_1,a_2,\ldots, _{1375}\}
{
a
1
,
a
2
,
…
,
1375
}
such that 2 \left( \sqrt{a_n - (n-1)}\right) \geq a_{n+1} - (n-1), \forall n \in \{1,2,\ldots,1374\}, and
2
(
a
1375
−
1374
)
≥
a
1
+
1.
2 \left( \sqrt{a_{1375}-1374} \right) \geq a_1 +1.
2
(
a
1375
−
1374
)
≥
a
1
+
1.
p=a_1 and (x-a1)(x-a2)...(x-an)≤x^n-a_1^n
Let
a
1
≥
a
2
≥
⋯
≥
a
n
a_1 \geq a_2 \geq \cdots \geq a_n
a
1
≥
a
2
≥
⋯
≥
a
n
be
n
n
n
real numbers such that
a
1
k
+
a
2
k
+
⋯
+
a
n
k
≥
0
a_1^k +a_2^k + \cdots + a_n^k \geq 0
a
1
k
+
a
2
k
+
⋯
+
a
n
k
≥
0
for all positive integers
k
k
k
. Suppose that
p
=
max
{
∣
a
1
∣
,
∣
a
2
∣
,
…
,
∣
a
n
∣
}
p=\max\{|a_1|,|a_2|, \ldots,|a_n|\}
p
=
max
{
∣
a
1
∣
,
∣
a
2
∣
,
…
,
∣
a
n
∣
}
. Prove that
p
=
a
1
p=a_1
p
=
a
1
, and
(
x
−
a
1
)
(
x
−
a
2
)
⋯
(
x
−
a
n
)
≤
x
n
−
a
1
n
∀
x
>
a
1
.
(x-a_1)(x-a_2)\cdots(x-a_n)\leq x^n-a_1^n \qquad \forall x>a_1.
(
x
−
a
1
)
(
x
−
a
2
)
⋯
(
x
−
a
n
)
≤
x
n
−
a
1
n
∀
x
>
a
1
.
Cyclic
Suppose that
10
10
10
points are given in the plane, such that among any five of them there are four lying on a circle. Find the minimum number of these points which must lie on a circle.
4
3
Hide problems
No such A, B sets exist
Show that there doesn't exist two infinite and separate sets
A
,
B
A,B
A
,
B
of points such that(i) There are no three collinear points in
A
∪
B
A \cup B
A
∪
B
,(ii) The distance between every two points in
A
∪
B
A \cup B
A
∪
B
is at least
1
1
1
, and(iii) There exists at least one point belonging to set
B
B
B
in interior of each triangle which all of its vertices are chosen from the set
A
A
A
, and there exists at least one point belonging to set
A
A
A
in interior of each triangle which all of its vertices are chosen from the set
B
B
B
.
Denote phi(n)=n/k , k is the greatest square with k|n
Let
n
n
n
be a positive integer and suppose that
ϕ
(
n
)
=
n
k
\phi(n)=\frac{n}{k}
ϕ
(
n
)
=
k
n
, where
k
k
k
is the greatest perfect square such that
k
∣
n
k \mid n
k
∣
n
. Let
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots,a_n
a
1
,
a
2
,
…
,
a
n
be
n
n
n
positive integers such that
a
i
=
p
1
a
1
i
⋅
p
2
a
2
i
⋯
p
n
a
n
i
a_i=p_1^{a_1i} \cdot p_2^{a_2i} \cdots p_n^{a_ni}
a
i
=
p
1
a
1
i
⋅
p
2
a
2
i
⋯
p
n
a
n
i
, where
p
i
p_i
p
i
are prime numbers and
a
j
i
a_{ji}
a
ji
are non-negative integers,
1
≤
i
≤
n
,
1
≤
j
≤
n
1 \leq i \leq n, 1 \leq j \leq n
1
≤
i
≤
n
,
1
≤
j
≤
n
. We know that
p
i
∣
ϕ
(
a
i
)
p_i\mid \phi(a_i)
p
i
∣
ϕ
(
a
i
)
, and if
p
i
∣
ϕ
(
a
j
)
p_i\mid \phi(a_j)
p
i
∣
ϕ
(
a
j
)
, then
p
j
∣
ϕ
(
a
i
)
p_j\mid \phi(a_i)
p
j
∣
ϕ
(
a
i
)
. Prove that there exist integers
k
1
,
k
2
,
…
,
k
m
k_1,k_2,\ldots,k_m
k
1
,
k
2
,
…
,
k
m
with
1
≤
k
1
≤
k
2
≤
⋯
≤
k
m
≤
n
1 \leq k_1 \leq k_2 \leq \cdots \leq k_m \leq n
1
≤
k
1
≤
k
2
≤
⋯
≤
k
m
≤
n
such that
ϕ
(
a
k
1
⋅
a
k
2
⋯
a
k
m
)
=
p
1
⋅
p
2
⋯
p
n
.
\phi(a_{k_{1}} \cdot a_{k_{2}} \cdots a_{k_{m}})=p_1 \cdot p_2 \cdots p_n.
ϕ
(
a
k
1
⋅
a
k
2
⋯
a
k
m
)
=
p
1
⋅
p
2
⋯
p
n
.
Function
Determine all functions
f
:
N
0
→
N
0
−
{
1
}
f : \mathbb N_0 \rightarrow \mathbb N_0 - \{1\}
f
:
N
0
→
N
0
−
{
1
}
such that
f
(
n
+
1
)
+
f
(
n
+
3
)
=
f
(
n
+
5
)
f
(
n
+
7
)
−
1375
,
∀
n
∈
N
.
f(n + 1) + f(n + 3) = f(n + 5)f(n + 7) - 1375, \qquad \forall n \in \mathbb N.
f
(
n
+
1
)
+
f
(
n
+
3
)
=
f
(
n
+
5
)
f
(
n
+
7
)
−
1375
,
∀
n
∈
N
.