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Problems
Contests
National and Regional Contests
Iran Contests
Pre-Preparation Course Examination
2007 Pre-Preparation Course Examination
2007 Pre-Preparation Course Examination
Part of
Pre-Preparation Course Examination
Subcontests
(22)
22
1
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exist a1,a2,...,an for n-1-Iran 3rd round-Number Theory 2007
Prove that for any positive integer
n
≥
3
n \geq 3
n
≥
3
there exist positive integers
a
1
,
a
2
,
⋯
,
a
n
a_1,a_2,\cdots , a_n
a
1
,
a
2
,
⋯
,
a
n
such that
a
1
a
2
⋯
a
n
≡
a
i
(
m
o
d
a
i
2
)
∀
i
∈
{
1
,
2
,
⋯
,
n
}
a_1a_2\cdots a_n \equiv a_i \pmod {a_i^2} \qquad \forall i \in \{1,2,\cdots ,n\}
a
1
a
2
⋯
a
n
≡
a
i
(
mod
a
i
2
)
∀
i
∈
{
1
,
2
,
⋯
,
n
}
21
1
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On equation p^q-q^p=pq^2-1-Iran 3rd round-Number Theory 2007
Find all primes
p
,
q
p,q
p
,
q
such that
p
q
−
q
p
=
p
q
2
−
19
p^q-q^p=pq^2-19
p
q
−
q
p
=
p
q
2
−
19
20
1
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n \leq 4m(2^m-1)-Iran 3rd round-Number Theory 2007
Let
m
,
n
m,n
m
,
n
be two positive integers and
m
≥
2
m \geq 2
m
≥
2
. We know that for every positive integer
a
a
a
such that
gcd
(
a
,
n
)
=
1
\gcd(a,n)=1
g
cd
(
a
,
n
)
=
1
we have
n
∣
a
m
−
1
n|a^m-1
n
∣
a
m
−
1
. Prove that
n
≤
4
m
(
2
m
−
1
)
n \leq 4m(2^m-1)
n
≤
4
m
(
2
m
−
1
)
.
19
1
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Find all N-to-N functions-Iran 3rd round-Number Theory 2007
Find all functions
f
:
N
→
N
f : \mathbb N \to \mathbb N
f
:
N
→
N
such that:i)
f
2000
(
m
)
=
f
(
m
)
f^{2000}(m)=f(m)
f
2000
(
m
)
=
f
(
m
)
for all
m
∈
N
m \in \mathbb N
m
∈
N
,ii)
f
(
m
n
)
=
f
(
m
)
f
(
n
)
f
(
gcd
(
m
,
n
)
)
f(mn)=\dfrac{f(m)f(n)}{f(\gcd(m,n))}
f
(
mn
)
=
f
(
g
cd
(
m
,
n
))
f
(
m
)
f
(
n
)
, for all
m
,
n
∈
N
m,n\in \mathbb N
m
,
n
∈
N
, andiii)
f
(
m
)
=
1
f(m)=1
f
(
m
)
=
1
if and only if
m
=
1
m=1
m
=
1
.
18
1
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On x^3+y^3+z^3=t^4
Prove that the equation
x
3
+
y
3
+
z
3
=
t
4
x^3+y^3+z^3=t^4
x
3
+
y
3
+
z
3
=
t
4
has infinitely many solutions in positive integers such that
gcd
(
x
,
y
,
z
,
t
)
=
1
\gcd(x,y,z,t)=1
g
cd
(
x
,
y
,
z
,
t
)
=
1
.Mihai Pitticari & Sorin Rǎdulescu
17
1
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N consecutive terms-Iran 3rd round-Number Theory 2007
For a positive integer
n
n
n
, denote
r
a
d
(
n
)
rad(n)
r
a
d
(
n
)
as product of prime divisors of
n
n
n
. And also
r
a
d
(
1
)
=
1
rad(1)=1
r
a
d
(
1
)
=
1
. Define the sequence
{
a
i
}
i
=
1
∞
\{a_i\}_{i=1}^{\infty}
{
a
i
}
i
=
1
∞
in this way:
a
1
∈
N
a_1 \in \mathbb N
a
1
∈
N
and for every
n
∈
N
n \in \mathbb N
n
∈
N
,
a
n
+
1
=
a
n
+
r
a
d
(
a
n
)
a_{n+1}=a_n+rad(a_n)
a
n
+
1
=
a
n
+
r
a
d
(
a
n
)
. Prove that for every
N
∈
N
N \in \mathbb N
N
∈
N
, there exist
N
N
N
consecutive terms of this sequence which are in an arithmetic progression.
16
1
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2^2^n+2^2^{n-1}+1-Iran 3rd round-Number Theory 2007
Prove that
2
2
n
+
2
2
n
−
1
+
1
2^{2^{n}}+2^{2^{{n-1}}}+1
2
2
n
+
2
2
n
−
1
+
1
has at least
n
n
n
distinct prime divisors.
15
1
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Existing a subset of N-Iran 3rd round-Number Theory 2007
Does there exists a subset of positive integers with infinite members such that for every two members
a
,
b
a,b
a
,
b
of this set
a
2
−
a
b
+
b
2
∣
(
a
b
)
2
a^2-ab+b^2|(ab)^2
a
2
−
ab
+
b
2
∣
(
ab
)
2
14
1
Hide problems
Find all a,b,c in N-Iran 3rd round-Number Theory 2007
Find all
a
,
b
,
c
∈
N
a,b,c \in \mathbb{N}
a
,
b
,
c
∈
N
such that
a
2
b
∣
a
3
+
b
3
+
c
3
,
b
2
c
∣
a
3
+
b
3
+
c
3
,
c
2
a
∣
a
3
+
b
3
+
c
3
.
a^2b|a^3+b^3+c^3,\qquad b^2c|a^3+b^3+c^3, \qquad c^2a|a^3+b^3+c^3.
a
2
b
∣
a
3
+
b
3
+
c
3
,
b
2
c
∣
a
3
+
b
3
+
c
3
,
c
2
a
∣
a
3
+
b
3
+
c
3
.
[PS: The original problem was this: Find all
a
,
b
,
c
∈
N
a,b,c \in \mathbb{N}
a
,
b
,
c
∈
N
such that
a
2
b
∣
a
3
+
b
3
+
c
3
,
b
2
c
∣
a
3
+
b
3
+
c
3
,
c
2
b
∣
a
3
+
b
3
+
c
3
.
a^2b|a^3+b^3+c^3,\qquad b^2c|a^3+b^3+c^3, \qquad \color{red}{c^2b}|a^3+b^3+c^3.
a
2
b
∣
a
3
+
b
3
+
c
3
,
b
2
c
∣
a
3
+
b
3
+
c
3
,
c
2
b
∣
a
3
+
b
3
+
c
3
.
But I think the author meant
c
2
a
∣
a
3
+
b
3
+
c
3
c^2a|a^3+b^3+c^3
c
2
a
∣
a
3
+
b
3
+
c
3
, just because of symmetry]
13
1
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The number is not in N-Iran 3rd round-Number Theory 2007
Let
{
a
i
}
i
=
1
∞
\{a_i\}_{i=1}^{\infty}
{
a
i
}
i
=
1
∞
be a sequence of positive integers such that
a
1
<
a
2
<
a
3
⋯
a_1<a_2<a_3\cdots
a
1
<
a
2
<
a
3
⋯
and all of primes are members of this sequence. Prove that for every
n
<
m
n<m
n
<
m
1
a
n
+
1
a
n
+
1
+
⋯
+
1
a
m
∉
N
\dfrac{1}{a_n} + \dfrac{1}{a_{n+1}} + \cdots + \dfrac{1}{a_m} \not \in \mathbb N
a
n
1
+
a
n
+
1
1
+
⋯
+
a
m
1
∈
N
12
1
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Find all subsets of N-Iran 3rd round-Number Theory 2007
Find all subsets of
N
\mathbb N
N
like
S
S
S
such that
∀
m
,
n
∈
S
⟹
m
+
n
gcd
(
m
,
n
)
∈
S
\forall m,n \in S \implies \dfrac{m+n}{\gcd(m,n)} \in S
∀
m
,
n
∈
S
⟹
g
cd
(
m
,
n
)
m
+
n
∈
S
11
1
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product of some of members-Iran 3rd round-Number Theory 2007
Let
p
≥
3
p \geq 3
p
≥
3
be a prime and
a
1
,
a
2
,
⋯
,
a
p
−
2
a_1,a_2,\cdots , a_{p-2}
a
1
,
a
2
,
⋯
,
a
p
−
2
be a sequence of positive integers such that for every
k
∈
{
1
,
2
,
⋯
,
p
−
2
}
k \in \{1,2,\cdots,p-2\}
k
∈
{
1
,
2
,
⋯
,
p
−
2
}
neither
a
k
a_k
a
k
nor
a
k
k
−
1
a_k^k-1
a
k
k
−
1
is divisible by
p
p
p
. Prove that product of some of members of this sequence is equivalent to
2
2
2
modulo
p
p
p
.
10
1
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A subset- Iran 3rd round-Number Theory 2007
Let
a
>
1
a >1
a
>
1
be a positive integer. Prove that the set
{
a
2
+
a
−
1
,
a
3
+
a
−
1
,
⋯
}
\{a^2+a-1,a^3+a-1,\cdots\}
{
a
2
+
a
−
1
,
a
3
+
a
−
1
,
⋯
}
have a subset
S
S
S
with infinite members and for any two members of
S
S
S
like
x
,
y
x,y
x
,
y
we have
gcd
(
x
,
y
)
=
1
\gcd(x,y)=1
g
cd
(
x
,
y
)
=
1
. Then prove that the set of primes has infinite members.
9
1
Hide problems
On equation 4xy-x-y=z^2- Iran 3rd round-Number Theory 2007
Solve the equation
4
x
y
−
x
−
y
=
z
2
4xy-x-y=z^2
4
x
y
−
x
−
y
=
z
2
in positive integers.
8
1
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m is perfect square- Iran 3rd round-Number Theory 2007
Let
m
,
n
,
k
m,n,k
m
,
n
,
k
be positive integers and
1
+
m
+
n
3
=
(
2
+
3
)
2
k
+
1
1+m+n \sqrt 3=(2+ \sqrt 3)^{2k+1}
1
+
m
+
n
3
=
(
2
+
3
)
2
k
+
1
. Prove that
m
m
m
is a perfect square.
7
1
Hide problems
Two sets- Iran 3rd round-Number Theory 2007
Let
p
p
p
be a prime such that
p
≡
3
(
m
o
d
4
)
p \equiv 3 \pmod 4
p
≡
3
(
mod
4
)
. Prove that we can't partition the numbers
a
,
a
+
1
,
a
+
2
,
⋯
,
a
+
p
−
2
a,a+1,a+2,\cdots,a+p-2
a
,
a
+
1
,
a
+
2
,
⋯
,
a
+
p
−
2
,(
a
∈
Z
a \in \mathbb Z
a
∈
Z
) in two sets such that product of members of the sets be equal.
6
1
Hide problems
b^2+a-1 has two divisors- Iran 3rd round-Number Theory 2007
Let
a
,
b
a,b
a
,
b
be two positive integers and
b
2
+
a
−
1
∣
a
2
+
b
−
1
b^2+a-1|a^2+b-1
b
2
+
a
−
1∣
a
2
+
b
−
1
. Prove that
b
2
+
a
−
1
b^2+a-1
b
2
+
a
−
1
has at least two prime divisors.
5
1
Hide problems
On equation y^3=x^2+5 - Iran 3rd round-Number Theory 2007
Prove that the equation
y
3
=
x
2
+
5
y^3=x^2+5
y
3
=
x
2
+
5
doesn't have any solutions in
Z
Z
Z
.
4
3
Hide problems
Prove that a=0 - Iran 3rd round-Number Theory 2007
a
,
b
∈
Z
a,b \in \mathbb Z
a
,
b
∈
Z
and for every
n
∈
N
0
n \in \mathbb{N}_0
n
∈
N
0
, the number
2
n
a
+
b
2^na+b
2
n
a
+
b
is a perfect square. Prove that
a
=
0
a=0
a
=
0
.
Constant point
Let
(
C
)
(C)
(
C
)
and
(
L
)
(L)
(
L
)
be a circle and a line.
P
1
,
…
,
P
2
n
+
1
P_{1},\dots,P_{2n+1}
P
1
,
…
,
P
2
n
+
1
are odd number of points on
(
L
)
(L)
(
L
)
.
A
1
A_{1}
A
1
is an arbitrary point on
(
C
)
(C)
(
C
)
.
A
k
+
1
A_{k+1}
A
k
+
1
is the intersection point of
A
k
P
k
A_{k}P_{k}
A
k
P
k
and
(
C
)
(C)
(
C
)
(
1
≤
k
≤
2
n
+
1
1\leq k\leq 2n+1
1
≤
k
≤
2
n
+
1
). Prove that
A
1
A
2
n
+
2
A_{1}A_{2n+2}
A
1
A
2
n
+
2
passes through a constant point while
A
1
A_{1}
A
1
varies on
(
C
)
(C)
(
C
)
.
Sum
Prove that
∑
i
=
−
2007
2007
∣
i
+
1
∣
(
2
)
∣
i
∣
>
∑
i
=
−
2007
2007
∣
i
∣
(
2
)
∣
i
∣
\sum_{i=-2007}^{2007}\frac{\sqrt{|i+1|}}{(\sqrt2)^{|i|}}>\sum_{i=-2007}^{2007}\frac{\sqrt{|i|}}{(\sqrt2)^{|i|}}
i
=
−
2007
∑
2007
(
2
)
∣
i
∣
∣
i
+
1∣
>
i
=
−
2007
∑
2007
(
2
)
∣
i
∣
∣
i
∣
3
3
Hide problems
Triangle and area
A
B
C
ABC
A
BC
is an arbitrary triangle.
A
′
,
B
′
,
C
′
A',B',C'
A
′
,
B
′
,
C
′
are midpoints of arcs
B
C
,
A
C
,
A
B
BC, AC, AB
BC
,
A
C
,
A
B
. Sides of triangle
A
B
C
ABC
A
BC
, intersect sides of triangle
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
at points
P
,
Q
,
R
,
S
,
T
,
F
P,Q,R,S,T,F
P
,
Q
,
R
,
S
,
T
,
F
. Prove that
S
P
Q
R
S
T
F
S
A
B
C
=
1
−
a
b
+
a
c
+
b
c
(
a
+
b
+
c
)
2
\frac{S_{PQRSTF}}{S_{ABC}}=1-\frac{ab+ac+bc}{(a+b+c)^{2}}
S
A
BC
S
PQRSTF
=
1
−
(
a
+
b
+
c
)
2
ab
+
a
c
+
b
c
Graph & Equation
This question is both combinatorics and Number Theory : a ) Prove that we can color edges of
K
p
K_{p}
K
p
with
p
p
p
colors which is proper, (
p
p
p
is an odd prime) and
K
p
K_{p}
K
p
can be partitioned to
p
−
1
2
\frac{p-1}2
2
p
−
1
rainbow Hamiltonian cycles. (A Hamiltonian cycle is a cycle that passes from all of verteces, and a rainbow is a subgraph that all of its edges have different colors.) b) Find all answers of
x
2
+
y
2
+
z
2
=
1
x^{2}+y^{2}+z^{2}=1
x
2
+
y
2
+
z
2
=
1
is
Z
p
\mathbb Z_{p}
Z
p
Divisibility
Prove that for each
a
∈
N
a\in\mathbb N
a
∈
N
, there are infinitely many natural
n
n
n
, such that n\mid a^{n \minus{} a \plus{} 1} \minus{} 1.
2
4
Show problems
1
4
Show problems