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Problems
Contests
National and Regional Contests
Iran Contests
Pre-Preparation Course Examination
1997 Pre-Preparation Course Examination
1997 Pre-Preparation Course Examination
Part of
Pre-Preparation Course Examination
Subcontests
(6)
6
3
Hide problems
There exists a "way" in which we can go to the room E from S
A building has some rooms and there is one or more than one doors between the rooms. We know that we can go from each room to another one. Two rooms
E
,
S
E,S
E
,
S
has been labeled, and the room
S
S
S
has exactly one door. Someone is in the room
S
S
S
and wants to move to the room
E
E
E
.http://s1.picofile.com/file/6475095570/image005.jpgA "way"
P
P
P
for moving between the rooms is an infinite sequence of
L
L
L
and
R
R
R
. We say that someone moves according to the "way"
P
P
P
, if he start moving from the room
S
S
S
, and after passing the
n
n
n
'th door, if
P
n
P_n
P
n
is
R
R
R
, then he goes to the first door which is in the right side, and if
P
n
P_n
P
n
is
L
L
L
, then he goes to the first door which is in the left side (obviously, if some room has exactly one door, then there is no difference between
L
L
L
and
R
R
R
), and when he arrives to the room
E
E
E
, he stops moving. Prove that there exists a "way" such that if the person move according to it, then he can arrive to the room
E
E
E
of any building.
There exists a subset of points
We have considered an arbitrary segment from each line in a plane. Show that the set of points of these segments have a subset such that the points of this subset form a triangle in the plane.
The polygon cannot be dissected into 100 triangles - OLD?
A polygon can be dissected into
100
100
100
rectangles, but it cannot be dissected into
99
99
99
rectangles. Prove that this polygon cannot be dissected into
100
100
100
triangles.
4
3
Hide problems
Infinitely many perfect squares of the form n. 2^k - 7
Let
n
n
n
and
k
k
k
be two positive integers. Prove that there exist infinitely many perfect squares of the form
n
⋅
2
k
−
7
n \cdot 2^k - 7
n
⋅
2
k
−
7
.
All possible values for k - Nice Problem
Let
n
≥
3
n \geq 3
n
≥
3
be an integer. Consider the set
A
=
{
1
,
2
,
3
,
…
,
n
}
A=\{1,2,3,\ldots,n\}
A
=
{
1
,
2
,
3
,
…
,
n
}
, in each move, we replace the numbers
i
,
j
i, j
i
,
j
by the numbers
i
+
j
i+j
i
+
j
and
∣
i
−
j
∣
|i-j|
∣
i
−
j
∣
. After doing such moves all of the numbers are equal to
k
k
k
. Find all possible values for
k
k
k
.
Prove that there exist infinitely many primes q
Let
f
:
N
→
N
f : \mathbb N \to \mathbb N
f
:
N
→
N
be an injective function such that there exists a positive integer
k
k
k
for which
f
(
n
)
≤
n
k
f(n) \leq n^k
f
(
n
)
≤
n
k
. Prove that there exist infinitely many primes
q
q
q
such that the equation
f
(
x
)
≡
0
(
m
o
d
q
)
f(x) \equiv 0 \pmod q
f
(
x
)
≡
0
(
mod
q
)
has a solution in prime numbers.
5
3
Hide problems
Show that EX is parallel with AP
Let
H
H
H
be the orthocenter of the triangle
A
B
C
ABC
A
BC
and
P
P
P
an arbitrary point on circumcircle of triangle.
B
H
BH
B
H
meets
A
C
AC
A
C
at
E
E
E
.
P
A
Q
B
PAQB
P
A
QB
and
P
A
R
C
PARC
P
A
RC
are two parallelograms and
A
Q
AQ
A
Q
meets
H
R
HR
H
R
at
X
X
X
. Show that
E
X
∥
A
P
EX \parallel AP
EX
∥
A
P
.
Geometric Inequality - Show that OA'.OB'.OC'≥8R^3
Let
A
B
C
ABC
A
BC
be an acute angled triangle,
O
O
O
be the circumcenter of
A
B
C
ABC
A
BC
, and
R
R
R
be the cicumradius.
A
O
AO
A
O
meets the circumcircle of
B
O
C
BOC
BOC
at
A
′
A'
A
′
,
B
O
BO
BO
meets the circumcircle of
C
O
A
COA
CO
A
, and
C
O
CO
CO
meets the circumcircle of
A
O
B
AOB
A
OB
at
C
′
C'
C
′
. Prove that
O
A
′
⋅
O
B
′
⋅
O
C
′
≥
8
R
3
.
OA' \cdot OB' \cdot OC' \geq 8R^3.
O
A
′
⋅
O
B
′
⋅
O
C
′
≥
8
R
3
.
When does inequality occur?
Show that D^2-H^2 \geq P^2/4
Let
O
O
O
be a point in the plane and let
F
F
F
be a (not necessary convex) polygon. Let
P
P
P
be the perimeter of
F
F
F
, let
D
D
D
be sum of the distances of the point
O
O
O
from the vertices of
F
F
F
, and let
H
H
H
be sum of the distances of the point
O
O
O
from the lines that pass through the vertices of
F
F
F
. Show that
D
2
−
H
2
≥
P
2
4
.
D^2-H^2 \geq \frac{P^2}{4}.
D
2
−
H
2
≥
4
P
2
.
2
2
Hide problems
Nice (circle pass through a constant point)
Let
P
P
P
be a variable point on arc
B
C
BC
BC
of the circumcircle of triangle
A
B
C
ABC
A
BC
not containing
A
A
A
. Let
I
1
I_1
I
1
and
I
2
I_2
I
2
be the incenters of the triangles
P
A
B
PAB
P
A
B
and
P
A
C
PAC
P
A
C
, respectively. Prove that:(a) The circumcircle of
?
P
I
1
I
2
?PI_1I_2
?
P
I
1
I
2
passes through a fixed point.(b) The circle with diameter
I
1
I
2
I_1I_2
I
1
I
2
passes through a fixed point.(c) The midpoint of
I
1
I
2
I_1I_2
I
1
I
2
lies on a fixed circle.
Show that <MKN=90
Two circles
O
,
O
′
O, O'
O
,
O
′
meet each other at points
A
,
B
A, B
A
,
B
. A line from
A
A
A
intersects the circle
O
O
O
at
C
C
C
and the circle
O
′
O'
O
′
at
D
D
D
(
A
A
A
is between
C
C
C
and
D
D
D
). Let
M
,
N
M,N
M
,
N
be the midpoints of the arcs
B
C
,
B
D
BC, BD
BC
,
B
D
, respectively (not containing
A
A
A
), and let
K
K
K
be the midpoint of the segment
C
D
CD
C
D
. Show that
∠
K
M
N
=
9
0
∘
\angle KMN = 90^\circ
∠
K
MN
=
9
0
∘
.
3
3
Hide problems
The permutation and the sum
Let
ω
1
,
ω
2
,
.
.
.
,
ω
k
\omega_1,\omega_2, . . . ,\omega_k
ω
1
,
ω
2
,
...
,
ω
k
be distinct real numbers with a nonzero sum. Prove that there exist integers
n
1
,
n
2
,
.
.
.
,
n
k
n_1, n_2, . . . , n_k
n
1
,
n
2
,
...
,
n
k
such that
∑
i
=
1
k
n
i
ω
i
>
0
\sum_{i=1}^k n_i\omega_i>0
∑
i
=
1
k
n
i
ω
i
>
0
, and for any non-identical permutation
π
\pi
π
of
{
1
,
2
,
…
,
k
}
\{1, 2,\dots, k\}
{
1
,
2
,
…
,
k
}
we have
∑
i
=
1
k
n
i
ω
π
(
i
)
<
0.
\sum_{i=1}^k n_i\omega_{\pi(i)}<0.
i
=
1
∑
k
n
i
ω
π
(
i
)
<
0.
Decreasing function - Show that f(x) = f^{-1}(x)
Suppose that
f
:
R
+
→
R
+
f : \mathbb R^+ \to \mathbb R^+
f
:
R
+
→
R
+
is a decreasing function such that f(x+y)+f(f(x)+f(y))=f(f(x+f(y))+f(y+f(x)), \forall x,y \in \mathbb R^+. Prove that
f
(
x
)
=
f
−
1
(
x
)
.
f(x) = f^{-1}(x).
f
(
x
)
=
f
−
1
(
x
)
.
Find a formula for f(n)
We say three sets
A
1
,
A
2
,
A
3
A_1, A_2, A_3
A
1
,
A
2
,
A
3
form a triangle if for each
1
≤
i
,
j
≤
3
1 \leq i,j \leq 3
1
≤
i
,
j
≤
3
we have
A
i
∩
A
j
≠
∅
A_i \cap A_j \neq \emptyset
A
i
∩
A
j
=
∅
, and
A
1
∩
A
2
∩
A
3
=
∅
A_1 \cap A_2 \cap A_3 = \emptyset
A
1
∩
A
2
∩
A
3
=
∅
. Let
f
(
n
)
f(n)
f
(
n
)
be the smallest positive integer such that any subset of
{
1
,
2
,
3
,
…
,
n
}
\{1,2,3,\ldots, n\}
{
1
,
2
,
3
,
…
,
n
}
of the size
f
(
n
)
f(n)
f
(
n
)
has at least one triangle. Find a formula for
f
(
n
)
f(n)
f
(
n
)
.
1
2
Hide problems
Show that f is a periodic function
Let
f
:
R
→
R
f: \mathbb R \to\mathbb R
f
:
R
→
R
be a function such that
∣
f
(
x
)
∣
≤
1
|f(x)| \leq 1
∣
f
(
x
)
∣
≤
1
for all
x
∈
R
x \in \mathbb R
x
∈
R
and f \left( x + \frac{13}{42} \right) + f(x) = f \left( x + \frac 17 \right) + f \left( x + \frac 16 \right), \forall x \in \mathbb R. Show that
f
f
f
is a periodic function.
Show that there exist polynomials f, g for each n
Let
n
n
n
be a positive integer. Prove that there exist polynomials
f
(
x
)
f(x)
f
(
x
)
and
g
(
x
g(x
g
(
x
) with integer coefficients such that
f
(
x
)
(
x
+
1
)
2
n
+
g
(
x
)
(
x
2
n
+
1
)
=
2.
f(x)\left(x + 1 \right)^{2^n}+ g(x) \left(x^{2^n}+ 1 \right) = 2.
f
(
x
)
(
x
+
1
)
2
n
+
g
(
x
)
(
x
2
n
+
1
)
=
2.