MathDB
Problems
Contests
National and Regional Contests
Iran Contests
Iran Team Selection Test
2014 Iran Team Selection Test
2014 Iran Team Selection Test
Part of
Iran Team Selection Test
Subcontests
(6)
6
3
Hide problems
question6
I
I
I
is the incenter of triangle
A
B
C
ABC
A
BC
. perpendicular from
I
I
I
to
A
I
AI
A
I
meet
A
B
AB
A
B
and
A
C
AC
A
C
at
B
′
{B}'
B
′
and
C
′
{C}'
C
′
respectively . Suppose that
B
′
′
{B}''
B
′′
and
C
′
′
{C}''
C
′′
are points on half-line
B
C
BC
BC
and
C
B
CB
CB
such that
B
B
′
′
=
B
A
B{B}''=BA
B
B
′′
=
B
A
and
C
C
′
′
=
C
A
C{C}''=CA
C
C
′′
=
C
A
. Suppose that the second intersection of circumcircles of
A
B
′
B
′
′
A{B}'{B}''
A
B
′
B
′′
and
A
C
′
C
′
′
A{C}'{C}''
A
C
′
C
′′
is
T
T
T
. Prove that the circumcenter of
A
I
T
AIT
A
I
T
is on the
B
C
BC
BC
.
Simple 2n-gon
Consider
n
n
n
segments in the plane which no two intersect and between their
2
n
2n
2
n
endpoints no three are collinear. Is the following statement true? Statement: There exists a simple
2
n
2n
2
n
-gon such that it's vertices are the
2
n
2n
2
n
endpoints of the segments and each segment is either completely inside the polygon or an edge of the polygon.
question 6
The incircle of a non-isosceles triangle
A
B
C
ABC
A
BC
with the center
I
I
I
touches the sides
B
C
BC
BC
at
D
D
D
. let
X
X
X
is a point on arc
B
C
BC
BC
from circumcircle of triangle
A
B
C
ABC
A
BC
such that if
E
,
F
E,F
E
,
F
are feet of perpendicular from
X
X
X
on
B
I
,
C
I
BI,CI
B
I
,
C
I
and
M
M
M
is midpoint of
E
F
EF
EF
we have
M
B
=
M
C
MB=MC
MB
=
MC
. prove that
B
A
D
^
=
C
A
X
^
\widehat{BAD}=\widehat{CAX}
B
A
D
=
C
A
X
5
3
Hide problems
question5
n
n
n
is a natural number. for every positive real numbers
x
1
,
x
2
,
.
.
.
,
x
n
+
1
x_{1},x_{2},...,x_{n+1}
x
1
,
x
2
,
...
,
x
n
+
1
such that
x
1
x
2
.
.
.
x
n
+
1
=
1
x_{1}x_{2}...x_{n+1}=1
x
1
x
2
...
x
n
+
1
=
1
prove that:
n
x
1
+
.
.
.
+
n
x
n
+
1
≥
n
x
1
n
+
.
.
.
+
n
x
n
+
1
n
\sqrt[x_{1}]{n}+...+\sqrt[x_{n+1}]{n} \geq n^{\sqrt[n]{x_{1}}}+...+n^{\sqrt[n]{x_{n+1}}}
x
1
n
+
...
+
x
n
+
1
n
≥
n
n
x
1
+
...
+
n
n
x
n
+
1
nice problem (Inequalities)(question 5)
if
x
,
y
,
z
>
0
x,y,z>0
x
,
y
,
z
>
0
are postive real numbers such that
x
2
+
y
2
+
z
2
=
x
2
y
2
+
y
2
z
2
+
z
2
x
2
x^{2}+y^{2}+z^{2}=x^{2}y^{2}+y^{2}z^{2}+z^{2}x^{2}
x
2
+
y
2
+
z
2
=
x
2
y
2
+
y
2
z
2
+
z
2
x
2
prove that
(
(
x
−
y
)
(
y
−
z
)
(
z
−
x
)
)
2
≤
2
(
(
x
2
−
y
2
)
2
+
(
y
2
−
z
2
)
2
+
(
z
2
−
x
2
)
2
)
((x-y)(y-z)(z-x))^{2}\leq 2((x^{2}-y^{2})^{2}+(y^{2}-z^{2})^{2}+(z^{2}-x^{2})^{2})
((
x
−
y
)
(
y
−
z
)
(
z
−
x
)
)
2
≤
2
((
x
2
−
y
2
)
2
+
(
y
2
−
z
2
)
2
+
(
z
2
−
x
2
)
2
)
Compact sets
Given a set
X
=
{
x
1
,
…
,
x
n
}
X=\{x_1,\ldots,x_n\}
X
=
{
x
1
,
…
,
x
n
}
of natural numbers in which for all
1
<
i
≤
n
1< i \leq n
1
<
i
≤
n
we have
1
≤
x
i
−
x
i
−
1
≤
2
1\leq x_i-x_{i-1}\leq 2
1
≤
x
i
−
x
i
−
1
≤
2
, call a real number
a
a
a
good if there exists
1
≤
j
≤
n
1\leq j \leq n
1
≤
j
≤
n
such that
2
∣
x
j
−
a
∣
≤
1
2|x_j-a|\leq 1
2∣
x
j
−
a
∣
≤
1
. Also a subset of
X
X
X
is called compact if the average of its elements is a good number. Prove that at least
2
n
−
3
2^{n-3}
2
n
−
3
subsets of
X
X
X
are compact.Proposed by Mahyar Sefidgaran
4
3
Hide problems
question4
Find the maximum number of Permutation of set {
1
,
2
,
3
,
.
.
.
,
2014
1,2,3,...,2014
1
,
2
,
3
,
...
,
2014
} such that for every 2 different number
a
a
a
and
b
b
b
in this set at last in one of the permutation
b
b
b
comes exactly after
a
a
a
The n-cubic Permutations
n
n
n
is a natural number. We shall call a permutation
a
1
,
…
,
a
n
a_1,\dots,a_n
a
1
,
…
,
a
n
of
1
,
…
,
n
1,\dots,n
1
,
…
,
n
a quadratic(cubic) permutation if
∀
1
≤
i
≤
n
−
1
\forall 1\leq i \leq n-1
∀1
≤
i
≤
n
−
1
we have
a
i
a
i
+
1
+
1
a_ia_{i+1}+1
a
i
a
i
+
1
+
1
is a perfect square(cube).
(
a
)
(a)
(
a
)
Prove that for infinitely many natural numbers
n
n
n
there exists a quadratic permutation.
(
b
)
(b)
(
b
)
Prove that for no natural number
n
n
n
exists a cubic permutation.
easy function equation(question 4)
Find all functions
f
:
R
+
→
R
+
f:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}
f
:
R
+
→
R
+
such that
x
,
y
∈
R
+
,
x,y\in \mathbb{R}^{+},
x
,
y
∈
R
+
,
f
(
y
f
(
x
+
1
)
)
+
f
(
x
+
1
x
f
(
y
)
)
=
f
(
y
)
f\left(\frac{y}{f(x+1)}\right)+f\left(\frac{x+1}{xf(y)}\right)=f(y)
f
(
f
(
x
+
1
)
y
)
+
f
(
x
f
(
y
)
x
+
1
)
=
f
(
y
)
3
3
Hide problems
question 3
prove for all
k
>
1
k> 1
k
>
1
equation
(
x
+
1
)
(
x
+
2
)
.
.
.
(
x
+
k
)
=
y
2
(x+1)(x+2)...(x+k)=y^{2}
(
x
+
1
)
(
x
+
2
)
...
(
x
+
k
)
=
y
2
has finite solutions.
question 3
we named a
n
∗
n
n*n
n
∗
n
table
s
e
l
f
i
s
h
selfish
se
l
f
i
s
h
if we number the row and column with
0
,
1
,
2
,
3
,
.
.
.
,
n
−
1
0,1,2,3,...,n-1
0
,
1
,
2
,
3
,
...
,
n
−
1
.(from left to right an from up to down) for every {
i
,
j
∈
0
,
1
,
2
,
.
.
.
,
n
−
1
i,j\in{0,1,2,...,n-1}
i
,
j
∈
0
,
1
,
2
,
...
,
n
−
1
} the number of cell
(
i
,
j
)
(i,j)
(
i
,
j
)
is equal to the number of number
i
i
i
in the row
j
j
j
. for example we have such table for
n
=
5
n=5
n
=
5
1 0 3 3 4 1 3 2 1 1 0 1 0 1 0 2 1 0 0 0 1 0 0 0 0 prove that for
n
>
5
n>5
n
>
5
there is no
s
e
l
f
i
s
h
selfish
se
l
f
i
s
h
table
hard question 3
let
m
,
n
∈
N
m,n\in \mathbb{N}
m
,
n
∈
N
and
p
(
x
)
,
q
(
x
)
,
h
(
x
)
p(x),q(x),h(x)
p
(
x
)
,
q
(
x
)
,
h
(
x
)
are polynomials with real Coefficients such that
p
(
x
)
p(x)
p
(
x
)
is Descending. and for all
x
∈
R
x\in \mathbb{R}
x
∈
R
p
(
q
(
n
x
+
m
)
+
h
(
x
)
)
=
n
(
q
(
p
(
x
)
)
+
h
(
x
)
)
+
m
p(q(nx+m)+h(x))=n(q(p(x))+h(x))+m
p
(
q
(
n
x
+
m
)
+
h
(
x
))
=
n
(
q
(
p
(
x
))
+
h
(
x
))
+
m
. prove that dont exist function
f
:
R
→
R
f:\mathbb{R}\rightarrow \mathbb{R}
f
:
R
→
R
such that for all
x
∈
R
x\in \mathbb{R}
x
∈
R
f
(
q
(
p
(
x
)
)
+
h
(
x
)
)
=
f
(
x
)
2
+
1
f(q(p(x))+h(x))=f(x)^{2}+1
f
(
q
(
p
(
x
))
+
h
(
x
))
=
f
(
x
)
2
+
1
2
3
Hide problems
question 2
find all polynomials with integer coefficients that
P
(
Z
)
=
P(\mathbb{Z})=
P
(
Z
)
=
{
p
(
a
)
:
a
∈
Z
p(a):a\in \mathbb{Z}
p
(
a
)
:
a
∈
Z
} has a Geometric progression.
MN goes through fixed point
Point
D
D
D
is an arbitary point on side
B
C
BC
BC
of triangle
A
B
C
ABC
A
BC
.
I
I
I
,
I
1
I_1
I
1
and
I
2
I_2
I
2
are the incenters of triangles
A
B
C
ABC
A
BC
,
A
B
D
ABD
A
B
D
and
A
C
D
ACD
A
C
D
respectively.
M
≠
A
M\not=A
M
=
A
and
N
≠
A
N\not=A
N
=
A
are the intersections of circumcircle of triangle
A
B
C
ABC
A
BC
and circumcircles of triangles
I
A
I
1
IAI_1
I
A
I
1
and
I
A
I
2
IAI_2
I
A
I
2
respectively. Prove that regardless of point
D
D
D
, line
M
N
MN
MN
goes through a fixed point.
nice question 2
is there a function
f
:
N
→
N
f:\mathbb{N}\rightarrow \mathbb{N}
f
:
N
→
N
such that
i
)
∃
n
∈
N
:
f
(
n
)
≠
n
i) \exists n\in \mathbb{N}:f(n)\neq n
i
)
∃
n
∈
N
:
f
(
n
)
=
n
i
i
)
ii)
ii
)
the number of divisors of
m
m
m
is
f
(
n
)
f(n)
f
(
n
)
if and only if the number of divisors of
f
(
m
)
f(m)
f
(
m
)
is
n
n
n
1
3
Hide problems
question1
suppose that
O
O
O
is the circumcenter of acute triangle
A
B
C
ABC
A
BC
. we have circle with center
O
O
O
that is tangent too
B
C
BC
BC
that named
w
w
w
suppose that
X
X
X
and
Y
Y
Y
are the points of intersection of the tangent from
A
A
A
to
w
w
w
with line
B
C
BC
BC
(
X
X
X
and
B
B
B
are in the same side of
A
O
AO
A
O
)
T
T
T
is the intersection of the line tangent to circumcirle of
A
B
C
ABC
A
BC
in
B
B
B
and the line from
X
X
X
parallel to
A
C
AC
A
C
.
S
S
S
is the intersection of the line tangent to circumcirle of
A
B
C
ABC
A
BC
in
C
C
C
and the line from
Y
Y
Y
parallel to
A
B
AB
A
B
. prove that
S
T
ST
ST
is tangent
A
B
C
ABC
A
BC
.
Tree and Permutations
Consider a tree with
n
n
n
vertices, labeled with
1
,
…
,
n
1,\ldots,n
1
,
…
,
n
in a way that no label is used twice. We change the labeling in the following way - each time we pick an edge that hasn't been picked before and swap the labels of its endpoints. After performing this action
n
−
1
n-1
n
−
1
times, we get another tree with its labeling a permutation of the first graph's labeling. Prove that this permutation contains exactly one cycle.
hard question 1
The incircle of a non-isosceles triangle
A
B
C
ABC
A
BC
with the center
I
I
I
touches the sides
B
C
,
A
C
,
A
B
BC,AC,AB
BC
,
A
C
,
A
B
at
A
1
,
B
1
,
C
1
A_{1},B_{1},C_{1}
A
1
,
B
1
,
C
1
. let
A
I
,
B
I
,
C
I
AI,BI,CI
A
I
,
B
I
,
C
I
meets
B
C
,
A
C
,
A
B
BC,AC,AB
BC
,
A
C
,
A
B
at
A
2
,
B
2
,
C
2
A_{2},B_{2},C_{2}
A
2
,
B
2
,
C
2
. let
A
′
A'
A
′
is a point on
A
I
AI
A
I
such that
A
1
A
′
⊥
B
2
C
2
A_{1}A'\perp B_{2}C_{2}
A
1
A
′
⊥
B
2
C
2
.
B
′
,
C
′
B',C'
B
′
,
C
′
respectively. prove that two triangle
A
′
B
′
C
′
,
A
1
B
1
C
1
A'B'C',A_{1}B_{1}C_{1}
A
′
B
′
C
′
,
A
1
B
1
C
1
are equal.