MathDB
Problems
Contests
National and Regional Contests
Iran Contests
Iran Team Selection Test
2011 Iran Team Selection Test
2011 Iran Team Selection Test
Part of
Iran Team Selection Test
Subcontests
(12)
12
1
Hide problems
af(a)+bf(b)+2ab=x^2 for all natural a, b - show that f(a)=a
Suppose that
f
:
N
→
N
f : \mathbb{N} \rightarrow \mathbb{N}
f
:
N
→
N
is a function for which the expression
a
f
(
a
)
+
b
f
(
b
)
+
2
a
b
af(a)+bf(b)+2ab
a
f
(
a
)
+
b
f
(
b
)
+
2
ab
for all
a
,
b
∈
N
a,b \in \mathbb{N}
a
,
b
∈
N
is always a perfect square. Prove that
f
(
a
)
=
a
f(a)=a
f
(
a
)
=
a
for all
a
∈
N
a \in \mathbb{N}
a
∈
N
.
11
1
Hide problems
All PP' lines pass through a fixed point
Let
A
B
C
ABC
A
BC
be a triangle and
A
′
,
B
′
,
C
′
A',B',C'
A
′
,
B
′
,
C
′
be the midpoints of
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
respectively. Let
P
P
P
and
P
′
P'
P
′
be points in plane such that
P
A
=
P
′
A
′
,
P
B
=
P
′
B
′
,
P
C
=
P
′
C
′
PA=P'A',PB=P'B',PC=P'C'
P
A
=
P
′
A
′
,
PB
=
P
′
B
′
,
PC
=
P
′
C
′
. Prove that all
P
P
′
PP'
P
P
′
pass through a fixed point.
10
1
Hide problems
The least value of k for which the inequality holds
Find the least value of
k
k
k
such that for all
a
,
b
,
c
,
d
∈
R
a,b,c,d \in \mathbb{R}
a
,
b
,
c
,
d
∈
R
the inequality
(
a
2
+
1
)
(
b
2
+
1
)
(
c
2
+
1
)
+
(
b
2
+
1
)
(
c
2
+
1
)
(
d
2
+
1
)
+
(
c
2
+
1
)
(
d
2
+
1
)
(
a
2
+
1
)
+
(
d
2
+
1
)
(
a
2
+
1
)
(
b
2
+
1
)
≥
2
(
a
b
+
b
c
+
c
d
+
d
a
+
a
c
+
b
d
)
−
k
\begin{array} c \sqrt{(a^2+1)(b^2+1)(c^2+1)} +\sqrt{(b^2+1)(c^2+1)(d^2+1)} +\sqrt{(c^2+1)(d^2+1)(a^2+1)} +\sqrt{(d^2+1)(a^2+1)(b^2+1)} \\ \ \\ \ge 2( ab+bc+cd+da+ac+bd)-k \end{array}
(
a
2
+
1
)
(
b
2
+
1
)
(
c
2
+
1
)
+
(
b
2
+
1
)
(
c
2
+
1
)
(
d
2
+
1
)
+
(
c
2
+
1
)
(
d
2
+
1
)
(
a
2
+
1
)
+
(
d
2
+
1
)
(
a
2
+
1
)
(
b
2
+
1
)
≥
2
(
ab
+
b
c
+
c
d
+
d
a
+
a
c
+
b
d
)
−
k
holds.
9
1
Hide problems
Infinitely many points in plane with at least n+1 good lines
We have
n
n
n
points in the plane such that they are not all collinear. We call a line
ℓ
\ell
ℓ
a 'good' line if we can divide those
n
n
n
points in two sets
A
,
B
A,B
A
,
B
such that the sum of the distances of all points in
A
A
A
to
ℓ
\ell
ℓ
is equal to the sum of the distances of all points in
B
B
B
to
ℓ
\ell
ℓ
. Prove that there exist infinitely many points in the plane such that for each of them we have at least
n
+
1
n+1
n
+
1
good lines passing through them.
8
1
Hide problems
Prove that there exist polynomials A0,A1,...,Ak
Let
p
p
p
be a prime and
k
k
k
a positive integer such that
k
≤
p
k \le p
k
≤
p
. We know that
f
(
x
)
f(x)
f
(
x
)
is a polynomial in
Z
[
x
]
\mathbb Z[x]
Z
[
x
]
such that for all
x
∈
Z
x \in \mathbb{Z}
x
∈
Z
we have
p
k
∣
f
(
x
)
p^k | f(x)
p
k
∣
f
(
x
)
. (a) Prove that there exist polynomials
A
0
(
x
)
,
…
,
A
k
(
x
)
A_0(x),\ldots,A_k(x)
A
0
(
x
)
,
…
,
A
k
(
x
)
all in
Z
[
x
]
\mathbb Z[x]
Z
[
x
]
such that
f
(
x
)
=
∑
i
=
0
k
(
x
p
−
x
)
i
p
k
−
i
A
i
(
x
)
,
f(x)=\sum_{i=0}^{k} (x^p-x)^ip^{k-i}A_i(x),
f
(
x
)
=
i
=
0
∑
k
(
x
p
−
x
)
i
p
k
−
i
A
i
(
x
)
,
(b) Find a counter example for each prime
p
p
p
and each
k
>
p
k > p
k
>
p
.
7
1
Hide problems
Locus of P in an equilateral triangle
Find the locus of points
P
P
P
in an equilateral triangle
A
B
C
ABC
A
BC
for which the square root of the distance of
P
P
P
to one of the sides is equal to the sum of the square root of the distance of
P
P
P
to the two other sides.
5
1
Hide problems
f(x+f(x)+2f(y))=f(2x)+f(2y); f is surjective
Find all surjective functions
f
:
R
→
R
f: \mathbb R \to \mathbb R
f
:
R
→
R
such that for every
x
,
y
∈
R
,
x,y\in \mathbb R,
x
,
y
∈
R
,
we have
f
(
x
+
f
(
x
)
+
2
f
(
y
)
)
=
f
(
2
x
)
+
f
(
2
y
)
.
f(x+f(x)+2f(y))=f(2x)+f(2y).
f
(
x
+
f
(
x
)
+
2
f
(
y
))
=
f
(
2
x
)
+
f
(
2
y
)
.
3
1
Hide problems
'n' points on a circle, with 'intervals' and 'sub-intervals'
There are
n
n
n
points on a circle (
n
>
1
n>1
n
>
1
). Define an "interval" as an arc of a circle such that it's start and finish are from those points. Consider a family of intervals
F
F
F
such that for every element of
F
F
F
like
A
A
A
there is almost one other element of
F
F
F
like
B
B
B
such that
A
⊆
B
A \subseteq B
A
⊆
B
(in this case we call
A
A
A
is sub-interval of
B
B
B
). We call an interval maximal if it is not a sub-interval of any other interval. If
m
m
m
is the number of maximal elements of
F
F
F
and
a
a
a
is number of non-maximal elements of
F
,
F,
F
,
prove that
n
≥
m
+
a
2
.
n\geq m+\frac a2.
n
≥
m
+
2
a
.
2
1
Hide problems
n>=2, a_i are natural numbers; and an arithmetic progression
Find all natural numbers
n
n
n
greater than
2
2
2
such that there exist
n
n
n
natural numbers
a
1
,
a
2
,
…
,
a
n
a_{1},a_{2},\ldots,a_{n}
a
1
,
a
2
,
…
,
a
n
such that they are not all equal, and the sequence
a
1
a
2
,
a
2
a
3
,
…
,
a
n
a
1
a_{1}a_{2},a_{2}a_{3},\ldots,a_{n}a_{1}
a
1
a
2
,
a
2
a
3
,
…
,
a
n
a
1
forms an arithmetic progression with nonzero common difference.
6
1
Hide problems
hardest!
The circle
ω
\omega
ω
with center
O
O
O
has given. From an arbitrary point
T
T
T
outside of
ω
\omega
ω
draw tangents
T
B
TB
TB
and
T
C
TC
TC
to it.
K
K
K
and
H
H
H
are on
T
B
TB
TB
and
T
C
TC
TC
respectively.a)
B
′
B'
B
′
and
C
′
C'
C
′
are the second intersection point of
O
B
OB
OB
and
O
C
OC
OC
with
ω
\omega
ω
respectively.
K
′
K'
K
′
and
H
′
H'
H
′
are on angle bisectors of
∠
B
C
O
\angle BCO
∠
BCO
and
∠
C
B
O
\angle CBO
∠
CBO
respectively such that
K
K
′
⊥
B
C
KK' \bot BC
K
K
′
⊥
BC
and
H
H
′
⊥
B
C
HH'\bot BC
H
H
′
⊥
BC
. Prove that
K
,
H
′
,
B
′
K,H',B'
K
,
H
′
,
B
′
are collinear if and only if
H
,
K
′
,
C
′
H,K',C'
H
,
K
′
,
C
′
are collinear.b) Consider there exist two circle in
T
B
C
TBC
TBC
such that they are tangent two each other at
J
J
J
and both of them are tangent to
ω
\omega
ω
.and one of them is tangent to
T
B
TB
TB
at
K
K
K
and other one is tangent to
T
C
TC
TC
at
H
H
H
. Prove that two quadrilateral
B
K
J
I
BKJI
B
K
J
I
and
C
H
J
I
CHJI
C
H
J
I
are cyclic (
I
I
I
is incenter of triangle
O
B
C
OBC
OBC
).
4
1
Hide problems
good set! sit! stay!
Define a finite set
A
A
A
to be 'good' if it satisfies the following conditions:
*
For every three disjoint element of
A
,
A,
A
,
like
a
,
b
,
c
a,b,c
a
,
b
,
c
we have
gcd
(
a
,
b
,
c
)
=
1
;
\gcd(a,b,c)=1;
g
cd
(
a
,
b
,
c
)
=
1
;
*
For every two distinct
b
,
c
∈
A
,
b,c\in A,
b
,
c
∈
A
,
there exists an
a
∈
A
,
a\in A,
a
∈
A
,
distinct from
b
,
c
b,c
b
,
c
such that
b
c
bc
b
c
is divisible by
a
.
a.
a
.
Find all good sets.
1
1
Hide problems
Show that TA=TM
In acute triangle
A
B
C
ABC
A
BC
angle
B
B
B
is greater than
C
C
C
. Let
M
M
M
is midpoint of
B
C
BC
BC
.
D
D
D
and
E
E
E
are the feet of the altitude from
C
C
C
and
B
B
B
respectively.
K
K
K
and
L
L
L
are midpoint of
M
E
ME
ME
and
M
D
MD
M
D
respectively. If
K
L
KL
K
L
intersect the line through
A
A
A
parallel to
B
C
BC
BC
in
T
T
T
, prove that
T
A
=
T
M
TA=TM
T
A
=
TM
.