MathDB
Problems
Contests
National and Regional Contests
Iran Contests
Iran Team Selection Test
2015 Iran Team Selection Test
2015 Iran Team Selection Test
Part of
Iran Team Selection Test
Subcontests
(6)
1
3
Hide problems
Easy polynomial
Find all polynomials
P
,
Q
∈
Q
[
x
]
P,Q\in \Bbb{Q}\left [ x \right ]
P
,
Q
∈
Q
[
x
]
such that
P
(
x
)
3
+
Q
(
x
)
3
=
x
12
+
1.
P(x)^3+Q(x)^3=x^{12}+1.
P
(
x
)
3
+
Q
(
x
)
3
=
x
12
+
1.
4 Variable Inequality
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
are positive numbers such that
∑
c
y
c
1
a
b
=
1
\sum_{cyc} \frac{1}{ab} =1
∑
cyc
ab
1
=
1
. Prove that :
a
b
c
d
+
16
≥
8
(
a
+
c
)
(
1
a
+
1
c
)
+
8
(
b
+
d
)
(
1
b
+
1
d
)
abcd+16 \geq 8 \sqrt{(a+c)(\frac{1}{a} + \frac{1}{c})}+8\sqrt{(b+d)(\frac{1}{b}+\frac{1}{d})}
ab
c
d
+
16
≥
8
(
a
+
c
)
(
a
1
+
c
1
)
+
8
(
b
+
d
)
(
b
1
+
d
1
)
easy geometry
Point
A
A
A
is outside of a given circle
ω
\omega
ω
. Let the tangents from
A
A
A
to
ω
\omega
ω
meet
ω
\omega
ω
at
S
,
T
S, T
S
,
T
points
X
,
Y
X, Y
X
,
Y
are midpoints of
A
T
,
A
S
AT, AS
A
T
,
A
S
let the tangent from
X
X
X
to
ω
\omega
ω
meet
ω
\omega
ω
at
R
≠
T
R\neq T
R
=
T
. points
P
,
Q
P, Q
P
,
Q
are midpoints of
X
T
,
X
R
XT, XR
XT
,
XR
let
X
Y
∩
P
Q
=
K
,
S
X
∩
T
K
=
L
XY\cap PQ=K, SX\cap TK=L
X
Y
∩
PQ
=
K
,
SX
∩
T
K
=
L
prove that quadrilateral
K
R
L
Q
KRLQ
K
R
L
Q
is cyclic.
5
3
Hide problems
Edges in a Table
Let
A
A
A
be a subset of the edges of an
n
×
n
n\times n
n
×
n
table. Let
V
(
A
)
V(A)
V
(
A
)
be the set of vertices from the table which are connected to at least on edge from
A
A
A
and
j
(
A
)
j(A)
j
(
A
)
be the number of the connected components of graph
G
G
G
which it's vertices are the set
V
(
A
)
V(A)
V
(
A
)
and it's edges are the set
A
A
A
. Prove that for every natural number
l
l
l
:
l
2
≤
m
i
n
∣
A
∣
≥
l
(
∣
V
(
A
)
∣
−
j
(
A
)
)
≤
l
2
+
l
2
+
1
\frac{l}{2}\leq min_{|A|\geq l}(|V(A)|-j(A)) \leq \frac{l}{2}+\sqrt{\frac{l}{2}}+1
2
l
≤
mi
n
∣
A
∣
≥
l
(
∣
V
(
A
)
∣
−
j
(
A
))
≤
2
l
+
2
l
+
1
Good permutations
We call a permutation
(
a
1
,
a
2
,
⋯
,
a
n
)
(a_1, a_2,\cdots , a_n)
(
a
1
,
a
2
,
⋯
,
a
n
)
of the set
{
1
,
2
,
⋯
,
n
}
\{ 1,2,\cdots, n\}
{
1
,
2
,
⋯
,
n
}
"good" if for any three natural numbers
i
<
j
<
k
i <j <k
i
<
j
<
k
,
n
∤
a
i
+
a
k
−
2
a
j
n\nmid a_i+a_k-2a_j
n
∤
a
i
+
a
k
−
2
a
j
find all natural numbers
n
≥
3
n\ge 3
n
≥
3
such that there exist a "good" permutation of a set
{
1
,
2
,
⋯
,
n
}
\{1,2,\cdots, n\}
{
1
,
2
,
⋯
,
n
}
.
Iran TST 2015 Polynomial
Prove that for each natural number
d
d
d
, There is a monic and unique polynomial of degree
d
d
d
like
P
P
P
such that
P
(
1
)
P(1)
P
(
1
)
≠
0
0
0
and for each sequence like
a
1
a_{1}
a
1
,
a
2
a_{2}
a
2
,
.
.
.
...
...
of real numbers that the recurrence relation below is true for them, there is a natural number
k
k
k
such that
0
=
a
k
=
a
k
+
1
=
.
.
.
0=a_{k}=a_{k+1}= ...
0
=
a
k
=
a
k
+
1
=
...
:
P
(
n
)
a
1
+
P
(
n
−
1
)
a
2
+
.
.
.
+
P
(
1
)
a
n
=
0
P(n)a_{1} + P(n-1)a_{2} + ... + P(1)a_{n}=0
P
(
n
)
a
1
+
P
(
n
−
1
)
a
2
+
...
+
P
(
1
)
a
n
=
0
n
>
1
n>1
n
>
1
3
3
Hide problems
Rectangles into each other
Find the maximum number of rectangles with sides equal to 1 and 2 and parallel to the coordinate axes such that each two have an area equal to 1 in common.
Sequence of numbers in form of a^2+b^2
Let
b
1
<
b
2
<
b
3
<
…
b_1<b_2<b_3<\dots
b
1
<
b
2
<
b
3
<
…
be the sequence of all natural numbers which are sum of squares of two natural numbers. Prove that there exists infinite natural numbers like
m
m
m
which
b
m
+
1
−
b
m
=
2015
b_{m+1}-b_m=2015
b
m
+
1
−
b
m
=
2015
.
iran TST
a
1
,
a
2
,
⋯
,
a
n
,
b
1
,
b
2
,
⋯
,
b
n
a_1,a_2,\cdots ,a_n,b_1,b_2,\cdots ,b_n
a
1
,
a
2
,
⋯
,
a
n
,
b
1
,
b
2
,
⋯
,
b
n
are
2
n
2n
2
n
positive real numbers such that
a
1
,
a
2
,
⋯
,
a
n
a_1,a_2,\cdots ,a_n
a
1
,
a
2
,
⋯
,
a
n
aren't all equal. And assume that we can divide
a
1
,
a
2
,
⋯
,
a
n
a_1,a_2,\cdots ,a_n
a
1
,
a
2
,
⋯
,
a
n
into two subsets with equal sums.similarly
b
1
,
b
2
,
⋯
,
b
n
b_1,b_2,\cdots ,b_n
b
1
,
b
2
,
⋯
,
b
n
have these two conditions. Prove that there exist a simple
2
n
2n
2
n
-gon with sides
a
1
,
a
2
,
⋯
,
a
n
,
b
1
,
b
2
,
⋯
,
b
n
a_1,a_2,\cdots ,a_n,b_1,b_2,\cdots ,b_n
a
1
,
a
2
,
⋯
,
a
n
,
b
1
,
b
2
,
⋯
,
b
n
and parallel to coordinate axises Such that the lengths of horizontal sides are among
a
1
,
a
2
,
⋯
,
a
n
a_1,a_2,\cdots ,a_n
a
1
,
a
2
,
⋯
,
a
n
and the lengths of vertical sides are among
b
1
,
b
2
,
⋯
,
b
n
b_1,b_2,\cdots ,b_n
b
1
,
b
2
,
⋯
,
b
n
.(simple polygon is a polygon such that it doesn't intersect itself)
6
3
Hide problems
Hard geometry
A
B
C
D
ABCD
A
BC
D
is a circumscribed and inscribed quadrilateral.
O
O
O
is the circumcenter of the quadrilateral.
E
,
F
E,F
E
,
F
and
S
S
S
are the intersections of
A
B
,
C
D
AB,CD
A
B
,
C
D
,
A
D
,
B
C
AD,BC
A
D
,
BC
and
A
C
,
B
D
AC,BD
A
C
,
B
D
respectively.
E
′
E'
E
′
and
F
′
F'
F
′
are points on
A
D
AD
A
D
and
A
B
AB
A
B
such that
A
E
^
E
′
=
E
′
E
^
D
A\hat{E}E'=E'\hat{E}D
A
E
^
E
′
=
E
′
E
^
D
and
A
F
^
F
′
=
F
′
F
^
B
A\hat{F}F'=F'\hat{F}B
A
F
^
F
′
=
F
′
F
^
B
.
X
X
X
and
Y
Y
Y
are points on
O
E
′
OE'
O
E
′
and
O
F
′
OF'
O
F
′
such that
X
A
X
D
=
E
A
E
D
\frac{XA}{XD}=\frac{EA}{ED}
X
D
X
A
=
E
D
E
A
and
Y
A
Y
B
=
F
A
F
B
\frac{YA}{YB}=\frac{FA}{FB}
Y
B
Y
A
=
FB
F
A
.
M
M
M
is the midpoint of arc
B
D
BD
B
D
of
(
O
)
(O)
(
O
)
which contains
A
A
A
. Prove that the circumcircles of triangles
O
X
Y
OXY
OX
Y
and
O
A
M
OAM
O
A
M
are coaxal with the circle with diameter
O
S
OS
OS
.
Inequality
If
a
,
b
,
c
a,b,c
a
,
b
,
c
are positive real numbers such that
a
+
b
+
c
=
a
b
c
a+b+c=abc
a
+
b
+
c
=
ab
c
prove that
a
b
c
3
2
(
∑
c
y
c
a
3
+
b
3
a
b
+
1
)
≥
∑
c
y
c
a
a
2
+
1
\frac{abc}{3\sqrt{2}}\left ( \sum_{cyc}\frac{\sqrt{a^3+b^3}}{ab+1} \right )\geq \sum_{cyc}\frac{a}{a^2+1}
3
2
ab
c
(
cyc
∑
ab
+
1
a
3
+
b
3
)
≥
cyc
∑
a
2
+
1
a
Iran TST 2015 Geometry
A
H
AH
A
H
is the altitude of triangle
A
B
C
ABC
A
BC
and
H
′
H^\prime
H
′
is the reflection of
H
H
H
trough the midpoint of
B
C
BC
BC
. If the tangent lines to the circumcircle of
A
B
C
ABC
A
BC
at
B
B
B
and
C
C
C
, intersect each other at
X
X
X
and the perpendicular line to
X
H
′
XH^\prime
X
H
′
at
H
′
H^\prime
H
′
, intersects
A
B
AB
A
B
and
A
C
AC
A
C
at
Y
Y
Y
and
Z
Z
Z
respectively, prove that
∠
Z
X
C
=
∠
Y
X
B
\angle ZXC=\angle YXB
∠
ZXC
=
∠
Y
XB
.
4
3
Hide problems
Find the least k
n
n
n
is a fixed natural number. Find the least
k
k
k
such that for every set
A
A
A
of
k
k
k
natural numbers, there exists a subset of
A
A
A
with an even number of elements which the sum of it's members is divisible by
n
n
n
.
x,y,z collinear
Let
△
A
B
C
\triangle ABC
△
A
BC
be an acute triangle. Point
Z
Z
Z
is on
A
A
A
altitude and points
X
X
X
and
Y
Y
Y
are on the
B
B
B
and
C
C
C
altitudes out of the triangle respectively, such that:
∠
A
Y
B
=
∠
B
Z
C
=
∠
C
X
A
=
90
\angle AYB=\angle BZC=\angle CXA=90
∠
A
Y
B
=
∠
BZC
=
∠
CX
A
=
90
Prove that
X
X
X
,
Y
Y
Y
and
Z
Z
Z
are collinear, if and only if the length of the tangent drawn from
A
A
A
to the nine point circle of
△
A
B
C
\triangle ABC
△
A
BC
is equal with the sum of the lengths of the tangents drawn from
B
B
B
and
C
C
C
to the nine point circle of
△
A
B
C
\triangle ABC
△
A
BC
.
Triangles into each other
Ali puts
5
5
5
points on the plane such that no three of them are collinear. Ramtin adds a sixth point that is not collinear with any two of the former points.Ali wants to eventually construct two triangles from the six points such that one can be placed inside another. Can Ali put the 5 points in such a manner so that he would always be able to construct the desired triangles? (We say that triangle
T
1
T_1
T
1
can be placed inside triangle
T
2
T_2
T
2
if
T
1
T_1
T
1
is congruent to a triangle that is located completely inside
T
2
T_2
T
2
.)
2
3
Hide problems
Easy Geometry
I
b
I_b
I
b
is the
B
B
B
-excenter of the triangle
A
B
C
ABC
A
BC
and
ω
\omega
ω
is the circumcircle of this triangle.
M
M
M
is the middle of arc
B
C
BC
BC
of
ω
\omega
ω
which doesn't contain
A
A
A
.
M
I
b
MI_b
M
I
b
meets
ω
\omega
ω
at
T
≠
M
T\not =M
T
=
M
. Prove that
T
B
⋅
T
C
=
T
I
b
2
.
TB\cdot TC=TI_b^2.
TB
⋅
TC
=
T
I
b
2
.
number theory
Assume that
a
1
,
a
2
,
a
3
a_1, a_2, a_3
a
1
,
a
2
,
a
3
are three given positive integers consider the following sequence:
a
n
+
1
=
lcm
[
a
n
,
a
n
−
1
]
−
lcm
[
a
n
−
1
,
a
n
−
2
]
a_{n+1}=\text{lcm}[a_n, a_{n-1}]-\text{lcm}[a_{n-1}, a_{n-2}]
a
n
+
1
=
lcm
[
a
n
,
a
n
−
1
]
−
lcm
[
a
n
−
1
,
a
n
−
2
]
for
n
≥
3
n\ge 3
n
≥
3
Prove that there exist a positive integer
k
k
k
such that
k
≤
a
3
+
4
k\le a_3+4
k
≤
a
3
+
4
and
a
k
≤
0
a_k\le 0
a
k
≤
0
. (
[
a
,
b
]
[a, b]
[
a
,
b
]
means the least positive integer such that
a
∣
[
a
,
b
]
,
b
∣
[
a
,
b
]
a\mid[a,b], b\mid[a, b]
a
∣
[
a
,
b
]
,
b
∣
[
a
,
b
]
also because
lcm
[
a
,
b
]
\text{lcm}[a, b]
lcm
[
a
,
b
]
takes only nonzero integers this sequence is defined until we find a zero number in the sequence)
easy geometry
In triangle
A
B
C
ABC
A
BC
(with incenter
I
I
I
) let the line parallel to
B
C
BC
BC
from
A
A
A
intersect circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
at
A
1
A_1
A
1
let
A
I
∩
B
C
=
D
AI\cap BC=D
A
I
∩
BC
=
D
and
E
E
E
is tangency point of incircle with
B
C
BC
BC
let
E
A
1
∩
⊙
(
△
A
D
E
)
=
T
EA_1\cap \odot (\triangle ADE)=T
E
A
1
∩
⊙
(
△
A
D
E
)
=
T
prove that
A
I
=
T
I
AI=TI
A
I
=
T
I
.