MathDB
Problems
Contests
International Contests
India Iran Friendly Math Competition
2024 India Iran Friendly Math Competition
2024 India Iran Friendly Math Competition
Part of
India Iran Friendly Math Competition
Subcontests
(6)
4
1
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Diophantine No Sols
Prove that there are no integers
x
,
y
,
z
x, y, z
x
,
y
,
z
satisfying the equation
x
2
+
y
2
−
z
2
=
x
y
z
−
2.
x^2+y^2-z^2=xyz-2.
x
2
+
y
2
−
z
2
=
x
yz
−
2.
Proposed by Navid Safaei
5
1
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Filling Boxes
Let
n
≥
k
n \geq k
n
≥
k
be positive integers and let
a
1
,
…
,
a
n
a_1, \dots, a_n
a
1
,
…
,
a
n
be a non-increasing list of positive real numbers. Prove that there exists
k
k
k
sets
B
1
,
…
,
B
k
B_1, \dots, B_k
B
1
,
…
,
B
k
which partition the set
{
1
,
2
,
…
,
n
}
\{1, 2, \dots, n\}
{
1
,
2
,
…
,
n
}
such that
min
1
≤
j
≤
k
(
∑
i
∈
B
j
a
i
)
≥
min
1
≤
j
≤
k
(
1
2
k
+
1
−
2
j
⋅
∑
i
=
j
n
a
i
)
.
\min_{1 \le j \le k} \left(\sum_{i \in B_j} a_i \right) \geq \min_{1 \le j \le k} \left(\frac{1}{2k+1-2j} \cdot \sum^n_{i=j} a_i\right).
1
≤
j
≤
k
min
i
∈
B
j
∑
a
i
≥
1
≤
j
≤
k
min
(
2
k
+
1
−
2
j
1
⋅
i
=
j
∑
n
a
i
)
.
Proposed by Navid Safaei
6
1
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Immaculate Geometry
Let
A
B
C
ABC
A
BC
be a triangle with midpoint
M
M
M
of
B
C
BC
BC
. A point
X
X
X
is called immaculate if the perpendicular line from
X
X
X
to line
M
X
MX
MX
intersects lines
A
B
AB
A
B
and
A
C
AC
A
C
at two points that are equidistant from
M
M
M
. Suppose
U
,
V
,
W
U, V, W
U
,
V
,
W
are three immaculate points on the circumcircle of triangle
A
B
C
ABC
A
BC
. Prove that
M
M
M
is the incentre of
△
U
V
W
\triangle UVW
△
U
VW
. Proposed by Pranjal Srivastava and Rohan Goyal
1
1
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Balanced Tournaments
A league consists of
2024
2024
2024
players. A round involves splitting the players into two different teams and having every member of one team play with every member of the other team. A round is called balanced if both teams have an equal number of players. A tournament consists of several rounds at the end of which any two players have played each other. The committee organised a tournament last year which consisted of
N
N
N
rounds. Prove that the committee can organise a tournament this year with
N
N
N
balanced rounds.Proposed by Anant Mudgal and Navilarekallu Tejaswi
3
1
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Polygon vanishing locus of polynomial
Let
n
≥
3
n \ge 3
n
≥
3
be an integer. Let
P
\mathcal{P}
P
denote the set of vertices of a regular
n
n
n
-gon on the plane. A polynomial
f
(
x
,
y
)
f(x, y)
f
(
x
,
y
)
of two variables with real coefficients is called
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
r
e
g
u
l
a
r
<
/
s
p
a
n
>
<span class='latex-italic'>regular</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
re
gu
l
a
r
<
/
s
p
an
>
if
P
=
{
(
u
,
v
)
∈
R
2
∣
f
(
u
,
v
)
=
0
}
.
\mathcal{P} = \{(u, v) \in \mathbb{R}^2 \, | \, f(u, v) = 0 \}.
P
=
{(
u
,
v
)
∈
R
2
∣
f
(
u
,
v
)
=
0
}
.
Find the smallest possible value of the degree of a regular polynomial.Proposed by Navid Safaei
2
1
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Incenters concyclic hence collinear
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral with circumcentre
O
1
O_1
O
1
. The diagonals
A
C
AC
A
C
and
B
D
BD
B
D
meet at point
P
P
P
. Suppose the four incentres of triangles
P
A
B
,
P
B
C
,
P
C
D
,
P
D
A
PAB, PBC, PCD, PDA
P
A
B
,
PBC
,
PC
D
,
P
D
A
lie on a circle with centre
O
2
O_2
O
2
. Prove that
P
,
O
1
,
O
2
P, O_1, O_2
P
,
O
1
,
O
2
are collinear.Proposed by Shantanu Nene