MathDB
Problems
Contests
National and Regional Contests
Myanmar Contests
Myanmar IMO Training
2024 Myanmar IMO Training
2024 Myanmar IMO Training
Part of
Myanmar IMO Training
Subcontests
(4)
6
1
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Number Theory Problem with Induction
Prove that for all integers
n
≥
3
n \geq 3
n
≥
3
, there exist odd positive integers
x
x
x
,
y
y
y
such that
7
x
2
+
y
2
=
2
n
7x^2 + y^2 = 2^n
7
x
2
+
y
2
=
2
n
.
8
1
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Excircle tangent to symmedian
Let
A
B
C
ABC
A
BC
be a triangle and let
X
X
X
and
Y
Y
Y
be points on the
A
A
A
-symmedian such that
A
X
=
X
B
AX = XB
A
X
=
XB
and
A
Y
=
Y
C
AY = YC
A
Y
=
Y
C
. Let
B
X
BX
BX
and
C
Y
CY
C
Y
meet at
Z
Z
Z
. Let the
Z
Z
Z
-excircle of triangle
X
Y
Z
XYZ
X
Y
Z
touch
Z
X
ZX
ZX
and
Z
Y
ZY
Z
Y
at
E
E
E
and
F
F
F
. Show that
A
A
A
,
E
E
E
,
F
F
F
are collinear.
2
1
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Minimum value of a ratio with given constraints
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive real numbers satisfying
a
+
b
+
c
=
a
2
+
b
2
+
c
2
.
a+b+c = a^2 + b^2 + c^2.
a
+
b
+
c
=
a
2
+
b
2
+
c
2
.
Let M = \max\left(\frac{2a^2}{b} + c, \frac{2b^2}{a} + c \right) \text{ and } N = \min(a^2 + b^2, c^2). Find the minimum possible value of
M
/
N
M/N
M
/
N
.
5
1
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Counting in a Fighting Game Club
A fighting game club has
2024
2024
2024
members. One day, a game of Smash is played between some pairs of members so that every member has played against exactly
3
3
3
other members. Each match has a winner and a loser. A member will be happy if they won in at least
2
2
2
of the matches. What is the maximum number of happy members over all possible match-ups and all possible outcomes?