MathDB
Problems
Contests
National and Regional Contests
India Contests
India EGMO TST
India EGMO 2021 TST
India EGMO 2021 TST
Part of
India EGMO TST
Subcontests
(3)
6
1
Hide problems
a^2+b^2-ab is not a square
Let
n
>
2
n>2
n
>
2
be a positive integer and
b
=
2
2
n
b=2^{2^n}
b
=
2
2
n
. Let
a
a
a
be an odd positive integer such that
a
≤
b
≤
2
a
a\le b \le 2a
a
≤
b
≤
2
a
. Show that
a
2
+
b
2
−
a
b
a^2+b^2-ab
a
2
+
b
2
−
ab
is not a square.
3
1
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Weird angle condition geo
In acute
△
A
B
C
\triangle ABC
△
A
BC
with circumcircle
Γ
\Gamma
Γ
and incentre
I
I
I
, the incircle touches side
A
B
AB
A
B
at
F
F
F
. The external angle bisector of
∠
A
C
B
\angle ACB
∠
A
CB
meets ray
A
B
AB
A
B
at
L
L
L
. Point
K
K
K
lies on the arc
C
B
CB
CB
of
Γ
\Gamma
Γ
not containing
A
A
A
, such that
∠
C
K
I
=
∠
I
K
L
\angle CKI=\angle IKL
∠
C
K
I
=
∠
I
K
L
. Ray
K
I
KI
K
I
meets
Γ
\Gamma
Γ
again at
D
≠
K
D\ne K
D
=
K
. Prove that
∠
A
C
F
=
∠
D
C
B
\angle ACF =\angle DCB
∠
A
CF
=
∠
D
CB
.
1
1
Hide problems
Mad scientist writes numbers
Mad scientist Kyouma writes
N
N
N
positive integers on a board. Each second, he chooses two numbers
x
,
y
x, y
x
,
y
written on the board with
x
>
y
x > y
x
>
y
, and writes the number
x
2
−
y
2
x^2-y^2
x
2
−
y
2
on the board. After some time, he sends the list of all the numbers on the board to Christina. She notices that all the numbers from 1 to 1000 are present on the list. Aid Christina in finding the minimum possible value of N.