MathDB
Problems
Contests
National and Regional Contests
India Contests
India STEMS
2024 India STEMS
STEMS 2024 Math Cat A
STEMS 2024 Math Cat A
Part of
2024 India STEMS
Subcontests
(6)
P3
1
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AI_A and (S,SB) intersect at X, Y. prove that BCX = ACY
Let
A
B
C
ABC
A
BC
be a triangle. Let
I
I
I
be the Incenter of
A
B
C
ABC
A
BC
and
S
S
S
be the midpoint of arc
B
A
C
BAC
B
A
C
. Define
I
A
IA
I
A
as the
A
A
A
-excenter wrt
A
B
C
ABC
A
BC
. Define
ω
\omega
ω
to be the circle centred at
S
S
S
with radius
S
B
SB
SB
. Let
A
I
A
∩
ω
=
X
AI_A \cap \omega = X
A
I
A
∩
ω
=
X
,
Y
Y
Y
. Show that
∠
B
C
X
=
∠
A
C
Y
\angle BCX = \angle ACY
∠
BCX
=
∠
A
C
Y
.
P1
1
Hide problems
find all fancy pairs of (a,b)
Let
n
n
n
be a positive integer and
S
=
{
m
∣
2
n
≤
m
<
2
n
+
1
}
S = \{ m \mid 2^n \le m < 2^{n+1} \}
S
=
{
m
∣
2
n
≤
m
<
2
n
+
1
}
. We call a pair of non-negative integers
(
a
,
b
)
(a, b)
(
a
,
b
)
fancy if
a
+
b
a + b
a
+
b
is in
S
S
S
and is a palindrome in binary. Find the number of fancy pairs
(
a
,
b
)
(a, b)
(
a
,
b
)
.
P5
1
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find the maximum value of t so that b_n becomes unbounded
Let
r
r
r
,
s
s
s
be real numbers, find maximum
t
t
t
so that if
a
1
,
a
2
,
…
a_1, a_2, \ldots
a
1
,
a
2
,
…
is a sequence of positive real numbers satisfying
a
1
r
+
a
2
r
+
⋯
+
a
n
r
≤
2023
⋅
n
t
a_1^r + a_2^r + \cdots + a_n^r \le 2023 \cdot n^t
a
1
r
+
a
2
r
+
⋯
+
a
n
r
≤
2023
⋅
n
t
for all
n
≥
2023
n \ge 2023
n
≥
2023
then the sum
b
n
=
1
a
1
s
+
⋯
+
1
a
n
s
b_n = \frac 1{a_1^s} + \cdots + \frac 1{a_n^s}
b
n
=
a
1
s
1
+
⋯
+
a
n
s
1
is unbounded, i.e for all positive reals
M
M
M
there is an
n
n
n
such that
b
n
>
M
b_n > M
b
n
>
M
.
P2
1
Hide problems
find all (p,q) that forms Z^2
Let
S
=
Z
×
Z
S = \mathbb Z \times \mathbb Z
S
=
Z
×
Z
. A subset
P
P
P
of
S
S
S
is called nice if[*]
(
a
,
b
)
∈
P
⟹
(
b
,
a
)
∈
P
(a, b) \in P \implies (b, a) \in P
(
a
,
b
)
∈
P
⟹
(
b
,
a
)
∈
P
[*]
(
a
,
b
)
(a, b)
(
a
,
b
)
,
(
c
,
d
)
∈
P
⟹
(
a
+
c
,
b
−
d
)
∈
P
(c, d)\in P \implies (a + c, b - d) \in P
(
c
,
d
)
∈
P
⟹
(
a
+
c
,
b
−
d
)
∈
P
Find all
(
p
,
q
)
∈
S
(p, q) \in S
(
p
,
q
)
∈
S
so that if
(
p
,
q
)
∈
P
(p, q) \in P
(
p
,
q
)
∈
P
for some nice set
P
P
P
then
P
=
S
P = S
P
=
S
.
P6
1
Hide problems
Orthocentre,mixed with median and symmedian
Let ABC with orthocenter
H
H
H
and circumcenter
O
O
O
be an acute scalene triangle satisfying
A
B
=
A
M
AB = AM
A
B
=
A
M
where
M
M
M
is the midpoint of
B
C
BC
BC
. Suppose
Q
Q
Q
and
K
K
K
are points on
(
A
B
C
)
(ABC)
(
A
BC
)
distinct from A satisfying
∠
A
Q
H
=
90
\angle AQH = 90
∠
A
Q
H
=
90
and
∠
B
A
K
=
∠
C
A
M
\angle BAK = \angle CAM
∠
B
A
K
=
∠
C
A
M
. Let
N
N
N
be the midpoint of
A
H
AH
A
H
. • Let
I
I
I
be the intersection of
B
-midline
B\text{-midline}
B
-midline
and
A
-altitude
A\text{-altitude}
A
-altitude
Prove that
I
N
=
I
O
IN = IO
I
N
=
I
O
. • Prove that there is point
P
P
P
on the symmedian lying on circle with center
B
B
B
and radius
B
M
BM
BM
such that
(
A
P
N
)
(APN)
(
A
PN
)
is tangent to
A
B
AB
A
B
.Proposed by Krutarth Shah
P4
1
Hide problems
maximum number of people in CMI
In CMI, each person has atmost
3
3
3
friends. A disease has infected exactly
2023
2023
2023
peoplein CMI . Each day, a person gets infected if and only if atleast two of their friends were infected on the previous day. Once someone is infected, they can neither die nor be cured. Given that everyone in CMI eventually got infected, what is the maximum possible number of people in CMI?