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Problems
Contests
National and Regional Contests
Sri Lanka Contests
Sri Lanka Mathematics Challenge Competition
Sri Lankan Mathematics Challenge Competition 2022
Sri Lankan Mathematics Challenge Competition 2022
Part of
Sri Lanka Mathematics Challenge Competition
Subcontests
(4)
P4
1
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Easy Radical Axis Problem
Problem 4 : A point
C
C
C
lies on a line segment
A
B
AB
A
B
between
A
A
A
and
B
B
B
and circles are drawn having
A
C
AC
A
C
and
C
B
CB
CB
as diameters. A common tangent line to both circles touches the circle with
A
C
AC
A
C
as diameter at
P
≠
C
P \neq C
P
=
C
and the circle with
C
B
CB
CB
as diameter at
Q
≠
C
.
Q \neq C.
Q
=
C
.
Prove that lines
A
P
,
B
Q
AP, BQ
A
P
,
BQ
and the common tangent line to both circles at
C
C
C
all meet at a single point which lies on the circle with
A
B
AB
A
B
as diameter.
P3
1
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Easiest Inequality ever
Problem 3 : Let
x
1
,
x
2
,
⋯
,
x
2022
x_1,x_2,\cdots,x_{2022}
x
1
,
x
2
,
⋯
,
x
2022
be non-negative real numbers such that
x
k
+
x
k
+
1
+
x
k
+
2
≤
2
x_k + x_{k+1}+x_{k+2} \leq 2
x
k
+
x
k
+
1
+
x
k
+
2
≤
2
for all
k
=
1
,
2
,
⋯
,
2020
k = 1,2,\cdots,2020
k
=
1
,
2
,
⋯
,
2020
. Prove that
∑
k
=
1
2020
x
k
x
k
+
2
≤
1010
\sum_{k=1}^{2020}x_kx_{k+2}\leq 1010
k
=
1
∑
2020
x
k
x
k
+
2
≤
1010
P2
1
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Go grid Go!
Problem 2 :
k
k
k
number of unit squares selected from a
99
×
99
99 \times 99
99
×
99
square grid are coloured using five colours Red, Blue, Yellow, Green and Black such that each colour appears the same number of times and on each row and on each column there are no differently coloured unit squares. Find the maximum possible value of
k
k
k
.
P1
1
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Integers and Square roots
Problem 1 : Find the smallest positive integer
n
n
n
, such that
5
n
5
\sqrt[5]{5n}
5
5
n
,
6
n
6
\sqrt[6]{6n}
6
6
n
,
7
n
7
\sqrt[7]{7n}
7
7
n
are integers.