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National and Regional Contests
Estonia Contests
SAFEST (South Africa - Estonia) Olympiad
2022 SAFEST Olympiad
2022 SAFEST Olympiad
Part of
SAFEST (South Africa - Estonia) Olympiad
Subcontests
(2)
2
1
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Integers and divisibility
Let
n
≥
2
n \geq 2
n
≥
2
be an integer. Prove that if
n
2
+
4
n
+
7
n
n
\frac{n^2+4^n+7^n}{n}
n
n
2
+
4
n
+
7
n
is an integer, then it is divisible by 11.
5
1
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Reciprocal tangencies
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral such that the circle with diameter
A
B
AB
A
B
is tangent to the line
C
D
CD
C
D
, and the circle with diameter
C
D
CD
C
D
is tangent to the line
A
B
AB
A
B
. Prove that the two intersection points of these circles and the point
A
C
∩
B
D
AC \cap BD
A
C
∩
B
D
are collinear.