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National and Regional Contests
Honduras Contests
Honduran Mathematical Olympiad
2021 Honduras National Mathematical Olympiad
2021 Honduras National Mathematical Olympiad
Part of
Honduran Mathematical Olympiad
Subcontests
(5)
Problem 5
1
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Honduran National MO (OHM) Intermediate Problem 5
A positive integer
m
m
m
is called growing if its digits, read from left to right, are non-increasing. Prove that for each natural number
n
n
n
there exists a growing number
m
m
m
with
n
n
n
digits such that the sum of its digits is a perfect square.
Problem 4
1
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Honduran National MO (OHM) Intermediate Problem 4
Consider parallelogram
A
B
C
D
ABCD
A
BC
D
and let
E
E
E
be the midpoint of
B
C
BC
BC
. In segment
D
E
DE
D
E
a point
F
F
F
is chosen such that
A
F
AF
A
F
is perpendicular to
D
E
DE
D
E
. Prove that
∠
C
D
E
=
∠
E
F
B
\angle CDE=\angle EFB
∠
C
D
E
=
∠
EFB
.
Problem 3
1
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Honduran National MO (OHM) Intermediate Problem 3
Let
a
a
a
and
b
b
b
be positive integers satisfying
a
a
−
2
=
b
+
2021
b
+
2008
\frac a{a-2} = \frac{b+2021}{b+2008}
a
−
2
a
=
b
+
2008
b
+
2021
Find the maximum value
a
b
\dfrac ab
b
a
can attain.
Problem 2
1
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Honduran National MO (OHM) Intermediate Problem 2
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be real numbers such that
a
2
+
b
2
=
1
,
c
2
+
d
2
=
1
a^2+b^2=1,c^2+d^2=1
a
2
+
b
2
=
1
,
c
2
+
d
2
=
1
and
a
c
+
b
d
=
0
ac+bd=0
a
c
+
b
d
=
0
. Determine all possible values of
a
b
+
c
d
ab+cd
ab
+
c
d
.
Problem 1
1
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Honduran National MO (OHM) 2021 Intermediate Problem 1
In a circle,
15
15
15
equally spaced points are drawn and arbitrary triangles are formed connecting
3
3
3
of these points. How many non-congruent triangles can be drawn?