MathDB
Problems
Contests
National and Regional Contests
Colombia Contests
Colombia National Olympiad
2020 Colombia National Olympiad
2020 Colombia National Olympiad
Part of
Colombia National Olympiad
Subcontests
(6)
6
1
Hide problems
n+i stones at P_i points on a circle
Let
k
k
k
be a positive integer and
n
1
,
n
2
,
.
.
.
,
n
k
n_1, n_2, ..., n_k
n
1
,
n
2
,
...
,
n
k
be non-negative integers. Points
P
1
,
P
2
,
.
.
.
,
P
k
P_1, P_2, ..., P_k
P
1
,
P
2
,
...
,
P
k
lie on a circle in such a way that at point
P
i
P_i
P
i
there are
n
i
n_i
n
i
stones. Leandro wishes to change the position of some of these stones in order to accomplish his objective which is to have the same number of stones at each point of the circle. He does this by repeating as many times as necessary the following operation: if there exists a point on the circle with at least
k
−
1
k - 1
k
−
1
stones, he can choose
k
−
1
k -1
k
−
1
of these and distribute them by giving one to each of the remaining
k
−
1
k - 1
k
−
1
points. For which values
n
1
,
n
2
,
.
.
.
,
n
k
n_1, n_2, ..., n_k
n
1
,
n
2
,
...
,
n
k
can Leandro accomplish his objective?In the figure below there is a configuration of stones for
k
=
4
k = 4
k
=
4
. On the right is the initial division of stones, while on the left there is the configuration obtained from the initial one by choosing
k
−
1
=
3
k - 1 = 3
k
−
1
=
3
stones from the top point on the circle and distributing one each to the other points.[figures missing]
4
1
Hide problems
a_{b_n} is arithmentic progression , b_n | b_{n+1}
Find all of the sequences
a
1
,
a
2
,
a
3
,
.
.
.
a_1, a_2, a_3, . . .
a
1
,
a
2
,
a
3
,
...
of real numbers that satisfy the following property: given any sequence
b
1
,
b
2
,
b
3
,
.
.
.
b_1, b_2, b_3, . . .
b
1
,
b
2
,
b
3
,
...
of positive integers such that for all
n
≥
1
n \ge 1
n
≥
1
we have
b
n
≠
b
n
+
1
b_n \ne b_{n+1}
b
n
=
b
n
+
1
and
b
n
∣
b
n
+
1
b_n | b_{n+1}
b
n
∣
b
n
+
1
, then the sub-sequence
a
b
1
,
a
b
2
,
a
b
3
,
.
.
.
a_{b_1}, a_{b_2}, a_{b_3}, . . .
a
b
1
,
a
b
2
,
a
b
3
,
...
is an arithmetic progression.
3
1
Hide problems
retriangular and triangular numbers , aT +b
A number is said to be triangular if it can be expressed in the form
1
+
2
+
.
.
.
+
n
1 + 2 +...+n
1
+
2
+
...
+
n
for some positive integer
n
n
n
. We call a positive integer
a
a
a
retriangular if there exists a fixed positive integer
b
b
b
such that
a
T
+
b
aT +b
a
T
+
b
is a triangular number whenever
T
T
T
is a triangular number. Determine all retriangular numbers.
2
1
Hide problems
max no of diagonals that Maria can draw in such a way inside a regular n-gon
Given a regular
n
n
n
-sided polygon with
n
≥
3
n \ge 3
n
≥
3
. Maria draws some of its diagonals in such a way that each diagonal intersects at most one of the other diagonals drawn in the interior of the polygon. Determine the maximum number of diagonals that Maria can draw in such a way.Note: Two diagonals can share a vertex of the polygon. Vertices are not part of the interior of the polygon.
1
1
Hide problems
2-digit sabroso numbers wanted, perfect square
A positive integer is called sabroso if when it is added to the number obtained when its digits are interchanged from one side of its written form to the other, the result is a perfect square. For example,
143
143
143
is sabroso, since
143
+
341
=
484
=
2
2
2
143 + 341 =484 = 22^2
143
+
341
=
484
=
2
2
2
. Find all two-digit sabroso numbers.
5
1
Hide problems
Geometry from Colombia
Given an acute-angled triangle
A
B
C
ABC
A
BC
with
D
D
D
is the foot of the altitude from
A
.
A.
A
.
The perpendicular lines to
B
C
BC
BC
through
B
B
B
and
C
C
C
intersect the altitudes from
C
C
C
and
B
B
B
at points
M
M
M
and
N
N
N
, respectively. Show that
A
D
AD
A
D
=
=
=
B
C
BC
BC
if and only if
A
,
M
,
N
A,M,N
A
,
M
,
N
and
D
D
D
lie on the same circle.