MathDB
Problems
Contests
National and Regional Contests
Chile Contests
Chile Classification NMO
2012 Chile Classification NMO Seniors
2012 Chile Classification NMO Seniors
Part of
Chile Classification NMO
Subcontests
(1)
1
Hide problems
2012 Chile Classification / Qualifying NMO Seniors XXIV
p1. Show that if
a
,
b
,
c
a, b, c
a
,
b
,
c
are odd integers then the equation
a
x
2
+
b
x
+
c
=
0
ax^2 + bx + c = 0
a
x
2
+
b
x
+
c
=
0
has no rational roots. p2. If
k
k
k
is a positive integer, find the greatest power of
3
3
3
that divides
1
0
k
−
1
10^k-1
1
0
k
−
1
. p3. The figure shows the triangle
A
B
C
ABC
A
BC
, right at
C
C
C
, its circumscribed circle and semicircles built on the two legs. Show that the sum of the areas of the two shaded regions is
1
2
A
C
⋅
C
B
\frac12 AC\cdot CB
2
1
A
C
⋅
CB
. https://cdn.artofproblemsolving.com/attachments/b/1/bb5d58b24d2dc68efcd6aa218489fd379f709c.png p4. Each vertex of a cube is assigned the value
+
1
+1
+
1
or
−
1
-1
−
1
, and each face the product of the values assigned to its vertices. What values can the sum of the
14
14
14
numbers thus obtained, have? p5. Consider the regular pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
in the figure. If
B
I
BI
B
I
is
1
1
1
, how long is
A
B
AB
A
B
? https://cdn.artofproblemsolving.com/attachments/d/0/ed3ed73bda0d3fc4a1fb62c3cef5c829ccc937.jpg p6. Let
n
≥
3
n\ge 3
n
≥
3
be an integer. A circle is divided into
2
n
2n
2
n
arcs by
2
n
2n
2
n
points. Each arch measures one of three possible lengths, and no two adjacent arches are the same length. The
2
n
2n
2
n
points are alternately colored red and blue. Show that the
n
n
n
-gon with blue vertices and the
n
n
n
-gon with red vertices have the same perimeter and the same area. PS. Seniors P3,P4 were also posted as [url=https://artofproblemsolving.com/community/c4h2690796p23355830]Juniors P1, P5.