MathDB
Problems
Contests
National and Regional Contests
Belgium Contests
Flanders Math Olympiad
2017 Flanders Math Olympiad
2017 Flanders Math Olympiad
Part of
Flanders Math Olympiad
Subcontests
(4)
3
1
Hide problems
S + V = H + 3 in a closed rectangular neighborhood
In a closed rectangular neighborhood there are:
S
S
S
streets (these are straight roads of maximum length),
V
V
V
four-arm intersections ( https://cdn.artofproblemsolving.com/attachments/e/4/6a5974a30dc182b59a519a8ef4eb4f1412e05e.png),
H
H
H
city blocks (these are rectangular areas bounded by four streets, which are no be intersected by another street) and
T
T
T
represents the number of
T
T
T
-intersections (https://cdn.artofproblemsolving.com/attachments/0/a/b390a30a0b27d83db681f70f633bdeed697163.png ). For example, in the neighborhood below, there are
15
15
15
streets,
8
8
8
four-arm intersections,
20
20
20
city blocks and
22
22
22
T
T
T
-intersections. https://cdn.artofproblemsolving.com/attachments/a/2/c1a5e463d0fb5671ac0702c91cfc2272d4e2c3.png Prove that in each district
S
+
V
=
H
+
3
S + V = H + 3
S
+
V
=
H
+
3
.
4
1
Hide problems
new sumbol, n'
For every natural number
n
n
n
we define the derived number
n
′
n'
n
′
as follows:
∙
\bullet
∙
0
′
=
1
′
=
0
0' = 1' = 0
0
′
=
1
′
=
0
∙
\bullet
∙
if
n
n
n
is prime, then
n
′
=
1
n' = 1
n
′
=
1
∙
\bullet
∙
if
n
=
a
⋅
b
n = a \cdot b
n
=
a
⋅
b
, then
n
′
=
a
′
b
+
a
b
′
n' = a' b + a b'
n
′
=
a
′
b
+
a
b
′
. For example:
1
5
′
=
3
′
5
+
3
5
′
=
1
⋅
5
+
3
⋅
1
=
8
15' = 3' 5 + 3 5' = 1\cdot 5 + 3\cdot 1 = 8
1
5
′
=
3
′
5
+
3
5
′
=
1
⋅
5
+
3
⋅
1
=
8
. Determine all natural numbers
n
n
n
for which
n
=
n
′
n = n'
n
=
n
′
.
1
1
Hide problems
equal projections from y=x^2 on x-axis, triangle area (2017 VWO Flanders MO p1)
On the parabola
y
=
x
2
y = x^2
y
=
x
2
lie three different points
P
,
Q
P, Q
P
,
Q
and
R
R
R
. Their projections
P
′
,
Q
′
P', Q'
P
′
,
Q
′
and
R
′
R'
R
′
on the
x
x
x
-axis are equidistant and equal to
s
s
s
, i.e.
∣
P
′
Q
′
∣
=
∣
Q
′
R
′
∣
=
s
| P'Q'| = | Q'R'| = s
∣
P
′
Q
′
∣
=
∣
Q
′
R
′
∣
=
s
. Determine the area of
△
P
Q
R
\vartriangle PQR
△
PQR
in terms of
s
s
s
2
1
Hide problems
angle chasing, 3 circumcircles, starting with a 50-60-70 triangle
In triangle
△
A
B
C
\vartriangle ABC
△
A
BC
,
∠
A
=
5
0
o
,
∠
B
=
6
0
o
\angle A = 50^o, \angle B = 60^o
∠
A
=
5
0
o
,
∠
B
=
6
0
o
and
∠
C
=
7
0
o
\angle C = 70^o
∠
C
=
7
0
o
. The point
P
P
P
is on the side
[
A
B
]
[AB]
[
A
B
]
(with
P
≠
A
P \ne A
P
=
A
and
P
≠
B
P \ne B
P
=
B
). The inscribed circle of
△
A
B
C
\vartriangle ABC
△
A
BC
intersects the inscribed circle of
△
A
C
P
\vartriangle ACP
△
A
CP
at points
U
U
U
and
V
V
V
and intersects the inscribed circle of
△
B
C
P
\vartriangle BCP
△
BCP
at points
X
X
X
and
Y
Y
Y
. The rights
U
V
UV
U
V
and
X
Y
XY
X
Y
intersect in
K
K
K
. Calculate the
∠
U
K
X
\angle UKX
∠
U
K
X
.