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Problems
Contests
National and Regional Contests
Belgium Contests
Flanders Math Olympiad
2014 Flanders Math Olympiad
2014 Flanders Math Olympiad
Part of
Flanders Math Olympiad
Subcontests
(4)
4
1
Hide problems
a + b=? remainder P(x):(x^3 + ax + b)=remainder P(x):(x^3 + ax^2 + b)
Let
P
(
x
)
P(x)
P
(
x
)
be a polynomial of degree
5
5
5
and suppose that a and b are real numbers different from zero. Suppose the remainder when
P
(
x
)
P(x)
P
(
x
)
is divided by
x
3
+
a
x
+
b
x^3 + ax + b
x
3
+
a
x
+
b
equals the remainder when
P
(
x
)
P(x)
P
(
x
)
is divided by
x
3
+
a
x
2
+
b
x^3 + ax^2 + b
x
3
+
a
x
2
+
b
. Then determine
a
+
b
a + b
a
+
b
.
2
1
Hide problems
no of liars in the class Miss Lies
In Miss Lies' class there are only students who never lie and students who always lie. All students know which category they belong to. During the day in a class discussion, every student in the class says about every other student or he or she a liar or not. In total, it is said
320
320
320
times that someone is not lying. The next day, one of the students who always lies is sick. There will be one again organize such a class discussion in which no mention is made of the sick pupil. Now it is said
300
300
300
times that someone does lie. How many liars are there in the Miss Lies' class ?
1
1
Hide problems
4(sum x^2 )=sum a^2, in tetrahedron, law of# (2014 VWO Flanders MO p1)
(a) Prove the parallelogram law that says that in a parallelogram the sum of the squares of the lengths of the four sides equals the sum of the squares of the lengths of the two diagonals. (b) The edges of a tetrahedron have lengths
a
,
b
,
c
,
d
,
e
a, b, c, d, e
a
,
b
,
c
,
d
,
e
and
f
f
f
. The three line segments connecting the centers of intersecting edges have lengths
x
,
y
x, y
x
,
y
and
z
z
z
. Prove that
4
(
x
2
+
y
2
+
z
2
)
=
a
2
+
b
2
+
c
2
+
d
2
+
e
2
+
f
2
4 (x^2 + y^2 + z^2) = a^2 + b^2 + c^2 + d^2 + e^2 + f^2
4
(
x
2
+
y
2
+
z
2
)
=
a
2
+
b
2
+
c
2
+
d
2
+
e
2
+
f
2
3
1
Hide problems
angle chasing in PQRS, PQ =QR = RS , <Q= 110^o, < R = 130^o .
Let
P
Q
R
S
PQRS
PQRS
be a quadrilateral with
∣
P
Q
∣
=
∣
Q
R
∣
=
∣
R
S
∣
| P Q | = | QR | = | RS |
∣
PQ
∣
=
∣
QR
∣
=
∣
RS
∣
,
∠
Q
=
11
0
o
\angle Q= 110^o
∠
Q
=
11
0
o
and
∠
R
=
13
0
o
\angle R = 130^o
∠
R
=
13
0
o
. Determine
∠
P
\angle P
∠
P
and
∠
S
\angle S
∠
S
.