MathDB
Problems
Contests
National and Regional Contests
North Macedonia Contests
North Macedonia National Olympiads
1998 North Macedonia National Olympiad
1998 North Macedonia National Olympiad
Part of
North Macedonia National Olympiads
Subcontests
(5)
5
1
Hide problems
a_{n+1} =\sqrt{2-\sqrt{4-a_n^2}} , b_n =2^{n+1}a_n
The sequence
(
a
n
)
(a_n)
(
a
n
)
is defined by
a
1
=
2
a_1 =\sqrt2
a
1
=
2
and
a
n
+
1
=
2
−
4
−
a
n
2
a_{n+1} =\sqrt{2-\sqrt{4-a_n^2}}
a
n
+
1
=
2
−
4
−
a
n
2
. Let
b
n
=
2
n
+
1
a
n
b_n =2^{n+1}a_n
b
n
=
2
n
+
1
a
n
. Prove that
b
n
≤
7
b_n \le 7
b
n
≤
7
and
b
n
<
b
n
+
1
b_n < b_{n+1}
b
n
<
b
n
+
1
for all
n
n
n
.
4
1
Hide problems
(ab+bc+ca)/4P \ge \sqrt3 where a,b,c sidelengths and P area
If
P
P
P
is the area of a triangle
A
B
C
ABC
A
BC
with sides
a
,
b
,
c
a,b,c
a
,
b
,
c
, prove that
a
b
+
b
c
+
c
a
4
P
≥
3
\frac{ab+bc+ca}{4P} \ge \sqrt3
4
P
ab
+
b
c
+
c
a
≥
3
3
1
Hide problems
\sum_{k=0}^{n-1}f(x+k,y+k,z+k) = n^3f(x/n,y/n,z/n) , ratio of triangle areas
A triangle
A
B
C
ABC
A
BC
is given. For every positive numbers
p
,
q
,
r
p,q,r
p
,
q
,
r
, let
A
′
,
B
′
,
C
′
A',B',C'
A
′
,
B
′
,
C
′
be the points such that
B
A
′
→
=
p
A
B
→
,
C
B
′
→
=
q
B
C
→
\overrightarrow{BA'} = p\overrightarrow{AB}, \overrightarrow{CB'}=q\overrightarrow{BC}
B
A
′
=
p
A
B
,
C
B
′
=
q
BC
, and
A
C
′
→
=
r
C
A
→
\overrightarrow{AC'}=r\overrightarrow{CA}
A
C
′
=
r
C
A
. Define
f
(
p
,
q
,
r
)
f(p,q,r)
f
(
p
,
q
,
r
)
as the ratio of the area of
△
A
′
B
′
C
′
\vartriangle A'B'C'
△
A
′
B
′
C
′
to that of
△
A
B
C
\vartriangle ABC
△
A
BC
. Prove that for all positive numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
and every positive integer
n
n
n
,
∑
k
=
0
n
−
1
f
(
x
+
k
,
y
+
k
,
z
+
k
)
=
n
3
f
(
x
n
,
y
n
,
z
n
)
\sum_{k=0}^{n-1}f(x+k,y+k,z+k) = n^3f\left(\frac{x}{n},\frac{y}{n},\frac{z}{n}\right)
∑
k
=
0
n
−
1
f
(
x
+
k
,
y
+
k
,
z
+
k
)
=
n
3
f
(
n
x
,
n
y
,
n
z
)
.
2
1
Hide problems
partition set 1,2,...,1998 into 3 sets with sums 2000, 3999 and 5998
Prove that the numbers
1
,
2
,
.
.
.
,
1998
1,2,...,1998
1
,
2
,
...
,
1998
cannot be separated into three classes whose sums of elements are divisible by
2000
,
3999
2000,3999
2000
,
3999
, and
5998
5998
5998
, respectively.
1
1
Hide problems
angle chasing in a pentagon ABCDE with AB = BC =CA , CD = DE = EC
Let
A
B
C
D
E
ABCDE
A
BC
D
E
be a convex pentagon with
A
B
=
B
C
=
C
A
AB = BC =CA
A
B
=
BC
=
C
A
and
C
D
=
D
E
=
E
C
CD = DE = EC
C
D
=
D
E
=
EC
. Let
T
T
T
be the centroid of
△
A
B
C
\vartriangle ABC
△
A
BC
, and
N
N
N
be the midpoint of
A
E
AE
A
E
. Compute
∠
N
T
D
\angle NT D
∠
NT
D