MathDB
Problems
Contests
National and Regional Contests
North Macedonia Contests
North Macedonia National Olympiads
1994 North Macedonia National Olympiad
1994 North Macedonia National Olympiad
Part of
North Macedonia National Olympiads
Subcontests
(5)
5
1
Hide problems
tiling with L tiles a board of 3^n x 3^n minus one 1x1 square
A square with the dimension
1
×
1
1 \times1
1
×
1
has been removed from a square board
3
n
×
3
n
3 ^n \times 3 ^n
3
n
×
3
n
(
n
∈
N
,
n \in \mathbb {N},
n
∈
N
,
n
>
1
n> 1
n
>
1
). a) Prove that any defective board with the dimension
3
n
×
3
n
3 ^ n \times3 ^ n
3
n
×
3
n
can be covered with shaped figures of shape 1 (the 3 squares' one) and of shape 2 (the 5 squares' one). Figures covering the board must not overlap each other and must not cross the edge of the board. Also the squares removed from the board must not be covered. (b) How many small figures in shape 2 must be used to cover the board? https://cdn.artofproblemsolving.com/attachments/4/7/e970fadd7acc7fd6f5897f1766a84787f37acc.png
4
1
Hide problems
98 of any 100 of 1994 points can be selected and be rounded by a circle
1994
1994
1994
points from the plane are given so that any
100
100
100
of them can be selected
98
98
98
that can be rounded (some points may be at the border of the circle) with a diameter of
1
1
1
. Determine the smallest number of circles with radius
1
1
1
, sufficient to cover all
1994
1994
1994
3
1
Hide problems
constant sum, max of sum x_ix_j
a) Let
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2, ..., x_n
x
1
,
x
2
,
...
,
x
n
(
n
>
2
n> 2
n
>
2
) be negative real numbers and
x
1
+
x
2
+
.
.
.
+
x
n
=
m
.
x_1 + x_2 + ... + x_n = m.
x
1
+
x
2
+
...
+
x
n
=
m
.
Determine the maximum value of the sum
S
=
x
1
x
2
+
x
1
x
3
+
⋯
+
x
1
x
n
+
x
2
x
3
+
x
2
x
4
+
⋯
+
x
2
x
n
+
⋯
+
x
n
−
1
x
n
.
S = x_1x_2 + x_1x_3 + \dots + x_1x_n + x_2x_3 + x_2x_4 + \dots + x_2x_n + \dots + x_ {n-1} x_n.
S
=
x
1
x
2
+
x
1
x
3
+
⋯
+
x
1
x
n
+
x
2
x
3
+
x
2
x
4
+
⋯
+
x
2
x
n
+
⋯
+
x
n
−
1
x
n
.
b) Let
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2, ..., x_n
x
1
,
x
2
,
...
,
x
n
(
n
>
2
n> 2
n
>
2
) be nonnegative natural numbers and
x
1
+
x
2
+
.
.
.
+
x
n
=
m
.
x_1 + x_2 + ... + x_n = m.
x
1
+
x
2
+
...
+
x
n
=
m
.
Determine the maximum value of the sum
S
=
x
1
x
2
+
x
1
x
3
+
⋯
+
x
1
x
n
+
x
2
x
3
+
x
2
x
4
+
⋯
+
x
2
x
n
+
⋯
+
x
n
−
1
x
n
.
S = x_1x_2 + x_1x_3 + \dots + x_1x_n + x_2x_3 + x_2x_4 + \dots + x_2x_n + \dots + x_ {n-1} x_n.
S
=
x
1
x
2
+
x
1
x
3
+
⋯
+
x
1
x
n
+
x
2
x
3
+
x
2
x
4
+
⋯
+
x
2
x
n
+
⋯
+
x
n
−
1
x
n
.
2
1
Hide problems
lattice triangle given, with 1 only lattice interior point, ratio of segments
Let
A
B
C
ABC
A
BC
be a triangle whose vertices have integer coordinates and inside of which there is exactly one point
O
O
O
with integer coordinates. Let
D
D
D
be the intersection of the lines
B
C
BC
BC
and
A
O
.
AO.
A
O
.
Find the largest possible value of
A
O
‾
O
D
‾
\frac {\overline{AO}} {\overline{OD}}
O
D
A
O
.
1
1
Hide problems
a_1+a_2+... a_{1994}=1994^{1994}, sum a_i^3 mod6
Let
a
1
,
a
2
,
.
.
.
,
a
1994
a_1, a_2, ..., a_ {1994}
a
1
,
a
2
,
...
,
a
1994
be integers such that
a
1
+
a
2
+
.
.
.
+
a
1994
=
199
4
1994
a_1 + a_2 + ... + a_{1994} = 1994 ^{1994}
a
1
+
a
2
+
...
+
a
1994
=
199
4
1994
. Determine the remainder of the division of
a
1
3
+
a
2
3
+
.
.
.
+
a
1994
3
a ^ 3_1 + a ^ 3_2 + ... + a ^ 3_{1994}
a
1
3
+
a
2
3
+
...
+
a
1994
3
with
6
6
6
.