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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1999 Kurschak Competition
1999 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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2^k+1 integers, select k+2 with given property
We are given more than
2
k
2^k
2
k
integers, where
k
∈
N
k\in\mathbb{N}
k
∈
N
. Prove that we can choose
k
+
2
k+2
k
+
2
of them such that if some of our selected numbers satisfy
x
1
+
x
2
+
⋯
+
x
m
=
y
1
+
y
2
+
⋯
+
y
m
x_1+x_2+\dots+x_m=y_1+y_2+\dots+y_m
x
1
+
x
2
+
⋯
+
x
m
=
y
1
+
y
2
+
⋯
+
y
m
where
x
1
<
⋯
<
x
m
x_1<\dots<x_m
x
1
<
⋯
<
x
m
and
y
1
<
⋯
<
y
m
y_1<\dots<y_m
y
1
<
⋯
<
y
m
, then
x
i
=
y
i
x_i=y_i
x
i
=
y
i
for any
1
≤
i
≤
m
1\le i\le m
1
≤
i
≤
m
.
2
1
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Centroid of projection points on the sides of a triangle
Given a triangle on the plane, construct inside the triangle the point
P
P
P
for which the centroid of the triangle formed by the three projections of
P
P
P
onto the sides of the triangle happens to be
P
P
P
.
1
1
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Number of even and odd divisors
For any positive integer
m
m
m
, denote by
d
i
(
m
)
d_i(m)
d
i
(
m
)
the number of positive divisors of
m
m
m
that are congruent to
i
i
i
modulo
2
2
2
. Prove that if
n
n
n
is a positive integer, then
∣
∑
k
=
1
n
(
d
0
(
k
)
−
d
1
(
k
)
)
∣
≤
n
.
\left|\sum_{k=1}^n \left(d_0(k)-d_1(k)\right)\right|\le n.
k
=
1
∑
n
(
d
0
(
k
)
−
d
1
(
k
)
)
≤
n
.