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Kyiv Mathematical Festival
2013 Kyiv Mathematical Festival
2013 Kyiv Mathematical Festival
Part of
Kyiv Mathematical Festival
Subcontests
(5)
5
1
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a \ne b such that (a+b)$ is a perfect square & (a^3 +b^3) is a fourth power
Do there exist positive integers
a
≠
b
a \ne b
a
=
b
such that
a
+
b
a+b
a
+
b
is a perfect square and
a
3
+
b
3
a^3 +b^3
a
3
+
b
3
is a fourth power of an integer?
4
1
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winning strategy in erazing cities from 2013 totally with N roads
Elza draws
2013
2013
2013
cities on the map and connects some of them with
N
N
N
roads. Then Elza and Susy erase cities in turns until just two cities left (first city is to be erased by Elza). If these cities are connected with a road then Elza wins, otherwise Susy wins. Find the smallest
N
N
N
for which Elza has a winning strategy.
2
3
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represent n^2 as a sum of several distinct positive integers <= 2n
For which positive integers
n
≥
2
n \ge 2
n
≥
2
it is possible to represent the number
n
2
n^2
n
2
as a sum of several distinct positive integers not exceeding
2
n
2n
2
n
?
represent n^2 as a sum of n distinct positive integers <= 3n / 2
For which positive integers
n
≥
2
n \ge 2
n
≥
2
it is possible to represent the number
n
2
n^2
n
2
as a sum of n distinct positive integers not exceeding
3
n
2
\frac{3n}{2}
2
3
n
?
prove ab/(a+b)+bc/(b+c)+cd/(c+d)+da/(d+a)\ge 4 if (a+c>=ac) &(b+d>=bd)
For every positive
a
,
b
,
c
,
d
a, b,c, d
a
,
b
,
c
,
d
such that
a
+
c
≤
a
c
a + c \le ac
a
+
c
≤
a
c
and
b
+
d
≤
b
d
b + d \le bd
b
+
d
≤
b
d
prove that
a
b
a
+
b
+
b
c
b
+
c
+
c
d
c
+
d
+
d
a
d
+
a
≥
4
\frac{ab}{a + b} +\frac{bc}{b + c} +\frac{cd}{c + d} +\frac{da}{d + a} \ge 4
a
+
b
ab
+
b
+
c
b
c
+
c
+
d
c
d
+
d
+
a
d
a
≥
4
1
2
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an optimistic worm can eat no more than half of 24 apples, who are in 4 boxes
There are
24
24
24
apples in
4
4
4
boxes. An optimistic worm is convinced that he can eat no more than half of the apples such that there will be
3
3
3
boxes with equal number of apples. Is it possible that he is wrong?
if a,b,c,d >0 with (a+c<= ac) & (b+d <= bd) prove that ab + cd >= 8
For every positive
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
such that
a
+
c
≤
a
c
a + c\le ac
a
+
c
≤
a
c
and
b
+
d
≤
b
d
b + d \le bd
b
+
d
≤
b
d
prove that
a
b
+
c
d
≥
8
ab + cd \ge 8
ab
+
c
d
≥
8
.
3
1
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perpendicularity starting with angle bisectors of parallelogram
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram (
A
B
<
B
C
AB < BC
A
B
<
BC
). The bisector of the angle
B
A
D
BAD
B
A
D
intersects the side
B
C
BC
BC
at the point K; and the bisector of the angle
A
D
C
ADC
A
D
C
intersects the diagonal
A
C
AC
A
C
at the point
F
F
F
. Suppose that
K
D
⊥
B
C
KD \perp BC
KD
⊥
BC
. Prove that
K
F
⊥
B
D
KF \perp BD
K
F
⊥
B
D
.