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Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
2013 Czech And Slovak Olympiad IIIA
2013 Czech And Slovak Olympiad IIIA
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
6
1
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\sqrt{a^2 + pb^2} +\sqrt{b^2 + pa^2} \ge a + b + (p - 1) \sqrt{ab}
Find all positive real numbers
p
p
p
such that
a
2
+
p
b
2
+
b
2
+
p
a
2
≥
a
+
b
+
(
p
−
1
)
a
b
\sqrt{a^2 + pb^2} +\sqrt{b^2 + pa^2} \ge a + b + (p - 1) \sqrt{ab}
a
2
+
p
b
2
+
b
2
+
p
a
2
≥
a
+
b
+
(
p
−
1
)
ab
holds for any pair of positive real numbers
a
,
b
a, b
a
,
b
.
4
1
Hide problems
N with last digit c, wipe c and name it as m, replace m with m-3c
On the board is written in decimal the integer positive number
N
N
N
. If it is not a single digit number, wipe its last digit
c
c
c
and replace the number
m
m
m
that remains on the board with a number
m
−
3
c
m -3c
m
−
3
c
. (For example, if
N
=
1
,
204
N = 1,204
N
=
1
,
204
on the board,
120
−
3
⋅
4
=
108
120 - 3 \cdot 4 = 108
120
−
3
⋅
4
=
108
.) Find all the natural numbers
N
N
N
, by repeating the adjustment described eventually we get the number
0
0
0
.
2
1
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100n coins to be shared by n thieves
Each of the thieves in the
n
n
n
-member party (
n
≥
3
n \ge 3
n
≥
3
) charged a certain number of coins. All the coins were
100
n
100n
100
n
. Thieves decided to share their prey as follows: at each step, one of the bandits puts one coin to the other two. Find them all natural numbers
n
≥
3
n \ge 3
n
≥
3
for which after a finite number of steps each outlaw can have
100
100
100
coins no matter how many coins each thug has charged.
5
1
Hide problems
|\angle BCA| + |\angle ABD| = |\angle BDA| + |\angle ACD| => KL//AC
Given the parallelogram
A
B
C
D
ABCD
A
BC
D
such that the feet
K
,
L
K, L
K
,
L
of the perpendiculars from point
D
D
D
on the sides
A
B
,
B
C
AB, BC
A
B
,
BC
respectively are internal points. Prove that
K
L
∥
A
C
KL \parallel AC
K
L
∥
A
C
when
∣
∠
B
C
A
∣
+
∣
∠
A
B
D
∣
=
∣
∠
B
D
A
∣
+
∣
∠
A
C
D
∣
|\angle BCA| + |\angle ABD| = |\angle BDA| + |\angle ACD|
∣∠
BC
A
∣
+
∣∠
A
B
D
∣
=
∣∠
B
D
A
∣
+
∣∠
A
C
D
∣
.
3
1
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parallel lines inside a parallelogram, incircle related
In the parallelolgram A
B
C
D
BCD
BC
D
with the center
S
S
S
, let
O
O
O
be the center of the circle of the inscribed triangle
A
B
D
ABD
A
B
D
and let
T
T
T
be the touch point with the diagonal
B
D
BD
B
D
. Prove that the lines
O
S
OS
OS
and
C
T
CT
CT
are parallel.
1
1
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(a^2+1)/(2b^2-3)=(a-1)/(2b-1) diophantine
Find all pairs of integers
a
,
b
a, b
a
,
b
for which equality holds
a
2
+
1
2
b
2
−
3
=
a
−
1
2
b
−
1
\frac{a^2+1}{2b^2-3}=\frac{a-1}{2b-1}
2
b
2
−
3
a
2
+
1
=
2
b
−
1
a
−
1