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Contests
National and Regional Contests
Greece Contests
Greece Team Selection Test
2011 Greece Team Selection Test
2011 Greece Team Selection Test
Part of
Greece Team Selection Test
Subcontests
(4)
4
1
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Similar cyclic quadrilateral
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral and let
K
,
L
,
M
,
N
,
S
,
T
K,L,M,N,S,T
K
,
L
,
M
,
N
,
S
,
T
the midpoints of
A
B
,
B
C
,
C
D
,
A
D
,
A
C
,
B
D
AB, BC, CD, AD, AC, BD
A
B
,
BC
,
C
D
,
A
D
,
A
C
,
B
D
respectively. Prove that the circumcenters of
K
L
S
,
L
M
T
,
M
N
S
,
N
K
T
KLS, LMT, MNS, NKT
K
L
S
,
L
MT
,
MNS
,
N
K
T
form a cyclic quadrilateral which is similar to
A
B
C
D
ABCD
A
BC
D
.
3
1
Hide problems
Find two functions from two conditions
Find all functions
f
,
g
:
Q
→
Q
f,g: \mathbb{Q}\to \mathbb{Q}
f
,
g
:
Q
→
Q
such that the following two conditions hold:
f
(
g
(
x
)
−
g
(
y
)
)
=
f
(
g
(
x
)
)
−
y
(
1
)
f(g(x)-g(y))=f(g(x))-y \ \ (1)
f
(
g
(
x
)
−
g
(
y
))
=
f
(
g
(
x
))
−
y
(
1
)
g
(
f
(
x
)
−
f
(
y
)
)
=
g
(
f
(
x
)
)
−
y
(
2
)
g(f(x)-f(y))=g(f(x))-y\ \ (2)
g
(
f
(
x
)
−
f
(
y
))
=
g
(
f
(
x
))
−
y
(
2
)
for all
x
,
y
∈
Q
x,y \in \mathbb{Q}
x
,
y
∈
Q
.
1
1
Hide problems
A Classic equation in primes
Find all prime numbers
p
,
q
p,q
p
,
q
such that:
p
4
+
p
3
+
p
2
+
p
=
q
2
+
q
p^4+p^3+p^2+p=q^2+q
p
4
+
p
3
+
p
2
+
p
=
q
2
+
q
2
1
Hide problems
Cover a 10X11 board by crosses
What is the maximal number of crosses than can fit in a
10
×
11
10\times 11
10
×
11
board without overlapping? Is this problem well-known? [asy] size(4.58cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -3.18, xmax = 1.4, ymin = -0.22, ymax = 3.38; /* image dimensions */ /* draw figures */ draw((-3.,2.)--(1.,2.)); draw((-2.,3.)--(-2.,0.)); draw((-2.,0.)--(-1.,0.)); draw((-1.,0.)--(-1.,3.)); draw((-1.,3.)--(-2.,3.)); draw((-3.,1.)--(1.,1.)); draw((1.,1.)--(1.,2.)); draw((-3.,2.)--(-3.,1.)); draw((0.,2.)--(0.,1.)); draw((-1.,2.)--(-1.,1.)); draw((-2.,2.)--(-2.,1.)); /* dots and labels */ clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]