MathDB
Problems
Contests
National and Regional Contests
Greece Contests
Greece Team Selection Test
2012 Greece Team Selection Test
2012 Greece Team Selection Test
Part of
Greece Team Selection Test
Subcontests
(4)
4
1
Hide problems
Colouring trapezoids in equilateral grid
Let
n
=
3
k
n=3k
n
=
3
k
be a positive integer (with
k
≥
2
k\geq 2
k
≥
2
). An equilateral triangle is divided in
n
2
n^2
n
2
unit equilateral triangles with sides parallel to the initial, forming a grid. We will call "trapezoid" the trapezoid which is formed by three equilateral triangles (one base is equal to one and the other is equal to two). We colour the points of the grid with three colours (red, blue and green) such that each two neighboring points have different colour. Finally, the colour of a "trapezoid" will be the colour of the midpoint of its big base. Find the number of all "trapezoids" in the grid (not necessarily disjoint) and determine the number of red, blue and green "trapezoids".
2
1
Hide problems
Angle Equality
Given is an acute triangle
A
B
C
ABC
A
BC
(
A
B
<
A
C
<
B
C
)
\left(AB<AC<BC\right)
(
A
B
<
A
C
<
BC
)
,inscribed in circle
c
(
O
,
R
)
c(O,R)
c
(
O
,
R
)
.The perpendicular bisector of the angle bisector
A
D
AD
A
D
(
D
∈
B
C
)
\left(D\in BC\right)
(
D
∈
BC
)
intersects
c
c
c
at
K
,
L
K,L
K
,
L
(
K
K
K
lies on the small arc \overarc{AB}).The circle
c
1
(
K
,
K
A
)
c_1(K,KA)
c
1
(
K
,
K
A
)
intersects
c
c
c
at
T
T
T
and the circle
c
2
(
L
,
L
A
)
c_2(L,LA)
c
2
(
L
,
L
A
)
intersects
c
c
c
at
S
S
S
.Prove that
∠
B
A
T
=
∠
C
A
S
\angle{BAT}=\angle{CAS}
∠
B
A
T
=
∠
C
A
S
.[asy]import graph; size(10cm); 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 = -6.94236331697463, xmax = 15.849400903703716, ymin = -5.002235438802758, ymax = 7.893104843949444; /* image dimensions */ pen aqaqaq = rgb(0.6274509803921569,0.6274509803921569,0.6274509803921569); pen uququq = rgb(0.25098039215686274,0.25098039215686274,0.25098039215686274); pen qqqqtt = rgb(0.,0.,0.2); draw((1.8318261909633622,3.572783369254345)--(0.,0.)--(6.,0.)--cycle, aqaqaq); draw(arc((1.8318261909633622,3.572783369254345),0.6426249310341638,-117.14497824050169,-101.88970202103212)--(1.8318261909633622,3.572783369254345)--cycle, qqqqtt); draw(arc((1.8318261909633622,3.572783369254345),0.6426249310341638,-55.85706977865775,-40.60179355918817)--(1.8318261909633622,3.572783369254345)--cycle, qqqqtt); /* draw figures */ draw((1.8318261909633622,3.572783369254345)--(0.,0.), uququq); draw((0.,0.)--(6.,0.), uququq); draw((6.,0.)--(1.8318261909633622,3.572783369254345), uququq); draw(circle((3.,0.7178452373968209), 3.0846882800136055)); draw((2.5345020274407277,0.)--(1.8318261909633622,3.572783369254345)); draw(circle((-0.01850947366601585,1.3533783539547308), 2.889550258039566)); draw(circle((5.553011501106743,2.4491551634556963), 3.887127532933951)); draw((-0.01850947366601585,1.3533783539547308)--(5.553011501106743,2.4491551634556963), linetype("2 2")); draw((1.8318261909633622,3.572783369254345)--(0.7798408954511686,-1.423695174396108)); draw((1.8318261909633622,3.572783369254345)--(5.22015910454883,-1.4236951743961088)); /* dots and labels */ dot((1.8318261909633622,3.572783369254345),linewidth(3.pt) + dotstyle); label("
A
A
A
", (1.5831274347452782,3.951671933606579), NE * labelscalefactor); dot((0.,0.),linewidth(3.pt) + dotstyle); label("
B
B
B
", (-0.6,0.05), NE * labelscalefactor); dot((6.,0.),linewidth(3.pt) + dotstyle); label("
C
C
C
", (6.188606107156787,0.07450151636712989), NE * labelscalefactor); dot((2.5345020274407277,0.),linewidth(3.pt) + dotstyle); label("
D
D
D
", (2.3,-0.7), NE * labelscalefactor); dot((-0.01850947366601585,1.3533783539547308),linewidth(3.pt) + dotstyle); label("
K
K
K
", (-0.3447473583572136,1.6382221818835927), NE * labelscalefactor); dot((5.553011501106743,2.4491551634556963),linewidth(3.pt) + dotstyle); label("
L
L
L
", (5.631664500260511,2.580738747400365), NE * labelscalefactor); dot((0.7798408954511686,-1.423695174396108),linewidth(3.pt) + dotstyle); label("
T
T
T
", (0.5977692071595602,-1.960477431907719), NE * labelscalefactor); dot((5.22015910454883,-1.4236951743961088),linewidth(3.pt) + dotstyle); label("
S
S
S
", (5.160406217502124,-1.8747941077698307), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy]
3
1
Hide problems
a+b+c=3
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers satisfying
a
+
b
+
c
=
3
a+b+c=3
a
+
b
+
c
=
3
.Prove that
∑
s
y
m
a
2
(
b
+
c
)
3
≥
3
8
\sum_{sym} \frac{a^{2}}{(b+c)^{3}}\geq \frac{3}{8}
∑
sy
m
(
b
+
c
)
3
a
2
≥
8
3
1
1
Hide problems
Equation with prime and natural numers
Find all triples
(
p
,
m
,
n
)
(p,m,n)
(
p
,
m
,
n
)
satisfying the equation
p
m
−
n
3
=
8
p^m-n^3=8
p
m
−
n
3
=
8
where
p
p
p
is a prime number and
m
,
n
m,n
m
,
n
are nonnegative integers.