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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
2002 Federal Competition For Advanced Students, Part 2
2002 Federal Competition For Advanced Students, Part 2
Part of
Austrian MO National Competition
Subcontests
(3)
3
2
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Prove that P lies on line BE.
Let
A
B
C
D
ABCD
A
BC
D
and
A
E
F
G
AEFG
A
EFG
be two similar cyclic quadrilaterals (with the vertices denoted counterclockwise). Their circumcircles intersect again at point
P
P
P
. Prove that
P
P
P
lies on line
B
E
BE
BE
.
Triangles have the same perimeter iff ABC is equilateral
Let
H
H
H
be the orthocenter of an acute-angled triangle
A
B
C
ABC
A
BC
. Show that the triangles
A
B
H
,
B
C
H
ABH,BCH
A
B
H
,
BC
H
and
C
A
H
CAH
C
A
H
have the same perimeter if and only if the triangle
A
B
C
ABC
A
BC
is equilateral.
2
2
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Find all 2002-tuples s.t. the sum equals 2002b^b
Let
b
b
b
be a positive integer. Find all
2002
2002
2002
−tuples
(
a
1
,
a
2
,
…
,
a
2002
)
(a_1, a_2,\ldots , a_{2002})
(
a
1
,
a
2
,
…
,
a
2002
)
, of natural numbers such that
∑
j
=
1
2002
a
j
a
j
=
2002
b
b
.
\sum_{j=1}^{2002} a_j^{a_j}=2002b^b.
j
=
1
∑
2002
a
j
a
j
=
2002
b
b
.
In how many ways can one reach the point 3n+1?
In the net drawn below, in how many ways can one reach the point
3
n
+
1
3n+1
3
n
+
1
starting from the point
1
1
1
so that the labels of the points on the way increase?[asy] import graph; size(12cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.3,xmax=12.32,ymin=-10.66,ymax=6.3; draw((1,2)--(xmax,0*xmax+2)); draw((1,0)--(xmax,0*xmax+0)); draw((0,1)--(1,2)); draw((1,0)--(0,1)); draw((1,2)--(3,0)); draw((1,0)--(3,2)); draw((3,2)--(5,0)); draw((3,0)--(5,2)); draw((5,2)--(7,0)); draw((5,0)--(7,2)); draw((7,2)--(9,0)); draw((7,0)--(9,2)); dot((1,0),linewidth(1pt)+ds); label("2",(0.96,-0.5),NE*lsf); dot((0,1),linewidth(1pt)+ds); label("1",(-0.42,0.9),NE*lsf); dot((1,2),linewidth(1pt)+ds); label("3",(0.98,2.2),NE*lsf); dot((2,1),linewidth(1pt)+ds); label("4",(1.92,1.32),NE*lsf); dot((3,2),linewidth(1pt)+ds); label("6",(2.94,2.2),NE*lsf); dot((4,1),linewidth(1pt)+ds); label("7",(3.94,1.32),NE*lsf); dot((6,1),linewidth(1pt)+ds); label("10",(5.84,1.32),NE*lsf); dot((3,0),linewidth(1pt)+ds); label("5",(2.98,-0.46),NE*lsf); dot((5,2),linewidth(1pt)+ds); label("9",(4.92,2.24),NE*lsf); dot((5,0),linewidth(1pt)+ds); label("8",(4.94,-0.42),NE*lsf); dot((8,1),linewidth(1pt)+ds); label("13",(7.88,1.34),NE*lsf); dot((7,2),linewidth(1pt)+ds); label("12",(6.8,2.26),NE*lsf); dot((7,0),linewidth(1pt)+ds); label("11",(6.88,-0.38),NE*lsf); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]
1
2
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What's the formula if “8 × 8” is replaced with “a × b”?
Consider all possible rectangles that can be drawn on a
8
×
8
8 \times 8
8
×
8
chessboard, covering only whole cells. Calculate the sum of their areas.What formula is obtained if “
8
×
8
8 \times 8
8
×
8
” is replaced with “
a
×
b
a \times b
a
×
b
”, where
a
,
b
a, b
a
,
b
are positive integers?
Find all polynomials P(x) of the smallest possible degree
Find all polynomials
P
(
x
)
P(x)
P
(
x
)
of the smallest possible degree with the following properties:(i) The leading coefficient is
200
200
200
; (ii) The coefficient at the smallest non-vanishing power is
2
2
2
; (iii) The sum of all the coefficients is
4
4
4
; (iv)
P
(
−
1
)
=
0
,
P
(
2
)
=
6
,
P
(
3
)
=
8
P(-1) = 0, P(2) = 6, P(3) = 8
P
(
−
1
)
=
0
,
P
(
2
)
=
6
,
P
(
3
)
=
8
.