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International Contests
Tuymaada Olympiad
2000 Tuymaada Olympiad
2000 Tuymaada Olympiad
Part of
Tuymaada Olympiad
Subcontests
(8)
5
1
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q \ p^2-1, p \ q^2-1, where p,q primes >3
Are there prime
p
p
p
and
q
q
q
larger than
3
3
3
, such that
p
2
−
1
p^2-1
p
2
−
1
is divisible by
q
q
q
and
q
2
−
1
q^2-1
q
2
−
1
divided by
p
p
p
?
6
1
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circumcircles are vertices of an equilateral triangle, right triangle wanted
Let
O
O
O
be the center of the circle circumscribed around the the triangle
A
B
C
ABC
A
BC
. The centers of the circles circumscribed around the squares
O
A
B
,
O
B
C
,
O
C
A
OAB,OBC,OCA
O
A
B
,
OBC
,
OC
A
lie at the vertices of a regular triangle. Prove that the triangle
A
B
C
ABC
A
BC
is right.
8
1
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2000 cities each of which has exactly 3 roads to other cities
There are
2000
2000
2000
cities in the country, each of which has exactly three roads to other cities. Prove that you can close
1000
1000
1000
roads, so that there is not a single closed route in the country, consisting of an odd number of roads.
7
1
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every 2 of 5 regular pentagons have a common point
Every two of five regular pentagons on the plane have a common point. Is it true that some of these pentagons have a common point?
2
3
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When a tangent line meets a rhombus..
A tangent
l
l
l
to the circle inscribed in a rhombus meets its sides
A
B
AB
A
B
and
B
C
BC
BC
at points
E
E
E
and
F
F
F
respectively. Prove that the product
A
E
⋅
C
F
AE\cdot CF
A
E
⋅
CF
is independent of the choice of
l
l
l
.
What if we connect the cities of Graphland?
There are 2000 cities in Graphland; some of them are connected by roads. For every city the number of roads going from it is counted. It is known that there are exactly two equal numbers among all the numbers obtained. What can be these numbers?
4 colors to paint the plane, inside any circle all 4 colors, is it possible?
Is it possible to paint the plane in
4
4
4
colors so that inside any circle are the dots of all four colors?
3
2
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Polynomial - TUYMAADA-2000
Polynomial
P
(
t
)
P(t)
P
(
t
)
is such that for all real
x
x
x
, P(\sin x) \plus{} P(\cos x) \equal{} 1. What can be the degree of this polynomial?
Tuymaada Yakut Olympiad 2000, Problem 3
Can the 'brick wall' (infinite in all directions) drawn at the picture be made of wires of length
1
,
2
,
3
,
…
1, 2, 3, \dots
1
,
2
,
3
,
…
(each positive integral length occurs exactly once)? (Wires can be bent but should not overlap; size of a 'brick' is
1
×
2
1\times 2
1
×
2
).[asy] unitsize(0.5 cm);for(int i = 1; i <= 9; ++i) { draw((0,i)--(10,i)); }for(int i = 0; i <= 4; ++i) { for(int j = 0; j <= 4; ++j) { draw((2*i + 1,2*j)--(2*i + 1,2*j + 1)); } }for(int i = 0; i <= 3; ++i) { for(int j = 0; j <= 4; ++j) { draw((2*i + 2,2*j + 1)--(2*i + 2,2*j + 2)); } } [/asy]
4
3
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Like vasc's one or cmo i don't remember
Prove for real
x
1
x_1
x
1
,
x
2
x_2
x
2
, .....,
x
n
x_n
x
n
,
0
<
x
k
≤
1
2
0 < x_k \leq {1\over 2}
0
<
x
k
≤
2
1
, the inequality
(
n
x
1
+
⋯
+
x
n
−
1
)
n
≤
(
1
x
1
−
1
)
…
(
1
x
n
−
1
)
.
\left( {n \over x_1 + \dots + x_n} - 1 \right)^n \leq \left( {1 \over x_1} - 1 \right) \dots \left( {1 \over x_n} - 1 \right).
(
x
1
+
⋯
+
x
n
n
−
1
)
n
≤
(
x
1
1
−
1
)
…
(
x
n
1
−
1
)
.
10^{-n} \neq \sum 1/a_i!
Prove that no number of the form
1
0
−
n
10^{-n}
1
0
−
n
,
n
≥
1
,
n\geq 1,
n
≥
1
,
can be represented as the sum of reciprocals of factorials of different positive integers.
1/a(a+1)+ 1/b(b+1)+1/c(c+1) >= 3/2 if abc=1 and a,b,c>0
Prove that if the product of positive numbers
a
,
b
a,b
a
,
b
and
c
c
c
equals one, then
1
a
(
a
+
1
)
+
1
b
(
b
+
1
)
+
1
c
(
c
+
1
)
≥
3
2
\frac{1}{a(a+1)}+\frac{1}{b(b+1)}+\frac{1}{c(c+1)}\ge \frac{3}{2}
a
(
a
+
1
)
1
+
b
(
b
+
1
)
1
+
c
(
c
+
1
)
1
≥
2
3
1
3
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Number of positive divisors and integer part.
Let
d
(
n
)
d(n)
d
(
n
)
denote the number of positive divisors of
n
n
n
and let
e
(
n
)
=
[
2000
n
]
e(n)=\left[2000\over n\right]
e
(
n
)
=
[
n
2000
]
for positive integer
n
n
n
. Prove that
d
(
1
)
+
d
(
2
)
+
⋯
+
d
(
2000
)
=
e
(
1
)
+
e
(
2
)
+
⋯
+
e
(
2000
)
.
d(1)+d(2)+\dots+d(2000)=e(1)+e(2)+\dots+e(2000).
d
(
1
)
+
d
(
2
)
+
⋯
+
d
(
2000
)
=
e
(
1
)
+
e
(
2
)
+
⋯
+
e
(
2000
)
.
colouring the plane
Can the plane be coloured in 2000 colours so that any nondegenerate circle contains points of all 2000 colors?
erasing digits from 188188...188, largest multiple of 7
Given the number
188188...188
188188...188
188188...188
(number
188
188
188
is written
101
101
101
times). Some digits of this number are crossed out. What is the largest multiple of
7
7
7
, that could happen?