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Problems
Contests
International Contests
Tuymaada Olympiad
1995 Tuymaada Olympiad
1995 Tuymaada Olympiad
Part of
Tuymaada Olympiad
Subcontests
(8)
4
1
Hide problems
mercant and customer debts, located along a ring road
It is known that the merchant’s
n
n
n
clients live in locations laid along the ring road. Of these,
k
k
k
customers have debts to the merchant for
a
1
,
a
2
,
.
.
.
,
a
k
a_1,a_2,...,a_k
a
1
,
a
2
,
...
,
a
k
rubles, and the merchant owes the remaining
n
−
k
n-k
n
−
k
clients, whose debts are
b
1
,
b
2
,
.
.
.
,
b
n
−
k
b_1,b_2,...,b_{n-k}
b
1
,
b
2
,
...
,
b
n
−
k
rubles, moreover,
a
1
+
a
2
+
.
.
.
+
a
k
=
b
1
+
b
2
+
.
.
.
+
b
n
−
k
a_1+a_2+...+a_k=b_1+b_2+...+b_{n-k}
a
1
+
a
2
+
...
+
a
k
=
b
1
+
b
2
+
...
+
b
n
−
k
. Prove that a merchant who has no money can pay all his debts and have paid all the customer debts, by starting a customer walk along the road from one of points and not missing any of their customers.
6
1
Hide problems
16 primitive Pythagorean triangles around a circle of radius 1995
Given a circle of radius
r
=
1995
r= 1995
r
=
1995
. Show that around it you can describe exactly
16
16
16
primitive Pythagorean triangles. The primitive Pythagorean triangle is a right-angled triangle, the lengths of the sides of which are expressed by coprime integers.
8
1
Hide problems
min MP+PQ+QR+RM where M inside ABC, P,Q,R in AB,BC,CA resp
Inside the triangle
A
B
C
ABC
A
BC
a point
M
M
M
is given . Find the points
P
,
Q
P,Q
P
,
Q
and
R
R
R
lying on the sides
A
B
,
B
C
AB,BC
A
B
,
BC
and
A
C
AC
A
C
respectively and such so that the sum
M
P
+
P
Q
+
Q
R
+
R
M
MP+PQ+QR+RM
MP
+
PQ
+
QR
+
RM
is the smallest.
7
1
Hide problems
f(x)-f(ax)=x^n-x^m where n,m\in N , 0<a<1
Find a continuous function
f
(
x
)
f(x)
f
(
x
)
satisfying the identity
f
(
x
)
−
f
(
a
x
)
=
x
n
−
x
m
f(x)-f(ax)=x^n-x^m
f
(
x
)
−
f
(
a
x
)
=
x
n
−
x
m
, where
n
,
m
∈
N
,
0
<
a
<
1
n,m\in N , 0<a<1
n
,
m
∈
N
,
0
<
a
<
1
5
1
Hide problems
find n so that any 3 points of n are vertices of an isosceles triangle in plane
A set consisting of
n
n
n
points of a plane is called an isosceles
n
n
n
-point if any three of its points are located in vertices of an isosceles triangle. Find all natural the numbers for which there exist isosceles
n
n
n
-points.
3
1
Hide problems
(\sqrt5 +1)^{2x}+ (\sqrt5 -1)^{2x}=2^x(y^2+2) , infinite natural solutions
Prove that the equation
(
5
+
1
)
2
x
+
(
5
−
1
)
2
x
=
2
x
(
y
2
+
2
)
(\sqrt5 +1)^{2x}+ (\sqrt5 -1)^{2x}=2^x(y^2+2)
(
5
+
1
)
2
x
+
(
5
−
1
)
2
x
=
2
x
(
y
2
+
2
)
has an infinite number of solutions in natural numbers.
2
1
Hide problems
max a wanted so that \lim_{n\to \infty} x_n exists where x_n=a^{x_{n-1}} , a>1
Let
x
1
=
a
,
x
2
=
a
x
1
,
.
.
.
,
x
n
=
a
x
n
−
1
x_1=a, x_2=a^{x_1}, ..., x_n=a^{x_{n-1}}
x
1
=
a
,
x
2
=
a
x
1
,
...
,
x
n
=
a
x
n
−
1
where
a
>
1
a>1
a
>
1
. What is the maximum value of
a
a
a
for which lim exists
lim
n
→
∞
x
n
\lim_{n\to \infty} x_n
lim
n
→
∞
x
n
and what is this limit?
1
1
Hide problems
fold line on a sheet of paper is straight, geo proof wanted
Give a geometric proof of the statement that the fold line on a sheet of paper is straight.