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Problems
Contests
International Contests
Tuymaada Olympiad
1995 Tuymaada Olympiad
4
4
Part of
1995 Tuymaada Olympiad
Problems
(1)
mercant and customer debts, located along a ring road
Source: Tuymaada 1995 p4
4/27/2019
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.
combinatorics
circle
Sum