MathDB
Problems
Contests
National and Regional Contests
Serbia Contests
Serbia Team Selection Test
1976 Yugoslav Team Selection Test
1976 Yugoslav Team Selection Test
Part of
Serbia Team Selection Test
Subcontests
(3)
Problem 3
1
Hide problems
min/max with four variables over R+0
Find the minimum and maximum values of the function
f
(
x
,
y
,
z
,
t
)
=
a
x
2
+
b
y
2
a
x
+
b
y
+
a
z
2
+
b
t
2
a
z
+
b
t
,
(
a
>
0
,
b
>
0
)
,
f(x,y,z,t)=\frac{ax^2+by^2}{ax+by}+\frac{az^2+bt^2}{az+bt},~(a>0,b>0),
f
(
x
,
y
,
z
,
t
)
=
a
x
+
b
y
a
x
2
+
b
y
2
+
a
z
+
b
t
a
z
2
+
b
t
2
,
(
a
>
0
,
b
>
0
)
,
given that
x
+
z
=
y
+
t
=
1
x+z=y+t=1
x
+
z
=
y
+
t
=
1
, and
x
,
y
,
z
,
t
≥
0
x,y,z,t\ge0
x
,
y
,
z
,
t
≥
0
.
Problem 2
1
Hide problems
set of integers with property
Assume that
2
n
+
1
2n+1
2
n
+
1
positive integers satisfy the following: If we remove any of these integers, the remaining
2
n
2n
2
n
integers can be partitioned in two groups of
n
n
n
numbers in each, such that the sum of the numbers in one group is equal to the sum of the numbers in the other. Prove that all of these numbers must be equal.
Problem 1
1
Hide problems
radius of circle in polygon
Prove that for a given convex polygon of area
A
A
A
and perimeter
P
P
P
there exists a circle of radius
A
P
\frac AP
P
A
that is contained in the interior of the polygon.