MathDB
Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
1989 Federal Competition For Advanced Students
1989 Federal Competition For Advanced Students
Part of
Austrian MO National Competition
Subcontests
(4)
4
1
Hide problems
radius
Prove that for any triangle each exradius is less than four times the circumradius.
3
1
Hide problems
real roots
Let
a
a
a
be a real number. Prove that if the equation x^2\minus{}ax\plus{}a\equal{}0 has two real roots
x
1
x_1
x
1
and
x
2
x_2
x
2
, then: x_1^2\plus{}x_2^2 \ge 2(x_1\plus{}x_2).
2
1
Hide problems
easy inequality
If
a
a
a
and
b
b
b
are nonnegative real numbers with a^2\plus{}b^2\equal{}4, show that: \frac{ab}{a\plus{}b\plus{}2} \le \sqrt{2}\minus{}1 and determine when equality occurs.
1
1
Hide problems
maximum value
Natural numbers
a
≤
b
≤
c
≤
d
a \le b \le c \le d
a
≤
b
≤
c
≤
d
satisfy a\plus{}b\plus{}c\plus{}d\equal{}30. Find the maximum value of the product P\equal{}abcd.