MathDB
Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
1997 Spain Mathematical Olympiad
1997 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(6)
6
1
Hide problems
n containers distribute gas to a car for a single loop along the track
The exact quantity of gas needed for a car to complete a single loop around a track is distributed among
n
n
n
containers placed along the track. Prove that there exists a position starting at which the car, beginning with an empty tank of gas, can complete a loop around the track without running out of gas. The tank of gas is assumed to be large enough.
5
1
Hide problems
sum of lengths of sides and diagonals >= 2(2+\sqrt2), when (ABCD)=1
Prove that in every convex quadrilateral of area
1
1
1
, the sum of the lengths of the sides and diagonals is not smaller than
2
(
2
+
2
)
2(2+\sqrt2)
2
(
2
+
2
)
.
4
1
Hide problems
find all integers k for which \sqrt{k^2 − pk} is a positive integer, if p prime
Let
p
p
p
be a prime number. Find all integers
k
k
k
for which
k
2
−
p
k
\sqrt{k^2 -pk}
k
2
−
p
k
is a positive integer.
2
1
Hide problems
max No. 3 of 16 vertices on 4x4 grid, do not form an isosceles right triangle
A square of side
5
5
5
is divided into
25
25
25
unit squares. Let
A
A
A
be the set of the
16
16
16
interior points of the initial square which are vertices of the unit squares. What is the largest number of points of
A
A
A
no three of which form an isosceles right triangle?
1
1
Hide problems
sum of squares of the first 100 terms of an arithmetic progression
Compute the sum of the squares of the first
100
100
100
terms of an arithmetic progression, given that their sum is
−
1
-1
−
1
and that the sum of those among them having an even index is
1
1
1
.
3
1
Hide problems
Parabolas producing Circumcircles with Common Point
For each parabola
y
=
x
2
+
p
x
+
q
y = x^2+ px+q
y
=
x
2
+
p
x
+
q
intersecting the coordinate axes in three distinct points, consider the circle passing through these points. Prove that all these circles pass through a single point, and find this point.