MathDB
Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
2014 Swedish Mathematical Competition
2014 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
5
1
Hide problems
20 finalists, 6 problems, 5 contestants vs 2 problems ?
In next year's finals in Schools Mathematics competition,
20
20
20
finalists will participate. The final exam contains six problems. Emil claims that regardless of results, there must be five contestants and two problems such that either all the five contestants solve both problems, or neither of them solve any of the two problems. Is he right?
4
1
Hide problems
sum of diameters of incircles of triangles > sidelength of square
A square is cut into a finitely number of triangles in an arbitrary way. Show the sum of the diameters of the inscribed circles in these triangles is greater than the side length of the square.
1
1
Hide problems
non-negative integer P(x), with p (1) = 7,p (10) = 2014
Determine all polynomials
p
(
x
)
p(x)
p
(
x
)
with non-negative integer coefficients such that
p
(
1
)
=
7
p (1) = 7
p
(
1
)
=
7
and
p
(
10
)
=
2014
p (10) = 2014
p
(
10
)
=
2014
.
6
1
Hide problems
find odd primes p,q such that x^p + y^q = pq has >=1 solution
Determine all odd primes
p
p
p
and
q
q
q
such that the equation
x
p
+
y
q
=
p
q
x^p + y^q = pq
x
p
+
y
q
=
pq
at least one solution
(
x
,
y
)
(x, y)
(
x
,
y
)
where
x
x
x
and
y
y
y
are positive integers.
3
1
Hide problems
f (f (x + y) - f (x - y)) = xy
Determine all functions
f
:
R
→
R
f: \mathbb R \to \mathbb R
f
:
R
→
R
, such that
f
(
f
(
x
+
y
)
−
f
(
x
−
y
)
)
=
x
y
f (f (x + y) - f (x - y)) = xy
f
(
f
(
x
+
y
)
−
f
(
x
−
y
))
=
x
y
for all real
x
x
x
and
y
y
y
.
2
1
Hide problems
total area of 3 ext. tangent circles with centers on fourth circle <4\pi R^2
Three circles that touch each other externally have all their centers on one fourth circle with radius
R
R
R
. Show that the total area of the three circle disks is smaller than
4
π
R
2
4\pi R^2
4
π
R
2
.