AoPS Community 2003 Austrian-Polish Competition
Austrian-Polish Competition 2003
www.artofproblemsolving.com/community/c1135994
by parmenides51, Beat, tmrfea
– Individual
1 Find all real polynomials p(x) such that p(x − 1)p(x + 1) = p(x2 − 1).
2 The sequence a0 , a1 , a2 , .. is defined by a0 = a, an+1 = an + L(an ), where L(m) is the last digit
of m (eg L(14) = 4). Suppose that the sequence is strictly increasing. Show that infinitely many
terms must be divisible by d = 3. For what other d is this true?
3 ABC is a triangle. Take a = BC etc as usual.
Take points T1 , T2 on the side AB so that AT1 = T1 T2 = T2 B. Similarly, take points T3 , T4 on the
side BC so that BT3 = T3 T4 = T4 C, and points T5 , T6 on the side CA so that CT5 = T5 T6 = T6 A.
Show that if a′ = BT5 , b′ = CT1 , c′ = AT3 , then there is a triangle A′ B ′ C ′ with sides a′ , b′ , c′
(a′ = B ′ C’ etc).
In the same way we take points Ti′ on the sides of A′ B ′ C ′ and put a′′ = B ′ T6′ , b′′ = C ′ T2′ , c′′ =
A′ T4′ .
Show that there is a triangle A′′ B ′′ C ′′ with sides a′′ b′′ , c′′ and that it is similar to ABC.
Find a′′ /a.
Austrian-Polish 2003
5 A triangle with sides a, b, c has area S. The distances of its entroid from the vertices are x, y, z.
Show that, if √
(x + y + z)2 ≤ (a2 + b2 + c2 )/2 + 2S 3,
then the triangle is equilateral.
posted for the latex, this hide shall be removed soon, problem comes from Austrian Polish 2003
4 A positive integer m is alpine if m divides 22n+1 + 1 for some positive integer n. Show that the
product of two alpine numbers is alpine.
6 ABCD is a tetrahedron such that we can find a sphere k(A, B, C) through A, B, C which meets
the plane BCD in the circle diameter BC, meets the plane ACD in the circle diameter AC,
and meets the plane ABD in the circle diameter AB. Show that there exist spheres k(A, B, D),
k(B, C, D) and k(C, A, D) with analogous properties.
Austrian-Polish 2003
– Team
© 2022 AoPS Incorporated 1
AoPS Community 2003 Austrian-Polish Competition
n
7 Put f (n) = nn−1
−1
. Show that n!f (n) divides (nn )!.
Find as many positive integers as possible for which n!f (n)+1 does not divide (nn )! .
9 Take any 26 distinct numbers from 1, 2, ... , 100. Show that there must be a non-empty subset
of the 26 whose product is a square.
I think that the upper limit for such subset is 37.
8 Given reals x1 ≥ x2 ≥ ... ≥ x2003 ≥ 0, show that
xn1 − xn2 + xn2 − ... − xn2002 + xn2003 ≥ (x1 − x2 + x3 − x4 + ... − x2002 + x2003 )n
for any positive integer n.
10 What is the smallest number of 5 × 1 tiles which must be placed on a 31 × 5 rectangle (each
covering exactly 5 unit squares) so that no further tiles can be placed? How many different
ways are there of placing the minimal number (so that further tiles are blocked)? What are the
answers for a 52 × 5 rectangle?
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