MathDB

1

Part of 2002 Estonia National Olympiad

Problems(4)

K,L on sides BC,CD of square <AKB = <AKL, <KAL ?

Source: 2002 Estonia National Olympiad Final Round grade 9 p1

3/16/2020
Points KK and LL are taken on the sides BCBC and CDCD of a square ABCDABCD so that ∠AKB=∠AKL\angle AKB = \angle AKL. Find ∠KAL\angle KAL.
equal anglesanglessquaregeometry
3m + n = 3u + d , where u =lcm (m,n) and d=gcd(m,n) , prove n | m

Source: 2002 Estonia National Olympiad Final Round grade 10 p1

3/16/2020
The greatest common divisor dd and the least common multiple uu of positive integers mm and nn satisfy the equality 3m+n=3u+d3m + n = 3u + d. Prove that mm is divisible by nn.
GCDLCMdivisibledividesnumber theory
x^8 +ax^4 +1 = 0 has 4 real roots that form an arithmetic progression

Source: 2002 Estonia National Olympiad Final Round grade 11 p1

3/14/2020
Find all real parameters aa for which the equation x8+ax4+1=0x^8 +ax^4 +1 = 0 has four real roots forming an arithmetic progression.
Arithmetic Progressionalgebrapolynomialarithmetic sequence
min time for 4 people to pass a dark tunnel, one torch, at most 2

Source: 2002 Estonia National Olympiad Final Round grade 12 p1

3/14/2020
Peeter, Juri, Kati and Mari are standing at the entrance of a dark tunnel. They have one torch and none of them dares to be in the tunnel without it, but the tunnel is so narrow that at most two people can move together. It takes 11 minute for Peeter, 22 minutes for Juri, 55 for Kati and 1010 for Mari to pass the tunnel. Find the minimum time in which they can all pass through the tunnel.
algebramin