MathDB

4

Part of 2010 Lithuania National Olympiad

Problems(2)

divisible by 37

Source: Lithuania NMO 2010

3/11/2012
Decimal digits a,b,ca,b,c satisfy 37∣(a0a0…a0b0c0c…0c)10 37\mid (a0a0\ldots a0b0c0c\ldots 0c)_{10} where there are 10011001 a's and 10011001 c's. Prove that b=a+cb=a+c.
modular arithmeticnumber theory proposednumber theory
residues modulo 5

Source: Lithuania NMO 2010

3/11/2012
Arrange arbitrarily 1,2,…,251,2,\ldots ,25 on a circumference. We consider all 2525 sums obtained by adding 55 consecutive numbers. If the number of distinct residues of those sums modulo 55 is dd (0≤d≤5)(0\le d\le 5),find all possible values of dd.
combinatorics proposedcombinatorics