MathDB

4

Part of 1996 Estonia National Olympiad

Problems(4)

can remainder of division of a prime p> 30 by 30 be a composite

Source: 1996 Estonia National Olympiad Final Round grade 9 p4

3/13/2020
Can the remainder of the division of a prime number p>30p> 30 by 3030 be a composite?
number theoryprime numbers
computational with areas of triangles inside a square

Source: 1996 Estonia National Olympiad Final Round grade 10

3/11/2020
Let K,L,MK, L, M, and NN be the midpoints of CD,DA,ABCD,DA,AB and BCBC of a square ABCDABCD respectively. Find the are of the triangles AKB,BLC,CMDAKB, BLC, CMD and DNADNA if the square ABCDABCD has area 11.
triangle areaareasgeometrysquare
for each prime p>5 exists n such p^n ends in 001 in decimal representation.

Source: 1996 Estonia National Olympiad Final Round grade 12 p4

3/11/2020
Prove that for each prime number p>5p > 5 there exists a positive integer n such that pnp^n ends in 001001 in decimal representation.
decimal representationprimesPower
1^n+2^n+...+15^n is divisible by 480 for each odd n >=5

Source: 1996 Estonia National Olympiad Final Round grade 11 p4

3/11/2020
Prove that, for each odd integer n≥5n \ge 5, the number 1n+2n+...+15n1^n+2^n+...+15^n is divisible by 480480.
number theorydivisibledividesSum of powersSum