MathDB

4

Part of 2016 Latvia National Olympiad

Problems(4)

2016 Latvia National Olympiad 3rd Round Grade9Problem4

Source:

7/22/2016
Find the least prime factor of the number 20162016−33\frac{2016^{2016}-3}{3}.
number theory
2016 Latvia National Olympiad 3rd Round Grade11Problem4

Source:

7/22/2016
The integer sequence (si)(s_i) "having pattern 2016'" is defined as follows: ∘\circ The first member s1s_1 is 2. ∘\circ The second member s2s_2 is the least positive integer exceeding s1s_1 and having digit 0 in its decimal notation. ∘\circ The third member s3s_3 is the least positive integer exceeding s2s_2 and having digit 1 in its decimal notation. ∘\circ The third member s3s_3 is the least positive integer exceeding s2s_2 and having digit 6 in its decimal notation. The following members are defined in the same way. The required digits change periodically: 2→0→1→6→2→0→…2 \rightarrow 0 \rightarrow 1 \rightarrow 6 \rightarrow 2 \rightarrow 0 \rightarrow \ldots. The first members of this sequence are the following: 2;10;11;16;20;30;31;36;42;502; 10; 11; 16; 20; 30; 31; 36; 42; 50.\\ Does this sequence contain a) 2001, b) 2006?
recursionnumber theory
2016 Latvia National Olympiad 3rd Round Grade10Problem4

Source:

7/22/2016
In a Pythagorean triangle all sides are longer than 5. Is it possible that (a) all three sides are prime numbers, (b) exactly two sides are prime numbers. (Note: We call a triangle "Pythagorean", if it is a right-angled triangle where all sides are positive integers.)
number theoryprime numbers
2016 Latvia National Olympiad 3rd Round Grade12Problem4

Source:

7/22/2016
Two functions are defined by equations: f(a)=a2+3a+2f(a) = a^2 + 3a + 2 and g(b,c)=b2−b+3c2+3cg(b, c) = b^2 - b + 3c^2 + 3c. Prove that for any positive integer aa there exist positive integers bb and cc such that f(a)=g(b,c)f(a) = g(b, c).
functionalgebra