MathDB

6

Part of 2011 Iran MO (3rd Round)

Problems(3)

bacteria filling cells of a table

Source: Iran 3rd round 2011-combinatorics exam-p6

9/4/2011
Every bacterium has a horizontal body with natural length and some nonnegative number of vertical feet, each with nonnegative (!) natural length, that lie below its body. In how many ways can these bacteria fill an m×nm\times n table such that no two of them overlap?
proposed by Mahyar Sefidgaran
combinatorics proposedcombinatorics
how many functions are there?

Source: Iran 3rd round 2011-number theory exam-p6

9/5/2011
aa is an integer and pp is a prime number and we have p≥17p\ge 17. Suppose that S={1,2,....,p−1}S=\{1,2,....,p-1\} and T={y∣1≤y≤p−1,ordp(y)<p−1}T=\{y|1\le y\le p-1,ord_p(y)<p-1\}. Prove that there are at least 4(p−3)(p−1)p−44(p-3)(p-1)^{p-4} functions f:S⟶Sf:S\longrightarrow S satisfying ∑x∈Txf(x)≡a\sum_{x\in T} x^{f(x)}\equiv a (mod(mod p)p). proposed by Mahyar Sefidgaran
functionnumber theory proposednumber theory
fighting circles

Source: Iran 3rd round 2011-final exam-p6

9/12/2011
We call two circles in the space fighting if they are intersected or they are clipsed. Find a good necessary and sufficient condition for four distinct points A,B,A′,B′A,B,A',B' such that each circle passing through A,BA,B and each circle passing through A′,B′A',B' are fighting circles.
proposed by Ali Khezeli
geometry proposedgeometry