1

Part of 1995 IberoAmerican

Problems(2)

10th ibmo - chile 1995/q1.

Source: Spanish Communities

Find all the possible values of the sum of the digits of all the perfect squares.
[Commented by djimenez] Comment: I would rewrite it as follows: Let

f: \mathbb{N}\rightarrow \mathbb{N}

f(n)

is the sum of all the digits of the number

n^2

. Find the image of

f

(where, by image it is understood the set of all

x

such that exists an

n

f(n)=x

number theorynumber theory unsolved

10th ibmo - chile 1995/q4.

Source: Spanish Communities

m\times{n}

grid are there are token. Every token dominates every square on its same row (

\leftrightarrow

), its same column (

\updownarrow

), and diagonal (

\searrow\hspace{-4.45mm}\nwarrow

)(Note that the token does not \emph{dominate} the diagonal (

\nearrow\hspace{-4.45mm}\swarrow

), determine the lowest number of tokens that must be on the board to dominate all the squares on the board.

ceiling functionsymmetrycombinatorics unsolvedcombinatorics