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 10 Given a three digit number $n$, let $f(n)$ be the sum of digits of $n$, their products in pairs, and the product of all digits. When does $n=f(n)$? 7 Prove Fibonacci Sequence Property: $x^2_n + x^2_{n+1}=x_{2n+1}$ 5 Find all functions $f:\Bbb{R} \to \Bbb{R}$ such that for all $x,y,z \in \Bbb{R}$ , $f(f(x)+yz)=x+f(y)f(z)$ 4 Cauchy equation with an additional condition 3 Find all functions for $f:\Bbb{N}\to\Bbb{N}$ such that $f\left(m^2+f(n)\right)=f\left(m^2\right) +n$

### Reputation (931)

 +55 Cauchy equation with an additional condition +10 Given a three digit number $n$, let $f(n)$ be the sum of digits of $n$, their products in pairs, and the product of all digits. When does $n=f(n)$? +2 Find “A” in this equation. +2 Squares of an $8\times8$ grid are coloured black or white. How many colourings of an $8\times8$ grid are there?

### Questions (10)

 19 Solving $f(yf(x)+x/y)=xyf(x^2+y^2)$ over the reals 5 Proving that there is an element common to all $35$ sets given certain set restrictions 4 If $pqr(p+q+r)$ is a square and $p,q,r$ are primes, then what's the maximum value of $p+q+r$? 2 Proving prime divisibility relation between $a^2-a+3$ and $b^2-b+25$. 2 Proof of polynomial divisibility without using complex numbers?

### Tags (46)

 23 functional-equations × 22 7 discrete-mathematics 18 contest-math × 19 6 real-analysis × 3 11 elementary-number-theory × 3 5 algebra-precalculus × 5 7 functions × 5 4 continuity 7 fibonacci-numbers × 2 2 geometry × 6

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