A function defined on the positive integers (and taking positive integers values) is given by: for all positive integers Determine with proof the number of positive integers for which
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A function $f$ defined on the positive integers (and taking positive integers values) is given by:
$\begin{matrix} f(1) = 1, f(3) = 3 \\ f(2n) = f(n) \\ f(4n + 1) = 2f(2n + 1) - f(n) \\ f(4n + 3) = 3f(2n + 1) - 2f(n)\text{,} \end{matrix}$
for all positive integers $n.$ Determine with proof the number of positive integers $\leq 1988$ for which $f(n) = n.$
Let be real numbers. Consider the quadratic equation in Using the numbers form a quadratic equation in whose roots are the same as those of the original equation. Compare the equation in and for , , .
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Let $a,b,c$ be real numbers. Consider the quadratic equation in $\cos{x}$ $$a \cos^2{x}+b \cos{x}+c=0.$$ Using the numbers $a,b,c$ form a quadratic equation in $\cos{2x}$ whose roots are the same as those of the original equation. Compare the equation in $\cos{x}$ and $\cos{2x}$ for $a=4$, $b=2$, $c=-1$.
The function is defined on the positive integers and takes non-negative integer values. and for all Determine .
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The function $f(n)$ is defined on the positive integers and takes non-negative integer values. $f(2)=0,f(3)>0,f(9999)=3333$ and for all $m,n:$ $$f(m+n)-f(m)-f(n)=0 \text{ or } 1.$$ Determine $f(1982)$.
Let be a function defined on the set of all positive integers and having its values in the same set. Suppose that for all positive integers Find the possible value for
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Let $f(n)$ be a function defined on the set of all positive integers and having its values in the same set. Suppose that $f(f(n) + f(m)) = m + n$ for all positive integers $n,m.$ Find the possible value for $f(1988).$
Let and be two integer-valued functions defined on the set of all integers such that
(a) for all integers and (b) is a polynomial function with integer coefficients and g(n) =
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Let $f$ and $g$ be two integer-valued functions defined on the set of all integers such that
(a) $f(m + f(f(n))) = -f(f(m+ 1) - n$ for all integers $m$ and $n;$
(b) $g$ is a polynomial function with integer coefficients and g(n) = $g(f(n))$ $\forall n \in \mathbb{Z}.$
Školjka 
defined on the positive integers (and taking positive integers values) is given by: 
Determine with proof the number of positive integers 
for which 
be real numbers. Consider the quadratic equation in 
Using the numbers 
whose roots are the same as those of the original equation. Compare the equation in 
, 
, 
. 
as a function of the parameter 
. Which pairs 
of integers satisfy this equation ? 
satisfies: 
for all non-negative integers 
. Find 
. 
is defined on the positive integers and takes non-negative integer values. 
and for all 
Determine 
. 
for all positive integers 
Find the possible value for 
be two integer-valued functions defined on the set of all integers such that 
for all integers 
