Functional Equations: Problems with Solutions

The following problems are related to functional equations. Many of the problems were given at national and international mathematical competitions and olympiads, and thus are challenging. You may want to read first an introductory text to Functional equations.

Problem 1
 
Find all functions \( f:\mathbb{Q}\rightarrow\mathbb{Q} \) such that \( f(1)=2 \) and \( f(xy)=f(x)f(y)-f(x+y)+1 \).

Problem 2 (Belarus 1997)
 
Find all functions \( g:\mathbb{R}\rightarrow\mathbb{R} \) such that for arbitrary real numbers \( x \) and \( y \): \[ g(x+y)+g(x)g(y)=g(xy)+g(x)+g(y).\]

Problem 3
 
The function \( f:\mathbb{R}\rightarrow\mathbb{R} \) satisfies \( x+f(x)=f(f(x)) \) for every \( x\in\mathbb{R} \). Find all solutions of the equation \( f(f(x))=0 \).

Problem 4
 
Find all injective functions \( f:\mathbb{N}\rightarrow\mathbb{R} \) that satisfy: \[ (a)\ f(f(m)+f(n))=f(f(m))+f(n), \quad (b)\ f(1)=2,\ f(2)=4.\]

Problem 5 (BMO 1997, 2000)
 
Solve the functional equation \[ f(xf(x)+f(y))=y+f(x)^2,\ x,y\in\mathbb{R}.\]

Problem 6 (IMO 1979, shortlist)
 
Given a function \( f:\mathbb{R}\rightarrow\mathbb{R} \), if for every two real numbers \( x \) and \( y \) the equality \( f(xy+x+y)=f(xy)+f(x)+f(y) \) holds, prove that \( f(x+y)=f(x)+f(y) \) for every two real numbers \( x \) and \( y \).

Problem 7
 
Does there exist a function \( f:\mathbb{R}\rightarrow\mathbb{R} \) such that \( f(f(x))=x^2-2 \) for every real number \( x \)?

Problem 8
 
Find all functions \( f:\mathbb{R}^+\rightarrow\mathbb{R}^+ \) such that \( f(x)f(yf(x))=f(x+y) \) for every two positive real numbers \( x,y \).

Problem 9
 
(IMO 2000, shortlist) Find all pairs of functions \( f:\mathbb{R}\rightarrow\mathbb{R} \) and \( g:\mathbb{R}\rightarrow\mathbb{R} \) such that for every two real numbers \( x,y \) the following relation holds: \[ f(x+g(y))=xf(y)-yf(x)+g(x).\]

Problem 10 (IMO 1992, shortlist)
 
Find all functions \( f:\mathbb{R}^+\rightarrow\mathbb{R}^+ \) which satisfy \[ f(f(x))+af(x)=b(a+b)x.\]

Problem 11 (Vietnam 2003)
 
Let \( F \) be the set of all functions \( f:\mathbb{R}^+\rightarrow\mathbb{R}^+ \) which satisfy the inequality \( f(3x)\geq f(f(2x))+x \), for every positive real number \( x \). Find the largest real number \( \alpha \) such that for all functions \( f\in F \): \( f(x)\geq \alpha\cdot x \).

Problem 12
 
Find all functions \( f,g,h:\mathbb{R}\rightarrow\mathbb{R} \) that satisfy \[ f(x+y)+g(x-y)=2h(x)+2h(y).\]

Problem 13
 
Find all functions \( f:\mathbb{Q}\rightarrow\mathbb{Q} \) for which \[ f(xy)=f(x)f(y)-f(x+y)+1.\] Solve the same problem for the case \( f:\mathbb{R}\rightarrow\mathbb{R} \).

Problem 14
 
(IMO 2003, shortlist) Let \( \mathbb{R}^+ \) denote the set of positive real numbers. Find all functions \( f:\mathbb{R}^+\rightarrow \mathbb{R}^+ \) that satisfy the following conditions:

  • (i) \( f(xyz)+f(x)+f(y)+f(z)=f(\sqrt{xy})f(\sqrt{yz})f(\sqrt{zx}) \)

  • (ii) \( f(x)< f(y) \) for all \( 1\leq x< y \).

Problem 15
 
Find all functions \( f:[1,\infty)\rightarrow [1,\infty) \) that satisfy:

  • (i) \( f(x)\leq 2(1+x) \) for every \( x\in [1,\infty) \);

  • (ii)\( xf(x+1)=f(x)^2-1 \) for every \( x\in [1,\infty) \).

Problem 16 (IMO 1999, probelm 6)
 
Find all functions \( f:\mathbb{R}\rightarrow \mathbb{R} \) such that \[ f(x-f(y))=f(f(y))+xf(y)+f(x)-1.\]

Problem 17
 
Given an integer \( n \), let \( f:\mathbb{R}\rightarrow\mathbb{R} \) be a continuous function satisfying \( f(0)=0 \), \( f(1)=1 \), and \( f^{(n)}(x)=x \), for every \( x\in[0,1] \). Prove that \( f(x)=x \) for each \( x\in[0,1] \).

Problem 18
 
Find all functions \( f: (0,+\infty)\rightarrow(0,+\infty) \) that satisfy \( f(f(x)+y)=xf(1+xy) \) for all \( x,y\in(0,+\infty) \).

Problem 19 (Bulgaria 1998)
 
Prove that there is no function \( f:\mathbb{R}^+\rightarrow\mathbb{R}^+ \) such that \( f(x)^2\geq f(x+y)(f(x)+y) \) for every two positive real numbers \( x \) and \( y \).

Problem 20
 
Let \( f:\mathbb{N}\rightarrow\mathbb{N} \) be a function satisfying \[ f(1)=2,\quad f(2)=1,\quad f(3n)=3f(n),\quad f(3n+1)=3f(n)+2,\quad f(3n+2)=3f(n)+1.\] Find the number of integers \( n\leq 2006 \) for which \( f(n)=2n \).

Problem 21 (BMO 2003, shortlist)
 
Find all possible values for \( f\Big( \frac{2004}{2003}\Big) \) if \( f:\mathbb{Q}\rightarrow[0,+\infty) \) is the function satisfying the conditions:

  • (i) \( f(xy)=f(x)f(y) \) for all \( x,y\in\mathbb{Q} \);

  • (ii) \( f(x)\leq 1\Rightarrow f(x+1)\leq 1 \) for all \( x\in\mathbb{Q} \);

  • (iii) \( f\Big( \frac{2003}{2002}\Big)=2 \).

Problem 22
 
Let \( I=[0,1] \), \( G=I\times I \) and \( k\in\mathbb{N} \). Find all \( f:G\rightarrow I \) such that for all \( x,y,z\in I \) the following statements hold:

  • (i) \( f(f(x,y),z)=f(x,f(y,z)) \);

  • (ii) \( f(x,1)=x \), \( f(x,y)=f(y,x) \);

  • (iii) \( f(zx,zy)=z^kf(x,y) \) for every \( x,y,z\in I \), where \( k \) is a fixed real number.

Problem 23 (APMO 1989)
 
Find all strictly increasing functions \( f:\mathbb{R}\rightarrow\mathbb{R} \) such that \[ f(x)+g(x)=2x,\] where \( g \) is the inverse of \( f \).

Problem 24
 
Find all functions \( h:\mathbb{N}\rightarrow\mathbb{N} \) that satisfy \[ h(h(n))+h(n+1)=n+2.\]

Problem 25 (IMO 2004, shortlist)
 
Find all functions \( f:\mathbb{R}\rightarrow\mathbb{R} \) satisfying the equality \[ f(x^2+y^2+2f(xy))=f(x+y)^2.\]


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