Functional Equations: Problems with Solutions

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 \(f:\mathbb{N}\rightarrow\mathbb{N}\) 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 \[f(x^2+y^2+2f(xy))=f(x+y)^2.\]