Problems for Independent Study

Most of the ideas for solving the problems below are already mentioned in the introduction or in the section with solved problems. The difficulty of the problems vary as well as the range of ideas used to solve them. Before solving the problems we highly encourage you to first solve (or look at the solutions) the problems from the previous section. Some of the problems are quite difficult.

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

Problem 2
 
Find all functions \( f:\mathbb{Z}\rightarrow\mathbb{Z} \) for which we have \( f(0)=1 \) and \( f(f(n))=f(f(n+2)+2)=n \), for every natural number \( n \).

Problem 3
 
Find all functions \( f:\mathbb{N}\rightarrow\mathbb{N} \) for which \( f(n) \) is a square of an integer for all \( n\in\mathbb{N} \), and that satisfy \( f(m+n)=f(m)+f(n)+2mn \) for all \( m,n\in\mathbb{N} \).

Problem 4
 
Find all functions \( f:\mathbb{R}\rightarrow\mathbb{R} \) that satisfy \( f((x-y)^2)=f(x)^2-2xf(y)+y^2. \)

Problem 5
 
Let \( n\in\mathbb{N} \). Find all monotone functions \( f:\mathbb{R}\rightarrow\mathbb{R} \) such that \[ f(x+f(y))=f(x)+y^n.\]

Problem 6 (USA 2002)
 
Find all functions \( f:\mathbb{R}\rightarrow\mathbb{R} \) which satisfy the equality \( f(x^2-y^2)=xf(x)-yf(y) \).

Problem 7 (Mathematical High Schol, Belgrade 2004)
 
Find all functions \( f:\mathbb{N}\rightarrow\mathbb{N} \) such that \( f(f(m)+f(n))=m+n \) for every two natural numbers \( m \) and \( n \).

Problem 8
 
Find all continuous functions \( f:\mathbb{R}\rightarrow\mathbb{R} \) such that \( f(xy)=xf(y)+yf(x). \)

Problem 9 (IMO 1983, problem 1)
 
Find all functions \( f:\mathbb{R}\rightarrow\mathbb{R} \) such that

  • (i) \( f(xf(y))=yf(x) \), for all \( x,y\in\mathbb{R} \);

  • (ii) \( f(x)\rightarrow 0 \) as \( x\rightarrow+\infty \).


Problem 10
 
Let \( f:\mathbb{N}\rightarrow\mathbb{N} \) be strictly increasing function that satisfies \( f(f(n))=3n \) for every natural number \( n \). Determine \( f(2006) \).

Problem 11 (IMO 1989, shortlist)
 
Let \( 0< a< 1 \) be a real number and \( f \) continuous function on \( [0,1] \) which satisfies \( f(0)=0 \), \( f(1)=1 \), and \[ f\Big( \frac {x+y}{2}\Big)=(1-a)f(x)+af(y),\] for every two real numbers \( x,y\in[0,1] \) such that \( x\leq y \). Determine \( f\Big( \frac 17\Big) \).

Problem 12 (IMO 1996, shortlist)
 
Let \( f:\mathbb{R}\rightarrow\mathbb{R} \) be the function such that \( |f(x)|\leq 1 \) and \[ f\Big(x+ \frac{13}{42}\Big)+f(x)=f\Big(x+\frac 16\Big)+f\Big(x+\frac 17\Big).\] Prove that \( f \) is periodic.

Problem 13 (BMO 2003, problem 3)
 
Find all functions \( f:\mathbb{Q}\rightarrow\mathbb{R} \) that satisfy:

  • (i) \( f(x+y)-yf(x)-xf(y)=f(x)f(y)-x-y+xy \) for every \( x \), \( y\in\mathbb{Q} \);

  • (ii) \( f(x)=2f(x+1)+2+x \), for every \( x\in\mathbb{Q} \);

  • (iii) \( f(1)+1> 0 \).


Problem 14 (IMO 1990, problem 4)
 
Determine the function \( f:\mathbb{Q}^+\rightarrow\mathbb{Q}^+ \) such that \[ f(xf(y))=\frac{f(x)}{y},\ \mbox{for all }x,y\in\mathbb{Q}^+.\]

Problem 15 (IMO 2002, shortlist)
 
Find all functions \( f:\mathbb{R}\rightarrow\mathbb{R} \) such that \[ f(f(x)+y)=2x+f(f(y)-x).\]

Problem 16 (Iran 1997)
 
Let \( f:\mathbb{R}\rightarrow\mathbb{R} \) be an increasing function such that for all positive real numbers \( x \) and \( y \): \[ f(x+y)+f(f(x)+f(y))=f(f(x+f(y))+f(y+f(x))).\] Prove that \( f(f(x))=x \).

Problem 17 (IMO 1992, problem 2)
 
Find all functions \( f:\mathbb{R}\rightarrow\mathbb{R} \), such that \( f(x^2+f(y))=y+f(x)^2 \) for all \( x,y\in\mathbb{R} \).

Problem 18 (IMO 1994, problem 5)
 
Let \( S \) be the set of all real numbers strictly greater than -1. Find all functions \( f:S\rightarrow S \) that satisfy the following two conditions:

  • (i) \( f(x+f(y)+xf(y))=y+f(x)+yf(x) \) for all \( x \), \( y\in S \);

  • (ii) \( \frac{f(x)}{x} \) is strictly increasing on each of the intervals \( -1< x< 0 \) and \( 0< x \).


Problem 19 (IMO 1994, shortlist)
 
Find all functions \( f:\mathbb{R}^+\rightarrow\mathbb{R} \) such that \[ f(x)f(y)=y^{\alpha}f(x/2)+x^{\beta}f(y/2), \;\mbox{ for all } x,y\in\mathbb{R}^+.\]

Problem 20 (IMO 2002, problem 5)
 
Find all functions \( f:\mathbb{R}\rightarrow\mathbb{R} \) such that \[ (f(x)+f(z))(f(y)+f(t))=f(xy-zt)+f(xt+yz).\]

Problem 21 (Vietnam 2005)
 
Find all values for a real parameter \( \alpha \) for which there exists exactly one function \( f:\mathbb{R}\rightarrow\mathbb{R} \) satisfying \[ f(x^2+y+f(y))=f(x)^2+\alpha\cdot y.\]

Problem 22 (IMO 1998, problem 3)
 
Find the least possible value for \( f(1998) \) where \( f:\mathbb{N}\rightarrow\mathbb{N} \) is a function that satisfies \[ f(n^2f(m))=mf(n)^2.\]

Problem 23
 
Does there exist a function \( f:\mathbb{N}\rightarrow\mathbb{N} \) such that \[ f(f(n-1))=f(n+1)-f(n)\] for each natural number \( n \)?

Problem 24 (IMO 1987, problem 4)
 
Does there exist a function \( f:\mathbb{N}_0\rightarrow\mathbb{N}_0 \) such that \( f(f(n))=n+1987 \)?

Problem 25
 
Assume that the function \( f:\mathbb{N}\rightarrow\mathbb{N} \) satisfies \( f(n+1)> f(f(n)) \), for every \( n\in\mathbb{N} \). Prove that \( f(n)=n \) for every \( n \).

Problem 26
 
Find all functions \( f:\mathbb{N}_0\rightarrow\mathbb{N}_0 \), that satisfy:

  • (i) \( 2f(m^2+n^2)=f(m)^2+f(n)^2 \), for every two natural numbers \( m \) and \( n \);

  • (ii) If \( m\geq n \) then \( f(m^2)\geq f(n^2) \).


Problem 27
 
Find all functions \( f:\mathbb{N}_0\rightarrow \mathbb{N}_0 \) that satisfy:

    (i) \( f(2)=2 \);

  • (ii) \( f(mn)=f(m)f(n) \) for every two relatively prime natural numbers \( m \) and \( n \);

  • (iii) \( f(m)< f(n) \) whenever \( m< n \).


Problem 28
 
Find all functions \( f:\mathbb{N}\rightarrow [1,\infty) \) that satisfy conditions (i) and (ii) of the previous problem and the condition (ii) is modified to require the equality for every two natural numbers \( m \) and \( n \).

Problem 29
 
Given a natural number \( k \), find all functions \( f: \mathbb N_0\rightarrow \mathbb N_0 \) for which \[ f(f(n))+f(n)=2n+3k,\] for every \( n\in\mathbb N_0 \).

Problem 30 (Vietnam 2005)
 
Find all functions \( f:\mathbb{R}\rightarrow\mathbb{R} \) that satisfy \( f(f(x-y))=f(x)f(y)-f(x)+f(y)-xy \).

Problem 31 (China 1996)
 
The function \( f:\mathbb{R}\rightarrow\mathbb{R} \) satisfy \( f(x^3+y^3)=(x+y)\Big(f(x)^2-f(x)f(y)+f(y)^2\Big) \), for all real numbers \( x \) and \( y \). Prove that \( f(1996x)=1996f(x) \) for every \( x\in\mathbb R \).

Problem 32
 
Find all functions \( f:\mathbb{R}\rightarrow\mathbb{R} \) that satisfy:

    (i) \( f(x+y)=f(x)+f(y) \) for every two real numbers \( x \) and \( y \);

  • (ii) \( f\Big( \frac{1}{x}\Big)= \frac{f(x)}{x^2} \) for \( x\neq 0. \)


Problem 33 (IMO 1989, shortlist)
 
A function \( f:\mathbb{Q}\rightarrow\mathbb{R} \) satisfy the following conditions:

  • (i) \( f(0)=0 \), \( f(\alpha)> 0 \) za \( \alpha\neq 0 \);

  • (ii) \( f(\alpha\beta)=f(\alpha)f(\beta) \) i \( f(\alpha+\beta)\leq f(\alpha)+f(\beta) \), for all \( \alpha,\beta\in\mathbb{Q} \);

  • (iii) \( f(m)\leq 1989 \) za \( m\in \mathbb{Z} \).


Prove that \( f(\alpha+\beta)=\max\{f(\alpha),f(\beta)\} \) whenever \( f(\alpha)\neq f(\beta) \).

Problem 34
 
Find all functions \( f:\mathbb{R}\rightarrow\mathbb{R} \) such that for every two real numbers \( x\neq y \) the equality \[ f \Big ( \frac{x+y}{x-y} \Big ) =\frac{f(x)+f(y)}{f(x)-f(y)}\] is satisfied.

Problem 35
 
Find all functions \( f:\mathbb{Q}^+\rightarrow\mathbb{Q}^+ \) satisfying:

    (i) \( f(x+1)=f(x)+1 \) for all \( x\in \mathbb{Q}^+ \);

  • (ii) \( f(x^3)=f(x)^3 \) for all \( x\in \mathbb{Q}^+ \).


Problem 36
 
Find all continuous functions \( f:\mathbb{R}\rightarrow\mathbb{R} \) that satisfy the equality \[ f(x+y)+f(xy)=f(x)+f(y)+f(xy+1).\]

Problem 37
 
Find all continuous functions \( f,g,h,k:\mathbb{R}\rightarrow\mathbb{R} \) that satisfy the equality \[ f(x+y)+g(x-y)=2h(x)+2k(y).\]

Problem 38 (IMO 1996, shortlist)
 
Find all functions \( f:\mathbb{N}_0\rightarrow\mathbb{N}_0 \) such that \[ f(m+f(n))=f(f(m))+f(n).\]

Problem 39 (IMO 1995, shortlist)
 
Does there exist a function \( f:\mathbb{R}\rightarrow\mathbb{R} \) satisfying the conditions:

  • (i) There exists a positive real number \( M \) such that \( -M\leq f(x)\leq M \) for all \( x\in\mathbb R \);

  • (ii) \( f(1)=1 \);

  • (iii) If \( x\neq 0 \) then \( f\Big(x+ \frac{1}{x^2}\Big)=f(x)+\left[f\Big( \frac 1x\Big)\right]^2 \)?


Problem 40 (Belarus)
 
Find all continuous functions \( f:\mathbb{R}\rightarrow\mathbb{R} \) that satisfy \[ f(f(x))=f(x)+2x.\]

Problem 41
 
Prove that if the function \( f:\mathbb{R}^+\rightarrow\mathbb{R} \) satisfy the equality \[ f\Big(\frac{x+y}{2}\Big)+f\Big(\frac{2xy}{x+y}\Big)= f(x)+f(y),\] the it satisfy the equality \( 2f(\sqrt{xy})=f(x)+f(y) \) as well.

Problem 42
 
Find all continuous functions \( f: (0,\infty)\rightarrow (0,\infty) \) that satisfy \[ f(x)f(y)=f(xy)+f(x/y).\]

Problem 43
 
Prove that there is no function \( f:\mathbb{R}\rightarrow\mathbb{R} \) that satisfy the inequality \( f(y)> (y-x)f(x)^2 \), for every two real numbers \( x \) and \( y \).

Problem 44 (IMC 2001)
 
Prove that there doesn\’t exist a function \( f:\mathbb{R}\rightarrow\mathbb{R} \) for which \( f(0)> 0 \) and \[ f(x+y)\geq f(x)+yf(f(x)).\]

Problem 45 (Romania 1998)
 
Find all functions \( u:\mathbb{R}\rightarrow\mathbb{R} \) for which there exists a strictly monotone function \( f:\mathbb{R}\rightarrow\mathbb{R} \) such that \[ f(x+y)=f(x)u(y)+f(y),\quad \forall x,y\in\mathbb{R}.\]

Problem 46 (Iran 1999)
 
Find all functions \( f:\mathbb{R}\rightarrow\mathbb{R} \) for which \[ f(f(x)+y)=f(x^2-y)+4f(x)y.\]

Problem 47 (IMO 1988, problem 3)
 
A function \( f:\mathbb{N}\rightarrow\mathbb{N} \) satisfies:

  • (i) \( f(1)=1 \), \( f(3)=3 \);

  • (ii) \( f(2n)=f(n) \);

  • (iii) \( f(4n+1)=2f(2n+1)-f(n) \) and \( f(4n+3)=3f(2n+1)-2f(n) \),


for every natural number \( n\in\mathbb{N} \). Find all natural numbers \( n\leq 1998 \) such that \( f(n)=n \).

Problem 48 (IMO 2000, shortlist)
 
Given a function \( F:\mathbb{N}_0\rightarrow\mathbb{N}_0 \), assume that for \( n\geq 0 \) the following relations hold:

  • (i) \( F(4n)=F(2n)+F(n) \);

  • (ii) \( F(4n+2)=F(4n)+1 \);

  • (iii) \( F(2n+1)=F(2n)+1 \).


Prove that for every natural number \( m \), the number of positive integers \( n \) such that \( 0\leq n< 2^m \) and \( F(4n)=F(3n) \) is equal to \( F(2^{m+1}) \).

Problem 49
 
Let \( f:\mathbb{Q}\times\mathbb{Q}\rightarrow\mathbb{Q}^+ \) be a function satisfying \[ f(xy,z)=f(x,z)f(y,z),\quad f(z,xy)=f(z,x)f(z,y),\quad f(x,1-x)=1,\] for all rational numbers \( x,y,z \). Prove that \( f(x,x)=1 \), \( f(x,-x)=1 \), and \( f(x,y)f(y,x)=1 \).

Problem 50
 
Find all functions \( f:\mathbb{N}\times\mathbb{N}\rightarrow\mathbb{R} \) that satisfy \[ f(x,x)=x,\quad f(x,y)=f(y,x),\quad (x+y)f(x,y)=yf(x,x+y).\]


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