Has anyone done this?

by SomeonecoolLovesMaths, Feb 21, 2025, 1:13 PM

1. Find all surjective functions \( f : \mathbb{N} \to \mathbb{N} \) such that
\[ f(n) \ge n + (-1)^n,\quad \forall n \in \mathbb{N}. \]
2. Find all functions \( g : \mathbb{R} \to \mathbb{R} \) such that for any real numbers \( x \) and \( y \)
\[ g(x+y) + g(x)g(y) = g(xy) + g(x) + g(y). \]
3. Find all real valued functions defined on \(\mathbb{R}\) such that for every real \( x, y \)
\[ f(x^2-y^2) = xf(x) - yf(y). \]
4. Find all real valued functions defined on \(\mathbb{R}\) such that for every real \( x, y \)
\[ f\bigl(xf(x) + f(y)\bigr) = f(x)^2 + y. \]
5. Find all functions \( f : \mathbb{N} \to \mathbb{N} \) such that
\[ f(f(n)) + (f(n))^2 = n^2 + 3n + 3,\quad \forall n\in\mathbb{N}. \]
6. Let \( n \) be a positive integer. Find all strictly increasing functions \( f : \mathbb{N}^* \to \mathbb{N}^* \) such that the equation
\[ \frac{f(x)}{n} = k - x \]has an integral solution \( x \) for all \( k \in \mathbb{N} \).

7. Find all functions \( f : \mathbb{R}^+ \to \mathbb{R}^+ \) such that
\[ f\Bigl(\frac{x+y}{2}\Bigr) = \frac{2f(x)f(y)}{f(x)+f(y)} \quad \forall x,y \in \mathbb{R}^+. \]
8. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f(1-x) = 1 - f(f(x)) \quad \forall x \in \mathbb{R}. \]
9. Find all functions \( f : \mathbb{R}^+ \to \mathbb{R}^+ \) such that
\[ f(1+xf(y)) = yf(x+y) \quad \forall x,y \in \mathbb{R}^+. \]
10. Find all functions \( f : \mathbb{R}^+ \to \mathbb{R}^+ \) such that
\[ f(xf(y)) = f(x+y) \quad \forall x,y \in \mathbb{R}^+. \]
11. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f\bigl(f(x)+y\bigr) = f\bigl(x^2-y\bigr) + 4yf(x) \quad \forall x,y \in \mathbb{R}. \]
12. Find all functions \( f,\, g,\, h : \mathbb{R} \to \mathbb{R} \) such that
\[ f(x+y) + g(x-y) = 2h(x) + 2h(y) \quad \forall x,y \in \mathbb{R}. \]
13. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f(x+y+z) = f(x) \cdot f(1-y) + f(y) \cdot f(1-z) + f(z) \cdot f(1-x) \quad \forall x,y,z \in \mathbb{R}. \]
14. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f\bigl(f(x)-f(y)\bigr) = (x-y)^2\,f(x+y) \quad \forall x,y \in \mathbb{R}. \]
15. Find all functions \( f,\, g : \mathbb{R} \to \mathbb{R} \) such that
If \( x < y \), then \( f(x) < f(y) \);
For all \( x,y \in \mathbb{R} \), we have
\[ f(xy) = g(y)f(x) + f(y). \]16. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f\bigl((x+z)(y+z)\bigr) = \bigl(f(x)+f(z)\bigr)\bigl(f(y)+f(z)\bigr) \quad \forall x,y,z \in \mathbb{R}. \]
17. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) that satisfy
\[ f(x^3+y^3) = x^2f(x) + yf(y^2) \quad \forall x,y \in \mathbb{R}. \]
18. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) that satisfy
\[ f(m+nf(m)) = f(m) + m\,f(n) \quad \forall m,n \in \mathbb{R}. \]
19. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f(x)\,f(y) = f(x+y) + xy \quad \forall x,y \in \mathbb{R}. \]
20. Find all functions \( f : \mathbb{N}\cup\{0\} \to \mathbb{N}\cup\{0\} \) such that
\[ x\cdot 3f(y) \text{ divides } f(x)\cdot 3y \quad \forall x,y \in \mathbb{N}\cup\{0\}. \]
21. Find all continuous functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f(x+y)\,f(x-y) = \bigl(f(x)f(y)\bigr)^2 \quad \forall x,y \in \mathbb{R}. \]
22. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ (x+y)\Bigl(f(x)-f(y)\Bigr) = (x-y)f(x+y) \quad \forall x,y \in \mathbb{R}. \]
23. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f\bigl(f(x)+y\bigr) = f\bigl(x^2-y\bigr) + 4f(x)y \quad \forall x,y \in \mathbb{R}. \]
24. Find all functions \( f : \mathbb{Z} \to \mathbb{R} \) such that
\[ f(m+n-mn) = f(m)+f(n)-f(mn) \quad \forall m,n \in \mathbb{Z}. \]
25. Find all functions \( f : (0,1) \to (0,1) \) such that \( f\Bigl(\frac{1}{2}\Bigr) = \frac{1}{2} \) and
\[ \Bigl(f(ab)\Bigr)^2 = \Bigl(af(b)+f(a)\Bigr)\Bigl(bf(a)+f(b)\Bigr) \quad \forall a,b \in (0,1). \]
26. Find all functions \( f : \mathbb{Q} \to \mathbb{Q} \) such that
\[ f(x+y+f(x)) = x+f(x)+f(y) \quad \forall x,y \in \mathbb{Q}. \]
27. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f(x^2+f(y)) = (x-y)^2\,f(x+y) \quad \forall x,y \in \mathbb{R}. \]
28. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ \begin{cases} f(x+y)=f(x)+f(y), & \forall x,y \in \mathbb{R},\\[1mm] f(x)=x^2\,f\Bigl(\frac{1}{x}\Bigr), & \forall x\in \mathbb{R}\setminus\{0\}. \end{cases} \]
29. Let \( a>\frac{3}{4} \) be a real number. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f(f(x)) + a = x^2 \quad \forall x \in \mathbb{R}. \]
30. Find all injective functions \( f : \mathbb{N} \to \mathbb{N} \) which satisfy
\[ f(f(n)) \le \frac{n+f(n)}{2} \quad \forall n \in \mathbb{N}. \]
31. Find all continuous functions \( f(x),\, g(x),\, q(x) : \mathbb{R} \to \mathbb{R} \) such that
\[ f(x^2)+f(y^2) = \Bigl[q(x)-q(y)\Bigr]\,g(x+y) \quad \forall x,y \in \mathbb{R}. \]
32. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) so that
\[ f(x+y)+f(x-y)=2f(x)\cos y \quad \forall x,y \in \mathbb{R}. \]
33. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f(x-f(y)) = f(x) + x\,f(y) + f(f(y)) \quad \forall x,y \in \mathbb{R}. \]
34. Find all functions \( f : \mathbb{R}^+ \to \mathbb{R}^+ \) such that
\[ f(f(x)) = 6x - f(x) \quad \forall x \in \mathbb{R}^+. \]
35. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f(x+y)+f(xy)+1 = f(x)+f(y)+f(xy+1) \quad \forall x,y \in \mathbb{R}. \]
36. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ x^2y^2\Bigl(f(x+y)-f(x)-f(y)\Bigr) = 3(x+y)f(x)f(y) \quad \forall x,y \in \mathbb{R}. \]
37. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f\bigl(x^3+y^3\bigr) = x\,f(x^2) + y\,f(y^2) \quad \forall x,y \in \mathbb{R}. \]
38. Find all functions \( f : \mathbb{Q} \to \mathbb{R} \) such that
\[ \lvert f(x)-f(y) \rvert \le (x-y)^2 \quad \forall x,y \in \mathbb{Q}. \]
39. Find all functions \( f : \mathbb{R} \to \mathbb{R}^+ \) such that
\[ f(x+y)=f(x^2+y^2) \quad \forall x \in \mathbb{R}^+. \]
40. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ x^2y^2\Bigl(f(x+y)-f(x)-f(y)\Bigr)=3(x+y)f(x)f(y) \quad \forall x,y \in \mathbb{R}. \]
41. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f\bigl(f(x)+f(y)+f(z)\bigr)=f\bigl(f(x)-f(y)\bigr)+f\bigl(2xy+f(z)\bigr)+2f(xz-yz) \quad \forall x,y \in \mathbb{R}. \]
42. Find all functions \( f : \mathbb{N} \to \mathbb{N} \) such that
\[ m^2+f(n) \mid f(m)^2+n \quad \forall m,n \in \mathbb{N}. \]
43. Let \( n \) be a positive integer. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f(x+f(y)) = f(x)+y^n \quad \forall x,y \in \mathbb{R}. \]
44. Find all functions \( f : \mathbb{N} \to \mathbb{N} \) such that
\[ 3f\bigl(f(f(n))\bigr) + 2f\bigl(f(n)\bigr) + f(n) = 6n \quad \forall n \in \mathbb{N}. \]
45. Find all functions \( f : \mathbb{N}^* \to \mathbb{N}^* \) satisfying
\[ \Bigl(f^2(m)+f(n)\Bigr) \mid \Bigl(m^2+n^2\Bigr)^2 \quad \forall m,n \in \mathbb{N}^*. \]
46. Find all functions \( f : \mathbb{R}^+ \to \mathbb{R}^+ \) such that
\[ f\Bigl(\frac{2xy}{x+y}\Bigr) = \frac{2f(x)f(y)}{f(x)+f(y)} \quad \forall x,y \in \mathbb{R}^+. \]
47. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f(xy) = \max\{f(x),y\} + \min\{f(y),x\} \quad \forall x,y \in \mathbb{R}. \]
48. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ \begin{cases} f(x+f(y)) = y+f(x) & \forall x,y \in \mathbb{R}, \\ \{f(x)/x : x \in \mathbb{R}\} \text{ is finite.} \end{cases} \]
49. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f\bigl(f(x)+f(y)\bigr)+f\bigl(f(x)\bigr)=2f(x)+y \quad \forall x,y \in \mathbb{R}. \]
50. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f\Bigl(x^2(z^2+1)+f(y)(z+1)\Bigr)=1-f(z)(x^2+f(y))-z\Bigl((1+z)x^2+2f(y)\Bigr) \quad \forall x,y,z \in \mathbb{R}. \]
51. Prove that there is no bijective function \( f : \{1,2,3,\dots\} \to \{0,1,2,3,\dots\} \) such that
\[ f(mn)=f(m)+f(n)+3f(m)f(n). \]
52. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f\bigl(x-f(y)\bigr)=f\bigl(f(y)\bigr)+x\,f(y)+f(x)-1 \quad \forall x,y \in \mathbb{R}. \]
53. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f\bigl(xf(x+y)\bigr)=f\bigl(yf(x)\bigr)+x^2 \quad \forall x,y \in \mathbb{R}. \]
54. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f\Bigl(x^2+\frac{x}{3}+\frac{1}{9}\Bigr)=f(x) \quad \forall x \in \mathbb{R}. \]
55. Given \(0<p<2\), find all continuous functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f\bigl(f(x)\bigr)=f(x)+px \quad \forall x \in \mathbb{R}. \]
56. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f\bigl(x+xy+f(y)\bigr)=\Bigl(f(x)+\frac{1}{2}\Bigr)\Bigl(f(y)+\frac{1}{2}\Bigr) \quad \forall x,y \in \mathbb{R}. \]
57. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f\bigl(f(x)+y\bigr)=f(x+y)+x\,f(y)-xy-x+1 \quad \forall x,y \in \mathbb{R}. \]
58. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ x\Bigl(f(x)+f(-x)+2\Bigr)+2f(-x)=0 \quad \forall x \in \mathbb{R}. \]
59. Find all non-decreasing functions \( f : \mathbb{R}^+ \cup \{0\} \to \mathbb{R}^+ \cup \{0\} \) such that for each \( x,y \in \mathbb{R}^+ \cup \{0\} \)
\[ f\Bigl(x+f(x)^2+y\Bigr)=2x-f(x)+f\bigl(f(y)\bigr). \]
60. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ (1+f(x)f(y))\,f(x+y)=f(x)+f(y) \quad \forall x,y \in \mathbb{R}. \]
61. For the function \( f : \mathbb{R} \to \mathbb{R} \) given by
\[ f(x^2+x+3)+2\,f(x^2-3x+5)=6x^2-10x+17, \]calculate \( f(2009) \).

62. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f\Bigl(f(x)y+\frac{x}{y}\Bigr)=(f(x))^n+y+f(y) \quad \forall x,y \in \mathbb{R},\; n\in\mathbb{Z}_{\ge2}. \]
63. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f(x^2+y^2)=f(x^2)+f(y^2)+2f(x)f(y) \quad \forall x,y \in \mathbb{R}. \]
64. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f(x+y)f(x-y) = \Bigl(f(x)+f(y)\Bigr)^2 - 4x^2f(y) \quad \forall x,y \in \mathbb{R}. \]
65. Find all injective functions \( f : \mathbb{N} \to \mathbb{R} \) such that
\[ f(1)=2,\quad f(2)=4,\quad \text{and}\quad f\bigl(f(m)+f(n)\bigr)=f\bigl(f(m)\bigr)+f(n) \quad \forall m,n \in \mathbb{N}. \]
66. Find all functions \( f : \mathbb{R}^+ \to \mathbb{R}^+ \) such that for any real numbers \( a,b,c,d>0 \) with \( abcd=1 \), we have
\[ \Bigl(f(a)+f(b)\Bigr)\Bigl(f(c)+f(d)\Bigr)=(a+b)(c+d). \]
67. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f(x^2)\Bigl(f(x)^2+f\Bigl(\frac{1}{y^2}\Bigr)\Bigr)=1+f\Bigl(\frac{1}{xy}\Bigr) \quad \forall x,y \in \mathbb{R}\setminus\{0\}. \]
68. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f\bigl(f(x)-f(y)\bigr)=f\bigl(f(x)\bigr)-2x^2f(y)+f(y^2) \quad \forall x,y \in \mathbb{R}. \]
69. Find all functions \( f : \mathbb{R}^+ \to \mathbb{R}^+ \) such that
\[ f(x+1)=f(x)+1 \quad \text{and} \quad f\Bigl(\frac{1}{f(x)}\Bigr)=\frac{1}{x} \quad \forall x \in \mathbb{R}^+. \]
70. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f(x+f(x)f(y))=f(x)+x\,f(y) \quad \forall x,y \in \mathbb{R}. \]
71. Find all continuous functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f(x)+f(y)-f(x+y)=xy \quad \forall x,y \in \mathbb{R}. \]72. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f(x^2+y^2) = f\bigl(f(x)\bigr) + f(xy) + f\bigl(f(y)\bigr) \quad \forall x,y \in \mathbb{R}. \]
73. Find all functions \( f : \mathbb{R}^+ \to \mathbb{R}^+ \) such that
\[ (x+y)f\bigl(f(x)y\bigr) = x^2f\bigl(f(x)+f(y)\bigr) \quad \forall x,y \in \mathbb{R}^+. \]
74. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f(x+y^2) \ge (y+1)f(x) \quad \forall x,y \in \mathbb{R}. \]
75. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f(x)f(y) \le f(xy) \quad \text{and} \quad f(x)+f(y) \le f(x+y) \quad \forall x,y \in \mathbb{R}. \]
76. Find all functions \( f : \mathbb{Q} \to \mathbb{R}^+ \) such that
\[ \begin{aligned} & f(x) \ge 0 \quad \forall x \in \mathbb{Q}, \quad f(x)=0 \iff x=0,\\[1mm] & f(xy) = f(x) \cdot f(y),\\[1mm] & f(x+y) \le \max\{f(x),f(y)\} \quad \forall x,y \in \mathbb{Q}. \end{aligned} \]
77. Determine all functions \( f : \mathbb{R} \to \mathbb{R} \) satisfying
\[ xf(y) - yf(x) = f\Bigl(\frac{y}{x}\Bigr) \quad \forall x,y \in \mathbb{R} \text{ with } x \neq 0. \]
78. Determine all functions \( f : \mathbb{N} \to \mathbb{N} \) such that
\[ \sum_{k=1}^{n} \frac{1}{f(k)\cdot f(k+1)} = \frac{f(f(n))}{f(n+1)} \quad \forall n \in \mathbb{N}. \]
79. Find all functions \( f : \mathbb{N} \to \mathbb{N} \) such that for all \( m,n \in \mathbb{N} \),
\[ (2m+1)f(n)f(2mn) = 2m f(n)^2 + f(2mn)^2 + (2m-1)^2 n. \]
80. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f(x - f(y)) = f\bigl(f(y)\bigr) - 2x f(y) + f(x) \quad \forall x,y \in \mathbb{R}. \]
81. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f\bigl(f(x)-y^2\bigr) = f(x)^2 - 2f(x)y^2 + f(y^2) \quad \forall x,y \in \mathbb{R}. \]
82. Find all functions \( f : [0,+\infty) \to [0,+\infty) \) such that
\[ f\bigl(x+f(x)+2y\bigr) = 2x + f\bigl(2f(y)\bigr) \quad \forall x,y \in [0,+\infty). \]
83. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f(x^2)+f(xy) = f(x)f(y) + yf(x) + x\,f(x+y) \quad \forall x,y \in \mathbb{R}. \]
84. Find all functions \( f : \mathbb{Q} \to \mathbb{Q} \) such that
\[ f\bigl(x+f(x)+2y\bigr) = 2x + 2f\bigl(f(y)\bigr) \quad \forall x,y \in \mathbb{Q}. \]
85. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ \begin{aligned} & f\bigl(x+f(x)^2+y+f(2z)\bigr) = 2x - f(x) + f\bigl(f(f(y))\bigr) + 2f\bigl(f(z)\bigr) \quad \forall x,y,z \in \mathbb{R},\\[1mm] & f\bigl(f(0)\bigr) = f(0). \end{aligned} \]
86. Find all functions \( f : \mathbb{R}^+ \to \mathbb{R}^+ \) which satisfy:
\[ \begin{aligned} & f(x+f(y)) = f(x)f(y) \quad \forall x,y>0,\\[1mm] & \text{there are at most finitely many } x \text{ with } f(x)=1. \end{aligned} \]
87. Find all functions \( f : \mathbb{N} \cup \{0\} \to \mathbb{N} \cup \{0\} \) such that for all \( m,n \in \mathbb{N} \cup \{0\} \),
\[ m f(n) + n f(m) = (m+n)f\bigl(m^2+n^2\bigr). \]
88. Find all functions \( f : (0,1) \to \mathbb{R} \) such that
\[ f(xyz) = x f(x) + y f(y) + z f(z) \quad \forall x,y,z \in (0,1). \]
89. Find all functions \( f : \mathbb{Z} \to \mathbb{Z} \) satisfying
\[ f(x^3+y^3+z^3) = f(x)^3 + f(y)^3 + f(z)^3. \]
90. Determine all real functions \( f(x) \) that are defined and continuous on the interval \((-1,1)\) and that satisfy the functional equation
\[ f(x+y)=\frac{f(x)+f(y)}{1-f(x)f(y)} \]for all \( x,y,x+y \in (-1,1) \).

91. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f\bigl(x^n+2f(y)\bigr) = (f(x))^n + y + f(y) \quad \forall x,y \in \mathbb{R},\; n\in \mathbb{Z}_{\ge2}. \]
92. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f(x^2+y^2)=f(x^2)+f(y^2)+2f(x)f(y) \quad \forall x,y \in \mathbb{R}. \]
93. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f(x+y)f(x-y)=\Bigl(f(x)+f(y)\Bigr)^2-4x^2f(y) \quad \forall x,y \in \mathbb{R}. \]
94. Find all injective functions \( f : \mathbb{N} \to \mathbb{R} \) such that
\[ f(1)=2,\quad f(2)=4,\quad \text{and}\quad f\bigl(f(m)+f(n)\bigr)=f\bigl(f(m)\bigr)+f(n) \quad \forall m,n \in \mathbb{N}. \]
95. Find all functions \( f : \mathbb{R}^+ \to \mathbb{R}^+ \) such that for any real numbers \( a,b,c,d>0 \) satisfying \( abcd=1 \), we have
\[ \Bigl(f(a)+f(b)\Bigr)\Bigl(f(c)+f(d)\Bigr) = (a+b)(c+d). \]
96. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f(x^2)\Bigl(f(x)^2+f\Bigl(\frac{1}{y^2}\Bigr)\Bigr)=1+f\Bigl(\frac{1}{xy}\Bigr) \quad \forall x,y \in \mathbb{R}\setminus\{0\}. \]
97. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f\bigl(f(x)-f(y)\bigr)=f\bigl(f(x)\bigr)-2x^2f(y)+f(y^2) \quad \forall x,y \in \mathbb{R}. \]
98. Find all functions \( f : \mathbb{R}^+ \to \mathbb{R}^+ \) such that
\[ f(x+1)=f(x)+1 \quad \text{and} \quad f\Bigl(\frac{1}{f(x)}\Bigr)=\frac{1}{x} \quad \forall x \in \mathbb{R}^+. \]
99. Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f\bigl(x+f(x)f(y)\bigr)=f(x)+x\,f(y) \quad \forall x,y \in \mathbb{R}. \]
100. Find all continuous functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f(x)+f(y)-f(x+y)=xy \quad \forall x,y \in \mathbb{R}. \]
This post has been edited 1 time. Last edited by SomeonecoolLovesMaths, Feb 21, 2025, 1:18 PM

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solution set for this?

by Levieee, Mar 17, 2025, 3:59 PM

My First Blog

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  • I will bookmarked this blog

    by giangtruong13, Monday at 7:37 AM

  • W blog op

    by quasar_lord, Mar 10, 2025, 5:32 PM

  • orz blog
    how so orzzzz

    by Erratum, Jan 31, 2025, 9:47 AM

  • shouts; your post is too short , it must be atleast 8 characters

    by mqoi_KOLA, Dec 5, 2024, 6:37 PM

  • Guys it took my like 10 hours to do this! Idk why did I even do this, but now it looks kinda satisfying ngl.

    by SomeonecoolLovesMaths, Nov 25, 2024, 5:07 PM

  • add this one new thing to the intergrals that is lim n tends to infinity b-a/n summation k=1 to n f(a+k(b-a)/n))= int_{a}^b f(x) dx

    by Levieee, Nov 21, 2024, 8:37 PM

  • woahh hi HSM main orz blog!!!

    by eg4334, Nov 17, 2024, 3:31 PM

  • Me in 4 years of Aops - 555 posts.

    This guy in 10 months of Aops - 2608 posts

    by HoRI_DA_GRe8, Oct 17, 2024, 10:22 AM

  • Remember; the one who falls and gets back up is stronger than the ones who never tried... Good luck for next year's IOQM

    by alexanderhamilton124, Oct 15, 2024, 7:03 AM

  • fourth shout

    by QueenArwen, Sep 2, 2024, 4:05 PM

  • Hii
    !!!!!!!!!!!

    by Yummo, Apr 2, 2024, 2:43 PM

  • i am shouting. it is very loud.

    by fruitmonster97, Mar 21, 2024, 7:49 PM

  • real!!!!!

    by SirAppel, Mar 17, 2024, 6:22 PM

13 shouts
Tags
geometry
About Owner
  • Posts: 3139
  • Joined: Dec 22, 2023
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  • Blog created: Mar 17, 2024
  • Total entries: 20
  • Total visits: 1249
  • Total comments: 19
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