Found problems: 2349
Find all functions $f : \mathbb R \to \mathbb R$ for which
\[f(a - b) f(c - d) + f(a - d) f(b - c) \leq (a - c) f(b - d),\]
for all real numbers $a, b, c$ and $d$. Note that there is only one occurrence of $f$ on the right hand side!
(a) Find all strictly monotone functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
\[f(x+f(y))=f(x)+y\quad\text{for all real}\ x,y. \]
(b) If $n>1$ is an integer, prove that there is no strictly monotone function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
\[ f(x+f(y))=f(x)+y^n\quad \text{for all real}\ x, y.\]
Find all pairs of positive integer $\alpha$ and function $f : N \to N_0$ that satisfies
(i) $f(mn^2) = f(mn) + \alpha f(n)$ for all positive integers $m, n$.
(ii) If $n$ is a positive integer and $p$ is a prime number with $p|n$, then $f(p) \ne 0$ and $f(p)|f(n)$.
Let $p$ a positive prime number and $\mathbb{F}_{p}$ the set of integers $mod \ p$. For $x\in \mathbb{F}_{p}$, define $|x|$ as the cyclic distance of $x$ to $0$, that is, if we represent $x$ as an integer between $0$ and $p-1$, $|x|=x$ if $x<\frac{p}{2}$, and $|x|=p-x$ if $x>\frac{p}{2}$ . Let $f: \mathbb{F}_{p} \rightarrow \mathbb{F}_{p}$ a function such that for every $x,y \in \mathbb{F}_{p}$
\[ |f(x+y)-f(x)-f(y)|<100 \]
Prove that exist $m \in \mathbb{F}_{p}$ such that for every $x \in \mathbb{F}_{p}$
\[ |f(x)-mx|<1000 \]
Find all functions $f,g: \mathbb{Q}\to \mathbb{Q}$ such that the following two conditions hold:
$$f(g(x)-g(y))=f(g(x))-y \ \ (1)$$
$$g(f(x)-f(y))=g(f(x))-y\ \ (2)$$
for all $x,y \in \mathbb{Q}$.
Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that
\[ m^2 + f(n) \mid mf(m) +n \]
for all positive integers $m$ and $n$.
Let ${R^* = R-\{0\}}$ be the set of non-zero real numbers. Find all functions ${f : R^* \rightarrow R^*}$ satisfying ${f(x) + f(y) = f(xy f(x + y))}$, for ${x, y \in R^*}$ and ${ x + y\ne 0 }$.
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that
$f(f(x)f(1-x))=f(x)$ and $f(f(x))=1-f(x)$,
for all real $x$.
Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\] for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$.
Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.
Let $\mathbb{R}^{*}$ be the set of non-zero real numbers. Find all functions $f : \mathbb{R}^{*} \to \mathbb{R}^{*}$ such that $f(x^{2}+y) = (f(x))^{2} + \frac{f(xy)}{f(x)}$, for all $x,y \in \mathbb{R}^{*}$ and $-x^{2} \not= y$.
Find all functions $f:\mathbb{Q}\to\mathbb{Q}$ such that the equation
\[f(xf(x)+y) = f(y) + x^2\]holds for all rational numbers $x$ and $y$.
Here, $\mathbb{Q}$ denotes the set of rational numbers.
Suppose that $f : \mathbb R^+ \to \mathbb R^+$ is a decreasing function such that
\[f(x+y)+f(f(x)+f(y))=f(f(x+f(y))+f(y+f(x)), \quad \forall x,y \in \mathbb R^+.\]
Prove that $f(x) = f^{-1}(x).$
Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the following conditions: $f(x+1)=f(x)+1$ and $f(x^2)=f(x)^2.$
[i]Based on a problem of Romanian Masters of Mathematics[/i]
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$,
$$(f(x)+xy)\cdot f(x-3y)+(f(y)+xy)\cdot f(3x-y)=(f(x+y))^2.$$
Find the function $f(x)$ such that :
\[f(x)=\cos x+\int_0^{2\pi} f(y)\sin (x-y)\ dy\]
Find all functions $f: (0,\infty)\rightarrow(0,\infty)$ with the following properties: $f(x+1)=f(x)+1$ and $f\left(\frac{1}{f(x)}\right)=\frac{1}{x}$.
[i]Proposed by P. Volkmann[/i]
Determine all functions $ f: \mathbb{R} \to \mathbb{R}$ satisfying the condition $ f(xy) \le xf(y)$ for all real numbers $ x$ and $ y$.
Find all functions $f:\mathbb{R} \to \mathbb{R}$, such that for any $x, y \in \mathbb{R}$ holds the following:
$$f(x)f(yf(x)) + yf(xy) = xf(xy) + y^2f(x)$$
[i]Proposed by Mykhailo Shtandenko[/i]
Find all functions $f:\mathbb{Z}_{\ge 1}\to \mathbb{R}_{>0}$ such that for all positive integers $n$ the following relation holds: $$\sum_{d|n} f(d)^3=\left (\sum_{d|n} f(d) \right )^2,$$
where both sums are taken over the positive divisors of $n$.
[i] (Vlad Robu) [/i]
Let $f: \mathbb{R}^{2} \to \mathbb{R}^{+}$such that for every rectangle $A B C D$ one has
$$
f(A)+f(C)=f(B)+f(D).
$$
Let $K L M N$ be a quadrangle in the plane such that $f(K)+f(M)=f(L)+f(N)$, for each such function. Prove that $K L M N$ is a rectangle.
[i]Proposed by Navid.[/i]
Determine all functions $f : \mathbb R \to \mathbb R$ such that for all real numbers $x$ and $y$,
$$f(x^2 + xy) = f(x)f(y) + yf(x) + xf(x+y).$$
[i]Proposed by Walther Janous, Austria[/i]
Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$,
\[xf(x+f(y))=(y-x)f(f(x)).\]
[i]Proposed by Nikola Velov, Macedonia[/i]
Find all function $f:\mathbb{N}\rightarrow\mathbb{N}$ such that for all $a,b\in\mathbb{N}$ ,
$(f(a)+b) f(a+f(b))=(a+f(b))^2$
Find all the functions $f: \mathbb R \rightarrow \mathbb R$ satisfying :
$(x+y)(f(x)-f(y))=(x-y)f(x+y)$ for all $x,y \in \mathbb R$
10) Call a string of letters $S$ an [i]almost-palindrome[/i] if $S$ and the reverse of $S$ differ in exactly $2$ places. Find the number of ways to order the letters in $HMMTTHEMETEAM$ to get an almost-palindrome.
11) Find all integers $n$, not necessarily positive, for which there exist positive integers ${a,b,c}$ satisfying $a^n + b^n = c^n$.
12) Let $a$ and $b$ be positive real numbers. Determine the minimum possible value of $\sqrt{a^2 + b^2} + \sqrt{a^2 + (b-1)^2} + \sqrt{(a-1)^2 + b^2} + \sqrt{(a-1)^2 + (b-1)^2}$.
13) Consider a $4$ x $4$ grid of squares, each originally colored red. Every minute, Piet can jump on any of the squares, changing the color of it and any adjacent squares to blue (two squares are adjacent if they share a side). What is the minimum number of minutes it will take Piet to change the entire grid to blue?
14) Let $ABC$ be an acute triangle with orthocenter $H$. Let ${D,E}$ be the feet of the ${A,B}$-altitudes, respectively. Given that $\overline{AH} = 20$ and $\overline{HD} =16$ and $\overline{BE} = 56$, find the length of $\overline{BH}$.
15) Find the smallest positive integer $b$ such that $1111 _b$ ($1111$ in base $b$) is a perfect square. If no such $b$ exists, write "No Solution"
16) For how many triples $( {x,y,z} )$ of integers between $-10$ and $10$, inclusive, do there exist reals ${a,b,c}$ that satisfy
$ab = x$
$ac = y$
$bc = z$?
17) Unit squares $ABCD$ and $EFGH$ have centers $O_1$ and $O_2$, respectively, and are originally oriented so that $B$ and $E$ are at the same position and $C$ and $H$ are at the same position. The squares then rotate clockwise around their centers at a rate of one revolution per hour. After $5$ minutes, what is the area of the intersection of the two squares?
18) A function $f$ satisfies, for all
nonnegative integers $x$ and $y$,
$f(x,0) = f(0,x) = x$
If $x \ge y \ge 0$, $f(x,y)=f(x-y,y)+1$
If $y \ge x \ge 0$, $f(x,y) = f(x,y-x)+1$
Find the maximum value of $f$ over $0 \le x,y \le 100$.