Found problems: 2349
We say that a triangle $ABC$ is great if the following holds: for any point $D$ on the side $BC$, if $P$ and $Q$ are the feet of the perpendiculars from $D$ to the lines $AB$ and $AC$, respectively, then the reflection of $D$ in the line $PQ$ lies on the circumcircle of the triangle $ABC$. Prove that triangle $ABC$ is great if and only if $\angle A = 90^{\circ}$ and $AB = AC$.
[i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]
Let \( \mathbb{Z^{+}} \) denote the set of all positive integers. Find all functions \( f: \mathbb{Z^{+}} \rightarrow \mathbb{Z^{+}} \) such that for every pair of positive integers \( a \) and \( b \), there exists a positive integer \( c \) satisfying:
$$
f(a)f(b) - ab = 2^{c-1} - 1.
$$
Proposed by Matin Yousefi
Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that
$\boxed{1} \ f(1) = 1$
$\boxed{2} \ f(m+n)(f(m)-f(n)) = f(m-n)(f(m)+f(n)) \ \forall \ m,n \in \mathbb{Z}$
Find all functions $f:\mathbb{N}\to\mathbb{R}$ such that
$f(km)+f(kn)-f(k)f(mn)\ge 1$
for all $k,m,n\in\mathbb{N}$.
Find all functions $f:\mathbb N\to\mathbb N$ such that
\begin{align*}
x+f(y)&\mid f(y+f(x))\\
f(x)-2017&\mid x-2017\end{align*}
Find all functions $f:\mathbb{N} \to \mathbb{N}$ so that for any distinct positive integers $x,y,z$ the value of $x+y+z$ is a perfect square if and only if $f(x)+f(y)+f(z)$ is a perfect square.
Find all functions $g:\mathbb{N}\rightarrow\mathbb{N}$ such that \[\left(g(m)+n\right)\left(g(n)+m\right)\] is a perfect square for all $m,n\in\mathbb{N}.$
[i]Proposed by Gabriel Carroll, USA[/i]
Let $\mathbb R$ be the set of real numbers. We denote by $\mathcal F$ the set of all functions $f\colon\mathbb R\to\mathbb R$ such that
$$f(x + f(y)) = f(x) + f(y)$$
for every $x,y\in\mathbb R$ Find all rational numbers $q$ such that for every function $f\in\mathcal F$, there exists some $z\in\mathbb R$ satisfying $f(z)=qz$.
Find all surjective functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(n) \ge n+(-1)^{n}.\]
Determine if there exists a function f: $N^*\to N^*$ that satisfies that for all $n \in N^*$, $$10^{f (n)} <10n + 1 <10^{f (n) +1}.$$ Justify your answer.
Note: $N^*$ denotes the set of positive integers.
Let $\otimes$ be a binary operation that takes two positive real numbers and returns a positive real number. Suppose further that $\otimes$ is continuous, commutative $(a\otimes b=b\otimes a)$, distributive across multiplication $(a\otimes(bc)=(a\otimes b)(a\otimes c))$, and that $2\otimes 2=4$. Solve the equation $x\otimes y=x$ for $y$ in terms of $x$ for $x>1$.
Functions $f,g:\mathbb{Z}\to\mathbb{Z}$ satisfy $$f(g(x)+y)=g(f(y)+x)$$ for any integers $x,y$. If $f$ is bounded, prove that $g$ is periodic.
Let be given natural numbers $a_1,a_2,...,a_n$ and a function $f : Z \to R$ such that $f(x) = 1$ for all integers $x < 0$ and $f(x) = 1- f(x-a_1)f(x-a_2)... f(x-a_n)$ for all integers $x \ge 0$. Prove that there exist natural numbers $s$ and $t$ such that for all integers $x > s$ it holds that $f(x+t) = f(x)$.
Determine all functions $ f : \mathbb{R} \to \mathbb{R} $ continuous at zero and such that for every real number $ x $ the equality holds $$ 2f(2x) = f(x) + x.$$
Solve the functional equation $$f(x+y)+f(x-y)=2f(x)\cos{y}$$ where $x,y \in \mathbb{R}$ and $f : \mathbb{R} \rightarrow \mathbb{R}$
Determine all functions $f : R \to R$ satisfying $f (f(x)f(y) + f(y)f(z) + f(z)f(x))= f(x) + f(y) + f(z)$ for all real numbers $x, y, z$.
Find all functions $f : \mathbb{Q} \to \mathbb{R}$ such that $f(x)f(y)f(x+y) = f(xy)(f(x) + f(y))$ for all $x,y\in\mathbb{Q}$.
[i]Sammy Luo and Alex Zhu.[/i]
Find all $f:\mathbb{R}\to\mathbb{R}$ satisfying:
$$f(xf(y)-y)+f(xy-x)+f(x+y)=2xy,\quad\forall x,y\in\mathbb{R}.$$
Solve the function $f: \Re \to \Re$:
\[ f( x^{3} )+ f(y^{3}) = (x+y)(f(x^{2} )+f(y^{2} )-f(xy))\]
The continuous function $f(x)$ satisfies $c^2f(x + y) = f(x)f(y)$ for all real numbers $x$ and $y,$ where $c > 0$ is a constant. If $f(1) = c$, find $f(x)$ (with proof).
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which satisfy conditions:
$f(x) + f(y) \ge xy$ for all real $x, y$ and
for each real $x$ exists real $y$, such that $f(x) + f(y) = xy$.
Find all functions $f:\mathbb{Z}_{>0}\mapsto\mathbb{Z}_{>0}$ such that
$$xf(x)+(f(y))^2+2xf(y)$$
is perfect square for all positive integers $x,y$.
**This problem was proposed by me for the BMO 2017 and it was shortlisted. We then used it in our TST.
A function $f : R \to R$ satisfy the functional equation $f(x + 2y) + 2f(y - 2x) = 3x -4y + 6$ for all reals $x, y$. Compute $f(2548)$.
Suppose that a function $f:\mathbb R\to\mathbb R$ satisfies the following conditions:
(i) $\left|f(a)-f(b)\right|\le|a-b|$ for all $a,b\in\mathbb R$;
(ii) $f(f(f(0)))=0$.
Prove that $f(0)=0$.
Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.