Found problems: 4776
2008 Bosnia And Herzegovina - Regional Olympiad, 4
Determine is there a function $a: \mathbb{N} \rightarrow \mathbb{N}$ such that:
$i)$ $a(0)=0$
$ii)$ $a(n)=n-a(a(n))$, $\forall n \in$ $ \mathbb{N}$.
If exists prove:
$a)$ $a(k)\geq a(k-1)$
$b)$ Does not exist positive integer $k$ such that $a(k-1)=a(k)=a(k+1)$.
2022 Austrian MO National Competition, 1
Find all functions $f : Z_{>0} \to Z_{>0}$ with $a - f(b) | af(a) - bf(b)$ for all $a, b \in Z_{>0}$.
[i](Theresia Eisenkoelbl)[/i]
2014 Tuymaada Olympiad, 3
Positive numbers $a,\ b,\ c$ satisfy $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=3$. Prove the inequality
\[\dfrac{1}{\sqrt{a^3+1}}+\dfrac{1}{\sqrt{b^3+1}}+\dfrac{1}{\sqrt{c^3+1}}\le \dfrac{3}{\sqrt{2}}. \]
[i](N. Alexandrov)[/i]
MIPT Undergraduate Contest 2019, 2.3
Let $A$ and $B$ be rectangles in the plane and $f : A \rightarrow B$ be a mapping which is uniform on the interior of $A$, maps the boundary of $A$ homeomorphically to the boundary of $B$ by mapping the sides of $A$ to corresponding sides in $B$. Prove that $f$ is an affine transformation.
2009 IMO, 3
Suppose that $ s_1,s_2,s_3, \ldots$ is a strictly increasing sequence of positive integers such that the sub-sequences \[s_{s_1},\, s_{s_2},\, s_{s_3},\, \ldots\qquad\text{and}\qquad s_{s_1+1},\, s_{s_2+1},\, s_{s_3+1},\, \ldots\] are both arithmetic progressions. Prove that the sequence $ s_1, s_2, s_3, \ldots$ is itself an arithmetic progression.
[i]Proposed by Gabriel Carroll, USA[/i]
2007 Nicolae Păun, 3
Let $ M $ be a finite set of integers, and let be a function $ \varphi :\mathbb{Z}\longrightarrow\mathbb{Z} $ whose restriction to $ \mathbb{Z}\setminus M $ evaluates to a constant $ c, $ such that
$$ 2\le |\varphi (M)|=|M|\neq \frac{1}{c}\cdot \sum_{\iota \in \varphi (M) } \iota . $$
Prove that $ \varphi $ is not a sum between an injective function and a surjective function.
[i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]
1968 AMC 12/AHSME, 5
If $f(n)=\tfrac{1}{3}n(n1)(n+2)$, then $f(r)-f(r-1)$ equals:
$\textbf{(A)}\ r(r+1) \qquad
\textbf{(B)}\ (r+1)(r+2) \qquad
\textbf{(C)}\ \tfrac{1}{3}r(r+1) \qquad\\
\textbf{(D)}\ \tfrac{1}{3}(r+1)(r+2) \qquad
\textbf{(E)}\ \tfrac{1}{3}r(r+1)(r+2) $
2006 IMO Shortlist, 4
Prove the inequality:
\[\sum_{i < j}{\frac {a_{i}a_{j}}{a_{i} \plus{} a_{j}}}\leq \frac {n}{2(a_{1} \plus{} a_{2} \plus{}\cdots \plus{} a_{n})}\cdot \sum_{i < j}{a_{i}a_{j}}\]
for positive reals $ a_{1},a_{2},\ldots,a_{n}$.
[i]Proposed by Dusan Dukic, Serbia[/i]
2004 Singapore Team Selection Test, 2
Let $0 < a, b, c < 1$ with $ab + bc + ca = 1$. Prove that
\[\frac{a}{1-a^2} + \frac{b}{1-b^2} + \frac{c}{1-c^2} \geq \frac {3 \sqrt{3}}{2}.\]
Determine when equality holds.
2024 Irish Math Olympiad, P10
Let $\mathbb{Z}_+=\{1,2,3,4...\}$ be the set of all positive integers. Find, with proof, all functions $f : \mathbb{Z}_+ \mapsto \mathbb{Z}_+$ with the property that $$f(x+f(y)+f(f(z)))=z+f(y)+f(f(x))$$ for all positive integers $x,y,z$.
2000 IMC, 5
Find all functions $\mathbb{R}^+\rightarrow\mathbb{R}^+$ for which we have for all $x,y\in \mathbb{R}^+$ that $f(x)f(yf(x))=f(x+y)$.
1996 Turkey MO (2nd round), 3
Show that there is no function $f:{{\mathbb{R}}^{+}}\to {{\mathbb{R}}^{+}}$ such that $f(x+y)>f(x)(1+yf(x))$
for all $x,y\in {{\mathbb{R}}^{+}}$.
2012 Romania Team Selection Test, 4
Let $k$ be a positive integer. Find the maximum value of \[a^{3k-1}b+b^{3k-1}c+c^{3k-1}a+k^2a^kb^kc^k,\] where $a$, $b$, $c$ are non-negative reals such that $a+b+c=3k$.
1976 Chisinau City MO, 130
Prove that the function $f (x)$ satisfying the relation $|f (x) - f (y) | \le | x - y|^a$ for any real numbers $x, y$ and some number $a> 1$ is constant.
2014 International Zhautykov Olympiad, 2
Does there exist a function $f: \mathbb R \to \mathbb R $ satisfying the following conditions:
(i) for each real $y$ there is a real $x$ such that $f(x)=y$ , and
(ii) $f(f(x)) = (x - 1)f(x) + 2$ for all real $x$ ?
[i]Proposed by Igor I. Voronovich, Belarus[/i]
2017 Korea National Olympiad, problem 7
Find all real numbers $c$ such that there exists a function $f: \mathbb{R}_{ \ge 0} \rightarrow \mathbb{R}$ which satisfies the following.
For all nonnegative reals $x, y$, $f(x+y^2) \ge cf(x)+y$.
Here $\mathbb{R}_{\ge 0}$ is the set of all nonnegative reals.
2012 Czech-Polish-Slovak Match, 2
Find all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying
\[f(x+f(y))-f(x)=(x+f(y))^4-x^4\]
for all $x,y \in \mathbb{R}$.
2014 Iran Team Selection Test, 2
is there a function $f:\mathbb{N}\rightarrow \mathbb{N}$ such that
$i) \exists n\in \mathbb{N}:f(n)\neq n$
$ii)$ the number of divisors of $m$ is $f(n)$ if and only if the number of divisors of $f(m)$ is $n$
India EGMO 2023 TST, 4
Let $f, g$ be functions $\mathbb{R} \rightarrow \mathbb{R}$ such that for all reals $x,y$, $$f(g(x) + y) = g(x + y)$$
Prove that either $f$ is the identity function or $g$ is periodic.
[i]Proposed by Pranjal Srivastava[/i]
2002 Italy TST, 3
Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ which satisfy the following conditions:
$(\text{i})$ $f(x+f(y))=f(x)f(y)$ for all $x,y>0;$
$(\text{ii})$ there are at most finitely many $x$ with $f(x)=1$.
2016 Puerto Rico Team Selection Test, 6
$N$ denotes the set of all natural numbers. Define a function $T: N \to N$ such that $T (2k) = k$ and $T (2k + 1) = 2k + 2$. We write $T^2 (n) = T (T (n))$ and in general $T^k (n) = T^{k-1} (T (n))$ for all $k> 1$.
(a) Prove that for every $n \in N$, there exists $k$ such that $T^k (n) = 1$.
(b) For $k \in N$, $c_k$ denotes the number of elements in the set $\{n: T^k (n) = 1\}$.
Prove that $c_{k + 2} = c_{k + 1} + c_k$, for $1 \le k$.
2008 Korean National Olympiad, 7
Prove that the only function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the following is $f(x)=x$.
(i) $\forall x \not= 0$, $f(x) = x^2f(\frac{1}{x})$.
(ii) $\forall x, y$, $f(x+y) = f(x)+f(y)$.
(iii) $f(1)=1$.
2018 Greece Team Selection Test, 3
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.
2010 China Girls Math Olympiad, 5
Let $f(x)$ and $g(x)$ be strictly increasing linear functions from $\mathbb R $ to $\mathbb R $ such that $f(x)$ is an integer if and only if $g(x)$ is an integer. Prove that for any real number $x$, $f(x)-g(x)$ is an integer.
1990 China Team Selection Test, 3
In set $S$, there is an operation $'' \circ ''$ such that $\forall a,b \in S$, a unique $a \circ b \in S$ exists. And
(i) $\forall a,b,c \in S$, $(a \circ b) \circ c = a \circ (b \circ c)$.
(ii) $a \circ b \neq b \circ a$ when $a \neq b$.
Prove that:
a.) $\forall a,b,c \in S$, $(a \circ b) \circ c = a \circ c$.
b.) If $S = \{1,2, \ldots, 1990\}$, try to define an operation $'' \circ ''$ in $S$ with the above properties.