Found problems: 4776
2017 Taiwan TST Round 2, 3
Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
2017 VJIMC, 1
Let $(a_n)_{n=1}^{\infty}$ be a sequence with $a_n \in \{0,1\}$ for every $n$. Let $F:(-1,1) \to \mathbb{R}$ be defined by
\[F(x)=\sum_{n=1}^{\infty} a_nx^n\]
and assume that $F\left(\frac{1}{2}\right)$ is rational. Show that $F$ is the quotient of two polynomials with integer coefficients.
2004 Postal Coaching, 4
In how many ways can a $2\times n$ grid be covered by
(a) 2 monominoes and $n-1$ dominoes
(b) 4 monominoes and $n-2$ dominoes.
2008 Harvard-MIT Mathematics Tournament, 10
Determine the number of $ 8$-tuples of nonnegative integers $ (a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)$ satisfying $ 0\le a_k\le k$, for each $ k \equal{} 1,2,3,4$, and $ a_1 \plus{} a_2 \plus{} a_3 \plus{} a_4 \plus{} 2b_1 \plus{} 3b_2 \plus{} 4b_3 \plus{} 5b_4 \equal{} 19$.
2014 Hanoi Open Mathematics Competitions, 14
Let be given $a < b < c$ and $f(x) =\frac{c(x - a)(x - b)}{(c - a)(c - b)}+\frac{a(x - b)(x - c)}{(a - b)(a -c)}+\frac{b(x -c)(x - a)}{(b - c)(b - a)}$.
Determine $f(2014)$.
2012 India IMO Training Camp, 3
Let $\mathbb{R}^{+}$ denote the set of all positive real numbers. Find all functions $f:\mathbb{R}^{+}\longrightarrow \mathbb{R}$ satisfying
\[f(x)+f(y)\le \frac{f(x+y)}{2}, \frac{f(x)}{x}+\frac{f(y)}{y}\ge \frac{f(x+y)}{x+y},\]
for all $x, y\in \mathbb{R}^{+}$.
2011 ELMO Shortlist, 1
Let $S$ be a finite set, and let $F$ be a family of subsets of $S$ such that
a) If $A\subseteq S$, then $A\in F$ if and only if $S\setminus A\notin F$;
b) If $A\subseteq B\subseteq S$ and $B\in F$, then $A\in F$.
Determine if there must exist a function $f:S\to\mathbb{R}$ such that for every $A\subseteq S$, $A\in F$ if and only if
\[\sum_{s\in A}f(s)<\sum_{s\in S\setminus A}f(s).\]
[i]Evan O'Dorney.[/i]
2016 Taiwan TST Round 3, 2
Determine all functions $f:\mathbb{R}^+\rightarrow \mathbb{R}^+$ satisfying
$f(x+y+f(y))=4030x-f(x)+f(2016y), \forall x,y \in \mathbb{R}^+$.
2025 Thailand Mathematical Olympiad, 6
Find all function $f: \mathbb{R}^+ \rightarrow \mathbb{R}$,such that the inequality $$f(x) + f\left(\frac{y}{x}\right) \leqslant \frac{x^3}{y^2} + \frac{y}{x^3}$$ holds for all positive reals $x,y$ and for every positive real $x$, there exist positive reals $y$, such that the equality holds.
2016 India National Olympiad, P3
Let $\mathbb{N}$ denote the set of natural numbers. Define a function $T:\mathbb{N}\rightarrow\mathbb{N}$ by $T(2k)=k$ 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 any $k>1$.
(i) Show that for each $n\in\mathbb{N}$, there exists $k$ such that $T^k(n)=1$.
(ii) For $k\in\mathbb{N}$, let $c_k$ denote the number of elements in the set $\{n: T^k(n)=1\}$. Prove that $c_{k+2}=c_{k+1}+c_k$, for $k\ge 1$.
2009 Tournament Of Towns, 4
Denote by $[n]!$ the product $ 1 \cdot 11 \cdot 111\cdot ... \cdot \underbrace{111...1}_{\text{n ones}}$.($n$ factors in total). Prove that $[n + m]!$ is divisible by $ [n]! \times [m]!$
[i](8 points)[/i]
2012 Pre-Preparation Course Examination, 1
Suppose that $X$ and $Y$ are two metric spaces and $f:X \longrightarrow Y$ is a continious function. Also for every compact set $K \subseteq Y$, it's pre-image $f^{pre}(K)$ is a compact set in $X$. Prove that $f$ is a closed function, i.e for every close set $C\subseteq X$, it's image $f(C)$ is a closed subset of $Y$.
2021 Korea Winter Program Practice Test, 8
For function $f:\mathbb Z^+ \to \mathbb R$ and coprime positive integers $p,q$ ; define $f_p,f_q$ as
$$f_p(x)=f(px)-f(x), f_q(x)=f(qx)-f(x) \space \space (x\in\mathbb Z^+)$$
$f$ satisfies following conditions.
$ $ $ $ $(i)$ $ $ for all $r$ that isn't multiple of $pq$, $f(r)=0$
$ $ $ $ $(ii)$ $ $ $\exists m\in \mathbb Z^+$ $ $ $s.t.$ $ $ $\forall x\in \mathbb Z^+, f_p(x+m)=f_p(x)$ and $f_q(x+m)=f_q(x)$
Prove that if $x\equiv y$ $ $ $(mod m)$, then $f(x)=f(y)$ $ $ ($x, y\in \mathbb Z^+$).
KoMaL A Problems 2018/2019, A. 735
For any function $f:[0,1]\to [0,1]$, let $P_n (f)$ denote the number of fixed points of the function $\underbrace{f(f(\dotsc f}_{n} (x)\dotsc )$, i.e., the number of points $x\in [0,1]$ satisfying $\underbrace{f(f(\dotsc f}_{n} (x)\dotsc )=x$. Construct a piecewise linear, continuous, surjective function $f:[0,1] \to [0,1]$ such that for a suitable $2<A<3$, the sequence $\frac{P_n(f)}{A^n}$ converges.
[i]Based on the 8th problem of the Miklós Schweitzer competition, 2018[/i]
1989 IMO Longlists, 13
Let $ n \leq 44, n \in \mathbb{N}.$ Prove that for any function $ f$ defined over $ \mathbb{N}^2$ whose images are in the set $ \{1, 2, \ldots , n\},$ there are four ordered pairs $ (i, j), (i, k), (l, j),$ and $ (l, k)$ such that \[ f(i, j) \equal{} f(i, k) \equal{} f(l, j) \equal{} f(l, k),\] in which $ i, j, k, l$ are chosen in such a way that there are natural numbers $ m, p$ that satisfy \[ 1989m \leq i < l < 1989 \plus{} 1989m\] and \[ 1989p \leq j < k < 1989 \plus{} 1989p.\]
2009 Ukraine National Mathematical Olympiad, 1
Solve the system of equations
\[\{\begin{array}{cc}x^3=2y^3+y-2\\ \text{ } \\ y^3=2z^3+z-2 \\ \text{ } \\ z^3 = 2x^3 +x -2\end{array}\]
2002 Moldova National Olympiad, 2
Let $ a,b,c\geq 0$ such that $ a\plus{}b\plus{}c\equal{}1$. Prove that:
$ a^2\plus{}b^2\plus{}c^2\geq 4(ab\plus{}bc\plus{}ca)\minus{}1$
2009 Today's Calculation Of Integral, 483
Let $ n\geq 2$ be natural number. Answer the following questions.
(1) Evaluate the definite integral $ \int_1^n x\ln x\ dx.$
(2) Prove the following inequality.
$ \frac 12n^2\ln n \minus{} \frac 14(n^2 \minus{} 1) < \sum_{k \equal{} 1}^n k\ln k < \frac 12n^2\ln n \minus{} \frac 14 (n^2 \minus{} 1) \plus{} n\ln n.$
(3) Find $ \lim_{n\to\infty} (1^1\cdot 2^2\cdot 3^3\cdots\cdots n^n)^{\frac {1}{n^2 \ln n}}.$
2003 Miklós Schweitzer, 10
Let $X$ and $Y$ be independent random variables with "Saint-Petersburg" distribution, i.e. for any $k=1,2,\ldots$ their value is $2^k$ with probability $\frac{1}{2^k}$. Show that $X$ and $Y$ can be realized on a sufficiently big probability space such that there exists another pair of independent "Saint-Petersburg" random variables $(X', Y')$ on this space with the property that $X+Y=2X'+Y'I(Y'\le X')$ almost surely (here $I(A)$ denotes the indicator function of the event $A$).
(translated by L. Erdős)
2002 National High School Mathematics League, 1
The increasing interval of $f(x)=\log_{\frac{1}{2}}(x^2-2x-3)$ is
$\text{(A)}(-\infty,-1)\qquad\text{(B)}(-\infty,1)\qquad\text{(C)}(1,+\infty)\qquad\text{(D)}(3,+\infty)$
1991 Balkan MO, 4
Prove that there is no bijective function $f : \left\{1,2,3,\ldots \right\}\rightarrow \left\{0,1,2,3,\ldots \right\}$ such that $f(mn)=f(m)+f(n)+3f(m)f(n)$.
1951 AMC 12/AHSME, 34
The value of $ 10^{\log_{10}7}$ is:
$ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ \log_{10} 7 \qquad\textbf{(E)}\ \log_7 10$
2017 CMIMC Computer Science, 7
You are presented with a mystery function $f:\mathbb N^2\to\mathbb N$ which is known to satisfy \[f(x+1,y)>f(x,y)\quad\text{and}\quad f(x,y+1)>f(x,y)\] for all $(x,y)\in\mathbb N^2$. I will tell you the value of $f(x,y)$ for \$1. What's the minimum cost, in dollars, that it takes to compute the $19$th smallest element of $\{f(x,y)\mid(x,y)\in\mathbb N^2\}$? Here, $\mathbb N=\{1,2,3,\dots\}$ denotes the set of positive integers.
2006 Tuymaada Olympiad, 4
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]
2003 SNSB Admission, 3
Let be the set $ \Lambda = \left\{ \lambda\in\text{Hol} \left[ \mathbb{C}\longrightarrow\mathbb{C} \right] |z\in\mathbb{C}\implies |\lambda (z)|\le e^{|\text{Im}(z)|} \right\} . $ Show that:
$ \text{(1)}\sin\in\Lambda $
$ \text{(2)}\sum_{p\in\mathbb{Z}}\frac{1}{(1+2p)^2} =\frac{\pi^2}{4} $
$ \text{(3)} f\in\Lambda\implies \left| f'(0) \right|\le 1 $