Found problems: 4776
2000 China Team Selection Test, 2
[b]a.)[/b] Let $a,b$ be real numbers. Define sequence $x_k$ and $y_k$ such that
\[x_0 = 1, y_0 = 0, x_{k+1} = a \cdot x_k - b \cdot y_l, \quad y_{k+1} = x_k - a \cdot y_k \text{ for } k = 0,1,2, \ldots \]
Prove that
\[x_k = \sum^{[k/2]}_{l=0} (-1)^l \cdot a^{k - 2 \cdot l} \cdot \left(a^2 + b \right)^l \cdot \lambda_{k,l}\]
where $\lambda_{k,l} = \sum^{[k/2]}_{m=l} \binom{k}{2 \cdot m} \cdot \binom{m}{l}$
[b]b.)[/b] Let $u_k = \sum^{[k/2]}_{l=0} \lambda_{k,l} $. For positive integer $m,$ denote the remainder of $u_k$ divided by $2^m$ as $z_{m,k}$. Prove that $z_{m,k},$ $k = 0,1,2, \ldots$ is a periodic function, and find the smallest period.
1994 China Team Selection Test, 2
An $n$ by $n$ grid, where every square contains a number, is called an $n$-code if the numbers in every row and column form an arithmetic progression. If it is sufficient to know the numbers in certain squares of an $n$-code to obtain the numbers in the entire grid, call these squares a key.
[b]a.) [/b]Find the smallest $s \in \mathbb{N}$ such that any $s$ squares in an $n-$code $(n \geq 4)$ form a key.
[b]b.)[/b] Find the smallest $t \in \mathbb{N}$ such that any $t$ squares along the diagonals of an $n$-code $(n \geq 4)$ form a key.
2001 Vietnam National Olympiad, 2
Find all real-valued continuous functions defined on the interval $(-1, 1)$ such that $(1-x^{2}) f(\frac{2x}{1+x^{2}}) = (1+x^{2})^{2}f(x)$ for all $x$.
2007 Today's Calculation Of Integral, 210
Evaluate $\int_{1}^{\pi}\left(x^{3}\ln x-\frac{6}{x}\right)\sin x\ dx$.
2008 239 Open Mathematical Olympiad, 6
Given a polynomial $P(x,y)$ with real coefficients, suppose that some real function $f:\mathbb R \to \mathbb R$ satisfies
$$P(x,y) = f(x+y)-f(x)-f(y)$$for all $x,y\in\mathbb R$. Show that some polynomial $q$ satisfies
$$P(x,y) = q(x+y)-q(x)-q(y)$$
PEN J Problems, 17
Show that $\phi(n)+\sigma(n) \ge 2n$ for all positive integers $n$.
1981 Miklós Schweitzer, 10
Let $ P$ be a probability distribution defined on the Borel sets of the real line. Suppose that $ P$ is symmetric with respect to the origin, absolutely continuous with respect to the Lebesgue measure, and its density function $ p$ is zero outside the interval $ [\minus{}1,1]$ and inside this interval it is between the positive numbers $ c$ and $ d$ ($ c < d$). Prove that there is no distribution whose convolution square equals $ P$.
[i]T. F. Mori, G. J. Szekely[/i]
2002 Austrian-Polish Competition, 7
Find all real functions $f$ definited on positive integers and satisying:
(a) $f(x+22)=f(x)$,
(b) $f\left(x^{2}y\right)=\left(f(x)\right)^{2}f(y)$
for all positive integers $x$ and $y$.
2009 ITAMO, 3
A natural number $k$ is said $n$-squared if by colouring the squares of a $2n \times k$ chessboard, in any manner, with $n$ different colours, we can find $4$ separate unit squares of the same colour, the centers of which are vertices of a rectangle having sides parallel to the sides of the board. Determine, in function of $n$, the smallest natural $k$ that is $n$-squared.
2014 District Olympiad, 4
Find all functions $f:\mathbb{N}^{\ast}\rightarrow\mathbb{N}^{\ast}$ with
the properties:
[list=a]
[*]$ f(m+n) -1 \mid f(m)+f(n),\quad \forall m,n\in\mathbb{N}^{\ast} $
[*]$ n^{2}-f(n)\text{ is a square } \;\forall n\in\mathbb{N}^{\ast} $[/list]
2012 Balkan MO Shortlist, N3
Let $\mathbb{Z}^+$ be the set of positive integers. Find all functions $f:\mathbb{Z}^+ \rightarrow\mathbb{Z}^+$ such that the following conditions both hold:
(i) $f(n!)=f(n)!$ for every positive integer $n$,
(ii) $m-n$ divides $f(m)-f(n)$ whenever $m$ and $n$ are different positive integers.
2006 Romania National Olympiad, 1
Find the maximal value of \[ \left( x^3+1 \right) \left( y^3 + 1\right) , \] where $x,y \in \mathbb R$, $x+y=1$.
[i]Dan Schwarz[/i]
Russian TST 2017, P3
Let $K=(V, E)$ be a finite, simple, complete graph. Let $\phi: E \to \mathbb{R}^2$ be a map from the edge set to the plane, such that the preimage of any point in the range defines a connected graph on the entire vertex set $V$, and the points assigned to the edges of any triangle are collinear. Show that the range of $\phi$ is contained in a line.
2023 Indonesia TST, A
Find all function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfied
\[f(x+y) + f(x)f(y) = f(xy) + 1 \]
$\forall x, y \in \mathbb{R}$
2012 France Team Selection Test, 1
Let $k>1$ be an integer. A function $f:\mathbb{N^*}\to\mathbb{N^*}$ is called $k$-[i]tastrophic[/i] when for every integer $n>0$, we have $f_k(n)=n^k$ where $f_k$ is the $k$-th iteration of $f$:
\[f_k(n)=\underbrace{f\circ f\circ\cdots \circ f}_{k\text{ times}}(n)\]
For which $k$ does there exist a $k$-tastrophic function?
2009 Kyrgyzstan National Olympiad, 8
Does there exist a function $ f: {\Bbb N} \to {\Bbb N}$ such that $ f(f(n \minus{} 1)) \equal{} f(n \plus{} 1) \minus{} f(n)$ for all $ n > 2$.
2015 SG Originals, N6
Let $\mathbb{Z}_{>0}$ denote the set of positive integers. Consider a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$. For any $m, n \in \mathbb{Z}_{>0}$ we write $f^n(m) = \underbrace{f(f(\ldots f}_{n}(m)\ldots))$. Suppose that $f$ has the following two properties:
(i) if $m, n \in \mathbb{Z}_{>0}$, then $\frac{f^n(m) - m}{n} \in \mathbb{Z}_{>0}$;
(ii) The set $\mathbb{Z}_{>0} \setminus \{f(n) \mid n\in \mathbb{Z}_{>0}\}$ is finite.
Prove that the sequence $f(1) - 1, f(2) - 2, f(3) - 3, \ldots$ is periodic.
[i]Proposed by Ang Jie Jun, Singapore[/i]
2004 Baltic Way, 8
Let $f\left(x\right)$ be a non-constant polynomial with integer coefficients, and let $u$ be an arbitrary positive integer. Prove that there is an integer $n$ such that $f\left(n\right)$ has at least $u$ distinct prime factors and $f\left(n\right) \neq 0$.
2009 Putnam, A6
Let $ f: [0,1]^2\to\mathbb{R}$ be a continuous function on the closed unit square such that $ \frac{\partial f}{\partial x}$ and $ \frac{\partial f}{\partial y}$ exist and are continuous on the interior of $ (0,1)^2.$ Let $ a\equal{}\int_0^1f(0,y)\,dy,\ b\equal{}\int_0^1f(1,y)\,dy,\ c\equal{}\int_0^1f(x,0)\,dx$ and $ d\equal{}\int_0^1f(x,1)\,dx.$ Prove or disprove: There must be a point $ (x_0,y_0)$ in $ (0,1)^2$ such that
$ \frac{\partial f}{\partial x}(x_0,y_0)\equal{}b\minus{}a$ and $ \frac{\partial f}{\partial y}(x_0,y_0)\equal{}d\minus{}c.$
1996 IMO Shortlist, 4
Let $ a_{1}, a_{2}...a_{n}$ be non-negative reals, not all zero. Show that that
(a) The polynomial $ p(x) \equal{} x^{n} \minus{} a_{1}x^{n \minus{} 1} \plus{} ... \minus{} a_{n \minus{} 1}x \minus{} a_{n}$ has preceisely 1 positive real root $ R$.
(b) let $ A \equal{} \sum_{i \equal{} 1}^n a_{i}$ and $ B \equal{} \sum_{i \equal{} 1}^n ia_{i}$. Show that $ A^{A} \leq R^{B}$.
1996 Tournament Of Towns, (521) 4
Prove that for any function $f(x)$, continuous or otherwise, $$f(f(x)) = x^2 - 1996$$ cannot hold for all real numbers $x$.
(S Bogatiy, M Smurov,)
1969 Miklós Schweitzer, 3
Let $ f(x)$ be a nonzero, bounded, real function on an Abelian group $ G$, $ g_1,...,g_k$ are given elements of $ G$ and $ \lambda_1,...,\lambda_k$ are real numbers. Prove that if \[ \sum_{i=1}^k \lambda_i f(g_ix) \geq 0\] holds for all $ x \in G$, then \[ \sum_{i=1}^k \lambda_i \geq 0.\]
[i]A. Mate[/i]
2000 National Olympiad First Round, 28
$$\begin{array}{ rlrlrl}
f_1(x)=&x^2+x & f_2(x)=&2x^2-x & f_3(x)=&x^2 +x \\
g_1(x)=&x-2 & g_2(x)=&2x \ \ & g_3(x)=&x+2 \\
\end{array}$$
If $h(x)=x$ can be get from $f_i$ and $g_i$ by using only addition, substraction, multiplication defined on those functions where $i\in\{1,2,3\}$, then $F_i=1$. Otherwise, $F_i=0$. What is $(F_1,F_2,F_3)$ ?
$ \textbf{(A)}\ (0,0,0)
\qquad\textbf{(B)}\ (0,0,1)
\qquad\textbf{(C)}\ (0,1,0)
\qquad\textbf{(D)}\ (0,1,1)
\qquad\textbf{(E)}\ \text{None}
$
2016 Turkey Team Selection Test, 5
Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that for all $m,n \in \mathbb{N}$ holds $f(mn)=f(m)f(n)$ and $m+n \mid f(m)+f(n)$ .
2004 Iran MO (2nd round), 2
Let $f:\mathbb{R}^{\geq 0}\to\mathbb{R}$ be a function such that $f(x)-3x$ and $f(x)-x^3$ are ascendant functions. Prove that $f(x)-x^2-x$ is an ascendant function, too.
(We call the function $g(x)$ ascendant, when for every $x\leq{y}$ we have $g(x)\leq{g(y)}$.)