Found problems: 1269
2006 China Team Selection Test, 2
The function $f(n)$ satisfies $f(0)=0$, $f(n)=n-f \left( f(n-1) \right)$, $n=1,2,3 \cdots$. Find all polynomials $g(x)$ with real coefficient such that
\[ f(n)= [ g(n) ], \qquad n=0,1,2 \cdots \]
Where $[ g(n) ]$ denote the greatest integer that does not exceed $g(n)$.
1998 All-Russian Olympiad, 1
Two lines parallel to the $x$-axis cut the graph of $y=ax^3+bx^2+cx+d$ in points $A,C,E$ and $B,D,F$ respectively, in that order from left to right. Prove that the length of the projection of the segment $CD$ onto the $x$-axis equals the sum of the lengths of the projections of $AB$ and $EF$.
2000 India National Olympiad, 5
Let $a,b,c$ be three real numbers such that $1 \geq a \geq b \geq c \geq 0$. prove that if $\lambda$ is a root of the cubic equation $x^3 + ax^2 + bx + c = 0$ (real or complex), then $| \lambda | \leq 1.$
2003 All-Russian Olympiad, 1
Suppose that $M$ is a set of $2003$ numbers such that, for any distinct $a, b, c \in M$, the number $a^2 + bc$ is rational. Prove that there is a positive integer $n$ such that $a\sqrt n$ is rational for all $a \in M.$
2003 China Team Selection Test, 3
The $ n$ roots of a complex coefficient polynomial $ f(z) \equal{} z^n \plus{} a_1z^{n \minus{} 1} \plus{} \cdots \plus{} a_{n \minus{} 1}z \plus{} a_n$ are $ z_1, z_2, \cdots, z_n$. If $ \sum_{k \equal{} 1}^n |a_k|^2 \leq 1$, then prove that $ \sum_{k \equal{} 1}^n |z_k|^2 \leq n$.
2007 South East Mathematical Olympiad, 1
Let $f(x)$ be a function satisfying $f(x+1)-f(x)=2x+1 (x \in \mathbb{R})$.In addition, $|f(x)|\le 1$ holds for $x\in [0,1]$. Prove that $|f(x)|\le 2+x^2$ holds for $x \in \mathbb{R}$.
2006 Bulgaria Team Selection Test, 1
Find all sequences of positive integers $\{a_n\}_{n=1}^{\infty}$, for which $a_4=4$ and
\[\frac{1}{a_1a_2a_3}+\frac{1}{a_2a_3a_4}+\cdots+\frac{1}{a_na_{n+1}a_{n+2}}=\frac{(n+3)a_n}{4a_{n+1}a_{n+2}}\]
for all natural $n \geq 2$.
[i]Peter Boyvalenkov[/i]
2012 Albania National Olympiad, 2
The trinomial $f(x)$ is such that $(f(x))^3-f(x)=0$ has three real roots. Find the y-coordinate of the vertex of $f(x)$.
2007 Hungary-Israel Binational, 3
Let $ t \ge 3$ be a given real number and assume that the polynomial $ f(x)$ satisfies $|f(k)\minus{}t^k|<1$, for $ k\equal{}0,1,2,\ldots ,n$. Prove that the degree of $f(x)$ is at least $n$.
2009 Germany Team Selection Test, 1
Consider cubes of edge length 5 composed of 125 cubes of edge length 1 where each of the 125 cubes is either coloured black or white. A cube of edge length 5 is called "big", a cube od edge length is called "small". A posititve integer $ n$ is called "representable" if there is a big cube with exactly $ n$ small cubes where each row of five small cubes has an even number of black cubes whose centres lie on a line with distances $ 1,2,3,4$ (zero counts as even number).
(a) What is the smallest and biggest representable number?
(b) Construct 45 representable numbers.
2002 China Team Selection Test, 3
Sequence $ \{ f_n(a) \}$ satisfies $ \displaystyle f_{n\plus{}1}(a) \equal{} 2 \minus{} \frac{a}{f_n(a)}$, $ f_1(a) \equal{} 2$, $ n\equal{}1,2, \cdots$. If there exists a natural number $ n$, such that $ f_{n\plus{}k}(a) \equal{} f_{k}(a), k\equal{}1,2, \cdots$, then we call the non-zero real $ a$ a $ \textbf{periodic point}$ of $ f_n(a)$.
Prove that the sufficient and necessary condition for $ a$ being a $ \textbf{periodic point}$ of $ f_n(a)$ is $ p_n(a\minus{}1)\equal{}0$, where $ \displaystyle p_n(x)\equal{}\sum_{k\equal{}0}^{\left[ \frac{n\minus{}1}{2} \right]} (\minus{}1)^k C_n^{2k\plus{}1}x^k$, here we define $ \displaystyle \frac{a}{0}\equal{} \infty$ and $ \displaystyle \frac{a}{\infty} \equal{} 0$.
1995 Canada National Olympiad, 5
$u$ is a real parameter such that $0<u<1$.
For $0\le x \le u$, $f(x)=0$.
For $u\le x \le n$, $f(x)=1-\left(\sqrt{ux}+\sqrt{(1-u)(1-x)}\right)^2$.
The sequence $\{u_n\}$ is define recursively as follows: $u_1=f(1)$ and $u_n=f(u_{n-1})$ $\forall n\in \mathbb{N}, n\neq 1$.
Show that there exists a positive integer $k$ for which $u_k=0$.
1989 Federal Competition For Advanced Students, P2, 5
Find all real solutions of the system:
$ x^2\plus{}2yz\equal{}x,$
$ y^2\plus{}2zx\equal{}y,$
$ z^2\plus{}2xy\equal{}z.$
1992 IMTS, 4
Prove that if $f$ is a non-constant real-valued function such that for all real $x$, $f(x+1) + f(x-1) = \sqrt{3} f(x)$, then $f$ is periodic. What is the smallest $p$, $p > 0$ such that $f(x+p) = f(x)$ for all $x$?
2011 Moldova Team Selection Test, 2
Find all pairs of real number $x$ and $y$ which simultaneously satisfy the following 2 relations:
$x+y+4=\frac{12x+11y}{x^2+y^2}$
$y-x+3=\frac{11x-12y}{x^2+y^2}$
1986 IMO Longlists, 61
Given a positive integer $n$, find the greatest integer $p$ with the property that for any function $f : \mathbb P(X) \to C$, where $X$ and $C$ are sets of cardinality $n$ and $p$, respectively, there exist two distinct sets $A,B \in \mathbb P(X)$ such that $f(A) = f(B) = f(A \cup B)$. ($\mathbb P(X)$ is the family of all subsets of $X$.)
2003 China Team Selection Test, 1
$m$ and $n$ are positive integers. Set $A=\{ 1, 2, \cdots, n \}$. Let set $B_{n}^{m}=\{ (a_1, a_2 \cdots, a_m) \mid a_i \in A, i= 1, 2, \cdots, m \}$ satisfying:
(1) $|a_i - a_{i+1}| \neq n-1$, $i=1,2, \cdots, m-1$; and
(2) at least three of $a_1, a_2, \cdots, a_m$ ($m \geq 3$) are pairwise distince.
Find $|B_n^m|$ and $|B_6^3|$.
2006 China Second Round Olympiad, 2
Let $x,y$ be real numbers. Define a sequence $\{a_n \}$ through the recursive formula
\[ a_0=x,a_1=y,a_{n+1}=\frac{a_na_{n-1}+1}{a_n+a_{n-1}},\]
Find $a_n$.
2014 Saudi Arabia BMO TST, 4
Let $f :\mathbb{N} \rightarrow\mathbb{N}$ be an injective function such that $f(1) = 2,~ f(2) = 4$ and \[f(f(m) + f(n)) = f(f(m)) + f(n)\] for all $m, n \in \mathbb{N}$. Prove that $f(n) = n + 2$ for all $n \ge 2$.
1994 Turkey MO (2nd round), 1
For $n\in\mathbb{N}$, let $a_{n}$ denote the closest integer to $\sqrt{n}$. Evaluate \[\sum_{n=1}^\infty{\frac{1}{a_{n}^{3}}}.\]
1990 China Team Selection Test, 2
Find all functions $f,g,h: \mathbb{R} \mapsto \mathbb{R}$ such that $f(x) - g(y) = (x-y) \cdot h(x+y)$ for $x,y \in \mathbb{R}.$
2006 China Northern MO, 4
Given a function $f(x)=x^{2}+ax+b$ with $a,b \in R$, if there exists a real number $m$ such that $\left| f(m) \right| \leq \frac{1}{4}$ and $\left| f(m+1) \right| \leq \frac{1}{4}$, then find the maximum and minimum of the value of $\Delta=a^{2}-4b$.
1997 South africa National Olympiad, 4
Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ which satisfy \[ f(m + f(n)) = f(m) + n \] for all $m,n \in \mathbb{Z}$.
2008 USA Team Selection Test, 5
Two sequences of integers, $ a_1, a_2, a_3, \ldots$ and $ b_1, b_2, b_3, \ldots$, satisfy the equation
\[ (a_n \minus{} a_{n \minus{} 1})(a_n \minus{} a_{n \minus{} 2}) \plus{} (b_n \minus{} b_{n \minus{} 1})(b_n \minus{} b_{n \minus{} 2}) \equal{} 0
\]
for each integer $ n$ greater than $ 2$. Prove that there is a positive integer $ k$ such that $ a_k \equal{} a_{k \plus{} 2008}$.
1997 Greece National Olympiad, 2
Let a function $f : \Bbb{R}^+ \to \Bbb{R}$ satisfy:
(i) $f$ is strictly increasing,
(ii) $f(x) > -1/x$ for all $x > 0$,
(iii)$ f(x)f (f(x) + 1/x) = 1$ for all $x > 0$.
Determine $f(1)$.