Found problems: 1311
2005 Germany Team Selection Test, 1
Given the positive numbers $a$ and $b$ and the natural number $n$, find the greatest among the $n + 1$ monomials in the binomial expansion of $\left(a+b\right)^n$.
2008 Poland - Second Round, 3
Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ for which the equality
\[f(f(x)\minus{}y)\equal{}f(x)\plus{}f(f(y)\minus{}f(\minus{}x))\plus{}x\]
holds for all real $x,y$.
2009 Kazakhstan National Olympiad, 6
Let $P(x)$ be polynomial with integer coefficients.
Prove, that if for any natural $k$ holds equality: $ \underbrace{P(P(...P(0)...))}_{n -times}=0$ then $P(0)=0$ or $P(P(0))=0$
2011 Bulgaria National Olympiad, 2
Let $f_1(x)$ be a polynomial of degree $2$ with the leading coefficient positive and $f_{n+1}(x) =f_1(f_n(x))$ for $n\ge 1.$ Prove that if the equation $f_2(x)=0$ has four different non-positive real roots, then for arbitrary $n$ then $f_n(x)$ has $2^n$ different real roots.
2006 Iran Team Selection Test, 2
Let $n$ be a fixed natural number.
[b]a)[/b] Find all solutions to the following equation :
\[ \sum_{k=1}^n [\frac x{2^k}]=x-1 \]
[b]b)[/b] Find the number of solutions to the following equation ($m$ is a fixed natural) :
\[ \sum_{k=1}^n [\frac x{2^k}]=x-m \]
2012 Singapore MO Open, 5
There are $2012$ distinct points in the plane, each of which is to be coloured using one of $n$ colours, so that the numbers of points of each colour are distinct. A set of $n$ points is said to be [i]multi-coloured [/i]if their colours are distinct. Determine $n$ that maximizes the number of multi-coloured sets.
1993 IberoAmerican, 3
Let $\mathbb{N}^*=\{1,2,\ldots\}$. Find al the functions $f: \mathbb{N}^*\rightarrow \mathbb{N}^*$ such that:
(1) If $x<y$ then $f(x)<f(y)$.
(2) $f\left(yf(x)\right)=x^2f(xy)$ for all $x,y \in\mathbb{N}^*$.
2012 Romania Team Selection Test, 2
Let $n$ be a positive integer. Find the value of the following sum \[\sum_{(n)}\sum_{k=1}^n {e_k2^{e_1+\cdots+e_k-2k-n}},\] where $e_k\in\{0,1\}$ for $1\leq k \leq n$, and the sum $\sum_{(n)}$ is taken over all $2^n$ possible choices of $e_1,\ldots ,e_n$.
2008 District Olympiad, 4
Determine $ x,y,z>0$ for which $ x^3y\plus{}3<\equal{}4z, y^3z\plus{}3<\equal{}4x,z^3x\plus{}3<\equal{}4y.$
2014 Mediterranean Mathematics Olympiad, 3
Prove that for every integer $S\ge100$ there exists an integer $P$ for which the following story could hold true:
The mathematician asks the shop owner: ``How much are the table, the cabinet and the bookshelf?'' The shop owner replies: ``Each item costs a positive integer amount of Euros. The table is more expensive than the cabinet, and the cabinet is more expensive than the bookshelf. The sum of the three prices is $S$ and their product is $P$.''
The mathematician thinks and complains: ``This is not enough information to determine the three prices!''
(Proposed by Gerhard Woeginger, Austria)
2004 Silk Road, 1
Find all $ f: \mathbb{R} \to \mathbb{R}$, such that $(x+y)(f(x)-f(y))=(x-y)f(x+y)$ for all real $x,y$.
2006 IMAR Test, 1
Consider the equation \[\frac{xy-C}{x+y}= k ,\] where all symbols used are positive integers.
1. Show that, for any (fixed) values $C, k$ this equation has at least a solution $x, y$;
2. Show that, for any (fixed) values $C, k$ this equation has at most a finite number of solutions $x, y$;
3. Show that, for any $C, n$ there exists $k = k(C,n)$ such that the equation has more than $n$ solutions $x, y$.
1998 Vietnam National Olympiad, 1
Does there exist an infinite sequence $\{x_{n}\}$ of reals satisfying the following conditions
i)$|x_{n}|\leq 0,666$ for all $n=1,2,...$
ii)$|x_{m}-x_{n}|\geq \frac{1}{n(n+1)}+\frac{1}{m(m+1)}$ for all $m\not = n$?
2015 China Team Selection Test, 4
Let $n$ be a positive integer, let $f_1(x),\ldots,f_n(x)$ be $n$ bounded real functions, and let $a_1,\ldots,a_n$ be $n$ distinct reals.
Show that there exists a real number $x$ such that $\sum^n_{i=1}f_i(x)-\sum^n_{i=1}f_i(x-a_i)<1$.
2000 Federal Competition For Advanced Students, Part 2, 3
Find all real solutions to the equation
\[| | | | | | |x^2 -x - 1| - 3| - 5| - 7| - 9| - 11| - 13| = x^2 - 2x - 48.\]
2005 Moldova Team Selection Test, 4
$n$ is a positive integer, $K$ the set of polynoms of real variables $x_1,x_2,...,x_{n+1}$ and $y_1,y_2,...,y_{n+1}$, function $f:K\rightarrow K$ satisfies
\[f(p+q)=f(p)+f(q),\quad f(pq)=f(p)q+pf(q),\quad (\forall)p,q\in K.\]
If $f(x_i)=(n-1)x_i+y_i,\quad f(y_i)=2ny_i$ for all $i=1,2,...,n+1$ and
\[\prod_{i=1}^{n+1}(tx_i+y_i)=\sum_{i=0}^{n+1}p_it^{n+1-i}\]
for any real $t$, prove, that for all $k=1,...,n+1$
\[f(p_{k-1})=kp_k+(n+1)(n+k-2)p_{k-1}\]
2007 Irish Math Olympiad, 5
Suppose that $ a$ and $ b$ are real numbers such that the quadratic polynomial $ f(x)\equal{}x^2\plus{}ax\plus{}b$ has no nonnegative real roots. Prove that there exist two polynomials $ g,h$ whose coefficients are nonnegative real numbers such that: $ f(x)\equal{}\frac{g(x)}{h(x)}$ for all real numbers $ x$.
2008 Hungary-Israel Binational, 1
Prove that: $ \sum_{i\equal{}1}^{n^2} \lfloor \frac{i}{3} \rfloor\equal{} \frac{n^2(n^2\minus{}1)}{6}$
For all $ n \in N$.
2009 China Team Selection Test, 3
Let $ f(x)$ be a $ n \minus{}$degree polynomial all of whose coefficients are equal to $ \pm 1$, and having $ x \equal{} 1$ as its $ m$ multiple root. If $ m\ge 2^k (k\ge 2,k\in N)$, then $ n\ge 2^{k \plus{} 1} \minus{} 1.$
1970 IMO Longlists, 37
Solve the set of simultaneous equations
\begin{align*}
v^2+ w^2+ x^2+ y^2 &= 6 - 2u, \\
u^2+ w^2+ x^2+ y^2 &= 6 - 2v, \\
u^2+ v^2+ x^2+ y^2 &= 6- 2w, \\
u^2+ v^2+ w^2+ y^2 &= 6 - 2x, \\
u^2+ v^2+ w^2+ x^2 &= 6- 2y.
\end{align*}
2007 Czech-Polish-Slovak Match, 1
Find all polynomials $P$ with real coefficients satisfying $P(x^2)=P(x)\cdot P(x+2)$ for all real numbers $x.$
2006 District Olympiad, 4
For each positive integer $n\geq 2$ we denote with $p(n)$ the largest prime number less than or equal to $n$, and with $q(n)$ the smallest prime number larger than $n$. Prove that \[ \sum^n_{k=2} \frac 1{p(k)q(k)} < \frac 12. \]
2006 Pre-Preparation Course Examination, 6
Suppose that $P_c(z)=z^2+c$. You are familiar with the Mandelbrot set: $M=\{c\in \mathbb{C} | \lim_{n\rightarrow \infty}P_c^n(0)\neq \infty\}$.
We know that if $c\in M$ then the points of the dynamical system $(\mathbb{C},P_c)$ that don't converge to $\infty$ are connected and otherwise they are completely disconnected. By seeing the properties of periodic points of $P_c$ prove the following ones:
a) Prove the existance of the heart like shape in the Mandelbrot set.
b) Prove the existance of the large circle next to the heart like shape in the Mandelbrot set.
[img]http://astronomy.swin.edu.au/~pbourke/fractals/mandelbrot/mandel1.gif[/img]
1996 Turkey Team Selection Test, 3
Determine all ordered pairs of positive real numbers $(a, b)$ such that every sequence $(x_{n})$ satisfying $\lim_{n \rightarrow \infty}{(ax_{n+1} - bx_{n})} = 0$ must have $\lim_{n \rightarrow \infty} x_n = 0$.
2006 Pre-Preparation Course Examination, 2
Show that there exists a continuos function $f: [0,1]\rightarrow [0,1]$ such that it has no periodic orbit of order $3$ but it has a periodic orbit of order $5$.