Found problems: 70
2007 QEDMO 5th, 6
Find all functions $ f: \mathbb{R}\to\mathbb{R}$ that satisfy the equation:
$ f\left(\left(f\left(x\right)\right)^2 \plus{} f\left(y\right)\right) \equal{} xf\left(x\right) \plus{} y$
for any two real numbers $ x$ and $ y$.
2007 JBMO Shortlist, 1
Let $a$ be positive real number such that $a^{3}=6(a+1)$. Prove that the equation $x^{2}+ax+a^{2}-6=0$ has no real solution.
2005 MOP Homework, 2
Find all real numbers $x$ such that
$\lfloor x^2-2x \rfloor+2\lfloor x \rfloor=\lfloor x \rfloor^2$.
(For a real number $x$, $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$.)
2005 MOP Homework, 6
Solve the system of equations:
$x^2=\frac{1}{y}+\frac{1}{z}$,
$y^2=\frac{1}{z}+\frac{1}{x}$,
$z^2=\frac{1}{x}+\frac{1}{y}$.
in the real numbers.
2004 Baltic Way, 5
Determine the range of the following function defined for integer $k$,
\[f(k)=(k)_3+(2k)_5+(3k)_7-6k\]
where $(k)_{2n+1}$ denotes the multiple of $2n+1$ closest to $k$
1996 Romania National Olympiad, 3
Prove that $ \forall x\in \mathbb{R} $ , $ \cos ^7x+\cos ^7(x+\frac {2\pi}{3})+\cos ^7(x+\frac {4\pi}{3})=\frac {63}{64}\cos 3x $
1986 Flanders Math Olympiad, 4
Given a cube in which you can put two massive spheres of radius 1.
What's the smallest possible value of the side - length of the cube?
Prove that your answer is the best possible.
2001 Flanders Math Olympiad, 4
A student concentrates on solving quadratic equations in $\mathbb{R}$. He starts with a first quadratic equation $x^2 + ax + b = 0$ where $a$ and $b$ are both different from 0. If this first equation has solutions $p$ and $q$ with $p \leq q$, he forms a second quadratic equation $x^2 + px + q = 0$. If this second equation has solutions, he forms a third quadratic equation in an identical way. He continues this process as long as possible. Prove that he will not obtain more than five equations.
2005 Taiwan TST Round 1, 2
The absolute value of every number in the sequence $\{a_n\}$ is smaller than 2005, and \[a_{n+6}=a_{n+4}+a_{n+2}-a_n.\] holds for all positive integers n. Prove that $\{a_n\}$ is periodic.
Incredibly, this was probably the most difficult problem of our independent study problems in the 1st TST (excluding the final exam).
2008 JBMO Shortlist, 2
Find all real numbers $ a,b,c,d$ such that \[ \left\{\begin{array}{cc}a \plus{} b \plus{} c \plus{} d \equal{} 20, \\
ab \plus{} ac \plus{} ad \plus{} bc \plus{} bd \plus{} cd \equal{} 150. \end{array} \right.\]
1985 Bundeswettbewerb Mathematik, 3
Starting with the sequence $F_1 = (1,2,3,4, \ldots)$ of the natural numbers further sequences are generated as follows: $F_{n+1}$ is created from $F_n$ by the following rule: the order of elements remains unchanged, the elements from $F_n$ which are divisible by $n$ are increased by 1 and the other elements from $F_n$ remain unchanged. Example: $F_2 = (2,3,4,5 \ldots)$ and $F_3 = (3,3,5,5, \ldots)$. Determine all natural numbers $n$ such that exactly the first $n-1$ elements of $F_n$ take the value $n.$
2010 Contests, 1
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all $x, y\in\mathbb{R}$, we have
\[f(x+y)+f(x)f(y)=f(xy)+(y+1)f(x)+(x+1)f(y).\]
2010 Middle European Mathematical Olympiad, 5
Three strictly increasing sequences
\[a_1, a_2, a_3, \ldots,\qquad b_1, b_2, b_3, \ldots,\qquad c_1, c_2, c_3, \ldots\]
of positive integers are given. Every positive integer belongs to exactly one of the three sequences. For every positive integer $n$, the following conditions hold:
(a) $c_{a_n}=b_n+1$;
(b) $a_{n+1}>b_n$;
(c) the number $c_{n+1}c_{n}-(n+1)c_{n+1}-nc_n$ is even.
Find $a_{2010}$, $b_{2010}$ and $c_{2010}$.
[i](4th Middle European Mathematical Olympiad, Team Competition, Problem 1)[/i]
2010 Iran MO (3rd Round), 4
For each polynomial $p(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ we define it's derivative as this and we show it by $p'(x)$:
\[p'(x)=na_nx^{n-1}+(n-1)a_{n-1}x^{n-2}+...+2a_2x+a_1\]
a) For each two polynomials $p(x)$ and $q(x)$ prove that:(3 points)
\[(p(x)q(x))'=p'(x)q(x)+p(x)q'(x)\]
b) Suppose that $p(x)$ is a polynomial with degree $n$ and $x_1,x_2,...,x_n$ are it's zeros. prove that:(3 points)
\[\frac{p'(x)}{p(x)}=\sum_{i=1}^{n}\frac{1}{x-x_i}\]
c) $p(x)$ is a monic polynomial with degree $n$ and $z_1,z_2,...,z_n$ are it's zeros such that:
\[|z_1|=1, \quad \forall i\in\{2,..,n\}:|z_i|\le1\]
Prove that $p'(x)$ has at least one zero in the disc with length one with the center $z_1$ in complex plane. (disc with length one with the center $z_1$ in complex plane: $D=\{z \in \mathbb C: |z-z_1|\le1\}$)(20 points)
2000 India National Olympiad, 3
If $a,b,c,x$ are real numbers such that $abc \not= 0$ and \[ \frac{xb + (1-x)c}{a} = \frac{xc + (1-x)a}{b} = \frac{xa + (1-x) b }{c}, \] then prove that $a = b = c$.
2005 Taiwan TST Round 1, 2
Does there exist an positive integer $n$, so that for any positive integer $m<1002$, there exists an integer $k$ so that \[\displaystyle \frac{m}{1002} < \frac{k}{n} < \frac {m+1}{1003}\] holds? If $n$ does not exist, prove it; if $n$ exists, determine the minimum value of it.
I know this problem was easy, but it still appeared on our TST, and so I posted it here.
2010 All-Russian Olympiad, 1
Let $a \neq b a,b \in \mathbb{R}$ such that $(x^2+20ax+10b)(x^2+20bx+10a)=0$ has no roots for $x$. Prove that $20(b-a)$ is not an integer.
1985 IMO Longlists, 79
Let $a, b$, and $c$ be real numbers such that
\[\frac{1}{bc-a^2} + \frac{1}{ca-b^2}+\frac{1}{ab-c^2} = 0.\]
Prove that
\[\frac{a}{(bc-a^2)^2} + \frac{b}{(ca-b^2)^2}+\frac{c}{(ab-c^2)^2} = 0.\]
2006 Silk Road, 1
Found all functions $f: \mathbb{R} \to \mathbb{R}$, such that for any $x,y \in \mathbb{R}$,
\[f(x^2+xy+f(y))=f^2(x)+xf(y)+y.\]
2012 Canadian Mathematical Olympiad Qualification Repechage, 6
Determine whether there exist two real numbers $a$ and $b$ such that both $(x-a)^3+ (x-b)^2+x$ and $(x-b)^3 + (x-a)^2 +x$ contain only real roots.