Found problems: 1311
2001 District Olympiad, 1
Consider the equation $x^2+(a+b+c)x+\lambda (ab+bc+ca)=0$ with $a,b,c>0$ and $\lambda\in \mathbb{R}$. Prove that:
a)If $\lambda\le \frac{3}{4}$, the equation has real roots.
b)If $a,b,c$ are the side lengths of a triangle and $\lambda\ge 1$, then the equation doesn't have real roots.
[i]***[/i]
2011 Switzerland - Final Round, 4
Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that for any real numbers $a, b, c, d >0$ satisfying $abcd=1$,\[(f(a)+f(b))(f(c)+f(d))=(a+b)(c+d)\] holds true.
[i](Swiss Mathematical Olympiad 2011, Final round, problem 4)[/i]
2002 Baltic Way, 3
Find all sequences $0\le a_0\le a_1\le a_2\le \ldots$ of real numbers such that
\[a_{m^2+n^2}=a_m^2+a_n^2 \]
for all integers $m,n\ge 0$.
2008 ISI B.Math Entrance Exam, 8
Let $a^2+b^2=1$ , $c^2+d^2=1$ , $ac+bd=0$
Prove that
$a^2+c^2=1$ , $b^2+d^2=1$ , $ab+cd=0$ .
2014 All-Russian Olympiad, 3
If the polynomials $f(x)$ and $g(x)$ are written on a blackboard then we can also write down the polynomials $f(x)\pm g(x)$, $f(x)g(x)$, $f(g(x))$ and $cf(x)$, where $c$ is an arbitrary real constant. The polynomials $x^3-3x^2+5$ and $x^2-4x$ are written on the blackboard. Can we write a nonzero polynomial of form $x^n-1$ after a finite number of steps?
2010 Contests, 3
prove that for each natural number $n$ there exist a polynomial with degree $2n+1$ with coefficients in $\mathbb{Q}[x]$ such that it has exactly $2$ complex zeros and it's irreducible in $\mathbb{Q}[x]$.(20 points)
2014 Iran MO (3rd Round), 2
Find all continuous function $f:\mathbb{R}^{\geq 0}\rightarrow \mathbb{R}^{\geq 0}$ such that :
\[f(xf(y))+f(f(y)) = f(x)f(y)+2 \: \: \forall x,y\in \mathbb{R}^{\geq 0}\]
[i]Proposed by Mohammad Ahmadi[/i]
2008 Hungary-Israel Binational, 2
The sequence $ a_n$ is defined as follows: $ a_0\equal{}1, a_1\equal{}1, a_{n\plus{}1}\equal{}\frac{1\plus{}a_{n}^2}{a_{n\minus{}1}}$.
Prove that all the terms of the sequence are integers.
1992 Hungary-Israel Binational, 6
We examine the following two sequences: The Fibonacci sequence: $F_{0}= 0, F_{1}= 1, F_{n}= F_{n-1}+F_{n-2 }$ for $n \geq 2$; The Lucas sequence: $L_{0}= 2, L_{1}= 1, L_{n}= L_{n-1}+L_{n-2}$ for $n \geq 2$. It is known that for all $n \geq 0$
\[F_{n}=\frac{\alpha^{n}-\beta^{n}}{\sqrt{5}},L_{n}=\alpha^{n}+\beta^{n}, \]
where $\alpha=\frac{1+\sqrt{5}}{2},\beta=\frac{1-\sqrt{5}}{2}$. These formulae can be used without proof.
The coordinates of all vertices of a given rectangle are Fibonacci numbers. Suppose that the rectangle is not such that one of its vertices is on the $x$-axis and another on the $y$-axis. Prove that either the sides of the rectangle are parallel to the axes, or make an angle of $45^{\circ}$ with the axes.
1971 IMO Longlists, 47
A sequence of real numbers $x_1,x_2,\ldots ,x_n$ is given such that $x_{i+1}=x_i+\frac{1}{30000}\sqrt{1-x_i^2},\ i=1,2,\ldots ,$ and $x_1=0$. Can $n$ be equal to $50000$ if $x_n<1$?
2010 All-Russian Olympiad, 1
ِDo there exist non-zero reals numbers $a_1, a_2, ....., a_{10}$ for which \[(a_1+\frac{1}{a_1})(a_2+\frac{1}{a_2}) \cdots(a_{10}+\frac{1}{a_{10}})= (a_1-\frac{1}{a_1})(a_2-\frac{1}{a_2})\cdots(a_{10}-\frac{1}{a_{10}}) \ ? \]
1991 Iran MO (2nd round), 3
Let $f : \mathbb R \to \mathbb R$ be a function such that $f(1)=1$ and
\[f(x+y)=f(x)+f(y)\]
And for all $x \in \mathbb R / \{0\}$ we have $f\left( \frac 1x \right) = \frac{1}{f(x)}.$ Find all such functions $f.$
2000 Iran MO (3rd Round), 3
Prove that for every natural number $ n$ there exists a polynomial $ p(x)$ with
integer coefficients such that$ p(1),p(2),...,p(n)$ are distinct powers of $ 2$ .
1994 Balkan MO, 2
Let $n$ be an integer. Prove that the polynomial $f(x)$ has at most one zero, where \[ f(x) = x^4 - 1994 x^3 + (1993+n)x^2 - 11x + n . \]
[i]Greece[/i]
2011 Baltic Way, 1
The real numbers $x_1,\ldots ,x_{2011}$ satisfy
\[x_1+x_2=2x_1',\ x_2+x_3=2x_2', \ \ldots, \ x_{2011}+x_1=2x_{2011}'\]
where $x_1',x_2',\ldots,x_{2011}'$ is a permutation of $x_1,x_2,\ldots,x_{2011}$. Prove that $x_1=x_2=\ldots =x_{2011}$ .
2012 Albania Team Selection Test, 3
It is given the equation $x^4-2ax^3+a(a+1)x^2-2ax+a^2=0$.
a) Find the greatest value of $a$, such that this equation has at least one real root.
b) Find all the values of $a$, such that the equation has at least one real root.
2024 All-Russian Olympiad, 2
Call a triple $(a,b,c)$ of positive numbers [i]mysterious [/i]if
\[\sqrt{a^2+\frac{1}{a^2c^2}+2ab}+\sqrt{b^2+\frac{1}{b^2a^2}+2bc}+\sqrt{c^2+\frac{1}{c^2b^2}+2ca}=2(a+b+c).\]
Prove that if the triple $(a,b,c)$ is mysterious, then so is the triple $(c,b,a)$.
[i]Proposed by A. Kuznetsov, K. Sukhov[/i]
2012 Hitotsubashi University Entrance Examination, 3
For constants $a,\ b,\ c,\ d$ consider a process such that the point $(p,\ q)$ is mapped onto the point $(ap+bq,\ cp+dq)$.
Note : $(a,\ b,\ c,\ d)\neq (1,\ 0,\ 0,\ 1)$. Let $k$ be non-zero constant. All points of the parabola $C: y=x^2-x+k$ are mapped onto $C$ by the process.
(1) Find $a,\ b,\ c,\ d$.
(2) Let $A'$ be the image of the point $A$ by the process. Find all values of $k$ and the coordinates of $A$ such that the tangent line of $C$ at $A$ and the tangent line of $C$ at $A'$ formed by the process are perpendicular at the origin.
1990 India National Olympiad, 3
Let $ f$ be a function defined on the set of non-negative integers and taking values in the same
set. Given that
(a) $ \displaystyle x \minus{} f(x) \equal{} 19\left[\frac{x}{19}\right] \minus{} 90\left[\frac{f(x)}{90}\right]$ for all non-negative integers $ x$;
(b) $ 1900 < f(1990) < 2000$,
find the possible values that $ f(1990)$ can take.
(Notation : here $ [z]$ refers to largest integer that is $ \leq z$, e.g. $ [3.1415] \equal{} 3$).
2012 European Mathematical Cup, 3
Are there positive real numbers $x$, $y$ and $z$ such that
$ x^4 + y^4 + z^4 = 13\text{,} $
$ x^3y^3z + y^3z^3x + z^3x^3y = 6\sqrt{3} \text{,} $
$ x^3yz + y^3zx + z^3xy = 5\sqrt{3} \text{?} $
[i]Proposed by Matko Ljulj.[/i]
2007 IMAR Test, 1
For real numbers $ x_{i}>1,1\leq i\leq n,n\geq 2,$ such that:
$ \frac{x_{i}^2}{x_{i}\minus{}1}\geq S\equal{}\displaystyle\sum^n_{j\equal{}1}x_{j},$ for all $ i\equal{}1,2\dots, n$
find, with proof, $ \sup S.$
1997 Baltic Way, 1
Determine all functions $f$ from the real numbers to the real numbers, different from the zero function, such that $f(x)f(y)=f(x-y)$ for all real numbers $x$ and $y$.
2012 Gulf Math Olympiad, 2
Prove that if $a, b, c$ are positive real numbers, then the least possible value of \[6a^3 + 9b^3 + 32c^3 + \frac{1}{4abc}\]
is $6$. For which values of $a, b$ and $c$ is equality attained?
2011 ELMO Shortlist, 2
Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that whenever $a>b>c>d>0$ and $ad=bc$,
\[f(a+d)+f(b-c)=f(a-d)+f(b+c).\]
[i]Calvin Deng.[/i]
1994 Baltic Way, 4
Is there an integer $n$ such that $\sqrt{n-1}+\sqrt{n+1}$ is a rational number?