Found problems: 1311
2014 Lithuania Team Selection Test, 5
Given real numbers $x$ and $y$. Let $s_{1}=x+y, s_{2}=x^2+y^2, s_{3}=x^3+y^3, s_{4}=x^4+y^4$ and $t=xy$.
[b]a)[/b] Prove, that number $t$ is rational, if $s_{2}, s_{3}$ and $s_{4}$ are rational numbers.
[b]b)[/b] Prove, that number $s_{1}$ is rational, if $s_{2}, s_{3}$ and $s_{4}$ are rational numbers.
[b]c)[/b] Can number $s_{1}$ be irrational, if $s_{2}$ and $s_{3}$ are rational numbers?
1997 Federal Competition For Advanced Students, P2, 6
For every natural number $ n$, find all polynomials $ x^2\plus{}ax\plus{}b$, where $ a^2 \ge 4b$, that divide $ x^{2n}\plus{}ax^n\plus{}b$.
2013 Iran Team Selection Test, 3
For nonnegative integers $m$ and $n$, define the sequence $a(m,n)$ of real numbers as follows. Set $a(0,0)=2$ and for every natural number $n$, set $a(0,n)=1$ and $a(n,0)=2$. Then for $m,n\geq1$, define \[ a(m,n)=a(m-1,n)+a(m,n-1). \] Prove that for every natural number $k$, all the roots of the polynomial $P_{k}(x)=\sum_{i=0}^{k}a(i,2k+1-2i)x^{i}$ are real.
1982 Vietnam National Olympiad, 1
Determine a quadric polynomial with intergral coefficients whose roots are $\cos 72^{\circ}$ and $\cos 144^{\circ}.$
2011 Vietnam National Olympiad, 2
Let $\langle x_n\rangle$ be a sequence of real numbers defined as
\[x_1=1; x_n=\dfrac{2n}{(n-1)^2}\sum_{i=1}^{n-1}x_i\]
Show that the sequence $y_n=x_{n+1}-x_n$ has finite limits as $n\to \infty.$
1990 IMO Longlists, 24
Find the real number $t$, such that the following system of equations has a unique real solution $(x, y, z, v)$:
\[ \left\{\begin{array}{cc}x+y+z+v=0\\ (xy + yz +zv)+t(xz+xv+yv)=0\end{array}\right. \]
1987 India National Olympiad, 4
If $ x$, $ y$, $ z$, and $ n$ are natural numbers, and $ n\geq z$ then prove that the relation $ x^n \plus{} y^n \equal{} z^n$ does not hold.
1977 IMO Longlists, 7
Prove the following assertion: If $c_1,c_2,\ldots ,c_n\ (n\ge 2)$ are real numbers such that
\[ (n-1)(c_1^2+c_2^2+\cdots +c_n^2)=(c_1+c_2+\cdots + c_n)^2,\]
then either all these numbers are nonnegative or all these numbers are nonpositive.
1992 Baltic Way, 12
Let $ N$ denote the set of natural numbers. Let $ \phi: N\rightarrow N$ be a bijective function and assume that there exists a finite limit
\[ \lim_{n\rightarrow\infty}\frac{\phi(n)}{n}\equal{}L.
\] What are the possible values of $ L$?
2009 Paraguay Mathematical Olympiad, 1
Find the value of the following expression:
$2 + 33 + 6 + 35 + 10 + 37 + \ldots + 1194 + 629 + 1198 + 631$
2014 Contests, 2
Find all $f$ functions from real numbers to itself such that for all real numbers $x,y$ the equation
\[f(f(y)+x^2+1)+2x=y+(f(x+1))^2\]
holds.
2011 Bosnia Herzegovina Team Selection Test, 3
Numbers $1,2, ..., 2n$ are partitioned into two sequences $a_1<a_2<...<a_n$ and $b_1>b_2>...>b_n$. Prove that number
\[W= |a_1-b_1|+|a_2-b_2|+...+|a_n-b_n|\]
is a perfect square.
2003 Iran MO (3rd Round), 23
Find all homogeneous linear recursive sequences such that there is a $ T$ such that $ a_n\equal{}a_{n\plus{}T}$ for each $ n$.
2014 Moldova Team Selection Test, 2
Find all functions $f:R \rightarrow R$, which satisfy the equality for any $x,y \in R$:
$f(xf(y)+y)+f(xy+x)=f(x+y)+2xy$,
1995 Vietnam National Olympiad, 1
Find all real solutions to $ x^3 \minus{} 3x^2 \minus{} 8x \plus{} 40 \minus{} 8\sqrt[4]{4x \plus{} 4} \equal{} 0$
2009 Moldova Team Selection Test, 2
[color=darkred]Determine all functions $ f : [0; \plus{} \infty) \rightarrow [0; \plus{} \infty)$, such that
\[ f(x \plus{} y \minus{} z) \plus{} f(2\sqrt {xz}) \plus{} f(2\sqrt {yz}) \equal{} f(x \plus{} y \plus{} z)\]
for all $ x,y,z \in [0; \plus{} \infty)$, for which $ x \plus{} y\ge z$.[/color]
2010 All-Russian Olympiad, 4
Given is a natural number $n \geq 3$. What is the smallest possible value of $k$ if the following statements are true?
For every $n$ points $ A_i = (x_i, y_i) $ on a plane, where no three points are collinear, and for any real numbers $ c_i$ ($1 \le i \le n$) there exists such polynomial $P(x, y)$, the degree of which is no more than $k$, where $ P(x_i, y_i) = c_i $ for every $i = 1, \dots, n$.
(The degree of a nonzero monomial $ a_{i,j} x^{i}y^{j} $ is $i+j$, while the degree of polynomial $P(x, y)$ is the greatest degree of the degrees of its monomials.)
2004 China National Olympiad, 1
For a given real number $a$ and a positive integer $n$, prove that:
i) there exists exactly one sequence of real numbers $x_0,x_1,\ldots,x_n,x_{n+1}$ such that
\[\begin{cases} x_0=x_{n+1}=0,\\ \frac{1}{2}(x_i+x_{i+1})=x_i+x_i^3-a^3,\ i=1,2,\ldots,n.\end{cases}\]
ii) the sequence $x_0,x_1,\ldots,x_n,x_{n+1}$ in i) satisfies $|x_i|\le |a|$ where $i=0,1,\ldots,n+1$.
[i]Liang Yengde[/i]
1986 India National Olympiad, 5
If $ P(x)$ is a polynomial with integer coefficients and $ a$, $ b$, $ c$, three distinct integers, then show that it is impossible to have $ P(a)\equal{}b$, $ P(b)\equal{}c$, $ P(c)\equal{}a$.
2014 ITAMO, 5
Prove that there exists a positive integer that can be written, in at least two ways, as a sum of $2014$-th powers of $2015$ distinct positive integers $x_1 <x_2 <\cdots <x_{2015}$.
2006 Peru IMO TST, 2
[color=blue][size=150]PERU TST IMO - 2006[/size]
Saturday, may 20.[/color]
[b]Question 02[/b]
Find all pairs $(a,b)$ real positive numbers $a$ and $b$ such that:
$[a[bn]]= n-1,$
for all $n$ positive integer.
Note: [x] denotes the integer part of $x$.
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$\text{\LaTeX}{}$ed by carlosbr
2014 All-Russian Olympiad, 1
Does there exist positive $a\in\mathbb{R}$, such that
\[|\cos x|+|\cos ax| >\sin x +\sin ax \]
for all $x\in\mathbb{R}$?
[i]N. Agakhanov[/i]
2008 Junior Balkan Team Selection Tests - Moldova, 9
Find all triplets $ (x,y,z)$, that satisfy:
$ \{\begin{array}{c}\ \ x^2 - 2x - 4z = 3\
y^2 - 2y - 2x = - 14 \
z^2 - 4y - 4z = - 18 \end{array}$
2011 ISI B.Math Entrance Exam, 1
Given $a,x\in\mathbb{R}$ and $x\geq 0$,$a\geq 0$ . Also $\sin(\sqrt{x+a})=\sin(\sqrt{x})$ . What can you say about $a$??? Justify your answer.
2007 Iran Team Selection Test, 3
Find all solutions of the following functional equation: \[f(x^{2}+y+f(y))=2y+f(x)^{2}. \]