Found problems: 1269
2008 Bundeswettbewerb Mathematik, 4
In a planar coordinate system we got four pieces on positions with coordinates. You can make a move according to the following rule: You can move a piece to a new position if there is one of the other pieces in the middle of the old and new position. Initially the four pieces have positions $ \{(0,0),(0,1),(1,0),(1,1)\}$. Given a finite number of moves can you yield the configuration $ \{(0,0), (1,1), (3,0), (2, \minus{} 1)\}$ ?
2011 International Zhautykov Olympiad, 2
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ which satisfy the equality,
\[f(x+f(y))=f(x-f(y))+4xf(y)\]
for any $x,y\in\mathbb{R}$.
1987 IMO Longlists, 59
It is given that $a_{11}, a_{22}$ are real numbers, that $x_1, x_2, a_{12}, b_1, b_2$ are complex numbers, and that $a_{11}a_{22}=a_{12}\overline{a_{12}}$ (Where $\overline{a_{12}}$ is he conjugate of $a_{12}$). We consider the following system in $x_1, x_2$:
\[\overline{x_1}(a_{11}x_1 + a_{12}x_2) = b_1,\]\[\overline{x_2}(a_{12}x_1 + a_{22}x_2) = b_2.\]
[b](a) [/b]Give one condition to make the system consistent.
[b](b) [/b]Give one condition to make $\arg x_1 - \arg x_2 = 98^{\circ}.$
2008 Mediterranean Mathematics Olympiad, 3
Let $n$ be a positive integer. Calculate the sum $\sum_{k=1}^n\ \ {\sum_{1\le i_1 < \ldots < i_k\le n}^{}{\frac {2^k}{(i_1 + 1)(i_2 + 1)\ldots (i_k + 1)}}}$
2006 China Western Mathematical Olympiad, 2
Find the smallest positive real $k$ satisfying the following condition: for any given four DIFFERENT real numbers $a,b,c,d$, which are not less than $k$, there exists a permutation $(p,q,r,s)$ of $(a,b,c,d)$, such that the equation $(x^{2}+px+q)(x^{2}+rx+s)=0$ has four different real roots.
1980 IMO, 7
Prove that $4x^3-3x+1=2y^2$ has at least $31$ solutions in positive integers $x,y$ with $x\le 1980$.
[i] Variant: [/i] Prove the equation $4x^3-3x+1=2y^2$ has infinitely many solutions in positive integers x,y.
2000 China Team Selection Test, 3
Let $n$ be a positive integer. Denote $M = \{(x, y)|x, y \text{ are integers }, 1 \leq x, y \leq n\}$. Define function $f$ on $M$ with the following properties:
[b]a.)[/b] $f(x, y)$ takes non-negative integer value; [b]
b.)[/b] $\sum^n_{y=1} f(x, y) = n - 1$ for $1 \eq x \leq n$;
[b]c.)[/b] If $f(x_1, y_1)f(x2, y2) > 0$, then $(x_1 - x_2)(y_1 - y_2) \geq 0.$
Find $N(n)$, the number of functions $f$ that satisfy all the conditions. Give the explicit value of $N(4)$.
1995 Kurschak Competition, 2
Consider a polynomial in $n$ variables with real coefficients. We know that if every variable is $\pm1$, the value of the polynomial is positive, or negative if the number of $-1$'s is even, or odd, respectively. Prove that the degree of this polynomial is at least $n$.
2012 France Team Selection Test, 2
Determine all non-constant polynomials $X^n+a_{n-1}X^{n-1}+\cdots +a_1X+a_0$ with integer coefficients for which the roots are exactly the numbers $a_0,a_1,\ldots ,a_{n-1}$ (with multiplicity).
1983 IMO Longlists, 38
Let $\{u_n \}$ be the sequence defined by its first two terms $u_0, u_1$ and the recursion formula
\[u_{n+2 }= u_n - u_{n+1}.\]
[b](a)[/b] Show that $u_n$ can be written in the form $u_n = \alpha a^n + \beta b^n$, where $a, b, \alpha, \beta$ are constants independent of $n$ that have to be determined.
[b](b)[/b] If $S_n = u_0 + u_1 + \cdots + u_n$, prove that $S_n + u_{n-1}$ is a constant independent of $n.$ Determine this constant.
1994 Dutch Mathematical Olympiad, 5
Three real numbers $ a,b,c$ satisfy the inequality $ |ax^2\plus{}bx\plus{}c| \le 1$ for all $ x \in [\minus{}1,1]$. Prove that $ |cx^2\plus{}bx\plus{}a| \le 2$ for all $ x \in [\minus{}1,1]$.
2010 Contests, 1
Find all functions $f$ from the reals into the reals such that \[ f(ab) = f(a+b) \] for all irrational $a, b$.
2010 Contests, 3
Given complex numbers $a,b,c$, we have that $|az^2 + bz +c| \leq 1$ holds true for any complex number $z, |z| \leq 1$. Find the maximum value of $|bc|$.
1978 Canada National Olympiad, 6
Sketch the graph of $x^3 + xy + y^3 = 3$.
2011 Postal Coaching, 3
Let $f : \mathbb{N} \longrightarrow \mathbb{N}$ be a function such that $(x + y)f (x) \le x^2 + f (xy) + 110$, for all $x, y$ in $\mathbb{N}$. Determine the minimum and maximum values of $f (23) + f (2011)$.
2006 Pan African, 3
For a real number $x$ let $\lfloor x\rfloor$ be the greatest integer less than or equal to $x$ and let $\{x\} = x - \lfloor x\rfloor$. If $a, b, c$ are distinct real numbers, prove that
\[\frac{a^3}{(a-b)(a-c)}+\frac{b^3}{(b-a)(b-c)}+\frac{c^3}{(c-a)(c-b)}\]
is an integer if and only if $\{a\} + \{b\} + \{c\}$ is an integer.
1991 China Team Selection Test, 1
Let real coefficient polynomial $f(x) = x^n + a_1 \cdot x^{n-1} + \ldots + a_n$ has real roots $b_1, b_2, \ldots, b_n$, $n \geq 2,$ prove that $\forall x \geq max\{b_1, b_2, \ldots, b_n\}$, we have
\[f(x+1) \geq \frac{2 \cdot n^2}{\frac{1}{x-b_1} + \frac{1}{x-b_2} + \ldots + \frac{1}{x-b_n}}.\]
2006 MOP Homework, 2
Let $a, b_1, b_2, \dots, b_n, c_1, c_2, \dots, c_n$ be real numbers such that \[x^{2n} + ax^{2n - 1} + ax^{2n - 2} + \dots + ax + 1 = \prod_{i = 1}^{n}{(x^2 + b_ix + c_i)}\]
Prove that $c_1 = c_2 = \dots = c_n = 1$.
As a consequence, all complex zeroes of this polynomial must lie on the unit circle.
2008 Vietnam National Olympiad, 1
Determine the number of solutions of the simultaneous equations $ x^2 \plus{} y^3 \equal{} 29$ and $ \log_3 x \cdot \log_2 y \equal{} 1.$
1989 IMO Longlists, 34
Prove the identity
\[ 1 \plus{} \frac{1}{2} \minus{} \frac{2}{3} \plus{} \frac{1}{4} \plus{} \frac{1}{5} \minus{} \frac{2}{6} \plus{} \ldots \plus{} \frac{1}{478} \plus{} \frac{1}{479} \minus{} \frac{2}{480}
\equal{} 2 \cdot \sum^{159}_{k\equal{}0} \frac{641}{(161\plus{}k) \cdot (480\minus{}k)}.\]
1985 Vietnam National Olympiad, 2
Find all real values of parameter $ a$ for which the equation in $ x$
\[ 16x^4 \minus{} ax^3 \plus{} (2a \plus{} 17)x^2 \minus{} ax \plus{} 16 \equal{} 0
\]
has four solutions which form an arithmetic progression.
1984 Iran MO (2nd round), 5
Suppose that
\[S_n=\frac 59 \times \frac{14}{20} \times \frac{27}{35} \times \cdots \times \frac{2n^2-n-1}{2n^2+n-1}\]
Find $\lim_{n \to \infty} S_n.$
2003 District Olympiad, 4
Let $\displaystyle a,b,c,d \in \mathbb R$ such that $\displaystyle a>c>d>b>1$ and $\displaystyle ab>cd$.
Prove that $\displaystyle f : \left[ 0,\infty \right) \to \mathbb R$, defined through
\[ \displaystyle f(x) = a^x+b^x-c^x-d^x, \, \forall x \geq 0 , \]
is strictly increasing.
2006 Brazil National Olympiad, 3
Find all functions $f\colon \mathbb{R}\to \mathbb{R}$ such that
\[f(xf(y)+f(x)) = 2f(x)+xy\]
for every reals $x,y$.
2006 Balkan MO, 4
Let $m$ be a positive integer and $\{a_n\}_{n\geq 0}$ be a sequence given by $a_0 = a \in \mathbb N$, and \[ a_{n+1} = \begin{cases} \displaystyle \frac{a_n}2 & \textrm { if } a_n \equiv 0 \pmod 2, \\ a_n + m & \textrm{ otherwise. } \end{cases} \]
Find all values of $a$ such that the sequence is periodical (starting from the beginning).