Found problems: 15925
2009 Romania Team Selection Test, 3
Given an integer $n\geq 2$ and a closed unit disc, evaluate the maximum of the product of the lengths of all $\frac{n(n-1)}{2}$ segments determined by $n$ points in that disc.
2010 Indonesia TST, 1
Find all functions $ f : R \to R$ that satisfies $$xf(y) - yf(x)= f\left(\frac{y}{x}\right)$$ for all $x, y \in R$.
2015 Vietnam National Olympiad, 1
Given a non negative real $a$ and a sequence $(u_n)$ defined by \[ \begin{cases} u_1=3\\ u_{n+1}=\frac{u_n}{2}+\frac{n^2}{4n^2+a}\sqrt{u_n^2+3} \end{cases} \]
a) Prove that for $a=0$, the sequence is convergent and find its limit.
b) For $a\in [0,1]$, prove that the sequence if convergent.
2006 All-Russian Olympiad, 5
Two sequences of positive reals, $ \left(x_n\right)$ and $ \left(y_n\right)$, satisfy the relations $ x_{n \plus{} 2} \equal{} x_n \plus{} x_{n \plus{} 1}^2$ and $ y_{n \plus{} 2} \equal{} y_n^2 \plus{} y_{n \plus{} 1}$ for all natural numbers $ n$. Prove that, if the numbers $ x_1$, $ x_2$, $ y_1$, $ y_2$ are all greater than $ 1$, then there exists a natural number $ k$ such that $ x_k > y_k$.
2016 Harvard-MIT Mathematics Tournament, 1
Let $z$ be a complex number such that $|z| = 1$ and $|z-1.45|=1.05$. Compute the real part of $z$.
1995 Singapore MO Open, 5
Let $a, b, c, d$ be four positive real numbers. Prove that
$$a^{10} + b^{10}+c^{10} + d^{10} \ge (0.1a + 0.2b + 0.3c + 0.4d)^{10} + (0.4a + 0.3b + 0.2c + 0.ld)^{10} + (0.2a + 0.4b + 0.1c + 0.3d)^{10} + (0.3a + 0.1b + 0.4c + 0.2d)^{10}$$
2013 South East Mathematical Olympiad, 3
A sequence $\{a_n\}$ , $a_1=1,a_2=2,a_{n+1}=\dfrac{a_n^2+(-1)^n}{a_{n-1}}$. Show that $a_m^2+a_{m+1}^2\in\{a_n\},\forall m\in\Bbb N$
OMMC POTM, 2023 2
Find all functions $f$ from the set of reals to itself so that for all reals $x,y,$
$$f(x)f(f(x)+y) = f(x^2) + f(xy).$$
[i]Proposed by Culver Kwan[/i]
2016 Korea Junior Math Olympiad, 5
$n \in \mathbb {N^+}$
Prove that the following equation can be expressed as a polynomial about $n$.
$$\left[2\sqrt {1}\right]+\left[2\sqrt {2}\right]+\left[2\sqrt {3}\right]+ . . . +\left[2\sqrt {n^2}\right]$$
2015 Belarus Team Selection Test, 3
Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively.
Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes.
[i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]
2021 Taiwan TST Round 2, A
[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x .
\]
[i]Version 2[/i]. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x .
\]
2002 China Team Selection Test, 2
Given an integer $k$. $f(n)$ is defined on negative integer set and its values are integers. $f(n)$ satisfies \[ f(n)f(n+1)=(f(n)+n-k)^2, \] for $n=-2,-3,\cdots$. Find an expression of $f(n)$.
1981 AMC 12/AHSME, 30
If $ a$, $ b$, $ c$, and $ d$ are the solutions of the equation $ x^4 \minus{} bx \minus{} 3 \equal{} 0$, then an equation whose solutions are
\[ \frac {a \plus{} b \plus{} c}{d^2}, \frac {a \plus{} b \plus{} d}{c^2}, \frac {a \plus{} c \plus{} d}{b^2}, \frac {b \plus{} c \plus{} d}{a^2}
\]is
$ \textbf{(A)}\ 3x^4 \plus{} bx \plus{} 1 \equal{} 0\qquad \textbf{(B)}\ 3x^4 \minus{} bx \plus{} 1 \equal{} 0\qquad \textbf{(C)}\ 3x^4 \plus{} bx^3 \minus{} 1 \equal{} 0$
$ \textbf{(D)}\ 3x^4 \minus{} bx^3 \minus{} 1 \equal{} 0\qquad \textbf{(E)}\ \text{none of these}$
2005 Greece Team Selection Test, 1
The side lengths of a triangle are the roots of a cubic polynomial with rational coefficients. Prove that the altitudes of this triangle are roots of a polynomial of sixth degree with rational coefficients.
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]
1994 Baltic Way, 1
Let $a\circ b=a+b-ab$. Find all triples $(x,y,z)$ of integers such that
\[(x\circ y)\circ z +(y\circ z)\circ x +(z\circ x)\circ y=0\]
2000 Turkey MO (2nd round), 3
Find all continuous functions $f:[0,1]\to [0,1]$ for which there exists a positive integer $n$ such that $f^{n}(x)=x$ for $x \in [0,1]$ where $f^{0} (x)=x$ and $f^{k+1}=f(f^{k}(x))$ for every positive integer $k$.
2010 Bundeswettbewerb Mathematik, 4
In the following, let $N_0$ denotes the set of non-negative integers.
Find all polynomials $P(x)$ that fulfill the following two properties:
(1) All coefficients of $P(x)$ are from $N_0$.
(2) Exists a function $f : N_0 \to N_0$ such as $f (f (f (n))) = P (n)$ for all $n \in N_0$.
2002 Austrian-Polish Competition, 8
Determine the number of real solutions of the system \[\left\{ \begin{aligned}\cos x_{1}&= x_{2}\\ &\cdots \\ \cos x_{n-1}&= x_{n}\\ \cos x_{n}&= x_{1}\\ \end{aligned}\right.\]
1995 USAMO, 1
Let $\, p \,$ be an odd prime. The sequence $(a_n)_{n \geq 0}$ is defined as follows: $\, a_0 = 0,$ $a_1 = 1, \, \ldots, \, a_{p-2} = p-2 \,$ and, for all $\, n \geq p-1, \,$ $\, a_n \,$ is the least positive integer that does not form an arithmetic sequence of length $\, p \,$ with any of the preceding terms. Prove that, for all $\, n, \,$ $\, a_n \,$ is the number obtained by writing $\, n \,$ in base $\, p-1 \,$ and reading the result in base $\, p$.
2020 GQMO, 1
Find all quadruples of real numbers $(a,b,c,d)$ such that the equalities
\[X^2 + a X + b = (X-a)(X-c) \text{ and } X^2 + c X + d = (X-b)(X-d)\]
hold for all real numbers $X$.
[i]Morteza Saghafian, Iran[/i]
2017 Harvard-MIT Mathematics Tournament, 2
Determine the sum of all distinct real values of $x$ such that $||| \cdots ||x|+x| \cdots |+x|+x|=1$ where there are $2017$ $x$s in the equation.
2002 Junior Balkan Team Selection Tests - Moldova, 11
Simultaneously from the same point of a circular route and in the same direction for two hours two bodies move evenly. The first body performs a complete rotation three minutes faster than the second body and exceeds it every $9$ minutes and $20$ seconds. Whenever the first body will overtake the other the second exactly at the starting point?
2006 Estonia Math Open Senior Contests, 7
A real-valued function $ f$ satisfies for all reals $ x$ and $ y$ the equality
\[ f (xy) \equal{} f (x)y \plus{} x f (y).
\]
Prove that this function satisfies for all reals $ x$ and $ y \ne 0$ the equality
\[ f\left(\frac{x}{y}\right)\equal{}\frac{f (x)y \minus{} x f (y)}{y^2}
\]
2018 Hong Kong TST, 1
Does there exist a polynomial $P(x)$ with integer coefficients such that $P(1+\sqrt[3]{2})=1+\sqrt[3]{2}$ and $P(1+\sqrt5)=2+3\sqrt5$?