A sequence ($a_n$) satisfies the following conditions: (i) For each $m \in N$ it holds that $a_{2^m} = 1/m$. (ii) For each natural $n \ge 2$ it holds that $a_{2n-1}a_{2n} = a_n$. (iii) For all integers $m,n$ with $2m > n \ge 1$ it holds that $a_{2n}a_{2n+1} = a_{2^m+n}$. Determine $a_{2000}$. You may assume that such a sequence exists.

Let $a, b, c$ be positive real numbers such that $abc=8$. Prove that \[ \frac{a^2}{\sqrt{(1+a^3)(1+b^3)}} +\frac{b^2}{\sqrt{(1+b^3)(1+c^3)}} +\frac{c^2}{\sqrt{(1+c^3)(1+a^3)}} \geq \frac{4}{3} \]

Let $x, y, z$ be real numbers each of whose absolute value is different from $\frac{1}{\sqrt 3}$ such that $x + y + z = xyz$. Prove that \[\frac{3x - x^3}{1-3x^2} + \frac{3y - y^3}{1-3y^2} + \frac{3z -z^3}{1-3z^2} = \frac{3x - x^3}{1-3x^2} \cdot \frac{3y - y^3}{1-3y^2} \cdot \frac{3z - z^3}{1-3z^2}\]

Let $P(x) = x^3 + (t - 1)x^2 - (t + 3)x + 1$. For what values of real $t$ the sum of the squares and the reciprocals of the roots of $ P(x)$ is minimum?

Let $n$ be a positive integer and let $k_0,k_1, \dots,k_{2n}$ be nonzero integers such that $k_0+k_1 +\dots+k_{2n}\neq 0$. Is it always possible to a permutation $(a_0,a_1,\dots,a_{2n})$ of $(k_0,k_1,\dots,k_{2n})$ so that the equation \begin{align*} a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_0=0 \end{align*} has not integer roots?

It is known that $\frac{7}{13} + \sin \phi = \cos \phi$ for some real $\phi$. What is sin $2\phi$?

Given the equation $x^4-x^3-1=0$ [b](a)[/b] Find the number of its real roots. [b](b)[/b] We denote by $S$ the sum of the real roots and by $P$ their product. Prove that $P< - \frac{11}{10}$ and $S> \frac {6}{11}$.

Let $a,b\in \mathbb{R}$ and $z\in \mathbb{C}\backslash \mathbb{R}$ so that $\left| a-b \right|=\left| a+b-2z \right|$. a) Prove that the equation ${{\left| z-a \right|}^{x}}+{{\left| \bar{z}-b \right|}^{x}}={{\left| a-b \right|}^{x}}$, with the unknown number $x\in \mathbb{R}$, has a unique solution. b) Solve the following inequation ${{\left| z-a \right|}^{x}}+{{\left| \bar{z}-b \right|}^{x}}\le {{\left| a-b \right|}^{x}}$, with the unknown number $x\in \mathbb{R}$. The Mathematical Gazette

Suppose that the integers $a_{1}$, $a_{2}$, $\cdots$, $a_{n}$ are distinct. Show that \[(x-a_{1})(x-a_{2}) \cdots (x-a_{n})-1\] cannot be expressed as the product of two nonconstant polynomials with integer coefficients.

[list=a] [*]Determine all real numbers $x{}$ satisfying $\lfloor x\rfloor^2-x=-0.99$. [*]Prove that if $a\leqslant -1$, the equation $\lfloor x\rfloor^2-x=a$ does not have real solutions. [/list]

Find the smallest positive integer $n$ for which $0 <\sqrt[4]{n}- [\sqrt[4]{n}]< 10^{-5}$ .

Find all functions $f: {\mathbb{R^\plus{}}}\to{\mathbb{R^\plus{}}}$ such that \[ f(1\plus{}xf(y))\equal{}yf(x\plus{}y)\] for all $x,y\in\mathbb{R^\plus{}}$.

Find all triples $(p,q,n)$ that satisfy \[q^{n+2} \equiv 3^{n+2} (\mod p^n) ,\quad p^{n+2} \equiv 3^{n+2} (\mod q^n)\] where $p,q$ are odd primes and $n$ is an positive integer.

If $a,b,c,d$ are Distinct Real no. such that $a = \sqrt{4+\sqrt{5+a}}$ $b = \sqrt{4-\sqrt{5+b}}$ $c = \sqrt{4+\sqrt{5-c}}$ $d = \sqrt{4-\sqrt{5-d}}$ Then $abcd = $

For every integer $ n \ge 3$ and any given angle $ \alpha$ with $ 0 < \alpha < \pi$, let $ P_n(x) \equal{} x^n \sin\alpha \minus{} x \sin n\alpha \plus{} \sin(n \minus{} 1)\alpha$. (a) Prove that there is a unique polynomial of the form $ f(x) \equal{} x^2 \plus{} ax \plus{} b$ which divides $ P_n(x)$ for every $ n \ge 3$. (b) Prove that there is no polynomial $ g(x) \equal{} x \plus{} c$ which divides $ P_n(x)$ for every $ n \ge 3$.

For what polynomials $P(n)$ with integer coefficients can a positive integer be assigned to every lattice point in $\mathbb{R}^3$ so that for every integer $n \ge 1$, the sum of the $n^3$ integers assigned to any $n \times n \times n$ grid of lattice points is divisible by $P(n)$? [i]Proposed by Andre Arslan[/i]

Let $p$ and $q$ be two given positive integers. A set of $p+q$ real numbers $a_1<a_2<\cdots <a_{p+q}$ is said to be balanced iff $a_1,\ldots,a_p$ were an arithmetic progression with common difference $q$ and $a_p,\ldots,a_{p+q}$ where an arithmetic progression with common difference $p$. Find the maximum possible number of balanced sets, so that any two of them have nonempty intersection. Comment: The intended problem also had "$p$ and $q$ are coprime" in the hypothesis. A typo when the problems where written made it appear like that in the exam (as if it were the only typo in the olympiad). Fortunately, the problem can be solved even if we didn't suppose that and it can be further generalized: we may suppose that a balanced set has $m+n$ reals $a_1<\cdots <a_{m+n-1}$ so that $a_1,\ldots,a_m$ is an arithmetic progression with common difference $p$ and $a_m,\ldots,a_{m+n-1}$ is an arithmetic progression with common difference $q$.

Let $S$ be a finite set of nonzero real numbers, and let $f : S\to S$ be a function with the following property: for each $x \in S$, either $f ( f (x)) = x+ f (x)$ or $f ( f (x)) = \frac{x+ f (x)}{2}$. Prove that $f (x) = x$ for all $x \in S$.

An integer $n>1$ is given . Find the smallest positive number $m$ satisfying the following conditions： for any set $\{a,b\}$ $\subset \{1,2,\cdots,2n-1\}$ ,there are non-negative integers $ x, y$ ( not all zero) such that $2n|ax+by$ and $x+y\leq m.$

Let $A=\{a_1,a_2,\cdots,a_{2010}\}$ and $B=\{b_1,b_2,\cdots,b_{2010}\}$ be two sets of complex numbers. Suppose \[\sum_{1\leq i<j\leq 2010} (a_i+a_j)^k=\sum_{1\leq i<j\leq 2010}(b_i+b_j)^k\] holds for every $k=1,2,\cdots, 2010$. Prove that $A=B$.

$14.$ If $3^x+2^y=985.$ and $3^x-2^y=473.$ What is the value of $xy ?$

Let $ a,b,c$ be the three sides of a triangle. Determine all possible values of $ \frac {a^2 \plus{} b^2 \plus{} c^2}{ab \plus{} bc \plus{} ca}$

Find the real numbers $ x,y,z $ that satisfy the following: $ \text{(i)} -2\le x\le y\le z $ $ \text{(ii)} x+y+z=2/3 $ $ \text{(iii)} \frac{1}{x^2} +\frac{1}{y^2} +\frac{1}{z^2} =\frac{1}{x} +\frac{1}{y} +\frac{1}{z} +\frac{3}{8} $ [i]Cristinel Mortici[/i]

Let $a$ be a rational number and let $n$ be a positive integer. Prove that the polynomial $X^{2^n}(X+a)^{2^n}+1$ is irreducible in the ring $\mathbb{Q}[X]$ of polynomials with rational coefficients. [i]Proposed by Vincent Jugé, École Polytechnique, Paris.[/i]

Given three functions $$P(x) = (x^2-1)^{2023}, Q(x) = (2x+1)^{14}, R(x) = \left(2x+1+\frac 2x \right)^{34}.$$ Initially, we pick a set $S$ containing two of these functions, and we perform some [i]operations[/i] on it. Allowed operations include: - We can take two functions $p,q \in S$ and add one of $p+q, p-q$, or $pq$ to $S$. - We can take a function $p \in S$ and add $p^k$ to $S$ for $k$ is an arbitrary positive integer of our choice. - We can take a function $p \in S$ and choose a real number $t$, and add to $S$ one of the function $p+t, p-t, pt$. Show that no matter how we pick $S$ in the beginning, there is no way we can perform finitely many operations on $S$ that would eventually yield the third function not in $S$.