Olympiad problems

2011 China Second Round Olympiad, 2

Find the range of the function $f(x)=\frac{\sqrt{x^2+1}}{x-1}$.

2008 Croatia Team Selection Test, 2

Tags: function , algebra unsolved , algebra

For which $ n\in \mathbb{N}$ do there exist rational numbers $ a,b$ which are not integers such that both $ a \plus{} b$ and $ a^n \plus{} b^n$ are integers?

1999 Hungary-Israel Binational, 3

Tags: function , induction , algebra unsolved , algebra

Find all functions $ f:\mathbb{Q}\to\mathbb{R}$ that satisfy $ f(x\plus{}y)\equal{}f(x)f(y)\minus{}f(xy)\plus{}1$ for every $x,y\in\mathbb{Q}$.

2006 Federal Competition For Advanced Students, Part 2, 3

Tags: algebra , system of equations , algebra unsolved

Let $ A$ be an integer not equal to $ 0$. Solve the following system of equations in $ \mathbb{Z}^3$. $ x \plus{} y^2 \plus{} z^3 \equal{} A$ $ \frac {1}{x} \plus{} \frac {1}{y^2} \plus{} \frac {1}{z^3} \equal{} \frac {1}{A}$ $ xy^2z^3 \equal{} A^2$

2018 Latvia Baltic Way TST, P1

Tags: algebra , inequalities , algebra unsolved

Let $p_1,p_2,...,p_n$ be $n\ge 2$ fixed positive real numbers. Let $x_1,x_2,...,x_n$ be nonnegative real numbers such that $$x_1p_1+x_2p_2+...+x_np_n=1.$$ Determine the [i](a)[/i] maximal; [i](b)[/i] minimal possible value of $x_1^2+x_2^2+...+x_n^2$.

2004 China Team Selection Test, 1

Tags: algebra , system of equations , algebra unsolved

Given integer $ n$ larger than $ 5$, solve the system of equations (assuming $x_i \geq 0$, for $ i=1,2, \dots n$): \[ \begin{cases} \displaystyle x_1+ \phantom{2^2} x_2+ \phantom{3^2} x_3 + \cdots + \phantom{n^2} x_n &= n+2, \\ x_1 + 2\phantom{^2}x_2 + 3\phantom{^2}x_3 + \cdots + n\phantom{^2}x_n &= 2n+2, \\ x_1 + 2^2x_2 + 3^2 x_3 + \cdots + n^2x_n &= n^2 + n +4, \\ x_1+ 2^3x_2 + 3^3x_3+ \cdots + n^3x_n &= n^3 + n + 8. \end{cases} \]

2009 Tuymaada Olympiad, 2

Tags: quadratics , algebra unsolved , algebra

$ P(x)$ is a quadratic trinomial. What maximum number of terms equal to the sum of the two preceding terms can occur in the sequence $ P(1)$, $ P(2)$, $ P(3)$, $ \dots?$ [i]Proposed by A. Golovanov[/i]

2012 Kazakhstan National Olympiad, 1

Tags: function , algebra unsolved , algebra , Kazakhstan

Function $ f:\mathbb{R}\rightarrow\mathbb{R} $ such that $f(xf(y))=yf(x)$ for any $x,y$ are real numbers. Prove that $f(-x) = -f(x)$ for all real numbers $x$.

2011 Croatia Team Selection Test, 1

Tags: modular arithmetic , pigeonhole principle , induction , algebra unsolved , algebra

We define a sequence $a_n$ so that $a_0=1$ and \[a_{n+1} = \begin{cases} \displaystyle \frac{a_n}2 & \textrm { if } a_n \equiv 0 \pmod 2, \\ a_n + d & \textrm{ otherwise. } \end{cases} \] for all postive integers $n$. Find all positive integers $d$ such that there is some positive integer $i$ for which $a_i=1$.

2003 Czech-Polish-Slovak Match, 1

Tags: algebra , system of equations , algebra unsolved

Given an integer $n \ge 2$, solve in real numbers the system of equations \begin{align*} \max\{1, x_1\} &= x_2 \\ \max\{2, x_2\} &= 2x_3 \\ &\cdots \\ \max\{n, x_n\} &= nx_1. \\ \end{align*}

1989 IMO Longlists, 15

Tags: function , algebra , rational function , algebra unsolved

A sequence $ a_1, a_2, a_3, \ldots$ is defined recursively by $ a_1 \equal{} 1$ and $ a_{2^k\plus{}j} \equal{} \minus{}a_j$ $ (j \equal{} 1, 2, \ldots, 2^k).$ Prove that this sequence is not periodic.

2012 Indonesia TST, 1

Tags: number theory , least common multiple , algebra , polynomial , algebra unsolved

Given a positive integer $n$. (a) If $P$ is a polynomial of degree $n$ where $P(x) \in \mathbb{Z}$ for every $x \in \mathbb{Z}$, prove that for every $a,b \in \mathbb{Z}$ where $P(a) \neq P(b)$, \[\text{lcm}(1, 2, \ldots, n) \ge \left| \dfrac{a-b}{P(a) - P(b)} \right|\] (b) Find one $P$ (for each $n$) such that the equality case above is achieved for some $a,b \in \mathbb{Z}$.

2006 Lithuania National Olympiad, 1

Tags: algebra , system of equations , algebra unsolved

Solve the system of equations: $\left\{ \begin{aligned} x^4+y^2-xy^3-\frac{9}{8}x = 0 \\ y^4+x^2-yx^3-\frac{9}{8}y=0 \end{aligned} \right.$

2013 South East Mathematical Olympiad, 5

Tags: floor function , calculus , algebra unsolved , algebra

$f(x)=\sum\limits_{i=1}^{2013}\left[\dfrac{x}{i!}\right]$. A integer $n$ is called [i]good[/i] if $f(x)=n$ has real root. How many good numbers are in $\{1,3,5,\dotsc,2013\}$?

2012 China Second Round Olympiad, 9

Tags: function , trigonometry , inequalities , algebra unsolved , algebra

Given a function $f(x)=a\sin x-\frac{1}{2}\cos 2x+a-\frac{3}{a}+\frac{1}{2}$, where $a\in\mathbb{R}, a\ne 0$. [b](1)[/b] If for any $x\in\mathbb{R}$, inequality $f(x)\le 0$ holds, find all possible value of $a$. [b](2)[/b] If $a\ge 2$, and there exists $x\in\mathbb{R}$, such that $f(x)\le 0$. Find all possible value of $a$.

2010 Postal Coaching, 2

Tags: induction , algebra unsolved , algebra

Let $a_1, a_2, \ldots, a_n$ be real numbers lying in $[-1, 1]$ such that $a_1 + a_2 + \cdots + a_n = 0$. Prove that there is a $k \in \{1, 2, \ldots, n\}$ such that $|a_1 + 2a_2 + 3a_3 + \cdots + k a_k | \le \frac{2k+1}4$ .

2010 Contests, 3

Tags: analytic geometry , algebra unsolved , algebra

One point of the plane is called $rational$ if both coordinates are rational and $irrational$ if both coordinates are irrational. Check whether the following statements are true or false: [b]a)[/b] Every point of the plane is in a line that can be defined by $2$ rational points. [b]b)[/b] Every point of the plane is in a line that can be defined by $2$ irrational points. This maybe is not algebra so sorry if I putted it in the wrong category!

2008 Turkey MO (2nd round), 1

Tags: function , algebra unsolved , algebra

$ f: \mathbb N \times \mathbb Z \rightarrow \mathbb Z$ satisfy the given conditions $ a)$ $ f(0,0)\equal{}1$ , $ f(0,1)\equal{}1$ , $ b)$ $ \forall k \notin \left\{0,1\right\}$ $ f(0,k)\equal{}0$ and $ c)$ $ \forall n \geq 1$ and $ k$ , $ f(n,k)\equal{}f(n\minus{}1,k)\plus{}f(n\minus{}1,k\minus{}2n)$ find the sum $ \displaystyle\sum_{k\equal{}0}^{\binom{2009}{2}}f(2008,k)$

1997 China Team Selection Test, 1

Tags: algebra , polynomial , inequalities , Vieta , induction , algebra unsolved , Polynomials

Find all real-coefficient polynomials $f(x)$ which satisfy the following conditions: [b]i.[/b] $f(x) = a_0 x^{2n} + a_2 x^{2n - 2} + \cdots + a_{2n - 2} x^2 + a_{2n}, a_0 > 0$; [b]ii.[/b] $\sum_{j=0}^n a_{2j} a_{2n - 2j} \leq \left( \begin{array}{c} 2n\\ n\end{array} \right) a_0 a_{2n}$; [b]iii.[/b] All the roots of $f(x)$ are imaginary numbers with no real part.

1988 IMO Longlists, 65

Tags: number theory , greatest common divisor , induction , algorithm , algebra unsolved , algebra

The Fibonacci sequence is defined by \[ a_{n+1} = a_n + a_{n-1}, n \geq 1, a_0 = 0, a_1 = a_2 = 1. \] Find the greatest common divisor of the 1960-th and 1988-th terms of the Fibonacci sequence.

2009 India National Olympiad, 2

Tags: function , algebra unsolved , algebra

Define a a sequence $ {<{a_n}>}^{\infty}_{n\equal{}1}$ as follows $ a_n\equal{}0$, if number of positive divisors of $ n$ is [i]odd[/i] $ a_n\equal{}1$, if number of positive divisors of $ n$ is [i]even[/i] (The positive divisors of $ n$ include $ 1$ as well as $ n$.)Let $ x\equal{}0.a_1a_2a_3........$ be the real number whose decimal expansion contains $ a_n$ in the $ n$-th place,$ n\geq1$.Determine,with proof,whether $ x$ is rational or irrational.

2007 Moldova National Olympiad, 10.3

Tags: algebra unsolved , algebra

Determine strictly positive real numbers $ a_{1},a_{2},...,a_{n}$ if for any $ n\in N^*$ takes place equality: $ a_{1}^2\plus{}a_{2}^2\plus{}...\plus{}a_{n}^2\equal{}a_{1}\plus{}a_{2}\plus{}...\plus{}a_{n}\plus{}\frac{n(n^2\plus{}6n\plus{}11)}{3}$

2004 Brazil National Olympiad, 3

Tags: induction , algebra unsolved , algebra

Let $x_1, x_2, ..., x_{2004}$ be a sequence of integer numbers such that $x_{k+3}=x_{k+2}+x_{k}x_{k+1}$, $\forall 1 \le k \le 2001$. Is it possible that more than half of the elements are negative?

1999 Tuymaada Olympiad, 3

Tags: algebra unsolved , algebra

A sequence of integers $a_0,\ a_1,\dots a_n \dots $ is defined by the following rules: $a_0=0,\ a_1=1,\ a_{n+1} > a_n$ for each $n\in \mathbb{N}$, and $a_{n+1}$ is the minimum number such that no three numbers among $a_0,\ a_1,\dots a_{n+1}$ form an arithmetical progression. Prove that $a_{2^n}=3^n$ for each $n \in \mathbb{N}.$

1995 Turkey Team Selection Test, 1

Tags: algebra , polynomial , algebra unsolved

Given real numbers $b \geq a>0$, find all solutions of the system \begin{align*} &x_1^2+2ax_1+b^2=x_2,\\ &x_2^2+2ax_2+b^2=x_3,\\ &\qquad\cdots\cdots\cdots\\ &x_n^2+2ax_n+b^2=x_1. \end{align*}