Olympiad problems

2005 MOP Homework, 6

Let $c$ be a fixed positive integer, and $\{x_k\}^{\inf}_{k=1}$ be a sequence such that $x_1=c$ and $x_n=x_{n-1}+\lfloor \frac{2x_{n-1}-2}{n} \rfloor$ for $n \ge 2$. Determine the explicit formula of $x_n$ in terms of $n$ and $c$. (Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)

2011 China Team Selection Test, 1

Tags: function , algebra unsolved , algebra

Let $n\geq 2$ be a given integer. Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that \[f(x-f(y))=f(x+y^n)+f(f(y)+y^n), \qquad \forall x,y \in \mathbb R.\]

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.

2003 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: arithmetic sequence , algebra unsolved , algebra

Let $n$ and $k$ be positive integers. Consider $n$ infinite arithmetic progressions of nonnegative integers with the property that among any $k$ consecutive nonnegative integers, at least one of $k$ integers belongs to one of the $n$ arithmetic progressions. Let $d_1,d_2,\ldots,d_n$ denote the differences of the arithmetic progressions, and let $d=\min\{d_1,d_2,\ldots,d_n\}$. In terms of $n$ and $k$, what is the maximum possible value of $d$?

2010 Malaysia National Olympiad, 2

Tags: logarithms , algebra unsolved , algebra

Find $x$ such that \[2010^{\log_{10}x}=11^{\log_{10}(1+3+5+\cdots +4019).}\]

2001 Macedonia National Olympiad, 2

Tags: function , algebra unsolved , algebra

Does there exist a function $f:\mathbb{N}\rightarrow\mathbb{N}$ such that \[f(f(n-1)=f(n+1)-f(n)\quad\text{for all}\ n\ge 2\text{?} \]

2001 India IMO Training Camp, 3

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

Let $P(x)$ be a polynomial of degree $n$ with real coefficients and let $a\geq 3$. Prove that \[\max_{0\leq j \leq n+1}\left | a^j-P(j) \right |\geq 1\]

2007 India Regional Mathematical Olympiad, 3

Tags: search , algebra unsolved , algebra

Find all pairs $ (a, b)$ of real numbers such that whenever $ \alpha$ is a root of $ x^{2} \plus{} ax \plus{} b \equal{} 0$, $ \alpha^{2} \minus{} 2$ is also a root of the equation. [b][Weightage 17/100][/b]

1976 IMO Longlists, 28

Tags: algebra unsolved , algebra

Let $Q$ be a unit square in the plane: $Q = [0, 1] \times [0, 1]$. Let $T :Q \longrightarrow Q$ be defined as follows: \[T(x, y) =\begin{cases} (2x, \frac{y}{2}) &\mbox{ if } 0 \le x \le \frac{1}{2};\\(2x - 1, \frac{y}{2}+ \frac{1}{2})&\mbox{ if } \frac{1}{2} < x \le 1.\end{cases}\] Show that for every disk $D \subset Q$ there exists an integer $n > 0$ such that $T^n(D) \cap D \neq \emptyset.$

2009 Nordic, 2

Tags: algebra , polynomial , algebra unsolved

On a faded piece of paper it is possible to read the following: \[(x^2 + x + a)(x^{15}- \cdots ) = x^{17} + x^{13} + x^5 - 90x^4 + x - 90.\] Some parts have got lost, partly the constant term of the first factor of the left side, partly the majority of the summands of the second factor. It would be possible to restore the polynomial forming the other factor, but we restrict ourselves to asking the following question: What is the value of the constant term $a$? We assume that all polynomials in the statement have only integer coefficients.

2001 Cono Sur Olympiad, 2

Tags: algebra unsolved , algebra

A sequence $a_1,a_2,\ldots$ of positive integers satisfies the following properties.[list][*]$a_1 = 1$ [*]$a_{3n+1} = 2a_n + 1$ [*]$a_{n+1}\ge a_n$ [*]$a_{2001} = 200$[/list]Find the value of $a_{1000}$. [i]Note[/i]. In the original statement of the problem, there was an extra condition:[list][*]every positive integer appears at least once in the sequence.[/list]However, with this extra condition, there is no solution, i.e., no such sequence exists. (Try to prove it.) The problem as written above does have a solution.

1990 Putnam, A4

Tags: Putnam , algebra unsolved , algebra

Consider a paper punch that can be centered at any point of the plane and that, when operated, removes from the plane precisely those points whose distance from the center is irrational. How many punches are needed to remove every point?

2010 Albania Team Selection Test, 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!

2013 Romanian Masters In Mathematics, 2

Tags: function , floor function , induction , algebra unsolved , functional equation

Does there exist a pair $(g,h)$ of functions $g,h:\mathbb{R}\rightarrow\mathbb{R}$ such that the only function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying $f(g(x))=g(f(x))$ and $f(h(x))=h(f(x))$ for all $x\in\mathbb{R}$ is identity function $f(x)\equiv x$?

2000 Irish Math Olympiad, 5

Tags: conics , parabola , analytic geometry , linear algebra , matrix , algebra unsolved , algebra

Consider all parabolas of the form $ y\equal{}x^2\plus{}2px\plus{}q$ for $ p,q \in \mathbb{R}$ which intersect the coordinate axes in three distinct points. For such $ p,q$, denote by $ C_{p,q}$ the circle through these three intersection points. Prove that all circles $ C_{p,q}$ have a point in common.

1991 Bundeswettbewerb Mathematik, 4

Tags: algebra unsolved , algebra

Given wo non-negative integers $a$ and $b$, one of them is odd and the other one even. By the following rule we define two sequences $(a_n),(b_n)$: \[ a_0 = a, \quad a_1 = b, \quad a_{n+1} = 2a_n - a_{n-1} + 2 \quad (n = 1,2,3, \ldots)\] \[ b_0 = b, \quad b_1 = a, \quad b_{n+1} = 2a_n - b_{n-1} + 2 \quad (n = 1,2,3, \ldots)\] Prove that none of these two sequences contain a negative element if and only if we have $|\sqrt{a} - \sqrt{b}| \leq 1$.

2008 Baltic Way, 1

Tags: algebra , polynomial , search , algebra unsolved

Determine all polynomials $p(x)$ with real coefficients such that $p((x+1)^3)=(p(x)+1)^3$ and $p(0)=0$.

2014 Postal Coaching, 3

Tags: quadratics , algebra , polynomial , algebra unsolved

Find all real numbers $p$ for which the equation $x^3+3px^2+(4p-1)x+p=0$ has two real roots with difference $1$.

1983 IMO Longlists, 32

Tags: function , inequalities , logarithms , algebra unsolved , algebra

Let $a, b, c$ be positive real numbers and let $[x]$ denote the greatest integer that does not exceed the real number $x$. Suppose that $f$ is a function defined on the set of non-negative integers $n$ and taking real values such that $f(0) = 0$ and \[f(n) \leq an + f([bn]) + f([cn]), \qquad \text{ for all } n \geq 1.\] Prove that if $b + c < 1$, there is a real number $k$ such that \[f(n) \leq kn \qquad \text{ for all } n \qquad (1)\] while if $b + c = 1$, there is a real number $K$ such that $f(n) \leq K n \log_2 n$ for all $n \geq 2$. Show that if $b + c = 1$, there may not be a real number $k$ that satisfies $(1).$

2002 Austrian-Polish Competition, 7

Tags: function , algebra unsolved , algebra , functional equation

Find all real functions $f$ definited on positive integers and satisying: (a) $f(x+22)=f(x)$, (b) $f\left(x^{2}y\right)=\left(f(x)\right)^{2}f(y)$ for all positive integers $x$ and $y$.

1992 APMO, 5

Tags: inequalities , algebra unsolved , algebra

Find a sequence of maximal length consisting of non-zero integers in which the sum of any seven consecutive terms is positive and that of any eleven consecutive terms is negative.

2003 Canada National Olympiad, 3

Tags: algebra unsolved , algebra

Find all real positive solutions (if any) to \begin{align*} x^3+y^3+z^3 &= x+y+z, \mbox{ and} \\ x^2+y^2+z^2 &= xyz. \end{align*}

2010 Romania National Olympiad, 2

Tags: inequalities , algebra unsolved , algebra

Consider $v,w$ two distinct non-zero complex numbers. Prove that \[|zw+\bar{w}|\le |zv+\bar{v}|,\] for any $z\in\mathbb{C},|z|=1$, if and only if there exists $k\in [-1,1]$ such that $w=kv$. [i]Dan Marinescu[/i]

1998 Greece JBMO TST, 5

Tags: ceiling function , algebra unsolved , algebra

Let $I$ be an open interval of length $\frac{1}{n}$, where $n$ is a positive integer. Find the maximum possible number of rational numbers of the form $\frac{a}{b}$ where $1 \le b \le n$ that lie in $I$.

2007 ISI B.Stat Entrance Exam, 6

Tags: function , algebra unsolved , algebra

Let $S=\{1,2,\cdots ,n\}$ where $n$ is an odd integer. Let $f$ be a function defined on $\{(i,j): i\in S, j \in S\}$ taking values in $S$ such that (i) $f(s,r)=f(r,s)$ for all $r,s \in S$ (ii) $\{f(r,s): s\in S\}=S$ for all $r\in S$ Show that $\{f(r,r): r\in S\}=S$