Olympiad problems

2004 Nordic, 1

Twenty-seven balls labelled from $1$ to $27$ are distributed in three bowls: red, blue, and yellow. What are the possible values of the number of balls in the red bowl if the average labels in the red, blue and yellow bowl are $15$, $3$, and $18$, respectively?

2009 BMO TST, 1

Tags: function , algebra unsolved , algebra

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 $S> \frac {6}{11}$.

2006 Bulgaria Team Selection Test, 2

Tags: algebra unsolved , algebra

a) Let $\{a_n\}_{n=1}^\infty$ is sequence of integers bigger than 1. Proove that if $x>0$ is irrational, then $\ds x_n>\frac{1}{a_{n+1}}$ for infinitely many $n$, where $x_n$ is fractional part of $a_na_{n-1}\dots a_1x$. b)Find all sequences $\{a_n\}_{n=1}^\infty$ of positive integers, for which exist infinitely many $x\in(0,1)$ such that $\ds x_n>\frac{1}{a_{n+1}}$ for all $n$. [i]Nikolai Nikolov, Emil Kolev[/i]

2010 Malaysia National Olympiad, 6

Tags: algebra unsolved , algebra

Find the number of different pairs of positive integers $(a,b)$ for which $a+b\le100$ and \[\dfrac{a+\frac{1}{b}}{\frac{1}{a}+b}=10\]

2001 Bundeswettbewerb Mathematik, 1

Tags: algebra unsolved , algebra

On a table there is a pile with $ T$ tokens which incrementally shall be converted into piles with three tokens each. Each step is constituted of selecting one pile removing one of its tokens. And then the remaining pile is separated into two piles. Is there a sequence of steps that can accomplish this process? a.) $ T \equal{} 1000$ (Cono Sur) b.) $ T \equal{} 2001$ (BWM)

1990 Dutch Mathematical Olympiad, 2

Tags: limit , induction , algebra unsolved , algebra

Consider the sequence $ a_1\equal{}\frac{3}{2}, a_{n\plus{}1}\equal{}\frac{3a_n^2\plus{}4a_n\minus{}3}{4a_n^2}.$ $ (a)$ Prove that $ 1<a_n$ and $ a_{n\plus{}1}<a_n$ for all $ n$. $ (b)$ From $ (a)$ it follows that $ \displaystyle\lim_{n\to\infty}a_n$ exists. Find this limit. $ (c)$ Determine $ \displaystyle\lim_{n\to\infty}a_1a_2a_3...a_n$.

1994 Polish MO Finals, 3

Tags: limit , LaTeX , function , algebra unsolved , algebra

$k$ is a fixed positive integer. Let $a_n$ be the number of maps $f$ from the subsets of $\{1, 2, ... , n\}$ to $\{1, 2, ... , k\}$ such that for all subsets $A, B$ of $\{1, 2, ... , n\}$ we have $f(A \cap B) = \min (f(A), f(B))$. Find $\lim_{n \to \infty} \sqrt[n]{a_n}$.

2010 Greece Team Selection Test, 4

Tags: function , algebra unsolved , algebra

Find all functions $ f:\mathbb{R^{\ast }}\rightarrow \mathbb{ R^{\ast }}$ satisfying $f(\frac{f(x)}{f(y)})=\frac{1}{y}f(f(x))$ for all $x,y\in \mathbb{R^{\ast }}$ and are strictly monotone in $(0,+\infty )$

2006 China Girls Math Olympiad, 1

Tags: function , algebra unsolved , algebra

Let $a>0$, the function $f: (0,+\infty) \to R$ satisfies $f(a)=1$, if for any positive reals $x$ and $y$, there is \[f(x)f(y)+f \left( \frac{a}{x}\right)f \left( \frac{a}{y}\right) =2f(xy)\] then prove that $f(x)$ is a constant.

2010 Contests, 3

Tags: algebra , function , domain , algebra unsolved

Find all the functions $f:\mathbb{N}\to\mathbb{R}$ that satisfy \[ f(x+y)=f(x)+f(y) \] for all $x,y\in\mathbb{N}$ satisfying $10^6-\frac{1}{10^6} < \frac{x}{y} < 10^6+\frac{1}{10^6}$. Note: $\mathbb{N}$ denotes the set of positive integers and $\mathbb{R}$ denotes the set of real numbers.

2002 China Team Selection Test, 1

Tags: algebra unsolved , algebra

Given a positive integer $ n$, for all positive integers $ a_1, a_2, \cdots, a_n$ that satisfy $ a_1 \equal{} 1$, $ a_{i \plus{} 1} \leq a_i \plus{} 1$, find $ \displaystyle \sum_{i \equal{} 1}^{n} a_1a_2 \cdots a_i$.

2014 China Girls Math Olympiad, 4

Tags: algebra , polynomial , geometric sequence , algebra unsolved

For an integer $m\geq 4,$ let $T_{m}$ denote the number of sequences $a_{1},\dots,a_{m}$ such that the following conditions hold: (1) For all $i=1,2,\dots,m$ we have $a_{i}\in \{1,2,3,4\}$ (2) $a_{1} = a_{m} = 1$ and $a_{2}\neq 1$ (3) For all $i=3,4\cdots, m, a_{i}\neq a_{i-1}, a_{i}\neq a_{i-2}.$ Prove that there exists a geometric sequence of positive integers $\{g_{n}\}$ such that for $n\geq 4$ we have that \[ g_{n} - 2\sqrt{g_{n}} < T_{n} < g_{n} + 2\sqrt{g_{n}}.\]

2008 Germany Team Selection Test, 1

Tags: induction , LaTeX , algebra unsolved , algebra

A sequence $ (S_n), n \geq 1$ of sets of natural numbers with $ S_1 = \{1\}, S_2 = \{2\}$ and \[{ S_{n + 1} = \{k \in }\mathbb{N}|k - 1 \in S_n \text{ XOR } k \in S_{n - 1}\}. \] Determine $ S_{1024}.$

1992 Polish MO Finals, 1

Tags: function , induction , inequalities , algebra unsolved , algebra

The functions $f_0, f_1, f_2, ...$ are defined on the reals by $f_0(x) = 8$ for all $x$, $f_{n+1}(x) = \sqrt{x^2 + 6f_n(x)}$. For all $n$ solve the equation $f_n(x) = 2x$.

2012 Kazakhstan National Olympiad, 2

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 $x$.

2006 Vietnam National Olympiad, 4

Tags: function , calculus , derivative , limit , algebra unsolved , algebra

Given is the function $f(x)=-x+\sqrt{(x+a)(x+b)}$, where $a$, $b$ are distinct given positive real numbers. Prove that for all real numbers $s\in (0,1)$ there exist only one positive real number $\alpha$ such that \[ f(\alpha)=\sqrt [s]{\frac{a^s+b^s}{2}} . \]

2012 Indonesia TST, 1

Tags: function , algebra , polynomial , induction , algebra unsolved

Let $P$ be a polynomial with real coefficients. Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that there exists a real number $t$ such that \[f(x+t) - f(x) = P(x)\] for all $x \in \mathbb{R}$.

1994 Canada National Olympiad, 1

Tags: factorial , algebra , polynomial , algebra unsolved

Evaluate $\sum_{n=1}^{1994}{\left((-1)^{n}\cdot\left(\frac{n^2 + n + 1}{n!}\right)\right)}$ .

2014 Romania Team Selection Test, 2

Tags: algebra , polynomial , algebra unsolved

Let $p$ be an[color=#FF0000] odd [/color]prime number. Determine all pairs of polynomials $f$ and $g$ from $\mathbb{Z}[X]$ such that \[f(g(X))=\sum_{k=0}^{p-1} X^k = \Phi_p(X).\]

1989 IMO Longlists, 26

Tags: algebra unsolved , algebra

Let $ b_1, b_2, \ldots, b_{1989}$ be positive real numbers such that the equations \[ x_{r\minus{}1} \minus{} 2x_r \plus{} x_{r\plus{}1} \plus{} b_rx_r \equal{} 0 \quad (1 \leq r \leq 1989)\] have a solution with $ x_0 \equal{} x_{1989} \equal{} 0$ but not all of $ x_1, \ldots, x_{1989}$ are equal to zero. Prove that \[ \sum^{1989}_{k\equal{}1} b_k \geq \frac{2}{995}.\]

2014 Contests, 3

Tags: algebra , polynomial , complex analysis , algebra unsolved

Let $p,q\in \mathbb{R}[x]$ such that $p(z)q(\overline{z})$ is always a real number for every complex number $z$. Prove that $p(x)=kq(x)$ for some constant $k \in \mathbb{R}$ or $q(x)=0$. [i]Proposed by Mohammad Ahmadi[/i]

1983 Dutch Mathematical Olympiad, 3

Tags: algebra unsolved , algebra

Suppose that $ a,b,c,p$ are real numbers with $ a,b,c$ not all equal, such that: $ a\plus{}\frac{1}{b}\equal{}b\plus{}\frac{1}{c}\equal{}c\plus{}\frac{1}{a}\equal{}p.$ Determine all possible values of $ p$ and prove that $ abc\plus{}p\equal{}0$.

2009 Romanian Master of Mathematics, 1

Tags: floor function , inequalities , number theory , greatest common divisor , least common multiple , triangle inequality , algebra unsolved

For $ a_i \in \mathbb{Z}^ \plus{}$, $ i \equal{} 1, \ldots, k$, and $ n \equal{} \sum^k_{i \equal{} 1} a_i$, let $ d \equal{} \gcd(a_1, \ldots, a_k)$ denote the greatest common divisor of $ a_1, \ldots, a_k$. Prove that $ \frac {d} {n} \cdot \frac {n!}{\prod\limits^k_{i \equal{} 1} (a_i!)}$ is an integer. [i]Dan Schwarz, Romania[/i]

2010 Postal Coaching, 6

Tags: algebra , polynomial , modular arithmetic , induction , number theory , relatively prime , algebra unsolved

Find all polynomials $P$ with integer coeﬃcients which satisfy the property that, for any relatively prime integers $a$ and $b$, the sequence $\{P (an + b) \}_{n \ge 1}$ contains an inﬁnite number of terms, any two of which are relatively prime.

2002 China Team Selection Test, 1

Tags: algebra , polynomial , inequalities , algebra unsolved

Let $P_n(x)=a_0 + a_1x + \cdots + a_nx^n$, with $n \geq 2$, be a real-coefficient polynomial. Prove that if there exists $a > 0$ such that \begin{align*} P_n(x) = (x + a)^2 \left( \sum_{i=0}^{n-2} b_i x^i \right), \end{align*} where $b_i$ are positive real numbers, then there exists some $i$, with $1 \leq i \leq n-1$, such that \[a_i^2 - 4a_{i-1}a_{i+1} \leq 0.\]