Olympiad problems

2014 Contests, 3

Let $a$, $b$ and $c$ be rational numbers for which $a+bc$, $b+ac$ and $a+b$ are all non-zero and for which we have \[\frac{1}{a+bc}+\frac{1}{b+ac}=\frac{1}{a+b}.\] Prove that $\sqrt{(c-3)(c+1)}$ is rational.

2001 Romania National Olympiad, 2

Tags: algebra proposed , algebra

Let $a$ and $b$ be real, positive and distinct numbers. We consider the set: \[M=\{ ax+by\mid x,y\in\mathbb{R},\ x>0,\ y>0,\ x+y=1\} \] Prove that: (i) $\frac{2ab}{a+b}\in M;$ (ii) $\sqrt{ab}\in M.$

2011 Kosovo National Mathematical Olympiad, 2

Tags: function , algebra proposed , algebra

It is given the function $f:\left( \mathbb{R} - \{0\} \right) \times \left( \mathbb{R}-\{0\} \right) \to \mathbb{R}$ such that $f(a,b)= \left| \frac{|b-a|}{|ab|}+\frac{b+a}{ab}-1 \right|+ \frac{|b-a|}{|ab|}+ \frac{b+a}{ab}+1$ where $a,b \not=0$. Prove that: \[ f(a,b)=4 \cdot \text{max} \left\{\frac{1}{a},\frac{1}{b},\frac{1}{2} \right\}\]

1999 Italy TST, 3

Tags: function , algebra proposed , algebra

(a) Find all strictly monotone functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that \[f(x+f(y))=f(x)+y\quad\text{for all real}\ x,y. \] (b) If $n>1$ is an integer, prove that there is no strictly monotone function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that \[ f(x+f(y))=f(x)+y^n\quad \text{for all real}\ x, y.\]

1996 Romania National Olympiad, 2

Tags: algebra , polynomial , algebra proposed

Find all polynomials $p_n(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ ($n\geq 2$) with real and non-zero coeficients s.t. $p_n(x)-p_1(x)p_2(x)...p_{n-1}(x)$ be a constant polynomial. ;)

1991 IberoAmerican, 3

Tags: function , algebra proposed , algebra

Let $f: \ [0,\ 1] \rightarrow \mathbb{R}$ be an increasing function satisfying the following conditions: a) $f(0)=0$; b) $f\left(\frac{x}{3}\right)=\frac{f(x)}{2}$; c) $f(1-x)=1-f(x)$. Determine $f\left(\frac{18}{1991}\right)$.

2007 Baltic Way, 3

Tags: algebra , polynomial , algebra proposed

Suppose that $F,G,H$ are polynomials of degree at most $2n+1$ with real coefficients such that: i) For all real $x$ we have $F(x)\le G(x)\le H(x)$. ii) There exist distinct real numbers $x_1,x_2,\ldots ,x_n$ such that $F(x_i)=H(x_i)\quad\text{for}\ i=1,2,3,\ldots ,n$. iii) There exists a real number $x_0$ different from $x_1,x_2,\ldots ,x_n$ such that $F(x_0)+H(x_0)=2G(x_0)$. Prove that $F(x)+H(x)=2G(x)$ for all real numbers $x$.

2013 Tuymaada Olympiad, 6

Tags: quadratics , absolute value , algebra proposed , algebra

Quadratic trinomials with positive leading coefficients are arranged in the squares of a $6 \times 6$ table. Their $108$ coefficients are all integers from $-60$ to $47$ (each number is used once). Prove that at least in one column the sum of all trinomials has a real root. [i]K. Kokhas & F. Petrov[/i]

2005 Romania Team Selection Test, 3

Tags: function , linear algebra , matrix , algebra proposed , algebra

Let $\mathbb{N}=\{1,2,\ldots\}$. Find all functions $f: \mathbb{N}\to\mathbb{N}$ such that for all $m,n\in \mathbb{N}$ the number $f^2(m)+f(n)$ is a divisor of $(m^2+n)^2$.

1991 Hungary-Israel Binational, 4

Tags: quadratics , algebra , system of equations , algebra proposed

Find all the real values of $ \lambda$ for which the system of equations $ x\plus{}y\plus{}z\plus{}v\equal{}0$ and $ \left(xy\plus{}yz\plus{}zv\right)\plus{}\lambda\left(xz\plus{}xv\plus{}yv\right)\equal{}0$, has a unique real solution.

2007 Iran Team Selection Test, 1

Tags: algebra , polynomial , inequalities , algebra proposed

Find all polynomials of degree 3, such that for each $x,y\geq 0$: \[p(x+y)\geq p(x)+p(y)\]

IMSC 2024, 6

Tags: algebra , polynomial , number theory , number theory proposed , algebra proposed , Lifting the Exponent , Divisibility

Let $a\equiv 1\pmod{4}$ be a positive integer. Show that any polynomial $Q\in\mathbb{Z}[X]$ with all positive coefficients such that $$Q(n+1)((a+1)^{Q(n)}-a^{Q(n)})$$ is a perfect square for any $n\in\mathbb{N}^{\ast}$ must be a constant polynomial. [i]Proposed by Vlad Matei, Romania[/i]

1997 Baltic Way, 3

Tags: floor function , algebra proposed , algebra

Let $x_1=1$ and $x_{n+1} =x_n+\left\lfloor \frac{x_n}{n}\right\rfloor +2$, for $n=1,2,3,\ldots $ where $x$ denotes the largest integer not greater than $x$. Determine $x_{1997}$.

2011 APMO, 5

Tags: function , algebra proposed , algebra

Determine all functions $f:\mathbb{R}\to\mathbb{R}$, where $\mathbb{R}$ is the set of all real numbers, satisfying the following two conditions: 1) There exists a real number $M$ such that for every real number $x,f(x)<M$ is satisfied. 2) For every pair of real numbers $x$ and $y$, \[ f(xf(y))+yf(x)=xf(y)+f(xy)\] is satisfied.

1997 Pre-Preparation Course Examination, 1

Tags: algebra , polynomial , greatest common divisor , algebra proposed

Let $n$ be a positive integer. Prove that there exist polynomials$f(x)$and $g(x$) with integer coefficients such that \[f(x)\left(x + 1 \right)^{2^n}+ g(x) \left(x^{2^n}+ 1 \right) = 2.\]

2014 Turkey MO (2nd round), 5

Tags: integration , algebra proposed , algebra

Find all natural numbers $n$ for which there exist non-zero and distinct real numbers $a_1, a_2, \ldots, a_n$ satisfying \[ \left\{a_i+\dfrac{(-1)^i}{a_i} \, \Big | \, 1 \leq i \leq n\right\} = \{a_i \mid 1 \leq i \leq n\}. \]

1997 Turkey Team Selection Test, 2

Tags: algebra proposed , algebra

The sequences $(a_{n})$, $(b_{n})$ are deﬁned by $a_{1} = \alpha$, $b_{1} = \beta$, $a_{n+1} = \alpha a_{n} - \beta b_{n}$, $b_{n+1} = \beta a_{n} + \alpha b_{n}$ for all $n > 0.$ How many pairs $(\alpha, \beta)$ of real numbers are there such that $a_{1997} = b_{1}$ and $b_{1997} = a_{1}$?

2013 Romania National Olympiad, 1

Tags: LaTeX , induction , arithmetic sequence , algebra proposed , algebra

A series of numbers is called complete if it has non-zero natural terms and any nonzero integer has at least one among multiple series. Show that the arithmetic progression is a complete sequence if and only if it divides the first term relationship.

2011 Kosovo National Mathematical Olympiad, 2

Tags: function , floor function , algebra proposed , algebra

Find all solutions to the equation: \[ \left(\left\lfloor x+\frac{7}{3} \right\rfloor \right)^2-\left\lfloor x-\frac{9}{4} \right\rfloor = 16 \]

2007 Danube Mathematical Competition, 1

Tags: algebra proposed , algebra

Let $ n\ge2$ be a positive integer and denote by $ S_n$ the set of all permutations of the set $ \{1,2,\ldots,n\}$. For $ \sigma\in S_n$ define $ l(\sigma)$ to be $ \displaystyle\min_{1\le i\le n\minus{}1}\left|\sigma(i\plus{}1)\minus{}\sigma(i)\right|$. Determine $ \displaystyle\max_{\sigma\in S_n}l(\sigma)$.

2009 Canadian Mathematical Olympiad Qualification Repechage, 3

Tags: algebra , polynomial , algebra proposed

Prove that there does not exist a polynomial $f(x)$ with integer coefficients for which $f(2008) = 0$ and $f(2010) = 1867$.

2010 Slovenia National Olympiad, 4

Tags: arithmetic sequence , algebra proposed , algebra

For real numbers $a, b$ and $c$ we have \[(2b-a)^2 + (2b-c)^2 = 2(2b^2-ac).\] Prove that the numbers $a, b$ and $c$ are three consecutive terms in some arithmetic sequence.

1993 All-Russian Olympiad, 1

Tags: geometry , geometric transformation , algebra proposed , algebra

Find all quadruples of real numbers such that each of them is equal to the product of some two other numbers in the quadruple.

1977 IMO Longlists, 31

Tags: function , algebra proposed , algebra

Let $f$ be a function defined on the set of pairs of nonzero rational numbers whose values are positive real numbers. Suppose that $f$ satisfies the following conditions: [b](1)[/b] $f(ab,c)=f(a,c)f(b,c),\ f(c,ab)=f(c,a)f(c,b);$ [b](2)[/b] $f(a,1-a)=1$ Prove that $f(a,a)=f(a,-a)=1,f(a,b)f(b,a)=1$.

2013 Saint Petersburg Mathematical Olympiad, 1

Tags: trigonometry , floor function , calculus , integration , algebra proposed , algebra

Find the minimum positive noninteger root of $ \sin x=\sin \lfloor x \rfloor $. F. Petrov