Olympiad problems

1985 Iran MO (2nd round), 1

Let $\alpha $ be an angle such that $\cos \alpha = \frac pq$, where $p$ and $q$ are two integers. Prove that the number $q^n \cos n \alpha$ is an integer.

2012 Greece National Olympiad, 2

Tags: algebra , polynomial , algebra proposed

Find all the non-zero polynomials $P(x),Q(x)$ with real coefficients and the minimum degree,such that for all $x \in \mathbb{R}$: \[ P(x^2)+Q(x)=P(x)+x^5Q(x) \]

2006 Taiwan National Olympiad, 3

Tags: quadratics , calculus , integration , algebra , algebra proposed

If positive integers $p,q,r$ are such that the quadratic equation $px^2-qx+r=0$ has two distinct real roots in the open interval $(0,1)$, find the minimum value of $p$.

2008 ISI B.Math Entrance Exam, 9

Tags: algebra proposed , algebra

For $n\geq 3$ , determine all real solutions of the system of n equations : $x_1+x_2+...+x_{n-1}=\frac{1}{x_n}$ ....................... $x_1+x_2+...+x_{i-1}+x_{i+1}+...+x_n=\frac{1}{x_i}$ ....................... $x_2+...+x_{n-1}+x_n=\frac{1}{x_1}$

2012 All-Russian Olympiad, 1

Tags: algebra , polynomial , algebra proposed

Given is the polynomial $P(x)$ and the numbers $a_1,a_2,a_3,b_1,b_2,b_3$ such that $a_1a_2a_3\not=0$. Suppose that for every $x$, we have \[P(a_1x+b_1)+P(a_2x+b_2)=P(a_3x+b_3)\] Prove that the polynomial $P(x)$ has at least one real root.

2006 Pre-Preparation Course Examination, 4

Tags: geometry , geometric transformation , rotation , algebra proposed , algebra

Show that $ \rho (f)$ changes continously over $ f$. It means for every bijection $ f: S^1\rightarrow S^1$ and $ \epsilon > 0$ there is $ \delta > 0$ such that if $ g: S^1\rightarrow S^1$ is a bijection such that $ \parallel{}f \minus{} g\parallel{} < \delta$ then $ |\rho(f) \minus{} \rho(g)| < \epsilon$. Note that $ \rho(f)$ is the rotatation number of $ f$ and $ \parallel{}f \minus{} g\parallel{} \equal{} \sup\{|f(x) \minus{} g(x)| | x\in S^1\}$.

2004 Germany Team Selection Test, 1

Tags: algebra , polynomial , linear algebra , matrix , complex numbers , system of equations , algebra proposed

Let n be a positive integer. Find all complex numbers $x_{1}$, $x_{2}$, ..., $x_{n}$ satisfying the following system of equations: $x_{1}+2x_{2}+...+nx_{n}=0$, $x_{1}^{2}+2x_{2}^{2}+...+nx_{n}^{2}=0$, ... $x_{1}^{n}+2x_{2}^{n}+...+nx_{n}^{n}=0$.

2013 Waseda University Entrance Examination, 2

Tags: trigonometry , algebra , polynomial , modular arithmetic , algebra proposed

For a complex number $z=1+2\sqrt{6}i$ and natural number $n=1,\ 2,\ 3,\ \cdots$, express the complex number $z^n$ in using real numbers $a_n,\ b_n$ as $z^n=a_n+b_ni$. Answer the following questions. (1) Show that $a_n^2+b_n^2=5^{2n}\ (n=1,\ 2,\ 3,\ \cdots).$ (2) Find the constants $p,\ q$ such that $a_{n+2}=pa_{n+1}+qa_n$ holds for all $n$. (3) Show that $a_n$ is not a multiple of $5$ for any $n$. (4) Show that $z^n\ (n=1,\ 2,\ 3,\ \cdots)$ is not a real number.

2011 Brazil Team Selection Test, 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.

2005 Taiwan National Olympiad, 3

Tags: quadratics , calculus , integration , algebra , algebra proposed

If positive integers $p,q,r$ are such that the quadratic equation $px^2-qx+r=0$ has two distinct real roots in the open interval $(0,1)$, find the minimum value of $p$.

2011 Romanian Master of Mathematics, 1

Tags: function , algebra proposed , algebra

Prove that there exist two functions $f,g \colon \mathbb{R} \to \mathbb{R}$, such that $f\circ g$ is strictly decreasing and $g\circ f$ is strictly increasing. [i](Poland) Andrzej Komisarski and Marcin Kuczma[/i]

1997 IMC, 5

Tags: algebra proposed , algebra

Let $X$ be an arbitrary set and $f$ a bijection from $X$ to $X$. Show that there exist bijections $g,\ g':X\to X$ s.t. $f=g\circ g',\ g\circ g=g'\circ g'=1_X$.

2005 Romania Team Selection Test, 1

Tags: function , induction , algebra proposed , algebra

Let $a\in\mathbb{R}-\{0\}$. Find all functions $f: \mathbb{R}\to\mathbb{R}$ such that $f(a+x) = f(x) - x$ for all $x\in\mathbb{R}$. [i]Dan Schwartz[/i]

2015 India Regional MathematicaI Olympiad, 3

Tags: algebra , polynomial , algebra proposed , Integer Polynomial , Divisibility

Let $P(x)$ be a polynomial whose coefficients are positive integers. If $P(n)$ divides $P(P(n)-2015)$ for every natural number $n$, prove that $P(-2015)=0$. [hide]One additional condition must be given that $P$ is non-constant, which even though is understood.[/hide]

1983 IMO Longlists, 68

Tags: trigonometry , function , algebra proposed , algebra

Three of the roots of the equation $x^4 -px^3 +qx^2 -rx+s = 0$ are $\tan A, \tan B$, and $\tan C$, where $A, B$, and $C$ are angles of a triangle. Determine the fourth root as a function only of $p, q, r$, and $s.$

2003 Silk Road, 4

Tags: algebra proposed , algebra

Find $ \sum_{k \in A} \frac{1}{k-1}$ where $A= \{ m^n : m,n \in \mathbb{Z} m,n \geq 2 \} $. Problem was post earlier [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=67&t=29456&hilit=silk+road]here[/url] , but solution not gives and olympiad doesn't indicate, so I post it again :blush: Official solution [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=125&t=365714&p=2011659#p2011659]here[/url]

2011 Kosovo National Mathematical Olympiad, 2

Tags: function , quadratics , algebra proposed , algebra

Is it possible that by using the following transformations: \[ f(x) \mapsto x^2 \cdot f \left(\frac{1}{x}+1 \right) \ \ \ \text{or} \ \ \ f(x) \mapsto (x-1)^2 \cdot f\left(\frac{1}{x-1} \right)\] the function $f(x)=x^2+5x+4$ is sent to the function $g(x)=x^2+10x+8$ ?

1991 Federal Competition For Advanced Students, 2

Tags: algebra proposed , algebra

Solve in real numbers the equation: $ \frac{1}{x}\plus{}\frac{1}{x\plus{}2}\minus{}\frac{1}{x\plus{}4}\minus{}\frac{1}{x\plus{}6}\minus{}\frac{1}{x\plus{}8}\minus{}\frac{1}{x\plus{}10}\plus{}\frac{1}{x\plus{}12}\plus{}\frac{1}{x\plus{}14}\equal{}0.$

1986 IMO Longlists, 13

Tags: graph theory , algebra proposed , algebra

Let $N = \{1, 2, \ldots, n\}$, $n \geq 3$. To each pair $i \neq j $ of elements of $N$ there is assigned a number $f_{ij} \in \{0, 1\}$ such that $f_{ij} + f_{ji} = 1$. Let $r(i)=\sum_{i \neq j} f_{ij}$, and write $M = \max_{i\in N} r(i)$, $m = \min_{i\in N} r(i)$. Prove that for any $w \in N$ with $r(w) = m$ there exist $u, v \in N$ such that $r(u) = M$ and $f_{uv}f_{vw} = 1$.

1990 IMO Longlists, 41

Tags: algebra proposed , algebra

Let $n$ be an arbitrary positive integer. Calculate $S_n = \sum_{r=0}^n 2^{r-2n} \binom{2n-r}{n}.$

2001 Junior Balkan Team Selection Tests - Romania, 2

Tags: algebra proposed , algebra

Let $A$ be a non-empty subset of $\mathbb{R}$ with the property that for every real numbers $x,y$, if $x+y\in A$ then $xy\in A$. Prove that $A=\mathbb{R}$.

2008 Moldova Team Selection Test, 1

Tags: algebra proposed , algebra

Find all solutions $ (x,y)\in \mathbb{R}\times\mathbb R$ of the following system: $ \begin{cases}x^3 \plus{} 3xy^2 \equal{} 49, \\ x^2 \plus{} 8xy \plus{} y^2 \equal{} 8y \plus{} 17x.\end{cases}$

1997 All-Russian Olympiad, 1

Tags: algebra proposed , algebra

Do there exist real numbers $b$ and $c$ such that each of the equations $x^2+bx+c = 0$ and $2x^2+(b+1)x+c+1 = 0$ have two integer roots? [i]N. Agakhanov[/i]

2013 China Team Selection Test, 3

Tags: induction , modular arithmetic , algebra proposed , algebra

Find all positive real numbers $r<1$ such that there exists a set $\mathcal{S}$ with the given properties: i) For any real number $t$, exactly one of $t, t+r$ and $t+1$ belongs to $\mathcal{S}$; ii) For any real number $t$, exactly one of $t, t-r$ and $t-1$ belongs to $\mathcal{S}$.

2014 District Olympiad, 2

Tags: logarithms , algebra proposed , algebra

Solve in real numbers the equation \[ x+\log_{2}\left( 1+\sqrt{\frac{5^{x}}{3^{x}+4^{x}}}\right) =4+\log_{1/2}\left(1+\sqrt{\frac{25^{x}}{7^{x}+24^{x}}}\right) \]