Olympiad problems

2011 IberoAmerican, 2

Let $x_1,\ldots ,x_n$ be positive real numbers. Show that there exist $a_1,\ldots ,a_n\in\{-1,1\}$ such that: \[a_1x_1^2+a_2x_2^2+\ldots +a_nx_n^2\ge (a_1x_1+a_2x_2+\ldots + a_n x_n)^2\]

2001 Baltic Way, 13

Tags: floor function , algebra proposed , algebra

Let $a_0, a_1, a_2,\ldots $ be a sequence of real numbers satisfying $a_0=1$ and $a_n=a_{\lfloor 7n/9\rfloor}+a_{\lfloor n/9\rfloor}$ for $n=1, 2,\ldots $ Prove that there exists a positive integer $k$ with $a_k<\frac{k}{2001!}$.

1992 Putnam, A1

Tags: function , Putnam , algebra proposed , algebra

Find all functions $ f : Z\rightarrow Z$ for which we have $ f (0) \equal{} 1$ and $ f ( f (n)) \equal{} f ( f (n\plus{}2)\plus{}2) \equal{} n$, for every natural number $ n$.

2002 Baltic Way, 1

Tags: algebra proposed , equation , Symmetric

Solve the system of simultaneous equations \[\begin{cases}a^3+3ab^2+3ac^2-6abc=1\\ b^3+3ba^2+3bc^2-6abc=1\\c^3+3ca^2+3cb^2-6abc=1\end{cases}\] in real numbers.

2004 Balkan MO, 1

Tags: induction , algebra proposed , algebra

The sequence $\{a_n\}_{n\geq 0}$ of real numbers satisfies the relation: \[ a_{m+n} + a_{m-n} - m + n -1 = \frac12 (a_{2m} + a_{2n}) \] for all non-negative integers $m$ and $n$, $m \ge n$. If $a_1 = 3$ find $a_{2004}$.

2010 ELMO Shortlist, 1

Tags: function , induction , algebra proposed , algebra

Determine all strictly increasing functions $f: \mathbb{N}\to\mathbb{N}$ satisfying $nf(f(n))=f(n)^2$ for all positive integers $n$. [i]Carl Lian and Brian Hamrick.[/i]

1997 Brazil National Olympiad, 3

Tags: function , algebra proposed , algebra

a) Show that there are no functions $f, g: \mathbb R \to \mathbb R$ such that $g(f(x)) = x^3$ and $f(g(x)) = x^2$ for all $x \in \mathbb R$. b) Let $S$ be the set of all real numbers greater than 1. Show that there are functions $f, g : S \to S$ satsfying the condition above.

2006 Australia National Olympiad, 2

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

Let $f$ be a function defined on the positive integers, taking positive integral values, such that $f(a)f(b) = f(ab)$ for all positive integers $a$ and $b$, $f(a) < f(b)$ if $a < b$, $f(3) \geq 7$. Find the smallest possible value of $f(3)$.

2006 IberoAmerican Olympiad For University Students, 3

Tags: algebra , polynomial , induction , trigonometry , algebra proposed

Let $p_1(x)=p(x)=4x^3-3x$ and $p_{n+1}(x)=p(p_n(x))$ for each positive integer $n$. Also, let $A(n)$ be the set of all the real roots of the equation $p_n(x)=x$. Prove that $A(n)\subseteq A(2n)$ and that the product of the elements of $A(n)$ is the average of the elements of $A(2n)$.

2006 Moldova National Olympiad, 11.1

Tags: limit , trigonometry , algebra proposed , algebra

Let $n\in\mathbb{N}^*$. Prove that \[ \lim_{x\to 0}\frac{ \displaystyle (1+x^2)^{n+1}-\prod_{k=1}^n\cos kx}{ \displaystyle x\sum_{k=1}^n\sin kx}=\frac{2n^2+n+12}{6n}. \]

2014 ELMO Shortlist, 7

Tags: algebra , polynomial , algebra proposed

Find all positive integers $n$ with $n \ge 2$ such that the polynomial \[ P(a_1, a_2, ..., a_n) = a_1^n+a_2^n + ... + a_n^n - n a_1 a_2 ... a_n \] in the $n$ variables $a_1$, $a_2$, $\dots$, $a_n$ is irreducible over the real numbers, i.e. it cannot be factored as the product of two nonconstant polynomials with real coefficients. [i]Proposed by Yang Liu[/i]

2002 India IMO Training Camp, 3

Tags: quadratics , algebra proposed , algebra

Let $X=\{2^m3^n|0 \le m, \ n \le 9 \}$. How many quadratics are there of the form $ax^2+2bx+c$, with equal roots, and such that $a,b,c$ are distinct elements of $X$?

2008 Peru IMO TST, 2

Tags: algebra , functional equation , algebra proposed

Determine all functions $f : \mathbb{R} \to \mathbb{R}$ such that $$ f(2f(x) + y) = f(f(x) - f(y)) + 2y + x, $$ for all $x,y \in \mathbb{R}.$

1999 Vietnam National Olympiad, 1

Tags: algebra , system of equations , algebra proposed

Solve the system of equations: $ (1\plus{}4^{2x\minus{}y}).5^{1\minus{}2x\plus{}y}\equal{}1\plus{}2^{2x\minus{}y\plus{}1}$ $ y^3\plus{}4x\plus{}ln(y^2\plus{}2x)\plus{}1\equal{}0$

2011 ELMO Problems, 4

Tags: function , induction , algebra , functional equation , algebra proposed

Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that whenever $a>b>c>d>0$ and $ad=bc$, \[f(a+d)+f(b-c)=f(a-d)+f(b+c).\] [i]Calvin Deng.[/i]

1998 Brazil National Olympiad, 1

Tags: quadratics , algebra , algebra proposed , Combinatorial games , game

Two players play a game as follows. The first player chooses two non-zero integers A and B. The second player forms a quadratic with A, B and 1998 as coefficients (in any order). The first player wins iff the equation has two distinct rational roots. Show that the first player can always win.

2011 Kosovo National Mathematical Olympiad, 1

Tags: algebra proposed , algebra

Suppose that the roots $p,q$ of the equation $x^2-x+c=0$ where $c \in \mathbb{R}$, are rational numbers. Prove that the roots of the equation $x^2+px-q=0$ are also rational numbers.

1990 Balkan MO, 1

Tags: modular arithmetic , algebra proposed , algebra

The sequence $ (a_{n})_{n\geq 1}$ is defined by $ a_{1} \equal{} 1, a_{2} \equal{} 3$, and $ a_{n \plus{} 2} \equal{} (n \plus{} 3)a_{n \plus{} 1} \minus{} (n \plus{} 2)a_{n}, \forall n \in \mathbb{N}$. Find all values of $ n$ for which $ a_{n}$ is divisible by $ 11$.

2014 India Regional Mathematical Olympiad, 2

Tags: calculus , integration , arithmetic sequence , algebra proposed , algebra

The roots of the equation \[ x^3-3ax^2+bx+18c=0 \] form a non-constant arithmetic progression and the roots of the equation \[ x^3+bx^2+x-c^3=0 \] form a non-constant geometric progression. Given that $a,b,c$ are real numbers, find all positive integral values $a$ and $b$.