Olympiad problems

2001 All-Russian Olympiad, 1

Two monic quadratic trinomials $f(x)$ and $g(x)$ take negative values on disjoint intervals. Prove that there exist positive numbers $\alpha$ and $\beta$ such that $\alpha f(x) + \beta g(x) > 0$ for all real $x$.

2009 Tuymaada Olympiad, 4

Tags: floor function , inequalities , function , algebra unsolved , algebra

The sum of several non-negative numbers is not greater than 200, while the sum of their squares is not less than 2500. Prove that among them there are four numbers whose sum is not less than 50. [i]Proposed by A. Khabrov[/i]

2010 Vietnam National Olympiad, 1

Tags: algebra , polynomial , algebra unsolved

Solve the system equations \[\left\{\begin{array}{cc}x^{4}-y^{4}=240\\x^{3}-2y^{3}=3(x^{2}-4y^{2})-4(x-8y)\end{array}\right.\]

2007 Bosnia Herzegovina Team Selection Test, 4

Tags: algebra , polynomial , inequalities , Vieta , algebra unsolved

Let $P(x)$ be a polynomial such that $P(x)=x^3-2x^2+bx+c$. Roots of $P(x)$ belong to interval $(0,1)$. Prove that $8b+9c \leq 8$. When does equality hold?

2012 USA Team Selection Test, 1

Tags: algebra , polynomial , algebra unsolved

Consider (3-variable) polynomials \[P_n(x,y,z)=(x-y)^{2n}(y-z)^{2n}+(y-z)^{2n}(z-x)^{2n}+(z-x)^{2n}(x-y)^{2n}\] and \[Q_n(x,y,z)=[(x-y)^{2n}+(y-z)^{2n}+(z-x)^{2n}]^{2n}.\] Determine all positive integers $n$ such that the quotient $Q_n(x,y,z)/P_n(x,y,z)$ is a (3-variable) polynomial with rational coefficients.

2011 Estonia Team Selection Test, 3

Tags: algebra unsolved , algebra

Does there exist an operation $*$ on the set of all integers such that the following conditions hold simultaneously: $(1)$ for all integers $x,y,z$, $(x*y)*z=x*(y*z)$; $(2)$ for all integers $x$ and $y$, $x*x*y=y*x*x=y$?

2014 India IMO Training Camp, 1

Tags: quadratics , algebra , quadratic formula , algebra unsolved

Let $x$ and $y$ be rational numbers, such that $x^{5}+y^{5}=2x^{2}y^{2}$. Prove that $1-xy$ is the square of a rational number.

1985 IMO Longlists, 60

Tags: trigonometry , algebra unsolved , algebra

The sequence $(s_n)$, where $s_n= \sum_{k=1}^n \sin k$ for $n = 1, 2,\dots$ is bounded. Find an upper and lower bound.

1985 Vietnam National Olympiad, 2

Tags: function , algebra unsolved , algebra

Find all functions $ f \colon \mathbb{Z} \mapsto \mathbb{R}$ which satisfy: i) $ f(x)f(y) \equal{} f(x \plus{} y) \plus{} f(x \minus{} y)$ for all integers $ x$, $ y$ ii) $ f(0) \neq 0$ iii) $ f(1) \equal{} \frac {5}{2}$

2009 Czech-Polish-Slovak Match, 2

Tags: induction , algebra unsolved , algebra

For positive integers $a$ and $k$, define the sequence $a_1,a_2,\ldots$ by \[a_1=a,\qquad\text{and}\qquad a_{n+1}=a_n+k\cdot\varrho(a_n)\qquad\text{for } n=1,2,\ldots\] where $\varrho(m)$ denotes the product of the decimal digits of $m$ (for example, $\varrho(413)=12$ and $\varrho(308)=0$). Prove that there are positive integers $a$ and $k$ for which the sequence $a_1,a_2,\ldots$ contains exactly $2009$ different numbers.

2006 MOP Homework, 3

Tags: algebra unsolved , algebra

Find the number of all infinite sequences $a_1$, $a_2$, ... of positive integers such that $a_n+a_{n+1}=2a_{n+2}a_{n+3}+2005$ for all positive integers $n$.

2005 India National Olympiad, 6

Tags: function , algebra , functional equation , algebra unsolved

Find all functions $f : \mathbb{R} \longrightarrow \mathbb{R}$ such that \[ f(x^2 + yf(z)) = xf(x) + zf(y) , \] for all $x, y, z \in \mathbb{R}$.

2011 Morocco National Olympiad, 2

Tags: linear algebra , matrix , inequalities , algebra , polynomial , algebra unsolved

Solve in $(\mathbb{R}_{+}^{*})^{4}$ the following system : $\left\{\begin{matrix} x+y+z+t=4\\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{t}=5-\frac{1}{xyzt} \end{matrix}\right.$

2009 India National Olympiad, 3

Tags: function , floor function , special factorizations , algebra unsolved , algebra

Find all real numbers $ x$ such that: $ [x^2\plus{}2x]\equal{}{[x]}^2\plus{}2[x]$ (Here $ [x]$ denotes the largest integer not exceeding $ x$.)

2000 Federal Competition For Advanced Students, Part 2, 3

Tags: function , algebra unsolved , algebra

Find all functions $f : \mathbb R \to \mathbb R$ such that for all reals $x, y, z$ it holds that \[f(x + f(y + z)) + f(f(x + y) + z) = 2y.\]

2003 China Team Selection Test, 3

Tags: algebra , polynomial , algebra unsolved , inequalities

The $ n$ roots of a complex coefficient polynomial $ f(z) \equal{} z^n \plus{} a_1z^{n \minus{} 1} \plus{} \cdots \plus{} a_{n \minus{} 1}z \plus{} a_n$ are $ z_1, z_2, \cdots, z_n$. If $ \sum_{k \equal{} 1}^n |a_k|^2 \leq 1$, then prove that $ \sum_{k \equal{} 1}^n |z_k|^2 \leq n$.

2011 Kazakhstan National Olympiad, 2

Tags: inequalities , limit , algebra unsolved , algebra , Kazakhstan

Given a positive integer $n$. Prove the inequality $\sum\limits_{i=1}^{n}\frac{1}{i(i+1)(i+2)(i+3)(i+4)}<\frac{1}{96}$

2008 South africa National Olympiad, 6

Tags: function , trigonometry , quadratics , algebra unsolved , algebra

Find all function pairs $(f,g)$ where each $f$ and $g$ is a function defined on the integers and with values, such that, for all integers $a$ and $b$, \[f(a+b)=f(a)g(b)+g(a)f(b)\\ g(a+b)=g(a)g(b)-f(a)f(b).\]

2011 Kazakhstan National Olympiad, 6

Tags: function , quadratics , trigonometry , algebra unsolved , algebra , Kazakhstan

Determine all pairs of positive real numbers $(a, b)$ for which there exists a function $ f:\mathbb{R^{+}}\rightarrow\mathbb{R^{+}} $ satisfying for all positive real numbers $x$ the equation $ f(f(x))=af(x)- bx $

2010 Germany Team Selection Test, 1

Tags: algebra , polynomial , algebra unsolved

Let $a \in \mathbb{R}.$ Show that for $n \geq 2$ every non-real root $z$ of polynomial $X^{n+1}-X^2+aX+1$ satisfies the condition $|z| > \frac{1}{\sqrt[n]{n}}.$

1963 Vietnam National Olympiad, 3

Tags: trigonometry , algebra unsolved , algebra

Solve the equation $ \sin^3x \cos 3x \plus{} \cos^3x \sin 3x \equal{} \frac{3}{8}$.

2011 Indonesia TST, 1

Tags: algebra unsolved , number theory unsolved , number theory , algebra

Find all real number $x$ which could be represented as $x = \frac{a_0}{a_1a_2 . . . a_n} + \frac{a_1}{a_2a_3 . . . a_n} + \frac{a_2}{a_3a_4 . . . a_n} + . . . + \frac{a_{n-2}}{a_{n-1}a_n} + \frac{a_{n-1}}{a_n}$ , with $n, a_1, a_2, . . . . , a_n$ are positive integers and $1 = a_0 \leq a_1 < a_2 < . . . < a_n$

2011 Albania Team Selection Test, 1

Tags: analytic geometry , conics , parabola , function , algebra unsolved , algebra

The given parabola $y=ax^2+bx+c$ doesn't intersect the $X$-axis and passes from the points $A(-2,1)$ and $B(2,9)$. Find all the possible values of the $x$ coordinates of the vertex of this parabola.

1993 Cono Sur Olympiad, 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)

2012 France Team Selection Test, 2

Tags: algebra , polynomial , algebra unsolved

Determine all non-constant polynomials $X^n+a_{n-1}X^{n-1}+\cdots +a_1X+a_0$ with integer coefficients for which the roots are exactly the numbers $a_0,a_1,\ldots ,a_{n-1}$ (with multiplicity).