Olympiad problems

2010 Tournament Of Towns, 2

Let $f(x)$ be a function such that every straight line has the same number of intersection points with the graph $y = f(x)$ and with the graph $y = x^2$. Prove that $f(x) = x^2.$

2008 Vietnam National Olympiad, 4

Tags: limit , algebra unsolved , algebra

he sequence of real number $ (x_n)$ is defined by $ x_1 \equal{} 0,$ $ x_2 \equal{} 2$ and $ x_{n\plus{}2} \equal{} 2^{\minus{}x_n} \plus{} \frac{1}{2}$ $ \forall n \equal{} 1,2,3 \ldots$ Prove that the sequence has a limit as $ n$ approaches $ \plus{}\infty.$ Determine the limit.

2013 CentroAmerican, 3

Tags: algebra , polynomial , algebra unsolved

Determine all pairs of non-constant polynomials $p(x)$ and $q(x)$, each with leading coefficient $1$, degree $n$, and $n$ roots which are non-negative integers, that satisfy $p(x)-q(x)=1$.

2010 Albania National Olympiad, 2

Tags: induction , algebra unsolved , algebra

We denote $N_{2010}=\{1,2,\cdots,2010\}$ [b](a)[/b]How many non empty subsets does this set have? [b](b)[/b]For every non empty subset of the set $N_{2010}$ we take the product of the elements of the subset. What is the sum of these products? [b](c)[/b]Same question as the [b](b)[/b] part for the set $-N_{2010}=\{-1,-2,\cdots,-2010\}$. Albanian National Mathematical Olympiad 2010---12 GRADE Question 2.

2009 Croatia Team Selection Test, 1

Tags: inequalities , algebra unsolved , algebra

Solve in the set of real numbers: \[ 3\left(x^2 \plus{} y^2 \plus{} z^2\right) \equal{} 1, \] \[ x^2y^2 \plus{} y^2z^2 \plus{} z^2x^2 \equal{} xyz\left(x \plus{} y \plus{} z\right)^3. \]

2003 Korea - Final Round, 3

Tags: algebra unsolved , algebra

Show that the equation, $2x^4+2x^2y^2+y^4=z^2$, does not have integer solution when $x \neq 0$.

2000 Vietnam Team Selection Test, 2

Tags: function , algebra unsolved , algebra

Let $a > 1$ and $r > 1$ be real numbers. (a) Prove that if $f : \mathbb{R}^{+}\to\mathbb{ R}^{+}$ is a function satisfying the conditions (i) $f (x)^{2}\leq ax^{r}f (\frac{x}{a})$ for all $x > 0$, (ii) $f (x) < 2^{2000}$ for all $x < \frac{1}{2^{2000}}$, then $f (x) \leq x^{r}a^{1-r}$ for all $x > 0$. (b) Construct a function $f : \mathbb{R}^{+}\to\mathbb{ R}^{+}$ satisfying condition (i) such that for all $x > 0, f (x) > x^{r}a^{1-r}$ .

2014 Iran Team Selection Test, 2

Tags: algebra , polynomial , number theory , algebra unsolved

find all polynomials with integer coefficients that $P(\mathbb{Z})= ${$p(a):a\in \mathbb{Z}$} has a Geometric progression.

1995 China Team Selection Test, 2

Tags: algebra , polynomial , algebra unsolved

$ A$ and $ B$ play the following game with a polynomial of degree at least 4: \[ x^{2n} \plus{} \_x^{2n \minus{} 1} \plus{} \_x^{2n \minus{} 2} \plus{} \ldots \plus{} \_x \plus{} 1 \equal{} 0 \] $ A$ and $ B$ take turns to fill in one of the blanks with a real number until all the blanks are filled up. If the resulting polynomial has no real roots, $ A$ wins. Otherwise, $ B$ wins. If $ A$ begins, which player has a winning strategy?

2000 China Team Selection Test, 1

Tags: function , algebra , polynomial , algebra unsolved

Let $F$ be the set of all polynomials $\Gamma$ such that all the coefficients of $\Gamma (x)$ are integers and $\Gamma (x) = 1$ has integer roots. Given a positive intger $k$, find the smallest integer $m(k) > 1$ such that there exist $\Gamma \in F$ for which $\Gamma (x) = m(k)$ has exactly $k$ distinct integer roots.

1995 APMO, 5

Tags: function , modular arithmetic , algebra unsolved , algebra

Find the minimum positive integer $k$ such that there exists a function $f$ from the set $\Bbb{Z}$ of all integers to $\{1, 2, \ldots k\}$ with the property that $f(x) \neq f(y)$ whenever $|x-y| \in \{5, 7, 12\}$.

2005 Postal Coaching, 14

Tags: algebra , polynomial , algebra unsolved

Let $f(z) = a_m z^n + a_{n-1} z^{n-1} + \cdots + a_1 z + a_0$ be a polynomial of degree $n \geq 3$ with real coefficients.Suppose all roots of $f(z) =0$ lie in the half plane ${\ z \in \mathbb{C} : Re(z) < 0 \}}$. Prove that $a_k a_{k+3} < a_{k+1}a_{k+2}$ for $k = 0,1,2,3,.... n-3$

2010 Tuymaada Olympiad, 3

Tags: quadratics , algebra , polynomial , Vieta , absolute value , algebra unsolved

Let $f(x) = ax^2+bx+c$ be a quadratic trinomial with $a$,$b$,$c$ reals such that any quadratic trinomial obtained by a permutation of $f$'s coefficients has an integer root (including $f$ itself). Show that $f(1)=0$.

1989 China National Olympiad, 6

Tags: function , inequalities , algebra unsolved , algebra

Find all functions $f:(1,+\infty) \rightarrow (1,+\infty)$ that satisfy the following condition: for arbitrary $x,y>1$ and $u,v>0$, inequality $f(x^uy^v)\le f(x)^{\dfrac{1}{4u}}f(y)^{\dfrac{1}{4v}}$ holds.

2013 Turkmenistan National Math Olympiad, 2

Tags: induction , algebra unsolved , algebra

Sequence $x_1 , x_2 , ..., $ with $x_1=20$ ; $x_2=12$ for all $n\geq 1$ such that $x_{n+2}=x_n+x_{n+1}+2\sqrt{x_{n}*x_{n+1}+121} $then prove that $x_{2013}$ is an integer number.

2014 Contests, 3

Tags: algebra unsolved , algebra

Let $n \ge 2$ be a positive integer, and write in a digit form \[\frac{1}{n}=0.a_1a_2\dots.\] Suppose that $n = a_1 + a_2 + \cdots$. Determine all possible values of $n$.

2001 Tournament Of Towns, 3

Tags: algebra unsolved , algebra

An $8\times8$ array consists of the numbers $1,2,...,64$. Consecutive numbers are adjacent along a row or a column. What is the minimum value of the sum of the numbers along the diagonal?

1996 China National Olympiad, 3

Tags: geometry , 3D geometry , function , algebra unsolved , algebra

Suppose that the function $f:\mathbb{R}\to\mathbb{R}$ satisfies \[f(x^3 + y^3)=(x+y)(f(x)^2-f(x)f(y)+f(y)^2)\] for all $x,y\in\mathbb{R}$. Prove that $f(1996x)=1996f(x)$ for all $x\in\mathbb{R}$.

1989 IMO Longlists, 77

Tags: algebra unsolved , algebra

Let $ a, b, c, r,$ and $ s$ be real numbers. Show that if $ r$ is a root of $ ax^2\plus{}bx\plus{}c \equal{} 0$ and s is a root of $ \minus{}ax^2\plus{}bx\plus{}c \equal{} 0,$ then \[ \frac{a}{2} x^2 \plus{} bx \plus{} c \equal{} 0\] has a root between $ r$ and $ s.$

2000 Hong kong National Olympiad, 2

Tags: induction , algebra unsolved , algebra

Define $a_1=1$ and $a_{n+1}=\frac{a_n}{n}+\frac{n}{a_n}$ for $n\in\mathbb{N}$. Find the greatest integer not exceeding $a_{2000}$ and prove your claim.

2008 Singapore Senior Math Olympiad, 3

Tags: function , algebra unsolved , algebra

Let there's a function $ f: \mathbb{R}\rightarrow\mathbb{R}$ Find all functions $ f$ that satisfies: a) $ f(2u)\equal{}f(u\plus{}v)f(v\minus{}u)\plus{}f(u\minus{}v)f(\minus{}u\minus{}v)$ b) $ f(u)\geq0$

2002 China Team Selection Test, 2

Tags: algebra unsolved , algebra

Let $ \left(a_{n}\right)$ be the sequence of reals defined by $ a_{1}=\frac{1}{4}$ and the recurrence $ a_{n}= \frac{1}{4}(1+a_{n-1})^{2}, n\geq 2$. Find the minimum real $ \lambda$ such that for any non-negative reals $ x_{1},x_{2},\dots,x_{2002}$, it holds \[ \sum_{k=1}^{2002}A_{k}\leq \lambda a_{2002}, \] where $ A_{k}= \frac{x_{k}-k}{(x_{k}+\cdots+x_{2002}+\frac{k(k-1)}{2}+1)^{2}}, k\geq 1$.

2010 Contests, 1

Tags: algebra unsolved , algebra

Solve in positive reals the system: $x+y+z+w=4$ $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{w}=5-\frac{1}{xyzw}$

1989 IMO Longlists, 17

Tags: function , limit , geometric series , algebra unsolved , algebra

Let $ a \in \mathbb{R}, 0 < a < 1,$ and $ f$ a continuous function on $ [0, 1]$ satisfying $ f(0) \equal{} 0, f(1) \equal{} 1,$ and \[ f \left( \frac{x\plus{}y}{2} \right) \equal{} (1\minus{}a) f(x) \plus{} a f(y) \quad \forall x,y \in [0,1] \text{ with } x \leq y.\] Determine $ f \left( \frac{1}{7} \right).$

2014 Iran Team Selection Test, 3

Tags: algebra , polynomial , function , algebra unsolved

let $m,n\in \mathbb{N}$ and $p(x),q(x),h(x)$ are polynomials with real Coefficients such that $p(x)$ is Descending. and for all $x\in \mathbb{R}$ $p(q(nx+m)+h(x))=n(q(p(x))+h(x))+m$ . prove that dont exist function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x\in \mathbb{R}$ $f(q(p(x))+h(x))=f(x)^{2}+1$