Olympiad problems

2015 India National Olympiad, 3

Find all real functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x^2+yf(x))=xf(x+y)$.

1978 IMO Longlists, 35

Tags: function , algebra unsolved , algebra

A sequence $(a_n)_0^N$ of real numbers is called concave if $2a_n\ge a_{n-1} + a_{n+1}$ for all integers $n, 1 \le n \le N - 1$. $(a)$ Prove that there exists a constant $C >0$ such that \[\left(\displaystyle\sum_{n=0}^{N}a_n\right)^2\ge C(N - 1)\displaystyle\sum_{n=0}^{N}a_n^2\:\:\:\:\:(1)\] for all concave positive sequences $(a_n)^N_0$ $(b)$ Prove that $(1)$ holds with $C = \frac{3}{4}$ and that this constant is best possible.

2010 Postal Coaching, 3

Tags: algebra unsolved , algebra

Determine the smallest odd integer $n \ge 3$, for which there exist $n$ rational numbers $x_1 , x_2 , . . . , x_n$ with the following properties: $(a)$ \[\sum_{i=1}^{n} x_i =0 , \ \sum_{i=1}^{n} x_i^2 = 1.\] $(b)$ \[x_i \cdot x_j \ge - \frac 1n \ \forall \ 1 \le i,j \le n.\]

1985 USAMO, 2

Tags: AMC , USA(J)MO , USAMO , quadratics , algebra unsolved , algebra , biquadratic

Determine each real root of \[x^4-(2\cdot10^{10}+1)x^2-x+10^{20}+10^{10}-1=0\] correct to four decimal places.

2003 All-Russian Olympiad, 1

Tags: trigonometry , calculus , integration , function , algebra unsolved , algebra

Let $\alpha , \beta , \gamma , \delta$ be positive numbers such that for all $x$, $\sin{\alpha x}+\sin {\beta x}=\sin {\gamma x}+\sin {\delta x}$. Prove that $\alpha =\gamma$ or $\alpha=\delta$.

2006 Czech-Polish-Slovak Match, 3

Tags: algebra unsolved , algebra

The sum of four real numbers is $9$ and the sum of their squares is $21$. Prove that these numbers can be denoted by $a, b, c, d$ so that $ab-cd \ge 2$ holds.

1978 IMO Longlists, 12

Tags: algebra unsolved , algebra

The equation $x^3 + ax^2 + bx + c = 0$ has three (not necessarily distinct) real roots $t, u, v$. For which $a, b, c$ do the numbers $t^3, u^3, v^3$ satisfy the equation $x^3 + a^3x^2 + b^3x + c^3 = 0$?

2011 Czech-Polish-Slovak Match, 1

Tags: algebra , polynomial , algebra unsolved

A polynomial $P(x)$ with integer coefficients satisfies the following: if $F(x)$, $G(x)$, and $Q(x)$ are polynomials with integer coefficients satisfying $P\Big(Q(x)\Big)=F(x)\cdot G(x)$, then $F(x)$ or $G(x)$ is a constant polynomial. Prove that $P(x)$ is a constant polynomial.

1995 Turkey Team Selection Test, 3

Tags: limit , algebra unsolved , algebra

The sequence $\{x_n\}$ of real numbers is defined by \[x_1=1 \quad\text{and}\quad x_{n+1}=x_n+\sqrt[3]{x_n} \quad\text{for}\quad n\geq 1.\] Show that there exist real numbers $a, b$ such that $\lim_{n \rightarrow \infty}\frac{x_n}{an^b} = 1$.

1980 Vietnam National Olympiad, 2

Tags: algebra , polynomial , calculus , integration , algebra unsolved

Can the equation $x^3-2x^2-2x+m = 0$ have three different rational roots?

2012 Indonesia TST, 1

Tags: algebra , polynomial , trigonometry , algebra unsolved

Suppose $P(x,y)$ is a homogenous non-constant polynomial with real coefficients such that $P(\sin t, \cos t) = 1$ for all real $t$. Prove that $P(x,y) = (x^2+y^2)^k$ for some positive integer $k$. (A polynomial $A(x,y)$ with real coefficients and having a degree of $n$ is homogenous if it is the sum of $a_ix^iy^{n-i}$ for some real number $a_i$, for all integer $0 \le i \le n$.)

2011 German National Olympiad, 1

Tags: inequalities , trigonometry , algebra unsolved , algebra , sine double angle

Prove for each non-negative integer $n$ and real number $x$ the inequality \[ \sin{x} \cdot(n \sin{x}-\sin{nx}) \geq 0 \]

2008 Junior Balkan Team Selection Tests - Moldova, 6

Tags: algebra unsolved , algebra

Solve the equation $ 2(x^2\minus{}3x\plus{}2)\equal{}3 \sqrt{x^3\plus{}8}$, where $ x\in R$

2010 Indonesia MO, 1

Tags: algebra unsolved , algebra

Let $a,b,c$ be three different positive integers. Show that the sequence \[a+b+c,ab+bc+ca,3abc\] could be neither an arithmetic nor geometric progression. [i]Fajar Yuliawan, Bandung[/i]

2004 South East Mathematical Olympiad, 3

Tags: logarithms , algebra , polynomial , inequalities , algebra unsolved

(1) Determine if there exists an infinite sequence $\{a_n\}$ with positive integer terms, such that $a^2_{n+1}\ge 2a_na_{n+2}$ for any positive integer $n$. (2) Determine if there exists an infinite sequence $\{a_n\}$ with positive irrational terms, such that $a^2_{n+1}\ge 2a_na_{n+2}$ for any positive integer $n$.

2012 ISI Entrance Examination, 8

Tags: function , invariant , algebra unsolved , algebra , combinatorics

Let $S = \{1,2,3,\ldots,n\}$. Consider a function $f\colon S\to S$. A subset $D$ of $S$ is said to be invariant if for all $x\in D$ we have $f(x)\in D$. The empty set and $S$ are also considered as invariant subsets. By $\deg (f)$ we define the number of invariant subsets $D$ of $S$ for the function $f$. [b]i)[/b] Show that there exists a function $f\colon S\to S$ such that $\deg (f)=2$. [b]ii)[/b] Show that for every $1\leq k\leq n$ there exists a function $f\colon S\to S$ such that $\deg (f)=2^{k}$.

2010 Moldova National Olympiad, 12.5

Tags: algebra unsolved , algebra

Prove that exists a infinity of triplets $a, b, c\in\mathbb{R}$ satisfying simultaneously the relations $a+b+c=0$ and $a^4+b^4+c^4=50$. Moldova National Math Olympiad 2010, 12th grade

2004 Poland - First Round, 1

Tags: algebra unsolved , algebra

1. Solve in real numbers x,y,z : $\{\begin{array}{ccc} x^2=yz+1 \\ y^2=zx+2 \\ z^2=xy+4 \\ \end{array}$

2005 IberoAmerican, 1

Tags: quadratics , algebra unsolved , algebra

Determine all triples of real numbers $(a,b,c)$ such that \begin{eqnarray*} xyz &=& 8 \\ x^2y + y^2z + z^2x &=& 73 \\ x(y-z)^2 + y(z-x)^2 + z(x-y)^2 &=& 98 . \end{eqnarray*}

1991 Vietnam National Olympiad, 1

Tags: function , algebra unsolved , algebra

Find all functions $f: \mathbb{R}\to\mathbb{R}$ satisfying: $\frac{f(xy)+f(xz)}{2} - f(x)f(yz) \geq \frac{1}{4}$ for all $x,y,z \in \mathbb{R}$

1988 IMO Longlists, 44

Tags: trigonometry , algebra unsolved , algebra

Let $-1 < x < 1.$ Show that \[ \sum^{6}_{k=0} \frac{1 - x^2}{1 - 2 \cdot x \cdot \cos \left( \frac{2 \cdot \pi \cdot k }{7} \right) + x^2} = \frac{7 \cdot \left( 1 + x^7 \right)}{\left( 1 - x^7 \right)}. \] Deduce that \[ \csc^2\left( x + \frac{\pi}{7} \right) + \csc^2\left(2 \cdot x + \frac{\pi}{7} \right) + \csc^2\left(3 \cdot x + \frac{\pi}{7} \right) = 8. \]

1982 IMO Longlists, 48

Tags: inequalities , induction , complex numbers , combinatorial geometry , algebra unsolved , algebra

Given a finite sequence of complex numbers $c_1, c_2, \ldots , c_n$, show that there exists an integer $k$ ($1 \leq k \leq n$) such that for every finite sequence $a_1, a_2, \ldots, a_n$ of real numbers with $1 \geq a_1 \geq a_2 \geq \cdots \geq a_n \geq 0$, the following inequality holds: \[\left| \sum_{m=1}^n a_mc_m \right| \leq \left| \sum_{m=1}^k c_m \right|.\]

1983 IMO Longlists, 46

Tags: function , limit , logarithms , algebra unsolved , algebra

Let $f$ be a real-valued function defined on $I = (0,+\infty)$ and having no zeros on $I$. Suppose that \[\lim_{x \to +\infty} \frac{f'(x)}{f(x)}=+\infty.\] For the sequence $u_n = \ln \left| \frac{f(n+1)}{f(n)} \right|$, prove that $u_n \to +\infty$ as $n \to +\infty.$

2014 Saudi Arabia BMO TST, 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 $n$.

2009 Hong Kong TST, 5

Tags: inequalities , quadratics , function , algebra unsolved , algebra

Let $ a,b,c$ be the three sides of a triangle. Determine all possible values of $ \frac {a^2 \plus{} b^2 \plus{} c^2}{ab \plus{} bc \plus{} ca}$