Olympiad problems

2005 Uzbekistan National Olympiad, 2

Solve in integer the equation $\frac{1}{2}(x+y)(y+z)(x+z)+(x+y+z)^{3}=1-xyz$

2018 IFYM, Sozopol, 2

Tags: algebra , functional equation , algebra unsolved

a) The real number $a$ and the continuous function $f : [a, \infty) \rightarrow [a, \infty)$ are such that $|f(x)-f(y)| < |x–y|$ for every two different $x, y \in [a, \infty)$. Is it always true that the equation $f(x)=x$ has only one solution in the interval $[a, \infty)$? b) The real numbers $a$ and $b$ and the continuous function $f : [a, b] \rightarrow [a, b]$ are such that $|f(x)-f(y)| < |x–y|$, for every two different $x, y \in [a, b]$. Is it always true that the equation $f(x)=x$ has only one solution in the interval $[a, b]$?

2011 China Second Round Olympiad, 3

Tags: logarithms , inequalities , algebra unsolved , algebra

Let $a,b$ be positive reals such that $\frac{1}{a}+\frac{1}{b}\leq2\sqrt2$ and $(a-b)^2=4(ab)^3$. Find $\log_a b$.

2002 Korea - Final Round, 1

Tags: algebra , polynomial , inequalities , algebra unsolved

For $n \ge 3$, let $S=a_1+a_2+\cdots+a_n$ and $T=b_1b_2\cdots b_n$ for positive real numbers $a_1,a_2,\ldots,a_n, b_1,b_2 ,\ldots,b_n$, where the numbers $b_i$ are pairwise distinct. (a) Find the number of distinct real zeroes of the polynomial \[f(x)=(x-b_1)(x-b_2)\cdots(x-b_n)\sum_{j=1}^n \frac{a_j}{x-b_j}\] (b) Prove the inequality \[\frac1{n-1}\sum_{j=1}^n \left(1-\frac{a_j}{S}\right)b_j > \left(\frac{T}{S}\sum_{j=1}^{n} \frac{a_j}{b_j}\right)^{\frac1{n-1}}\]

1980 USAMO, 2

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

Determine the maximum number of three-term arithmetic progressions which can be chosen from a sequence of $n$ real numbers \[a_1<a_2<\cdots<a_n.\]

2007 All-Russian Olympiad, 1

Tags: quadratics , algebra unsolved , algebra

Unitary quadratic trinomials $ f(x)$ and $ g(x)$ satisfy the following interesting condition: $ f(g(x)) \equal{} 0$ and $ g(f(x)) \equal{} 0$ do not have real roots. Prove that at least one of equations $ f(f(x)) \equal{} 0$ and $ g(g(x)) \equal{} 0$ does not have real roots too. [i]S. Berlov [/i]

2003 China Team Selection Test, 1

Tags: function , induction , logarithms , algebra unsolved , algebra

Find all functions $f: \mathbb{Z}^+\to \mathbb{R}$, which satisfies $f(n+1)\geq f(n)$ for all $n\geq 1$ and $f(mn)=f(m)f(n)$ for all $(m,n)=1$.

1985 IberoAmerican, 1

Tags: Columbia , algebra unsolved , algebra

If $ x\neq1$, $ y\neq1$, $ x\neq y$ and \[ \frac{yz\minus{}x^{2}}{1\minus{}x}\equal{}\frac{xz\minus{}y^{2}}{1\minus{}y}\] show that both fractions are equal to $ x\plus{}y\plus{}z$.

2004 Unirea, 1

Tags: algebra unsolved , algebra

Let $a,b,c$ be real numbers. Show that $\sqrt[3]{a} + \sqrt[3]{b} +\sqrt[3]{c} = \sqrt[3]{a+b+c}$ if and only if $ a^3 + b^3 + c^3 = (a + b + c)^3 $

2001 Korea - Final Round, 1

Tags: algebra , polynomial , algebra unsolved

For given positive integers $n$ and $N$, let $P_n$ be the set of all polynomials $f(x)=a_0+a_1x+\cdots+a_nx^n$ with integer coefficients such that: [list] (a) $|a_j| \le N$ for $j = 0,1, \cdots ,n$; (b) The set $\{ j \mid a_j = N\}$ has at most two elements. [/list] Find the number of elements of the set $\{f(2N) \mid f(x) \in P_n\}$.

2014 Romania Team Selection Test, 2

Tags: function , algebra unsolved , algebra

Let $a$ be a real number in the open interval $(0,1)$. Let $n\geq 2$ be a positive integer and let $f_n\colon\mathbb{R}\to\mathbb{R}$ be defined by $f_n(x) = x+\frac{x^2}{n}$. Show that \[\frac{a(1-a)n^2+2a^2n+a^3}{(1-a)^2n^2+a(2-a)n+a^2}<(f_n \circ\ \cdots\ \circ f_n)(a)<\frac{an+a^2}{(1-a)n+a}\] where there are $n$ functions in the composition.

2007 Moldova National Olympiad, 11.1

Tags: algebra unsolved , algebra

Define the sequence $(x_{n})$: $x_{1}=\frac{1}{3}$ and $x_{n+1}=x_{n}^{2}+x_{n}$. Find $\left[\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\dots+\frac{1}{x_{2007}+1}\right]$, wehere $[$ $]$ denotes the integer part.

2013 Indonesia MO, 5

Tags: algebra , polynomial , quadratics , algebra unsolved

Let $P$ be a quadratic (polynomial of degree two) with a positive leading coefficient and negative discriminant. Prove that there exists three quadratics $P_1, P_2, P_3$ such that: - $P(x) = P_1(x) + P_2(x) + P_3(x)$ - $P_1, P_2, P_3$ have positive leading coefficients and zero discriminants (and hence each has a double root) - The roots of $P_1, P_2, P_3$ are different

2014 European Mathematical Cup, 1

Tags: algebra unsolved , algebra

Which of the following claims are true, and which of them are false? If a fact is true you should prove it, if it isn't, find a counterexample. a) Let $a,b,c$ be real numbers such that $ a^{2013} + b^{2013} + c^{2013} = 0 $. Then $ a^{2014} + b^{2014} + c^{2014} = 0 $. b) Let $a,b,c$ be real numbers such that $ a^{2014} + b^{2014} + c^{2014} = 0 $. Then $ a^{2015} + b^{2015} + c^{2015} = 0 $. c) Let $a,b,c$ be real numbers such that $ a^{2013} + b^{2013} + c^{2013} = 0 $ and $ a^{2015} + b^{2015} + c^{2015} = 0 $. Then $ a^{2014} + b^{2014} + c^{2014} = 0 $. [i]Proposed by Matko Ljulj[/i]

2011 Morocco National Olympiad, 3

Tags: function , search , algebra unsolved , algebra

Find all functions $f : \mathbb{R} \to \mathbb{R} $ which verify the relation \[(x-2)f(y)+f(y+2f(x))= f(x+yf(x)), \qquad \forall x,y \in \mathbb R.\]

1989 China Team Selection Test, 1

Tags: function , algebra unsolved , algebra

Let $\mathbb{N} = \{1,2, \ldots\}.$ Does there exists a function $f: \mathbb{N} \mapsto \mathbb{N}$ such that $\forall n \in \mathbb{N},$ $f^{1989}(n) = 2 \cdot n$ ?

2005 Federal Competition For Advanced Students, Part 1, 3

Tags: algebra unsolved , algebra

For 3 real numbers $a,b,c$ let $s_n=a^{n}+b^{n}+c^{n}$. It is known that $s_1=2$, $s_2=6$ and $s_3=14$. Prove that for all natural numbers $n>1$, we have $|s^2_n-s_{n-1}s_{n+1}|=8$

2009 Ukraine National Mathematical Olympiad, 1

Tags: algebra , system of equations , algebra unsolved

Find all possible real values of $a$ for which the system of equations \[\{\begin{array}{cc}x +y +z=0\\\text{ } \\ xy+yz+azx=0\end{array}\] has exactly one solution.

2005 All-Russian Olympiad, 1

Tags: function , inequalities , algebra unsolved , algebra

Do there exist a bounded function $f: \mathbb{R}\to\mathbb{R}$ such that $f(1)>0$ and $f(x)$ satisfies an inequality $f^2(x+y)\ge f^2(x)+2f(xy)+f^2(y)$?

2004 India National Olympiad, 5

Tags: quadratics , number theory , algebra unsolved , algebra

S is the set of all ($a$, $b$, $c$, $d$, $e$, $f$) where $a$, $b$, $c$, $d$, $e$, $f$ are integers such that $a^2 + b^2 + c^2 + d^2 + e^2 = f^2$. Find the largest $k$ which divides abcdef for all members of $S$.

2011 Morocco National Olympiad, 3

Tags: linear algebra , matrix , algebra unsolved , algebra

Solve in $\mathbb{R}^{3}$ the following system \[\left\{\begin{matrix} \sqrt{x^{2}-y}=z-1\\ \sqrt{y^{2}-z}=x-1\\ \sqrt{z^{2}-x}=y-1 \end{matrix}\right.\]

2012 Poland - Second Round, 1

Tags: algebra unsolved , algebra

$a,b,c,d\in\mathbb{R}$, solve the system of equations: \[ \begin{cases} a^3+b=c \\ b^3+c=d \\ c^3+d=a \\ d^3+a=b \end{cases} \]

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.

2006 China Western Mathematical Olympiad, 3

Tags: trigonometry , algebra , polynomial , algebra unsolved

Let $k$ be a positive integer not less than 3 and $x$ a real number. Prove that if $\cos (k-1)x$ and $\cos kx$ are rational, then there exists a positive integer $n>k$, such that both $\cos (n-1)x$ and $\cos nx$ are rational.

1994 Vietnam Team Selection Test, 3

Tags: algebra , polynomial , algebra unsolved

Let $P(x)$ be given a polynomial of degree 4, having 4 positive roots. Prove that the equation \[(1-4 \cdot x) \cdot \frac{P(x)}{x^2} + (x^2 + 4 \cdot x - 1) \cdot \frac{P'(x)}{x^2} - P''(x) = 0\] has also 4 positive roots.