Olympiad problems

1993 Irish Math Olympiad, 4

Let $ f(x)\equal{}x^n\plus{}a_{n\minus{}1} x^{n\minus{}1}\plus{}...\plus{}a_0$ $ (n \ge 1)$ be a polynomial with real coefficients such that $ |f(0)|\equal{}f(1)$ and each root $ \alpha$ of $ f$ is real and lies in the interval $ [0,1]$. Prove that the product of the roots does not exceed $ \frac{1}{2^n}$.

2021 Baltic Way, 5

Tags: algebra , algebra proposed , system of equations

Let $x,y\in\mathbb{R}$ be such that $x = y(3-y)^2$ and $y = x(3-x)^2$. Find all possible values of $x+y$.

1987 Iran MO (2nd round), 1

Tags: trigonometry , algebra proposed , algebra

Calculate the product: \[A=\sin 1^\circ \times \sin 2^\circ \times \sin 3^\circ \times \cdots \times \sin 89^\circ\]

2016 Bulgaria JBMO TST, 1

Tags: algebra proposed , algebra

$ a,b,c,d,e,f $ are real numbers. It is true that: $ a+b+c+d+e+f=20 $ $ (a-2)^2+(b-2)^2+...+(f-2)^2=24 $ Find the maximum value of d.

2004 Bulgaria Team Selection Test, 3

Tags: algebra proposed , algebra , number theory

Prove that among any $2n+1$ irrational numbers there are $n+1$ numbers such that the sum of any $k$ of them is irrational, for all $k \in \{1,2,3,\ldots, n+1 \}$.

2008 China Team Selection Test, 2

Tags: algebra , polynomial , modular arithmetic , absolute value , algebra proposed

Prove that for all $ n\geq 2,$ there exists $ n$-degree polynomial $ f(x) \equal{} x^n \plus{} a_{1}x^{n \minus{} 1} \plus{} \cdots \plus{} a_{n}$ such that (1) $ a_{1},a_{2},\cdots, a_{n}$ all are unequal to $ 0$; (2) $ f(x)$ can't be factorized into the product of two polynomials having integer coefficients and positive degrees; (3) for any integers $ x, |f(x)|$ isn't prime numbers.

2010 Baltic Way, 4

Tags: algebra , polynomial , algebra proposed

Find all polynomials $P(x)$ with real coefficients such that \[(x-2010)P(x+67)=xP(x) \] for every integer $x$.

2012 Junior Balkan Team Selection Tests - Moldova, 4

Tags: linear algebra , matrix , algebra proposed , algebra

How many solutions does the system have: $ \{\begin{matrix}&(3x+2y) *(\frac{3}{x}+\frac{1}{y})=2\\ & x^2+y^2\leq 2012\\ \end{matrix} $ where $ x,y $ are non-zero integers

2024 Baltic Way, 5

Tags: inequalities , inequalities proposed , algebra , algebra proposed

Find all positive real numbers $\lambda$ such that every sequence $a_1, a_2, \ldots$ of positive real numbers satisfying \[ a_{n+1}=\lambda\cdot\frac{a_1+a_2+\ldots+a_n}{n} \] for all $n\geq 2024^{2024}$ is bounded. [i]Remark:[/i] A sequence $a_1,a_2,\ldots$ of positive real numbers is \emph{bounded} if there exists a real number $M$ such that $a_i<M$ for all $i=1,2,\ldots$

2010 Spain Mathematical Olympiad, 2

Tags: function , floor function , induction , algebra proposed , algebra

Let $\mathbb{N}_0$ and $\mathbb{Z}$ be the set of all non-negative integers and the set of all integers, respectively. Let $f:\mathbb{N}_0\rightarrow\mathbb{Z}$ be a function defined as \[f(n)=-f\left(\left\lfloor\frac{n}{3}\right\rfloor \right)-3\left\{\frac{n}{3}\right\} \] where $\lfloor x \rfloor$ is the greatest integer smaller than or equal to $x$ and $\{ x\}=x-\lfloor x \rfloor$. Find the smallest integer $n$ such that $f(n)=2010$.

2009 Indonesia TST, 2

Tags: parameterization , logarithms , inequalities , algebra proposed , algebra

Find the value of real parameter $ a$ such that $ 2$ is the smallest integer solution of \[ \frac{x\plus{}\log_2 (2^x\minus{}3a)}{1\plus{}\log_2 a} >2.\]

2010 Junior Balkan MO, 1

Tags: algebra , polynomial , function , algebra proposed

The real numbers $a$, $b$, $c$, $d$ satisfy simultaneously the equations \[abc -d = 1, \ \ \ bcd - a = 2, \ \ \ cda- b = 3, \ \ \ dab - c = -6.\] Prove that $a + b + c + d \not = 0$.

1998 All-Russian Olympiad, 6

Tags: algebra proposed , algebra

A binary operation $*$ on real numbers has the property that $(a * b) * c = a+b+c$ for all $a$, $b$, $c$. Prove that $a * b = a+b$.

2001 Romania National Olympiad, 1

Tags: algebra proposed , algebra

Let $A$ be a set of real numbers which verifies: \[ a)\ 1 \in A \\ b) \ x\in A\implies x^2\in A\\ c)\ x^2-4x+4\in A\implies x\in A \] Show that $2000+\sqrt{2001}\in A$.

2006 Switzerland Team Selection Test, 2

Tags: geometry , algebra proposed , algebra

Let $n\ge5$ be an integer. Find the biggest integer $k$ such that there always exists a $n$-gon with exactly $k$ interior right angles. (Find $k$ in terms of $n$).

2013 India IMO Training Camp, 3

Tags: geometry , geometric transformation , algebra , polynomial , algebra proposed

For a positive integer $n$, a cubic polynomial $p(x)$ is said to be [i]$n$-good[/i] if there exist $n$ distinct integers $a_1, a_2, \ldots, a_n$ such that all the roots of the polynomial $p(x) + a_i = 0$ are integers for $1 \le i \le n$. Given a positive integer $n$ prove that there exists an $n$-good cubic polynomial.

2010 Romanian Master of Mathematics, 2

Tags: function , inequalities , Cauchy Inequality , absolute value , algebra proposed , algebra

For each positive integer $n$, find the largest real number $C_n$ with the following property. Given any $n$ real-valued functions $f_1(x), f_2(x), \cdots, f_n(x)$ defined on the closed interval $0 \le x \le 1$, one can find numbers $x_1, x_2, \cdots x_n$, such that $0 \le x_i \le 1$ satisfying \[|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n\] [i]Marko Radovanović, Serbia[/i]

1998 Turkey Team Selection Test, 2

Tags: induction , trigonometry , algebra proposed , algebra

Let the sequence $(a_{n})$ be deﬁned by $a_{1} = t$ and $a_{n+1} = 4a_{n}(1 - a_{n})$ for $n \geq 1$. How many possible values of t are there, if $a_{1998} = 0$?

2005 Georgia Team Selection Test, 9

Tags: induction , algebra proposed , algebra

Let $ a_{0},a_{1},\ldots,a_{n}$ be integers, one of which is nonzero, and all of the numbers are not less than $ \minus{} 1$. Prove that if \[ a_{0} \plus{} 2a_{1} \plus{} 2^{2}a_{2} \plus{} \cdots \plus{} 2^{n}a_{n} \equal{} 0,\] then $ a_{0} \plus{} a_{1} \plus{} \cdots \plus{} a_{n} > 0$.

2024 German National Olympiad, 6

Tags: algebra proposed , algebra , inequalities , inequalities proposed , Mean Inequality Chain

Decide whether there exists a largest positive integer $n$ such that the inequality \[\frac{\frac{a^2}{b}+\frac{b^2}{a}}{2} \ge \sqrt[n]{\frac{a^n+b^n}{2}}\] holds for all positive real numbers $a$ and $b$. If such a largest positive integer $n$ exists, determine it.

2010 Iran MO (3rd Round), 1

Tags: algebra , polynomial , limit , algebra proposed

suppose that polynomial $p(x)=x^{2010}\pm x^{2009}\pm...\pm x\pm 1$ does not have a real root. what is the maximum number of coefficients to be $-1$?(14 points)

2023 German National Olympiad, 1

Tags: number theory , Diophantine equation , number theory proposed , algebra proposed

Determine all pairs $(m,n)$ of integers with $n \ge m$ satisfying the equation \[n^3+m^3-nm(n+m)=2023.\]

2010 Iran MO (3rd Round), 3

Tags: group theory , abstract algebra , algebra proposed , algebra

suppose that $G<S_n$ is a subgroup of permutations of $\{1,...,n\}$ with this property that for every $e\neq g\in G$ there exist exactly one $k\in \{1,...,n\}$ such that $g.k=k$. prove that there exist one $k\in \{1,...,n\}$ such that for every $g\in G$ we have $g.k=k$.(20 points)

2007 Moldova National Olympiad, 12.5

Tags: algebra , polynomial , algebra proposed

Find all polynomials $P\in \mathbb C[X]$ such that \[P(X^{2})=P(X)^{2}+2P(X)\]

2015 Iran Team Selection Test, 1

Tags: algebra , polynomial , algebra proposed

Find all polynomials $P,Q\in \Bbb{Q}\left [ x \right ]$ such that $$P(x)^3+Q(x)^3=x^{12}+1.$$