Olympiad problems

2013 Romania National Olympiad, 3

Tags: algebra , function

Find all injective functions$f:\mathbb{Z}\to \mathbb{Z}$ that satisfy: $\left| f\left( x \right)-f\left( y \right) \right|\le \left| x-y \right|$ ,for any $x,y\in \mathbb{Z}$.

2019 Baltic Way, 1

Tags: inequality , algebra

For all non-negative real numbers $x,y,z$ with $x \geq y$, prove the inequality $$\frac{x^3-y^3+z^3+1}{6}\geq (x-y)\sqrt{xyz}.$$

2015 BMT Spring, 18

Tags: algebra

Evaluate $\sum_{n=1}^{\infty}\frac{1}{(2n - 1)(3n - 1)}$.

2014 All-Russian Olympiad, 3

Tags: algebra , polynomial , derivative , calculus , function

If the polynomials $f(x)$ and $g(x)$ are written on a blackboard then we can also write down the polynomials $f(x)\pm g(x)$, $f(x)g(x)$, $f(g(x))$ and $cf(x)$, where $c$ is an arbitrary real constant. The polynomials $x^3-3x^2+5$ and $x^2-4x$ are written on the blackboard. Can we write a nonzero polynomial of form $x^n-1$ after a finite number of steps?

2023 IFYM, Sozopol, 5

Tags: algebra

Let $n \geq 4$ be a natural number. The polynomials $x^{n+1} + x$, $x^n$, and $x^{n-3}$ are written on the board. In one move, you can choose two polynomials $f(x)$ and $g(x)$ (not necessarily distinct) and add the polynomials $f(x)g(x)$, $f(x) + g(x)$, and $f(x) - g(x)$ to the board. Find all $n$ such that after a finite number of operations, the polynomial $x$ can be written on the board.

2017 USA TSTST, 6

Tags: number theory , fibonacci , algebra

A sequence of positive integers $(a_n)_{n \ge 1}$ is of [i]Fibonacci type[/i] if it satisfies the recursive relation $a_{n + 2} = a_{n + 1} + a_n$ for all $n \ge 1$. Is it possible to partition the set of positive integers into an infinite number of Fibonacci type sequences? [i]Proposed by Ivan Borsenco[/i]

2022 Romania National Olympiad, P4

Tags: functional equation , algebra , combinatorics

Let $X$ be a set with $n\ge 2$ elements. Define $\mathcal{P}(X)$ to be the set of all subsets of $X$. Find the number of functions $f:\mathcal{P}(X)\mapsto \mathcal{P}(X)$ such that $$|f(A)\cap f(B)|=|A\cap B|$$ whenever $A$ and $B$ are two distinct subsets of $X$. [i] (Sergiu Novac)[/i]

2024 China Team Selection Test, 15

Tags: polynomial , algebra

$n>1$ is an integer. Let real number $x>1$ satisfy $$x^{101}-nx^{100}+nx-1=0.$$ Prove that for any real $0<a<b<1$, there exists a positive integer $m$ so that $a<\{x^m\}<b.$ [i]Proposed by Chenjie Yu[/i]

1995 Canada National Olympiad, 1

Tags: algebra

Let $f(x)=\frac{9^x}{9^x + 3}$. Evaluate $\sum_{i=1}^{1995}{f\left(\frac{i}{1996}\right)}$.

2014 Brazil Team Selection Test, 4

Tags: sequence , algebra , inequalities

Let $n$ be a positive integer, and consider a sequence $a_1 , a_2 , \dotsc , a_n $ of positive integers. Extend it periodically to an infinite sequence $a_1 , a_2 , \dotsc $ by defining $a_{n+i} = a_i $ for all $i \ge 1$. If \[a_1 \le a_2 \le \dots \le a_n \le a_1 +n \] and \[a_{a_i } \le n+i-1 \quad\text{for}\quad i=1,2,\dotsc, n, \] prove that \[a_1 + \dots +a_n \le n^2. \]

2006 Belarusian National Olympiad, 1

Tags: algebra , vector , inequalities

Let $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ be unit vectors in $R^3$. Prove that $$\sqrt{1-\overrightarrow{a}\cdot\overrightarrow{b}}\le \sqrt{1-\overrightarrow{a}\cdot\overrightarrow{c}}+\sqrt{1-\overrightarrow{c}\cdot\overrightarrow{b}}$$ (A.Mirotin)

2003 Silk Road, 4

Tags: algebra

Find $ \sum_{k \in A} \frac{1}{k-1}$ where $A= \{ m^n : m,n \in \mathbb{Z} m,n \geq 2 \} $. Problem was post earlier [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=67&t=29456&hilit=silk+road]here[/url] , but solution not gives and olympiad doesn't indicate, so I post it again :blush: Official solution [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=125&t=365714&p=2011659#p2011659]here[/url]

2025 Turkey Team Selection Test, 3

Tags: functional equation , algebra

Find all $f: \mathbb{R} \rightarrow \mathbb{R}$ such that, for all $x,y \in \mathbb{R}-\{0\}$, $$ f(x) \neq 0 \text{ and } \frac{f(x)}{f(y)} + \frac{f(y)}{f(x)} - f \left( \frac{x}{y}-\frac{y}{x} \right) =2 $$

2013 Turkmenistan National Math Olympiad, 2

Tags: algebra , induction

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.

1983 Spain Mathematical Olympiad, 8

Tags: algebra , system of equations , number theory

In $1960$, the oldest of three brothers has an age that is the sum of the of his younger siblings. A few years later, the sum of the ages of two of brothers is double that of the other. A number of years have now passed since $1960$, which is equal to two thirds of the sum of the ages that the three brothers were at that year, and one of them has reached $21$ years. What is the age of each of the others two?

2010 Contests, 1

Tags: system of equations , algebra

Find all quadruples of real numbers $(a,b,c,d)$ satisfying the system of equations \[\begin{cases}(b+c+d)^{2010}=3a\\ (a+c+d)^{2010}=3b\\ (a+b+d)^{2010}=3c\\ (a+b+c)^{2010}=3d\end{cases}\]

2014 Saudi Arabia Pre-TST, 1.1

Tags: inequalities , algebra

Let $a_1, a_2,...,a_{2n}$ be positive real numbers such that $a_i + a_{n+i} = 1$, for all $i = 1,...,n$. Prove that there exist two different integers $1 \le j, k \le 2n$ for which $$\sqrt{a^2_j-a^2_k} < \frac{1}{\sqrt{n} +\sqrt{n - 1}}$$

2003 AIME Problems, 15

Tags: algebra , polynomial , function

Let \[P(x)=24x^{24}+\sum_{j=1}^{23}(24-j)(x^{24-j}+x^{24+j}). \] Let $z_{1},z_{2},\ldots,z_{r}$ be the distinct zeros of $P(x),$ and let $z_{k}^{2}=a_{k}+b_{k}i$ for $k=1,2,\ldots,r,$ where $i=\sqrt{-1},$ and $a_{k}$ and $b_{k}$ are real numbers. Let \[\sum_{k=1}^{r}|b_{k}|=m+n\sqrt{p}, \] where $m,$ $n,$ and $p$ are integers and $p$ is not divisible by the square of any prime. Find $m+n+p.$

2017 Iran Team Selection Test, 1

Tags: algebra , inequalities

Let $a,b,c,d$ be positive real numbers with $a+b+c+d=2$. Prove the following inequality: $$\frac{(a+c)^{2}}{ad+bc}+\frac{(b+d)^{2}}{ac+bd}+4\geq 4\left ( \frac{a+b+1}{c+d+1}+\frac{c+d+1}{a+b+1} \right).$$ [i]Proposed by Mohammad Jafari[/i]

2003 Moldova Team Selection Test, 1

Tags: algebra , vieta , polynomial

Let $ n>0$ be a natural number. Determine all the polynomials of degree $ 2n$ with real coefficients in the form $ P(X)\equal{}X^{2n}\plus{}(2n\minus{}10)X^{2n\minus{}1}\plus{}a_2X^{2n\minus{}2}\plus{}...\plus{}a_{2n\minus{}2}X^2\plus{}(2n\minus{}10)X\plus{}1$, if it is known that all the roots of them are positive reals. [i]Proposer[/i]: [b]Baltag Valeriu[/b]

VI Soros Olympiad 1999 - 2000 (Russia), 11.6

Tags: integer polynomial , polynomial , algebra

Let $P(x)$ be a polynomial with integer coefficients. It is known that the number $\sqrt2+\sqrt3$ is its root. Prove that the number $\sqrt2-\sqrt3$ is also its root.

2016 Regional Olympiad of Mexico West, 1

Tags: algebra

Indra has a bag for bringing flowers for her grandmother. The first day she brings $n$ flowers. From the second day Indra tries to bring three times plus one with respect to the number of flowers of the previous day. However, if this number is greater or equal to $40$, Indra substracts multiples of $40$ until the remainder is less than this number, since her bag cannot containt so many flowers. For which value of $n$ Indra will bring $30$ flowers the day $2016$?

2015 Turkmenistan National Math Olympiad, 4

Tags: inequality , algebra , inequalities

Find the max and minimum without using dervivate: $\sqrt{x} +4 \cdot \sqrt{\frac{1}{2} - x}$

2015 Thailand TSTST, 2

Tags: algebra , functional equation

Let $\mathbb{N} = \{1, 2, 3, \dots\}$ and let $f : \mathbb{N}\to\mathbb{R}$. Prove that there is an infinite subset $A$ of $\mathbb{N}$ such that $f$ is increasing on $A$ or $f$ is decreasing on $A$.

2021 IMO Shortlist, A7

Tags: algebra , inequality , inequalities

Let $n\geqslant 1$ be an integer, and let $x_0,x_1,\ldots,x_{n+1}$ be $n+2$ non-negative real numbers that satisfy $x_ix_{i+1}-x_{i-1}^2\geqslant 1$ for all $i=1,2,\ldots,n.$ Show that \[x_0+x_1+\cdots+x_n+x_{n+1}>\bigg(\frac{2n}{3}\bigg)^{3/2}.\][i]Pakawut Jiradilok and Wijit Yangjit, Thailand[/i]