Olympiad problems

2017 Junior Regional Olympiad - FBH, 1

It is given function $f(x)=3x-2$ $a)$ Find $g(x)$ if $f(2x-g(x))=-3(1+2m)x+34$ $b)$ Solve the equation: $g(x)=4(m-1)x-4(m+1)$, $m \in \mathbb{R}$

V Soros Olympiad 1998 - 99 (Russia), 11.7

Tags: inequalities , algebra , trigonometry

Prove that for all positive and admissible values of $x$ the following inequality holds: $$\sin x + arc \sin x>2x$$

1970 Swedish Mathematical Competition, 3

Tags: algebra , polynomial , integer polynomial

A polynomial with integer coefficients takes the value $5$ at five distinct integers. Show that it does not take the value $9$ at any integer.

2015 IMO Shortlist, A1

Tags: algebra

Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.

2016 CMIMC, 2

Tags: algebra

Determine the value of the sum \[\left|\sum_{1\leq i<j\leq 50}ij(-1)^{i+j}\right|.\]

2021 All-Russian Olympiad, 6

Tags: algebra , polynomial

Given is a polynomial $P(x)$ of degree $n>1$ with real coefficients. The equation $P(P(P(x)))=P(x)$ has $n^3$ distinct real roots. Prove that these roots could be split into two groups with equal arithmetic mean.

2006 Hanoi Open Mathematics Competitions, 9

Tags: minimum , inequalities , algebra

What is the smallest possible value of $x^2 + y^2 - x -y - xy$?

2016 China Northern MO, 5

Tags: algebra

Let $\theta_{i}\in(0,\frac{\pi}{2})(i=1,2,\cdots,n)$. Prove: $$(\sum_{i=1}^n\tan\theta_{i})(\sum_{i=1}^n\cot\theta_{i})\geq(\sum_{i=1}^n\sin\theta_{i})^2+(\sum_{i=1}^n\cos\theta_{i})^2.$$

2015 Switzerland - Final Round, 3

Tags: function , algebra , functional equation

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$, such that for arbitrary $x,y \in \mathbb{R}$: \[ (y+1)f(x)+f(xf(y)+f(x+y))=y.\]

1981 Romania Team Selection Tests, 3.

Tags: binomial coefficient , algebra , inequalities

Let $n>r\geqslant 3$ be two integers and $d$ be a positive integer such that $nd\geqslant \dbinom{n+r}{r+1}$. Show that \[(n-t)(d-t)>\dbinom{n-t+r}{r+1},\] for $t=1,2,\ldots,n-1$ [i]Vasile Brânzănescu[/i]

Kvant 2021, M2671

Tags: algebra , number theory

Let $x_1$ and $x_2$ be the roots of the equation $x^2-px+1=0$ where $p>2$ is a prime number. Prove that $x_1^p+x_2^p$ is an integer divisible by $p^2$. [i]From the folklore[/i]

2012 Bosnia Herzegovina Team Selection Test, 4

Tags: function , induction , algebra , polynomial , modular arithmetic , number theory

Define a function $f:\mathbb{N}\rightarrow\mathbb{N}$, \[f(1)=p+1,\] \[f(n+1)=f(1)\cdot f(2)\cdots f(n)+p,\] where $p$ is a prime number. Find all $p$ such that there exists a natural number $k$ such that $f(k)$ is a perfect square.

2017 European Mathematical Cup, 4

Tags: algebra , polynomial

Find all polynomials $P$ with integer coefficients such that $P (0)\ne 0$ and $$P^n(m)\cdot P^m(n)$$ is a square of an integer for all nonnegative integers $n, m$. [i]Remark:[/i] For a nonnegative integer $k$ and an integer $n$, $P^k(n)$ is defined as follows: $P^k(n) = n$ if $k = 0$ and $P^k(n)=P(P(^{k-1}(n))$ if $k >0$. Proposed by Adrian Beker.

2007 USA Team Selection Test, 2

Tags: inequalities , function , algebra , induction

Let $n$ be a positive integer and let $a_1 \le a_2 \le \dots \le a_n$ and $b_1 \le b_2 \le \dots \le b_n$ be two nondecreasing sequences of real numbers such that \[ a_1 + \dots + a_i \le b_1 + \dots + b_i \text{ for every } i = 1, \dots, n \] and \[ a_1 + \dots + a_n = b_1 + \dots + b_n. \] Suppose that for every real number $m$, the number of pairs $(i,j)$ with $a_i-a_j=m$ equals the numbers of pairs $(k,\ell)$ with $b_k-b_\ell = m$. Prove that $a_i = b_i$ for $i=1,\dots,n$.

2000 Austria Beginners' Competition, 2

Tags: inequalities , algebra

Let $a,b$ positive real numbers. Prove that $$\frac{(a+b)^3}{a^2b}\ge \frac{27}{4}.$$ When does equality occur?

2004 Gheorghe Vranceanu, 3

Tags: floor function , inequalities , algebra

Let $ a,b,c $ be real numbers satisfying $ \left\lfloor a^2+b^2+c^2 \right\rfloor \le\lfloor ab+bc+ca \rfloor . $ Show that: $$ 2 >\max\left\{ \left| -2a+b+c \right| ,\left| a-2b+c \right| ,\left| a+b-2c \right| \right\} $$ [i]Merticaru[/i]

1997 Singapore MO Open, 4

Tags: inequalities , algebra

Let $n \ge 2$ be a positive integer. Suppose that $a_1,a_2,...,a_n$ and $b_1,b_2,...,b_n$ are 2n numbers such that $\sum_{i=1}^n a_i =\sum_{i=1}^n n_i= 1$ and $a_i\ge 0, 0 \le b_i\le \frac{n-1}{n}, i = 1, 2,..., n$. Show that $$b_1a_2a_3...a_n+a_1b_2a_3...a_n+...+a_1a_2...a_{k-1}b_ka_{k+1}...a_n+ ...+ a_1a_2...a_{n-1}b_n \le \frac{1}{n(n-1)^{n-2}}$$

2008 Bulgarian Autumn Math Competition, Problem 12.1

Tags: parameterization , inequalities , algebra , interval

Determine the values of the real parameter $a$, such that the solutions of the system of inequalities $\begin{cases} \log_{\frac{1}{3}}{(3^{x}-6a)}+\frac{2}{\log_{a}{3}}<x-3\\ \log_{\frac{1}{3}}{(3^{x}-18)}>x-5\\ \end{cases}$ form an interval of length $\frac{1}{3}$.

2004 AMC 12/AHSME, 23

Tags: algebra , polynomial , vieta

The polynomial $ x^3\minus{}2004x^2\plus{}mx\plus{}n$ has integer coefficients and three distinct positive zeros. Exactly one of these is an integer, and it is the sum of the other two. How many values of $ n$ are possible? $ \textbf{(A)}\ 250,\!000 \qquad \textbf{(B)}\ 250,\!250 \qquad \textbf{(C)}\ 250,\!500 \qquad \textbf{(D)}\ 250,\!750 \qquad \textbf{(E)}\ 251,\!000$

2020 Thailand TST, 3

Tags: algebra , product , determinant

Let $x_1, x_2, \dots, x_n$ be different real numbers. Prove that \[\sum_{1 \leqslant i \leqslant n} \prod_{j \neq i} \frac{1-x_{i} x_{j}}{x_{i}-x_{j}}=\left\{\begin{array}{ll} 0, & \text { if } n \text { is even; } \\ 1, & \text { if } n \text { is odd. } \end{array}\right.\]

2006 Thailand Mathematical Olympiad, 7

Tags: algebra , min , inequalities

Let $x, y, z$ be reals summing to $1$ which minimizes $2x^2 + 3y^2 + 4z^2$. Find $x$.

1987 Vietnam National Olympiad, 2

Tags: algorithm , polynomial , modular arithmetic , algebra

Sequences $ (x_n)$ and $ (y_n)$ are constructed as follows: $ x_0 \equal{} 365$, $ x_{n\plus{}1} \equal{} x_n\left(x^{1986} \plus{} 1\right) \plus{} 1622$, and $ y_0 \equal{} 16$, $ y_{n\plus{}1} \equal{} y_n\left(y^3 \plus{} 1\right) \minus{} 1952$, for all $ n \ge 0$. Prove that $ \left|x_n\minus{} y_k\right|\neq 0$ for any positive integers $ n$, $ k$.

1983 IMO Longlists, 53

Tags: trigonometry , complex number , algebra

Let $a \in \mathbb R$ and let $z_1, z_2, \ldots, z_n$ be complex numbers of modulus $1$ satisfying the relation \[\sum_{k=1}^n z_k^3=4(a+(a-n)i)- 3 \sum_{k=1}^n \overline{z_k}\] Prove that $a \in \{0, 1,\ldots, n \}$ and $z_k \in \{1, i \}$ for all $k.$

2000 IMO Shortlist, 7

Tags: coefficient , algebra , polynomial , modular arithmetic

For a polynomial $ P$ of degree 2000 with distinct real coefficients let $ M(P)$ be the set of all polynomials that can be produced from $ P$ by permutation of its coefficients. A polynomial $ P$ will be called [b]$ n$-independent[/b] if $ P(n) \equal{} 0$ and we can get from any $ Q \in M(P)$ a polynomial $ Q_1$ such that $ Q_1(n) \equal{} 0$ by interchanging at most one pair of coefficients of $ Q.$ Find all integers $ n$ for which $ n$-independent polynomials exist.

2010 Vietnam National Olympiad, 2

Tags: algebra

Let $\{a_{n}\}$ be a sequence which satisfy $a_{1}=5$ and $a_{n=}\sqrt[n]{a_{n-1}^{n-1}+2^{n-1}+2.3^{n-1}} \qquad \forall n\geq2$ [b](a)[/b] Find the general fomular for $a_{n}$ [b](b)[/b] Prove that $\{a_{n}\}$ is decreasing sequences