Olympiad problems

1996 Austrian-Polish Competition, 8

Show that there is no polynomial $P(x)$ of degree $998$ with real coefficients which satisfies $P(x^2 + 1) = P(x)^2 - 1$ for all $x$.

2005 Slovenia National Olympiad, Problem 2

Tags: sequence , algebra

Let $(a_n)$ be a geometrical progression with positive terms. Define $S_n=\log a_1+\log a_2+\ldots+\log a_n$. Prove that if $S_n=S_m$ for some $m\ne n$, then $S_{n+m}=0$.

1996 Tuymaada Olympiad, 6

Tags: algebra , sequence , limit of sequence , limit , trigonometric

Given the sequence $f_1(a)=sin(0,5\pi a)$ $f_2(a)=sin(0,5\pi (sin(0,5\pi a)))$ $...$ $f_n(a)=sin(0,5\pi (sin(...(sin(0,5\pi a))...)))$ , where $a$ is any real number. What limit aspire the members of this sequence as $n \to \infty$?

1989 USAMO, 5

Tags: function , geometric series , algebra

Let $u$ and $v$ be real numbers such that \[ (u + u^2 + u^3 + \cdots + u^8) + 10u^9 = (v + v^2 + v^3 + \cdots + v^{10}) + 10v^{11} = 8. \] Determine, with proof, which of the two numbers, $u$ or $v$, is larger.

2018 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: algebra , equation , floor function , frac function

Solve equation $x \lfloor{x}\rfloor+\{x\}=2018$, where $x$ is real number

1983 Putnam, A5

Tags: number theory , algebra

Prove or disprove that there exists a positive real $u$ such that $\lfloor u^n\rfloor-n$ is an even integer for all positive integers $n$.

2001 Federal Competition For Advanced Students, Part 2, 1

Tags: function , algebra

Find all functions $f :\mathbb R \to \mathbb R$ such that for all real $x, y$ \[f(f(x)^2 + f(y)) = xf(x) + y.\]

2011 Poland - Second Round, 1

Tags: algebra

For $x,y\in\mathbb{R}$, solve the system of equations \[ \begin{cases} (x-y)(x^3+y^3)=7 \\ (x+y)(x^3-y^3)=3 \end{cases} \]

2019 Ecuador Juniors, 2

Tags: algebra , polynomial

Find how many integer values $3\le n \le 99$ satisfy that the polynomial $x^2 + x + 1$ divides $x^{2^n} + x + 1$.

2009 Moldova Team Selection Test, 1

Tags: algebra

[color=darkred]For any $ m \in \mathbb{N}^*$ solve the ecuation \[ \left\{\left( x \plus{} \frac {1}{m}\right) ^3\right\} \equal{} x^3 \] [/color]

2022 Auckland Mathematical Olympiad, 1

Tags: algebra

Each of the $10$ dwarfs either always tells the truth or always lies. It is known that each of them loves exactly one type of ice cream: vanilla, chocolate or fruit. First, Snow White asked those who like the vanilla ice cream to raise their hands, and everyone raised their hands, then those who like chocolate ice cream - and half of the dwarves raised their hands, then those who like the fruit ice cream - and only one dwarf raised his hand. How many of the gnomes are truthful?

1995 French Mathematical Olympiad, Problem 5

Tags: function , algebra

Let $f$ be a bijection from $\mathbb N$ to itself. Prove that one can always find three natural number $a,b,c$ such that $a<b<c$ and $f(a)+f(c)=2f(b)$.

1998 Denmark MO - Mohr Contest, 4

Tags: algebra , inequalities

Let $a$ and $b$ be positive real numbers with $a + b =1$. Show that $$\left(a+\frac{1}{a}\right)^2 + \left(b+\frac{1}{b}\right)^2 \ge \frac{25}{2}.$$

1949-56 Chisinau City MO, 1

Tags: algebra

The numbers $1, 2, ..., 1000$ are written out in a row along a circle. Starting from the first, every fifteenth number in the circle is crossed out $(1, 16, 31, ...)$, in this case, the crossed out numbers are still taken into account at each new round of the circle. How many numbers are left uncrossed?

2016 Swedish Mathematical Competition, 2

Tags: algebra , inequalities

Determine whether the inequality $$ \left|\sqrt{x^2+2x+5}-\sqrt{x^2-4x+8}\right|<3$$ is valid for all real numbers $x$.

I Soros Olympiad 1994-95 (Rus + Ukr), 10.6

Tags: algebra , functional equation

Find all functions $f:R\to R$ such that for any real $x, y$ , $$f(x+2^y)=f(2^x)+f(y)$$

2018 Regional Olympiad of Mexico Southeast, 6

Tags: algebra , polynomial

Find all polynomials $p(x)$ such that for all reals $a, b$ and $c$, with $a+b+c=0$, satisfies $$p(a^3)+p(b^3)+p(c^3)=3p(abc)$$

2018 Brazil Undergrad MO, 15

Tags: probability , algebra

A real number $ to $ is randomly and uniformly chosen from the $ [- 3,4] $ interval. What is the probability that all roots of the polynomial $ x ^ 3 + ax ^ 2 + ax + 1 $ are real?

2018 India IMO Training Camp, 3

Tags: algebra , functional equation

Find all functions $f: \mathbb{R} \mapsto \mathbb{R}$ such that $$f(x)f\left(yf(x)-1\right)=x^2f(y)-f(x),$$for all $x,y \in \mathbb{R}$.

2017 Balkan MO Shortlist, N1

Tags: algebra , equation

Find all ordered pairs of positive integers$ (x, y)$ such that:$$x^3+y^3=x^2+42xy+y^2.$$

1983 Spain Mathematical Olympiad, 4

Tags: algebra , equation , parameter

Determine the number of real roots of the equation $$16x^5 - 20x^3 + 5x + m = 0.$$

1989 IMO Longlists, 98

Tags: linear algebra , matrix , algebra

Let $ A$ be an $ n \times n$ matrix whose elements are non-negative real numbers. Assume that $ A$ is a non-singular matrix and all elements of $ A^{\minus{}1}$ are non-negative real numbers. Prove that every row and every column of $ A$ has exactly one non-zero element.

2019 Estonia Team Selection Test, 4

Tags: algebra , polynomial

Let us call a real number $r$ [i]interesting[/i], if $r = a + b\sqrt2$ for some integers a and b. Let $A(x)$ and $B(x)$ be polynomial functions with interesting coefficients for which the constant term of $B(x)$ is $1$, and $Q(x)$ be a polynomial function with real coefficients such that $A(x) = B(x) \cdot Q(x)$. Prove that the coefficients of $Q(x)$ are interesting.

2013 Saudi Arabia GMO TST, 2

Tags: polynomial , algebra , irreducible

Let $f(X) = a_nX^n + a_{n-1}X^{n-1} + ...+ a_1X + p$ be a polynomial of integer coefficients where $p$ is a prime number. Assume that $p >\sum_{i=1}^n |a_i|$. Prove that $f(X)$ is irreducible.

2019 Azerbaijan Senior NMO, 1

Tags: algebra , square root

Solve the following equation $$\sqrt{\frac{x^2}3-ax+a^2}+\sqrt{\frac{x^2}3-bx+b^2}=\sqrt{a^2-ab+b^2}$$ where $a;b\in\mathbb{R^+}$