Olympiad problems

2007 Nicolae Coculescu, 3

Determine all sets of natural numbers $ A $ that have at least two elements, and satisfying the following proposition: $$ \forall x,y\in A\quad x>y\implies \frac{x-y}{\text{gcd} (x,y)} \in A. $$ [i]Marius Perianu[/i]

2005 Slovenia Team Selection Test, 6

Tags: algebra , inequalities

Let $a,b,c > 0$ and $ab+bc+ca = 1$. Prove the inequality $3\sqrt[3]{\frac{1}{abc} +6(a+b+c) }\le \frac{\sqrt[3]3}{abc}$

2022 International Zhautykov Olympiad, 1

Tags: algebra , polynomial

Non-zero polynomials $P(x)$, $Q(x)$, and $R(x)$ with real coefficients satisfy the identities $$ P(x) + Q(x) + R(x) = P(Q(x)) + Q(R(x)) + R(P(x)) = 0. $$ Prove that the degrees of the three polynomials are all even.

2019 Philippine TST, 4

Tags: algebra , functional equation

Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$

2005 Cuba MO, 4

Tags: functional , algebra

Determine all functions $f : R_+ \to R$ such that:$$f(x)f(y) = f(xy) + \frac{1}{x} + \frac{1}{y}$$ for all $x, y$ positive reals.

2011 AMC 12/AHSME, 20

Tags: algebra , polynomial

Let $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are integers. Suppose that $f(1)=0$, $50 < f(7) < 60$, $70 < f(8) < 80$, and $5000k < f(100) < 5000(k+1)$ for some integer $k$. What is $k$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5 $

2024-IMOC, A4

Tags: algebra , functional equation

find all function $f:\mathbb{R} \to \mathbb{R}$ such that \[f(x^3-xf(y)^2)=xf(x+y)f(x-y)\] holds for all real number $x$, $y$. [i]Proposed by chengbilly[/i]

2006 Pre-Preparation Course Examination, 2

Tags: algebra , function , combinatorics , polynomial

If $f(x)$ is the generating function of the sequence $a_1,a_2,\ldots$ and if $f(x)=\frac{r(x)}{s(x)}$ holds such that $r(x)$ and $s(x)$ are polynomials show that $a_n$ has a homogenous recurrence.

2009 Abels Math Contest (Norwegian MO) Final, 4b

Tags: algebra , sum , inequalities

Let $x = 1 - 2^{-2009}$. Show that $x + x^2 + x^4 + x^8 +... + x^{2^m}< 2010$ for all positive integers $m$.

1967 IMO Longlists, 48

Tags: equation , root , exponential equation , algebra

Determine all positive roots of the equation $ x^x = \frac{1}{\sqrt{2}}.$

2024 IFYM, Sozopol, 2

Tags: algebra , polynomial

Let $m,n$ and $a$ be positive integers. Lumis has $m$ cards, each with the number $n$ written on it, and an infinite number of cards with each of the symbols addition, subtraction, multiplication, division, opening, and closing brackets. Umbra has composed an arithmetic expression with them, whose value is a positive integer less than $\displaystyle\frac{n}{2^m}$. Prove that if $n$ is replaced everywhere by $a$, then the resulting expression will have the same value as before or will be undefined due to division by zero.

1984 IMO Longlists, 58

Tags: limit , algebra , inequalities

Let $(a_n)_1^{\infty}$ be a sequence such that $a_n \le a_{n+m} \le a_n + a_m$ for all positive integers $n$ and $m$. Prove that $\frac{a_n}{n}$ has a limit as $n$ approaches infinity.

2023 Portugal MO, 1

Tags: algebra , combinatorics

Ana, Bruno and Carolina played table tennis with each other. In each game, only two of the friends played, with the third one resting. Every time one of the friends won a game, they rested during the next game. Ana played $12$ games, Bruno played $21$ games and Carolina rested for$ 8$ games. Who rested in the last game?

2006 Junior Balkan Team Selection Tests - Moldova, 3

Tags: polynomial , algebra

Determine all 2nd degree polynomials with integer coefficients of the form $P(X)=aX^{2}+bX+c$, that satisfy: $P(a)=b$, $P(b)=a$, with $a\neq b$.

1967 IMO Shortlist, 2

Tags: polynomial , algebra , system of equations

Find all real solutions of the system of equations: \[\sum^n_{k=1} x^i_k = a^i\] for $i = 1,2, \ldots, n.$

2023 VN Math Olympiad For High School Students, Problem 8

Tags: polynomial , algebra

Prove that: for all positive integers $n\ge 2,$ the polynomial$$(x^2-1)^2(x^2-1)^2...(x^2-2023)^2+1$$ is irreducible in $\mathbb{Q}[x].$

1969 Swedish Mathematical Competition, 2

Tags: irrational , algebra , trigonometry

Show that $\tan \frac{\pi}{3n}$ is irrational for all positive integers $n$.

1999 Brazil Team Selection Test, Problem 2

Tags: polynomial , algebra

If $a,b,c,d$ are Distinct Real no. such that $a = \sqrt{4+\sqrt{5+a}}$ $b = \sqrt{4-\sqrt{5+b}}$ $c = \sqrt{4+\sqrt{5-c}}$ $d = \sqrt{4-\sqrt{5-d}}$ Then $abcd = $

2011 AMC 10, 25

Tags: function , geometry , domain , algebra

Let $R$ be a square region and $n\ge4$ an integer. A point $X$ in the interior of $R$ is called $n\text{-}ray$ partitional if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are 100-ray partitional but not 60-ray partitional? $\textbf{(A)}\,1500 \qquad\textbf{(B)}\,1560 \qquad\textbf{(C)}\,2320 \qquad\textbf{(D)}\,2480 \qquad\textbf{(E)}\,2500$

2019 CCA Math Bonanza, I12

Tags: algebra , polynomial

Let $f\left(x,y\right)=x^2\left(\left(x+2y\right)^2-y^2+x-1\right)$. If $f\left(a,b+c\right)=f\left(b,c+a\right)=f\left(c,a+b\right)$ for distinct numbers $a,b,c$, what are all possible values of $a+b+c$? [i]2019 CCA Math Bonanza Individual Round #12[/i]

2001 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: floor function , function , sequence , radical , algebra

For every integer $n > 1$, the sequence $\left( {{S}_{n}} \right)$ is defined by ${{S}_{n}}=\left\lfloor {{2}^{n}}\underbrace{\sqrt{2+\sqrt{2+...+\sqrt{2}}}}_{n\ radicals} \right\rfloor $ where $\left\lfloor x \right\rfloor$ denotes the floor function of $x$. Prove that ${{S}_{2001}}=2\,{{S}_{2000}}+1$. .

KoMaL A Problems 2021/2022, A. 824

Tags: algebra

An infinite set $S$ of positive numbers is called thick, if in every interval of the form $\left [1/(n+1),1/n\right]$ (where $n$ is an arbitrary positive integer) there is a number which is the difference of two elements from $S$. Does there exist a thick set such that the sum of its elements is finite? Proposed by [i]Gábor Szűcs[/i], Szikszó

2016 Indonesia TST, 3

Tags: algebra , trigonometry

Let $n$ be a positive integer greater than $1$. Evaluate the following summation: \[ \sum_{k=0}^{n-1} \frac{1}{1 + 8 \sin^2 \left( \frac{k \pi}{n} \right)}. \]

1969 IMO Shortlist, 54

Tags: divisibility , algebra , number theory , modular arithmetic , polynomial

$(POL 3)$ Given a polynomial $f(x)$ with integer coefficients whose value is divisible by $3$ for three integers $k, k + 1,$ and $k + 2$. Prove that $f(m)$ is divisible by $3$ for all integers $m.$

2015 Germany Team Selection Test, 1

Tags: real number , integer , root , algebra , polynomial

Find the least positive integer $n$, such that there is a polynomial \[ P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0 \] with real coefficients that satisfies both of the following properties: - For $i=0,1,\dots,2n$ it is $2014 \leq a_i \leq 2015$. - There is a real number $\xi$ with $P(\xi)=0$.