Olympiad problems

1969 Canada National Olympiad, 7

Tags: abstract algebra , polynomial , quadratic , domain , modular arithmetic , algebra , function

Show that there are no integers $a,b,c$ for which $a^2+b^2-8c=6$.

2018 NZMOC Camp Selection Problems, 10

Tags: functional equation , functional , algebra

Find all functions $f : R \to R$ such that $$f(x)f(y) = f(xy + 1) + f(x - y) - 2$$ for all $x, y \in R$.

2023 Bulgaria JBMO TST, 4

Tags: algebra

The numbers $2, 2, ..., 2$ are written on a blackboard (the number $2$ is repeated $n$ times). One step consists of choosing two numbers from the blackboard, denoting them as $a$ and $b$, and replacing them with $\sqrt{\frac{ab + 1}{2}}$. $(a)$ If $x$ is the number left on the blackboard after $n - 1$ applications of the above operation, prove that $x \ge \sqrt{\frac{n + 3}{n}}$. $(b)$ Prove that there are infinitely many numbers for which the equality holds and infinitely many for which the inequality is strict.

2010 IMO, 3

Tags: function , perfect square , number theory , algebra

Find all functions $g:\mathbb{N}\rightarrow\mathbb{N}$ such that \[\left(g(m)+n\right)\left(g(n)+m\right)\] is a perfect square for all $m,n\in\mathbb{N}.$ [i]Proposed by Gabriel Carroll, USA[/i]

2021 Korea Winter Program Practice Test, 7

Tags: polynomial , algebra

Find all pair of constants $(a,b)$ such that there exists real-coefficient polynomial $p(x)$ and $q(x)$ that satisfies the condition below. [b]Condition[/b]: $\forall x\in \mathbb R,$ $ $ $p(x^2)q(x+1)-p(x+1)q(x^2)=x^2+ax+b$

1981 Canada National Olympiad, 1

Tags: function , algebra , floor function

For any real number $t$, denote by $[t]$ the greatest integer which is less than or equal to $t$. For example: $[8] = 8$, $[\pi] = 3$, and $[-5/2] = -3$. Show that the equation \[[x] + [2x] + [4x] + [8x] + [16x] + [32x] = 12345\] has no real solution.

2009 Austria Beginners' Competition, 2

Tags: algebra , inequalities

Let $x$ and $y$ be nonnegative real numbers. Prove that $(x +y^3) (x^3 +y) \ge 4x^2y^2$. When does equality holds? (Task committee)

2006 Pre-Preparation Course Examination, 4

Tags: algebra , polynomial , geometry

Find a 3rd degree polynomial whose roots are $r_a$, $r_b$ and $r_c$ where $r_a$ is the radius of the outer inscribed circle of $ABC$ with respect to $A$.

2017 Azerbaijan Senior National Olympiad, A1

Tags: system of equations , algebra

Solve the system of equation for $(x,y) \in \mathbb{R}$ $$\left\{\begin{matrix} \sqrt{x^2+y^2}+\sqrt{(x-4)^2+(y-3)^2}=5\\ 3x^2+4xy=24 \end{matrix}\right.$$ \\ Explain your answer

2006 USAMO, 4

Tags: quadratic , algebra

Find all positive integers $n$ such that there are $k \geq 2$ positive rational numbers $a_1, a_2, \ldots, a_k$ satisfying $a_1 + a_2 + \ldots + a_k = a_1 \cdot a_2 \cdots a_k = n.$

2004 Cuba MO, 1

Tags: system , algebra , system of equations

Determine all real solutions to the system of equations: $$x_1 + x_2 +...+ x_{2004 }= 2004$$ $$x^4_1+ x^4_2+ ... + x^4_{2004} = x^3_1+x^3_2+... + x^3_{2004}$$

KoMaL A Problems 2023/2024, A. 880

Tags: function , algebra

Find all triples $(a,b,c)$ of real numbers for which there exists a function $f:\mathbb{Z}^{+}\rightarrow\mathbb{Z}^{+}$ satisfying $af(n)+bf(n+1)+cf(n+2)<0$ for every $n\in\mathbb{Z}^{+}$ ($\mathbb{Z}^{+}$ denotes the set of positive integers). Proposed by [i]András Imolay[/i], Budapest

1966 IMO Shortlist, 26

Tags: n-variable inequality , algebra , inequalities

Prove the inequality [b]a.)[/b] $ \left( a_{1}+a_{2}+...+a_{k}\right) ^{2}\leq k\left( a_{1}^{2}+a_{2}^{2}+...+a_{k}^{2}\right) , $ where $k\geq 1$ is a natural number and $a_{1},$ $a_{2},$ $...,$ $a_{k}$ are arbitrary real numbers. [b]b.)[/b] Using the inequality (1), show that if the real numbers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ satisfy the inequality \[ a_{1}+a_{2}+...+a_{n}\geq \sqrt{\left( n-1\right) \left( a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}\right) }, \] then all of these numbers $a_{1},$ $a_{2},$ $\ldots,$ $a_{n}$ are non-negative.

2003 Switzerland Team Selection Test, 10

Tags: functional equation , algebra , functional

Find all strictly monotonous functions $f : N \to N$ that satisfy $f(f(n)) = 3n$ for all $n \in N$.

2022 Bulgarian Spring Math Competition, Problem 8.3

Tags: inequality , algebra , inequalities

Given the inequalities: $a)$ $\left(\frac{2a}{b+c}\right)^2+\left(\frac{2b}{a+c}\right)^2+\left(\frac{2c}{a+b}\right)^2\geq \frac{a}{c}+\frac{b}{a}+\frac{c}{b}$ $b)$ $\left(\frac{a+b}{c}\right)^2+\left(\frac{b+c}{a}\right)^2+\left(\frac{c+a}{b}\right)^2\geq \frac{a}{b}+\frac{b}{c}+\frac{c}{a}+9$ For each of them either prove that it holds for all positive real numbers $a$, $b$, $c$ or present a counterexample $(a,b,c)$ which doesn't satisfy the inequality.

1996 AIME Problems, 5

Tags: polynomial , algebra

Suppose that the roots of $x^3+3x^2+4x-11=0$ are $a, b,$ and $c,$ and that the roots of $x^3+rx^2+sx+t=0$ are $a+b, b+c,$ and $c+a.$ Find $t.$

2007 Korea - Final Round, 6

Tags: function , algebra

Let f:N→N be a function satisfying $ kf(n)\le f(kn)\le kf(n) \plus{} k \minus{} 1$ for all $ k, n\in N$. (a)Prove that $ f(a) \plus{} f(b)\le f(a \plus{} b)\le f(a) \plus{} f(b) \plus{} 1$ for all $ a, b\in N$. (b)If $ f$ satisfies $ f(2007n)\le 2007f(n) \plus{} 200$ for every $ n\in N$, show that there exists $ c\in N$ such that $ f(2007c) \equal{} 2007f(c)$.

2011 AMC 8, 22

Tags: binomial theorem , algebra

What is the tens digit of $7^{2011}$? $ \textbf{(A)}0\qquad\textbf{(B)}1\qquad\textbf{(C)}3\qquad\textbf{(D)}4\qquad\textbf{(E)}7 $

2022 Bulgaria EGMO TST, 1

Tags: closure , binary operation , addition , algebra

The finite set $M$ of real numbers is such that among any three of its elements there are two whose sum is in $M$. What is the maximum possible cardinality of $M$? [hide=Remark about the other problems] Problem 2 is UK National Round 2022 P2, Problem 3 is UK National Round 2022 P4, Problem 4 is Balkan MO 2021 Shortlist N2 (the one with Bertrand), Problem 5 is IMO Shortlist 2021 A1 and Problem 6 is USAMO 2002/1. Hence neither of these will be posted here. [/hide]

2005 Thailand Mathematical Olympiad, 20

Tags: max , algebra , inequalities

Let $a, b, c, d > 0$ satisfy $36a + 4b + 4c + 3d = 25$. What is the maximum possible value of $ab^{1/2}c^{1/3}d^{1/4}$ ?

2024 IFYM, Sozopol, 6

Tags: geometry , algebra , combinatorics , number theory

Prove that for some positive integer $N$, $N$ points can be chosen on a circle such that there are at least $1000N^2$ unordered quadruples $(A,B,C,D)$ of distinct selected points for which $\displaystyle \frac{AC}{BC} = \frac{AD}{BD}$.

2020 Nigerian MO round 3, #2

Tags: algebra

a sequence $(a_n)$ $n$ $\geq 1$ is defined by the following equations; $a_1=1$, $a_2=2$ ,$a_3=1$, $a_{2n-1}$$a_{2n}$=$a_2$$a_{2n-3}$+$(a_2a_{2n-3}+a_4a_{2n-5}.....+a_{2n-2}a_1)$ for $n$ $\geq 2$ $na_{2n}$$a_{2n+1}$=$a_2$$a_{2n-2}$+$(a_2a_{2n-2}+a_4a_{2n-4}.....+a_{2n-2}a_2)$ for $n$ $\geq 2$ find $a_{2020}$

2013 India IMO Training Camp, 3

Tags: induction , algebra

Let $h \ge 3$ be an integer and $X$ the set of all positive integers that are greater than or equal to $2h$. Let $S$ be a nonempty subset of $X$ such that the following two conditions hold: [list] [*]if $a + b \in S$ with $a \ge h, b \ge h$, then $ab \in S$; [*]if $ab \in S$ with $a \ge h, b \ge h$, then $a + b \in S$.[/list] Prove that $S = X$.

2010 Iran MO (3rd Round), 2

Tags: domain , function , integration , algebra , number theory , abstract algebra , calculus

$R$ is a ring such that $xy=yx$ for every $x,y\in R$ and if $ab=0$ then $a=0$ or $b=0$. if for every Ideal $I\subset R$ there exist $x_1,x_2,..,x_n$ in $R$ ($n$ is not constant) such that $I=(x_1,x_2,...,x_n)$, prove that every element in $R$ that is not $0$ and it's not a unit, is the product of finite irreducible elements.($\frac{100}{6}$ points)

1993 Brazil National Olympiad, 2

Tags: linear algebra , absolute value , matrix , algebra

A real number with absolute value less than $1$ is written in each cell of an $n\times n$ array, so that the sum of the numbers in each $2\times 2$ square is zero. Show that for odd $n$ the sum of all the numbers is less than $n$.