Olympiad problems

2024 IMO, 6

Tags: algebra

Let $\mathbb{Q}$ be the set of rational numbers. A function $f: \mathbb{Q} \to \mathbb{Q}$ is called aquaesulian if the following property holds: for every $x,y \in \mathbb{Q}$, \[ f(x+f(y)) = f(x) + y \quad \text{or} \quad f(f(x)+y) = x + f(y). \] Show that there exists an integer $c$ such that for any aquaesulian function $f$ there are at most $c$ different rational numbers of the form $f(r) + f(-r)$ for some rational number $r$, and find the smallest possible value of $c$.

2023 Philippine MO, 6

Tags: function , algebra , functional equation

Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(2f(x)) = f(x - f(y)) + f(x) + y$$ for all $x, y \in \mathbb{R}$.

2019 JBMO Shortlist, A1

Tags: algebra , inequality

Real numbers $a$ and $b$ satisfy $a^3+b^3-6ab=-11$. Prove that $-\frac{7}{3}<a+b<-2$. [i]Proposed by Serbia[/i]

2007 Harvard-MIT Mathematics Tournament, 6

Tags: algebra , polynomial

Consider the polynomial $P(x)=x^3+x^2-x+2$. Determine all real numbers $r$ for which there exists a complex number $z$ not in the reals such that $P(z)=r$.

2015 CHMMC (Fall), 8

Tags: number theory , floor function , algebra

Let $f(n) = \sum^n_{d=1} \left\lfloor \frac{n}{d} \right\rfloor$ and $g(n) = f(n) -f(n - 1)$. For how many $n$ from $1$ to $100$ inclusive is $g(n)$ even?

2005 Switzerland - Final Round, 9

Tags: functional , algebra , functional equation

Find all functions $f : R^+ \to R^+$ such that for all $x, y > 0$ $$f(yf(x))(x + y) = x^2(f(x) + f(y)).$$

2020 China National Olympiad, 6

Tags: algebra , polynomial

Does there exist positive reals $a_0, a_1,\ldots ,a_{19}$, such that the polynomial $P(x)=x^{20}+a_{19}x^{19}+\ldots +a_1x+a_0$ does not have any real roots, yet all polynomials formed from swapping any two coefficients $a_i,a_j$ has at least one real root?

1994 Greece National Olympiad, 3

Tags: inequalities , algebra

If $a^2+b^2+c^2+d^2=1$, prove that $$(a-b)^2+(b-c)^2+(c-d)^2+(a-c)^2+(a-d)^2+(b-d)^2\leq 4$$ When does equality holds?

2010 Contests, 3

Tags: function , algebra , domain

Find all the functions $f:\mathbb{N}\to\mathbb{R}$ that satisfy \[ f(x+y)=f(x)+f(y) \] for all $x,y\in\mathbb{N}$ satisfying $10^6-\frac{1}{10^6} < \frac{x}{y} < 10^6+\frac{1}{10^6}$. Note: $\mathbb{N}$ denotes the set of positive integers and $\mathbb{R}$ denotes the set of real numbers.

2022 Serbia National Math Olympiad, P6

Tags: algebra , prime

Let $p$ and $q$ be different primes, and $\alpha\in (0, 3)$ a real number. Prove that in sequence $$\left[ \alpha \right] , \left[ 2\alpha \right] , \left[ 3\alpha \right] \dots$$ exists number less than $2pq$, divisible by $p$ or $q$.

1973 Putnam, B6

Tags: trigonometry , inequalities , function , algebra , domain

On the domain $0\leq \theta \leq 2\pi:$ (a) Prove that $\sin^{2}\theta \cdot \sin 2\theta$ takes its maximum at $\frac{\pi}{3}$ and $\frac{4 \pi}{3}$ (and hence its minimum at $\frac{2 \pi}{3}$ and $\frac{ 5 \pi}{3}$). (b) Show that $$| \sin^{2} \theta \cdot \sin^{3} 2\theta \cdot \sin^{3} 4 \theta \cdots \sin^{3} 2^{n-1} \theta \cdot \sin 2^{n} \theta |$$ takes its maximum at $\frac{4 \pi}{3}$ (the maximum may also be attained at other points). (c) Derive the inequality: $$ \sin^{2} \theta \cdot \sin^{2} 2\theta \cdots \sin^{2} 2^{n} \theta \leq \left( \frac{3}{4} \right)^{n}.$$

2014 Online Math Open Problems, 13

Tags: polynomial , algebra

Suppose that $g$ and $h$ are polynomials of degree $10$ with integer coefficients such that $g(2) < h(2)$ and \[ g(x) h(x) = \sum_{k=0}^{10} \left( \binom{k+11}{k} x^{20-k} - \binom{21-k}{11} x^{k-1} + \binom{21}{11}x^{k-1} \right) \] holds for all nonzero real numbers $x$. Find $g(2)$. [i]Proposed by Yang Liu[/i]

1998 Singapore Senior Math Olympiad, 3

Tags: algebra , inequalities , radical

Prove that $\sqrt1+ \sqrt2+\sqrt3+...+ \sqrt{n^2-1}+\sqrt{n^2} \ge \frac{2n^3+n}{3}$ for any positive integer $n$.

2017 International Zhautykov Olympiad, 2

Tags: functional equation , function , algebra

Find all functions $f:R \rightarrow R$ such that $$(x+y^2)f(yf(x))=xyf(y^2+f(x))$$, where $x,y \in \mathbb{R}$

2013 India IMO Training Camp, 1

Tags: function , limit , induction , algebra , cauchy equation

Find all functions $f$ from the set of real numbers to itself satisfying \[ f(x(1+y)) = f(x)(1 + f(y)) \] for all real numbers $x, y$.

1991 Austrian-Polish Competition, 9

Tags: functional , algebra , composition , function

For a positive integer $n$ denote $A = \{1,2,..., n\}$. Suppose that $g : A\to A$ is a fixed function with $g(k) \ne k$ and $g(g(k)) = k$ for $k \in A$. How many functions $f: A \to A$ are there such that $f(k)\ne g(k)$ and $f(f(f(k))= g(k)$ for $k \in A$?

2019 Argentina National Olympiad, 2

Tags: inequalities , sum , algebra

Let $n\geq1$ be an integer. We have two sequences, each of $n$ positive real numbers $a_1,a_2,\ldots ,a_n$ and $b_1,b_2,\ldots ,b_n$ such that $a_1+a_2+\ldots +a_n=1$ and $ b_1+b_2+\ldots +b_n=1$. Find the smallest possible value that the sum can take $$\frac{a_1^2}{a_1+b_1}+\frac{a_2^2}{a_2+b_2}+\ldots +\frac{a_n^2}{a_n +b_n}.$$

2011 Harvard-MIT Mathematics Tournament, 6

Tags: algebra , hmmt , polynomial

How many polynomials $P$ with integer coefficients and degree at most $5$ satisfy $0 \le P(x) < 120$ for all $x \in \{0,1,2,3,4,5\}$?

2021 Azerbaijan EGMO TST, 2

Tags: algebra , sequence , unbounded

Given a non-decreasing unbounded sequence $a_n,$ construct a new sequence $b_n$ as follows $$b_n = \frac{a_2 - a_1}{a_2} + \frac{a_3 - a_2}{a_3} + ... + \frac{a_n - a_{n-1}}{a_n}$$ Prove that $b_n$ is also unbounded.

2014 China Team Selection Test, 3

Tags: algebra , number theory , function , polynomial

Let the function $f:N^*\to N^*$ such that [b](1)[/b] $(f(m),f(n))\le (m,n)^{2014} , \forall m,n\in N^*$; [b](2)[/b] $n\le f(n)\le n+2014 , \forall n\in N^*$ Show that: there exists the positive integers $N$ such that $ f(n)=n $, for each integer $n \ge N$. (High School Affiliated to Nanjing Normal University )

1979 Swedish Mathematical Competition, 5

Tags: algebra , trinomial

Find the smallest positive integer $a$ such that for some integers $b$, $c$ the polynomial $ax^2 - bx + c$ has two distinct zeros in the interval $(0,1)$.

2023 BMT, 12

Tags: algebra

Find the greatest integer less than $$\sqrt{10}+ \sqrt{80}.$$

2022 AMC 10, 11

Tags: algebra

Ted mistakenly wrote $2^m \cdot \sqrt{\frac{1}{4096}}$ as $2\cdot \sqrt[m]{\frac{1}{4096}}$. What is the sum of all real numbers $m$ for which these two expressions have the same value? $\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$

2007 IMO Shortlist, 2

Tags: algebra , functional inequality

Consider those functions $ f: \mathbb{N} \mapsto \mathbb{N}$ which satisfy the condition \[ f(m \plus{} n) \geq f(m) \plus{} f(f(n)) \minus{} 1 \] for all $ m,n \in \mathbb{N}.$ Find all possible values of $ f(2007).$ [i]Author: Nikolai Nikolov, Bulgaria[/i]

2005 Austrian-Polish Competition, 7

Tags: algebra , system of equations , number theory

For each natural number $n\geq 2$, solve the following system of equations in the integers $x_1, x_2, ..., x_n$: $$(n^2-n)x_i+\left(\prod_{j\neq i}x_j\right)S=n^3-n^2,\qquad \forall 1\le i\le n$$ where $$S=x_1^2+x_2^2+\dots+x_n^2.$$