Olympiad problems

2006 Junior Tuymaada Olympiad, 4

The sum of non-negative numbers $ x $, $ y $ and $ z $ is $3$. Prove the inequality $$ {1 \over x ^ 2 + y + z} + {1 \over x + y ^ 2 + z} + {1 \over x + y + z ^ 2} \leq 1. $$

2015 Harvard-MIT Mathematics Tournament, 5

Tags: algebra , inequalities , maximization

Let $a,b,c$ be positive real numbers such that $a+b+c=10$ and $ab+bc+ca=25$. Let $m=\min\{ab,bc,ca\}$. Find the largest possible value of $m$.

2011 NIMO Summer Contest, 7

Tags: algebra , polynomial , number theory , greatest common divisor

Let $P(x) = x^2 - 20x - 11$. If $a$ and $b$ are natural numbers such that $a$ is composite, $\gcd(a, b) = 1$, and $P(a) = P(b)$, compute $ab$. Note: $\gcd(m, n)$ denotes the greatest common divisor of $m$ and $n$. [i]Proposed by Aaron Lin [/i]

2023 Federal Competition For Advanced Students, P2, 1

Tags: algebra , functional equation

Given is a nonzero real number $\alpha$. Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $$f(f(x+y))=f(x+y)+f(x)f(y)+\alpha xy$$ for all $x, y \in \mathbb{R}$.

2013 Junior Balkan Team Selection Tests - Romania, 1

Tags: inequalities , algebra

If $a, b, c > 0$ satisfy $a + b + c = 3$, then prove that $$\frac{a^2(b + 1)}{ ab + a + b} + \frac{b^2(c + 1)}{ bc + b + c} + \frac{c^2(a + 1)}{ ca + c + a} \ge 2$$ Mathematical Excalibur P322/Vol.14, no.2

2011 India IMO Training Camp, 2

Tags: calculus , integration , algebra

Suppose $a_1,\ldots,a_n$ are non-integral real numbers for $n\geq 2$ such that ${a_1}^k+\ldots+{a_n}^k$ is an integer for all integers $1\leq k\leq n$. Prove that none of $a_1,\ldots,a_n$ is rational.

2011 HMNT, 10

Tags: hmmt , algebra , polynomial

Let $r_1, r_2, \cdots, r_7$ be the distinct complex roots of the polynomial $P(x) = x^7 - 7$ Let \[K = \prod_{1 \leq i < j \leq 7} (r_i + r_j)\] that is, the product of all the numbers of the form $r_i + r_j$, where $i$ and $j$ are integers for which $1 \leq i < j \leq 7$. Determine the value of $K^2$.

2010 Thailand Mathematical Olympiad, 5

Tags: functional equation , functional , algebra

Determine all functions $f : R \times R \to R$ satisfying the equation $f(x - t, y) + f(x + t, y) + f(x, y - t) + f(x, y + t) = 2010$ for all real numbers $x, y$ and for all nonzero $t$

2000 South africa National Olympiad, 5

Tags: function , algebra

Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ (where $\mathbb{Z}$ is the set of all integers) such that \[ 2000f(f(x)) - 3999f(x) + 1999x = 0\textrm{ for all }x \in \mathbb{Z}. \]

2000 Switzerland Team Selection Test, 12

Tags: functional equation , functional , algebra

Find all functions $f : R \to R$ such that for all real $x,y$, $f(f(x)+y) = f(x^2 -y)+4y f(x)$

2016 Baltic Way, 8

Tags: algebra , functional equation

Find all real numbers $a$ for which there exists a non-constant function $f :\Bbb R \to \Bbb R$ satisfying the following two equations for all $x\in \Bbb R:$ i) $f(ax) = a^2f(x)$ and ii) $f(f(x)) = a f(x).$

2021 China Team Selection Test, 3

Tags: number theory , diophantine equation , algebra

Given positive integers $a,b,c$ which are pairwise coprime. Let $f(n)$ denotes the number of the non-negative integer solution $(x,y,z)$ to the equation $$ax+by+cz=n.$$ Prove that there exists constants $\alpha, \beta, \gamma \in \mathbb{R}$ such that for any non-negative integer $n$, $$|f(n)- \left( \alpha n^2+ \beta n + \gamma \right) | < \frac{1}{12} \left( a+b+c \right).$$

2024 CMI B.Sc. Entrance Exam, 5

Tags: diophantine equation , number theory , algebra

Find all solutions for positive integers $(x,y,k,m)$ such that \[ 20x^k+24y^m = 2024\] with $k, m > 1$

2021 Balkan MO Shortlist, A5

Tags: shortlist , algebra , functional equation

Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that $$f(xf(x + y)) = yf(x) + 1$$ holds for all $x, y \in \mathbb{R}^{+}$. [i]Proposed by Nikola Velov, North Macedonia[/i]

2014 JBMO Shortlist, 1

Tags: algebra , floor function , floor , equation , positive real

Solve in positive real numbers: $n+ \lfloor \sqrt{n} \rfloor+\lfloor \sqrt[3]{n} \rfloor=2014$

2004 IMO Shortlist, 3

Tags: function , number theory , continued fraction , algebra , functional equation

Does there exist a function $s\colon \mathbb{Q} \rightarrow \{-1,1\}$ such that if $x$ and $y$ are distinct rational numbers satisfying ${xy=1}$ or ${x+y\in \{0,1\}}$, then ${s(x)s(y)=-1}$? Justify your answer. [i]Proposed by Dan Brown, Canada[/i]

2019 Federal Competition For Advanced Students, P2, 1

Tags: functional equation , algebra

Determine all functions $f: R\to R$, such that $f (2x + f (y)) = x + y + f (x)$ for all $x, y \in R$. (Gerhard Kirchner)

1984 Austrian-Polish Competition, 8

Tags: function , inequalities , algebra

The functions $f_0,f_1 : (1,\infty) \to (1,\infty)$ are given by $ f_0(x) = 2x$ and$ f_1(x) =\frac{x}{x-1}$. Show that for any real numbers $a, b$ with $1 \le a < b$ there exist a positive integer $k$ and indices $i_1,i_2,...,i_k \in \{0,1\}$ such that $a <f_{i_k}(f_{i_{k-1}}(...(f_{i_j}(2))...))< b$.

2000 Bulgaria National Olympiad, 1

Tags: polynomial , algebra

Find all polynomials $P(x)$ with real coefficients such that \[P(x)P(x + 1) = P(x^2), \quad \forall x \in \mathbb R.\]

2020 LIMIT Category 1, 5

Tags: algebra , polynomial , algorithm

Let $P(x),Q(x)$ be monic polynomials with integer coeeficients. Let $a_n=n!+n$ for all natural numbers $n$. Show that if $\frac{P(a_n)}{Q(a_n)}$ is an integer for all positive integer $n$ then $\frac{P(n)}{Q(n)}$ is an integer for every integer $n\neq0$. \\ [i]Hint (given in question): Try applying division algorithm for polynomials [/i]

2010 India IMO Training Camp, 11

Tags: function , algebra

Find all functions $f:\mathbb{R}\longrightarrow\mathbb{R}$ such that $f(x+y)+xy=f(x)f(y)$ for all reals $x, y$

2018 CMIMC Number Theory, 10

Tags: algebra , polynomial

Let $a_1 < a_2 < \cdots < a_k$ denote the sequence of all positive integers between $1$ and $91$ which are relatively prime to $91$, and set $\omega = e^{2\pi i/91}$. Define \[S = \prod_{1\leq q < p\leq k}\left(\omega^{a_p} - \omega^{a_q}\right).\] Given that $S$ is a positive integer, compute the number of positive divisors of $S$.

2018 Auckland Mathematical Olympiad, 5

Tags: algebra

There is a sequence of numbers $+1$ and $-1$ of length $n$. It is known that the sum of every $10$ neighbouring numbers in the sequence is $0$ and that the sum of every $12$ neighbouring numbers in the sequence is not zero. What is the maximal value of $n$?

2016 Iran Team Selection Test, 1

Tags: algebra

A real function has been assigned to every cell of an $n \times n$ table. Prove that a function can be assigned to each row and each column of this table such that the function assigned to each cell is equivalent to the combination of functions assigned to the row and the column containing it.

1967 Swedish Mathematical Competition, 5

Tags: algebra , inequalities

$a_1, a_2, a_3, ...$ are positive reals such that $a_n^2 \ge a_1 + a_2 +... + a_{n-1}$. Show that for some $C > 0$ we have $a_n \ge C n$ for all $n$.