Olympiad problems

2000 Turkey Team Selection Test, 1

$(a)$ Prove that for every positive integer $n$, the number of ordered pairs $(x, y)$ of integers satisfying $x^2-xy+y^2 = n$ is divisible by $3.$ $(b)$ Find all ordered pairs of integers satisfying $x^2-xy+y^2=727.$

2008 Princeton University Math Competition, B1

Tags: algebra

Find all pairs of positive real numbers $(a,b)$ such that $\frac{n-2}{a} \leq \left\lfloor bn \right\rfloor < \frac{n-1}{a}$ for all positive integes $n$.

2004 Tournament Of Towns, 3

Tags: algebra , polynomial

P(x) and Q(x) are polynomials of positive degree such that for all x P(P(x))=Q(Q(x)) and P(P(P(x)))=Q(Q(Q(x))). Does this necessarily mean that P(x)=Q(x)?

2023 Chile TST Ibero., 2

Tags: algebra

Consider a function $ n \mapsto f(n) $ that satisfies the following conditions: $ f(n) $ is an integer for each $ n $. $ f(0) = 1 $. $ f(n+1) > f(n) + f(n-1) + \cdots + f(0) $ for each $ n = 0, 1, 2, \dots $. Determine the smallest possible value of $ f(2023) $.

2018 JBMO Shortlist, NT1

Tags: number theory , algebra

Find all integers $m$ and $n$ such that the fifth power of $m$ minus the fifth power of $n$ is equal to $16mn$.

2018 India Regional Mathematical Olympiad, 2

Tags: algebra

Let $n$ be a natural number. Find all real numbers $x$ satisfying the equation $$\sum^n_{k=1}\frac{kx^k}{1+x^{2k}}=\frac{n(n+1)}4.$$

2024 Moldova EGMO TST, 1

Tags: parabola , algebra , parameterization

Let $P$ be the set of all parabolas with the equation of the form $$y=(a-1)x^2-2(a+2)x+a+1$$ where $a$ is a real parameter and $a\neq1$. Prove that there exists an unique point $M$ such that all parabolas in $P$ pass through $M$.

2023 Kyiv City MO Round 1, Problem 2

Tags: algebra

You are given $n\geq 4$ positive real numbers. Consider all $\frac{n(n-1)}{2}$ pairwise sums of these numbers. Show that some two of these sums differ in at most $\sqrt[n-2]{2}$ times. [i]Proposed by Anton Trygub[/i]

2003 Croatia National Olympiad, Problem 2

Tags: algebra , floor function , function

For every integer $n>2$, prove the equality $$\left\lfloor\frac{n(n+1)}{4n-2}\right\rfloor=\left\lfloor\frac{n+1}4\right\rfloor.$$

2007 ITest, 55

Tags: floor function , algebra

Let $T=\text{TNFTPP}$, and let $R=T-914$. Let $x$ be the smallest real solution of \[3x^2+Rx+R=90x\sqrt{x+1}.\] Find the value of $\lfloor x\rfloor$.

2011 Pre - Vietnam Mathematical Olympiad, 1

Tags: induction , algebra

Let a sequence $\left\{ {{x_n}} \right\}$ defined by: \[\left\{ \begin{array}{l} {x_0} = - 2 \\ {x_n} = \frac{{1 - \sqrt {1 - 4{x_{n - 1}}} }}{2},\forall n \ge 1 \\ \end{array} \right.\] Denote $u_n=n.x_n$ and ${v_n} = \prod\limits_{i = 0}^n {\left( {1 + x_i^2} \right)} $. Prove that $\left\{ {{u_n}} \right\}$, $\left\{ {{v_n}} \right\}$ have finite limit.

2009 District Olympiad, 2

Tags: abstract algebra , algebra , polynomial , function , superior algebra

Prove that in an abelian ring $ A $ in which $ 1\neq 0, $ every element is idempotent if and only if the number of polynomial functions from $ A $ to $ A $ is equal to the square of the cardinal of $ A. $

2025 Benelux, 1

Tags: algebra , functional equation

Does there exist a function $f:\mathbb{R}\to \mathbb{R}$ such that $$f(x^2+f(y))=f(x)^2-y$$ for all $x, y\in \mathbb{R}$?

2022 Baltic Way, 1

Tags: function , algebra

Let $\mathbb{R^+}$ denote the set of positive real numbers. Assume that $f:\mathbb{R^+} \to \mathbb{R^+}$ is a function satisfying the equations: $$ f(x^3)=f(x)^3 \quad \text{and} \quad f(2x)=f(x) $$ for all $x \in \mathbb{R^+}$. Find all possible values of $f(\sqrt[2022]{2})$.

2020 Bulgaria EGMO TST, 2

Tags: function , algebra , functional equation , injectivity , bijection , surjectivity

The function $f:\mathbb{R} \to \mathbb{R}$ is such that $f(f(x+1)) = x^3+1$ for all real numbers $x$. Prove that the equation $f(x) = 0 $ has exactly one real root.

2024 Simon Marais Mathematical Competition, B2

Tags: algebra

Determine all continuous functions $f : \mathbb{R} \setminus \{1\} \to \mathbb{R}$ that satisfy \[ f(x) = (x+1) f(x^2), \] for all $x \in \mathbb{R} \setminus \{-1, 1\}$.

2010 Greece JBMO TST, 2

Tags: system of equations , algebra

Find all real $x,y,z$ such that $\frac{x-2y}{y}+\frac{2y-4}{x}+\frac{4}{xy}=0$ and $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2$.

2025 Serbia Team Selection Test for the IMO 2025, 5

Tags: algebra

Determine the smallest positive real number $\alpha$ such that there exists a sequence of positive real numbers $(a_n)$, $n \in \mathbb{N}$, with the property that for every $n \in \mathbb{N}$ it holds that: \[ a_1 + \cdots + a_{n+1} < \alpha \cdot a_n. \] [i]Proposed by Pavle Martinović[/i]

2021 Iran Team Selection Test, 3

Tags: algebra , polynomial , inequalities , number theory , relatively prime

Prove there exist two relatively prime polynomials $P(x),Q(x)$ having integer coefficients and a real number $u>0$ such that if for positive integers $a,b,c,d$ we have: $$|\frac{a}{c}-1|^{2021} \le \frac{u}{|d||c|^{1010}}$$ $$| (\frac{a}{c})^{2020}-\frac{b}{d}| \le \frac{u}{|d||c|^{1010}}$$ Then we have : $$bP(\frac{a}{c})=dQ(\frac{a}{c})$$ (Two polynomials are relatively prime if they don't have a common root) Proposed by [i]Navid Safaii[/i] and [i]Alireza Haghi[/i]

2009 Indonesia TST, 1

Tags: algebra , polynomial , quadratic , number theory

Find the smallest odd integer $ k$ such that: for every $ 3\minus{}$degree polynomials $ f$ with integer coefficients, if there exist $ k$ integer $ n$ such that $ |f(n)|$ is a prime number, then $ f$ is irreducible in $ \mathbb{Z}[n]$.

2010 BAMO, 3

Tags: algebra , inequalities

Suppose $a,b,c$ are real numbers such that $a+b \ge 0, b+c \ge 0$, and $c+a \ge 0$. Prove that $a+b+c \ge \frac{|a|+|b|+|c|}{3}$ . (Note: $|x|$ is called the absolute value of $x$ and is defined as follows. If $x \ge 0$ then $|x|= x$, and if $x < 0$ then $|x| = -x$. For example, $|6|= 6, |0| = 0$ and $|-6| = 6$.)

2018 Malaysia National Olympiad, A3

Tags: algebra

Danial went to a fruit stall that sells apples, mangoes, and papayas. Each apple costs $3$ RM ,each mango costs $4$ RM , and each papaya costs $5$ RM . He bought at least one of each fruit, and paid exactly $50$ RM. What is the maximum number of fruits that he could have bought?

2013 District Olympiad, 3

Tags: linear algebra , matrix , abstract algebra , complex number , algebra

Let $A$ be an non-invertible of order $n$, $n>1$, with the elements in the set of complex numbers, with all the elements having the module equal with 1 a)Prove that, for $n=3$, two rows or two columns of the $A$ matrix are proportional b)Does the conclusion from the previous exercise remains true for $n=4$?

2020 LIMIT Category 1, 17

Tags: algebra , limit

The sum of $k$ consecutive integers is $90$. Then the sum of all possible values of $k$ is? (A)$89$ (B)$179$ (C)$168$ (D)$119$

1996 China Team Selection Test, 3

Tags: complex number , algebra

Does there exist non-zero complex numbers $a, b, c$ and natural number $h$ such that if integers $k, l, m$ satisfy $|k| + |l| + |m| \geq 1996$, then $|ka + lb + mc| > \frac {1}{h}$ is true?