Olympiad problems

1973 Poland - Second Round, 4

Tags: limit , algebra

Let $ x_n = (p + \sqrt{q})^n - [(p + \sqrt{q})^n] $ for $ n = 1, 2, 3, \ldots $. Prove that if $ p $, $ q $ are natural numbers satisfying the condition $ p - 1 < \sqrt{q} < p $, then $ \lim_{n\to \infty} x_n = 1 $. Attention. The symbol $ [a] $ denotes the largest integer not greater than $ a $.

2010 Poland - Second Round, 1

Tags: algebra

Solve in the real numbers $x, y, z$ a system of the equations: \[ \begin{cases} x^2 - (y+z+yz)x + (y+z)yz = 0 \\ y^2 - (z + x + zx)y + (z+x)zx = 0 \\ z^2 - (x+y+xy)z + (x+y)xy = 0. \\ \end{cases} \]

2016 Korea Summer Program Practice Test, 1

Tags: system of equations , algebra

Find all real numbers $x_1, \dots, x_{2016}$ that satisfy the following equation for each $1 \le i \le 2016$. (Here $x_{2017} = x_1$.) \[ x_i^2 + x_i - 1 = x_{i+1} \]

2011 IMAR Test, 3

Tags: functional equation , function , algebra

Given an integer number $n \ge 2$, show that there exists a function $f : R \to R$ such that $f(x) + f(2x) + ...+ f(nx) = 0$, for all $x \in R$, and $f(x) = 0$ if and only if $x = 0$.

2020 Miklós Schweitzer, 8

Tags: algebra , functional equation , number theory , function

Let $\mathbb{F}_{p}$ denote the $p$-element field for a prime $p>3$ and let $S$ be the set of functions from $\mathbb{F}_{p}$ to $\mathbb{F}_{p}$. Find all functions $D\colon S\to S$ satisfying \[D(f\circ g)=(D(f)\circ g)\cdot D(g)\] for all $f,g \in {S}$. Here, $\circ$ denotes the function composition, so $(f\circ g)(x)$ is the function $f(g(x))$, and $\cdot$ denotes multiplication, so $(f\cdot g)=f(x)g(x)$.

1987 Nordic, 3

Tags: algebra , minimum value , increasing function , functional equation in n

Let $f$ be a strictly increasing function defined in the set of natural numbers satisfying the conditions $f(2) = a > 2$ and $f(mn) = f(m)f(n)$ for all natural numbers $m$ and $n$. Determine the smallest possible value of $a$.

2013 Korea National Olympiad, 2

Tags: inequalities , algebra

Let $ a, b, c>0 $ such that $ ab+bc+ca=3 $. Prove that \[ \sum_{cyc} { \frac{ (a+b)^{3} }{ {(2(a+b)(a^2 + b^2))}^{\frac{1}{3}}} \ge 12 }\]

2012 Iran Team Selection Test, 3

Tags: algebra , induction , combinatorics , vector , function , polynomial

Let $n$ be a positive integer. Let $S$ be a subset of points on the plane with these conditions: $i)$ There does not exist $n$ lines in the plane such that every element of $S$ be on at least one of them. $ii)$ for all $X \in S$ there exists $n$ lines in the plane such that every element of $S - {X} $ be on at least one of them. Find maximum of $\mid S\mid$. [i]Proposed by Erfan Salavati[/i]

III Soros Olympiad 1996 - 97 (Russia), 9.8

Tags: algebra , trigonometry , geometric inequality

The two sides of the triangle are equal to $1$ and $x$, and $ x \ge 1$. The values $a$ and $b$ are the largest and smallest angles of this triangle, respectively. Find the greatest value of $\cos a$ and the smallest value of $\cos b$.

2018 IFYM, Sozopol, 2

Tags: algebra , inequalities

$x$, $y$, and $z$ are positive real numbers satisfying the equation $x+y+z=\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$. Prove the following inequality: $xy + yz + zx \geq 3$.

2019 Hong Kong TST, 3

Tags: floor function , algebra

Find an integral solution of the equation \[ \left \lfloor \frac{x}{1!} \right \rfloor + \left \lfloor \frac{x}{2!} \right \rfloor + \left \lfloor \frac{x}{3!} \right \rfloor + \dots + \left \lfloor \frac{x}{10!} \right \rfloor = 2019. \] (Note $\lfloor u \rfloor$ stands for the greatest integer less than or equal to $u$.)

1999 Finnish National High School Mathematics Competition, 2

Tags: arithmetic sequence , algebra

Suppose that the positive numbers $a_1, a_2,.. , a_n$ form an arithmetic progression; hence $a_{k+1}- a_k = d,$ for $k = 1, 2,... , n - 1.$ Prove that \[\frac{1}{a_1a_2}+\frac{1}{a_2a_3}+...+\frac{1}{a_{n-1}a_n}=\frac{n-1}{a_1a_n}.\]

2018 Harvard-MIT Mathematics Tournament, 2

Tags: algebra

Consider the addition problem: \begin{tabular}{ccccc} &C&A&S&H\\ +&&&M&E\\ \hline O&S&I&D&E \end{tabular} where each letter represents a base-ten digit, and $C,M,O \ne 0.$ (Distinct letters are allowed to represent the same digit.) How many ways are there to assign values to the letters so that the addition problem is true?

2006 Grigore Moisil Urziceni, 3

Tags: algebra , inequalities

Let be three positive real numbers $ x,y,z, $ whose product is $ 1. $ Prove that: $$ \sum_{\text{cyc}} \frac{3}{\sqrt{1+x+xy}} \le \sqrt 3<3\sqrt 3\le \sum_{\text{cyc}} \sqrt{1+x+xy} $$

2019 Saudi Arabia JBMO TST, 3

Tags: algebra

Given are 10 quadric equations $x^2+a_1x+b_1=0$, $x^2+a_2x+b_2=0$,..., $x^2+a_{10}x+b_{10}=0$. It is known that each of these equations has two distinct real roots and the set of all solutions is ${1,2,...10,-1,-2...,-10}$. Find the minimum value of $b_1+b_2+...+b_{10}$

2022 Korea -Final Round, P3

Tags: functional equation , algebra

A function $g \colon \mathbb{R} \to \mathbb{R}$ is given such that its range is a finite set. Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ that satisfies $$2f(x+g(y))=f(2g(x)+y)+f(x+3g(y))$$ for all $x, y \in \mathbb{R}$.

2011 Morocco National Olympiad, 3

Tags: algebra , function

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x,y, \in \mathbb{R}$, \[xf(x+xy)=xf(x)+f(x^{2})\cdot f(y).\]

2004 Vietnam Team Selection Test, 2

Tags: search , integration , calculus , function , algebra

Find all real values of $\alpha$, for which there exists one and only one function $f: \mathbb{R} \mapsto \mathbb{R}$ and satisfying the equation \[ f(x^2 + y + f(y)) = (f(x))^2 + \alpha \cdot y \] for all $x, y \in \mathbb{R}$.

2017 Hong Kong TST, 3

Tags: polynomial , algebra

Let $f(x)$ be a monic cubic polynomial with $f(0)=-64$, and all roots of $f(x)$ are non-negative real numbers. What is the largest possible value of $f(-1)$? (A polynomial is monic if its leading coefficient is 1.)

2008 Junior Balkan Team Selection Tests - Moldova, 6

Tags: algebra

Solve the equation $ 2(x^2\minus{}3x\plus{}2)\equal{}3 \sqrt{x^3\plus{}8}$, where $ x\in R$

2025 India STEMS Category B, 1

Tags: algebra , polynomial

Let $\mathcal{P}$ be the set of all polynomials with coefficients in $\{0, 1\}$. Suppose $a, b$ are non-zero integers such that for every $f \in \mathcal{P}$ with $f(a)\neq 0$, we have $f(a) \mid f(b)$. Prove that $a=b$. [i]Proposed by Shashank Ingalagavi and Krutarth Shah[/i]

2005 Miklós Schweitzer, 8

Tags: real analysis , functional equation , algebra

Determine all continuous, strictly monotone functions $\phi : \mathbb{R}^+\to\mathbb{R}$ such that $$F(x,y)=\phi^{-1} \left(\frac{x\phi(x)+y\phi(y)}{x+y}\right) + \phi^{-1} \left(\frac{y\phi(x)+x\phi(y)}{x+y}\right) $$ is homogeneous of degree 1, ie $F(tx,ty)=tF(x,y) , \forall x,y,t\in\mathbb{R}^+$ [hide=Note]F(x,y)=F(y,x) and F(x,x)=2x[/hide]

2018 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: algebra , sum , sequence

Let $a_1, a_2,...,a_{2018}$ be a sequence of numbers such that all its elements are elements of a set $\{-1,1\}$. Sum $$S=\sum \limits_{1 \leq i < j \leq 2018} a_i a_j$$ can be negative and can also be positive. Find the minimal value of this sum

2024 Bulgaria National Olympiad, 4

Tags: algebra

Do there exist $2024$ non-zero reals $a_1, a_2, \ldots, a_{2024}$, such that $$\sum_{i=1}^{2024}(a_i^2+\frac{1}{a_i^2})+2\sum_{i=1}^{2024} \frac{a_i} {a_{i+1}}+2024=2\sum_{i=1}^{2024}(a_i+\frac{1}{a_i})?$$

1950 Moscow Mathematical Olympiad, 177

Tags: combinatorics , time , distance , algebra

In a country, one can get from some point $A$ to any other point either by walking, or by calling a cab, waiting for it, and then being driven. Every citizen always chooses the method of transportation that requires the least time. It turns out that the distances and the traveling times are as follows: $1$ km takes $10$ min, $2$ km takes $15$ min, $3$ km takes $17.5 $ min. We assume that the speeds of the pedestrian and the cab, and the time spent waiting for cabs, are all constants. How long does it take to reach a point which is $6$ km from $A$?