Olympiad problems

2006 IMAR Test, 1

Tags: algebra

Consider the equation \[\frac{xy-C}{x+y}= k ,\] where all symbols used are positive integers. 1. Show that, for any (fixed) values $C, k$ this equation has at least a solution $x, y$; 2. Show that, for any (fixed) values $C, k$ this equation has at most a finite number of solutions $x, y$; 3. Show that, for any $C, n$ there exists $k = k(C,n)$ such that the equation has more than $n$ solutions $x, y$.

2020-IMOC, A2

Tags: functional equation , function , algebra

Find all function $f:\mathbb{R}^+$ $\rightarrow \mathbb{R}^+$ such that: $f(f(x) + y)f(x) = f(xy + 1) \forall x, y \in \mathbb{R}^+$

2020 Centroamerican and Caribbean Math Olympiad, 5

Tags: algebra , polynomial , inequality , inequalities , n-variable inequality

Let $P(x)$ be a polynomial with real non-negative coefficients. Let $k$ be a positive integer and $x_1, x_2, \dots, x_k$ positive real numbers such that $x_1x_2\cdots x_k=1$. Prove that $$P(x_1)+P(x_2)+\cdots+P(x_k)\geq kP(1).$$

1900 Eotvos Mathematical Competition, 3

Tags: algebra

A cliff is $300$ meters high. Consider two free-falling raindrops such that the second one leaves the top of the cliff when the first one has already fallen $0.001$ millimeters. What is the distance between the drops at the moment the first hits the ground? (Compute the answer to within $0.1$ mm. Neglect air resistance, etc.)

2007 Balkan MO, 3

Tags: algebra , polynomial , floor function , induction , number theory

Find all positive integers $n$ such that there exist a permutation $\sigma$ on the set $\{1,2,3, \ldots, n\}$ for which \[\sqrt{\sigma(1)+\sqrt{\sigma(2)+\sqrt{\ldots+\sqrt{\sigma(n-1)+\sqrt{\sigma(n)}}}}}\] is a rational number.

2011 Kazakhstan National Olympiad, 1

Tags: logarithm , algebra

Given a real number $a> 0$. How many positive real solutions of the equation is $ a^{x}=x^{a} $

2013 Kazakhstan National Olympiad, 2

Tags: algorithm , algebra

a)Does there exist for any rational number $\frac{a}{b}$ some rational numbers $x_1,x_2,....x_n$ such that $x_1*x_2*....*x_n=1$ and $x_1+x_2+....+x_n=\frac{a}{b}$ a)Does there exist for any rational number $\frac{a}{b}$ some rational numbers $x_1,x_2,....x_n$ such that $x_1*x_2*....*x_n=\frac{a}{b}$ and $x_1+x_2+....+x_n=1$

2005 Chile National Olympiad, 4

Tags: number theory , algebra

Let $f(a)$ be the largest integer less than or equal to the fourth root of " $a$". Calculate $$f(1)+f(2)+...+f(2005).$$

2016 Hanoi Open Mathematics Competitions, 14

Tags: algebra , polynomial , trinomial

Let $f (x) = x^2 + px + q$, where $p, q$ are integers. Prove that there is an integer $m$ such that $f (m) = f (2015) \cdot f (2016)$.

1978 Germany Team Selection Test, 3

Tags: algebra , recurrence relation , sequence , equation , floor function

Let $n$ be an integer greater than $1$. Define \[x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i = 1, 2, \ldots\ ,\] where $[z]$ denotes the largest integer less than or equal to $z$. Prove that \[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]

1955 Polish MO Finals, 1

Tags: algebra , polynomial

What conditions must the real numbers $ a $, $ b $, and $ c $ satisfy so that the equation $$ x^3 + ax^2 + bx + c = 0$$ has three distinct real roots forming a geometric progression?

2018 Iran MO (1st Round), 14

Tags: system of equations , algebra

For how many integers $k$ does the following system of equations has a solution other than $a=b=c=0$ in the set of real numbers? \begin{align*} \begin{cases} a^2+b^2=kc(a+b),\\ b^2+c^2 = ka(b+c),\\ c^2+a^2=kb(c+a).\end{cases}\end{align*}

1981 Romania Team Selection Tests, 4.

Tags: algebra , inequalities

Consider $x_1,\ldots,x_n>0$. Show that there exists $a_1,a_2,\ldots,a_n\in \{-1,1\}$ such that \[a_1x_1^2+a_2x_2^2+\ldots +a_nx_n^2\geqslant (a_1x_1+a_2x_2+\ldots +a_nx_n)^2.\] [i]Laurențiu Panaitopol[/i]

2014 Turkey MO (2nd round), 5

Tags: integration , algebra

Find all natural numbers $n$ for which there exist non-zero and distinct real numbers $a_1, a_2, \ldots, a_n$ satisfying \[ \left\{a_i+\dfrac{(-1)^i}{a_i} \, \Big | \, 1 \leq i \leq n\right\} = \{a_i \mid 1 \leq i \leq n\}. \]

2018 Istmo Centroamericano MO, 3

Tags: algebra , inequalities , integer sequence , sequence

Determine all sequences of integers $a_1, a_2,. . .,$ such that: (i) $1 \le a_i \le n$ for all $1 \le i \le n$. (ii) $| a_i - a_j| = | i - j |$ for any $1 \le i, j \le n$

2020 Brazil Cono Sur TST, 3

Tags: algebra

Let $a_0,a_1,a_2,\dots$ be a periodic sequence of real numbers(that is, there is a fixed positive integer $k$ such that $a_n=a_{n+k}$ for every integer $n\geq 0$). The following equality is true, for all $n\geq 0$: $a_{n+2}=\frac{1}{n+2} (a_n - \frac{n+1}{a_{n+1}})$ if $a_0=2020$, determine the value of $a_1$.

2009 Indonesia TST, 1

Tags: algebra , polynomial , search , combinatorics

Let $ n \ge 1$ and $ k \ge 3$ be integers. A circle is divided into $ n$ sectors $ a_1,a_2,\dots,a_n$. We will color the $ n$ sectors with $ k$ different colors such that $ a_i$ and $ a_{i \plus{} 1}$ have different color for each $ i \equal{} 1,2,\dots,n$ where $ a_{n \plus{} 1}\equal{}a_1$. Find the number of ways to do such coloring.

2017 Peru MO (ONEM), 3

Tags: algebra , rational number

The infinity sequence $r_{1},r_{2},...$ of rational numbers it satisfies that: $\prod_{i=1}^ {k}r_{i}=\sum_{i=1}^{k} r_{i}$. For all natural k. Show that $\frac{1}{r_{n}}-\frac{3}{4}$ is a square of rationale number for all natural $n\geq3$

1999 Abels Math Contest (Norwegian MO), 1b

Tags: inequalities , algebra

If $a,b,c,d,e$ are real numbers, prove the inequality $a^2 +b^2 +c^2 +d^2+e^2 \ge a(b+c+d+e)$.

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 $x, y \in R$.

2018 Dutch BxMO TST, 5

Tags: equation , algebra

Let $n$ be a positive integer. Determine all positive real numbers $x$ satisfying $nx^2 +\frac{2^2}{x + 1}+\frac{3^2}{x + 2}+...+\frac{(n + 1)^2}{x + n}= nx + \frac{n(n + 3)}{2}$

2018 Silk Road, 2

Tags: functional equation , algebra

Find all functions $f:\ \mathbb{R}\rightarrow\mathbb{R}$ such that for any real number $x$ the equalities are true: $f\left(x+1\right)=1+f(x)$ and $f\left(x^4-x^2\right)=f^4(x)-f^2(x).$ [url=http://matol.kz/comments/3373/show]source[/url]

2023 Macedonian Team Selection Test, Problem 3

Tags: function , algebra

Let $f:\mathbb{N} \rightarrow \mathbb{N}$ be a monotonically increasing function over the natural numbers, such that $f(f(n))=n^{2}$. What is the smallest, and what is the largest value that $f(2023)$ can take? [i]Proposed by Ilija Jovcheski[/i]

2005 IMO Shortlist, 2

Tags: function , algebra , functional equation

We denote by $\mathbb{R}^\plus{}$ the set of all positive real numbers. Find all functions $f: \mathbb R^ \plus{} \rightarrow\mathbb R^ \plus{}$ which have the property: \[f(x)f(y)\equal{}2f(x\plus{}yf(x))\] for all positive real numbers $x$ and $y$. [i]Proposed by Nikolai Nikolov, Bulgaria[/i]

2024 Nordic, 1

Tags: algebra

Let $T(a)$ be the sum of digits of $a$. For which positive integers $R$ does there exist a positive integer $n$ such that $\frac{T(n^2)}{T(n)}=R$?