Olympiad problems

2021 Saudi Arabia IMO TST, 1

For a non-empty set $T$ denote by $p(T)$ the product of all elements of $T$. Does there exist a set $T$ of $2021$ elements such that for any $a\in T$ one has that $P(T)-a$ is an odd integer? Consider two cases: 1) All elements of $T$ are irrational numbers. 2) At least one element of $T$ is a rational number.

2019 Teodor Topan, 1

Tags: algebra , irrational number

[b]a)[/b] Give example of two irrational numbers $ a,b $ having the property that $ a^3,b^3,a+b $ are all rational. [b]b)[/b] Prove that if $ x,y $ are two nonnegative real numbers having the property that $ x^3,y^3,x+y $ are rational, then $ x $ and $ y $ are both rational. [i]Mihai Piticari[/i] and [i]Vladimir Cerbu[/i]

PEN F Problems, 10

Tags: irrational number , rational number

The set $ S$ is a finite subset of $ [0,1]$ with the following property: for all $ s\in S$, there exist $ a,b\in S\cup\{0,1\}$ with $ a, b\neq s$ such that $ s \equal{}\frac{a\plus{}b}{2}$. Prove that all the numbers in $ S$ are rational.

2007 Croatia Team Selection Test, 2

Tags: floor function , irrational number , number theory

Prove that the sequence $a_{n}=\lfloor n\sqrt 2 \rfloor+\lfloor n\sqrt 3 \rfloor$ contains infintely many even and infinitely many odd numbers.

2004 Junior Tuymaada Olympiad, 1

Tags: algebra , irrational number

A positive rational number is written on the blackboard. Every minute Vasya replaces the number $ r $ written on the board with $ \sqrt {r + 1} $. Prove that someday he will get an irrational number.

2005 Serbia Team Selection Test, 1

Tags: algebra , trigonometry , complex number , irrational number

Prove that there is n rational number $r$ such that $cosr\pi=\frac{3}{5}$

PEN G Problems, 28

Tags: irrational number

Do there exist real numbers $a$ and $b$ such that [list=a][*] $a+b$ is rational and $a^n +b^n $ is irrational for all $n \in \mathbb{N}$ with $n \ge 2$? [*] $a+b$ is irrational and $a^n +b^n $ is rational for all $n \in \mathbb{N}$ with $n \ge 2$?[/list]

1959 AMC 12/AHSME, 34

Tags: vieta , irrational number , complex number , imaginary numbers , quadratic

Let the roots of $x^2-3x+1=0$ be $r$ and $s$. Then the expression $r^2+s^2$ is: $ \textbf{(A)}\ \text{a positive integer} \qquad\textbf{(B)}\ \text{a positive fraction greater than 1}\qquad\textbf{(C)}\ \text{a positive fraction less than 1}$ $\textbf{(D)}\ \text{an irrational number}\qquad\textbf{(E)}\ \text{an imaginary number}$

1967 IMO Longlists, 14

Tags: number theory , approximation , decimal representation , rational number , irrational number

Which fractions $ \dfrac{p}{q},$ where $p,q$ are positive integers $< 100$, is closest to $\sqrt{2} ?$ Find all digits after the point in decimal representation of that fraction which coincide with digits in decimal representation of $\sqrt{2}$ (without using any table).

PEN G Problems, 5

Tags: irrational number

Let $ a, b, c$ be integers, not all equal to $ 0$. Show that \[ \frac{1}{4a^{2}\plus{}3b^{2}\plus{}2c^{2}}\le\vert\sqrt[3]{4}a\plus{}\sqrt[3]{2}b\plus{}c\vert.\]

2007 Croatia Team Selection Test, 2

Tags: number theory , floor function , irrational number

Prove that the sequence $a_{n}=\lfloor n\sqrt 2 \rfloor+\lfloor n\sqrt 3 \rfloor$ contains infintely many even and infinitely many odd numbers.

PEN G Problems, 21

Tags: inequalities , floor function , irrational number

Prove that if $ \alpha$ and $ \beta$ are positive irrational numbers satisfying $ \frac{1}{\alpha}\plus{}\frac{1}{\beta}\equal{} 1$, then the sequences \[ \lfloor\alpha\rfloor,\lfloor 2\alpha\rfloor,\lfloor 3\alpha\rfloor,\cdots\] and \[ \lfloor\beta\rfloor,\lfloor 2\beta\rfloor,\lfloor 3\beta\rfloor,\cdots\] together include every positive integer exactly once.

2016 Nigerian Senior MO Round 2, Problem 7

Tags: induction , irrational number , algebra

Prove that $(2+\sqrt{3})^{2n}+(2-\sqrt{3})^{2n}$ is an even integer and that $(2+\sqrt{3})^{2n}-(2-\sqrt{3})^{2n}=w\sqrt{3}$ for some positive integer $w$, for all integers $n \geq 1$.

PEN G Problems, 15

Tags: inequalities , irrational number , logarithm

Prove that for any $ p, q\in\mathbb{N}$ with $ q > 1$ the following inequality holds: \[ \left\vert\pi\minus{}\frac{p}{q}\right\vert\ge q^{\minus{}42}.\]

1967 IMO Shortlist, 2

Tags: number theory , approximation , decimal representation , rational number , irrational number

Which fractions $ \dfrac{p}{q},$ where $p,q$ are positive integers $< 100$, is closest to $\sqrt{2} ?$ Find all digits after the point in decimal representation of that fraction which coincide with digits in decimal representation of $\sqrt{2}$ (without using any table).

2018 District Olympiad, 1

Tags: algebra , irrational number

Show that $$\sqrt{n + \left[ \sqrt{n} +\frac12\right]}$$ is an irrational number, for every positive integer $n$.

2021 Brazil National Olympiad, 3

Tags: number theory , perfect square , floor function , irrational number , power

Find all positive integers $k$ for which there is an irrational $\alpha>1$ and a positive integer $N$ such that $\left\lfloor\alpha^{n}\right\rfloor$ is a perfect square minus $k$ for every integer $n$ with $n>N$.

1987 National High School Mathematics League, 3

Tags: irrational number

In rectangular coordinate system, define that if and only if both $x$-axis and $y$-axis of a point are rational numbers, we call it rational point. If $a$ is an irrational number, then in all lines that passes $(a,0)$, $\text{(A)}$There are infinitely many lines, on which there are at least two rational points. $\text{(B)}$There are exactly $n(n\geq2)$ lines, on which there are at least two rational points. $\text{(C)}$There are exactly 1 line, on which there are at least two rational points. $\text{(D)}$Every line passes at least one rational point.

1999 Kazakhstan National Olympiad, 1

Tags: algebra , irrational number , irrational

Prove that for any real numbers $ a_1, a_2, \dots, a_ {100} $ there exists a real number $ b $ such that all numbers $ a_i + b $ ($ 1 \leq i \leq 100 $) are irrational.

2001 Poland - Second Round, 1

Tags: modular arithmetic , number theory , arithmetic sequence , irrational number

Find all integers $n\ge 3$ for which the following statement is true: Any arithmetic progression $a_1,\ldots ,a_n$ with $n$ terms for which $a_1+2a_2+\ldots+na_n$ is rational contains at least one rational term.

PEN G Problems, 24

Tags: irrational number

Let $\{a_{n}\}_{n \ge 1}$ be a sequence of positive numbers such that \[a_{n+1}^{2}= a_{n}+1, \;\; n \in \mathbb{N}.\] Show that the sequence contains an irrational number.

1967 AMC 12/AHSME, 9

Tags: trapezoid , irrational number , geometry

Let $K$, in square units, be the area of a trapezoid such that the shorter base, the altitude, and the longer base, in that order, are in arithmetic progression. Then: $\textbf{(A)}\ K \; \text{must be an integer} \qquad \textbf{(B)}\ K \; \text{must be a rational fraction} \\ \textbf{(C)}\ K \; \text{must be an irrational number} \qquad \textbf{(D)}\ K\; \text{must be an integer or a rational fraction} \qquad$ $\textbf{(E)}\ \text{taken alone neither} \; \textbf{(A)} \; \text{nor} \; \textbf{(B)} \; \text{nor} \; \textbf{(C)} \; \text{nor} \; \textbf{(D)} \; \text{is true}$

2012 District Olympiad, 2

Tags: number theory , algebra , irrational number

Let $a, b$ and $c$ be positive real numbers such that $$a^2+ab+ac-bc = 0.$$ a) Show that if two of the numbers $a, b$ and $c$ are equal, then at least one of the numbers $a, b$ and $c$ is irrational. b) Show that there exist infinitely many triples $(m, n, p)$ of positive integers such that $$m^2 + mn + mp -np = 0.$$

2008 Junior Balkan Team Selection Tests - Romania, 4

Tags: floor function , number theory , irrational number

Let $ a,b$ be real nonzero numbers, such that number $ \lfloor an \plus{} b \rfloor$ is an even integer for every $ n \in \mathbb{N}$. Prove that $ a$ is an even integer.

PEN G Problems, 2

Tags: inequalities , irrational number

Prove that for any positive integers $ a$ and $ b$ \[ \left\vert a\sqrt{2}\minus{}b\right\vert >\frac{1}{2(a\plus{}b)}.\]