Olympiad problems

2008 Postal Coaching, 1

Tags: number theory , irrational , fraction , natural , algebra , floor function

For each positive $ x \in \mathbb{R}$, define $ E(x)=\{[nx]: n\in \mathbb{N}\}$ Find all irrational $ \alpha >1$ with the following property: If a positive real $ \beta$ satisfies $ E(\beta) \subset E(\alpha)$. then $ \frac{\beta}{\alpha}$ is a natural number.

1992 Czech And Slovak Olympiad IIIA, 5

Tags: irrational , max , rational , algebra , function

The function $f : (0,1) \to R$ is defined by $f(x) = x$ if $x$ is irrational, $f(x) = \frac{p+1}{q}$ if $x =\frac{p}{q}$ , where $(p,q) = 1$. Find the maximum value of $f$ on the interval $(7/8,8/9)$.

2012 Belarus Team Selection Test, 1

Tags: sum , min , irrational , cubic , cubic equation , polynomial , algebra

A cubic trinomial $x^3 + px + q$ with integer coefficients $p$ and $q$ is said to be [i]irrational [/i] if it has three pairwise distinct real irrational roots $a_1,a_2, a_3$ Find all irrational cubic trinomials for which the value of $|a_1| + [a_2| + |a_3|$ is the minimal possible. (E. Barabanov)

1989 All Soviet Union Mathematical Olympiad, 499

Tags: irrational , sum , sum of powers , rational , algebra

Do there exist two reals whose sum is rational, but the sum of their $n$ th powers is irrational for all $n > 1$? Do there exist two reals whose sum is irrational, but the sum of whose $n$ th powers is rational for all $n > 1$?

2024 ITAMO, 1

Tags: irrational , easy , absolute value , algebra

Let $x_0=2024^{2024}$ and $x_{n+1}=|x_n-\pi|$ for $n \ge 0$. Show that there exists a value of $n$ such that $x_{n+2}=x_n$.

2000 All-Russian Olympiad Regional Round, 8.1

Tags: algebra , irrational

Non-zero numbers $a$ and $b$ satisfy the equality $$a^2b^2(a^2b^2 + 4) = 2(a^6 + b^6).$$ Prove that at least one of them is irrational.

1976 Poland - Second Round, 5

Tags: irrational , rational , trigonometry , algebra

Prove that if $ \cos \pi x =\frac{1}{3} $ then $ x $ is an irrational number.

1980 Swedish Mathematical Competition, 1

Tags: irrational , logarithm , algebra

Show that $\log_{10} 2$ is irrational.

2011 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: combinatorics , irrational , irrational number

Prove that among any $6$ irrational numbers you can pick three numbers $a$, $b$ and $c$ such that numbers $a+b$, $b+c$ and $c+a$ are irrational

2004 BAMO, 5

Tags: integer polynomial , irrational , algebra , polynomial

Find (with proof) all monic polynomials $f(x)$ with integer coefficients that satisfy the following two conditions. 1. $f (0) = 2004$. 2. If $x$ is irrational, then $f (x)$ is also irrational. (Notes: Apolynomial is monic if its highest degree term has coefficient $1$. Thus, $f (x) = x^4-5x^3-4x+7$ is an example of a monic polynomial with integer coefficients. A number $x$ is rational if it can be written as a fraction of two integers. A number $x$ is irrational if it is a real number which cannot be written as a fraction of two integers. For example, $2/5$ and $-9$ are rational, while $\sqrt2$ and $\pi$ are well known to be irrational.)

2002 All-Russian Olympiad Regional Round, 11.1

Tags: rational , irrational , number theory , algebra

The real numbers $x$ and $y$ are such that for any distinct odd primes $p$ and $q$ the number $x^p + y^q$ is rational. Prove that $x$ and $y$ are rational numbers.

1969 Swedish Mathematical Competition, 2

Tags: irrational , algebra , trigonometry

Show that $\tan \frac{\pi}{3n}$ is irrational for all positive integers $n$.

1998 Estonia National Olympiad, 3

Tags: area , geometry , ratio , triangle area , irrational , perpendicular

In a triangle $ABC$, the bisector of the largest angle $\angle A$ meets $BC$ at point $D$. Let $E$ and $F$ be the feet of perpendiculars from $D$ to $AC$ and $AB$, respectively. Let $R$ denote the ratio between the areas of triangles $DEB$ and $DFC$. (a) Prove that, for every real number $r > 0$, one can construct a triangle ABC for which $R$ is equal to $r$. (b) Prove that if $R$ is irrational, then at least one side length of $\vartriangle ABC$ is irrational. (c) Give an example of a triangle $ABC$ with exactly two sides of irrational length, but with rational $R$.

1974 Putnam, B3

Tags: irrational , cosine

Prove that if $a$ is a real number such that $$\cos \pi a= \frac{1}{3},$$ then $a$ is irrational.

2012 Thailand Mathematical Olympiad, 10

Tags: number theory , inequalities , algebra , irrational

Let $x$ be an irrational number. Show that there are integers $m$ and $n$ such that $\frac{1}{2555}< mx + n <\frac{1}{2012}$

1978 Chisinau City MO, 166

Tags: rational , analytic geometry , circles , irrational

It is known that at least one coordinate of the center $(x_0, y_0)$ of the circle $(x -x_0)^2+ (y -y_0)^2 = R^2$ is irrational. Prove that on the circle itself there are at most two points with rational coordinates.

1998 Estonia National Olympiad, 4

Tags: algebra , irrational , number theory , floor function

A real number $a$ satisfies the equality $\frac{1}{a} = a - [a]$. Prove that $a$ is irrational.

2001 Cuba MO, 5

Tags: irrational , algebra

Let $p$ and $q$ be two positive integers such that $1 \le q \le p$. Also let $a = \left( p +\sqrt{p^2 + q} \right)^2$. a) Prove that the number $a$ is irrational. b) Show that $\{a\} > 0.75$.

2002 Junior Balkan Team Selection Tests - Romania, 1

Tags: algebra , rational , radical , irrational

Let $a$ be an integer. Prove that for any real number $x, x^3 < 3$, both the numbers $\sqrt{3 -x^2}$ and $\sqrt{a - x^3}$ cannot be rational.

2020 LIMIT Category 1, 1

Tags: limit , irrational

If $a$ is a rational number and $b$ is an irrational number such that $ab$ is rational, then which of the following is false? (A)$ab^2$ is irrational (B)$a^2b$ is rational (C)$\sqrt{ab}$ is rational (D)$a+b$ is irrational

1983 Polish MO Finals, 2

Tags: irrational , inequalities , algebra

Let be given an irrational number $a$ in the interval $(0,1)$ and a positive integer $N$. Prove that there exist positive integers $p,q,r,s$ such that $\frac{p}{q} < a <\frac{r}{s}, \frac{r}{s} -\frac{p}{q}<\frac{1}{N}$, and $rq- ps = 1$.

2015 China Northern MO, 2

Tags: geometry , irrational

As shown in figure , a circle of radius $1$ passes through vertex $A$ of $\vartriangle ABC$ and is tangent to the side $BC$ at the point $D$ , intersect sides $AB$ and $AC$ at points $E$ and $F$ respectively . Also$ EF$ bisects $\angle AFD$, and $\angle ADC = 80^o$ , Is there a triangle that satisfies the condition, so that $\frac{AB+BC+CA}{AD^2}$ is an irrational number, and the irrational number is the root of a quadratic equation with integral coefficients? If it does not exist, please prove it; if it exists, find the quadratic equation that satisfies the condition. [img]https://cdn.artofproblemsolving.com/attachments/b/9/9e3b955b6d6df35832dd0c0a2d1d2a1e1cce94.png[/img]

2012 Ukraine Team Selection Test, 6

Tags: algebra , irrational number , irrational , digit , decimal representation

For the positive integer $k$ we denote by the $a_n$ , the $k$ from the left digit in the decimal notation of the number $2^n$ ($a_n = 0$ if in the notation of the number $2^n$ less than the digits). Consider the infinite decimal fraction $a = \overline{0, a_1a_2a_3...}$. Prove that the number $a$ is irrational.

2006 Tournament of Towns, 3

Tags: algebra , digit , integer part , irrational

The $n$-th digit of number $a = 0.12457...$ equals the first digit of the integer part of the number $n\sqrt2$. Prove that $a$ is irrational number. (6)

2017 Thailand Mathematical Olympiad, 1

Tags: irrational number , irrational , algebra

Let $p$ be a prime. Show that $\sqrt[3]{p} +\sqrt[3]{p^5} $ is irrational.