Olympiad problems

2012 Thailand Mathematical Olympiad, 10

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

1998 Estonia National Olympiad, 3

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

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 $R$.

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.

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.

2002 Junior Balkan Team Selection Tests - Romania, 1

Tags: rational , algebra , 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.

1980 All Soviet Union Mathematical Olympiad, 303

Tags: limit , irrational , sequence , algebra , digit

The number $x$ from $[0,1]$ is written as an infinite decimal fraction. Having rearranged its first five digits after the point we can obtain another fraction that corresponds to the number $x_1$. Having rearranged five digits of $x_k$ from $(k+1)$-th till $(k+5)$-th after the point we obtain the number $x_{k+1}$. a) Prove that the sequence $x_i$ has limit. b) Can this limit be irrational if we have started with the rational number? c) Invent such a number, that always produces irrational numbers, no matter what digits were transposed.

2019 Dürer Math Competition (First Round), P4

Tags: irrational , n-tuples , combinatorics

An $n$-tuple $(x_1, x_2,\dots, x_n)$ is called unearthly if $q_1x_1 +q_2x_2 +\dots+q_nx_n$ is irrational for any non-negative rational coefficients $q_1, q_2, \dots, q_n$ where $q_i$’s are not all zero. Prove that it is possible to select an unearthly $n$-tuple from any $2n-1$ distinct irrational numbers.

2017 Thailand Mathematical Olympiad, 1

Tags: algebra , irrational number , irrational

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

1978 Chisinau City MO, 166

Tags: analytic geometry , irrational , rational , circles

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.

2024 ITAMO, 1

Tags: absolute value , easy , irrational , 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 $x_{n+2}=x_n$.

2002 All-Russian Olympiad Regional Round, 11.1

Tags: number theory , algebra , rational , irrational

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.

1980 Swedish Mathematical Competition, 1

Tags: irrational , logarithm , algebra

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

1995 Poland - Second Round, 3

Tags: irrational , floor function , algebra

Let $a,b,c,d$ be positive irrational numbers with $a+b = 1$. Show that $c+d = 1$ if and only if $[na]+[nb] = [nc]+[nd]$ for all positive integers $n$.

1998 Estonia National Olympiad, 4

Tags: floor function , number theory , algebra , irrational

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

1976 Poland - Second Round, 5

Tags: algebra , trigonometry , rational , irrational

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

1992 Czech And Slovak Olympiad IIIA, 5

Tags: function , rational , max , algebra , irrational

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 $(7/8,8/9)$.

2006 Tournament of Towns, 3

Tags: integer part , algebra , irrational , digit

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)

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 $n > 1$?

1977 Chisinau City MO, 153

Tags: irrational , trigonometry , algebra

Prove that the number $\tan \frac{\pi}{3^n}$ is irrational for any natural $n$.

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

2012 Belarus Team Selection Test, 1

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

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)

2020 LIMIT Category 1, 1

Tags: irrational , limit

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

2004 BAMO, 5

Tags: polynomial , algebra , integer polynomial , irrational

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.)

2008 Postal Coaching, 1

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

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.

2001 Cuba MO, 5

Tags: algebra , irrational

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 $\{a\} > 0.75$.