Olympiad problems

1975 IMO Shortlist, 15

Tags: trigonometry , algebra , point set , rational , roots of unity

Can there be drawn on a circle of radius $1$ a number of $1975$ distinct points, so that the distance (measured on the chord) between any two points (from the considered points) is a rational number?

2013 Dutch IMO TST, 2

Tags: number theory , rational , perfect square

Determine all integers $n$ for which $\frac{4n-2}{n+5}$ is the square of a rational number.

2015 Hanoi Open Mathematics Competitions, 13

Tags: algebra , rational , equation

Give rational numbers $x, y$ such that $(x^2 + y^2 - 2) (x + y)^2 + (xy + 1)^2 = 0 $ Prove that $\sqrt{1 + xy}$ is a rational number.

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.

Estonia Open Senior - geometry, 2011.2.3

Tags: ratio , geometry , rational , area

Let $ABC$ be a triangle with integral side lengths. The angle bisector drawn from $B$ and the altitude drawn from $C$ meet at point $P$ inside the triangle. Prove that the ratio of areas of triangles $APB$ and $APC$ is a rational number.

2006 IMO Shortlist, 2

Tags: number theory , rational , decimal representation

For $ x \in (0, 1)$ let $ y \in (0, 1)$ be the number whose $ n$-th digit after the decimal point is the $ 2^{n}$-th digit after the decimal point of $ x$. Show that if $ x$ is rational then so is $ y$. [i]Proposed by J.P. Grossman, Canada[/i]

2015 Latvia Baltic Way TST, 8

Tags: algebra , rational , number theory

Given a fixed rational number $q$. Let's call a number $x$ [i]charismatic [/i] if we can find a natural number $n$ and integers $a_1, a_2,.., a_n$ such that $$x = (q + 1)^{a_1} \cdot (q + 2)^{a_2} \cdot ... \cdot(q + n)^{a_n} .$$ i) Prove that one can find a $q$ such that all positive rational numbers are charismatic. ii) Is it true that for all $q$, if the number $x$ is charismatic, then $x + 1$ is also charismatic?

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

2015 Estonia Team Selection Test, 3

Tags: number theory , rational

Let $q$ be a fixed positive rational number. Call number $x$ [i]charismatic [/i] if there exist a positive integer $n$ and integers $a_1, a_2, . . . , a_n$ such that $x = (q + 1)^{a_1} \cdot (q + 2)^{a_2} ...(q + n)^{a_n}$. a) Prove that $q$ can be chosen in such a way that every positive rational number turns out to be charismatic. b) Is it true for every $q$ that, for every charismatic number $x$, the number $x + 1$ is charismatic, too?

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.

2008 Postal Coaching, 2

Tags: altitude , sidelenghts , geometry , rational

Does there exist a triangle $ABC$ whose sides are rational numbers and $BC$ equals to the altitude from $A$?

2011 QEDMO 8th, 3

Tags: number theory , rational

Show that every rational number $r$ can be written as the sum of numbers in the form $\frac{a}{p^k}$ where $p$ is prime, $a$ is an integer and $k$ is natural.

2013 Nordic, 3

Tags: sequence , rational , number theory

Define a sequence ${(n_k)_{k\ge 0}}$ by ${n_{0 }= n_{1} = 1}$, and ${n_{2k} = n_k + n_{k-1} }$ and ${n_{2k+1} = n_k}$ for ${k \ge 1}$. Let further ${q_k = n_k }$ / ${ n_{k-1} }$ for each ${k \ge 1}$. Show that every positive rational number is present exactly once in the sequence ${(q_k)_{k\ge 1}}$

1949-56 Chisinau City MO, 10

Tags: root , rational , algebra

Get rid of irrationality in the denominator of a fraction $$\frac{1}{\sqrt[3]{4}+\sqrt[3]{2}+2}$$.

2007 Germany Team Selection Test, 3

Tags: number theory , rational , decimal representation

For $ x \in (0, 1)$ let $ y \in (0, 1)$ be the number whose $ n$-th digit after the decimal point is the $ 2^{n}$-th digit after the decimal point of $ x$. Show that if $ x$ is rational then so is $ y$. [i]Proposed by J.P. Grossman, Canada[/i]

2009 Dutch IMO TST, 5

Tags: geometry , rational , polygon , equal segments , analytic geometry

Suppose that we are given an $n$-gon of which all sides have the same length, and of which all the vertices have rational coordinates. Prove that $n$ is even.

1976 Bundeswettbewerb Mathematik, 3

Tags: tree , rational , algebra

A set $S$ of rational numbers is ordered in a tree-diagram in such a way that each rational number $\frac{a}{b}$ (where $a$ and $b$ are coprime integers) has exactly two successors: $\frac{a}{a+b}$ and $\frac{b}{a+b}$. How should the initial element be selected such that this tree contains the set of all rationals $r$ with $0 < r < 1$? Give a procedure for determining the level of a rational number $\frac{p}{q}$ in this tree.

2000 Regional Competition For Advanced Students, 4

Tags: rational , number theory , algebra , sequence , recurrence relation

We consider the sequence $\{u_n\}$ defined by recursion $u_{n+1} =\frac{u_n(u_n + 1)}{n}$ for $n \ge 1$. (a) Determine the terms of the sequence for $u_1 = 1$. (b) Show that if a member of the sequence is rational, then all subsequent members are also rational numbers. (c) Show that for every natural number $K$ there is a $u_1 > 1$ such that the first $K$ terms of the sequence are natural numbers.

2013 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: number theory , rational , square root

Find all integers $a$ such that $\sqrt{\frac{9a+4}{a-6}}$ is rational number

1955 Moscow Mathematical Olympiad, 316

Tags: algebra , polynomial , divisor , integer polynomial , rational

Prove that if $\frac{p}{q}$ is an irreducible rational number that serves as a root of the polynomial $f(x) = a_0x^n + a_1x^{n-1} + ... + a_n$ with integer coefficients, then $p - kq$ is a divisor of $f(k)$ for any integer $k$.

2013 Balkan MO Shortlist, N9

Tags: number theory , rational

Let $n\ge 2$ be a given integer. Determine all sequences $x_1,...,x_n$ of positive rational numbers such that $x_1^{x_2}=x_2^{x_3}=...=x_{n-1}^{x_n}=x_n^{x_1}$

2017 Federal Competition For Advanced Students, P2, 3

Tags: number theory , rational , recurrence relation , number theory with sequences

Let $(a_n)_{n\ge 0}$ be the sequence of rational numbers with $a_0 = 2016$ and $a_{n+1} = a_n + \frac{2}{a_n}$ for all $n \ge 0$. Show that the sequence does not contain a square of a rational number. Proposed by Theresia Eisenkölbl

1989 Chile National Olympiad, 5

Tags: geometry , rational , altitude

The lengths of the three sides of a $ \triangle ABC $ are rational. The altitude $ CD $ determines on the side $AB$ two segments $ AD $ and $ DB $. Prove that $ AD, DB $ are rational.

2013 Dutch IMO TST, 2

Tags: number theory , rational , perfect square

Determine all integers $n$ for which $\frac{4n-2}{n+5}$ is the square of a rational number.

2004 District Olympiad, 1

Tags: algebra , rational

We say that the real numbers $a$ and $b$ have property $P$ if: $a^2+b \in Q$ and $b^2 + a \in Q$.Prove that: a) The numbers $a= \frac{1+\sqrt2}{2}$ and $b= \frac{1-\sqrt2}{2}$ are irrational and have property $P$ b) If $a, b$ have property $P$ and $a+b \in Q -\{1\}$, then $a$ and $b$ are rational numbers c) If $a, b$ have property $P$ and $\frac{a}{b} \in Q$, then $a$ and $b$ are rational numbers.