Olympiad problems

2010 Ukraine Team Selection Test, 12

Tags: algebra , rational , sum , reciprocal sum , number theory

Is there a positive integer $n$ for which the following holds: for an arbitrary rational $r$ there exists an integer $b$ and non-zero integers $a _1, a_2, ..., a_n$ such that $r=b+\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}$ ?

2012 Junior Balkan Team Selection Tests - Romania, 2

Tags: algebra , rational , equation

Let $x$ and $y$ be two rational numbers and $n$ be an odd positive integer. Prove that, if $x^n - 2x = y^n - 2y$, then $x = y$.

2009 Greece JBMO TST, 3

Tags: algebra , rational , integration , radical

Given are the non zero natural numbers $a,b,c$ such that the number $\frac{a\sqrt2+b\sqrt3}{b\sqrt2+c\sqrt3}$ is rational. Prove that the number $\frac{a^2+b^2+c^2}{a+b+c}$ is an integer .

2013 Dutch IMO TST, 2

Tags: perfect square , number theory , rational

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

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.

1988 All Soviet Union Mathematical Olympiad, 469

Tags: rational , square , number theory

If rationals $x, y$ satisfy $x^5 + y^5 = 2 x^2 y^2$, show that $1-x y$ is the square of a rational.

1995 Tournament Of Towns, (456) 1

Tags: geometry , analytic geometry , rational , combinatorial geometry , combinatorics

Does there exist a sphere passing through only one rational point? (A rational point is a point whose Cartesian coordinates are all rational numbers.) (A Rubin)

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

1940 Putnam, B5

Tags: polynomial , rational , algebra , root

Suppose that the rational numbers $a, b$ and $c$ are the roots of the equation $x^3+ax^2 + bx + c = 0$. Find all such rational numbers $a, b$ and $c$. Justify your answer

1981 Austrian-Polish Competition, 5

Tags: polynomial , rational , algebra

Let $P(x) = x^4 + a_1x^3 + a_2x^2 + a_3x + a_4$ be a polynomial with rational coefficients. Show that if $P(x)$ has exactly one real root $\xi$, then $\xi$ is a rational number.

2013 Romania National Olympiad, 4

Tags: rational , algebra

A set $M$ of real numbers will be called [i]special [/i] if it has the properties: (i) for each $x, y \in M, x\ne y$, the numbers $x + y$ and $xy$ are not zero and exactly one of them is rational; (ii) for each $x \in M, x^2$ is irrational. Find the maximum number of elements of a [i]special [/i] set.

1996 Tuymaada Olympiad, 7

Tags: algebra , equation , exponential , rational , exponential equation

In the set of all positive real numbers define the operation $a * b = a^b$ . Find all positive rational numbers for which $a * b = b * a$.

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.

2017 Latvia Baltic Way TST, 13

Tags: rational , radical , algebra

Prove that the number $$\sqrt{1 + \frac{1}{n^2} + \frac{1}{(n+1)^2}}$$ is rational for all natural $n$.

2011 Junior Balkan Team Selection Tests - Romania, 4

Tags: rational , sum , algebra

Let $k$ and $n$ be integer numbers with $2 \le k \le n - 1$. Consider a set $A$ of $n$ real numbers such that the sum of any $k$ distinct elements of $A$ is a rational number. Prove that all elements of the set $A$ are rational numbers.

1969 Czech and Slovak Olympiad III A, 1

Tags: rational , algebra , equation , quadratic polynomial

Find all rational numbers $x,y$ such that \[\left(x+y\sqrt5\right)^2=7+3\sqrt5.\]

2013 Hanoi Open Mathematics Competitions, 15

Tags: algebra , rational

Denote by $Q$ and $N^*$ the set of all rational and positive integer numbers, respectively. Suppose that $\frac{ax + b}{cx + d} \in Q$ for every $x \in N^*$: Prove that there exist integers $A,B,C,D$ such that $\frac{ax + b}{cx + d}= \frac{Ax + B}{Cx+D}$ for all $x \in N^* $

1995 Yugoslav Team Selection Test, Problem 1

Tags: rational number , number theory , rational

Determine all triples $(x,y,z)$ of positive rational numbers with $x\le y\le z$ such that $x+y+z,\frac1x+\frac1y+\frac1z$, and xyz are natural numbers.

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]

2013 Dutch IMO TST, 2

Tags: perfect square , number theory , rational

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

2019 Ecuador NMO (OMEC), 6

Tags: geometry , rational , area

Let $n\ge 3$ be a positive integer. Danielle draws a math flower on the plane Cartesian as follows: first draw a unit circle centered on the origin, then draw a polygon of $n$ vertices with both rational coordinates on the circumference so that it has two diametrically opposite vertices, on each side draw a circumference that has the diameter of that side, and finally paints the area inside the $n$ small circles but outside the unit circle. If it is known that the painted area is rational, find all possible polygons drawn by Danielle.

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

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?

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 $n > 1$?

VMEO II 2005, 10

Tags: algebra , rational

a) Prove that for any positive integer $m > 2$, the equation $$y^3 = x^3_1 + x^3_2 + ... + x^3_m$$ always has a positive integer solution. b) Given a positive integer $n > 1$ and suppose $n \ne 3$. Prove that every rational number $x > 0$ can be expressed as $$x =\frac{a^3_1 + a^3_2 + ... + a^3_n}{b^3_1 + b^3_2 + ... + b^3_n}$$ where $a_i, b_i$ $(i = 1, . . . , n)$ are positive integers.