Olympiad problems

1966 IMO Longlists, 35

Let $ax^{3}+bx^{2}+cx+d$ be a polynomial with integer coefficients $a,$ $b,$ $c,$ $d$ such that $ad$ is an odd number and $bc$ is an even number. Prove that (at least) one root of the polynomial is irrational.

PEN G Problems, 25

Tags: irrational number , trigonometry , algebra , induction , polynomial

Show that $\tan \left( \frac{\pi}{m} \right)$ is irrational for all positive integers $m \ge 5$.

1953 Putnam, B7

Tags: irrational number , expansion

Let $w\in (0,1)$ be an irrational number. Prove that $w$ has a unique convergent expansion of the form $$w= \frac{1}{p_0} - \frac{1}{p_0 p_1 } + \frac{1}{ p_0 p_1 p_2 } - \frac{1}{p_0 p_1 p_2 p_3 } +\ldots,$$ where $1\leq p_0 < p_1 < p_2 <\ldots $ are integers. If $w= \frac{1}{\sqrt{2}},$ find $p_0 , p_1 , p_2.$

1994 Miklós Schweitzer, 4

Tags: real analysis , irrational number

For a given irrational number $\alpha$ , $y_{1,\alpha} = \alpha$. If $y_{n-1, \alpha}$ is given, let $y_{n, \alpha}$ be the first member of the sequence $\big (\{k \alpha \} \big) ^ \infty_{k = 1}$ to fall in the interval $(0, y_{n-1,\alpha})$ ({ x } denotes the fraction of the number x ). Show that there exists an open set $G\subset (0,1)$ , which has a limit point 0 and for all irrational $\alpha$ , infinitely many members of the $(y_{n,\alpha})$ sequence do not belong to G.

1995 Tournament Of Towns, (478) 2

Tags: irrational number , algebra , number theory

Let $p$ be the product of $n$ real numbers $x_1$, $x_2$,$...$, $x_n$. Prove that if $p - x_k$ is an odd integer for $k = 1, 2,..., n$, then each of the numbers $x_1$, $x_2$,$...$, $x_n$is irrational. (G Galperin)

2016 India IMO Training Camp, 1

Tags: irrational number , algebra

Suppose $\alpha, \beta$ are two positive rational numbers. Assume for some positive integers $m,n$, it is known that $\alpha^{\frac 1n}+\beta^{\frac 1m}$ is a rational number. Prove that each of $\alpha^{\frac 1n}$ and $\beta^{\frac 1m}$ is a rational number.

PEN G Problems, 12

Tags: trigonometry , number theory , irrational number , relatively prime

An integer-sided triangle has angles $ p\theta$ and $ q\theta$, where $ p$ and $ q$ are relatively prime integers. Prove that $ \cos\theta$ is irrational.

PEN G Problems, 4

Tags: irrational number , absolute value

Let $a, b, c$ be integers, not all zero and each of absolute value less than one million. Prove that \[\left\vert a+b\sqrt{2}+c\sqrt{3}\right\vert > \frac{1}{10^{21}}.\]

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

2002 District Olympiad, 2

Tags: irrational number , rational , algebra

a) Let $x$ be a real number such that $x^2+x$ and $x^3+2x$ are rational numbers. Show that $x$ is a rational number. b) Show that there exist irrational numbers $x$ such that $x^2+x$and $x^3-2x$ are rational.

PEN G Problems, 10

Tags: trigonometry , irrational number

Show that $\frac{1}{\pi} \arccos \left( \frac{1}{\sqrt{2003}} \right)$ is irrational.

2005 District Olympiad, 4

Tags: irrational number , polynomial , real analysis , algebra , function

Let $f:\mathbb{Q}\rightarrow \mathbb{Q}$ a monotonic bijective function. a)Prove that there exist a unique continuous function $F:\mathbb{R}\rightarrow \mathbb{R}$ such that $F(x)=f(x),\ (\forall)x\in \mathbb{Q}$. b)Give an example of a non-injective polynomial function $G:\mathbb{R}\rightarrow \mathbb{R}$ such that $G(\mathbb{Q})\subset \mathbb{Q}$ and it's restriction defined on $\mathbb{Q}$ is injective.

1993 Taiwan National Olympiad, 6

Tags: trigonometry , algebra , induction , irrational number

Let $m$ be equal to $1$ or $2$ and $n<10799$ be a positive integer. Determine all such $n$ for which $\sum_{k=1}^{n}\frac{1}{\sin{k}\sin{(k+1)}}=m\frac{\sin{n}}{\sin^{2}{1}}$.

2023 Indonesia Regional, 4

Tags: irrational number , algebra

Find all irrational real numbers $\alpha$ such that \[ \alpha^3 - 15 \alpha \text{ and } \alpha^4 - 56 \alpha \] are both rational numbers.

PEN G Problems, 18

Tags: arithmetic sequence , irrational number , 3d geometry , geometry

Show that the cube roots of three distinct primes cannot be terms in an arithmetic progression.

PEN G Problems, 22

Tags: floor function , irrational number

For a positive real number $\alpha$, define \[S(\alpha)=\{ \lfloor n\alpha\rfloor \; \vert \; n=1,2,3,\cdots \}.\] Prove that $\mathbb{N}$ cannot be expressed as the disjoint union of three sets $S(\alpha)$, $S(\beta)$, and $S(\gamma)$.

2014 South East Mathematical Olympiad, 6

Tags: irrational number , number theory

Let $a,b$ and $c$ be integers and $r$ a real number such that $ar^2+br+c=0$ with $ac\not =0$.Prove that $\sqrt{r^2+c^2}$ is an irrational number

2018 Brazil Team Selection Test, 1

Tags: irrational number , combinatorics , algebra , rational number , blackboard

The numbers $1- \sqrt{2}$, $\sqrt{2}$ and $1+\sqrt{2}$ are written on a blackboard. Every minute, if $x, y, z$ are the numbers written, then they are erased and the numbers, $x^2 + xy + y^2$, $y^2 + yz + z^2$ and $z^2 + zx + x^2$ are written. Determine whether it is possible for all written numbers to be rational numbers after a finite number of minutes.

2015 Poland - Second Round, 1

Tags: algebra , sum , rational , irrational number , polynomial

Real numbers $x_1, x_2, x_3, x_4$ are roots of the fourth degree polynomial $W (x)$ with integer coefficients. Prove that if $x_3 + x_4$ is a rational number and $x_3x_4$ is a irrational number, then $x_1 + x_2 = x_3 + x_4$.

PEN G Problems, 23

Tags: calculus , irrational number , derivative

Let $f(x)=\prod_{n=1}^{\infty} \left( 1 + \frac{x}{2^n} \right)$. Show that at the point $x=1$, $f(x)$ and all its derivatives are irrational.

1960 AMC 12/AHSME, 24

Tags: integration , geometry , calculus , logarithm , 3d geometry , irrational number , function

If $\log_{2x}216 = x$, where $x$ is real, then $x$ is: $ \textbf{(A)}\ \text{A non-square, non-cube integer} \qquad$ $\textbf{(B)}\ \text{A non-square, non-cube, non-integral rational number} \qquad$ $\textbf{(C)}\ \text{An irrational number} \qquad$ $\textbf{(D)}\ \text{A perfect square}\qquad$ $\textbf{(E)}\ \text{A perfect cube} $