PEN G Problems

Part of PEN Problems

Subcontests

(30)

1

G 30

\alpha=0.d_{1}d_{2}d_{3} \cdots

be a decimal representation of a real number between

0

1

r

be a real number with

\vert r \vert<1

\alpha

r

are rational, must

\sum_{i=1}^{\infty} d_{i}r^{i}

be rational? [*] If

\sum_{i=1}^{\infty} d_{i}r^{i}

r

\alpha

must be rational?

1

G 29

p(x)=x^{3}+a_{1}x^{2}+a_{2}x+a_{3}

have rational coefficients and have roots

r_{1}

r_{2}

r_{3}

r_{1}-r_{2}

is rational, must

r_{1}

r_{2}

r_{3}

1

G 28

Do there exist real numbers

a

b

a+b

is rational and

a^n +b^n

is irrational for all

n \in \mathbb{N}

n \ge 2

a+b

is irrational and

a^n +b^n

is rational for all

n \in \mathbb{N}

n \ge 2

1

G 27

1<a_{1}<a_{2}<\cdots

be a sequence of positive integers. Show that

\frac{2^{a_{1}}}{{a_{1}}!}+\frac{2^{a_{2}}}{{a_{2}}!}+\frac{2^{a_{3}}}{{a_{3}}!}+\cdots

1

G 26

g \ge 2

is an integer, then two series

\sum_{n=0}^{\infty}\frac{1}{g^{n^{2}}}\;\; \text{and}\;\; \sum_{n=0}^{\infty}\frac{1}{g^{n!}}

both converge to irrational numbers.

1

G 25

\tan \left( \frac{\pi}{m} \right)

is irrational for all positive integers

m \ge 5

1

G 24

\{a_{n}\}_{n \ge 1}

be a sequence of positive numbers such that

a_{n+1}^{2}= a_{n}+1, \;\; n \in \mathbb{N}.

Show that the sequence contains an irrational number.

1

G 23

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.

1

G 22

For a positive real number

\alpha

S(\alpha)=\{ \lfloor n\alpha\rfloor \; \vert \; n=1,2,3,\cdots \}.

\mathbb{N}

cannot be expressed as the disjoint union of three sets

S(\alpha)

S(\beta)

S(\gamma)

1

G 21

\alpha

\beta

are positive irrational numbers satisfying \frac{1}{\alpha}\plus{}\frac{1}{\beta}\equal{} 1, then the sequences

\lfloor\alpha\rfloor,\lfloor 2\alpha\rfloor,\lfloor 3\alpha\rfloor,\cdots

\lfloor\beta\rfloor,\lfloor 2\beta\rfloor,\lfloor 3\beta\rfloor,\cdots

together include every positive integer exactly once.

1

G 20

You are given three lists A, B, and C. List A contains the numbers of the form

10^{k}

in base 10, with

k

any integer greater than or equal to 1. Lists B and C contain the same numbers translated into base 2 and 5 respectively:

\begin{array}{lll}A & B & C \\ 10 & 1010 & 20 \\ 100 & 1100100 & 400 \\ 1000 & 1111101000 & 13000 \\ \vdots & \vdots & \vdots \end{array}.

Prove that for every integer

n > 1

, there is exactly one number in exactly one of the lists B or C that has exactly

n

1

G 19

n

be an integer greater than or equal to 3. Prove that there is a set of

n

points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with a rational area.

1

G 18

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

1

G 17

p, q \in \mathbb{N}

satisfy the inequality

\exp(1)\cdot( \sqrt{p+q}-\sqrt{q})^{2}<1.

\ln \left(1+\frac{p}{q}\right)

1

G 16

For each integer

n \ge 1

, prove that there is a polynomial

P_{n}(x)

with rational coefficients such that

x^{4n}(1-x)^{4n}=(1+x)^{2}P_{n}(x)+(-1)^{n}4^{n}

. Define the rational number

a_{n}

a_{n}= \frac{(-1)^{n-1}}{4^{n-1}}\int_{0}^{1}P_{n}(x) \; dx,\; n=1,2, \cdots.

a_{n}

satisfies the inequality

\left\vert \pi-a_{n}\right\vert < \frac{1}{4^{5n-1}}, \; n=1,2, \cdots.

1

G 15

Prove that for any

p, q\in\mathbb{N}

q > 1

the following inequality holds: \left\vert\pi\minus{}\frac{p}{q}\right\vert\ge q^{\minus{}42}.

1

G 14

For which angles

\theta

\theta

a rational number of degrees, is {\tan}^{2}\theta\plus{}{\tan}^{2}2\theta is irrational?

1

G 13

It is possible to show that \csc\frac{3\pi}{29}\minus{}\csc\frac{10\pi}{29}\equal{} 1.999989433.... Prove that there are no integers

j

k

n

n

satisfying \csc\frac{j\pi}{n}\minus{}\csc\frac{k\pi}{n}\equal{} 2.

1

G 12

An integer-sided triangle has angles

p\theta

q\theta

p

q

are relatively prime integers. Prove that

\cos\theta

1

G 11

\cos 1^{\circ}

1

G 10

\frac{1}{\pi} \arccos \left( \frac{1}{\sqrt{2003}} \right)

1

G 9

\cos \frac{\pi}{7}

1

G 8

e=\sum^{\infty}_{n=0} \frac{1}{n!}

1

G 7

\pi

1

G 6

Prove that for any irrational number

\xi

, there are infinitely many rational numbers

\frac{m}{n}

\left( (m,n) \in \mathbb{Z}\times \mathbb{N}\right)

\left\vert \xi-\frac{n}{m}\right\vert < \frac{1}{\sqrt{5}m^{2}}.

1

G 5

a, b, c

be integers, not all equal to

0

. Show that \frac{1}{4a^{2}\plus{}3b^{2}\plus{}2c^{2}}\le\vert\sqrt[3]{4}a\plus{}\sqrt[3]{2}b\plus{}c\vert.

1

G 4

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}}.

1

G 3

Prove that there exist positive integers

m

n

such that \left\vert\frac{m^{2}}{n^{3}}\minus{}\sqrt{2001}\right\vert <\frac{1}{10^{8}}.

1

G 2

Prove that for any positive integers

a

b

\left\vert a\sqrt{2}\minus{}b\right\vert >\frac{1}{2(a\plus{}b)}.

1

G 1

Find the smallest positive integer

n

0< \sqrt[4]{n}-\lfloor \sqrt[4]{n}\rfloor < 0.00001.