MathDB

Hard Inequality

Source: 2009 József Wildt International Mathematical Competition

4/15/2020

Let

a

,

b

,

c

be positive real numbers such that

a + b + c = 1

. Prove that

\sqrt[3]{\left (\frac{1+a}{b+c}\right )^{\frac{1-a}{bc}}\left (\frac{1+b}{c+a}\right )^{\frac{1-b}{ca}}\left (\frac{1+c}{a+b}\right )^{\frac{1-c}{ab}}} \geq 64

inequalities

Find the area of the set

Source: 2009 József Wildt International Mathematical Competition

4/15/2020

Find the area of the set

A = \{(x, y)\ |\ 1 \leq x \leq e,\ 0 \leq y \leq f (x)\}

, where \begin{tabular}{ c| c c c c |} &1 & 1& 1 & 1\\

f(x)

=&

\ln x

& 2

\ln x

& 3

\ln x

& 4

\ln x

\\ &

{(\ln x)}^2

&

4{(\ln x)}^2

&

9{(\ln x)}^2

&

16{(\ln x)}^2

\\ &

{(\ln x)}^3

&

8{(\ln x)}^3

&

27{(\ln x)}^3

&

64{(\ln x)}^3

\end{tabular}

algebra

Prove this hard inequality on triangles

Source: 2009 Jozsef Wildt International Mathematical Competition

4/26/2020

Let a triangle

\triangle ABC

and the real numbers

x

,

y

,

z>0

. Prove that

x^n\cos\frac{A}{2}+y^n\cos\frac{B}{2}+z^n\cos\frac{C}{2}\geq (yz)^{\frac{n}{2}}\sin A +(zx)^{\frac{n}{2}}\sin B +(xy)^{\frac{n}{2}}\sin C

trigonometryinequalities

Find all such numbers

Source: 2009 József Wildt International Mathematical Competition

4/15/2020

Let

\Phi

and

\Psi

denote the Euler totient and Dedekind‘s totient respectively. Determine all

n

such that

\Phi(n)

divides

n +\Psi (n)

.

number theory

Problem on euler's totient fumction

Source: 2009 József Wildt International Mathematical Competition

4/15/2020

Let

\Phi

denote the Euler totient function. Prove that for infinitely many

k

we have

\Phi (2^k+1) < 2^{k-1}

and that for infinitely many

m

one has

\Phi (2^m+1) > 2^{m-1}

number theorytotient functionfunction

Prove that this does not have any integer solutions

Source: 2009 Jozsef Wildt International Mathematical Competition

4/15/2020

Let

p_1

,

p_2

be two odd prime numbers and

\alpha

,

n

be positive integers with

\alpha >1

,

n>1

. Prove that if the equation

\left (\frac{p_2 -1}{2} \right )^{p_1} + \left (\frac{p_2 +1}{2} \right )^{p_1} = \alpha^n

does not have integer solutions for both

p_1 =p_2

and

p_1 \neq p_2

.

number theoryprime numbers

Problem related to partitions

Source: 2009 József Wildt International Mathematical Competition

4/15/2020

Prove that

p (n)= 2+ \left (p (1) + \cdots + p\left ( \left [\frac {n}{2} \right ] + \chi_1 (n)\right ) + \left (p'_2(n) + \cdots + p' _{ \left [\frac {n}{2} \right ] - 1}(n)\right )\right )

for every

n \in \mathbb {N}

with

n>2

where

\chi

denotes the principal character Dirichlet modulo 2, i.e.

\chi _1 (n) = \begin{cases} 1 & \text{if } (n,2)=1 \\ 0 &\text{if } (n,2)>1 \end{cases}

with

p (n)

we denote number of possible partitions of

n

and

p' _m(n)

we denote the number of partitions of

n

in exactly

m

sumands.

number theorypartitionsfunctions

Prove this integration inequality

Source: 2009 Jozsef Wildt International Mathematical Competition

4/17/2020

If

0<a<b

then

\int \limits_a^b \frac{\left (x^2-\left (\frac{a+b}{2} \right )^2\right )\ln \frac{x}{a} \ln \frac{x}{b}}{(x^2+a^2)(x^2+b^2)} dx > 0

integrationlogarithmsinequalitiescalculus

Prove this combinatorial formula

Source: 2009 József Wildt International Mathematical Competition

4/17/2020

If

n,p,q \in \mathbb{N}, p<q

then

{{(p+q)n}\choose{n}} \sum \limits_{k=0}^n (-1)^k {{n}\choose{k}} {{(p+q-1)n}\choose{pn-k}}= {{(p+q)n}\choose{pn}} \sum \limits_{k=0}^{\left [\frac{n}{2} \right ]} (-1)^k {{pn}\choose{k}} {{(q-p)n}\choose{n-2k}}

combinatoricsalgebra

Find a real set on which this series is convergent

Source: 2009 J&oacute;zsef Wildt International Math Competition

4/17/2020

Let the series

s(n,x)=\sum \limits_{k= 0}^n \frac{(1-x)(1-2x)(1-3x)\cdots(1-nx)}{n!}

Find a real set on which this series is convergent, and then compute its sum. Find also

\lim \limits_{(n,x)\to (\infty ,0)} s(n,x)

limitsseries

Find all real numbers m

Source: 2009 Jozsef Wildt International Math Competition

4/17/2020

Find all real numbers

m

such that

\frac{1-m}{2m} \in \{x\ |\ m^2x^4+3mx^3+2x^2+x=1\ \forall \ x\in \mathbb{R} \}

Setsnumber theory

Good functional and set problem

Source: 2009 Jozsef Wildt International Math Competition

4/17/2020

Let consider the following function set

F=\{f\ |\ f:\{1,\ 2,\ \cdots,\ n\}\to \{1,\ 2,\ \cdots,\ n\} \}

[*] Find

|F|

[*] For

n=2k

prove that

|F|< e{(4k)}^{k}

[*] Find

n

, if

|F|=540

and

n=2k

Setsfunction

Find all functions

Source: 2009 Jozsef Wildt International Math Competition

4/17/2020

Find all functions

f: (0, +\infty)\cap\mathbb{Q}\to (0, +\infty)\cap\mathbb{Q}

satisfying thefollowing conditions:
[*]

f(ax) \leq (f(x))^a

, for every

x\in (0, +\infty)\cap\mathbb{Q}

and

a \in (0, 1)\cap\mathbb{Q}

[*]

f(x+y) \leq f(x)f(y)

, for every

x,y\in (0, +\infty)\cap\mathbb{Q}

functionfunctional equation

Prove this inequality

Source: 2009 Jozsef Wildt International Math Competition

4/27/2020

If

x_k \in \mathbb{R}

(

k=1, 2, \cdots , n

) and

m \in \mathbb{N}

then
[*]

\sum \limits_{cyc} \left (x_1^2 -x_1x_2+x_2^2 \right )^m \leq 3^m \sum \limits_{k=1}^n x_k^{2m}

[*]

\prod \limits_{cyc} \left (x_1^2 -x_1x_2+x_2^2 \right )^m \leq \left (\frac{3^m}{n}\right )^m \left (\sum \limits_{k=1}^n x_k^{2m}\right )^n

inequalities

Prove this inequality

Source: 2009 Jozsef Wildt International Math Competition

4/22/2020

If

a_k >0

[

k=

1, 2,

\cdots

,

n

], then prove the following inequality

\left (\sum \limits_{k=1}^n a_k^5 \right )^4 \geq \frac{1}{n} \left (\frac{2}{n-1} \right )^5 \left (\sum \limits_{1\leq i<j\leq n} a_i^2a_j^2 \right )^5

inequalities

Prove this integral inequality

Source: 2009 Jozsef Wildt International Mathematical Competition

4/26/2020

If the function

f:[0,1]\to (0.+\infty)

is increasing and continuous, then for every

a\geq 0

the following inequality holds:

\int \limits_0^1 \frac{x^{a+1}}{f(x)}dx \leq \frac{a+1}{a+2} \int \limits_0^1 \frac{x^{a}}{f(x)}dx

integrationinequalitiescalculus

Prove this inequality on number of divisors of a number

Source: 2009 Jozsef Wildt International Mathematical Competition

4/26/2020

Prove that

\sum \limits_{k=1}^n \frac{1}{d(k)}>\sqrt{n+1}-1

For every

n\geq 1

,

d(n)

is the number of divisors of

n

number theoryinequalities

Prove this inequality on two functions

Source: 2009 Jozsef Wildt International Mathematical Competition

4/26/2020

If

a

,

b

,

c>0

and

abc=1

,

\alpha = max\{a,b,c\}

;

f,g : (0, +\infty )\to \mathbb{R}

, where

f(x)=\frac{2(x+1)^2}{x}

and

g(x)= (x+1)\left (\frac{1}{\sqrt{x}}+1\right )^2

, then

(a+1)(b+1)(c+1)\geq min\{ \{f(x),g(x) \}\ |\ x\in\{a,b,c\} \backslash \{ \alpha \}\}

functioninequalities

prove this cyclic inequality

Source: 2009 Jozsef Wildt International Mathematical Competition

4/26/2020

If

a

,

b

,

c>0

and

abc=1

, then

\sum \limits^{cyc} \frac{a+b+c^n}{a^{2n+3}+b^{2n+3}+ab} \leq a^{n+1}+b^{n+1}+c^{n+1}

for all

n\in \mathbb{N}

inequalities

Prove this inequality

Source: 2009 Jozsef Wildt International Math Competition

4/26/2020

If

x_k >0

(

k=1, 2, \cdots , n

), then

\sum \limits_{k=1}^n \left ( \frac{x_k}{1+x_1^2+x_2^2+\cdots +x_k^2} \right )^2 \leq \frac{\sum \limits_{k=1}^n x_k^2}{1+\sum \limits_{k=1}^n x_k^2}

inequalities

Prove this statement about riemann zeta function

Source: 2009 Jozsef Wildt International Mathematical Competition

4/26/2020

If

\zeta

denote the Riemann Zeta Function, and

s>1

then

\sum \limits_{k=1}^{\infty} \frac{1}{1+k^s}\geq \frac{\zeta (s)}{1+\zeta (s)}

functionRiemann Zeta Function

Prove this inequality

Source: 2009 Jozse Wildt International Mathematical Competition

4/26/2020

If

a_i >0

(

i=1, 2, \cdots , n

), then

\left (\frac{a_1}{a_2} \right )^k + \left (\frac{a_2}{a_3} \right )^k + \cdots + \left (\frac{a_n}{a_1} \right )^k \geq \frac{a_1}{a_2}+\frac{a_2}{a_3}+\cdots + \frac{a_n}{a_1}

for all

k\in \mathbb{N}

inequalities

Prove this trigonometric equation

Source: 2009 Jozsef Wildt International Mathematical Competition

4/26/2020

If

x \in \mathbb{R}\backslash \left \{\frac{k\pi}{2}\ |\ k\in \mathbb{Z} \right \}

, then

\left (\sum \limits_{0\leq j<k\leq n} \sin (2(j+k)x)\right )^2 + \left (\sum \limits_{0\leq j<k\leq n} \cos (2(j+k)x)\right )^2 = \frac{\sin ^2 nx \sin ^2 (n+1)x}{\sin ^2x \sin^22x}

trigonometry

Prove this inequality on perfect number

Source: 2009 Jozsef Wildt International Math Competition

4/27/2020

Let

a

,

n

be positive integers such that

a^n

is a perfect number. Prove that

a^{\frac{n}{\mu}}> \frac{\mu}{2}

where

\mu

denotes the number of distinct prime divisors of

a^n

number theoryinequalitiesPerfect Numbers

Prove this inequality for any point in the plane of triangle

Source: 2009 Jozsef Wildt International Math Competition

4/27/2020

If

K

,

L

,

M

denote the midpoints of the sides

AB

,

BC

,

CA

in triangle

\triangle ABC

, then for all

P

in the plane of triangle

\triangle ABC

, we have

\frac{AB}{PK}+\frac{BC}{PL}+\frac{CA}{PM} \geq \frac{AB\cdot BC \cdot CA}{4\cdot PK\cdot PL\cdot PM}

Trianglegeometryinequalities