MathDB

Find the set of distinct real numbers

Source: 2019 Jozsef Wildt International Math Competition-W. 5

5/18/2020

Let

n \geq 1

. Find a set of distincts real numbers

\left(x_j\right)_{1\leq j\leq n}

such that for any bijections

f : \{1, 2,\cdots ,n\}^2 \to \{1, 2,\cdots ,n\}^2

the matrix

\left(x_{f(i,j)}\right)_{1\leq i,j\leq n}

is invertible.

matrix

Prove this inequality on logarithms

Source: 2019 Jozsef Wildt International Math Competition-W. 4

5/18/2020

If

x, y, z, t > 1

then:

\left(\log _{zxt}x\right)^2+\left(\log _{xyt}y\right)^2+\left(\log _{xyz}z\right)^2+\left(\log _{yzt}t\right)^2>\frac{1}{4}

inequalitieslogarithms

Solve this integration from Jozsef competition 2019

Source: 2019 Jozsef Wildt International Math Competition-W. 3

5/18/2020

Compute

\int \limits_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\frac{\cos x+1-x^2}{(1+x\sin x)\sqrt{1-x^2}}dx

integrationtrigonometrycalculus

Find the value of this integration

Source: 2019 Jozsef Wildt International Math Competition-W. 6

5/18/2020

Compute

\int \limits_{\frac{\pi}{6}}^{\frac{\pi}{4}}\frac{(1+\ln x)\cos x+x\sin x\ln x}{\cos^2 x + x^2 \ln^2 x}dx

integrationtrigonometrylogarithmscalculus

Find the value of this limit

Source: 2019 Jozsef Wildt International Math Competition-W. 7

5/18/2020

If

\Omega_n=\sum \limits_{k=1}^n \left(\int \limits_{-\frac{1}{k}}^{\frac{1}{k}}(2x^{10} + 3x^8 + 1)\cos^{-1}(kx)dx\right)

Then find

\Omega=\lim \limits_{n\to \infty}\left(\Omega_n-\pi H_n\right)

limitintegrationSummationSequencesHarmonic Numbers

Find the limit

Source: 2019 Jozsef Wildt International Math Competition-W. 8

5/18/2020

Let

(a_n)_{n\geq 1}

be a positive real sequence given by

a_n=\sum \limits_{k=1}^n \frac{1}{k}

. Compute

\lim \limits_{n \to \infty}e^{-2a_n} \sum \limits_{k=1}^n \left \lfloor \left(\sqrt[2k]{k!}+\sqrt[2(k+1)]{(k+1)!}\right)^2 \right \rfloor

where we denote by

\lfloor x\rfloor

the integer part of

x

.

limitSummationHarmonic NumbersSequencesgreatest integer funtion

Find the limit of this saequence

Source: 2019 Jozsef Wildt International Math Competition-W. 9

5/18/2020

Let

\alpha > 0

be a real number. Compute the limit of the sequence

\{x_n\}_{n\geq 1}

defined by

x_n=\begin{cases} \sum \limits_{k=1}^n \sinh \left(\frac{k}{n^2}\right),& \text{when}\ n>\frac{1}{\alpha}\\ 0,& \text{when}\ n\leq \frac{1}{\alpha}\end{cases}

limitSequences

Prove this identity on integration

Source: 2019 Jozsef Wildt International Math Competition-W. 10

5/18/2020

If

{si}(x) =- \int \limits_{x}^{\infty}\left(\frac{\sin t}{t}\right)dt; x>0

then

\int \limits_{e}^{e^2} \left(\frac{1}{x}\left(si\left(e^4x\right)-si\left(e^3x\right)\right)\right)\,dx=\int \limits_{3}^{e^4} \left(\frac{1}{x}\left(\operatorname{si}\left(e^2x\right)-si\left(ex\right)\right)\right)dx

integrationtrigonometryExponentscalculus

Find the limit of the expression made by 3 sequences

Source: 2019 Jozsef Wildt International Math Competition-W. 11

5/18/2020

Let

(s_n)_{n\geq 1}

be a sequence given by

s_n=-2\sqrt{n}+\sum \limits_{k=1}^n\frac{1}{\sqrt{k}}

with

\lim \limits_{n \to \infty}s_n=s=

Ioachimescu constant and

(a_n)_{n\geq 1}

,

(b_n)_{n\geq 1}

be a positive real sequences such that

\lim \limits_{n\to \infty}\frac{a_{n+1}}{na_n}=a\in \mathbb{R}^*_+, \lim \limits_{n\to \infty}\frac{b_{n+1}}{b_n\sqrt{n}}=b\in \mathbb{R}^*_+

Compute

\lim \limits_{n\to \infty}\left(1+e^{s_n}-e^{s_{n+1}}\right)^{\sqrt[n]{a_nb_n}}

Sequenceslimit

Prove this integral inequality

Source: 2019 Jozsef Wildt International Math Competition-W. 12

5/18/2020

If

0 < a < b

then:

\frac{\int \limits^{\frac{a+b}{2}}_{a}\left(\tan^{-1}t\right)dt}{\int \limits_{a}^{b}\left(\tan^{-1}t\right)dt}<\frac{1}{2}

integrationinequalitiescalculus

Find the cube root of this polynomial

Source: 2019 Jozsef Wildt International Math Competition-W. 13

5/18/2020

Let

a

,

b

and

c

be complex numbers such that

abc = 1

. Find the value of the cubic root of
\begin{tabular}{|ccc|}

b + n^3c

&

n(c - b)

&

n^2(b - c)

\\

n^2(c - a)

&

c + n^3a

&

n(a - c)

\\

n(b - a)

&

n^2(a - b)

&

a + n^3b

\end{tabular}

determinantpolynomialcomplex numbersalgebra

Find if the number is irrational

Source: 2019 Jozsef Wildt International Math Competition-W. 15

5/18/2020

It is possible to partition the set

\{100, 101,\cdots , 1000\}

into two subsets so that for any two distinct elements

x

and

y

belonging to the same subset

\sqrt[3]{x + y}

is irrational?

number theory

Prove this inequality on trigonometric function tan inverse

Source: 2019 Jozsef Wildt International Math Competition-W. 14

5/18/2020

If

a

,

b

,

c > 0

;

ab + bc + ca = 3

then:

4\left(\tan^{-1} 2\right)\left(\tan^{-1}\left(\sqrt[3]{abc}\right)\right) \leq \pi \tan^{-1}\left(1 + \sqrt[3]{abc}\right)

trigonometryinequalitiesfunction

Prove this integral trigonometric inequality

Source: 2019 Jozsef Wildt International Math Competition-W. 16

5/18/2020

If

f : [a, b] \to (0,\infty)

;

0 < a \leq b

;

f

derivable;

f'

continuous then:

\int \limits_{a}^{b}\frac{f'(x)\sqrt{f(x)}}{f^3(x) + 1}\leq \tan^{-1}\left(\frac{f(b)-f(a)}{1 + f(a)f(b)}\right)

integrationtrigonometryinequalitiescalculus

Find the limit

Source: 2019 Jozsef Wildt International Math Competition-W. 35

5/19/2020

Calculate

\lim \limits_{n \to \infty}\frac{n!\left(1+\frac{1}{n}\right)^{n^2+n}}{n^{n+\frac{1}{2}}}

limit

Prove these properties on Fibonacci and Lucas Sequences

Source: 2019 Jozsef Wildt International Math Competition-W. 19

5/18/2020

Let

\{F_n\}_{n\in\mathbb{Z}}

and

\{L_n\}_{n\in\mathbb{Z}}

denote the Fibonacci and Lucas numbers, respectively, given by

F_{n+1} = F_n + F_{n-1}\ \text{and}\ L_{n+1} = L_n + L_{n-1}\ \text{for all}\ n \geq 1

with

F_0 = 0

,

F_1 = 1

,

L_0 = 2

, and

L_1 = 1

. Prove that for integers

n \geq 1

and

j \geq 0

[*]

\sum \limits_{k=1}^n F_{k\pm j}L_{k\mp j}=F_{2n+1}-1+\begin{cases} 0, & \text{if}\ n\ \text{is even}\\ \left(-1\right)^{\pm j}F_{\pm 2j}, & \text{if}\ n\ \text{is odd} \end{cases}

[*]

\sum \limits_{k=1}^nF_{k+j}F_{k-j}L_{k+j}L_{k-j}=\frac{F_{4n+2}-1-nL_{4j}}{5}

number theoryFibonacci sequenceLucas sequence

Draw the two graphs and find their relation

Source: 2019 Jozsef Wildt International Math Competition-W. 20

5/18/2020

[*] Let

G

be a

(4, 4)

unoriented graph, 2-regulate, containing a cycle with the length 3. Find the characteristic polynomial

P_G (\lambda)

, its spectrum

Spec (G)

and draw the graph

G

. [*] Let

G'

be another 2-regulate graph, having its characteristic polynomial

P_{G'} (\lambda) = \lambda^4 - 4\lambda^2 + \alpha, \alpha \in \mathbb{R}

. Find the spectrum

Spec(G')

and draw the graph

G'

. [*] Are the graphs

G

and

G'

cospectral or isomorphic?

polynomialgraphgraph theory

Prove the value of the limit is 0

Source: 2019 Jozsef Wildt International Math Competition-W. 18

5/18/2020

Let

\{c_k\}_{k\geq1}

be a sequence with

0 \leq c_k \leq 1

,

c_1 \neq 0

,

\alpha > 1

. Let

C_n = c_1 + \cdots + c_n

. Prove

\lim \limits_{n \to \infty}\frac{C_1^{\alpha}+\cdots+C_n^{\alpha}}{\left(C_1+\cdots +C_n\right)^{\alpha}}=0

limitSequences

Find the limit in terms of A and B

Source: 2019 Jozsef Wildt International Math Competition-W. 21

5/18/2020

Let

f

be a continuously differentiable function on

[0, 1]

and

m \in \mathbb{N}

. Let

A = f(1)

and let

B=\int \limits_{0}^1 x^{-\frac{1}{m}}f(x)dx

. Calculate

\lim \limits_{n \to \infty} n\left(\int \limits_{0}^1 f(x)dx-\sum \limits_{k=1}^n \left(\frac{k^m}{n^m}-\frac{(k-1)^m}{n^m}\right)f\left(\frac{(k-1)^m}{n^m}\right)\right)

in terms of

A

and

B

.

integrationlimit

Find if A/B is irrational

Source: 2019 Jozsef Wildt International Math Competition-W. 22

5/18/2020

Let

A

and

B

the series:

A=\sum \limits_{n=1}^{\infty}\frac{C_{2n}^1}{C_{2n}^0+C_{2n}^1+\cdots +C_{2n}^{2n}},\ B=\sum \limits_{n=1}^{\infty}\frac{\Gamma \left(n+\frac{1}{2}\right) }{\Gamma \left(n+\frac{5}{2}\right)}

Study if

\frac{A}{B}

is irrational number.

Gamma functionnumber theory

Proove this inequality

Source: 2019 Jozsef Wildt International Math Competition-W. 24

5/18/2020

If

a

,

b

,

c > 0

, prove that

\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\geq \frac{a+b}{a+b+2c}+\frac{b+c}{2a+b+c}+\frac{c+a}{a+2b+c}

inequalities

Proof this inequality on triangle

Source: 2019 Jozsef Wildt International Math Competition-W. 23

5/18/2020

If

b

,

c

are the legs, and

a

is the hypotenuse of a right triangle, prove that

\left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\geq 5+3\sqrt{2}

inequalitiesTriangle

Prove this inequality on gamma function

Source: 2019 Jozsef Wildt International Math Competition-W. 25

5/18/2020

Let

x_i

,

y_i

,

z_i

,

w_i \in \mathbb{R}^+, i = 1, 2,\cdots n

, such that

\sum \limits_{i=1}^nx_i=nx,\ \sum \limits_{i=1}^ny_i=ny,\ \sum \limits_{i=1}^nw_i=nw

\Gamma \left(z_i\right)\geq \Gamma \left(w_i\right),\ \sum \limits_{i=1}^n\Gamma \left(z_i\right)=n\Gamma^* (z)

Then

\sum \limits_{i=1}^n \frac{\left(\Gamma \left(x_i\right)+\Gamma \left(y_i\right)\right)^2}{\Gamma \left(z_i\right)-\Gamma \left(w_i\right)}\geq n\frac{\left(\Gamma \left(x\right)+\Gamma \left(y\right)\right)^2}{\Gamma^* \left(z\right)-\Gamma \left(w\right)}

Gamma functionSummationinequalitiesfunction

Find all continuous functions

Source: 2019 Jozsef Wildt International Math Competition-W. 27

5/18/2020

Find all continuous functions

f : \mathbb{R} \to \mathbb{R}

such that

f(-x)+\int \limits_0^xtf(x-t)dt=x,\ \forall\ x\in \mathbb{R}

integrationfunctional equationfunction

Prove these two expression

Source: 2019 Jozsef Wildt International Math Competition-W. 26

5/18/2020

Let

n \in \mathbb{N}

,

n \geq 2

,

a_1, a_2, \cdots , a_n \in \mathbb{R}

and

a_n = max \{a_1, a_2,\cdots , a_n\}

[*]If

t_k

,

t'_k \in \mathbb{R}

,

k \in \{1, 2,\cdots , n\}

,

t_k \leq t'_k

, for any

k \in \{1, 2, \cdots, n - 1\}

and

\sum \limits_{k=1}^nt_k=\sum \limits_{k=1}^nt'_k

Prove that

\sum \limits_{k=1}^nt_ka_k\geq \sum \limits_{k=1}^nt'_ka_k

[*] If

b_k

,

c_k \in \mathbb{R}^*_+

,

k \in \{1, 2,\cdots , n\}

,

b_k \leq c_k

for any

k \in \{1, 2,\cdots, k - 1\}

and

b_1b_2\cdots b_n=c_1c_2\cdots c_n

Prove that

\prod \limits_{k=1}^n b_k^{a_k}\geq \prod \limits_{k=1}^nc_k^{a_k}

Summation