2019 Jozsef Wildt International Math Competition

Part of Jozsef Wildt International Math Competition

Subcontests

(70)

1

Prove this trigonometric inequality

x \in \left(0,\frac{\pi}{2}\right)

\left(\frac{\sin \left(\frac{\pi}{2}\sin x\right)}{\sin x}\right)^2+\left(\frac{\sin \left(\frac{\pi}{2}\cos x\right)}{\cos x}\right)^2\geq 3

1

Prove this geometric inequality for external and internal bisector

\overline{w_a}, \overline{w_b}, \overline{w_c}

the external angle-bisectors in triangle

ABC

\sum \limits_{cyc} \frac{1}{w_a}\leq \sqrt{\frac{(s^2 - r^2 - 4Rr)(8R^2 - s^2 - r^2 - 2Rr)}{8s^2R^2r}}

1

Prove these two inequalities of tetrahedron

In all tetrahedron

ABCD

\displaystyle{\sum \limits_{cyc}\frac{h_a-r}{h_a+r}\geq \sum \limits_{cyc}\frac{h_a^t-r^t}{(h_a+r)^t}}

\displaystyle{\sum \limits_{cyc}\frac{2r_a-r}{2r_a+r}\geq \sum \limits_{cyc}\frac{2r_a^t-r^t}{(2r_a+r)^t}}

t\in [0,1]

1

Prove this geometric inequality

T

the Toricelli point of the triangle

ABC

AB^2 \times BC^2 \times CA^2 \geq 3(TA^2\times TB + TB^2 \times TC + TC^2 \times TA)(TA\times TB^2 + TB \times TC^2 + TC \times TA^2)

1

Prove this inequality as the bound of the integration

0 < a \leq b

\frac{2}{\sqrt{3}}\tan^{-1}\left(\frac{2(b^2 - a^2)}{(a^2+2)(b^2+2)}\right)\leq \int \limits_a^b \frac{(x^2+1)(x^2+x+1)}{(x^3 + x^2 + 1) (x^3 + x + 1)}dx\leq \frac{4}{\sqrt{3}}\tan^{-1}\left(\frac{(b-a)\sqrt{3}}{a+b+2(1+ab)}\right)

1

Prove this 8 variable inequality

a

b

c \geq 1

y \geq x \geq 1

p

q

r > 0

\left(\frac{1+y\left(a^pb^qc^r\right)^{\frac{1}{p+q+r}}}{1+x\left(a^pb^qc^r\right)^{\frac{1}{p+q+r}}}\right)^{\frac{p+q+r}{\left(a^pb^qc^r\right)^{\frac{1}{p+q+r}}}}\left(\frac{1+xa}{1+ya}\right)^{\frac{p}{a}}\left(\frac{1+xb}{1+yb}\right)^{\frac{q}{b}}\left(\frac{1+xc}{1+yc}\right)^{\frac{r}{c}}

\geq \prod \limits_{cyc}\left(\frac{1+y\left(a^pb^q\right)^{\frac{1}{p+q}}}{1+x\left(a^pb^q\right)^{\frac{1}{p+q}}}\right)^{\frac{p+q}{\left(a^pb^q\right)^{\frac{1}{p+q}}}}

1

Solve for x, y, z, t

Prove that exist different natural numbers

x

y

z

t

256\times 2019^{180n+1}=2x^9-2y^6+z^5-t^4

n\in \mathbb{N}^*

1

Prove this 6 variable inequality

b_k \geq a_k \geq 0

(k = 1, 2, 3)

\alpha \geq 1

(\alpha+3)\sum \limits_{cyc}(b_1-a_1)\left((b_2+b_3)^{\alpha+2}+(a_2+a_3)^{\alpha+2}-(a_2+b_3)^{\alpha+1}-(b_2+a_3)^{\alpha+1}\right)

\leq (\alpha+2)(\alpha+3)\sum \limits_{cyc}(b_1-a_1)(b_2-a_2)(b_3^{\alpha+1}-a_3^{\alpha+1})

+ (b_3 + b_2 + a_1)^{\alpha+3}+(b_3 + a_2 + a_1)^{\alpha+3}+(a_3 + b_2 + a_1)^{\alpha+3}+(a_3 + a_2 + b_1)^{\alpha+3}

-(b_3 + b_2 + b_1)^{\alpha+3}-(b_3 + a_2 + a_1)^{\alpha+3}-(a_3 + b_2 + b_1)^{\alpha+3}-(a_3 + a_2 + a_1)^{\alpha+3}

1

Prove this integral inequality involving pi

\int \limits_0^{\frac{\pi}{2}}(\cos x)^{1+\sqrt{2n+1}}dx\leq \frac{2^{n-1}n!\sqrt{\pi}}{\sqrt{2(2n+1)!}}

n\in \mathbb{N}^*

1

Prove this 3 real number variable inequality

a

b

c \in \mathbb{R}

\sum \limits_{cyc} \sqrt{(c+a)^2b^2+c^2a^2}+\sqrt{5}\left |\sum \limits_{cyc} \sqrt{ab}\right |\geq \sum \limits_{cyc}\sqrt{(ab+2bc+ca)^2+(b+c)^2a^2}

1

Prove these two inequalities on heights, inradius and exradius of a tetrahedon

In all tetrahedron

ABCD

(n(n+2))^{\frac{1}{n}} \sum \limits_{cyc} \left(\frac{(h_a-r)^2}{(h_a^n-r^n)(h_a^{n+2}-r^{n+2})}\right)^{\frac{1}{n}}\leq \frac{1}{r^2}

(n(n+2))^{\frac{1}{n}} \sum \limits_{cyc} \left(\frac{(r_a-r)^2}{(r_a^n-r^n)(r_a^{n+2}-r^{n+2})}\right)^{\frac{1}{n}}\leq \frac{1}{r^2}

n\in \mathbb{N}^*

1

Prove this inequality of radii of cicumsphere and insphere of a tetrahedron

[ABCD]

tetrahedron we denote with

\alpha

\beta

\gamma

the measures, in radians, of the angles of the three pairs of opposite edges and with

r

R

the lengths of the rays of the sphere inscribed and respectively circumscribed the tetrahedron. Demonstrate inequality

\left(\frac{3r}{R}\right)^3\leq \sin \frac{\alpha +\beta +\gamma}{3}

(A refinement of inequality

R \geq 3r

1

Prove this inequality on radius of spheres of a tetrahedron

[ABCD]

tetrahedron having all the faces acute angled triangles, is denoted by

r_X

R_X

the radius lengths of the circle inscribed and circumscribed respectively on the face opposite to the

X \in \{A,B,C,D\}

R

the length of the radius of the sphere circumscribed to the tetrahedron. Show that inequality occurs

8R^2 \geq (r_A + R_A)^2 + (r_B + R_B)^2 + (r_C + R_C)^2 + (r_D + R_D)^2

1

Prove that the sequence is convergent and find its limit

x_1=\frac{1}{\sqrt[n+1]{n!}}

x_2=\frac{1}{\sqrt[n+1]{(n-1)!}}

n\in \mathbb{N}^*

f:\left(\left .\frac{1}{\sqrt[n+1]{(n+1)!}},1\right.\right] \to \mathbb{R}

f(x)=\frac{n+1}{x\ln (n+1)!+(n+1)\ln \left(x^x\right)}

Prove that the sequence

(a_n)_{n\geq1}

a_n=\int \limits_{x_1}^{x_2}f(x)dx

is convergent and compute

\lim \limits_{n \to \infty}a_n

1

Prove that integration of this two functions from a to b is 1

f

g

h : [a, b] \to \mathbb{R}

, three integrable functions such that:

\int \limits_a^b fgdx=\int \limits_a^bghdx=\int \limits_a^bhfdx=\int \limits_a^bg^2dx\int \limits_a^bh^2dx=1

\int \limits_a^bg^2dx=\int \limits_a^bh^2dx=1

1

Prove this algebraic inequality

a_1,a_2,\cdots ,a_n

n

positive numbers such that

\sum \limits_{i=1}^n\sqrt{a_i}=\sqrt{n}

\prod \limits_{i=1}^{n-1}\left(1+\frac{1}{a_i}\right)^{a_{i+1}}\left(1+\frac{1}{a_n}\right)^{a_1}\geq 1+\frac{n}{\sum \limits_{i=1}^na_i}

1

Prove this inequality

x_1, x_2,\geq , x_n

be a positive numbers,

k \geq 1

. Then the following inequality is true:

\left(x_1^k+x_2^k+\cdots +x_n^k\right)^{k+1}\geq \left(x_1^{k+1}+x_2^{k+1}\cdots +x_n^{k+1}\right)^k+2\left(\sum \limits_{1\leq i<j\leq n}x_i^kx_j\right)^k

1

Find the limit of this summation

\lim \limits_{n \to \infty}\frac{1}{n}\sum \limits_{k=1}^n\frac{\sqrt[n+k+1]{n+1}-\sqrt[n+k]{n}}{\sqrt[n+k]{n+1}-\sqrt[n+k]{n}}

1

Prove that the value of this integration is 0

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

a periodic and continue function with period

T

F : \mathbb{R} \to \mathbb{R}

antiderivative of

f

\int \limits_0^T \left[F(nx)-F(x)-f(x)\frac{(n-1)T}{2}\right]dx=0

1

Prove this 5 variable inequality

a

b

c

d

e

be real strictly positive real numbers such that

abcde = 1

. Then is true the following inequality:

\frac{de}{a(b+1)}+\frac{ea}{b(c+1)}+\frac{ab}{c(d+1)}+\frac{bc}{d(e+1)}+\frac{cd}{e(a+1)}\geq \frac{5}{2}

1

Prove this 3 variable inequality

x

y

z > 0

\lambda \in (-\infty, 0) \cup (1,+\infty)

x + y + z = 1

\sum \limits_{cyc} x^{\lambda}y^{\lambda}\sum \limits_{cyc}\frac{1}{(x+y)^{2\lambda}}\geq 9\left(\frac{1}{4}-\frac{1}{9}\sum \limits_{cyc}\frac{1}{(x+1)^2} \right)^{\lambda}

1

Prove this cyclic inequality

a

b

c \in (0,+\infty)

. Then the following inequality is true:

\sqrt{(a+b)(b+c)}+\sqrt{(b+c)(c+a)}+\sqrt{(c+a)(a+b)}+a+b+c\leq \left(ab+bc+ca\right)\left(\frac{1}{\sqrt{ab}}+\frac{1}{\sqrt{bc}}+\frac{1}{\sqrt{ca}}\right)

1

Prove this integral inequality of the convex function

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

a convex function and

\alpha, \beta, \gamma > 0

\frac{1}{6\alpha}\int \limits_0^{6\alpha}f(x)dx\ +\ \frac{1}{6\beta}\int \limits_0^{6\beta}f(x)dx\ +\ \frac{1}{6\gamma}\int \limits_0^{6\gamma}f(x)dx

\geq \frac{1}{3\alpha +2\beta +\gamma}\int \limits_0^{3\alpha +2\beta +\gamma}f(x)dx\ +\ \frac{1}{\alpha +3\beta +2\gamma}\int \limits_0^{\alpha +3\beta +2\gamma}f(x)dx\

+\ \frac{1}{2\alpha +\beta +3\gamma}\int \limits_0^{2\alpha +\beta +3\gamma}f(x)dx

1

Prove the inequality and find the limit

a

b

c

d > 0

, show inequality:

\left(\tan^{-1}\left(\frac{ad-bc}{ac+bd}\right)\right)^2\geq 2\left(1-\frac{ac+bd}{\sqrt{\left(a^2+b^2\right)\left(c^2+d^2\right)}}\right)

\lim \limits_{n \to \infty}n^{\alpha}\left(n- \sum \limits_{k=1}^n\frac{n^+k^2-k}{\sqrt{\left(n^2+k^2\right)\left(n^2+(k-1)^2\right)}}\right)

\alpha \in \mathbb{R}

1

Prove this trigonometric inequality

x

y

z > 0

x^2 + y^2 + z^2 = 3

x^3\tan^{-1}\frac{1}{x}+y^3\tan^{-1}\frac{1}{y}+z^3\tan^{-1}\frac{1}{z}<\frac{\pi \sqrt{3}}{2}

1

Prove this trigonometric inequality with complex numbers

Consider the complex numbers

a_1, a_2,\cdots , a_n

n \geq 2

. Which have the following properties:
[*]

|a_i|=1

\forall

i=1,2,\cdots , n

\sum \limits_{k=1}^n arg(a_k)\leq \pi

Show that the inequality

\left(n^2\cot \left(\frac{\pi}{2n}\right)\right)^{-1}\left |\sum \limits_{k=0}^n(-1)^k\left[3n^2-(8k+5)n+4k(k+1)\sigma_k\right]\right |\geq \sqrt{\left(1+\frac{1}{n}\right)^2\cot^2 \left(\frac{\pi}{2n}\right)}+16\left |\sum \limits_{k=0}^n(-1)^k\sigma_k\right |

\sigma_0=1

\sigma_k=\sum \limits_{1\leq i_1\leq i_2\leq \cdots \leq i_k\leq n}a_{i_1}a_{i_2}\cdots a_{i_k}

\forall

k=1,2,\cdots , n

1

Prove this property of the matrix

We consider a natural number

n

n \geq 2

and the matrices
\begin{tabular}{cc}

A= \begin{pmatrix} 1 & 2 & 3 & \cdots & n\\ n & 1 & 2 & \cdots & n - 1\\ n - 1 & n & 1 & \cdots & n - 2\\ \cdots & \cdots & \cdots & \cdots & \cdots\\2 & 3 & 4 & \cdots & 1 \end{pmatrix}

\end{tabular}
Show that

\epsilon^ndet\left(I_n-A^{2n}\right)+\epsilon^{n-1}det\left(\epsilon I_n-A^{2n}\right)+\epsilon^{n-2}det\left(\epsilon^2 I_n-A^{2n}\right)+\cdots +det\left(\epsilon^n I_n-A^{2n}\right)

=n(-1)^{n-1}\left[\frac{n^n(n+1)}{2}\right]^{2n^2-4n}\left(1+(n+1)^{2n}\left(2n+(-1)^n{{2n}\choose{n}}\right)\right)

\epsilon \in \mathbb{C}\backslash \mathbb{R}

\epsilon^{n+1}=1

1

Prove for the expression based on limit of sequence of polynomial

Consider the sequence of polynomials

P_0(x) = 2

P_1(x) = x

P_n(x) = xP_{n-1}(x) - P_{n-2}(x)

n \geq 2

x_n

be the greatest zero of

P_n

in the the interval

|x| \leq 2

\lim \limits_{n \to \infty}n^2\left(4-2\pi +n^2\int \limits_{x_n}^2P_n(x)dx\right)=2\pi - 4-\frac{\pi^3}{12}

1

Prove this property

p

q

l

strictly positive real numbers, consider the following problem: for

y \geq 0

fixed, determine the values

x \geq 0

x^p - lx^q \leq y

S(y)

the set of solutions of this problem. Prove that if one has

p < q

\epsilon \in (0, l^\frac{1}{p-q})

0 \leq x \leq \epsilon

x \in S(y)

x\leq ky^{\delta},\ \text{where}\ k=\epsilon\left(\epsilon^p-l\epsilon^q\right)^{-\frac{1}{p}}\ \text{and}\ \delta=\frac{1}{p}

1

Show that number of points inside the tetrahedron

n \in \mathbb{N}

\mathbb{R}^3

the regular tetrahedron with vertices

O(0, 0, 0)

A(n, 9n, 4n)

B(9n, 4n, n)

C(4n, n, 9n)

. Show that the number

N

(x, y, z)

[x, y, z \in \mathbb{Z}]

inside or on the boundary of the tetrahedron

OABC

N=\frac{343n^3}{3}+\frac{35n^2}{2}+\frac{7n}{6}+1

1

Prove this expression on fibonacci numbers is divisible by 150 for any n

f_n

n

th Fibonacci number defined by recurrence

f_{n+1} - f_n - f_{n-1} = 0

n \in \mathbb{N}

and initial conditions

f_0 = 0

f_1 = 1

. Prove that for any

n \in \mathbb{N}

(n - 1) (n + 1) (2nf_{n+1} - (n + 6) f_n)

is divisible by 150 for any

n \in \mathbb{N}

1

Prove this inequality on area of the triangle

a

b

c

be the sides of an acute triangle

\triangle ABC

x, y, z \geq 0

xy+yz+zx=1

holds inequality:

a^2x + b^2y + c^2z \geq 4F

F

is the area of the triangle

\triangle ABC

1

Prove these 3 properties

u

v

w

complex numbers such that:

u + v + w = 1

u^2 + v^2 + w^2 = 3

uvw = 1

. Prove that
[*]

u

v

w

are distinct numbers two by two [*] If

S(k)= u^k + v^k + w^k

S(k)

is an odd natural number [*] The expression

\frac{u^{2n+1} - v^{2n+1}}{u-v}+\frac{v^{2n+1}-w^{2n+1}}{v-w}+\frac{w^{2n+1}-u^{2n+1}}{w-u}

is an integer number.

1

Find the limit of this expression

a > 1

\lim \limits_{n \to \infty}\sqrt[n]{\prod \limits_{k=2}^n \left(a-a^{\frac{1}{k}}\right)}

1

Prove this inequalitiy 2

a

b

c > 0

n \in \mathbb{N}^*

, prove the inequality

(a - b)\left(\frac{a}{b}\right)^n+(b - c)\left(\frac{b}{c}\right)^n+(c - a)\left(\frac{c}{a}\right)^n\geq (a - b)\frac{a}{b}+(b - c)\frac{b}{c}+(c - a)\frac{c}{a}

1

Find the limit

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

1

Prove this inequality

a

b

c

be positive real numbers and let

m

n

(m \geq n)

be positive integers. Prove that

\frac{a^{n-1}b^{n-1}c^{m-n-1}}{a^{m+n}+b^{m+n}+a^nb^nc^{m-n}}+\frac{b^{n-1}bc^{n-1}a^{m-n-1}}{b^{m+n}+c^{m+n}+b^nc^na^{m-n}}+\frac{c^{n-1}a^{n-1}b^{m-n-1}}{c^{m+n}+a^{m+n}+c^na^nb^{m-n}}\leq \frac{1}{abc}

1

Prove this inequality made by 4 sequences

0 < \frac{1}{q} \leq \frac{1}{p} < 1

\frac{1}{p}+\frac{1}{q}=1

u_k

v_k

a_k

b_k

be non-negative real sequences such as

u^2_k > a^p_k

v_k > b^q_k

k = 1, 2,\cdots , n

0 < m_1\leq u_k \leq M_1

0 < m_2 \leq v_k \leq M_2

\left(\sum \limits_{k=1}^n\left(l^p\left(u_k+v_k\right)^2-\left(a_k+b_k\right)^p\right)\right)^{\frac{1}{p}}\geq \left(\sum \limits_{k=1}^n\left(u_k^2-a_k^p\right)\right)^{\frac{1}{p}}\left(\sum \limits_{k=1}^n\left(v_k^2-b_k^p\right)\right)^{\frac{1}{p}}

l=\frac{M_1M_2+m_1m_2}{2\sqrt{m_1M_1m_2M_2}}

1

Prove this inequality based on 4 squences

u_k

v_k

a_k

b_k

be non-negative real sequences such as

u_k > a_k

v_k > b_k

k = 1, 2,\cdots , n

0 < m_1 \leq u_k \leq M_1

0 < m_2 \leq v_k \leq M_2

\sum \limits_{k=1}^n(lu_kv_k-a_kb_k)\geq \left(\sum \limits_{k=1}^n\left(u_k^2-a_k^2\right)\right)^\frac{1}{2}\left(\sum \limits_{k=1}^n\left(v_k^2-b_k^2\right)\right)^\frac{1}{2}

l=\frac{M_1M_2+m_1m_2}{2\sqrt{m_1M_1m_2M_2}}

1

Prove this integral inequality

a, b \in \Gamma

a < b

and the differentiable function

f : [a, b] \to \Gamma

f (a) = a

f (b) = b

\int \limits_{a}^{b} \left(f'(x)\right)^2dx \geq b-a

1

Prove the limit and find the value on zeta function

\lim \limits_{n \to \infty} \left(n+\frac{1}{4}-\zeta(3)-\zeta(5)-\cdots -\zeta(2n+1)\right)=0

\sum \limits_{n=1}^{\infty} \left(n+\frac{1}{4}-\zeta(3)-\zeta(5)-\cdots -\zeta(2n+1)\right)

1

Prove this expression

\int \limits_0^{\infty} e^{3t}\frac{4e^{4t}(3t - 1) + 2e^{2t}(15t - 17) + 18(1 - t)}{\left(1 + e^{4t} - e^{2t}\right)^2}=12\sum \limits_{k=0}^{\infty}\frac{(-1)^k}{(2k + 1)^2}-10

1

Find the number of switched remained on

In a room, we have 2019 aligned switches, connected to 2019 light bulbs, all initially switched on. Then, 2019 people enter the room one by one, performing the operation: The first, uses all the switches; the second, every second switch; the third, every third switch, and so on. How many lightbulbs remain switched on, after all the people entered ?

1

Find all continuous functions

Find all continuous functions

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

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

1

Prove these two expression

n \in \mathbb{N}

n \geq 2

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

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

t_k

t'_k \in \mathbb{R}

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

t_k \leq t'_k

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

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

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

b_k

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

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

b_k \leq c_k

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

b_1b_2\cdots b_n=c_1c_2\cdots c_n

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

1

Prove this inequality on gamma function

x_i

y_i

z_i

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

\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)

\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)}

1

Proove this inequality

a

b

c > 0

\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}

1

Proof this inequality on triangle

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}

1

Find if A/B is irrational

A

B

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)}

\frac{A}{B}

is irrational number.

1

Find the limit in terms of A and B

f

be a continuously differentiable function on

[0, 1]

m \in \mathbb{N}

A = f(1)

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

\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)

A

B

1

Draw the two graphs and find their relation

G

(4, 4)

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

P_G (\lambda)

Spec (G)

and draw the graph

G

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

G'

cospectral or isomorphic?

1

Prove these properties on Fibonacci and Lucas Sequences

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

\{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

F_0 = 0

F_1 = 1

L_0 = 2

L_1 = 1

. Prove that for integers

n \geq 1

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}

1

Prove the value of the limit is 0

\{c_k\}_{k\geq1}

be a sequence with

0 \leq c_k \leq 1

c_1 \neq 0

\alpha > 1

C_n = c_1 + \cdots + c_n

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

1

Find the limit on fibonacci numbers

f_n=\left(1+\frac{1}{n}\right)^n\left((2n-1)!F_n\right)^{\frac{1}{n}}

\lim \limits_{n \to \infty}(f_{n+1} - f_n)

F_n

n

th Fibonacci number (given by

F_0 = 0

F_1 = 1

F_{n+1} = F_n + F_{n-1}

n \geq 1

1

Prove this integral trigonometric inequality

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

0 < a \leq b

f

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)

1

Find if the number is irrational

It is possible to partition the set

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

into two subsets so that for any two distinct elements

x

y

belonging to the same subset

\sqrt[3]{x + y}

1

Prove this inequality on trigonometric function tan inverse

a

b

c > 0

ab + bc + ca = 3

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)

1

Find the cube root of this polynomial

a

b

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

1

Prove this integral inequality

0 < a < b

\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}

1

Find the limit of the expression made by 3 sequences

(s_n)_{n\geq 1}

be a sequence given by

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

\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}^*_+

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

1

Prove this identity on integration

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

\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

1

Find the limit of this saequence

\alpha > 0

be a real number. Compute the limit of the sequence

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

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}

1

Find the limit

(a_n)_{n\geq 1}

be a positive real sequence given by

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

\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

1

Find the value of this limit

\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)

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

1

Find the value of this integration

\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

1

Find the set of distinct real numbers

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

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

1

Prove this inequality on logarithms

x, y, z, t > 1

\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}

1

Solve this integration from Jozsef competition 2019

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

1

Prove this inequality

0<a\leq c\leq b

\frac{(b^{30}-a^{30})(b^{30}-c^{30})}{36b^{10}}\leq \frac{(b^{25}-a^{25})(b^{25}-c^{25})}{25}\leq \frac{(b^{30}-a^{30})(b^{30}-c^{30})}{36(ac)^{10}}

1

Find the value of this infinte summation about pell numbers

The Pell numbers

P_n

P_0 = 0

P_1 = 1

P_n=2P_{n-1}+P_{n-2}

n\geq 2

\sum \limits_{n=1}^{\infty} \left (\tan^{-1}\frac{1}{P_{2n}}+\tan^{-1}\frac{1}{P_{2n+2}}\right )\tan^{-1}\frac{2}{P_{2n+1}}