### Archive

Posts Tagged ‘Inequalities’

## IMC 2016 – Day 2 – Problem 7

Problem 7. Today, Ivan the Confessor prefers continuous functions ${f:[0,1]\rightarrow \Bbb{R}}$ satisfying ${f(x)+f(y) \geq |x-y|}$ for all ${x,y \in [0,1]}$. Fin the minimum of ${\int_0^1 f}$ over all preferred functions.

## IMC 2016 – Day 2 – Problem 6

July 28, 2016 1 comment

Problem 6. Let ${(x_1,x_2,...)}$ be a sequence of positive real numbers satisfying ${\displaystyle \sum_{n=1}^\infty \frac{x_n}{2n-1}=1}$. Prove that

$\displaystyle \sum_{k=1}^\infty \sum_{n=1}^k \frac{x_n}{k^2} \leq 2.$

## IMC 2016 – Day 1 – Problem 3

Problem 3. Let ${n}$ be a positive integer. Also let ${a_1,a_2,...,a_n}$ and ${b_1,b_2,...,b_n}$ be reap numbers such that ${a_i+b_i >0}$ for ${i = 1,2,...,n}$. Prove that

$\displaystyle \sum_{i=1}^n \frac{a_ib_i -b_i^2}{a_i+b_i} \leq \frac{\sum_{i=1}^n a_i \cdot \sum_{i=1}^n b_i - \left(\sum_{i=1}^n b_i \right)^2 }{\sum_{i=1}^n (a_i+b_i)}.$

## IMO 2015 Day 2

Problem 4. Triangle ${ABC}$ has circumcircle ${\Omega}$ and circumcenter ${O}$. A circle ${\Gamma}$ with center ${A}$ intersects the segment ${BC}$ at points ${D}$ and ${E}$, such that ${B}$, ${D}$, ${E}$, and ${C}$ are all different and lie on line ${BC}$ in this order. Let ${F}$ and ${G}$ be the points of intersection of ${\Gamma}$ and ${\Omega}$, such that ${A}$, ${F}$, ${B}$, ${C}$, and ${G}$ lie on ${\Omega}$ in this order. Let ${K}$ be the second point of intersection of the circumcircle of triangle ${BDF}$ and the segment ${AB}$. Let ${L}$ be the second point of intersection of the circumcircle of triangle ${CGE}$ and the segment ${CA}$.

Suppose that the lines ${FK}$ and ${GL}$ are different and intersect at the point ${X}$. Prove that ${X}$ lies on the line ${AO}$.

Problem 5. Let ${\mathbb R}$ be the set of real numbers. Determine all functions ${f:\mathbb R\rightarrow\mathbb R}$ that satisfy the equation

$\displaystyle f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)$

for all real numbers ${x}$ and ${y}$.

Problem 6. The sequence ${a_1,a_2,\dots}$ of integers satisfies the conditions:

(i) ${1\le a_j\le2015}$ for all ${j\ge1}$, (ii) ${k+a_k\neq \ell+a_\ell}$ for all ${1\le k<\ell}$. Prove that there exist two positive integers ${b}$ and ${N}$ for which

$\displaystyle \left\vert\sum_{j=m+1}^n(a_j-b)\right\vert\le1007^2$

for all integers ${m}$ and ${n}$ such that ${n>m\ge N}$.

## Nice characterization side-lengths of a triangle

Find the greatest ${k}$ such that ${a,b,c>0}$ and ${kabc > a^3+b^3+c^3}$ implies that ${a,b,c}$ are the side-lengths of a triangle.

## Best approximation of a certain square root

Let ${\lambda}$ be a real number such that the inequality

$\displaystyle 0 < \sqrt{2002}-\frac{a}{b} < \frac{\lambda}{ab}$

holds for an infinity of pairs ${(a,b)}$ of natural numbers. Prove that ${\lambda\geq 5}$.

## Agreg 2012 Analysis Part 1

Part 1. Finite dimension

The goal is to prove the following theorem:

Theorem 1. Let ${A \in M_n(\Bbb{R})}$ be a square matrix with non-negative coefficients. Suppose that for every ${x \in \Bbb{R}^n\setminus \{0\}}$ with non-negative coordinates, the vector ${Ax}$ has strictly positive components. Then

• (i) the spectral radius ${\rho = \sup \{ |\lambda | : \lambda \in \Bbb{C} \text{ is an eigenvalue for }A\}}$ is a simple eigenvalue for ${A}$;
• (ii) there exists an eigenvector ${v}$ of ${A}$ associated to ${\rho}$ with strictly positive coordinates.
• (iii) any other eigenvalue of ${A}$ verifies ${|\lambda|<\rho}$;
• (iv) there exists an eigenvector of ${A^T}$ associated to ${\rho}$ with strictly positive components.

1. Consider ${(w_1,..,w_n) \in \Bbb{C}^n}$ such that ${|w_1+..+w_n|=|w_1|+...+|w_n|}$. Prove that for distinct ${j,l \in \{1,..,n\}}$ we have ${\text{re}(\overline{w_j}w_l)=|w_j||w_l|}$. Deduce that there exists ${\theta \in [0,2\pi)}$ such that ${w_j=e^{i\theta}|w_j|,\ j=1..n}$.

2. Prove that the coefficients of ${A}$ are strictly positive.

3. For ${z \in \Bbb{C}^n}$ we denote ${|z|=(|z_1|,..,|z_n|)}$. Prove that ${A|z|=|Az|}$ if and only if there exists ${\theta \in [0,2\pi)}$ such that ${z_j=e^{i\theta}|z_j|,\ j=1..n}$.

4. Denote ${\mathcal{C}= \{x \in \Bbb{R}^n : x_i \geq 0, i=1..n\}}$. Consider ${x \in \mathcal{C}}$ and denote ${e=(1,1,..,1) \in \Bbb{R}^n}$. Prove that

$\displaystyle 0 \leq (Ax|e)\leq (x|e)\max_j \sum_{k=1}^n a_{kj}.$

5. Denote ${\mathcal{E}= \{ t \geq 0 : \text{ there exists }x \in \mathcal{C} \setminus \{0\} \text{ such that } Ax-tx \in \mathcal{C}\}}$. Prove that ${\mathcal{E}}$ is an interval which does not reduces to ${\{0\}}$, it is bounded and closed.

6. Denote ${\rho=\max \mathcal{E}>0}$. Prove that if ${x \in \mathcal{C}\setminus \{0\}}$ verifies ${Ax-\rho x \in \mathcal{C}}$ then we have ${Ax=\rho x}$. Deduce that ${\rho}$ is an eigenvalue of ${A}$ and that for this eigenvalue there exists an eigenvector ${v}$ with coordinates strictly positive.

7. Consider ${z \in \Bbb{C}^n}$. Prove that ${Az=\rho z}$ and ${(z|v)=0}$ implies ${z=0}$. Deduce that ${\ker(A-\rho I)=\text{span}\{v\}}$ and every other eigenvalue of ${A}$ verifies ${|\lambda | <\rho}$.

8. Prove that every eigenvector of ${A}$ which has positive coordinates is proportional to ${v}$.