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Posts Tagged ‘Inequalities’

IMC 2016 – Day 2 – Problem 7

July 28, 2016 Leave a comment

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.

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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.

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IMC 2016 – Day 1 – Problem 3

July 27, 2016 Leave a comment

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)}.

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IMO 2015 Day 2

July 11, 2015 Leave a comment

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}.

Categories: Olympiad Tags: , ,

Nice characterization side-lengths of a triangle

November 29, 2013 Leave a comment

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.

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Best approximation of a certain square root

November 29, 2013 Leave a comment

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}.

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Agreg 2012 Analysis Part 1

October 14, 2013 Leave a comment

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}.

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