MathDB

3

Part of 2008 Romania National Olympiad

Problems(6)

Find three prime numbers given conditions

Source: RMO 2008, Grade 7, Problem 3

4/30/2008
Let p,q,r p,q,r be 3 prime numbers such that 5≤p<q<r 5\leq p <q<r. Knowing that 2p^2\minus{}r^2 \geq 49 and 2q^2\minus{}r^2\leq 193, find p,q,r p,q,r.
inequalitiesnumber theoryprime numbers
Simple inequality with variables in [0,1]

Source: RMO 2008, Grade 8, Problem 3

4/30/2008
Let a,b∈[0,1] a,b \in [0,1]. Prove that \frac 1{1\plus{}a\plus{}b} \leq 1 \minus{} \frac {a\plus{}b}2 \plus{} \frac {ab}3.
inequalitiesalgebra
Interesting

Source: Romania NMO 2008, 9 form, Problem 3

4/30/2008
Let n n be a positive integer and let ai a_i be real numbers, i \equal{} 1,2,\ldots,n such that ∣ai∣≤1 |a_i|\leq 1 and \sum_{i\equal{}1}^n a_i \equal{} 0. Show that \sum_{i\equal{}1}^n |x \minus{} a_i|\leq n, for every x∈R x\in \mathbb{R} with ∣x∣≤1 |x|\le 1.
inequalities proposedinequalities
Set with 2008 elements

Source: RMO 2008, Grade 10, Problem 3

4/30/2008
Let A\equal{}\{1,2,\ldots, 2008\}. We will say that set X X is an r r-set if ∅≠X⊂A \emptyset \neq X \subset A, and ∑x∈Xx≡r(mod3) \sum_{x\in X} x \equiv r \pmod 3. Let Xr X_r, r∈{0,1,2} r\in\{0,1,2\} be the set of r r-sets. Find which one of Xr X_r has the most elements.
modular arithmeticcombinatorics proposedcombinatorics
Two times derivable real function

Source: RMO 2008, 11th Grade, Problem 3

4/30/2008
Let f:R→R f: \mathbb R \to \mathbb R be a function, two times derivable on R \mathbb R for which there exist c∈R c\in\mathbb R such that \frac { f(b)\minus{}f(a) }{b\minus{}a} \neq f'(c) , for all a≠b∈R a\neq b \in \mathbb R. Prove that f''(c)\equal{}0.
functionalgebradomainreal analysiscalculusintegrationanalytic geometry
Finite ring with unique solution

Source: RMO 2008, Grade 12, Problem 3

4/30/2008
Let A A be a unitary finite ring with n n elements, such that the equation x^n\equal{}1 has a unique solution in A A, x\equal{}1. Prove that a) 0 0 is the only nilpotent element of A A; b) there exists an integer k≥2 k\geq 2, such that the equation x^k\equal{}x has n n solutions in A A.
Ring Theorylinear algebramatrixnumber theoryleast common multiplesearchgroup theory