MathDB

5

Part of 2011 Postal Coaching

Problems(6)

Inequality with sequence of non negative real numbers

Source:

12/31/2011
Let <an><a_n> be a sequence of non-negative real numbers such that am+n≤am+ana_{m+n} \le a_m +a_n for all m,n∈Nm,n \in \mathbb{N}. Prove that ∑k=1Nakk2≥aN4Nln⁡N\sum_{k=1}^{N} \frac{a_k}{k^2}\ge \frac{a_N}{4N}\ln N for any N∈NN \in \mathbb{N}, where ln⁡\ln denotes the natural logarithm.
inequalitieslogarithmsinequalities unsolved
Prove that n exists satisfying divisibility

Source:

12/31/2011
Let (an)n≥1(a_n )_{n\ge 1} be a sequence of integers that satisfies an=an−1−min(an−2,an−3)a_n = a_{n-1} -\text{min}(a_{n-2} , a_{n-3} ) for all n≥4n \ge 4. Prove that for every positive integer kk, there is an nn such that ana_n is divisible by 3k3^k .
modular arithmeticnumber theory unsolvednumber theory
Area inequality with brocard point

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12/31/2011
Let PP be a point inside a triangle ABCABC such that ∠PAB=∠PBC=∠PCA\angle P AB = \angle P BC = \angle P CA Suppose AP,BP,CPAP, BP, CP meet the circumcircles of triangles PBC,PCA,PABP BC, P CA, P AB at X,Y,ZX, Y, Z respectively (≠P)(\neq P) . Prove that [XBC]+[YCA]+[ZAB]≥3[ABC][XBC] + [Y CA] + [ZAB] \ge 3[ABC]
geometryinequalitiescircumcircletrigonometrygeometry unsolved
Maximum possible number of MPs

Source:

12/31/2011
The seats in the Parliament of some country are arranged in a rectangle of 1010 rows of 1010 seats each. All the 100100 MPMPs have different salaries. Each of them asks all his neighbours (sitting next to, in front of, or behind him, i.e. 44 members at most) how much they earn. They feel a lot of envy towards each other: an MPMP is content with his salary only if he has at most one neighbour who earns more than himself. What is the maximum possible number of MPMPs who are satisfied with their salaries?
geometryrectangleinequalitiescombinatorics unsolvedcombinatorics
Inequality involving square roots

Source:

12/31/2011
Let a,ba, b and cc be positive real numbers. Prove that a2+bcb+c+b2+cac+a+c2+aba+b≥ab+c+bc+a+ca+b\frac{\sqrt{a^2+bc}}{b+c}+\frac{\sqrt{b^2+ca}}{c+a}+\frac{\sqrt{c^2+ab}}{a+b}\ge\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\sqrt{\frac{c}{a+b}}
inequalitiesinequalities unsolved
K, L, M are collinear iff X is cirumcentre of EOD

Source:

12/31/2011
Let HH be the orthocentre and OO be the circumcentre of an acute triangle ABCABC. Let ADAD and BEBE be the altitudes of the triangle with DD on BCBC and EE on CACA. Let K=OD∩BE,L=OE∩ADK =OD \cap BE, L = OE \cap AD. Let XX be the second point of intersection of the circumcircles of triangles HKDHKD and HLEHLE, and let MM be the midpoint of side ABAB. Prove that points K,L,MK, L, M are collinear if and only if XX is the circumcentre of triangle EODEOD.
geometrycircumcirclegeometry unsolved