MathDB

4

Part of 2006 Romania Team Selection Test

Problems(5)

Combinatorial inequality

Source: Romanian TST 1 2006, Problem 4

4/19/2006
The real numbers a1,a2,…,ana_1,a_2,\dots,a_n are given such that ∣ai∣≤1|a_i|\leq 1 for all i=1,2,…,ni=1,2,\dots,n and a1+a2+⋯+an=0a_1+a_2+\cdots+a_n=0.
a) Prove that there exists k∈{1,2,…,n}k\in\{1,2,\dots,n\} such that ∣a1+2a2+⋯+kak∣≤2k+14. |a_1+2a_2+\cdots+ka_k|\leq\frac{2k+1}{4}.
b) Prove that for n>2n > 2 the bound above is the best possible.
Radu Gologan, Dan Schwarz
inequalitiesinductioninequalities proposed
Another modulus inequality

Source: Romanian IMO TST 2006, day 2, problem 4

4/22/2006
Let xix_i, 1≤i≤n1\leq i\leq n be real numbers. Prove that ∑1≤i<j≤n∣xi+xj∣≥n−22∑i=1n∣xi∣. \sum_{1\leq i<j\leq n}|x_i+x_j|\geq\frac{n-2}{2}\sum_{i=1}^n|x_i|. Discrete version by Dan Schwarz of a Putnam problem
inequalitiesPutnaminequalities proposed
Not very nice

Source: Romanian IMO TST 2006, day 3, problem 4

5/16/2006
Let a,b,ca,b,c be positive real numbers such that a+b+c=3a+b+c=3. Prove that: 1a2+1b2+1c2≥a2+b2+c2. \frac 1{a^2}+\frac 1{b^2}+\frac 1{c^2} \geq a^2+b^2+c^2.
inequalitiescalculusalgebrathree variable inequalityromania
Inequality with semiconvex sequences

Source: Romanian IMO TST 2006, day 4, problem 4

5/19/2006
Let pp, qq be two integers, q≥p≥0q\geq p\geq 0. Let n≥2n \geq 2 be an integer and a0=0,a1≥0,a2,…,an−1,an=1a_0=0, a_1 \geq 0, a_2, \ldots, a_{n-1},a_n = 1 be real numbers such that ak≤ak−1+ak+12, ∀ k=1,2,…,n−1. a_{k} \leq \frac{ a_{k-1} + a_{k+1} } 2 , \ \forall \ k=1,2,\ldots, n-1 . Prove that (p+1)∑k=1n−1akp≥(q+1)∑k=1n−1akq. (p+1) \sum_{k=1}^{n-1} a_k^p \geq (q+1) \sum_{k=1}^{n-1} a_k^q .
inequalitiesfunctionintegrationcalculusderivativeinequalities unsolved
Metric relation describing the position of the incenter

Source: Romanian IMO TST 2006, day 5, problem 4

5/23/2006
Let ABCABC be an acute triangle with AB≠ACAB \neq AC. Let DD be the foot of the altitude from AA and ω\omega the circumcircle of the triangle. Let ω1\omega_1 be the circle tangent to ADAD, BDBD and ω\omega. Let ω2\omega_2 be the circle tangent to ADAD, CDCD and ω\omega. Let ℓ\ell be the interior common tangent to both ω1\omega_1 and ω2\omega_2, different from ADAD.
Prove that ℓ\ell passes through the midpoint of BCBC if and only if 2BC=AB+AC2BC = AB + AC.
geometryincentercircumcircleparallelogramtrigonometrygeometry proposed