MathDB

2

Part of 2005 IMAR Test

Problems(2)

Find the image of the function M:X->R

Source: GMB-IMAR 2005, Seniors, Problem 2

10/10/2005
Let n≥3n \geq 3 be an integer and let a,b∈Ra,b\in\mathbb{R} such that nb≥a2nb\geq a^2. We consider the set X={(x1,x2,…,xn)∈Rn∣∑k=1nxk=a, ∑k=1nxk2=b}. X = \left\{ (x_1,x_2,\ldots,x_n)\in\mathbb{R}^n \mid \sum_{k=1}^n x_k = a, \ \sum_{k=1}^n x_k^2 = b \right\} . Find the image of the function M:X→RM: X\to \mathbb{R} given by M(x1,x2,…,xn)=max⁡1≤k≤nxk. M(x_1,x_2,\ldots,x_n) = \max_{1\leq k\leq n} x_k . Dan Schwarz
functioninequalities unsolvedinequalities
$Q$ and $R$ are respectively the incenters in the triangles

Source: GMB-IMAR 2005, Juniors, Problem 2

10/10/2005
Let PP be an arbitrary point on the side BCBC of triangle ABCABC and let DD be the tangency point between the incircle of the triangle ABCABC and the side BCBC. If QQ and RR are respectively the incenters in the triangles ABPABP and ACPACP, prove that ∠QDR\angle QDR is a right angle. Prove that the triangle QDRQDR is isosceles if and only if PP is the foot of the altitude from AA in the triangle ABCABC.
geometryincentergeometry proposed