ML + NL> AL , 2 midpoints and 1 angle bisector, acute

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ML + NL> AL , 2 midpoints and 1 angle bisector, acute

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Source: 2001 Ukraine NMO 10.7

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#1 h Jul 8, 2021, 7:50 AM

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Given an acute triangle $ABC$ . Let $AL$ be its bisector, $M$ and $N$ be the midpoints of the sides $AC$ and $AB$ , respectively. Prove that $ML + NL> AL$ .

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#2 h Jul 9, 2021, 8:30 AM

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$\definecolor{A}{RGB}{255,0,205}\color{A}\fbox{Solution}$
WLOG $AB\le AC$ . Let $K$ be the midpoint of $BC$ . Then $B,L,K,C$ lie on one line in this order.
$\definecolor{A}{RGB}{61,148,134}\color{A}KM||AB\implies \angle MKC=\angle ABC<90^\circ\implies \angle MKB>90^\circ\implies ML\ge MK=\frac12\cdot AB=AN$ Triangle inequality yields
$\definecolor{A}{RGB}{61,148,134}\color{A}ML+NL\ge AN+NL>AL.\blacksquare$ #1850

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#3 h Jul 9, 2021, 8:55 AM

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Given a chessboard with size n rows and 9 columns. Each box
filled with the number 0 or 1 such that
i. Every two lines is different.
ii. Each row contains 3 squares containing the number 1.
iii. For every three rows, there is a column such that the intersection of
between that column and those three rows (three squares) all contain numbers
0.
Let N be the largest possible value of n.
a. Determine N.
b. A pair of squares is said to be "matched" if the two squares share a side. Count
the number of matching pairs on a chessboard of size N × 9.

anybody can you help me

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#4 h Jul 9, 2021, 8:59 AM

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akubisa wrote:

Given a chessboard with size n rows and 9 columns. Each box
filled with the number 0 or 1 such that
i. Every two lines is different.
ii. Each row contains 3 squares containing the number 1.
iii. For every three rows, there is a column such that the intersection of
between that column and those three rows (three squares) all contain numbers
0.
Let N be the largest possible value of n.
a. Determine N.
b. A pair of squares is said to be "matched" if the two squares share a side. Count
the number of matching pairs on a chessboard of size N × 9.

anybody can you help me

@above please post your problem, as a new post in this forum (High School Olympiads)

This post has been edited 2 times. Last edited by parmenides51, Jul 9, 2021, 9:38 AM

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