MathDB

3

Part of 2015 India IMO Training Camp

Problems(4)

Trigonometric inequality

Source: IMOTC 2015 Practice Test 1 Problem 3

7/11/2015
Prove that for any triangle ABCABC, the inequality ∑cycliccos⁡A≤∑cyclicsin⁡(A/2)\displaystyle\sum_{\text{cyclic}}\cos A\le\sum_{\text{cyclic}}\sin (A/2) holds.
trigonometryinequalities
Coloring of a board by two colours

Source: IMOTC 2015 Practice Test 2 Problem 3

7/11/2015
Every cell of a 3×33\times 3 board is coloured either by red or blue. Find the number of all colorings in which there are no 2×22\times 2 squares in which all cells are red.
combinatorics
10-colorable graph

Source: Indian Team Selection Test 2015 Day 2 Problem 3

7/11/2015
Let GG be a simple graph on the infinite vertex set V={v1,v2,v3,…}V=\{v_1, v_2, v_3,\ldots\}. Suppose every subgraph of GG on a finite vertex subset is 1010-colorable, Prove that GG itself is 1010-colorable.
graph theorycombinatorics
Lamps with soft-buttons

Source: Indian Team Selection Test 2015 Day 3 Problem 3

7/11/2015
There are n≥2n\ge 2 lamps, each with two states: <spanclass=′latex−bold′>on</span><span class='latex-bold'>on</span> or <spanclass=′latex−bold′>off</span><span class='latex-bold'>off</span>. For each non-empty subset AA of the set of these lamps, there is a <spanclass=′latex−italic′>soft−button</span><span class='latex-italic'>soft-button</span> which operates on the lamps in AA; that is, upon <spanclass=′latex−italic′>operating</span><span class='latex-italic'>operating</span> this button each of the lamps in AA changes its state(on to off and off to on). The buttons are identical and it is not known which button corresponds to which subset of lamps. Suppose all the lamps are off initially. Show that one can always switch all the lamps on by performing at most 2n−1+12^{n-1}+1 operations.
combinatorics