MathDB

2

Part of 2010 IMO Shortlist

Problems(3)

36 <= 4*sum(a^3) - sum(a^4) <= 48

Source: IMO Shortlist 2010, Algebra 2

7/17/2011
Let the real numbers a,b,c,da,b,c,d satisfy the relations a+b+c+d=6a+b+c+d=6 and a2+b2+c2+d2=12.a^2+b^2+c^2+d^2=12. Prove that 36≤4(a3+b3+c3+d3)−(a4+b4+c4+d4)≤48.36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.
Proposed by Nazar Serdyuk, Ukraine
inequalitiesalgebrapolynomialIMO ShortlistHi
There exist N flags forming a diverse set

Source: IMO Shortlist 2010, Combinatorics 2

7/17/2011
On some planet, there are 2N2^N countries (N≥4).(N \geq 4). Each country has a flag NN units wide and one unit high composed of NN fields of size 1×1,1 \times 1, each field being either yellow or blue. No two countries have the same flag. We say that a set of NN flags is diverse if these flags can be arranged into an N×NN \times N square so that all NN fields on its main diagonal will have the same color. Determine the smallest positive integer MM such that among any MM distinct flags, there exist NN flags forming a diverse set.
Proposed by Tonći Kokan, Croatia
combinatoricsIMO ShortlistHall s marriage theoremperfect matchinggraph theorymatchingHi
IMO Shortlist 2010 - Problem N2

Source:

7/17/2011
Find all pairs (m,n)(m,n) of nonnegative integers for which m2+2⋅3n=m(2n+1−1).m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).
Proposed by Angelo Di Pasquale, Australia
modular arithmeticnumber theoryequationIMO ShortlistLTE Lemma