MathDB

2

Part of 2011 IMO Shortlist

Problems(4)

IMO Shortlist 2011, Algebra 2

Source: IMO Shortlist 2011, Algebra 2

7/11/2012
Determine all sequences (x1,x2,…,x2011)(x_1,x_2,\ldots,x_{2011}) of positive integers, such that for every positive integer nn there exists an integer aa with ∑j=12011jxjn=an+1+1\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1
Proposed by Warut Suksompong, Thailand
algebraIMO ShortlistequationSequence
IMO Shortlist 2011, Combinatorics 2

Source: IMO Shortlist 2011, Combinatorics 2

7/12/2012
Suppose that 10001000 students are standing in a circle. Prove that there exists an integer kk with 100≤k≤300100 \leq k \leq 300 such that in this circle there exists a contiguous group of 2k2k students, for which the first half contains the same number of girls as the second half.
Proposed by Gerhard Wöginger, Austria
combinatoricsIMO ShortlistHi
IMO Shortlist 2011, G2

Source: IMO Shortlist 2011, G2

7/13/2012
Let A1A2A3A4A_1A_2A_3A_4 be a non-cyclic quadrilateral. Let O1O_1 and r1r_1 be the circumcentre and the circumradius of the triangle A2A3A4A_2A_3A_4. Define O2,O3,O4O_2,O_3,O_4 and r2,r3,r4r_2,r_3,r_4 in a similar way. Prove that 1O1A12−r12+1O2A22−r22+1O3A32−r32+1O4A42−r42=0.\frac{1}{O_1A_1^2-r_1^2}+\frac{1}{O_2A_2^2-r_2^2}+\frac{1}{O_3A_3^2-r_3^2}+\frac{1}{O_4A_4^2-r_4^2}=0.
Proposed by Alexey Gladkich, Israel
geometrycircumcirclealgebrapolynomialquadraticscomplex numbersIMO Shortlist
IMO Shortlist 2011, Number Theory 2

Source: IMO Shortlist 2011, Number Theory 2

7/11/2012
Consider a polynomial P(x)=∏j=19(x+dj),P(x) = \prod^9_{j=1}(x+d_j), where d1,d2,…d9d_1, d_2, \ldots d_9 are nine distinct integers. Prove that there exists an integer N,N, such that for all integers x≥Nx \geq N the number P(x)P(x) is divisible by a prime number greater than 20.
Proposed by Luxembourg
algebrapolynomialpigeonhole principlenumber theoryIMO Shortlistcombinatorics