MathDB

2

Part of 2011 Mongolia Team Selection Test

Problems(4)

Summation congruency

Source: Mongolia TST 2011 Test 1 #2

11/7/2011
Mongolia TST 2011 Test 1 #2
Let pp be a prime number. Prove that:
∑k=0p(−1)k(pk)(p+kk)≡−1(mod  p3)\sum_{k=0}^p (-1)^k \dbinom{p}{k} \dbinom{p+k}{k} \equiv -1 (\mod p^3)
(proposed by B. Batbayasgalan, inspired by Putnam olympiad problem)
Note: I believe they meant to say p>2p>2 as well.
Putnamfunctionnumber theory unsolvednumber theory
Mongolia TST 2011 Test 2 #2

Source: Mongolia TST 2011 Test 2 #2

11/8/2011
Let ABCABC be a scalene triangle. The inscribed circle of ABCABC touches the sides BCBC, CACA, and ABAB at the points A1A_1, B1B_1, C1C_1 respectively. Let II be the incenter, OO be the circumcenter, and lines OIOI and BCBC meet at point DD. The perpendicular line from A1A_1 to B1C1B_1 C_1 intersects ADAD at point EE. Prove that B1C1B_1 C_1 passes through the midpoint of EA1EA_1.
geometryincentercircumcirclegeometry unsolved
Mongolia TST 2011 Test 3 #2

Source: Mongolia TST 2011 Test 3 #2

11/8/2011
Given a triangle ABCABC, the internal and external bisectors of angle AA intersect BCBC at points DD and EE respectively. Let FF be the point (different from AA) where line ACAC intersects the circle ww with diameter DEDE. Finally, draw the tangent at AA to the circumcircle of triangle ABFABF, and let it hit ww at AA and GG. Prove that AF=AGAF=AG.
geometrycircumcircleincenterangle bisectorgeometry unsolved
Mongolia TST 2011 Test 4 #2

Source: Mongolia TST 2011 Test 4 #2

11/8/2011
Let rr be a given positive integer. Is is true that for every rr-colouring of the natural numbers there exists a monochromatic solution of the equation x+y=3zx+y=3z?
(proposed by B. Batbaysgalan, folklore)
algebralinear equationcombinatorics unsolvedcombinatorics