MathDB

1

Part of 2018 India IMO Training Camp

Problems(4)

In-touch points and then things get ugly

Source: IMOTC Practice Test 1 2018 P1, India

7/18/2018
Let ΔABC\Delta ABC be an acute triangle. D,E,FD,E,F are the touch points of incircle with BC,CA,ABBC,CA,AB respectively. AD,BE,CFAD,BE,CF intersect incircle at K,L,MK,L,M respectively. If,σ=AKKD+BLLE+CMMF\sigma = \frac{AK}{KD} + \frac{BL}{LE} + \frac{CM}{MF} τ=AKKD.BLLE.CMMF\tau = \frac{AK}{KD}.\frac{BL}{LE}.\frac{CM}{MF} Then prove that τ=R16r\tau = \frac{R}{16r}. Also prove that there exists integers u,v,wu,v,w such that, uvw≠0uvw \neq 0, uσ+vτ+w=0u\sigma + v\tau +w=0.
geometry
Circumcircle of triangle lies on diagonal

Source: IMOTC PT2 2018 P1, 2018

7/18/2018
Let ABCDABCD be a convex quadrilateral inscribed in a circle with center OO which does not lie on either diagonal. If the circumcentre of triangle AOCAOC lies on the line BDBD, prove that the circumcentre of triangle BODBOD lies on the line ACAC.
geometrycircumcircle
Find k so that S_k is finite

Source: India TST 2018, D2 P1

7/18/2018
For a natural number k>1k>1, define SkS_k to be the set of all triplets (n,a,b)(n,a,b) of natural numbers, with nn odd and gcd⁡(a,b)=1\gcd (a,b)=1, such that a+b=ka+b=k and nn divides an+bna^n+b^n. Find all values of kk for which SkS_k is finite.
number theory
Bicentric quads formed by cevians

Source: India TST 2018, D4 P1

7/18/2018
Let ABCABC be a triangle and AD,BE,CFAD,BE,CF be cevians concurrent at a point PP. Suppose each of the quadrilaterals PDCE,PEAFPDCE,PEAF and PFBDPFBD has both circumcircle and incircle. Prove that ABCABC is equilateral and PP coincides with the center of the triangle.
geometrycircumcircle