MathDB

3

Part of 2010 Mexico National Olympiad

Problems(2)

Tangent Circles

Source: OMM 2010 3

7/15/2014
Let C1\mathcal{C}_1 and C2\mathcal{C}_2 be externally tangent at a point AA. A line tangent to C1\mathcal{C}_1 at BB intersects C2\mathcal{C}_2 at CC and DD; then the segment ABAB is extended to intersect C2\mathcal{C}_2 at a point EE. Let FF be the midpoint of \overarc{CD} that does not contain EE, and let HH be the intersection of BFBF with C2\mathcal{C}_2. Show that CDCD, AFAF, and EHEH are concurrent.
geometry
(pq)^r+(qr)^p+(rp)^q

Source: OMM 2010 6

7/15/2014
Let pp, qq, and rr be distinct positive prime numbers. Show that if
pqr∣(pq)r+(qr)p+(rp)q−1,pqr\mid (pq)^r+(qr)^p+(rp)^q-1,
then
(pqr)3∣3((pq)r+(qr)p+(rp)q−1).(pqr)^3\mid 3((pq)^r+(qr)^p+(rp)^q-1).
number theory unsolvednumber theory