MathDB

1

Part of 2014 Danube Mathematical Competition

Problems(2)

natural a =(p+q)/r+(q+r)/p+(r+p)/q where p, q,r primes

Source: Danube 2014 junior p1

7/22/2019
Determine the natural number a=p+qr+q+rp+r+pqa =\frac{p+q}{r}+\frac{q+r}{p}+\frac{r+p}{q} where p,qp, q and rr are prime positive numbers.
number theorySumprimesnatural
2 intersecting circles, tangents, intersections, equal segments, tangent wanted

Source: Danube 2014 Seniors P1

9/8/2018
Two circles γ1\gamma_1 and γ2\gamma_2 cross one another at two points; let AA be one of these points. The tangent to γ1\gamma_1 at AA meets again γ2\gamma_2 at BB, the tangent to γ2\gamma_2 at AA meets again γ1\gamma_1 at CC, and the line BCBC meets again γ1\gamma_1 and γ2\gamma_2 at D1D_1 and D2D_2, respectively. Let E1E_1 and E2E_2 be interior points of the segments AD1AD_1 and AD2AD_2, respectively, such that AE1=AE2AE_1 = AE_2. The lines BE1BE_1 and ACAC meet at MM, the lines CE2CE_2 and ABAB meet at NN, and the lines MNMN and BCBC meet at PP. Show that the line PAPA is tangent to the circle ABCABC.
geometrycirclesTangents