MathDB

2

Part of 2010 Slovenia National Olympiad

Problems(4)

Find all primes p, q, r [Slovenia 2010]

Source:

11/14/2010
Find all prime numbers p,q,rp, q, r such that 15p+7pq+qr=pqr.15p+7pq+qr=pqr.
number theory proposednumber theory
p divides at least one of the numbers [Slovenia 2010]

Source:

11/14/2010
Let a,ba, b and cc be nonzero digits. Let pp be a prime number which divides the three digit numbers abc‾\overline{abc} and cba‾.\overline{cba}. Show that pp divides at least one of the numbers a+b+c,a−b+ca+b+c, a-b+c and a−c.a-c.
number theory proposednumber theory
Find all real x in the interval [0, 2pi) [Slovenia 2010]

Source:

11/16/2010
Find all real xx in the interval [0,2π)[0, 2\pi) such that 27⋅33sin⁡x=9cos⁡2x.27 \cdot 3^{3 \sin x} = 9^{\cos^2 x}.
trigonometryalgebra proposedalgebra
Points C_1, C_2, B, D lie on the same circle [Slovenia 2010]

Source:

11/16/2010
Let K1\mathfrak K_1 and K2\mathfrak K_2 be circles centered at O1O_1 and O2,O_2, respectively, meeting at the points AA and B.B. Let pp be the line through the point AA meeting the circles K1\mathfrak K_1 and K2\mathfrak K_2 again at C1C_1 and C2.C_2. Assume that AA lies between C1C_1 and C2.C_2. Denote the intersection of the lines C1O1C_1O_1 and C2O2C_2O_2 by D.D. Prove that the points C1,C2,BC_1, C_2, B and DD lie on the same circle.
geometry proposedgeometry