MathDB

9

Part of 2008 JBMO Shortlist

Problems(3)

1-1024, deleting 4k+3 numbers, successively for 5 times

Source: JBMO 2008 Shortlist A9

10/14/2017
Consider an integer n≥4n \ge 4 and a sequence of real numbers x1,x2,x3,...,xnx_1, x_2, x_3,..., x_n. An operation consists in eliminating all numbers not having the rank of the form 4k+34k + 3, thus leaving only the numbers x3.x7.x11,...x_3. x_7. x_{11}, ...(for example, the sequence 4,5,9,3,6,6,1,84,5,9,3,6, 6,1, 8 produces the sequence 9,19,1). Upon the sequence 1,2,3,...,10241, 2, 3, ..., 1024 the operation is performed successively for 55 times. Show that at the end only one number remains and fi nd this number.
JBMOalgebra
2008 JBMO Shortlist G9

Source: 2008 JBMO Shortlist G9

10/10/2017
Let OO be a point inside the parallelogram ABCDABCD such that ∠AOB+∠COD=∠BOC+∠AOD\angle AOB + \angle COD = \angle BOC + \angle AOD. Prove that there exists a circle kk tangent to the circumscribed circles of the triangles △AOB,△BOC,△COD\vartriangle AOB, \vartriangle BOC, \vartriangle COD and △DOA\vartriangle DOA.
JBMOgeometry
(4a + p)/b+\(4b + p)/a , a^2/b+b^2/a \in Z

Source: JBMO 2008 Shortlist N9

10/14/2017
Let pp be a prime number. Find all positive integers aa and bb such that: 4a+pb+4b+pa\frac{4a + p}{b}+\frac{4b + p}{a} and a2b+b2a \frac{a^2}{b}+\frac{b^2}{a} are integers.
JBMOnumber theory