MathDB
2007 El Salvador Correspondence / Qualifying NMO VII

Source:

October 16, 2021
algebrageometrynumber theorycombinatoricsel salvador NMO

Problem Statement

p1. Let ABCABC be a right triangle at BB, side BC is 7272 cm and side AC is 7878 cm. Consider DD on side ABAB such that 2AD=BD2AD = BD. If OO is the center of the circle that is tangent to side BCBC and passes through AA and DD. Find the measure of OBOB.
p2.The increasing sequence 3,15,24,48,...3, 15, 24, 48, ... consists of all positive multiples of 33 which are one less than a perfect square number. Determine the remainder of the division per 10001000 of the term that occupies position 20072007 in the sequence.
p3. Given x2+y2x2−y2+x2−y2x2+y2=k\frac{x^2+y^2}{x^2-y^2}+\frac{x^2-y^2}{x^2+y^2}=k, express x8+y8x8−y8+x8−y8x8+y2\frac{x^8+y^8}{x^8-y^8}+\frac{x^8-y^8}{x^8+y^2} in terms of k.
p4. Two of the squares on a 7×77\times 7 board are painted white, and the rest are painted blue. We will say that two colorings are equal if one can be obtained from another by applying a rotation to the board. Determine the number of different colorings that exist.
p5. In each square of a giant board there is written a natural number, according to the following rules: the numbers in the first column form an arithmetic sequence of first term 66 and difference 33, that is, 6,9,12,15,...6, 9, 12, 15, ... The numbers of the first row form an arithmetic sequence of difference 33, the numbers in the second row form an arithmetic progression of difference 55, and in general, the numbers in row number kk they form an arithmetic progression of difference 2k+12k + 1. Find all the cells that they have the number 20072007 written on them.
Back to ProblemsView on AoPS