MathDB

3

Part of 2008 Postal Coaching

Problems(6)

(a_1 - 1)(a_2 - 1)(a_3 -1)...(a_k - 1) divides a_1a_2a_3 ...a_k + 1

Source: Indian Postal Coaching 2008 set 1 p3

5/25/2020
Prove that there exists an in nite sequence <an><a_n> of positive integers such that for each k≥1k \ge 1
(a1−1)(a2−1)(a3−1)...(ak−1)(a_1 - 1)(a_2 - 1)(a_3 -1)...(a_k - 1) divides a1a2a3...ak+1a_1a_2a_3 ...a_k + 1.
number theoryProductdividesdivisible
3^m divides n^3 + 17 but 3^{m+1}does not divide it.

Source: Indian Postal Coaching 2008 set 2 p3

5/25/2020
Prove that for each natural number m≥2m \ge 2, there is a natural number nn such that 3m3^m divides n3+17n^3 + 17 but 3m+13^{m+1} does not divide it.
number theorydividesdivisible
XX' has max length iff AX lies between median and internal angle bisector

Source: Indian Postal Coaching 2008 set 3 p3

5/25/2020
Let ABCABC be a triangle. For any point XX on BCBC, let AXAX meet the circumcircle of ABCABC in X′X'. Prove or disprove: XX′XX' has maximum length if and only if AXAX lies between the median and the internal angle bisector from AA.
geometrymaxmedianangle bisector
P(x+y, x-y) = 2P(x, y)

Source: Indian Postal Coaching 2008 set 4 p3

5/25/2020
Find all real polynomials P(x,y)P(x, y) such that P(x+y,x−y)=2P(x,y)P(x+y, x-y) = 2P(x, y), for all x,yx, y in RR.
polynomialalgebra
14 teams in 799 such that first 7 teams have each defeated remaining ones

Source: Indian Postal Coaching 2008 set 5 p3

5/25/2020
Show that in a tournament of 799799 teams (every team plays with every other team for a win or loss), there exist 1414 teams such that the first seven teams have each defeated the remaining teams.
combinatorics
|1 + ab| + |a + b| &gt;= \sqrt{|a^2 - 1| \cdot |b^2 - 1|} , complex

Source: Indian Postal Coaching 2008 set 6 p3

5/25/2020
Let aa and bb be two complex numbers. Prove the inequality ∣1+ab∣+∣a+b∣≥∣a2−1∣⋅∣b2−1∣|1 + ab| + |a + b| \ge \sqrt{|a^2 - 1| \cdot |b^2 - 1|}
complexinequalitiesalgebra