MathDB

4

Part of 2008 Indonesia TST

Problems(4)

partition 15 people into 6 groups of 3, 3, 3, 2, 2, and 2

Source: 2008 Indonesia TST stage 2 test 2 p4

12/14/2020
There are 1515 people, including Petruk, Gareng, and Bagong, which will be partitioned into 66 groups, randomly, that consists of 3,3,3,2,23, 3, 3, 2, 2, and 22 people (orders are ignored). Determine the probability that Petruk, Gareng, and Bagong are in a group.
probabilitypartitioncombinatorics
gcd and positive divisors

Source:

8/9/2010
Let a a and b b be natural numbers with property gcd(a,b)=1 gcd(a,b)=1 . Find the least natural number k k such that for every natural number r≥k r \ge k , there exist natural numbers m,n>1 m,n >1 in such a way that the number manb m^a n^b has exactly r+1 r+1 positive divisors.
number theorygreatest common divisornumber theory unsolved
sum (a/(a+b)) ^2 >= 3/4

Source: 2008 Indonesia TST stage 2 test 3 p4

12/15/2020
Let a,b,ca, b, c be positive reals. Prove that (aa+b)2+(bb+c)2+(cc+a)2≥34\left(\frac{a}{a+b}\right)^2+\left(\frac{b}{b+c}\right)^2+\left(\frac{c}{c+a}\right)^2\ge \frac34
algebrainequalities
f(mn^2) = f(mn) +a f(n), if prime p|n, then f(p) \ne 0 and f(p)|f(n)

Source: 2008 Indonesia TST stage 2 test 4 p4

12/15/2020
Find all pairs of positive integer α\alpha and function f:N→N0f : N \to N_0 that satisfies (i) f(mn2)=f(mn)+αf(n)f(mn^2) = f(mn) + \alpha f(n) for all positive integers m,nm, n. (ii) If nn is a positive integer and pp is a prime number with p∣np|n, then f(p)≠0f(p) \ne 0 and f(p)∣f(n)f(p)|f(n).
number theoryfunctional equationfunctionalfunctional equation in N