MathDB

2

Part of 2011 Saudi Arabia BMO TST

Problems(4)

a_n is the no of pairs (x,y) of integers satisfying |x^2-y^2| = n

Source: 2011 Saudi Arabia BMO TST 1.2 - Balkan MO

12/29/2021
For any positive integer nn, let ana_n be the number of pairs (x,y)(x,y) of integers satisfying ∣x2−y2∣=n|x^2-y^2| = n. (a) Find a1432a_{1432} and a1433a_{1433}. (b) Find ana_n .
number theory
set A_n consist of all numbers \pm 1 \pm 2 \pm ...\pm n

Source: 2011 Saudi Arabia BMO TST 2.2 - Balkan MO

12/29/2021
For each positive integer nn let the set AnA_n consist of all numbers ±1±2±...±n\pm 1 \pm 2 \pm ...\pm n. For example, A1={−1,1},A2={−3,−1,1,3},A3={−6,−4,−2,0,2,4,6}.A_1 = \{-1,1\}, A_2 = \{ -3 ,-1 ,1 ,3 \} , A_3 = \{ -6 ,-4 ,-2 ,0 ,2 ,4 ,6 \}. Find the number of elements in AnA_n .
algebracombinatoricsnumber theory
(x + 2) (x + 4 )... (x + 2n) + (x +1) (x + 3 )... (x + 2n - 1) = 0

Source: 2011 Saudi Arabia BMO TST 3.2 - Balkan MO

12/29/2021
Let nn be a positive integer. Prove that all roots of the equation x(x+2)(x+4)...(x+2n)+(x+1)(x+3)...(x+2n−1)=0x(x + 2) (x + 4 )... (x + 2n) + (x +1) (x + 3 )... (x + 2n - 1) = 0 are real and irrational.
algebrapolynomial
|a_1 + 2a_2 + ... + na_n | <= (n-1)/2

Source: 2011 Saudi Arabia BMO TST 4.2 - Balkan MO

12/29/2021
Let a1,a2,...,ana_1,a_2,..., a_n be real numbers such that a1+a2+...+an=0a_1 + a_2 + ... + a_n = 0 and ∣a1∣+∣a2∣+...+∣an∣=1|a_1| + |a_2 | + ... + |a_n | = 1. Prove that ∣a1+2a2+...+nan∣≤n−12 |a_1 + 2a_2 + ... + na_n | \le \frac{n-1}{2}
algebrainequalities