MathDB

2

Part of 2013 Saudi Arabia IMO TST

Problems(3)

n + f(f(n)) <= 2f(n) , strictly increasing

Source: 2013 Saudi Arabia IMO TST I p2

7/23/2020
Let S=f{0.1.2.3,...}S = f\{0.1. 2.3,...\} be the set of the non-negative integers. Find all strictly increasing functions f:S→Sf : S \to S such that n+f(f(n))≤2f(n)n + f(f(n)) \le 2f(n) for every nn in SS
Increasingfunctioninequalitiesalgebra
concyclic wanted, altitudes related

Source: 2013 Saudi Arabia IMO TST II p2

7/23/2020
Let ABCABC be an acute triangle, and let AA1,BB1AA_1, BB_1, and CC1CC_1 be its altitudes. Segments AA1AA_1 and B1C1B_1C_1 meet at point KK. The perpendicular bisector of segment A1KA_1K intersects sides ABAB and ACAC at LL and MM, respectively. Prove that points A,A1,LA,A_1, L, and MM lie on a circle.
geometryConcyclicaltitudes
a_1 + a_2 + .. + a_n >= n^2 and a_1^2 + a_2^2 + ... + a_n^2 <= n^3 + 1

Source: 2013 Saudi Arabia IMO TST III p2

7/23/2020
Given an integer n≥2n \ge 2, determine the number of ordered nn-tuples of integers (a1,a2,...,an)(a_1, a_2,...,a_n) such that (a) a1+a2+..+an≥n2a_1 + a_2 + .. + a_n \ge n^2 and (b) a12+a22+...+an2≤n3+1a_1^2 + a_2^2 + ... + a_n^2 \le n^3 + 1
inequalitiesalgebracombinatorics