MathDB

2

Part of 2016 Korea Winter Program Practice Test

Problems(4)

Isolated squares

Source: 2016 Korea Winter Program Test1 Day1 #2

1/27/2016
Given an integer n≥3n\geq 3. For each 3×33\times3 squares on the grid, call this 3×33\times3 square isolated if the center unit square is white and other 8 squares are black, or the center unit square is black and other 8 squares are white.
Now suppose one can paint an infinite grid by white or black, so that one can select an a×ba\times b rectangle which contains at least n2−nn^2-n isolated 3×33\times 3 square. Find the minimum of a+ba+b that such thing can happen.
(Note that a,ba,b are positive reals, and selected a×ba\times b rectangle may have sides not parallel to grid line of the infinite grid.)
rectanglecombinatoricssquare grid
Easy inequality

Source: 2016 Korea Winter Camp 1st Test #6

1/25/2016
Let ai,bia_i, b_i (1≤i≤n1 \le i \le n, n≥2n \ge 2) be positive real numbers such that ∑i=1nai=∑i=1nbi\sum_{i=1}^n a_i = \sum_{i=1}^n b_i.
Prove that ∑i=1n(ai+1+bi+1)2n(ai−bi)2+4(n−1)∑j=1najbj≥1n−1\sum_{i=1}^n \frac{(a_{i+1}+b_{i+1})^2}{n(a_i-b_i)^2+4(n-1)\sum_{j=1}^n a_jb_j} \ge \frac{1}{n-1}
inequalities
Colinear Points

Source: 2016 Korea Winter Camp 2nd Test #2

1/25/2016
Let there be an acute triangle ABCABC, such that ∠ABC<∠ACB\angle ABC < \angle ACB. Let the perpendicular from AA to BCBC hit the circumcircle of ABCABC at DD, and let MM be the midpoint of ADAD. The tangent to the circumcircle of ABCABC at AA hits the perpendicular bisector of ADAD at EE, and the circumcircle of MDEMDE hits the circumcircle of ABCABC at FF. Let GG be the foot of the perpendicular from AA to BDBD, and NN be the midpoint of AGAG. Prove that B,N,FB, N, F are collinear.
geometry
Similar to that IMO problem

Source: 2016 Korea Winter Camp 2nd Test #6

1/25/2016
Find all pairs of positive integers (n,t)(n,t) such that 6n+1=n2t6^n+1=n^2t, and (n,29×197)=1(n,29 \times 197)=1
number theory