MathDB

1

Part of 2013 Saudi Arabia IMO TST

Problems(3)

Concurrent Lines

Source:

3/5/2016
Triangle ABCABC is inscribed in circle ω\omega. Point PP lies inside triangle ABCABC.Lines AP,BPAP,BP and CPCP intersect ω\omega again at points A1A_1, B1B_1 and C1C_1 (other than A,B,CA, B, C), respectively. The tangent lines to ω\omega at A1A_1 and B1B_1 intersect at C2C_2.The tangent lines to ω\omega at B1B_1 and C1C_1 intersect at A2A_2. The tangent lines to ω\omega at C1C_1 and A1A_1 intersect at B2B_2. Prove that the lines AA2,BB2AA_2,BB_2 and CC2CC_2 are concurrent.
geometryconcurrencycollineardesarguePascalincircletangent
min, max of (1-x_1)(1-y_1)+(1-x_2)(1-y_2) when x_1^2+x_2^2=y_1^2+y_2^2=2013

Source: 2013 Saudi Arabia IMO TST II p1

7/23/2020
Find the maximum and the minimum values of S=(1−x1)(1−y1)+(1−x2)(1−y2)S = (1 - x_1)(1 -y_1) + (1 - x_2)(1 - y_2) for real numbers x1,x2,y1,y2x_1, x_2, y_1,y_2 with x12+x22=y12+y22=2013x_1^2 + x_2^2 = y_1^2 + y_2^2 = 2013.
minmaxalgebrainequalities
colsed path along lattice points on a m x n grid of points

Source: 2013 Saudi Arabia IMO TST III p1

7/23/2020
Adel draws an m×nm \times n grid of dots on the coordinate plane, at the points of integer coordinates (a,b)(a,b) where 1≤a≤m1 \le a \le m and 1≤b≤n1 \le b \le n. He proceeds to draw a closed path along kk of these dots, (a1,b1)(a_1, b_1),(a2,b2)(a_2,b_2),...,(ak,bk)(a_k,b_k), such that (ai,bi)(a_i,b_i) and (ai+1,bi+1)(a_{i+1}, b_{i+1}) (where (ak+1,bk+1)=(a1,b1)(a_{k+1}, b_{k+1}) = (a_1, b_1)) are 11 unit apart for each 1≤i≤k1 \le i \le k. Adel makes sure his path does not cross itself, that is, the kk dots are distinct. Find, with proof, the maximum possible value of kk in terms of mm and nn.
combinatoricslatticegrid