MathDB

1

Part of 2016 Saudi Arabia IMO TST

Problems(7)

P156. Saudi Arabia IMO TST

Source:

5/19/2018
Let kk be a positive integer. Prove that there exist integers xx and yy, neither of which divisible by 77, such that \begin{align*} x^2 + 6y^2 = 7^k. \end{align*}
SAUDiophantine Equations
P263. Saudi Arabia IMO TST

Source:

8/3/2018
Let n≥3 n \geq 3 be an integer and let \begin{align*} x_1,x_2, \ldots, x_n \end{align*} be n n distinct integers. Prove that \begin{align*} (x_1 - x_2)^2 + (x_2 - x_3)^2 + \ldots + (x_n - x_1)^2 \geq 4n - 6. \end{align*}
saSequences
a_1 = 1, a_n = n - 2 if $a_{n-1} = 0 and a_n = a_{n-1} - 1 , otherwise

Source: 2016 Saudi Arabia IMO TST , level 4, III p1

7/29/2020
Define the sequence a1,a2,...a_1, a_2,... as follows: a1=1a_1 = 1, and for every n≥2n \ge 2, an=n−2a_n = n - 2 if an−1=0a_{n-1} = 0 and an=an−1−1a_n = a_{n-1} - 1, otherwise. Find the number of 1≤k≤20161 \le k \le 2016 such that there are non-negative integers r,sr, s and a positive integer nn satisfying k=r+sk = r + s and an+r=an+sa_{n+r} = a_n + s.
Sequencealgebrarecurrence relation
orthocenter and collinear wanted, incircle, circumcircle related

Source: 2016 Saudi Arabia IMO TST , level 4, IV p1

7/27/2020
Let ABCABC be a triangle whose incircle (I)(I) touches BC,CA,ABBC, CA, AB at D,E,FD, E, F, respectively. The line passing through AA and parallel to BCBC cuts DE,DFDE, DF at M,NM, N, respectively. The circumcircle of triangle DMNDMN cuts (I)(I) again at LL. a) Let KK be the intersection of NEN E and MFM F. Prove that KK is the orthocenter of the triangle DMNDMN. b) Prove that A,K,LA, K, L are collinear.
geometrycircumcircleincircleorthocentercollinear
x_{n+1} =\sqrt{\frac{x_n^2+x_{n+2}^2}{2}}, sequence of points, collinear

Source: 2016 Saudi Arabia IMO TST , level 4+, IV p1

7/29/2020
On the Cartesian coordinate system OxyOxy, consider a sequence of points An(xn,yn)A_n(x_n, y_n) in which (xn)n=1∞(x_n)^{\infty}_{n=1},(yn)n=1∞(y_n)^{\infty}_{n=1} are two sequences of positive numbers satisfing the following conditions: xn+1=xn2+xn+222,yn+1=(yn+yn+22)2  ∀n≥1x_{n+1} =\sqrt{\frac{x_n^2+x_{n+2}^2}{2}}, y_{n+1} =\big( \frac{\sqrt{y_n}+\sqrt{y_{n+2}}}{2} \big)^2 \,\, \forall n \ge 1 Suppose that O,A1,A2016O, A_1, A_{2016} belong to a line dd and A1,A2016A_1, A_{2016} are distinct. Prove that all the points A2,A3,...,A2015A_2, A_3,. .. , A_{2015} lie on one side of dd.
combinatorial geometryrecurrence relationpoints
circumcircle of HAB passes through orthocenter of HAC

Source: 2016 Saudi Arabia IMO TST , level 4+, II p1

7/27/2020
Let ABCABC be a triangle inscribed in the circle (O)(O). The bisector of ∠BAC\angle BAC cuts the circle (O)(O) again at DD. Let DEDE be the diameter of (O)(O). Let GG be a point on arc ABAB which does not contain CC. The lines GDGD and BCBC intersect at FF. Let HH be a point on the line AGAG such that FH∥AEFH \parallel AE. Prove that the circumcircle of triangle HABHAB passes through the orthocenter of triangle HACHAC.
geometryorthocentercircumcircle
N= sum of k positive integers that are relatively prime to N

Source: 2016 Saudi Arabia IMO TST , level 4+, I p1

7/29/2020
Call a positive integer N≥2N \ge 2 special if for every k such that 2≤k≤N,N2 \le k \le N, N can be expressed as a sum of kk positive integers that are relatively prime to NN (although not necessarily relatively prime to each other). Find all special positive integers.
number theorySumcoprime