MathDB

3

Part of 2016 Saudi Arabia GMO TST

Problems(6)

permutation of of (1, 2,3,..., n) such that a_1+a2+... +a_k is divisible by k

Source: 2016 Saudi Arabia GMO TST level 4, I p3

8/1/2020
Find all positive integer nn such that there exists a permutation (a1,a2,...,an)(a_1, a_2,...,a_n) of (1,2,3,...,n)(1, 2,3,..., n) satisfying the condition: a1+a2+...+aka_1 + a_2 +... + a_k is divisible by kk for each k=1,2,3,...,nk = 1, 2,3,..., n.
number theorypermutationdivisible
collinear, perpendicular and midpoint wanted, 2 circles related

Source: 2016 Saudi Arabia GMO TST level 4+, I p3

8/1/2020
Let ABCABC be an acute, non-isosceles triangle with the circumcircle (O)(O). Denote D,ED, E as the midpoints of AB,ACAB,AC respectively. Two circles (ABE)(ABE) and (ACD)(ACD) intersect at KK differs from AA. Suppose that the ray AKAK intersects (O)(O) at LL. The line LBLB meets (ABE)(ABE) at the second point MM and the line LCLC meets (ACD)(ACD) at the second point NN. a) Prove that M,K,NM, K, N collinear and MNMN perpendicular to OLOL. b) Prove that KK is the midpoint of MNMN
geometrycollinearmidpointperpendicular bisector
if incenter I in (KDE), prove that BD+CE=BC, symmetric lines, perpendicular

Source: 2016 Saudi Arabia GMO TST level 4, II p3

8/1/2020
Let ABCABC be a triangle with incenter II . Let CI,BICI, BI intersect AB,ACAB, AC at D,ED, E respectively. Denote by Δb,Δc\Delta_b,\Delta_c the lines symmetric to the lines AB,ACAB, AC with respect to CD,BECD, BE correspondingly. Suppose that Δb,Δc\Delta_b,\Delta_c meet at KK. a) Prove that IK⊥BCIK \perp BC. b) If I∈(KDE)I \in (K DE), prove that BD+CE=BCBD + C E = BC.
geometryincenterperpendicularincircle
n classes in a school, students in each class wear hats of the same color

Source: 2016 Saudi Arabia GMO TST level 4, III p3

8/1/2020
In a school there are totally n>2n > 2 classes and not all of them have the same numbers of students. It is given that each class has one head student. The students in each class wear hats of the same color and different classes have different hat colors. One day all the students of the school stand in a circle facing toward the center, in an arbitrary order, to play a game. Every minute, each student put his hat on the person standing next to him on the right. Show that at some moment, there are 22 head students wearing hats of the same color.
Coloringcombinatorics
in a school n students take part in m clubs

Source: 2016 Saudi Arabia GMO TST level 4+, II p3

8/1/2020
In a school, there are totally nn students, with n≥2n \ge 2. The students take part in mm clubs and in each club, there are at least 22 members (a student may take part in more than 11 club). Eventually, the Principal notices that: If 22 clubs share at least 22 common members then they have different numbers of members. Prove that m≤(n−1)2m \le (n - 1)^2
combinatorics
x_{2n+1} = P(x_{2n}), x_{2n+2} = Q(x_{2n+1}) ,every pos. integer divides x_n,

Source: 2016 Saudi Arabia GMO TST level 4+, III p3

8/1/2020
Find all polynomials P,Q∈Z[x]P,Q \in Z[x] such that every positive integer is a divisor of a certain nonzero term of the sequence (xn)n=0∞(x_n)_{n=0}^{\infty} given by the conditions:
x0=2016x_0 = 2016, x2n+1=P(x2n)x_{2n+1} = P(x_{2n}), x2n+2=Q(x2n+1)x_{2n+2} = Q(x_{2n+1}) for all n≥0n \ge 0
dividesdivisoralgebrapolynomial