MathDB

2

Part of 2016 Saudi Arabia IMO TST

Problems(6)

a set of 2^{2016} cards with the numbers 1,2, ..., 2^{2016} written

Source: 2016 Saudi Arabia IMO TST , level 4, I p2

7/29/2020
Given a set of 220162^{2016} cards with the numbers 1,2,...,220161,2, ..., 2^{2016} written on them. We divide the set of cards into pairs arbitrarily, from each pair, we keep the card with larger number and discard the other. We now again divide the 220152^{2015} remaining cards into pairs arbitrarily, from each pair, we keep the card with smaller number and discard the other. We now have 220142^{2014} cards, and again divide these cards into pairs and keep the larger one in each pair. We keep doing this way, alternating between keeping the larger number and keeping the smaller number in each pair, until we have just one card left. Find all possible values of this final card.
combinatorics
P71. Saudi Arabia IMO TST

Source:

7/21/2018
Let aa be a positive integer. Find all prime numbers p p with the following property: there exist exactly p p ordered pairs of integers (x,y) (x, y), with 0 \leq  x, y \leq p - 1 , such that p p divides y2−x3−a2x y^2 - x^3 - a^2x .
SAUDivisibility
perpendicular wanted, circumcircle and perpendicular related

Source: 2016 Saudi Arabia IMO TST , level 4, II p2

7/27/2020
Let ABCABC be a triangle inscribed in the circle (O)(O) and PP is a point inside the triangle ABCABC. Let DD be a point on (O)(O) such that AD⊥APAD \perp AP. The line CDCD cuts the perpendicular bisector of BCBC at MM. The line ADAD cuts the line passing through BB and is perpendicular to BPBP at QQ. Let NN be the reflection of QQ through MM. Prove that CN⊥CPCN \perp CP.
geometrycircleperpendicularcircumcircle
P(Q(x)) = (x - 1)(x - 2)...(x - 9), integer polynomials

Source: 2016 Saudi Arabia IMO TST , level 4, III p2

7/29/2020
Find all pairs of polynomials P(x),Q(x)P(x),Q(x) with integer coefficients such that P(Q(x))=(x−1)(x−2)...(x−9)P(Q(x)) = (x - 1)(x - 2)...(x - 9) for all real numbers xx
algebraInteger Polynomialpolynomial
pairs of equal angles in hexagon, if AB =CD=EF, BC=DE=FA , <A+<B =<C <D=<E+<F

Source: 2016 Saudi Arabia IMO TST , level 4+, IV p2

7/27/2020
Let ABCDEFABCDEF be a convex hexagon with AB=CD=EFAB = CD = EF, BC=DE=FABC =DE = FA and ∠A+∠B=∠C+∠D=∠E+∠F\angle A+\angle B = \angle C +\angle D = \angle E +\angle F. Prove that ∠A=∠C=∠E\angle A=\angle C=\angle E and ∠B=∠D=∠F\angle B=\angle D=\angle F.
Tran Quang Hung
convexhexagonequal segmentsanglesequal anglesgeometry
f (x + 1) >= f (x) + 1, f (x y) >=ge f (x)f (y)

Source: 2016 Saudi Arabia IMO TST , level 4+, II p2

7/29/2020
Find all functions f:R→Rf : R \to R satisfying the conditions: 1. f(x+1)≥f(x)+1f (x + 1) \ge f (x) + 1 for all x∈Rx \in R 2. f(xy)≥f(x)f(y)f (x y) \ge f (x)f (y) for all x,y∈Rx, y \in R
algebraFunctional inequality