MathDB

1

Part of 2013 Saudi Arabia GMO TST

Problems(3)

some distinct numbers from the set S = {2,...,111}

Source: 2013 Saudi Arabia GMO TST I p1

7/26/2020
Tarik wants to choose some distinct numbers from the set S={2,...,111}S = \{2,...,111\} in such a way that each of the chosen numbers cannot be written as the product of two other distinct chosen numbers. What is the maximum number of numbers Tarik can choose ?
combinatorics
f(x)f(y) = f(xy) + f (x/y)

Source: 2013 Saudi Arabia GMO TST III p1

7/26/2020
Find all functions f:R→Rf : R \to R which satisfy f(33x)=3f(x)−233xf \left(\frac{\sqrt3}{3} x\right) = \sqrt3 f(x) - \frac{2\sqrt3}{3} x
and f(x)f(y)=f(xy)+f(xy)f(x)f(y) = f(xy) + f \left(\frac{x}{y} \right) for all x,y∈Rx, y \in R, with y≠0y \ne 0
functionalalgebrafunctional equation
AB = AC if AP = AQ, circumcircle related

Source: 2013 Saudi Arabia GMO TST day II p1

7/31/2020
An acute triangle ABCABC is inscribed in circle ω\omega centered at OO. Line BOBO and side ACAC meet at B1B_1. Line COCO and side ABAB meet at C1C_1. Line B1C1B_1C_1 meets circle ω\omega at PP and QQ. If AP=AQAP = AQ, prove that AB=ACAB = AC.
geometrycircumcircleequal segments