MathDB

2

Part of 2020 Dutch IMO TST

Problems(3)

P (x^2) + 2P (x) = P (x)^2 + 2

Source: 2020 Dutch IMO TST 1.2

11/21/2020
Determine all polynomials P(x)P (x) with real coefficients that apply P(x2)+2P(x)=P(x)2+2P (x^2) + 2P (x) = P (x)^2 + 2.
algebrapolynomial
999 integers 2 player game

Source: 2020 Dutch IMO TST 2.2

11/22/2020
Ward and Gabrielle are playing a game on a large sheet of paper. At the start of the game, there are 999999 ones on the sheet of paper. Ward and Gabrielle each take turns alternatingly, and Ward has the first turn. During their turn, a player must pick two numbers a and b on the sheet such that gcd(a,b)=1gcd(a, b) = 1, erase these numbers from the sheet, and write the number a+ba + b on the sheet. The first player who is not able to do so, loses. Determine which player can always win this game.
combinatoricsgamewinning strategygame strategy
fixed point wanted, point on circumcircle, reflection wrt angle bisector

Source: 2020 Dutch IMO TST 3.2

11/22/2020
Given is a triangle ABCABC with its circumscribed circle and ∣AC∣<∣AB∣| AC | <| AB |. On the short arc ACAC, there is a variable point D≠AD\ne A. Let EE be the reflection of AA wrt the inner bisector of ∠BDC\angle BDC. Prove that the line DEDE passes through a fixed point, regardless of point DD.
geometryreflectionFixed pointfixedcircumcircleangle bisector