MathDB

3

Part of 2024 Dutch IMO TST

Problems(3)

A game with binary numbers on board, can we force divisibility?

Source: Dutch TST 2024, 1.3

6/28/2024
Player Zero and Player One play a game on a n×nn \times n board (n≥1n \ge 1). The columns of this n×nn \times n board are numbered 1,2,4,…,2n−11,2,4,\dots,2^{n-1}. Turn my turn, the players put their own number in one of the free cells (thus Player Zero puts a 00 and Player One puts a 11). Player Zero begins. When the board is filled, the game ends and each row yields a (reverse binary) number obtained by adding the values of the columns with a 11 in that row. For instance, when n=4n=4, a row with 01010101 yields the number 0⋅1+1⋅2+0⋅4+1⋅8=100 \cdot1+1 \cdot 2+0 \cdot 4+1 \cdot 8=10.
a) For which natural numbers nn can Player One always ensure that at least one of the row numbers is divisible by 44? b) For which natural numbers nn can Player One always ensure that at least one of the row numbers is divisible by 33?
combinatoricscombinatorics proposedgamewinning strategybinary representation
A strangely asymmetric inequality with square roots

Source: Dutch TST 2024, 2.3

6/28/2024
Let a,b,ca,b,c be real numbers such that 0≤a≤b≤c0 \le a \le b \le c and a+b+c=1a+b+c=1. Show that abb−a+bcc−b+acc−a<14.ab\sqrt{b-a}+bc\sqrt{c-b}+ac\sqrt{c-a}<\frac{1}{4}.
inequalitiesinequalities proposedalgebra proposedalgebra
Polynomial

Source: ARO 2021 10.6

4/20/2021
Given is a polynomial P(x)P(x) of degree n>1n>1 with real coefficients. The equation P(P(P(x)))=P(x)P(P(P(x)))=P(x) has n3n^3 distinct real roots. Prove that these roots could be split into two groups with equal arithmetic mean.
algebrapolynomial