MathDB

1

Part of 2023 Tuymaada Olympiad

Problems(2)

3-variable inequality for variables in [0;1]

Source: Tuymaada 2023 Junior P1

7/7/2023
Prove that for a,b,c∈[0;1]a, b, c \in [0;1], (1−a)(1+ab)(1+ac)(1−abc)≤(1+a)(1−ab)(1−ac)(1+abc).(1-a)(1+ab)(1+ac)(1-abc) \leq (1+a)(1-ab)(1-ac)(1+abc).
algebraInequalityTuymaada
Infinite Spiral of integers

Source: Tuymaada 2023 Senior P1

7/12/2023
The numbers 1,2,3,…1, 2, 3, \ldots are arranged in a spiral in the vertices of an infinite square grid (see figure). Then in the centre of each square the sum of the numbers in its vertices is placed. Prove that for each positive integer n the centres of the squares contain infinitely many multiples of nn.
number theorycombinatorics